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Social influence occurs when one's emotions, opinions, or behaviors are affected by others.[1] Social influence takes many forms and can be seen in conformity, socialization, peer pressure, obedience, leadership, persuasion, sales, and marketing. In 1958, Harvard psychologist, Herbert Kelman identified three broad varieties of social influence
ABSTRACT
Title of Dissertation:
THE INFLUENCE OF HETEROGENEOUS RISK
PREFERENCES ON WATER MARKET
ACTIVITY: AN APPLICATION TO THE PALOMA
SYSTEM OF THE LIMARÍ WATER BASIN,
CHILE
Oscar E Cristi, Doctor of Philosophy, 2007
Dissertation directed by:
Professor Lars Olson,
Department of Agricultural and Resource Economics
This dissertation contributes to our knowledge about water markets by
analyzing the factors that explain market transactions of water rights when there is
also a spot market for water volumes. I hypothesize that risk heterogeneity among
farmers can explain those transactions. To test the aforementioned hypothesis I
model farmers’ decisions on investment in water rights each season under the
assumptions that they face output risk and that uncertainty is generated by future
water availability and price. The first order condition to this problem, which is
represented by the Euler Equation, indicates that the current period reservation value
of a water right depends on the current value of the amount of water accorded to
water rights in the spot market, the stochastic discount factor, and the expected future
prices of water rights. Using the relationship between the reservation value of a water
right and the stochastic discount factor I show analytically how heterogeneous
preferences are a sufficient condition for an active market for water rights. Then, I
test for heterogeneous preferences by allowing them to be a function of specific
characteristics of farmers. That requires the estimation of a system of equations that
includes a parametric specification of the Euler Equation and the first order
conditions for optimal input quantities. For that, I use an exponential utility function
and a production function of the Just-Pope type. I jointly estimate the parameters that
describe a farmer’s utility function along with production function parameters. The
empirical application uses farmer micro-level data from a two-round survey that I
conducted on a sample of Limarí Basin farmers. That Basin is located in the northern
part of Chile and is characterized by an active water market that has existed since
1981. Evidence rejects the hypothesis of homogeneity among farmers and suggests
that those better educated and more experienced Limarí Basin farmers are less risk-
averse. Results also show that water, labor and fertilizers have a positive impact on
mean output per hectare but their effect on yield variability implies that those inputs
are risk increasing.
THE INFLUENCE OF HETEROGENEOUS RISK PREFERENCES ON WATER
MARKET ACTIVITY: AN APPLICATION TO THE PALOMA SYSTEM OF THE
LIMARÍ WATER BASIN, CHILE.
By
Oscar E. Cristi
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2007
Advisory Committee:
Professor Lars Olson
Professor Maureen Cropper
Professor Bruce Gardner
Professor Richard Just
Professor Erik Lichtenberg
© Copyright by
Oscar E. Cristi
2007
ii
Dedication
To my mother and Gislaine, Yuviza and Francois
iii
Acknowledgements
This dissertation would not have seen the light of day without the help of a
number of individuals and institutions. I thank the chair of my committee, Professor
Lars Olson, for his permanent encouragement and the valuable insights and advice
that he provided. I thank Professor Richard Just for the detailed comments and
valuable guidelines on previous versions which led to substantial improvements. I
am also appreciative of the input from committee members, especially Professor Erik
Lichtenberg whose careful reading of the manuscript yielded many suggestions for
clarifications and improvements.
I express my deepest gratitude to all the members of AREC for their
permanent support and enormous kindness and professionalism.
I thank the Universidad de los Andes, Chile, and the World Bank for the
financial support. Their generosity allowed me to gather the data for this dissertation.
I have had many fruitful discussions about the dissertation process with
Professor Ivar Strand and my colleagues. Especially important have been
conversations with Fernando Díaz, William Foster, Jean Sepulveda, Eugenio
Bobenrieth and Ricardo Bórquez.
Finally, I would like to thank Abel Mejía, from the World Bank, who not only
suggested me to study the economic aspects of water markets, but also provided
permanent support for this dissertation.
iv
Table of Contents
Dedication..................................................................................................................... ii
Acknowledgements...................................................................................................... iii
Table of Contents......................................................................................................... iv
List of Tables ............................................................................................................... vi
List of Figures ............................................................................................................. vii
Chapter 1: Introduction................................................................................................. 1
Chapter 2: Literature Review........................................................................................ 7
2.1 Water management ....................................................................................... 7
2.1.1 Water markets in Chile and the Chilean Limarí Basin ....................... 13
2.2 Asset pricing models................................................................................... 21
2.3 Identifying risk preferences ........................................................................ 25
Chapter 3: The Limarí River Basin’s Water Market .................................................. 29
3.1 Mechanism for the Allocation of Waters in the Paloma System................ 31
3.2 Market Activity........................................................................................... 34
3.2.1 Spot Market......................................................................................... 35
3.2.2 Permanent Market ............................................................................... 38
3.3 Price Behavior in the Water Market ........................................................... 41
3.3.1 Spot Market......................................................................................... 41
3.3.2 Permanent Market ............................................................................... 42
3.4 Market Expectations and Access to Information........................................ 44
3.5 Summary..................................................................................................... 46
Chapter 4: Theoretical Model ..................................................................................... 48
4.1 A Simple Model.......................................................................................... 48
4.2 The Farmer’s Decision Rule under Water and Output Uncertainty ........... 52
4.3 Market Frictions and Consumption-Based Asset Pricing for Water Rights66
4.4 Summary..................................................................................................... 72
Chapter 5: The Econometric Estimation..................................................................... 74
5.1 Parametric specification of the system of equations................................... 74
5.2 Data............................................................................................................. 89
Chapter 6: Estimation and results ............................................................................... 98
Appendix: A Hausman test for endogeneity........................................................ 119
v
Chapter 7: Conclusions and Suggestions for Future Research ................................. 122
Annex: Survey instrument ........................................................................................ 127
References................................................................................................................. 149
vi
List of Tables
Table 3.1 Average water accorded to water rights between 1980 and
2000…………………………………………………..…..
33
Table 5.1: Basic statistics of the main variables…………………….. 93
Table 6.1: Descriptive statistics for variables used in the estimation
of the system of equations…………………………......….
104
Table 6.2: Estimates of the parameters of the equation system using
FIML………………………………..…………………….
110
Table 6.3: List of instruments to test for possible endogenous
regressors…………………………………………………
120
vii
List of Figures
Figure 3.1: Limarí Basin and the Paloma System……............................. 31
Figure 3.2: Water accorded to water rights between 1980 and 2000…… 33
Figure 3.3: Transfers of water in the spot market…………………......... 38
Figure 3.4: Number of water rights transferred by each association
during the period 1980-2000………………………………..
39
Figure 3.5: Real average prices of water rights in each WUA…….......... 42
Figure 3.6: Reservation value for the right to one cubic meter of water
in ACEC……………………………………………………..
43
Figure 3.7: Reservation value for the right to one cubic meter of water
in ACER……………………………………………………..
44
Figure 3.8: Transfers of water through the spot market……………….... 45
Figure 4.1: Water right transactions…………………………………….. 70
Figure 6.1: Actual value of standardized output income per hectare
against the imputed values for that variable………………...
113
Figure 6.2: Actual value of standardized investment in water rights
against the imputed values for that variable………………...
113
Figure 6.3: Plot of the residuals from the FIML regression based on
estimates in Table 6.2………………………………….........
118
1
Chapter 1: Introduction
The study of water resources is a passionate task as their increasing value and
unique characteristics are appealing for economists working on natural resources and
have important implications for peoples’ well-being.
Almost any paper or document that tackles water management will begin by
pointing out that worldwide, even countries and regions where water is abundant face
increasing water scarcity (Rosegrant et al., 1997, Tsur, 2004, Saleth and Dinar, 2004).
Demands for water from all sectors – agriculture, industry, households, and even
environmental conservation – combined with increasing difficulty in developing new
structural solutions to increase water supply, explain growing water scarcity.
Projections of water withdrawals by sectors show the dramatic increase in pressure on
water resources over the next three decades (Rosegrant et al., 1997).
Water has special characteristics because it is not a resource like others that
can be easily appropriated, traded and used without affecting others. Water is a
mobile resource that generates multiple levels of physical inter-dependence among
users, while farmers’ water use and transfer decisions create externalities. Water
diversion requires expensive devices for volumetric or flow measurements and costly
conveyance systems. Finally, water for irrigation is characterized by the randomness
of supply and the high cost of reducing variability through storage capacity.
Since the 1950s, the literature has debated the merits of water allocation
institutions, and now wise water management policies are among the most crucial
challenges of nearly all countries. If Julian Simon – whom I met for the first time in
1986 and who encouraged me to come to study at the University of Maryland – were
2
still alive, surely he would have seen this problem as a challenge for the people.
Likely he would have said that this new challenge could best be resolved by allowing
people to deal freely with this problem, as he extensively documented in his works on
decreasing scarcity of several natural resources. The public-good approach with
public water ownership and state involvement in its development and distribution
does not work well in the present context of water scarcity. Thus, as Saleth and Dinar
(2004) point out, “The current trend is toward an alternative system that can allow
private decision-making in water resource development, allocation, and management.
For the alternative system to function effectively and equitably, legal changes are
needed to facilitate private and transferable water-rights system that ensures full legal,
physical and tenure certainty of water rights.”
Worldwide, irrigation continues to be, by far, the largest sector of water
consumption accounting for nearly 70% of water withdrawals worldwide and over
90% in low-income developing countries. Another 23 percent of water is used in
industry and the remainder is consumed by households. These numbers indicate that
the increasing demand for water will need to be met from water savings in irrigated
agriculture by improving efficiency. Traditionally, economists have argued that
efficiency can be achieved through a water pricing system that reflects water scarcity.
This approach has some problems. First, water users have been able to use their
political power to prevent major increases in water prices, especially for irrigation.
Second, the water authority needs to define a mechanism to value water, which may
differ from water users’ willingness to pay for water.
3
Given those problems, water markets are receiving increasing attention from
policymakers in an attempt to improve the efficiency of water allocation. Although,
in spite this and the fact that numerous informal water markets have evolved around
the world, very few countries have implemented formal and legal free markets for
water. This is a result in part from the difficulty that researchers face in studying
rights that are attached to a mobile resource like water. Therefore, relevant policy
questions about water markets have not been addressed with empirical evidence.
Moreover, policymakers have tended to claim state “ownership” of water and have
been reluctant to develop tradable water rights by separating them from land rights,
which would allow the transfer of the former. They recognize the theoretical value of
water market institutions, but some think that the number of recorded instances where
water is reallocated by market transactions is far too limited due to physical
constraints and third-party effects associated with water exchanges. There are also
doubts about whether the market can reallocate water to its optimal social use.
Hence, still the main question about water markets, as Saliba (1987) pointed out is
“Do water markets ‘work’?”
Chile has formal water markets that have been operating for more than 24
years, and has become, along with Australia, an example of how institutional reforms
that treat water as an economic good improve water-use efficiency and water
allocation. Nonetheless, more empirical research based on extended data is required
to support those hypotheses (Bauer, 2004). Thus, Chilean water markets offer an
excellent opportunity for researchers who seek to answer relevant policy questions
about water markets.
4
The purpose of this dissertation is to contribute to the understanding of water
right transactions. I attempt to answer a basic question: What explains water rights
trading when farmers can exchange water in the spot market, which has lower
transaction costs? In answering this question, I emphasize the link between the spot
and water rights markets where differences in marginal returns to water among
farmers are solved in the spot market while the water rights market address
differences in the stochastic discount factor among farmers.
In this dissertation, I develop a theoretical model in which I characterize
optimal decision making by farmers faced with the decision of investment in water
rights. This model assumes that farmers face water and output uncertainty, and that
they may trade water in the market for water rights and/or the spot market. Because
investment decisions affect future levels of consumption and farmers face
uncertainty, the theoretical model for farmer decisions is modeled as a stochastic
dynamic problem. This results in a Consumption Capital Asset Price Model
(CCAPM) whose solution is described by an Euler Equation that ties asset returns
(water right returns in this case) to marginal rates of substitution of consumption at
different points in time. This model provides insights into how farmers determine
their reservation value for water rights when it is considered an asset, often the main
asset for farmers, and it allows me to emphasize the role of farmers’ heterogeneous
risk preferences on their reservation values and on marketing activity.
This theoretical analysis provides the foundation for a case study of water
transfers for irrigation in the Limarí Basin, an important agricultural region in the
northern part of Chile, which has one of the most active Chilean irrigation water
5
markets. With farmer-level data obtained from a survey among farmers for two
different agricultural seasons, the case study allows me to estimate jointly the
parameters that describe a farmer’s utility function and production function. The
estimation is based on the Euler Equation and the first order conditions for input
quantities. Using observed economic behavior I test for heterogeneous preferences
among farmers. The use of an asset price model to jointly determine farmers’
preferences and production technology in developing countries is a contribution to the
literature on agricultural finance.
The present analysis of a Chilean water market and its empirical application
provide insights into how water markets work in a developing country. It is hoped
that the results of this dissertation will supplement our knowledge of the outcomes
and experiences of developing countries with active, but undocumented, water
markets, and help move the debate beyond principles to empirical results on the
operation of water markets. This analysis will serve countries that are contemplating
adopting market-based reallocation systems where policymakers wish to inform
themselves of other experiences in water markets in different cultures and geographic
regions before they make a decision.
This dissertation proceeds in 7 sections. In Chapter 2, I present a brief survey
of the literature on water markets. I then proceed to provide a detailed description of
the operation of the water market in the Limarí Basin. In Chapter 4, I develop a
theoretical model for optimal decision making by farmers who must decide on
investment in water rights and input quantities in every season. Data availability and
the econometric model are discussed in Chapter 5. In Chapter 6, I describe the
6
estimation procedures and I report estimation results. Finally, Chapter 7 concludes
with a summary of the main results and suggestions for future research.
7
Chapter 2: Literature Review
In this chapter I review the relevant portions of the extensive literature on
water management and water markets. I also review the part of the literature on asset
pricing models and the estimation of heterogeneous risk preferences that is related
with the analysis and methodology of this dissertation.
2.1 Water management
There is a broad consensus in the literature on water management that
increasing water scarcity requires a shift from supply-oriented approaches focused on
technical and hydrological solutions towards allocation-oriented approaches centered
on economic and institutional solutions that provide the right incentives for water
savings. These efforts focus primarily on agricultural irrigation, the main use of
water all around the world
1
(Saleth, 2004). This is the starting point for a wide
literature on alternative institutional arrangements to promote efficient use (Bruns and
Meinzen-Dick, 2000, Saleth and Dinar, 2004)
Since at least the 1950s, there has been a debate in the literature over socially
appropriate water allocation institutions (for an excellent summary of that debate, see
Lynne and Saarinen, 1993). Traditionally, economists and policymakers have argued
that what is needed is a centralized water pricing system
2
that reflects the opportunity
cost of water. Centralized water pricing systems have two main problems. First,
water users have been able to use their political power to prevent major increases in
1
Irrigation is the largest sector of water consumption; accounting for nearly 70% of
total water withdrawals worldwide and over 90% in low income developing countries.
2
In such a system the water authority determines the value of water in different uses
and fixes a price for the use of that resource.
8
water prices, especially for irrigation (Easter et al., 1998). Second, under a
centralized system of water pricing, the water authority needs to define a mechanism
to value water, which may differ from water users’ willingness to pay for water.
Non-market techniques to value water rights include the farm-budget residual
valuation for water
3
as in Hearne and Easter (1995 and 1997). Another approach is to
estimate the value of marginal productivity of water using a crop-water production
function. Marginal values of water in municipal uses or in instream and recreational
uses are usually derived by estimating consumer willingness-to-pay through
contingent valuation and travel cost methods. Another procedure that has been used
to evaluate willingness to pay for water is the least cost alternative technique as in
Hearne and Easter (1995 and 1997)
4
. Person and Michelsen (1994) offer a good
review of different methods for estimating water values and they summarize the
willingness to pay estimates for different water uses in studies until 1994.
The current trend of water allocation institutions is toward an alternative
system that allows private decision-making in water resource development,
allocation, and management (Saleth and Dinar, 2004). Thus, market-type allocation
institutions such as water markets and water banks, that recognize water as an
economic good rather than a social good, are receiving increasing policy attention in
attempts to improve the efficiency of water allocation (Rosengrant and Binswanger,
1994, Vermillion, 2000, Brookshire and Ganderton, 2004). In that framework, the
3
In the residual method, subtracting all non water and land input cost from the total
revenue yields a residual value, which can be viewed as the maximum price that the operator
could pay for land and water and still break even. The researcher then allocates the residual
value between the two components, land and water.
4
In this case, the technique is used to compare the present value of the cost of buying
water rights against the cost of building a new storage capacity to increase water availability
for municipal uses.
9
market provides the mechanism by which water is valued. This new paradigm has
been promoted by the United Nations at the Rio Convention on Environment and
Development in 1992 and by the World Bank (1993).
The literature on the design of water markets within the variety of legal
settings that exist around the world has focused on the distinction between formal and
informal markets. In the formal market a variety of transactions take place, such as
the rental of water rights, water volume sales for a specific time period, and water
entitlement transferences, whereas in the informal market only short term transactions
are observed (Bjornlund, 2004).
Formal water markets have been implemented in the western United States in
Colorado (early 1960’s) and California (since 1982), and in Chile (since 1981),
Australia (since 1983), South Africa (since 1998), New Zealand (since 1991) and
Mexico (since 1994). Peru, Bolivia, Argentina, Nicaragua are among the countries
that are discussing policy reforms oriented towards water markets (Bauer, 2004).
Saleth (2004) and Bruns and Meinzen-Dick (2000), advocate that for formal
water markets to function effectively and equitably, legal changes are needed to
ensure full legal, physical, and tenure certainty of water rights separated from rights
to land. The costs associated with these institutional changes necessary to move to
market mechanisms explain in part the reduced number of countries that have formal
markets (Coward, 2000). Those costs have been addressed by McCann and Easter
(2004), but as Saleth and Dinar (2004) point out, “A study of the full transaction cost
associated with the change to an alternative water allocation mechanism has not been
attempted to our knowledge.”
10
With regard to informal markets, those markets have been widely
implemented in a number of countries such as India (Saleth 1998), Pakistan
(Meinzen-Dick, 1998) and Jordan (Shatanawi and Orabi, 1994).
Most critics of water markets argue that they do not work at all or that
transactions are too few due to market failures, and that they are not compatible with
integrated water resource management because they do not jointly solve critical
economic, environmental and social issues (Bauer, 1995 and 2004, Crase et al., 2000
and 2003, Gleeson, 2003). Supporters emphasize the benefits of water markets and
how these markets are actively working in various parts of the world (Rosengrant and
Binswanger ,1994; Holden and Tobani ,1995; Briscoe, 1996)
The emergence of water markets as allocation institutions has led economic
analysis to focus on what type of market is likely to appear, how water prices may be
formed, and whether water markets improve efficiency of use by reallocating water to
its highest use value.
Up to now, the description of how water markets function has focused
primarily on developed regions including the Western States of the USA (Michelsen
and Young, 1993, Israel and Lund, 1995, Susan M Burke et al., 2004) and, more
recently, Australia (Bjornlund and McKay, 2002, Bjornlund, 2004, Crase et al.,
2004). Attention is now shifting toward the developing regions of Africa, Asia, and
Latin America. Examples of empirical research on Latin American countries are the
works done for Chile by Rios and Quiroz (1995) and Bauer (1995 and 2004).
Rosengrant and Binswanger (1994) describe markets in tradable water rights in Chile
and Mexico. A major contribution to the literature on the outcomes of water
11
negotiation is the book edited by Bruns and Meinzen-Dick (2000), which documents
cases primarily from South Asia and Indonesia. These cases show how negotiation is
frequently used by water users, and the successful outcomes that have resulted from
the process.
Understanding the factors that determine water right prices and their variation
has become important for establishing whether water markets reallocate water to its
most efficient use. Initial studies explain price formation as a result of a bargaining
process between farmers for which the value of marginal product of water differs.
Several studies use hedonic price function to estimate the value of water rights and
the factors that influence prices of water rights and its fluctuations, such as water
right characteristics, institutional constraints, physical transferability of water,
bargaining power of sellers and buyers, and speculative behavior over water right
prices (Colby et al., 1993, Person and Michelsen, 1994, Bjornlund and Mckay, 1998).
Numerous authors have constructed the theoretical arguments that some type of
market based trading mechanism would greatly increase the efficient use of water (for
a good summary of the state of art on water markets see Brookshire and Ganderton,
2004). The basic argument is well established: water is a natural resource with
varying value in different uses and with clearly defined social and political
constraints. In terms of applied research, qualitative analyses as well as increasingly
sophisticated empirical studies have been published to verify whether water markets
allocate water to its highest valued use. To make this determination several studies
simulate market performance (Saleth et al., 1991, Dinar and Latey, 1991, Tisdell et
al., 2004, Dinar et al., 1998, Murphy et al., 2000). Also, an increasing number of
12
studies provide empirical analysis for assessing water right market efficiency with
data on existing water markets, mainly in some states in the USA (Brown et al., 1982,
Saliva, 1987, Crouter, 1987, Michelsen, 1994, Rosegrant and Binswanger, 1994,
Brookshire at al., 2004) and Australia (Crase et al, 2000, Bjornlund ,2004). For the
case of developing countries such as Chile, Rios and Quiroz (1995) and Bauer (1995
and 2004) provide qualitative analysis of water market performance whereas Hearne
and Easter (1995 and 1997) provide quantitative analysis of water market efficiency.
A good number of these studies analyze whether water allocated through the market
moves from its lower to its highest value by empirically identifying who are the
buyers and the sellers. This is the case of Nieuwoudt and Armitage (2004), Bjornlund
(2004) and Crase et al. (2004) for developed countries as South Africa and Australia,
and Hadjigeorgalis (2000) and Zegarra (2002) for the water market in the Limarí
Valley, Chile. In general, studies that analyze water market efficiency conclude that
the major benefits of the formal market are associated with a reallocation of water to
1) more productive soils, 2) more efficient water users, 3) higher-value uses, and 4)
new developments and the consolidation of water into larger more viable units.
The work by Bjornlund (2004) is quite interesting because he measures and
compares temporary trade with permanent trade in the Goulburn System and Murray
System, Australia. Bjornlund (2004) finds that the temporary market has by far the
highest amount of traded water, and that the practice of using both markets has been
widely adopted to shift an irrigator’s risk position and to manage increased supply
uncertainty. He also indicates that trade in water rights surged after farmers became
familiar with water trading and aware of the potential benefits. This took around 7
13
years in the area he studied. Finally, he presents evidence that shows how liquidity
constraints cause farmers to participate as buyers in the temporary market because
they cannot afford to buy water rights. Crase et al. (2004) found similar results for the
water markets in the Murray Darling Basin of Australia. In New South Wales
permanent and temporary transactions took 10 years to be significant, with temporary
transactions always much more important. In Victoria’s water market trade in water
rights surged after 7 or 8 years. These results of Bjornlund and Crase et al. on the
relative size of the temporary market with respect to the permanent market and on the
time that the permanent market takes to become established are very similar to the
ones that I obtain for the Limarí Basin, and which are reported in Chapter 3 of this
dissertation.
Nevertheless, there still exists a real need for more applied work on water
markets. Brookshire and Ganderton (2004) point out that “it is necessary to
understand beyond theoretical considerations how well these alternative institutions
perform from an empirical standpoint and what are some of the institutional design
issues that remain” and that “It is also needed to move beyond the simple description
of markets to identify the forces operating within those markets.”
2.1.1 Water markets in Chile and the Chilean Limarí Basin
Chile, together with Australia, has become one of the world’s leading
examples of how institutional reforms that treat water as an economic good improve
water use efficiency and water allocation. Yet more empirical research based on
extended data needs to be done to support those hypotheses. Bauer (2004) points out
14
that “much of the discussion about Chilean water markets has been long on
theoretical or ideological argument and short on reliable information”.
The first real empirical study of water markets in Chile was done by Hearne
and Easter (1995 and 1997), followed by Hadjigeorgalis (2000) and Zegarra (2002).
All these studies examine the water market in the Limarí Basin. Hearne and Easter
(1995 and 1997) also analyze water markets in the adjacent Elqui Basin, and Cristi et
al. (2003) also use that basin in their case study.
The Limarí River Basin, in north central Chile, has attracted national and
international attention through the 1990s and first half of the 2000s. The Limarí
River Basin is the one example that is widely agreed to have an active and successful
agricultural water market, including both temporary and permanent sales, and even
local real estate agents broker and facilitate water rights trading. Hearne and Easter
(1995 and 1997) estimate economic gains (net return to society) and financial gains
(individual net benefits) from trade in that basin. Economic gains correspond to the
difference between the value of water to the buyer after a purchase and the value of
water to seller before a sale minus transaction costs of the transfer. Financial gains
for a seller equal the sale price less both the value of water to the seller and the
seller’s transaction cost, whereas for a buyer it is the difference between the value of
water to the buyer and the sum of the buyer’s purchase price and transaction costs.
They conclude that market transfers of water rights produce substantial economic and
financial gains from trade in the Limarí Basin.
While their results are interesting they do not consider key features of water
used in irrigation that affect individual values for water rights, such as uncertainty
15
about water availability and future water prices, output uncertainty, and farmers’
attitudes towards risk. They also do not consider that in the Limarí Basin there are
two markets for water that coexist and interact: the market for water rights sales
(permanent transactions or permanent water rights sales) and the spot market for
water (short-term or temporary transactions). As a consequence, they fail to
recognize the relationship between the value of a water right and the price of water in
the spot market.
Hadjigeorgalis (2000) provides the first empirical analysis of actual trading
outcomes in both spot and water rights markets in the Limarí Basin. For the spot
market, she measures the number of transactions, volumes sold, and the number of
participants - separated by buyers and sellers - for the period 1994-1997. She also
analyzes price behavior for the 95/96 and 96/97 seasons, using field data for around
332 farmers. She concludes that there exists an active spot water market with prices
highly sensitive to water scarcity, and that the facility to transfer water volumes
between sectors has resulted in an equalization of water prices for water volumes
between geographically segmented sectors. With respect to market activity in the
market for water rights she identifies the existence of physical constraints that prevent
transferring rights between different reservoirs and institutional constraints that
prevent trading rights that are stored within the same reservoir, but that have different
legal locations (i.e. farmers with water rights in different Water User Associations)
5
.
These constraints produce segmentation into local market sectors below the dams and
this segmentation allows for water right price differences between local markets. She
5
Cortés, M (1997) offers a lucid explanation of the legal constraints to water right
trades in the Paloma System.
16
also presents the first formal, theoretical analyses of the impact of risk and
uncertainty on water market trading and water decisions on the amount of water to be
used in the production process
6
. In her theoretical approach she allows for output
price and spot market price uncertainty, water endowment uncertainty and capital
production risk for farmers that produce perennial crops. She presents formal
expressions for reservation spot market prices and reservation values for water rights.
The former are a function of the net value of the marginal product of water in
irrigation, irrigation efficiency, risk aversion and uncertainty cost associated with
selling and buying water volumes. Reservation values for water rights – which she
derives by emphasizing that water rights are an asset – are a function of the sum over
time of the discounted per period net values of the marginal product of water in
irrigation (benefits from water use in irrigation less the cost of holding water rights),
irrigation efficiency, risk aversion, and uncertainty cost associated with stochastic
water supplies. For perennial crop producers, reservation value is also a function of
the risk of future loss of their stock of perennial crops from a water supply shortfall.
In the empirical application she analyzes market participation and the probability that
a farmer participates in either the spot market or the water right market, and whether
the farmer will buy or sell water volumes and/or water rights. Among the
explanatory variables, she includes risk aversion proxied by farmer’s wealth. She
shows that trades occur from farms with low irrigation efficiency to farms with high
irrigation efficiency and that transaction costs in the spot market as well as in the
6
Howitt (1998) provides a first theoretical analysis of the impact of risk and
uncertainty on water markets and water decisions in a case study for the existing water
market in California.
17
market for water rights are minimal. The main limitation of Hadjigeorgalis’s work is
that, although in the theoretical model she clearly addresses the effect of risk and risk
preferences in the farmers’ reservation value for a water right, in the empirical
application that relationship vanishes and is replaced by a set of prior assumptions
regarding what type of farmers are more or less risk averse. Thus she cannot clearly
show how risk affects farmers’ water trading decisions. An empirical test of the
relationship among farmers’ characteristics and risk aversion would have helped her
to explain what she called unexpected results. One of these unexpected results is that
perennial crop producers are not exclusively buyers of water rights but appear
consistently on both sides of the market. An empirical estimation of heterogeneous
risk preferences may show how differences in risk preferences explain differences in
reservation values for water rights among perennial crop producers. If such
differences exist then it would explain why water right trades occur as more risk-
averse perennial crop producers would buy water rights from those perennial crop
producers with lower risk aversion.
Zegarra (2002) focuses his research on the operation of the spot market in the
Limarí Valley in the face of an extremely negative shock: the severe drought of
96/97. He models farmers’ decisions about the amount of water to be used in
production and the amount of water to be sold in the spot market. Farmers reach
equilibrium when the water’s marginal of value product equals the spot market price
for water. Thus, farmers decide not to grow crops in those seasons in which the water
return for selling water in the spot market is greater than their expected income from
production. Farmers are risk neutral and with production functions characterized by a
18
minimum water requirement constraint, which results in a non-convexity of the
production technology. With production non-convexity one of the main assumptions
for Pareto efficient allocation through market transactions is broken. Heterogeneity
among farmers is given by their crop mix and this heterogeneity makes spot markets
work. By simulating expected income in different scenarios he tests the hypotheses
that the increasing presence of permanent crops creates demand rigidities that reduce
the effectiveness of spot water markets. He finds that as crops become more
concentrated in permanent crops the spot market water prices exhibit a higher average
value and a greater dispersion. He also analyzes a farmer’s participation in the spot
market, i.e. if a farmer trades in the spot market and if so whether he is a buyer a
seller or both. The main results from Zegarra are that the spot market for water
solves differences in the marginal return of water among farmers, promoting the
allocation of water from low value annual crops to high value permanent crops. He
finds that in the context of severe drought, the water market starts to be less effective
in allocating the resource, with greater water price dispersion. Unlike Hadjigeorgalis
(2000), he concludes that water rights are heterogeneous with statistically significant
differences in both the mean water per share and the standard deviation. He suggests
that there are low transaction costs in the spot market. The main limitation of
Zegarra’s work is that it does not take into account farmers’ risk aversion. This
omission weakens one of his main results: that the spot market price at which the
supply of water volumes starts to be greater than zero is $30 pesos. If farmers are risk
averse his model overestimates that value because a risk-averse farmer will be willing
19
to sell his seasonal amount of water – and obtain a sure income – at a price lesser than
his expected marginal return of water use in irrigation.
From the above literature review, it is possible to infer some guidelines for
economic water research. 1) There is a need to understand how well water markets
perform and the forces operating within those markets from an empirical standpoint,
especially in non developed countries. 2) There is a need to empirically estimate the
impact of risk and uncertainty on water market trading and water decisions on the
amount of water to be used in production. In the process it is advisable to infer
reservation values for water rights from a model that recognizes that water rights are
one of the farmer’s main assets. 3) When a market for water rights and a spot market
for water coexist there is a need to account for the link between the two markets in
order to understand how spot prices affect water right prices over time. It also needs
to be emphasized that the spot market resolves differences in the marginal return of
water while the market for water rights resolves differences in farmers’ reservation
values for a water right. The latter are due to farmers’ differences on risk of future
loss on their stock of perennial crops from a water supply shortfall as well as
heterogeneous risk preferences. 4) Finally, there is a need to move from institutional
constraints that explain price differences in the market for water rights across sectors,
to factors that explain differences in reservation values between farmers within the
same Water Users Association.
The present dissertation contributes to the literature on water markets by
providing new insights on several issues. It measures market activity and provides
the first estimation of the size of the temporary water trades (spot market) in relation
20
to the permanent markets (water right trades) in the most active Chilean water market.
The analysis shows that the volume of water traded on the spot market is several
times greater than on the permanent market. Contrary to what other researchers such
as Hadjigeorgalis (2000) believe, it illustrates how the spot market is active not only
during drought years but also in years with average water availability. The theoretical
model that I develop infers reservation values for a water right from an asset pricing
model that assumes heterogeneous risk preferences among farmers, incomplete asset
markets and uncertainty about output, future water availability and future water
prices. It also incorporates the interaction between the spot market and the market for
water rights where the spot market mainly resolves differences in the marginal return
to water and the market for water rights mainly resolves differences in the stochastic
discount factor among farmers. The model explains differences in reservation values
between farmers within the same Water Users Association as a function of farmers’
risk preferences. The empirical application estimates an asset price model for water
rights and input demands that allows testing for heterogeneous risk preferences
among farmers. In addition, the effect of water on the mean and variance of yields is
estimated using detailed farm data on output and input quantities for each crop. This
is an improvement from previous studies, such as Hearne and Easter (1995 and 1997)
and Hadjigeorgalis (2000), that rely on standard crop budgets to proxy the marginal
revenue of water use in irrigation. The role of risk differences due to different types
of crops or distance from the reservoirs is not included in this dissertation and it
represents an important future extension of this work.
21
2.2 Asset pricing models
In this dissertation I model water right reservation values using a capital asset
pricing model (CAPM). The empirical use of a CAPM requires choosing between the
conventional consumption-based capital asset-pricing model (CCAPM) and a
production-based capital asset-pricing model (PCAPM). Next I briefly review some
of the literature related to these approaches and some of the literature related to the
different issues embedded in the use of an asset pricing model to value water rights.
The CCAPM ties asset returns to marginal rates of substitution for
consumption at different points in time and so must use a utility function defined on
consumption over time. Alternatively, the PCAPM emphasizes the linkages between
asset returns and investment and production variables. In it, production is used
instead of consumption and so the production function is modeled instead of the
utility function. A production model is proposed by Cochrane (1991 and 1996),
where asset returns are tied to marginal rates of transformation (the rate at which the
firm can transform goods from date t to date t+1, i.e. the rate of return on
investment). Then he empirically tests the relationship between stock and investment
returns in which the investment/capital ratio is a key variable. Arroyo (1996)
explains asset returns as a function of capital productivity and the adjustment cost of
capital proxied by the investment/capital ratio. Those who propose the use of
PCAPM usually mention the mounting evidence against standard consumption-based
models of asset returns. The empirical evidence indicates that returns on equity seem
to be too high to be consistent with observed consumption behavior unless investors
are extremely risk averse: a risk aversion often too large to be credible (Arroyo, 1996,
22
Campbell et al., 1997). Cochrane (1996) and Campbell et al. (1977) point out that the
poor empirical performance of CCAPM in explaining asset returns may be due,
among other reasons, to measurement error in aggregate consumption and/or because
growth of aggregate consumption is very smooth. Another source of criticism of the
CCAPM arises from transactions costs, borrowing constraints and other market
frictions that may invalidate the condition that discounted expected marginal utilities
should be equilibrated across time, which is the heart of the consumption-based
capital asset pricing model. In spite of the potential advantages of PCAPM over
CCAPM, as I explain in Chapter 4, Section 4.2., I have chosen a consumption-based
model because it emphasizes the role that preferences over consumption have in the
determination of the reservation value for water rights, and as Moschini and
Hennessy (2001) point out “…one should keep in mind that farmers ultimately likely
care about their consumption, itself the result of an intertemporal decision”. In that
same line, Cochrane (2005, Chapter 9.1: 157) points out that “…good economists are
unhappy about a utility function that has wealth in it. Few of us are like Disney’s
Uncle Scrooge, who got pure enjoyment out of a daily swim in the coins in his vault.
Wealth is only valuable because it gives us access to more consumption. Utility
functions should always be written over consumption. One of the few real rules in
economics to keep our theories from being vacuous is that ad “hoc utility functions”
over other objects like wealth should eventually be defended as arising from a more
fundamental desire for consumption or leisure.”
In this dissertation the CCAPM is derived from farmers’ optimal decisions
about investment in water rights in each season. The optimality conditions of the
23
model are described by Euler Equations. Empirically, the Euler Equations are
estimated together with the first order conditions for input quantities, using farm-level
data.
A number of requirements for high quality empirical production research in
agriculture, or what Just (2000) calls guiding principles, are addressed by the way in
which farmers’ decisions are modeled in this dissertation. First, it deals with the need
to focus on long run considerations of investment and cost adjustments, and the need
to consider the role of serial correlation of farm income and the intertemporal
dependence of farmers’ marginal utilities. I model intertemporal decisions on
investment which emphasizes the long-run nature of farmers’ decisions. Although I
model farmers’ decisions assuming non-serial correlation of farmer’s consumption
and a time-additively separable utility function over consumption, the model
indirectly links consumption and utilities over different periods. This link arises
because in this model the optimal consumption path depends on the stock of water
rights which is related both to present and past investment in water rights (Bossaerts,
2002). Second, I identify risk preferences using an asset pricing model for water
rights, which arises from farmers’ investment decisions that reflect the greatest
consequences of risk on farmers’ decisions (Just, 2000, Just and Pope, 2003).
Usually, the problem is that data on asset choices are very limited, thus the data on
water right choices gathered for this dissertation provides an important piece of
information to be able to build an asset pricing model for these rights and from there
to analyze the effect of risk and risk preferences. Third, I use data at the individual
farmer level and I incorporate farmers’ heterogeneity, which helps to improve the
24
empirical quality of the CCAPM (Campbell et al., 1997, Heaton and Lucas, 1996,
Constantinides and Duffie, 1996). Studies that estimate Euler Equations or more
general first-order conditions with data at individual level include, among others,
Zeldes ( 1989), Langemeier and Patrick (1993) and Phimister (1995), all of them in
the context of testing for liquidity constraints in the permanent income/life cycle
model for consumption. Blundell et al. (1994) estimate an Euler Equation using
micro data in order to estimate the parameters of household preferences that
determine the allocation of goods within the period and over the life cycle. Most
models of asset pricing assume homogeneous preferences among individuals or,
equivalently, the existence of a representative agent. Allowing farmers to differ in
their utility functions is a contribution to the empirical literature on asset pricing. The
assumption of heterogeneous preferences is also a sufficient condition for the
occurrence of asset trading among individuals. Niehaus, (2001) considers a simple
economy, where only a riskless bond, shares of a stock and an option written on the
stock are available in the financial market, and shows that differences in investors’
preferences have an impact on asset prices and the amount of trading in the market.
He finds that the amount of trading and the price of the option grow with increasing
divergence in risk aversion, and the agents with a higher degree of risk aversion sell
shares and options and buy the riskless bond. The agents with a lower degree of risk
aversion take the opposite position: they buy shares and options and sell bonds. This
is the same approach that I use in my dissertation, where each farmer has one asset –
a water right – and the following options: to sell the water right and buy a riskless
asset (or just put the money in bank at a riskless interest rate), to take more risk by
25
selling the water right and buying water in the spot market for use in farming
activities, or to keep the water right and use it in a risky farming activity. These are
the relevant contributions of my dissertation to the literature on the analysis of
farmers’ behavior along time, subject to limitations associated with the extent of the
data which is at the farmer level for two agricultural seasons.
2.3 Identifying risk preferences
The inclusion of heterogeneous risk preferences in the CCAPM requires
attention to the literature on identifying risk preferences for agricultural producers. A
comprehensive review of the large literature on this issue exceeds the scope of this
dissertation. Thus I limit discussion to the main issues identified in the review by
Moschini and Hennessy (2001). I then review some of the studies that specify risk
aversion as a function of socioeconomic characteristics such as age, education level
and family size.
Moschini and Hennessy (2001) show how early empirical studies of
agricultural decision making under risk elicited risk preferences from choices
between hypothetical lotteries. Later, using an econometric approach, studies
imputed a measure of risk aversion from the divergence between actual farmers’
production decisions and optimal decisions under risk neutrality. Due to the
limitations of inferring risk from observed production decisions and because
hypothetical payout surveys can give unstable results, Binswanger (1980) made real
payments to peasants farmers in India to elicit risk preferences. Antle (1987)
described the optimality conditions of expected utility maximizing choices in terms of
a given individual’s absolute risk aversion and downside risk aversion, and as an
26
econometric procedure he used the generalized method of moments (GMM). Later
on, Antle (1989) developed a method to estimate risk preference structures separately
from the production technology. Myers (1989) assumed constant relative risk
aversion (CRRA) and joint lognormality of the distributions of output prices and
producer consumption, and developed a reduced-form rational expectations approach
to test for the aggregate level of relative risk aversion for US producers who store
crops. Exploiting technical attributes of CRRA and of constant partial relative risk
aversion (CPRRA), Pope (1988) developed implications for optimal choices by
individuals expressing such preferences. Several studies have followed Pope’s
approach or variations of it. Another characteristic of research that attempts to
determine farmers’ risk preference structures is that most of them are based on
aggregate data (Just, 2000). Exceptions to this are the already mentioned
Binswanger’s lottery experiment and the Bar-Shira, Just and Zilberman (1997) study.
For this dissertation it is relevant to make a brief review of the literature that
identifies risk preferences by assuming that farmer’s risk aversion is a function of
socioeconomic characteristics such as age, education level and family size, among
others. This is necessary because in my dissertation farmers’ risk preference
heterogeneity is tested by estimating risk aversion for each farmer as a function of his
socioeconomic characteristics. Moscardi and Janvry (1977) analyze the relationship
between risk aversion and a number of socioeconomic variables that characterize
Mexican peasant households, their access to income-generating opportunities, and
their relation to public institutions. Binswanger (1980) analyzes the effect of wealth,
education level, more progressive farmers, and off-farm salaries on farmer’s risk
27
aversion. Zeldes (1989) allows a household’s utility function to be influenced
linearly by tastes that may differ across families and shift across time. Tastes differ
due to observable (for the econometrician) and unobservable factors. The observable
factors, which vary across families and time, are family size, age and age squared.
This linear specification for family utility function is included in the Euler Equation
which is estimated with data at the family-level. Blundell et al. (1994), who also
estimate an Euler Equation, allow the parameters that describe individual preferences
over consumption to be a linear function of variables such as the number and age of
children and labor market status: whether the head of the house and/or the wife are in
paid employments, and the level of consumption itself. Dubois (2001) also
parameterizes agent preferences by specifying a linear function for the absolute risk
aversion coefficient as a function of observable individual characteristics (age,
household size, number of children, etc.).
At the end of this section it is worthwhile to mention that the approach
followed in this dissertation, where water rights are assets and farmers’ optimal
decisions about water rights are treated as an investment problem that affect present
and future income and consumption, is closely related to the so called literature on
Agriculture Finance. Barry and Robinson (2001) offer a good review of the main
issues in Agriculture Finance. One of those issues is intertemporal farm-level
analysis in the context of life cycle planning and performance models of farm
business, where production and consumption are linked. Intertemporal analysis is
expressed as the maximization of the utility of multiperiod consumption, constrained
by the present value of wealth and the available investment alternatives, including
28
both productive investments and lending and borrowing in a perfect or imperfect
financial market. A second issue is the effect of farmers’ risk attitudes on their
portfolio decisions. In this dissertation those two elements: intertemporal farm-level
analysis where production and consumption are linked and the effect of farmers’ risk
attitudes on their decisions are carefully considered.
Up to now, research on Agriculture Finance has focused on real estate as the
dominant asset for farmers. But now, due to the increasing interest on establishing
transferable water rights not married to land rights, research on Agriculture Finance
should also consider water rights as a primary asset in dry areas. Due to the special
characteristics of water resources, this new challenge offers a significant opportunity
for future research. This dissertation is an effort to contribute toward this goal.
29
Chapter 3: The Limarí River Basin’s Water Market
The coexistence of a market for water rights and a spot market for water
volumes is analyzed for four water user associations in the Limarí Basin in Chile’s
IVth Region. The existence of a legal framework that permits the transfer of water
rights independent of land rights has contributed to the development of a very active
water market with a variety of exchange mechanisms over the last 20 years. The
Limarí River basin is a semi-arid zone with approximately 65,000 hectares of
irrigated land used mainly in traditional crops such as maize, beans or potatoes,
horticultural production (artichokes, peppers and tomatoes), grains, grasses and other
valuable perennial crops such as avocados, export grapes and grapes used for pisco
7
.
The farmer base is diverse and consists of orchard owners, medium-sized farms
established by past land reform programs, and a few large multinational fruit
exporters. Each irrigation district possesses distinct climatic characteristics that favor
certain types of crops. The hydrologic system of the Limarí basin is characterized as
being primarily niveous, that is to say that it is fed from the snow-covered Andes
Mountains. The basin has an average annual precipitation of 140 ml. One essential
characteristic of this basin is the existence of three interconnected dams: Cogotí,
Recoleta, and the Paloma Dams. Together, these dams form the subbasin called the
Paloma System, which has a storage capacity of one billion cubic meters and
possesses a flexible physical system for the distribution of water based on floodgates
and a network of siphons and canals that allow interconnection to different irrigation
districts within this subbasin. The current Paloma System has six Water User
7
Grapes used in making local liquor.
30
Associations (WUAs), four of which are analyzed in this study: i) Junta de Vigilancia
del Río Limarí and its tributaries (JVRL); ii) Asociación de Canalistas del Canal
Camarico (ACCC); iii) Asociación de Canalistas del Embalse Recoleta (ACER); and
iv) Asociación de Canalistas del Embalse Cogotí (ACEC).
The data used in this section comes from a variety of sources. The series of
prices
8
and transferred water rights for the period 1981-1992 were reported by
Zegarra (2002) and, for the period 1992-2000, by Cristi et al. (2002) and Vicuña
(2000). These authors obtained this information through Conservador de Bienes
Raíces of Ovalle and the records of the WUAs. The series of prices and volumes of
water exchanged in the spot market were constructed using the records of the WUAs,
information obtained from the Direccion de Riego, and a farmer survey. This survey
was applied to a sample of farmers in the Limarí Basin on three occasions, and
information was collected for each of the five growing seasons between 1995 and
2000
9
. The surveyed sample was designed by Zegarra (2002)
10
who conducted the
first round survey. I conducted the second and third rounds (a detailed description of
the data is included in Chapter 5).
8
Unless otherwise noted, all prices are expressed in 1990 Chilean pesos. The
average peso-dollar exchange rate in 1990 was $304.9 pesos to the American dollar.
9
This survey was applied to 195 farmers of the region and contains production data
and figures on land and water use, among other information.
10
He did not develop a list of farmers to interview based on a random sample due to
the expense of finding each sampled individual; instead, he simulated random sampling for
farmers who were present at their farm when he conducted the survey. He began at some
point inside the irrigated area (stratum), interviewing farmers using a systematic round
skipping for close neighbors. This results in a sample, which is geographically representative
for each irrigation organization. The main limitation of this sampling procedure is that
farmers who were not present at the moment of the survey had zero probability of being
selected. The procedure also excludes farmers who, at the moment of the survey, had
abandoned production.
31
The following diagram illustrates the Limarí Basin and the Paloma System.
Figure 3.1: Limarí Basin and the Paloma System
3.1 Mechanism for the Allocation of Waters in the Paloma System
The allocation process determines the amount of water to be received by each
user. Conceptually, allocation is a distinct task from that of distribution. The latter is
defined as delivering water in accordance with allocations (Bruns and Meinzen,
2000). Legally, the Paloma system divides the Limarí basin into two districts: the
irrigation district that is located above the dams, and that below them, known as the
Paloma System. The Paloma System is the subject of this study. In this system,
water rights are defined in terms of cubic meters of stored water, and water is
COGOTÍ
DAM
Huatulame River
Grande River
Hurtado River
Limarí River
Grande River
Hurtado River
RECOLETA
DAM
LA PALOMA
DAM
OVALLE
Cogotí
Canal
Recoleta
main canal
OCEAN PACIFIC
Cogotí
distribution
canal
Camarico
Canal
Recoleta
distribution
Canal
Feeder canal
for Recoleta
Cogotí River
DAM
Artificial
canal
Natural
channel
32
distributed simultaneously to the farmers’ plots directly from the dams through its
associated canal network except during severe droughts when a rotating system of
distribution (also known as shifts) is implemented.
In the Paloma System, the responsibility of water allocation lies with the
Junta de Vigilancia del Sistema Paloma, which reports to the Dirección de Riego
(Irrigation Administration). Every year, the board adds up the amount of stored water
in the system and establishes a quantity of water for each irrigated area (and therefore
for each WUA) based on holder’s historical shares. The total volume of water to be
allocated in the system depends on existing levels. When the volume stored in the
system is less than 500 million m
3
, no more than half the stored water volume may be
assigned. When the system contains more than 500 million m
3
but less than 1 billion
m
3
, the maximum global assignment is 320 million m
3
. Lastly, when the volume
exceeds 1 billion m
3
, free use is granted to all WUAs.
Once water has been allocated among the WUAs, they assign the water
volume to their members. To do this, every season each WUA determines the
amount of water accorded to water rights. Then, by multiplying this amount by the
number of water rights owned by a farmer, it determines each farmer’s endowment of
water (expressed in cubic meters). During most seasons, the amount of water
accorded to water rights is determined by dividing the total allocated water to the
WUA by the total water rights that exist within it. Nevertheless, the expectation of
water availability for the next season can motivate the adoption of different criteria.
This distribution of water by the WUAs is at a farm or user level, and the WUA is
33
charged with the task of billing its associates for water use and administrative
expenses.
Table 3.1 and Figure 3.2 illustrate the variation in water accorded to water
rights for the associations under study between 1980 and 2000.
Table 3.1: Average water accorded to water rights between 1980 and 2000
WUA
AVERAGE
(m
3
)
STANDARD
DEVIATION
(m3)
ACCC 4,800 941
ACEC 5,193 1,182
ACER 3,955 1,168
JVRL 6,450 1,172
Source: Water User Associations
Figure 3.2: Water accorded to water rights between 1980 and 2000
0
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
8
0
-
8
1
8
2
-
8
3
8
4
-
8
5
8
6
-
8
7
8
8
-
8
9
9
0
-
9
1
9
2
-
9
3
9
4
-
9
5
9
6
-
9
7
9
8
-
9
9
Season
m3
Camarico Limarí Cogoti Recoleta
Source: Water user associations
11
11
In those seasons of free endowment, the highest amount of water accorded to water
rights for the period was recorded as that for those seasons.
34
3.2 Market Activity
The Limarí Basin beneath the dams is characterized as having an active water
market in which a spot market and a permanent transaction market coexist. In the
former, volumes of water are exchanged. In the latter, the purchase and sale of water
rights take place over time. After the annual allocation of water among water users
associations based on historical criteria and the posterior distribution of it among
farmers according to their number of water rights, the spot market reallocates this
resource to equalize differences in the marginal return to water among farmers
12
.
This process is facilitated by the existence of a significant number of farmers with
non-perennial crops that can, with relative ease, modify their water consumption by
varying the percentage of land used or the type of crops according to their water use
intensity. Moreover, water volume transactions are relatively easy due to the
existence of significant storage capacity. The use of flexible floodgates and the
proper operation of the water users associations also facilitate short term transactions.
Together, these factors support the existence of an active spot market. This
dissertation examines reasons for simultaneous water rights market and spot market
activity.
12
Differences in the marginal return to water are measured by Hearne and Easter
(1995). They estimated an average value for the marginal return to water rights in the case of
table grapes of US$ 856.7 and US$ 865.7 for the case of grapes used in pisco. This compares
to US$ 33.5 for potatoes and of US$317.5 for peppers, two of the main non-perennial crops
of the basin.
35
3.2.1 Spot Market
Because the allocation of water in each season among the WUAs in the
Paloma System does not necessarily coincide with the water demands of each WUA,
significant volumes of water are transferred among associations. As such, the
volumes of transfers received
13
by the four associations studied reached 24,189,000
m
3
in the 99/00 growing season representing 7% of the total amount of water assigned
to these associations. This number reached 16% for the 95/96 growing season, a year
of drought. As I analyze each association individually, I observe that, except for
ACER, which owns a volume of entries very similar to its actual outflows or debits,
the rest are net claimants of water (entries greater than debits) or net sellers (debits
greater than entries). ACCC and ACEC are examples of net claimants, and JVRL
stands out as an example of a net supplier. Thus, in the 99/00 growing season, the
irrigators of the ACCC and the ACEC obtained additional water rights from another
district equivalent to 34% and 7% of their water consumption, respectively. Such
figures reached 35% and 12%, respectively, in the 95/96 season. On the other hand,
during the 99/00 season the JVRL transferred 20% of its water assignment to other
associations while in the 95/96 season, transfers from that same association reached
40%.
That some associations are net water claimants or suppliers is explained by
differences in the marginal return to water and its availability. The fact that the
13
I can distinguish two types of water transfers: inter-association transfers and intra-
association transfers. The first group includes those transfers among WUAs, which can be
both entrances (when water is received from another association) and exits (when water is
transferred from one association to another). The second group includes those transfers
among irrigators within the same association.
36
ACCC and the ACEC are net claimants is explained by the presence of highly
profitable perennial crops such as grapes, and a significant share of irrigators who
belong to the ACEC develop crops on the land located above the Paloma Dam and
below the Cogotí Dam with excellent weather conditions and, thus, higher marginal
returns to water. In the case of JVRL, the practice of river water recovery by farmers
that increases their supply of water beyond that accorded to their water rights, partly
explains its condition as a net water seller.
Inside each association, irrigators produce different crops resulting in varying
marginal returns. This generates a significant level of internal water transfers
between those farmers with lower marginal returns to water and those with higher
marginal returns. During the 99/00 growing season, the total volume of internal
transfers for the WUAs under study reached 26,633,000 m
3
, which represents 8% of
the total amount of water assigned to these associations during this season. During
the period 1995-2000, the highest level of internal transfers occurred inside the
ACCC and ACEC associations accounting for 24% and 13%, respectively, of the total
amount of water assigned to each.
As the focus of this research is the water market, it is necessary to be precise
about the definition of water transfer in the spot market. First, it is necessary to
distinguish between the volumes of water transferred in the spot market from those
outside the spot market. In the former, transactions may occur between farmers in
different WUAs associations or among farmers within an association. Transfers that
the associations register as entries and debits have two different sources. The first
consists of transfers of water volumes made among different farmers and is counted
37
as water transferred in the spot market. The second are the so-called intra-farmer
water transfers. This is the case for farmers with two plots of land that are irrigated
with water from different WUAs, and the transfer reallocates water from one plot to
another. This is registered as a water transfer by the WUAs, but it occurs outside the
market, and is not a spot market activity. Transfers of water volumes among farmers
in different WUAs and transfers of water volumes among farmers within a same
WUA are considered spot market activities.
Unfortunately, it is not possible from the analysis of the WUAs records to
determine the exact magnitude of transfers between one farmer who has different
plots of land. Nevertheless, based on their experience, the executive directors of the
WUAs estimate that approximately 75% of the total transfers that imply an entry or
debit from one water association to another are made among different farmers, and
should be considered spot market transfers. For those transfers between an individual
WUA, records are clear on the type of transfer that occurs. Figure 3.3 illustrates spot
market activity obtained by adding together the total amount of transfers among
farmers within each WUA with 75% of the transfers that occur between different
WUAs
14
.
14
The evolution of spot market activity shows that the volume of water exchanged
reaches a low in the 97/98 growing season, an occurrence that is explained by a drought
during the first half of the season, and by heavy rains during winter that led the WUAs to
declare free river. This is to say, during that same season, there was a lack of supply at the
beginning of the season and a surplus later decreasing market activity for the season.
38
Figure 3.3: Transfers of water in the spot market
0
10
20
30
40
50
60
95-96 96-97 97-98 98-99 99-00
Season
Millions of m
3
0%
5%
10%
15%
20%
25%
Total transfers
Percentage of total transfers over the total amount of water
Source: Direccion de Riego, the Paloma System Administration.
3.2.2 Permanent Market
The spot market for water coexists with a separate market for water rights
(henceforth referred to as the permanent market). Beginning from an initial
assignment of water rights by the government, these rights are reallocated to farmers
through the market. When analyzing the activity of this market, I observe that the
total percentage of reallocated water rights, independent of land transactions during
the period 1980-2000, varies from 20% to 50% by WUA. Since the approval of the
water law in 1981, and until 2000, more than 27% of the total water rights have been
transferred through the permanent transactions market
15
.
15
Adding together the water rights of the different associations is imprecise because
the volumetric specification of these water rights is different for each association.
39
The analysis of the behavior of the permanent transactions market over time
shows how the activity in this market has grown. Figure 3.4 shows how the earliest
transactions had lower volumes of trade than later periods reflecting a market that
matured in its first decade.
Figure 3.4: Number of water rights transferred by each association during the
period 1980-2000.
0
200
400
600
800
1
9
8
0
-
8
1
1
9
8
2
-
8
3
1
9
8
4
-
8
5
1
9
8
6
-
8
7
1
9
8
8
-
8
9
1
9
9
0
-
9
1
1
9
9
2
-
9
3
1
9
9
4
-
9
5
1
9
9
6
-
9
7
1
9
9
8
-
9
9
Season
Water Rights
ACCC
ACEC
ACER
JVRL
Comparing the size of the spot market with the permanent market is difficult
because the former trades in volumes of water during a specific growing season while
a water right transaction implies the transfer of variable volumes of water over time.
In spite of these restrictions, I have compared the relative size of both markets
through the following steps: i) expressing the trade in rights in volumetric terms for
each season using the average amount of water accorded to water rights (see Table
3.1); ii) assuming that the sale (purchase) of a water right in any season is equivalent
to the sale (purchase) of the amount of water accorded to that right in the
40
spot market in all following seasons
16
; iii) assuming that only a portion of the
transfers among WUAs are conducted through the spot market due to the accounting
conventions discussed above in Section 3.2.1. These conventions require that we
consider the following two scenarios. In one, only transfers among different farmers
within a WUA are counted as spot market transactions (scenario 1); and in the other
transfers among different farmers within a WUA plus 75% of the transfers among
WUAs are counted as spot market transactions (scenario 2)
17
. A possible drawback
of this method is that it could lead to an over estimation of the relative size of the
permanent market because rights may be transferred several times in the period under
study resulting in the volume of water associated with those rights being double
counted. However, the history of water right transfers independent of land
transactions shows that few farmers who buy water rights sell them later. The
exceptions are a few water right holders who do not own land, but buy water rights to
sell them when prices rise. The detailed operation of the market indicates that
overestimation of the permanent market is not significant. As a result, I conclude that
16
One possible scenario is as follows:
Association XX 95-96 96-97 97-98 98-99 99-00
Exchanged water
rights (in cubic
meters)
100 200 100 50 100
Total cubic
meters
100 300 400 450 550
17
As I already mentioned, 75% of the total transfers that involve a change of water
association are assumed to be transfers among different farmers; therefore, they are
considered to spot market transactions. This assumption is based on interviews of the
executive directors of the four water user associations under study. Moreover, the content in
this chapter was presented in the in the city of Ovalle (Limarí Valley) in a seminar with more
than 20 attendees that included the executive directors and the water engineers of the WUAs.
During the presentation, I emphasized the assumption that few farmers that buy water rights
sell them later independent of land sales, and that approximately 75% of the transfers among
WUAs are done through the spot market. However, many non-attendees disagreed with those
assumptions.
41
for the period 1995-2000, the water transferred in the spot market was 3.8 times
greater than transfers in the permanent market under scenario 1 and nearly 7 times
that in scenario 2.
3.3 Price Behavior in the Water Market
To verify if a market behaves in an efficient manner, I test to determine
whether prices reflect the relative scarcity of water. In the following sections, I will
show that water prices in the spot market reflect its relative seasonal scarcity. In
regards to the permanent market, the systematic rise in the real prices of water rights
reflects a sustained expansion of the demand for water over time.
3.3.1 Spot Market
Evidence exists that the behavior of prices in the spot market reflects the
relative scarcity of the resource. Analysis of the period 1995-2000 illustrates that the
maximum real price per cubic meter of water is reached in all the WUAs in the 96/97
growing season, which suffered a drought. When comparing the real prices between
the 96/97 season and a normal season such as 99/00, prices during the drought season
are between 3 and 12 times greater than those of a normal season, varying by
association. The capacity of the spot market to reflect the relative scarcity of water
can also be observed through the correlation coefficient of water availability and the
average real price per cubic meter, which is –0.921 for the five growing seasons
between 1995 and 2000
18
.
18
This coefficient correlates the average real price of water in the spot market with
the sum of the water accorded to water rights in the WUAs between 1995 and 2000. The
price in the spot market is estimated as a weighted average of the prices as reported by
surveyed farmers that exchanged water. Overall, 123 observations of temporal transfers in
five growing seasons were used.
42
3.3.2 Permanent Market
The increasing scarcity of water due to a sustained expansion of the demand
over time has produced a systematic rise in the real prices of water rights. Thus,
during the period 1986-2000, the price of water rights experienced real increases
ranging from 41% to 240% depending upon the association. Figure 3.5 illustrates the
evolution of these prices over time.
Figure 3.5: Real average prices of water rights in each WUA
0
100.000
200.000
300.000
400.000
500.000
600.000
700.000
800.000
900.000
8
0
-
8
1
8
2
-
8
3
8
4
-
8
5
8
6
-
8
7
8
8
-
8
9
9
0
-
9
1
9
2
-
9
3
9
4
-
9
5
9
6
-
9
7
9
8
-
9
9
Season
$
1
9
9
0
Camarico
Cogot i
Recolet a
Junt a
Heterogeneity in the prices of water rights among associations tends to
diminish over time with exception of ACEC. The higher water rights prices in some
associations are largely explained by differences in the alternative mechanisms to
purchase water rights. More specifically, I can distinguish between two markets: The
first is restricted to the area below the Cogotí Dam and above the Paloma Dam, which
is characterized by limited suppliers of water rights due to the relatively small number
of farmers. In addition, several farmers in this area grow export grapes and have a
43
higher marginal return on water than farmers located in other zones in the basin. The
second one includes the area below the Paloma Dam, and is characterized by the
existence of a relatively large number of irrigators. Increasing the number of farmers
increases the dispersion of reservation values improving the probability of water right
sales. Figures 3.6 and 3.7 show the reservation values of a permanent right to one
cubic meter of water as reported by farmers from the ACEC and ACER WUAs. As
expected, water right reservation values are higher in the former market.
Figure 3.6: Reservation value for the right to one cubic meter of water in ACEC
0
1
2
3
4
5
D
o
e
s
n
'
t
k
n
o
w
0
-
1
2
5
1
2
5
-
1
7
5
1
7
5
-
2
2
5
2
2
5
-
2
7
5
2
7
5
-
3
2
5
3
2
5
-
4
5
0
4
5
0
-
5
5
0
5
5
0
-
6
5
0
g
r
e
a
t
e
r
t
h
a
n
$ (pesos)
F
r
e
q
u
e
n
c
y
0%
20%
40%
60%
80%
100%
120%
A
c
c
u
m
u
l
a
t
e
d
P
e
r
c
e
n
t
a
g
e
44
Figure 3.7: Reservation value for the right to one cubic meter of water in ACER
0
2
4
6
8
10
12
D
o
e
s
n
'
t
k
n
o
w
1
0
0
-
1
2
5
1
2
5
-
1
5
0
1
5
0
-
1
7
5
1
7
5
-
2
0
0
2
0
0
-
2
2
5
2
2
5
-
2
7
5
2
7
5
-
3
0
0
3
0
0
-
5
2
5
$ (pesos)
F
r
e
q
u
e
n
c
y
0%
20%
40%
60%
80%
100%
120%
A
c
c
u
m
u
l
a
t
e
d
P
e
r
c
e
n
t
a
g
e
3.4 Market Expectations and Access to Information
By observing water transactions in the spot market in each season, it is
possible to see that in most seasons the market does not become active until after the
passage of winter when the volume of rainfall is known and the need for water
increases due to the hot, dry summer weather. This seasonality provides some
support to the statement that the farmers have homogeneous information with respect
to the future; therefore, no additional benefits are associated with advance purchases
and sales of water volumes. The exception to this behavior is observed in the 97/98
growing season, which was preceded by a severe drought. The experience of three
previous years of drought and the relatively high prices in the spot market at the
beginning of that season could have motivated advance water purchases by farmers
45
who expected water prices to increase and advance water sales from sellers who
expected water prices to decrease. This seasonal behavior of the spot water market
and the advance sales of water in the 97/98 growing season are illustrated in Figure
3.8 for ACEC.
Figure 3.8: Transfers of water through the spot market
0
1.000.000
2.000.000
3.000.000
4.000.000
5.000.000
6.000.000
M
a
y
J
u
n
J
u
l
A
g
o
S
e
p
O
c
t
N
o
v
D
i
c
E
n
e
F
e
b
M
a
r
A
b
r
Month
m3:
96-97, 98-99, 99-00
seasons
0
50.000
100.000
150.000
200.000
250.000
300.000
m3: 97-98 season
96-97 season 98-99 season 99-00 season 97-98 season
With respect to rain expectations and the expected amount of water accorded
to water rights, I found agreement among surveyed farmers. When asked during the
third round survey about rain expectations for the following season, 70% responded
that it would be a normal year. In addition, farmers were asked in that same survey to
estimate the amount of water accorded to water rights in the following year as a share
of what they currently receive. Almost 50% of surveyed farmers estimated that the
water accorded to their water rights in the following period would be between 90%
and 120% of their current level. The majority of the remaining farmers (47%) admit
that they do not have an estimate of the future level of water accorded to water rights.
46
With regard to information access, I observe that the majority of the farmers are well
informed about water availability in the dams. When asked about the level of the
dams, 85% of them correctly answered that the dam was half-full or full
19
.
3.5 Summary
The current water law in Chile has been flexible enough to allow the Paloma
System, in the Limarí Basin, to develop not only an active market for water rights, but
also an active market for volumes of water, i.e. a spot market. The volume of water
exchanged in this market indicates that this mechanism of water allocation is highly
important. In the permanent market, more than 27% of existing water rights were
exchanged independently of land transfers between 1981 and 2000.
A flexible water market such as the spot market in the Paloma System allows
for the reallocation of water to those areas in which the water acquires its greatest
value. This market -contrary to what other researches believe- is active not only in
drought years but also in years with average water availability. Thus, in the 99/00
season characterized by the average water availability, approximately 14% of the
water allocated to the four main WUAs of the basin was reallocated through
exchange in the spot market. During the severe drought of the 95/96 season that
figure reached a value of 21%.
The Water User Associations are a primary factor in determining the correct
functioning of the market. Thanks to good organization and efficient management of
their records of water allocation, it has been possible to develop an active spot
market. Together with the Water Use Associations, it is also clear that the existence
19
At the time of the third round survey the amount of stored water stood at 70% of its
total capacity.
47
of a safe supply source, such as the three dams that form the Paloma System, is a
necessary condition for the existence of a spot market.
The correlation between the growth in the relative scarcity of water over time
and the prices of water rights, as well the correlation between the scarcity of water per
season and the price of water in the spot market, indicates that water markets operate
correctly. At the same time, for the market for water rights there is some
heterogeneity in prices that tends to diminish over time with exception of ACEC.
This heterogeneity is also captured when farmers are asked about their reservation
price for a water right. As expected, water right reservation values are higher for the
farmers that belong to ACEC.
Finally, survey questions regarding access to relevant information and water
market expectations supports the hypothesis that farmers’ access to information is
homogeneous and farmers form similar expectations with regard to future water
prices and the availability of water supplies. From this, I assume in the coming
chapters that farmers have homogeneous expectations with respect to the main
stochastic variables in the water markets.
48
Chapter 4: Theoretical Model
In this chapter I pursue an explanation for water rights trading, which arises
whenever there are differences in the reservation value of the water right asset. I use
this reservation value as a proxy for private valuation, and hypothesize that difference
in reservation values arise from heterogeneity in farmers’ preferences. To establish
the importance of heterogeneity in farmer preferences, I first consider a simple model
where investors are risk neutral. Under this assumption, the price of a water right is
equal to the expected discount sum over time of the spot market values of the amount
of water accorded to that water right.
4.1 A Simple Model
Let us define R
t
as the return on a water right in period t. It is a function of
the change in the water right price between t+1 and t, where
t
? denotes the water
right price in period t, the spot market price per cubic meter, s
t
, and the total water
(in cubic meters) accorded to each water right in season t, denoted by v
t
. I assume
that ? , s and v are stochastic variables that are revealed during each growing season.
Thus, at t the values of
t
? , s
t
and v
t
are known. Hence, the return on a water right is
what can be earned by buying a water right in period t, selling the assigned volume of
water on the spot market in period t+1, and then re-selling the water right in period
t+1:
1 1 1
.
1
s v
t t t t
R
t
t
? ?
?
? +
+ + +
=
+
(4.1.1)
49
In order to simplify the analysis I can assume that the expected water right
return is equal to a constant, R, and:
[ ] E R R
t t l
=
+
, for l= 0,1,2,.. (4.1.2)
where E
t
stands for the expectation operator conditional on the information set
available at time t, which is assumed to be known by all farmers.
The assumption that the expected return remains constant is sometimes known
as the Martingale Model of stock prices. Although that assumption of constant stock
returns contradicts the empirical evidence of returns behavior over time, it is
analytically convenient for the goal of this section, which is to show the importance
of heterogeneity in farmer preferences in the reservation values of water rights.
Developing the model with time-varying expected returns will lead to the same
conclusion, but the analysis is cumbersome because the relationship between prices
and returns becomes nonlinear (Campbell, et al., 1997, Chapter 7.1).
By taking expectations in (4.1.1) over
1 t
?
+
and
1 1
s v
t t + +
, imposing (4.1.2),
and rearranging terms, I obtain an equation that links the current water right price
with the next period’s expected water right price and the water price in the spot
market:
1 1 1
1
s v
t t t
E
t t
R
?
?
+ (
+ + +
=
(
+
¸ ¸
(4.1.3)
Recursively iterating forward the future prices of water rights and using the
Law of Iterated Expectations, the following equation is obtained:
1
1
1
l
L
E s v
t t t l t l
R
l
?
(
| |
(
= ¿
+ + |
(
+
\ .
=
¸ ¸
+
1
1
L
E
t t L
R
?
(
| |
(
+ |
+ ( \ .
¸ ¸
(4.1.4)
50
Ruling out the possibility of a rational bubble in the market, i.e. the water
right price
t
? is not expected to grow forever at a rate R or faster, the second term on
the right-hand side of this equation shrinks to zero as the time horizon, L, increases.
As such, equation (4.1.4) may be written as:
1
1
1
l
E s v
t t t l t l
R
l
?
(
?
| |
(
= ¿
+ + |
(
+
\ .
=
¸ ¸
(4.1.5)
If all farmers have identical expectations about water right returns, R , water
prices in the spot market, s
t l +
, and water quantities accorded to each water
right, v
t l +
, then they would have identical private water right valuations (or
reservation values), and no trading would take place.
To explain why water rights are exchanged in the Limarí Valley water market,
some type of heterogeneity among farmers must be assumed. As I mentioned in
Chapter 3, farmers’ market expectations and access to information indicate that
farmers have homogeneous information regarding the future of water prices and
water quantities accorded to water rights and, thus, homogeneous expectations. The
analysis of rain expectations and the expected amount of water accorded to water
rights for the coming season revealed a high coincidence in answers among surveyed
farmers who were also well informed about water availability in the dams.
Consequently, I focus on other types of heterogeneity among farmers to explain
differences in reservation values for a water right. One source for farmer
heterogeneity arises from differences in the marginal productivity of water due to
differences in soil quality, equipment (for example tractor use), and irrigation
systems, among others. These differences in marginal productivity are resolved in the
51
spot market for water, where farmers with higher marginal productivity will buy
water from farmers with lower marginal productivity until the differences vanish in
each agricultural season. Hadjigeorgalis, (2000) provides a theoretical and empirical
proof for this statement as well as empirical evidence of a unique price of water in the
spot market for the whole area below dams in the Limarí Valley. Accordingly,
differences in marginal productivity are not sufficient to explain trading in the market
for water rights.
Another potential source of heterogeneity is a variety of idiosyncratic random
shocks that affect farmers’ income. Constantinides (1996) and Heaton and Lucas
(1996), among others, consider the effect of idiosyncratic random shocks on asset
prices. Farmers in the Limarí Valley face independent shocks to their incomes as
revealed by the second survey when some farmers reported important frost damage to
their crops in just one night. Pest infestations are another potential source of shock,
although the farmers under study use pesticides. If farmers face incomplete asset
markets these shocks affect consumption and hence the reservation values for a water
right if farmers are not risk neutral. In practice, that is the case because most farmers
do not insure against transitory idiosyncratic shocks. The spot market for water helps
farmers to smooth household consumption, but does not totally preclude them from
the need to modify consumption due to idiosyncratic shocks.
Finally, heterogeneity of farmers’ preferences is a sufficient condition for
differences in the reservation value of water rights among farmers. In the next
section, I develop a model of a farmer’s decision for optimal consumption and
52
investment in water rights and use it to illustrate the role of incomplete asset
markets
20
and farmers’ preferences upon reservation values for water rights
4.2 The Farmer’s Decision Rule under Water and Output
Uncertainty
The goal of this section is to analyze the role of farmers’ preferences in
determining reservation values for water rights. I consider the case of irrigated
agricultural land in a region with stochastic water availability. The farmer has
income that can either be consumed or invested in water rights. In this problem
uncertainty comes from different sources: output uncertainty in season t, future spot
market and water right prices, and future amounts of water accorded to water rights.
The particular model I develop for farmer consumption and investment in
water rights is based on the conventional consumption-based capital asset-pricing
model (CCAPM) where the asset under study is a water right. In this model, water
right returns are linked to marginal rates of substitution for consumption at different
points in time. Alternatively, a production-based asset-pricing model (PCAPM) may
be developed emphasizing the linkages between asset returns and investment and
production variables. Data availability is key when it comes to deciding between the
CCAPM and the PCAPM. The first requires consumption data and the second
investment and capital data. Unfortunately, my data is not ideally suited to either of
these. First, while it reports farm equipment, it differs broadly in regards to numbers,
quality and age, and any effort to value farmer capital is unreliable. Second, it does
not contain information on consumption. I do have reliable data on current net
20
Incomplete asset markets occurs when some insurance markets are absent so that
farmers can not insure against each state of nature.
53
income which, in a framework with incomplete asset markets, is correlated with
consumption
21
. For this reason I have chosen the CCAPM.
I assume that the farmer’s optimal decision rule regarding consumption and
investment in each season is the solution to the maximization of the expectation of his
intertemporal utility of consumption subject to a budget constraint. I consider the
case of a farmer for whom the intertemporal utility of consumption takes the form of
a time-additively separable function, such that the expected present value of his utility
is given by:
( )
1
T
t
E U C
t it
t
? ¿
=
(4.2.1)
with i=1…,n and t=1…,T. Here n is the total number of farmers, T is the total
number of seasons, C
it
denotes farmer i’s consumption in season t and ? is a
constant discount factor measuring the rate of time preferences. Given that 0< ? <1,
one unit of utility tomorrow is valued less than one unit of utility today. The utility
function ( ) U is assumed to be twice continuously differentiable, with
( ) dU C
t
U
t
dC
t
? = >0 and
( )
2
2
d U C
t
U
t
dC
t
?? = <0, i.e. the marginal utility of consumption is
21
Under the permanent-income hypothesis, proposed by Milton Friedman in 1957,
increases and decreases in income that farmers see as temporary have little effect on their
consumption spending. Nevertheless, that independence between consumption and current
income requires the capacity of farmers to counteract specific random shocks on their current
income. They are able to do that if they have access to a complete asset market or have assets
that deliver wealth when they face unexpected reductions on their current income. If none of
those conditions is fulfilled then consumption is highly correlated with current income.
54
positive, but it decreases with consumption. Expectations in (4.2.1) are taken
conditional on information available at t.
The assumption of a time-additively separable utility is standard in the
literature, but it is not without restriction. It implies that farmers’ current utility is not
affected by the timing of the resolution of uncertainty (Duffie et al., 1997). In other
words, it rules out preferences for early or for late resolution of uncertainty. A
second drawback of this utility function is that the elasticity of intertemporal
substitution is inversely linked with the coefficient of relative risk aversion (Epstein
and Zin, 1981 and Just, 2000). Finally, farmers’ current utility is assumed to be
unaffected by past consumption. This rules out habit formation in consumption
where the marginal utility of future consumption is increasing with the level of past
consumption. Just (2000), demonstrates that assuming additive separability of utility
may be inadequate for studying longer-term agricultural production problems. This is
so because the expected utility over a time horizon with an additive separability
function does not take into account possible correlations across time of the variable
over which utility is specified. In my setting, the role of habit formation is mitigated
for two reasons: i) survey field experience indicates that durable goods are only a
small share of farmers’ consumption bundles in the Limarí Valley; and ii) optimal
consumption depends on state variables that are related to the past history of
consumption and investment. This indirectly links utilities of different periods
(Bossaerts, 2002). A primary advantage of time-additive utility is that preferences are
recursive and this allows dynamic programming methods to be used to analyze
optimal decisions. Chavas (2004) points out that nonadditive preferences are more
55
difficult to specify and evaluate, and he concludes by stating that “The reader should
keep in mind that the time-additive model remains a popular framework for the
analysis of dynamic behavior”.
Before developing a model of farmers’ consumption and investment
decisions, two main features of the water market under analysis require discussion.
First, the aggregate demand curve for water volumes in the spot market is assumed to
be deterministic. Consequently, s
jt
which is the price of water in the spot market of
WUA j in season t, varies from one season to another as water supply changes. Thus,
I have:
( )
s s v
jt jt
= . (4.2.2)
Because the number of water rights is fixed in every WUA, changes in water
supply are explained only by changes in v
jt
.
The second important feature of this case study is that water supply depends
on the level of water in the dams, which is known at the beginning of the agricultural
season in April. At this time each WUA determines the amount of water accorded to
each water right, v
jt
. Hence, uncertainty about water availability and the price of
water in the spot market is eliminated at the beginning of each season although it
remains unresolved for future seasons. Nevertheless, the WUAs may change the
amount of water accorded to water rights after April. This can be due to high levels
of rain in the winter or to the extraordinary size of snow packs, the main source of
water for the dams formed in the surrounding mountains during the winter months of
May, June and July, which could generate a water flow that overwhelms the storage
56
capacity of the dams during the summer. In these cases, the WUAs declare “free
river” so each farmer can take freely whatever amount of water he requires
22
. The
analysis of the time series of v for ACEC over 46 seasons shows that this WUA
declared free river in only 8 seasons and there were no seasons for which it modified
the amount of v announced at the beginning of the season. The JVRL announced a
modification on v at the beginning of a season in 5 out of the 22 observed seasons,
with an average change of 17%, and declared free river in four seasons. For the other
WUAs, either this information is not available or they did not modify v except when
declaring free river
23
. For the purposes of my analysis I assume that the normal case
is when water accorded to water rights is known at the beginning of the season and
does not change. This means that v
jt
and
( )
s s v
jt jt =
are known by the farmers at
that time.
The farmer’s stochastic problem is to choose a sequence of consumption over
time that maximizes the farmer’s expected utility in equation (4.2.1) subject to: an
22
In theory, the amount of water accorded to water rights announced at the beginning
of the season may be reduced although this never occurred in the years under study. In
addition, the WUA of the Cogotí dam permitted the dam to run dry in seven seasons between
1954 and 2000.
23
Number of seasons in which the WUAs declared free river.
WUA Number of growing seasons in the
available series for v
jt
Number of years declared
free river
ACCC 20 7
ACEC 46 8
JVRL 22 4
ACER 23 5
57
initial stock of water rights
0
N , a budget constraint, and a water availability
constraint.
Due to the fact that most of the farmers in the Limarí Valley face incomplete
markets
24
, it is reasonable to assume that farmers’ decisions over consumption
depend on net revenue from current production and current income from water
transactions. Thus, the budget constraint for farmer i in WUA j is defined by:
( ) ( ) C , ,
1
N N s z P T f w X r X I
it jt it it jt it it it it it it it it it
? ? ? = ? + + ? +
?
(4.2.3)
The first two terms of the right hand side of the budgets constraint represent
net income from water transactions, where
jt
? denotes the price of a water right in
WUAj at period t and is the same for all farmers in a WUAj; N
it
is the number of
water rights held by farmer i during season t; ( )
1
N N
jt it it
? ?
?
is the net cost of
investment in water rights at time t; s
jt
denotes the spot market price per cubic meter
in WUAj in season t, and z
it
is the volume of water exchanged in the spot market by
farmer i in season t, where z
it
>0 implies sale of water volumes and z
it
<0 implies
the purchase of water volumes. The third and fourth terms in the budget constraint
represent net revenue from production, where T
it
is the total amount of cultivated
land; ( ) , , f X w
it it it
? denotes a per hectare production function of an aggregate
output; w
it
denotes the volume of water used as an input in production and X
it
is a
vector of inputs different from water (all inputs are expressed in terms of per hectare);
it
? represents the value of a vector of shocks that are partially determined by nature;
24
Farmers face incomplete markets because only few of them have crop insurance
and access to credit is limited.
58
P
it
is the price of the aggregate output faced by farmer i in season t; and r
it
is the
vector of input prices for X
it
. Output and input prices are assumed to be known.
Finally, I
it
represents farmer’s “other net income” sources including consumption or
production credit, land transactions, livestock exchanges and work off-farm.
The water availability constraint is:
N v z T w
it jt it it it
? +
25
. (4.2.4)
It limits the amount of water that a farmer can use for irrigation, T w
it it
, and the total
volume that he can exchange in the spot market, z
it
.
Without violating the rule that utility is function of consumption
26
, I can
substitute the budget and the water constraints inside the utility function and express
the farmer’s expected present value of utility as:
( )
( )
( )
( ) ( )
, ,
1
1
T
t
E U N N s N v T w P T f w X r X I
jt it it jt it jt it it it it it it it it it it
t
? ? ? ? ? + ? + ? + ¿
?
=
(4.2.5)
The farmer’s problem now involves choosing investment in water rights and
input quantities to maximize his expected discounted sum of utility in equation
(4.2.5), subject to an initial stock of water rights. Hence, the consumption decision
problem has been transformed to an investment and production decision problem.
25
This assumption is an oversimplification because farmers in some WUAs may
occasionally use more water than the volume they obtain for the season by asking for an
advance on water from the next season. Farmers also may save water from one season to the
next, but there is a penalty of 15 to 20% of the endowment due to projected evaporation
losses making this practice rare (Zegarra 2002).
26
“..good economists are unhappy about a utility function that has wealth in it. Few
of us are like Disney’s Uncle Scrooge, who got pure enjoyment out of a daily swim in the
coins in his vault. Wealth is only valuable because it gives us access to more consumption.
Utility functions should always be written over consumption” (Cochrane, 2005, pg. 157).
59
The overall problem of finding the optimal sequence of water rights
transactions can be solved by using a standard dynamic programming approach, i.e.
by finding the optimal amount of water rights,
it
N for t=1,… T, and input quantities
( , w X
it it
), that satisfy the recursive functional equation:
( )
( )
( )
[ ]
1
( , )
, , 1 , ,
( , )
1
N N s N v T w
jt it it jt it jt it it
E U
t
V N Max
P T f w X r X I it it w X N
it it it it it it it it it it it
E V N
t it it
?
µ
?
? µ
¦ ¹
( | |
? + ? +
?
¦ ¦
( |
¦ ¦
( |
=
? ´ ` ? + ?
\ . ¸ ¸
¦ ¦
¦ ¦ +
+
¹ )
(4.2.6)
where
it
µ is the vector
it
µ =
( )
, , , , , v s T P r
jt jt jt it it it
? ? . For notational simplicity, the
dependence of the value function on the term
it
µ will be suppressed in what follows
unless it is needed to avoid confusion. The value function, ( )
1
V N
it ?
, denotes the
value of the stock of water rights held from t-1 to t,
1
N
it ?
, after the optimal sequence
of
it
N , t=1,…T has been determined and is the indirect utility function of that same
stock of water rights. In the first term the expectation is over
it
? (its value is resolved
at harvest time) and the random variable I
it
. The values of , , s v
jt jt jt
? are known
at t but their future values are unknown. The values of T
it
and
1
T
it +
are assumed to
be known. Thus, the expectation in the second term is over
1 it
?
+
, as well
as , ,
1 1 1
s v
jt j jt
?
+ + +
, and
1
I
it +
. Output prices, P
it
, and input prices, r
it
, are
certain. The control variables are N
it
, w
it
and X
it
. The state variables are
1
N
it ?
and
it
µ . I assume that ( ) U and ( ) f are continuously differentiable, the value
60
function is differentiable in N
it
, and that the optimization problem has an interior
solution.
The first order condition with respect to N
t
(omitting the indices i and j for
convenience) is:
( ) [ ]
1
E U s v E V
t t t t t t t
? ? ? ? ( ? ? =
+
¸ ¸
(4.2.7)
where ( )
1
V V N N
t t t
? = ? ?
+
is the marginal value of a water right held at the
beginning of period t+1.
Using the envelope theorem, the right hand term in (4.2.7) can be expressed in
terms of the discount factor, marginal utility, water right prices and the spot market
value of water accorded to each water right. I begin with the value function for
period t+1 evaluated at the optimal level of each one of the variables that are included
in it:
( ) ( )
( )
[ ]
1 1 1 1 1 1
1
1
( ) , ,
1 1 1 1 1 1
( )
1 1
N N s N v T w
t t t t t it t
t
E U
t
V N P f w X r X
t t t t t t t
E V N
t t
?
?
?
¦ ¹ (
| | ? + ? +
+ + + + + +
+
¦ ¦ ( |
+ ¦ ¦
|
( = ? ?
´ `
+ + + + + +
\ . ¸ ¸
¦ ¦
+
¦ ¦
+ +
¹ )
(4.2.8)
Then I differentiate equation (4.2.8) with respect to N
t
, to obtain:
[ ]
1 1 1 1
V E U
t t t t
? ? ? =
+ + + +
. (4.2.9)
Next I substitute equation (4.2.9) into equation (4.2.7), and use the Law of
Iterated Expectations, to obtain the first order condition with respect to N
t
:
( ) [ ] s v E U
t t t t t
? ? ? = [ ]
1 1
E U
t t t
? ? ?
+ +
(4.2.10)
Equation (4.2.10) is the so-called Euler Equation. The left hand side term
indicates how much the farmer’s utility increases if the available income for
61
consumption rises due to a sale of a water right at the price of
t
? . The deduction of
s v
t t
from
t
? , to get the net income from selling a water right, is due to the fact that
whenever a farmer sells a water right he will not be able to sell the amount of water
accorded to that water right, v
t
, on the spot market at a value of s v
t t
. The right hand
side term is the farmer’s expected discounted utility of keeping the water right from
season t to t+1, which may also be interpreted as the farmer’s expected forgone
discounted utility if he sells a water right in season t. In other words, this term can be
understood as the marginal cost of selling a water right in period t. Thus, (4.2.10)
reflects the optimality condition that the marginal benefit of selling a water right (the
left hand side term) is equal to its marginal cost (the right hand side term). This
allows me to interpret equation (4.2.10) as giving the marginal value or willingness to
pay for a water right (Cochrane, 2005, Chapter 2.1). More specifically, I can rewrite
the Euler Equation as:
( ) ( )
( )
( ) ( ) ( ) ( )
1 1 1 1 1 1 1
1 1
, ,
1 1 1 1 1 1 1 1
, ,
1
N N s N v T w
t t t t t t t t
U
t t
P T f w X r X I
t t t t t t t t
s v E
t t t t
E U N N s N v T w PT f w X r X I
t t t t t t t t t t t t t t t t t t
?
?
?
? ?
? ?
( | | ? + ? +
+ + + + + + +
? ( |
+ +
|
? ? +
(
+ + + + + + + +
\ .
= +
(
? ? ? + ? + ? +
?
(
(
¸ ¸
(4.2.11)
where the term on the right hand side of (4.2.11) is the willingness to pay or the
reservation value for a water right.
Furthermore, the stochastic discount factor in this problem is defined by the
random variable
1
M
t +
, where:
62
( ) ( )
( )
( ) ( ) ( ) ( )
1 1 1 1 1 1 1
1
, ,
1 1 1 1 1 1 1 1
1
, ,
1
N N s N v T w
t t t t t t t t
U
t
P T f w X r X I
t t t t t t t t
M
t
E U N N s N v T w PT f w X r X I
t t t t t t t t t t t t t t t t t t
?
?
?
? ?
| | ? + ? +
+ + + + + + +
?
|
+
|
? ? +
+ + + + + + + +
\ .
=
+
? ? ? + ? + ? +
?
(4.2.12)
Thus, the Euler Equation becomes:
[ ]
1 1
s v E M
t t t t t t
? ? = +
+ +
(4.2.13)
Following the tradition of the literature on asset pricing, the Euler Equation
can be written as:
1
1
1
s v
t t t
E M
t t
t t
?
? ?
(
+
= +
( +
¸ ¸
. (4.2.14)
The right hand side term in (4.2.14) is the discounted expected value of the
marginal rate of return to holding a water right from time t to time t+1. This marginal
return is comprised of two terms: the first term is s v
t t t
? , and it represents the rate of
return to a water right when the water accorded to it is sold in the spot market; and the
second term is
1 t t
? ?
+
, which represents the rate of change in the water right price
from season t to season t+1. Finally, expression (4.2.14) indicates that the optimal
investment in a water right occurs when the discounted expected value of the
marginal return to holding an extra water right is equal to 1. This formula is the
foundation of the consumption-based capital asset pricing model (CCAPM).
The Euler Equation can also be written in terms of the value of water accorded
to each water right in the spot market,
1 1 t t
s v
+ +
, for t = 0,1,2…T. To do this I solve
equation (4.2.13) forward by repeatedly substituting out the future price of water
rights,
1 t
?
+
, for l = 1,2…T, and I obtain:
63
[ ]
1 1
E M
t t t t t t
v s ? ? = +
+ +
[ ]
1 1 1 1 2 2
E M
t t t t t t
v s ? ? = +
+ + + + + +
[ ]
2 2 2 2 3 3
E M
t t t t t t
v s ? ? = +
+ + + + + +
[ ] [ ]
2 2 1 1 1 1 1 2
E M E E M M
t t t t t t t t t t t t t
v v v s s s ? ( = + +
+ + + + + + + +
¸ ¸
which, by the Law of Iterated Expectations, becomes:
[ ]
2 2 2 1 1 1 1
E M v M M v
t t t t t t t t t t t
v s s s ? = + +
+ + + + + + +
(4.2.15)
This equation states that the reservation value of a water right in period t is
determined by adding together the present and expected future values of the amount
of water accorded to a water right in the spot market, discounted by the stochastic
discount factor. In order to explain the economics behind this statement, consider a
representative farmer who lives only two periods, t and t+1, and has only one water
right, which is accorded one m
3
of water. If that farmer decides to use his cubic
meter of water as an input in crop production, he does so because he obtains a greater
return than what is expected from selling that water in the spot market. In
equilibrium, he would then buy water in the spot market until the marginal benefit of
water used as an input equals the expected present benefit of selling water in the spot
market
27
.
27
This is the case for a costless spot market for water such as that of the Limarí
Valley. Up to now, nobody has measured the real transaction cost on the spot market.
Zegarra (2002) indicates costs should be significant because when he conducted his survey in
the Limarí Valley, farmers were facing a severe drought and complaining about the difficulty
in purchasing water in the spot market. However, that is not in itself a proof of high
transaction costs. What happened is that, due to the severe drought, prices were quite high at
that point, and several farmers were not willing to pay those prices. Indeed an active water
market existed at that time in Ovalle’s plaza with water being sold to the highest bidder.
64
In order to examine the role of risk preferences and incomplete asset markets
in the reservation value for water rights as well as in water rights trading, it is
convenient to write the Euler Equation in terms of the covariance between the
stochastic discount factor and the price of a water right. Using the definition of
covariance [ ] [ ] [ ] [ ] ,
1 1 1 1 1 1
E M Cov M E M E
t t t t t t t t t t
? ? ? = +
+ + + + + +
, I can write
(4.2.13) with the sub index i and j as:
[ ]
,
1 1 1 1
s v E M E Cov M
jt jt jt t it t jt t it jt
? ? ?
( (
= + +
+ + + +
¸ ¸ ¸ ¸
(4.2.16)
Equation (4.2.16) splits the farmer’s reservation value of a water right into the
expected discounted marginal rate of return to holding a water right from time t to
time t+1,
[ ]
1 1
s v E M E
jt jt t it t jt
?
(
+
+ +
¸ ¸
, and a risk premium, ,
1 1
Cov M
t it jt
?
(
+ +
¸ ¸
.
Thus, differences in reservation values for a water right, among farmers in a same
water association and under the assumption of farmers’ identical expectations on
1 jt
?
+
, require differences in the expected value of the random variables
1
M
it +
and/or in the covariance between
1
M
it +
and
1 jt
?
+
.
If farmers are risk neutral then the expected value of
1
M
it +
will be the same
for all farmers and risk premium is zero. Risk neutral farmers utility function over
consumption can be represented by: ( )
0 1
U C C
it it
? ? = + . Since ( )
1
U C
it
? ? = , the
stochastic discount factor becomes
( )
( )
1
1
U C
it
M
it
E U C
t it
? ?
+
=
+
? (
¸ ¸
=
[ ]
1
1
E
t
??
?
?
= (Chavas,
2004, Chapter 10). In this case the covariance term, [ ] ,
1 1
Cov M
t it it
?
+ +
, would
65
vanish and the optimality condition reduces to
1
s v E
jt jt jt t jt
? ?? = +
+
, which is the
same as derived from the simple model in equation (4.1.5).
If farmers face complete asset markets that provide full insurance, then a
farmer’s specific random shocks to current income will not affect his consumption.
In this case the stochastic discount factor
1
M
it +
depends on aggregate temporal
shocks. If it is also assumed that farmers have homogenous preferences, then they are
all characterized by the same, unique stochastic discount factor, i.e.
1
M
it +
=
1
M
t +
(Campbell, 1997, Chapter 8.1). In this case the expected value of the random
variables
1
M
it +
and the covariance between
1
M
it +
and
1 jt
?
+
will not differ among
farmers and water right reservation values will be the same for all farmers.
To summarize, under the assumption that farmers have identical expectations
regarding
1 jt
?
+
, differences in reservation values for a water right among farmers in
the same water association can only arise if there are incomplete markets or if farmers
have heterogeneous preferences that are not risk neutral. In that case there are
multiple stochastic discount factors that satisfy equation (4.2.16).
How risk aversion affects water rights transactions depends on the sign of the
risk premium, ,
1 1
Cov M
t it jt
?
(
+ +
¸ ¸
. The risk premium increases with risk aversion,
so if the covariance between
1
M
it +
and
1 jt
?
+
is positive then more risk averse
farmers will place a higher value on water rights than less risk averse farmers, and the
former will buy water rights from the latter. In the case under study, with incomplete
asset markets, water rights are used not only to obtain water but also to smooth
consumption. This is because water rights allow farmers to insure themselves against
66
bad shocks to production income and hence consumption. Thus, water right prices
covary negatively with consumption and because a farmer’s marginal utility of
consumption is decreasing in consumption, the ,
1 1
Cov M
t it jt
?
(
+ +
¸ ¸
is positive.
Finally, optimal variable input quantities are obtained from the following first
order conditions:
( )
( )
* , , 0 E U P T f w X s
t t it it w it it it jt
?
(
? ? =
¸ ¸
(for water) (4.2.17)
( ) ( )
* , , 0 E U P T f w X r
t t it it x it it it it
?
( ? ? =
¸ ¸
(for other inputs) (4.2.18)
The first order condition for water in (4.2.17) can be written as:
( )
( ) ( )
[ ]
, , ,
, ,
Cov U P T f w X
t t it it w it it it
E P T f w X s
t it it w it it it jt
E U
t t
?
?
?
( = ?
¸ ¸
?
. This provides
an intuitive interpretation of the first order condition for water: at the optimal input
use, the expected marginal value product,
( )
, , E P T f w X
t it it w it it it
? (
¸ ¸
, is equal to the
input price, plus the marginal risk premium,
( ) ( )
[ ]
, , , Cov U P T f w X
t t it it w it it it
E U
t t
? ?
?
?
. An
identical interpretation applies to the first order conditions for the other inputs
(Chavas, 2004, Chapter 8).
4.3 Market Frictions and Consumption-Based Asset Pricing for
Water Rights
Before concluding this chapter, I discuss market frictions that may affect the
consumption-based capital asset pricing model upon which the expressions for the
farmer’s private reservation value of a water right are based. At the heart of the
consumption-based capital asset pricing model is the condition that discounted
expected marginal utilities should be equilibrated across time, as implied by equation
67
(4.2.14). In the presence of market frictions this equality may become an inequality
such as
1
1
1
t
E M
t t
s v
t t t
?
?
(
+
?
( +
?
¸ ¸
or as
1
1
1
t
E M
t t
s v
t t t
?
?
(
+
?
( +
?
¸ ¸
. He and Modest (1995)
review some of the market frictions that may produce such inequalities including: i) a
non-short sales constraint, which prevents the short selling of some assets; ii)
solvency constraints, which restrict the wealth process at some future date from
falling below some predetermined level; iii) transaction costs that include bid-ask
spreads and commissions; and iv) borrowing constraints, which preclude investor’s
current consumption from exceeding their current wealth.
The two-round survey I conducted among farmers in the Limarí Valley shows
that they have low levels of education, and most own less than 6 hectares. These
figures suggest that sophisticated market activities, such as short sales or solvency
constraints, are not relevant for this study. Excess liquidity and an intermediary
institution are necessary for short selling to work properly and the development of
more sophisticated markets where speculators can take short positions. Such is not
the case for water markets in Chile where the size of the market greatly reduces the
possibility that a market maker has a counterpart. The difficulty in hedging against
production shocks and the lack of crop insurance creates uncertainty with respect to
future income. The spot market for water plays a role in smoothing income, but not
nearly enough to avoid distress land sales by insolvent farmers. This suggests that for
the case under study there are two constraints that may be important: transaction costs
and borrowing constraints.
A large literature on the impact of transaction costs on asset prices has
developed. Some authors emphasize the cost in terms of bid-ask spreads (Luttmer,
68
1996), the existence of illiquid assets as a consequence of transaction costs (Vayanos
and Jean-Luc Vila, 1999), or the presence of transaction costs that are endogenously
determined because investors adjust trading frequencies (Constantinides, 1986).
Heaton and Lucas (1996) interpret transaction costs not as a trading cost, but rather as
a wedge between the borrowing and lending rates due to monitoring and other costs
incurred in each period. Gollier (2001) emphasizes that risk reduces long-term
savings on assets if the trading cost of the assets is high. In spite of differences in the
definition of transaction costs among authors, they do agree that for short periods,
transaction costs are important, but over time, they become insignificant. Thus, a
one-period transaction cost will be less significant if traders that buy an asset hold it
for many periods.
To show that transaction costs do not have much effect in the present study, I
start with Gollier’s (2001) findings on illiquid assets. I also assume that the existence
of a costless spot market helps to smooth income variations and minimize the
liquidity problem for water rights. For other types of transaction costs, both theory
and empirical evidence support the hypothesis that transaction costs are not relevant
for this study. In general, water right buyers want to keep them for long periods
minimizing the impact of transaction costs on farmer decisions, and on the price of
water rights or the degree of market activity. As mentioned in the previous chapter,
Hearne and Easter (1995) have estimated that the transaction costs associated with the
purchase and sale of water rights in the Limarí Valley range from 5% to 2% of the
price for buyers and sellers, respectively. Although these costs do not appear to be
high, problems arise when trade implies a change in the source of water because the
69
Direccion General de Aguas, a central governmental office, must approve the change
of source, a process which may take several months because it must ensure that the
petitioners meet certain requirements and evaluate possible negative externalities to
other farmers. The length of this process explains why most of the water rights
transactions are among farmers who obtain their water from the same source.
Secondly, the size (measured by the number of water rights exchanged) of each water
right transaction
28
is relatively small with little dispersion indicating that transaction
costs are insignificant. The following figure presents the frequency of the number of
water rights trades in each one of the 778 transactions included in the dataset.
28
The database contains information for 778 transactions of water rights
(independent of land transactions), in the four WUAs under study representing trade of
11,910 water rights.
70
Figure 4.1: Water Rights Transactions
0
20
40
60
80
100
120
0
,
0
9 3 6 9
1
2
1
5
1
8
2
5
4
0
5
5
7
0
8
5
2
0
0
Number of Traded Water Rights in each Transaction
F
r
e
q
u
e
n
c
y
As the Figure 4.1 illustrates, most transactions involve small amounts of water
rights, indicative of low transaction costs. In fact, 50% of all transactions involve 6
or fewer water rights, and 80% involve less than 20 while the mean value is 15 and
the mode is equal to 2. In dollar terms, a water right in 1999 had a minimum average
price of US$ 927.40 (in ACER) and a maximum average price of US$ 2,766.4 (in
ACEC). If I use the mode (2 water rights per transaction) and the upper bound of the
transaction cost estimated by Hearne and Easter (1995), i.e. 5% of the transaction
price, I arrive at an estimate for transaction cost that ranges from a minimum of $46.4
to a maximum of $138.32 U.S. dollars.
Borrowing or liquidity constraints may explain outcomes in
which
1
1
1
t
E M
t t
s v
t t t
?
?
(
+
?
( +
?
¸ ¸
. This implies that [ ]
1 1
s v E M
t t t t t t
? ? + ?
+ +
, i.e. the
71
marginal expected benefit of buying a water right (the left hand side of the
inequality) is greater than the marginal cost of buying a water right (right hand side
term of the inequality). In such a case the farmer has an incentive to buy water rights
and reduce his present consumption. With decreasing marginal utility of
consumption ( ( ) U C
t
?? <0), that will reduce the stochastic discount factor,
1
M
t +
moving the inequality toward an equality. But if the farmer faces liquidity or credit
constraints then he may not be able to buy all the water rights he desires and the
inequality may not vanish. Accordingly to the Chilean Water Code, water rights are
divorced from land. As a consequence water rights are accepted as collateral by most
banking institutions. This allows farmers to obtain loans to buy water rights. Thus,
credit constraints are mainly for consumption. The effect of this latter constraint is
included in my model through the budget constraint for consumption.
I also consider the impact of liquidity constraints on farmer savings decisions
and risk aversion. Deaton (1991), Carroll (1997), and Gollier (2001), among others,
show that the risk of facing a liquidity constraint in the future introduces an important
motive to save, since savings act as a buffer stock that reduces the probability that the
liquidity constraint will be binding in the future. Agents accumulate assets to insulate
themselves from a temporary drop in income that cannot be compensated by short-
term debts. Nevertheless, in the case under study farmers can smooth revenues from
production by trading in the spot market for water.
Now, if the inequality is such that [ ]
1 1
s v E M
t t t t t t
? ? + ?
+ +
, then the
marginal expected cost of selling a water right (the left hand side of the inequality) is
less than the marginal benefit of selling a water right in period t (the right hand side
72
term of the inequality). In this case the farmer wishes to sell water rights and increase
his consumption. A farmer without water rights is not able to do this leaving him at a
corner solution where the inequality is strict. From a theoretical point of view, such
farmers may exist, so I review the data to see how many farmers in the sample have
no water rights. I find that only 5% of farmers had no water rights.
4.4 Summary
In this chapter, I have developed a framework that models a farmer’s optimal
decision making in regards to investment in water rights and input quantities for each
growing season. This model is a consumption-based asset price model, and shows
that a private farmer’s reservation value for a water right depends on the value of his
stochastic discount factor. Further analysis of that discount factor allows me to relate
reservation values for water rights to the underlying preferences of farmers and risk
aversion. I have shown how heterogeneous preferences generate differences in the
stochastic discount factor, which creates differences in farmers’ private reservation
values for water rights and helps to explain water rights transactions among farmers.
Moreover, I show how more risk averse farmers have incentives to buy water rights
from farmers with lesser risk aversion. Lastly, I have discussed the effect of market
frictions on the water rights market and on the validity of my model. Among the
frictions that the literature addresses, I have focused on transaction costs and
borrowing constrains. The analysis of the data for the case under study indicates that
transaction costs do not affect the validity of the model and that the likelihood of a
corner solution with a farmer holding zero water rights and with a reservation value
less than water right market price is quite low. Moreover, borrowing constraints
73
affect consumption decisions but not water right purchases because the rights can be
used as collateral.
74
Chapter 5: The Econometric Estimation
In this section, I develop an approach to jointly estimate the parameters that
describe a farmer’s utility and production functions, based on observed economic
behavior. Joint estimation preserves estimation consistency and allows exploiting
cross-equations error correlations that might improve efficiency (Love and Buccola,
1991). I also summarize the data sources and the main characteristics of the data that
will be used in later estimations.
5.1 Parametric specification of the system of equations
The joint estimation of the parameters that describe a farmer’s utility and
production functions is performed through the simultaneous estimation of the
equation system described by the Euler Equation (4.2.10) and the first order
conditions that solve for the optimal variable input quantities (4.2.17) and (4.2.18).
That procedure requires a parametric specification of the instantaneous utility of
consumption, the stochastic production technology and the variable that represents
other net income, I
it
.
For the utility function, I have chosen an exponential function:
( ) ( ) exp U C C
i i i
? = ? ? (5.1.1)
where
i
? is restricted to be non-negative. The specification (5.1.1) has three main
features. First, the marginal utility of consumption is positive,
( ) ( ) exp U C C
i i i i
? ? ? = ? >0. Second, the utility function is concave,
( ) ( )
2
exp U C C
i i i
i
? ? ?? = ? ? <0, which implies aversion to risk. Third, it assumes that
75
the absolute Arrow-Pratt risk aversion coefficient for each farmer, denoted by
i
R
A
, is
constant (CARA) and equal to
i
? , i.e.
i
R
A
=
i
?
The CARA preferences embodied in the use of the negative exponential utility
function is a drawback of this specification because few decision-makers have the
implied characteristic that their attitude towards risk remains the same regardless their
wealth or asset position (Chavas, 2004, Chapter 4; Saha et al., 1994). Nevertheless,
risk preferences displaying constant risk aversion are extremely easy to deal with
analytically (Hammond, 1974). Moreover, the exponential function it is very suitable
as a local approximation to anyone’s utility function for evaluating small to moderate
gambles (Pratt, 1964). For these reasons the CARA preferences have been widely
used in applied decision analysis (Hammond, 1974, Keeney and Raffia, 1976,
Gregory, 1978, Love and Buccola, 1991).
To describe the stochastic production technology I use a Just-Pope Cobb-
Douglas (Just and Pope, 1978) production function that facilitates the estimation of
production risk endogenous to inputs. Moreover, the Just-Pope form holds the best
potential for mutual inference of preferences and technology (Love and Buccola,
1991). This function is:
( ) ( )
0 0 1 2 1 2
, , ; , , ; y f w L F h w L F w L F w L F
it it it
it it it it it it it it it it it it
? ? ? ? ? ?
? ? ? ? = + = +
(5.1.2)
where y is output per-hectare, w is water, L is labor, F is fertilizers (all inputs are
divided by the number of cultivated hectares), ? represents output uncertainty. The
vector of parameters of the non-stochastic component of the production function is
76
represented by ( ) , ,
1 2 3
? ? ? ? = , and ( ) , ,
1 2 3
? ? ? ? = is the vector of parameters of
the stochastic component of the production function. Output uncertainty,
it
? , is
assumed to be independent and identically distributed (i.i.d.) normal with
[ ] 0 E
it
? = and [ ] 1 V
it
? =
29
. Pope and Just (1977) point out that no generality is lost
in assuming [ ] 1 V
it
? = , since if
[ ]
2
V
it
? ?
?
= then the ( ) , , ; h w L F
it it it
? could simply
be modified by a multiplicative factor
2
?
?
. Moreover, as in Just and Pope (1977 and
1979) and Love and Buccola (1991) I assume that
0 1 2
0 E w L F
t it
it it it
? ? ?
?
(
=
(
¸ ¸
. That
assumption implies that input quantities are either not stochastic or stochastic, but
independent of ? .
Finally contemporaneous correlation among errors of the production function
is not excluded i.e. , 0 E
it jt
? ?
(
?
¸ ¸
From this production function specification, I obtain
[ ]
3 1 2
E y L w F
it
it it it
? ? ?
= and
[ ]
2
3 1 2
V y L w F
it
it it it
? ? ? | |
=
|
\ .
, which implies that the mean
and variance of output are endogenous to input decisions.
For the “other net income” variable, I
it
, I assume the following linear model:
29
The equation system described by the Euler equation and the first order conditions
that solve for the optimal variable input quantities are expressed in terms of conditional
moments to the set of information in t. However, because
it
? is assumed to be i.i.d. its
conditional and unconditional moments are the same (Cochrane, 2005, Chapter 1). Thus, if I
have that [ ] [ ] 0 E E
t it it
? ? = = I also can say that [ ] [ ] 0
1 1
E E
t it it
? ? = =
+ +
.
77
( )
( )
3 1 2
1
*
3 1 2
N N s N v T w P T L w F
jt it it jt it jt it it it it
it it it
I a
it it
P T L w F r X c
it it it it it i
it it it
? ? ?
?
? ? ?
?
| |
? + ? + +
?
|
= +?
|
|
? ? ?
\ .
(5.1.3)
where a is an unknown parameter, c
i
is “normal” consumption, and
it
? is a random
error that represents random aggregate shocks plus idiosyncratic shocks different
from production uncertainty that affect I
it
.
The variable I
it
represents farmer’s “other net income” sources as
consumption and production credit, land transactions, livestock exchanges and work
off-farm that can be used to afford consumption. Equation (5.1.3) assumes that part
of I
it
is endogenous and depends on the differences between net income from
production and water trading and some “normal” consumption level, c
i
, that is
farmer specific. For instance, a farmer that owns livestock and who faces a situation
in which her net production revenues plus water trading income is lesser than c
i
may
sell cattle.
Equation (5.1.3) also states that I
it
is a function of the deterministic part of
the net income, ( )
( )
3 1 2
1
N N s N v T w P T L w F r X
jt it it jt it jt it it it it it it
it it it
? ? ?
?
? ? + ? + ?
?
,
and two stochastic components: (1) farmer’s output specific or idiosyncratic random
shocks,
3 1 2
P T L w F
it it it
it it it
? ? ?
? , and (2) the random term
it
? .
I assume that output specific random shocks,
it
? , and
it
? are distributed
independently. That assumption follows from the fact that
it
? represents output
uncertainty, for instance pests, while
it
? represents uncertainty over “other net
78
income sources” such as random variations in bank interest rates. If asset markets
were complete then farmers’ specific random shocks would be aggregate temporal
shocks and the assumption of independent distributions may not be valid any more.
Nevertheless, because in the case under study asset markets are incomplete the above
independence assumption should hold.
Incomplete asset markets may also imply heterogeneous variances for I
it
among farmers. Different investment decisions of a surplus between “normal”
consumption level, c
i
, and production and water trading incomes may lead to
different variances in I
it
. Whether or not a farmer faces credit constraints may also
affect the variance of I
it
. These differences in the variance of I
it
among farmers
lead me to assume that
it
? is heteroskedastic. Thus I assume that
it
? distributes
normal with conditional expected mean zero and non constant conditional
variance
2
i
?
?
.
Replacing the Euler Equation (4.2.10) with these structural forms, I get:
79
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i jt
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
s v E
jt jt jt t
?
? ? ?
? ?
? ? ?
?
? ?
| | | | ? +
+ +
| |
|
|
? +
+ + + + +
|
|
+
| |
?
? ? + +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
|
|
? +?
+ \ .
= +
( )
( )
( )
1
exp 1
3 1 2
3 1 2
N N
jt it it
s N v T w
jt it jt it it
E a ac
t i i it
P T L W F r X
it it it it
it it it
P T L W F
it it it
it it it
?
?
? ? ?
? ? ?
?
(
(
(
(
(
(
(
(
(
(
(
( | | ( | | ? +
?
( | | (
( | | (
? +
( |
| (
? + ? +?
( |
| (
? ? +
( | | (
( | | (
( | |
(
\ .
\ . ¸ ¸
(
¸ ¸
(
(5.1.4)
Further simplification of equation (5.1.4) can be achieved by taking logs on both
sides
( )
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
ln ln ln
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
s v E
jt jt jt t
?
? ? ?
?
? ? ?
?
? ?
| | | | ? +
+ +
| |
| |
? +
+ + + + +
| |
+
| |
?
? ? +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
? +?
+ \ .
? = +
( )
( )
( )
1
1
exp 1
3 1 2
3 1 2
jt
N N
jt it it
s N v T w
jt it jt it it
E a ac
t i i it
P T L W F r X
it it it it
it it it
P T L W F
it it it
it it it
?
?
?
? ? ?
? ? ?
?
(
(
(
(
(
+ (
(
(
( |
|
(
(
(
| | ( | | ? +
?
( |
| (
( |
|
? +
( |
|
? + ? +?
( |
|
? ? +
( | |
( | |
( | |
\ . \ .
¸ ¸
¸ ¸
(
(
(
(
(
(
(
(
(5.1.5)
80
A convenient expression for the second term in the right hand of (5.1.5) can be obtained with the Jensen’s
inequality that states that:
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
ln
exp 1
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i jt
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
E
t
E
t i
?
? ? ?
? ?
? ? ?
?
?
| | | | ? +
+ +
| |
|
|
? +
+ + + + +
|
|
+
| |
?
? ? + +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
|
|
? +?
+ \ .
? + ( )
( )
( )
( )
1
1
exp
ln
1
3 1 2
3 1 2
N N
jt it it
a
i
E
t
N N
jt it it
s N v T w
jt it jt it it
a ac
i it
PT L W F r X
it it it it
it it it
PT L W F
it it it
it it it
?
?
?
? ? ?
? ? ?
?
(
?
+ +
(
(
(
+
(
?
(
(
(
(
(
( =
( | | ? + (
?
( |
(
( |
(
? +
( |
(
? +?
( |
(
? ? +
( |
(
( |
(
| (
(
\ . ¸ ¸
(
(
¸ ¸
( )
( )
( )
( )
1
1 1 1 1 1
3 1 2
1
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
1
exp 1
s N v T w
jt it jt it it
P T L W F r X jt
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
N N
jt it it
s N v T
jt it jt i
E a
t i
? ? ?
?
? ? ?
?
?
?
| | | | +
| |
|
|
? +
+ + + + +
|
|
| |
? ? + +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
|
|
? +?
+ \ .
? +
?
?
? +
( )
1
3 1 2
3 1 2
it
w
t it
ac
i it
PT L W F r X
it it it it
it it it
PT L W F
it it it
it it it
?
? ? ?
? ? ?
?
| |
|
|
|
|
|
|
|
|
|
|+
+
( | | | | |
( | | |
( | | |
+
( | | |
? +?
( | | |
? ? +
( | | |
( | | |
( | | |
\ .
\ . ¸ ¸
|
|
\ .
(5.1.6)
81
where
1 + it
? is a positive expression that corresponds to a Jensen’s inequality
adjustment. The economic intuition for the adjustment can be seen by noticing that
(using notation of Chapter 4)
1 + it
? =ln ln
1 1 1 1
E M E M
t it jt t it jt
? ?
( (
?
+ + + +
¸ ¸ ¸ ¸
,
where the difference on the right hand side of the expression is a measure of
conditional volatility of the discounted water right price for each farmer (Alvarez and
Jermann, 2005). As a special case, if the discounted price is distributed lognormal,
then the volatility measure
1 + it
? =
1
2
( )
ln 1
1
Var M jt
t it
? ( +
+
¸ ¸
(Campbell et al.
(1997, Chapter 8).
Substituting the right hand term of equation (5.1.6) in the Euler Equation
(5.1.5) I get:
( )
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
ln ln ln
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
s v E
jt jt jt t
?
? ? ?
?
? ? ?
?
? ?
| | | | ? +
+ +
|
|
| |
? +
+ + + + +
| |
+
|
|
?
? ? +
|
| + + + +
+ + +
| |
|
|
+ + +
+ + + \ .
? +?
+ \ .
? = +
( )
( )
( )
1
exp 1
3 1 2
3 1 2
N N
jt it it
s N v T w
jt it jt it it
E a ac
t i i it
P T L W F r X
it it it it
it it it
P T L W F
it it it
it it it
?
?
? ? ?
? ? ?
?
| |
|
|
|
|
|
|
|
| |
| |
|
(
| | | | | ? +
?
( | | |
( | | |
? +
( | | |
? + ? +?
( | | |
? ? +
( | | |
( | | |
( | |
\ .
\ . ¸ ¸
\ .
ln
1 1
E
t jt it
? ?
+
|
|
|
(
+
+ +
¸ ¸
(5.1.7)
82
Because
3 1 2
0
1 1 1
1 1 1
E P T L W F
t it it it
it it it
? ? ?
?
(
=
+ + +
( + + +
¸ ¸
and [ ] 0
1
E
t it
? =
+
, equation
(5.1.7) simplifies to:
30
( )
( )
( )
( )
( )
( )
( )
1 1
ln ln 1
1 1 1 1 1
3 1 2
1 1 1 1
1 1 1
1
ln exp 1
1
N N
jt it it
s v E a s N v T w ac
jt jt jt i t jt it jt it it i
P T L W F r X
it it it it
it it it
N N
jt it it
s N v T w
jt it jt it it
E a
t i
P T L W
it it
it it
?
? ? ?
? ? ?
?
?
? ?
( | |
? +
+ +
( |
( |
? = ? + ? + ? ?
( | + + + + +
( |
( |
? ?
+ + + +
+ + + \ . ¸ ¸
? +
?
? +
? + ln
1 1
3 2
3 1 2
ac E
i it t jt it
F r X
it it
it
P T L W F
it it it
it it it
? ?
?
? ? ?
?
( | | | |
( | |
( | |
( | |
(
? +? + +
( + + | |
¸ ¸
? ? +
( | |
( | |
( | |
\ . \ . ¸ ¸
(5.1.8)
Equations (4.2.3), (5.1.2) and (5.1.3) imply that the second term of the right
hand side of equation (5.1.8) represents the expectation in t of the expected
consumption in t+1 conditional to the information in that same period, i.e.
30
I have assumed that input quantities are not stochastic in the current period or
stochastically independent from? , i.e.
3 1 2
0 E P T L W F
t it it it
it it it
? ? ?
?
(
=
(
¸ ¸
. Now, to explore the
value of
3 1 2
1 1 1
1 1 1
E P T L W F
t it it it
it it it
? ? ?
?
(
+ + +
( + + +
¸ ¸
I use that assumption and the law of
iterated expectations to obtain:
3 3 1 2 1 2
1 1 1 1 1 1 1
1 1 1 1 1 1
E P T L W F E E P T L W F
t it it it t t it it it
it it it it it it
? ? ? ? ? ?
? ?
(
( (
=
+ + + + + + +
( ( ( + + + + + +
¸ ¸ ¸ ¸ ¸ ¸
=
[ ]
3 1 2
1 1 1 1 1
1 1 1
E E P T L W F E
t t it it t it
it it it
? ? ?
?
(
(
=
+ + + + +
( ( + + +
¸ ¸ ¸ ¸
3 1 2
0 0
1 1 1
1 1 1
E E P T L W F
t t it it
it it it
? ? ? (
(
=
+ + +
( ( + + +
¸ ¸ ¸ ¸
.
The result [ ] 0
1
E
t it
? =
+
follows from the law of iterated expectations and the
assumption that [ ] 0
1 1
E
t it
? =
+ +
. In fact the law of iterated expectations implies that
[ ] [ ]
1 1 1
E E E
t it t t it
( ? = ?
+ + +
¸ ¸
, and [ ] 0
1 1
E
t it
? =
+ +
implies
[ ]
0
1 1
E E
t t it
( ? =
+ +
¸ ¸
.
83
[ ]
1 1
E E C
t t it
(
+ +
¸ ¸
31
. I then write that expression in a more conventional time-series
notation as
[ ]
1 1 1 1
E E C C
t t it it it
? ( = +
+ + + +
¸ ¸
, where
1 it
?
+
is random error with
conditional expectation equal to 0 (Cochrane, 2005, Chapter 9).
Assuming [ ] 0
1
E
t it
? =
+
implies that the unconditional mean [ ] 0
1
E
it
? =
+
and that
1 it
?
+
is not correlated with the information set at time t (Wooldridge, 2002,
Chapter 2.2.3). Thus, the proposed transformation presumes that, based on the set of
information in period t, farmers can predict part of random consumption in t+1, but
there is another part of future consumption represented by
1 it
?
+
that can not be
predicted because is not related with the set of information at time t.
An identical argument can be used to transform the expectation of the natural
log of the price of a water right in the right hand side of equation (5.1.8) as:
ln ln
1 1 1 1
E E
t t jt jt jt
? ? ?
( (
= +
+ + + +
¸ ¸ ¸ ¸
. As in the case of the transformation for
consumption and for the same reasons that I have indicated above I assume
that 0
1
E
jt
t
?
(
=
+
¸ ¸
.
Substituting these transformations,
[ ]
1 1 1 1
E E C C
t t it it it
? ( = +
+ + + +
¸ ¸
and
ln ln
1 1 1 1
E E
t t jt jt jt
? ? ?
( (
= +
+ + + +
¸ ¸ ¸ ¸
, in equation (5.1.8) gives:
31
Because production uncertainty realized its value at the end of the season,
consumption in t+1 is not known until the end of that season. Thus, in t+1 the farmer only
has an expectation of his consumption level for that season.
84
( )
( )
( )
( )
( )
( )
( )
1 1
ln ln 1
1 1 1 1 1 1
3 1 2
1 1 1 1
1 1 1
1
ln exp 1
1
N N
jt it it
s v a s N v T w ac
jt jt jt i jt it jt it it i it
P T L W F r X
it it it it
it it it
N N
jt it it
s N v T w
jt it jt it it
E a
t i
P T L
it it
it
?
? ? ? ?
? ? ?
?
?
?
( | |
? +
+ +
( |
( |
? = ? + ? + ? + ?
( | + + + + + +
( |
( |
? ?
+ + + +
+ + + \ .
¸ ¸
? +
?
? +
? + ln
1 1 1
3 2
3 1 2
ac
i it jt jt it
W F r X
it it
it it
P T L W F
it it it
it it it
? ? ?
? ?
? ? ?
?
(
| | | |
( |
|
( |
|
( |
|
? +? + + +
( + + + |
|
? ? +
( |
|
( |
|
( | |
\ .
\ .
¸ ¸
(5.1.9)
In equation (5.1.9) the term ac
i
cancels out. Then, reordering terms, equation
(5.1.9) can be written as:
( )
( )
( )
( )
( )
( )
( )
1 1
ln ln 1 ln
1 1 1 1 1 1
3 1 2
1 1 1 1
1 1 1
1
ln exp 1
3 1 2
N N
jt it it
s v a s N v T w
jt jt jt i jt it jt it it jt
P T L W F r X
it it it it
it it it
N N
jt it it
s N v T w
jt it jt it it
E a
t i
P T L W F r X
it it it
it it it
?
? ? ? ?
? ? ?
?
?
? ? ?
| |
? +
+ +
|
|
? = ? + ? + + ?
| + + + + + +
|
|
? ?
+ + + +
+ + + \ .
? +
?
? +
? +
? ?
1 1
3 1 2
it it
it
P T L W F
it it it
it it it
?
? ? ?
?
(
| | | |
( |
|
( |
|
( |
|
+? +
( + |
|
+
( |
|
( | |
( | |
\ .
\ .
¸ ¸
(5.1.10)
where
1 1 1 1 1 it it it jt
? ? ? ? ? + +
+ + + +
is a composite error with
[ ] [ ]
1 1 1
E E
t it t it
? ? =
+ +
.
85
The structural forms proposed in equations (5.1.1), (5.1.2) and (5.1.3) implies
that the first order conditions for optimal quantities of water, labor and fertilizer,
represented by equations (4.2.17) and (4.2.18) are, respectively:
( )
( )
( )
1
exp 1
3 1 2
3 1 2
3 3 1 2 1 2
1 1
N N
jt it it
s N v T w
jt it jt it it
a ac
i i it
P T L W F r X
it it it it
it it it
E
t
P T L W F
it it it
it it it
P T w L F P T w L F
it it it it it
it it it it it it
s
jt
w
it
?
?
? ? ?
? ? ?
?
? ? ? ? ? ?
? ? ?
(
| | ? +
?
( |
( |
? +
( |
? + ? + ?
( |
? ? +
( |
( |
| (
\ . ¸ ¸
+
?
¸
0
(
(
(
(
(
(
=
(
(
(
(
(
(
¸
(5.1.11)
( )
( )
( )
1
exp 1
3 1 2
3 1 2
3 3 1 2 1 2
2 2
1
N N
jt it it
s N v T w
jt it jt it it
a ac
i i it
P T L W F r X
it it it it
it it it
E
t
P T L W F
it it it
it it it
P T w L F P T w L F
it it it it it
it it it it it it
r
it
L
it
?
?
? ? ?
? ? ?
?
? ? ? ? ? ?
? ? ?
( | | ? +
?
( |
( |
? +
( |
? + ? + ?
( |
? ? +
( |
( |
| (
\ .
¸ ¸
|
+
?
\
0
(
(
(
(
(
(
(
=
(
(
(
|
(
|
(
|
|
(
.
¸ ¸
(5.1.12)
( )
( )
( )
1
exp 1
3 1 2
3 1 2
3 3 1 2 1 2
3 3
2
N N
jt it it
s N v T w
jt it jt it it
a ac
i i it
P T L W F r X
it it it it
it it it
E
t
P T L W F
it it it
it it it
P T w L F P T w L F
it it it it it
it it it it it it
r
it
F
it
?
?
? ? ?
? ? ?
?
? ? ? ? ? ?
? ? ?
( | | ? +
?
( |
( |
? +
( |
? + ? + ?
( |
? ? +
( |
( |
| (
\ .
¸ ¸
|
+
?
\
0
(
(
(
(
(
(
(
=
(
(
(
|
(
|
(
|
|
(
.
¸ ¸
(5.1.13)
In the above equations the price for labor is denoted by
1
r , and for fertilizer by
2
r .
86
Using the already mentioned distributional assumptions for
it
? and
it
? , and
the assumption of distributional independence between
3 1 2
P T L W F
it it it
it it it
? ? ?
?
(
(
¸ ¸
and
it
? (discussed above), plus the properties of the moment generating functions of the
normal distributed variables
it
? and
it
?
32
, the system of equation (5.1.10) to (5.1.13)
simplifies to:
The Euler Equation:
( )
( )
( )
( )
( )
( )
( )
ln ln ln
1
1 1 1 1 1 1 1
1
3 1 2
1 1 1 1
1 1 1
1
1
3 1 2
s v
jt jt jt jt
N N s N v T w
jt it it jt it jt it it
a
i
P T L W F r X
it it it it
it it it
N N
jt it it
a s N v T w r X
i jt it jt it it it it
P T L W F
it it
it it it
? ? ?
?
?
? ? ?
?
?
? ? ?
? = + ?
+
| |
| |
? + ? +
+ + + + + + +
|
|
+ +
|
|
| ? | ?
+ + + +
+ + + \ .
\ .
? +
?
? + ? ? +
( )
2
2
1
2 2 2
3 1 2
1
1 1
2 2
i
a P T L W F
i it it i it
it it it
?
? ? ?
? ? ?
| |
|
|
?
|
|
|
\ .
| |
| |
?
|
+ ? +
| +
|
\ .
\ .
(5.1.14)
32
For a random variable X distributed normal with mean µ and variance
2
? the
generation moment function is: ( )
2 2
exp exp
2
t
E tx t
?
µ
| |
|
= +
|
\ .
. That implies
( )
( )
( )
2 2
exp
2
exp exp
2
E tx
t
E x tx t t
t
?
µ ? µ
| |
?
|
= = + +
| ?
\ .
or
( )
( )
2 2
2
exp exp
2
t
E x tx t t
?
µ ? µ
| |
|
? = ? ? +
|
\ .
87
The equations for the optimal variable input quantities:
( )
2
0 1 2
1
0 1 2 0
0
0
2it
a P T w L F
i it it
it it it P T w L F
it it
it it it
s
jt
w w
it it
? ? ?
? ? ? ? ?
?
?
| |
| |
| +
|
\ . |
? ? + =
|
|
|
\ .
(5.1.15)
( )
2
0 1 2
1
0 1 2 1
1
0
1 3it
a P T w L F
i it it
it it it P T w L F
it it
it it it
r
it
L L
it it
? ? ?
? ? ? ? ?
?
?
| |
| |
| +
|
\ . |
? ? + =
|
|
|
\ .
(5.1.16)
2
0 1 2
(1 )
0 1 2 2
2
0
2 4it
a P T w L F
i it it
it it it P T w L F
it it
it it it
r
it
F F
it it
? ? ?
? ? ? ? ?
?
?
| |
| |
| +
|
\ . |
? ? + =
|
|
|
\ .
(5.1.17)
Those first order conditions indicate that input decisions depend on the
marginal productivity of inputs, input prices, risk aversion and the effect of each input
on the variance of output. The last two effects on optimal input quantities are
captured by the second term within brackets in each of those equations. Given that
i
? is positive, that term is negative (positive) for risk decreasing (increasing) inputs.
Thus for a risk-averse farmer, when an input is risk decreasing (increasing), he has an
incentive to increase (decrease) the demand for this input.
It is important to notice that in equations (5.1.14) to (5.1.17) if the parameter a
is equal to -1, then risk aversion does not affect farmer decisions on either water
rights or input quantities. As it can be seen from equations (4.2.3) and (5.1.13) a
value of -1 for a implies that the farmer’s expected consumption in any season is
equal to his “normal” consumption level. Moreover, with a = -1 farmer’s
88
consumption is not affected by random shocks in production, i.e. C c
it i it
= + ? . This
implies that the farmer has production full insurance or unlimited access to “other
income sources” such as credits. Nevertheless, because neither production full
neither insurance nor unlimited access to “other income sources are feasible I rule out
the possibility of a = -1.
As in Love and Buccola (1991), Saha et al. (1994), Chavas an Holt (1996) and
Kumbhakar (2002), among others, I added to equations (5.1.15), (5.1.16) and (5.1.17)
the additive disturbances, , ,
2 3 4 it it it
? ? ? , associated with errors in optimization.
Pope and Just (2003) find credible evidence in U.S. agricultural production of errors
in optimization. I assume that these optimization mistakes occur in form of random
failures, which support a stochastic structure to the equation system. That stochastic
structure is needed to achieve an econometric estimation of the parameters of interest.
In addition, I assume that these
it
?
?
s have conditional expected value equal to zero. I
do not restrict the error terms of the first order condition for input quantities to be
independent among equations for each farmer. Only the error term of the Euler
Equation,
1 1 it
?
+
, is assumed to be independent of the error terms , ,
2 3 4 it it it
? ? ? .
This assumption about the independence of the error term of the Euler Equation with
respect to the error terms of the input equations is based on the structure of
1 1 it
?
+
.
That structure indicates that
1 1 it
?
+
is a function of the random variable
1 it
?
+
that
realizes its value in t+1 and which I expect not to be correlated with errors in
optimization for input quantities in period t.
89
I also assume that disturbances are correlated across farmers within equations.
Therefore, [ ] , 0 Cov
t hit hmt
? ? ? for i m ? and for h=1,2,3,4. I only exclude
correlations between errors associated with different farmers and across equations.
Given these assumptions regarding the error terms, ( ) , ,
1 2 3
? ? ? ? = , the
variance- covariance matrix of ? , which is denoted by ? is specified as follows:
?= ( ) E
t
??? =
0 0 0
11 * * *
0
* 22 23 * 24 *
0
* 32 * 33 34 *
0
* 42 * 43 * 44
I I I
N N N N N N
I I I
N N N N N N
I I I
N N N N N N
I I I
N N N N N N
? ?
? ?
? ?
? (
(
?
(
( ?
(
?
¸ ¸
Here ? is a matrix of order N*H x N*H with N the number of farmers, H the number
of equations, [ ] E
hh t h h
? ?? ? = and
( )
,
gh Cov git hit
t
? ? ?
=
, for g=2,3,4 and h=2,3,4.
5.2 Data
The data set has two parts and contains data for the four main WUAs
associations in the Limarí Valley
33
. The first part contains a cross-section time series
sample on farmers over two agricultural seasons (98/99 and 99/00). It includes micro
level data for farming activity. The second part of the data contains time series for: i)
water right prices and water right transactions for the period 1981 to 2001; ii) spot
market water prices and water transactions for the period 1995 to 2000; and iii) water
accorded to water rights for the period 1980 to 2000.
The farmer micro-level data is obtained from a two-round survey that I
conducted. This survey was performed in the Limarí Valley (see the survey
33
Those WUAs are: Asociación de Canalistas del Canal Camarico (ACCC);
Asociación de analistas del Embalse Cogotí (ACEC); Junta de Vigilancia del Río Limarí
(JVRL); and Asociación de Canalistas del Embalse Recoleta (ACER).
90
instrument in the Annex). Farmers from the main irrigated areas within this key
agricultural region were interviewed. The sample was designed by Zegarra (2002)
34
who conducted a previous survey on that same valley. In the first round (SI),
surveyed in 1999, I collected information for the 98/99 season from 161 farmers. The
second round (SII) was conducted in 2000 and I collected information for the 99/00
season from 151 farmers.
The farm level data set includes seasonal information on crop production,
input use for each crop, output and input prices, irrigation methods, water right
transactions, volume of water bought or sold in the spot market, land transactions,
livestock inventory, renting in or out of machinery, asset ownership, farmer’s
liabilities, household characteristics, family labor, well access and water storage
capacity, marketing, governmental subsidies to improve irrigation systems and water
expectations for the coming season.
As is usual in surveys that collect data, I have gaps in the data due to attrition
and survey non-response. Balgati (2001) presents the rate of attrition for a sample of
studies that use panel data. In my sample, attrition from the first to the second round
surveys that I conducted is comparable to that obtained in other empirical works.
Non-response is caused by farmers that have sold or have decided to abandon their
land, farmers previously interviewed that were not subsequently located and farmers
34
Zegarra did not develop a list of farmers to interview based on a random sample
due to the expense of finding each sampled individual; instead, he simulated random
sampling for farmers who were present at their farm when he conducted the survey. He
began at some point inside the irrigated area (stratum), interviewing farmers using a
systematic round skipping for close neighbors. This results in a sample, which is
geographically representative for each irrigation organization. The main limitation of this
sampling procedure is that farmers who were not present at the moment of the survey had
zero probability of being selected. The procedure also excludes farmers who, at the moment
of the survey, had abandoned production.
91
who refused to answer (7 and 4 farmers in the first and second rounds refused to
answer, respectively). The causes of non-response suggest few if any behavioral
reasons behind this problem; hence, the consequences of attrition appear to be
minimal.
There is missing data arising from partial response to survey questions. This
is mainly due to the fact that most farmers do not keep written records of the
information requested in the survey. In fact, only five among all surveyed farmers
had written records on most of the surveyed data. Thus, in most cases, a partial
response occurs when the respondent fails to answer a question because he has
difficulty recalling events that occurred in the past.
One way to handle the missing data problem is by imputing missing
observations. Nevertheless, as Cameron and Trivedi (2005) point out “there is a cost
of imputing missing data that comes from having to make assumptions to support a
procedure for generating proxies for the missing observations, and from the
approximation error inherent in such a procedure.”
Alternatively, it is possible to handle missing data by deleting them and
analyze only the reduced sample of “complete” observations. That procedure is
called listwise deletion. Its consequences for the econometric estimation depend on
the missing data mechanism. If the probability of missing data of the variables in the
data set depends neither on its own values nor on the values of other variables in the
data set, then missing data process is completely at random. In that case the
remaining set after listwise deletion remains a random sample from the original
population and the estimates based on it are consistent (Cameron and Trivedi, 2005).
92
If the probability of missing data on a variable does not depend on its value but may
depend on the values of other variables in the data set, then data is missing at random.
If the data set has gaps due to data missing at random and the parameters for the
missing data-generation process are unrelated to the parameters that one wants to
estimate, then the missing data problem is ignorable and the complete data set after
listwise deletion allows for consistent estimation of the parameters of interest
(Cameron and Trivedi, 2005). Nonetheless, under either the missing data complete at
random or just missing data at random assumptions, listwise deletion still reduces
efficiency in the estimation.
For the case of the missing data problem in my survey, the causes of non-
response suggest few if any behavioral reasons behind this problem. Furthermore,
missing data due to a partial response to survey questions is mainly related to the
difficulty of some farmers to recall past data. Thus, it seems reasonable to assume
my data set is characterized by missing data complete at random or at least missing at
random. Therefore I handle the missing data problem using listwise deletion.
The Euler Equation (5.1.14) links farmers’ decisions in two seasons. This
requires full data for each farmer in the two seasons under study. That requirement
causes substantial sample loss and after the listwise deletion process the number of
farmers that fulfill that requirement is 32.
93
The following table provides some statistics on the most important variables in the model for the whole sample as
well as for sub-sample of 32 farmers used in this dissertation (referred to in what follows as the sub-sample).
Table 5.1: Basic statistics of the main variables
Whole sample Sub-sample (3 2 farmers)
Variable SI
SII SI SII
Number of
observations.
mean Standard
deviation
Number of
observations
mean Standard
deviation
mean Standard
deviation
mean Standard
deviation
Cultivated land (hectare) 161 12.1 25.2 148 10.3 17.9 12 21.3 11.4 18.5
Labor (hours per hectare) 148 1242 2115 137 1021 1674 1099 1437 987 1201
Nitrogen (kilograms per
hectare) 117 258.2
458.6
137 109.8 151.5 320 591 125.7 170.9
Input water (cubic meters
per hectare)
144 15766 20763 132 9145 9997 6958 5129 8674 16081
Number of water rights 155 20.7 37 141 20.2 36 15.6 14.7 17.3 19.0
Education (years of
schooling)
151 8.3 4.7 139 8.0 4.7 8.6 4.7 8.7 4.6
Experience (years) 134 30.4 14.3 118 25.6 13.0 28.6 14.3 30.5 13.6
Household size 131 4.7 3.3 102 4.8 3.2 4.9 3.6 4.9 3.0
Percentage of multioutput
producers
53.7% 52.11% 31.3% 34.3%
94
Table 5.1 shows that sample mean values from the whole samples are close to
the sample mean values from the sub-samples for all the reported variables but input
water per hectare in survey I. The percentages of multioutput producers in the sub-
samples are lower than in the whole sample. If farmers’ decisions on the number of
crops they grow are related to their risk aversion, then differences in the percentage of
multioutput producers may cause a bias problem. That may be the case if an
estimation of risk aversion for a “representative farmer” is intended with the sub-
samples. In this dissertation I test for differences in risk aversion among farmers. As
I explain in next chapter, that test is based on the effect of farmer’s specific
characteristics upon his risk aversion and, as a consequence, it is not subject to the
above mentioned bias problem.
Annual average price for water rights and water right transaction time series
for the period 1981 to 2001 were obtained through the Conservador de Bienes Raíces
of Ovalle and the records kept by the WUAs
35
. Water prices and water transactions
in the spot market time series for the period 1995 to 2000 were constructed using the
records of the WUAs, information obtained from the Direccion de Riego and the two-
round survey. For water accorded to water rights I use data from the WUAs.
In the equation system (5.1.14) to (5.1.17), the land input is measured as the
total hectares of cultivated land. No distinction is made as to whether land is owned,
rented or sharecropped
36
.
35
The series for water right prices and transactions between 1981 and 1992 were
collected by Zegarra (2002) and between 1992 and 2000 by Cristi et al. (2002) and Vicuña
(2000).
36
Sharecroppers were considered as single producers.
95
Production input water is measured as the farmer’s total number of water
rights times the amount of water accorded to each right plus net sales of water
volumes
37
. The amount of water obtained through that formula was weighted by
farmer’s average irrigation efficiency
38
.
Labor is total hours per growing season and is obtained by grouping together
three different types of labor: family workers, permanent hired workers, and hired
workers for specific activities (temporary workers). The survey data show that some
farmers only use family work, others have permanent hired workers and others hire
workers for specific activities and time periods (temporary workers). For those
farmers with a mix of workers it is not possible to infer from the available data how
many hours worked correspond to each type of worker. Thus I restrict my
econometric results by making no distinction among type of workers.
Fertilizer is measured in kilograms of nitrogen.
37
Farmers may decide not to use all the water accorded to their water rights, but
saving water from one season to the next has a penalty of 15% to 20% of the endowment
which makes this practice rare (Zegarra 2002). Moreover, the existence of a price greater
than zero for water in the spot market implies that rational farmers will not resign to any
amount of water accorded to their water rights.
38
Because irrigation systems vary among land plots within the same farm I use
farmer’s average irrigation efficiency. This was calculated as the arithmetic mean of the
farmer’s irrigation efficiency in his different plots within the same farm:
( )
1
irrigation eficiency in plot q
q
total cultivated land
¿
=
,
where irrigation efficiency varies accordingly to the following table:
Irrigation
system/
Efficiency
Drip Sprinkler Furrow Flood
90% 75% 65% 45%
Source: Comisión Nacional de Riego, Gobierno de Chile
96
Some farmers in the sample produce more than one crop. For those farmers I
represent output price by a farmer-specific weighted average of all the farmgate
prices of the farmer’s crops. As in Saha et al. (1994) I use the product specific
income shares as weights for each crop price. Furthermore, because prices depend on
the arbitrary output units they value
39
I have divided each farmgate price by is sample
mean
40
. This scaling procedure does not affect the relations that I intend to estimate
econometrically with my model.
For wages paid to labor,
1
r , I use the per hour payments to temporary workers.
Those payments are a good proxy of the labor price that each farmer faces on the
labor market. I also estimate total labor cost by multiplying wages by total hours
worked. That procedure values the work of permanent hired workers and family
workers as equal with the work done by temporary workers. Three farmers in the
sample do not hire temporal workers. For them, I use the daily payment to their
permanent workers as the wage rate. Finally, three other farmers use only unpaid
family workers so I use the sample average of the per hour payments to temporary
39
As an example, potatoes can be measured in kilograms and there is a price for the
kilogram of potatoes. Nevertheless, potatoes are also exchanged in sack units of 50
kilograms and the price of a sack is higher than the price for a kilogram. In the estimation of
an average price the price of potatoes will be given more weight if I arbitrarily use the price
of a sack of potatoes. This problem can be eliminated by dividing potato prices by their
sample mean. The sample mean for the price of each crop has to be calculated over farmgate
prices that value identical units of that crop.
40
The sample average price of crop k in season t, p
kt
, is calculated as:
n
k
p p n
kt kit k
i
= ¿ where p
kit
is the farmgate price of crop k for farmer i in period t, and
n
k
is the total number of surveyed farmers that produce crop k in season t.
97
workers as proxy of their labor cost. Here I am assuming that family workers can at
least get the average wages for temporary workers by working off-farm.
The nitrogen price aggregate,
2
r , is computed as an arithmetic mean using
expenditure shares as the weights. Different nitrogen prices are obtained by dividing
unit fertilizer prices by kilograms of nitrogen per unit of fertilizer:
PF
fit
UN
f
(5.2.1)
where PF
fit
is the unit price of fertilizer f paid by farmer i in period t and UN
f
equals kilograms of nitrogen contained in one unit of fertilizer f. In the sample data
there is one farmer that reports a value of zero for nitrogen. Because the underlying
assumption of this dissertation is that observed farmers decisions are optimal, I take
that amount of nitrogen as the farmer’s optimal decision for the quantity of that input.
The nitrogen price for this observation is set at the arithmetic sample average of
nitrogen prices. The underlying assumption is that the farmer can do as well as the
average farmer in buying fertilizer input.
98
Chapter 6: Estimation and results
The estimation of the parameters , , a ? ? and the parameter vectors , ? ? , ?
is based on Full Information Maximum Likelihood (FIML) assuming a multivariate
normal distribution in the residuals of the system of equations. The likelihood of the
sample is:
( ) ( )
( )
( ) ( )
1
1
1
, , , , , 2 exp , , , , , , , , ) 2 2
2
NT
L a Z F Z F Z J
t
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? = ? ? ?
(6.1)
where Z=(Z
1
,Z
2
) is the vector of variables in the model. Z
1
is the set of endogenous
variables in the model, i.e. Z
1
=( ) , , N w X . Z
2
represent the set of exogenous
variables, i.e.
, , , , , , , , , , , ,
1, 1 1 1 1 1 1 1 2 , 2 1 1,
2
, ,
1 1 1, 1
s s v v P P T T r r r r w
jt jt jt jt jt jt it it it it i t i t i t i t it
Z
L F N N
it it it it
? ? | |
+ + + + + + + +
= |
|
+ + ? +
\ .
.
( ) , , , ,
det
1
F Z
J
Z
? ? ? ? ( ?
?
(
?
¸ ¸
is the Jacobian of the transformation from ? to Z
1
.
Consistent estimation of the system of equations (5.1.14) to (5.1.18) requires
that the exogenous variables in the model are not correlated with the error term of the
Euler Equation,
1 1 1 1 1 it it it jt
? ? ? ? ? + +
+ + + +
. Nevertheless, because that error term
contains an omitted variable,
1 it
?
+
, some of the regressors may be correlated with
1 1 it
?
+
. If so, the explanatory variables that are correlated with
1 1 it
?
+
are
endogenous (Kapetanios, 2004). Hence, I need to test for endogeneity of the
regressors in the Euler Equation. For that purpose I use a test proposed by Hausman
(1978), which I describe in detail in the Appendix at the end of this chapter. Also I
99
test whether the conditional mean of the error term in the Euler Equation is zero, i.e.
0
1 1 2
E Z
it
? ( =
+
¸ ¸
. The latter test procedure is as follows. First I obtain the FIML
estimated residuals for the Euler Equation. Then I estimate a linear regression of
those errors on a constant. If the constant is not statistically significant I do no reject
the hypothesis of a conditional mean value of zero for the error terms in the Euler
Equation.
Due to the lack of a longitudinal data, the estimation of the parameter for
preferences,
i
? , for each farmer is addressed by assuming that his utility function is
based on known farmer characteristics (Zeldes, 1989, Blundell et al., 1994, Dubois,
2001). Thus, the parameter that represents a farmer’s preferences is parameterized as
an exponential function of that farmer’s education (ED), experience (EXP) and
household size (HS)
i
? = ( ) exp
0 1 2 3
ED EXP HS
i i i
? ? ? ? + + + (6.2)
where , , ,
0 1 2 3
? ? ? ? are unknown parameters. The exponential form ensures that
i
? is positive. Hence, the right hand side of (6.2) replaces
i
? in the system of
equations (5.1.14) to (5.1.18).
For the variance of the random error
it
? in the equation for the variable I
it
,
equation (5.1.3), I propose the following specification:
( )
2
1 2
exp
0
D D
ic i il
? ?
? ? =
?
(6.3)
where , ,
0 1 2
? ? ? are unknown parameters, and D
il
and D
ic
are dummy variables
that indicate whether the farmer has livestock and whether the farmer has access to
100
consumption or production credit, respectively
41
. The exponential form for the
constant
0
? ensures that the variance of
it
? is positive.
Sample data indicates that a good number of farmers invest in livestock and
that cattle are sold and bought quiet often by the farmers, probably to mitigate
consumption volatility (Rosenzweig and Wolpin, 1993). The investment decision in
livestock should affect the variance of I
it
because asset returns do not have the same
variances in a real situation. Moreover, farmers without credit restrictions can
experience higher changes in I
it
because when it is needed they can substitute for
present net income from production and water transactions with market loans.
Furthermore, for those farmers, when income is greater than their “normal”
consumption levels they probably devote an important part of that difference to pay
their debts. Additionally, farmers with access to credit can face riskier activities and
therefore higher expected incomes precisely because they can solve consumption
smoothing through indebtedness. As a consequence, it is expected that farmers
without credit restrictions exhibit a higher variance in I
it
.
That structure for the variance of
it
? also allows me to control for the effect
of two main other income sources (credits and sales of cattle) when choosing optimal
decisions for consumption smoothing. In the present context of incomplete markets,
farmers maintain water rights not only for production but also for consumption
smoothing and this justifies the necessity for controlling from other income sources.
41
The variable D
il
takes the value of 2 if farmers have livestock and the value of 1
other wise. In the case of D
ic
it takes the value of 2 if farmers have access to credit and the
value of one other wise. I have used the value of 1 and 2 instead of 0 and 1 to allow
convergence in the estimation procedure.
101
This allows me to better explain within my model, decisions upon the number of
water rights held by a farmer.
In the construction of the dummy variable for credit, those farmers that have
received a credit in any of the agricultural seasons that go from the 95-96 to 99-00
seasons where classified as farmers with access to credit, otherwise they were labeled
as farmers without access to credit.
The parameter a in equation (5.1.3), the parameter
0
? in equation (6.3), as
well as the constants terms of the Just-Pope production function are not separately
identifiable from the parameter
0
? in the equation system (5.1.14) to (5.1.17). The
latter is the constant term in the specification for risk aversion in equation (6.2). Thus
the equation (5.1.14) should be written as:
( )
( )
( )
( )
( ) ( )
( )
ln ln ln
1
1 1 1 1 1 1 1
exp
0 1 2 3
3 1 2
1 1 1 1
1 1 1
1
exp
0 1 2 3
s v
jt jt jt jt
N N s N v T w
jt it it jt it jt it it
a ED EXP HS
i i i
P T L W F r X
it it it it
it it it
N N
jt it it
a ED EXP HS s
i i i j
? ? ?
?
? ? ? ?
? ? ?
?
? ? ? ?
? = + ?
+
| |
| |
? + ? +
+ + + + + + +
| |
+ + + + +
|
|
| ? | ?
+ + + +
+ + + \ . \ .
? +
?
+ + + +
( )
( )
( )
3 1 2
2
1
3 1 2
exp 2 2 2 2 2
0 1 2 3
2
1 2
exp 2 2 2 2
0 0 1 2 3 1 1
2
N v T w r X
t it jt it it it it
P T L W F
it it
it it it
a ED EXP HS P T L W F
it it
it it it
D D
ic il
ED EXP HS
it
? ? ?
? ? ?
? ? ? ?
? ?
? ? ? ? ? ?
| |
|
|
? ? ? + ?
|
|
|
\ .
| |
| |
|
+ + + + ?
|
|
\ .
\ .
+ + + + +
+
(6.4)
where ( ) ( ) exp 1 a a = + . Thus I estimate the parameters ln , , , , ,
0 1 2 3 0
? ? ? ? ? ?
plus the parameter vectors ? and ? . With
0
? =
0
a ? + and 2
0 0 0
? ? ? = +
. In the
102
input equations I also estimate
0
? instead of
0
? . As a consequence, I can not
estimate a specific value for the coefficient of risk aversion. Nevertheless, in spite of
the identification problem of
0
? , I can still test whether risk aversion is
heterogeneous among farmers.
FIML procedure requires an initial value for each of the parameters of the
system. Love and Buccola (1991) and Saha et al. (1994) estimate farmers’
preferences and production technology jointly in the presence of risk. They estimate
parameters for preferences and production technology from the first order conditions
of the maximization of the expect utility with respect to inputs. In both studies,
starting values for the production function parameters are provided by a prior
estimation of the Just-Pope production function. Nevertheless, if inputs are
endogenous, they should be correlated with the error term in the production function.
The Just-Pope parameter estimates are then inconsistent (Love and Buccola, 1991).
Due to that problem with the estimation of the Just-Pope production function, I have
chosen to obtain starting values for the production function parameters from a prior
estimation of the input demand system in my model. The estimates from those
equations provide the set of starting values for a new estimation of the whole system,
i.e. the input equations together with the Euler Equation.
For the constant discount factor, ? , I use as a starting value the reciprocal of
one plus the market annual interest rate on year 1998. For the constant in the
equation for the variance of
it
? (equation (6.3)) I use a value of zero as starting value
and a value of one for the parameters of livestock and credit.
103
Data used in the estimation presents large differences in the scale of the
variables. Therefore, I have re-scaled the values of the variables dividing each
variable by its sample standard deviation (with the only exception of dummy
variables of the equation (6.3)). Scaling the data facilitates convergence of the
estimation and does not affect the measurement of the underlying relationship among
the variables in the model neither the t-statistics, but it does affect the interpretation
of the parameter estimates (Carter et al., 1997, Chapter 6). The latter is not a problem
for the analysis developed in this dissertation that focus on the significance and the
sign of the parameter estimates rather than on their numerical values. Finally, a
constant term was added to each input equation in order to ensure a zero mean value
for the error terms.
The descriptive statistics for the variables used in the estimation of the system
of equations are reported in Table 6.1.
104
Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations.
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Number of water
shares (96-97)
Units 65.5 0.25 18.2 17.5
Number of water
shares (97-98)
Units 65 0.0 17.8 17.9
Number of water
shares (98-99)
Units 65 0.25 15.6 14.7
Number of water
shares (99-00)
Units 80 0.0 17.3 19
Water accorded to
water rights (97-98)
Cubic meters 6633 5000 6039 526.0
Water accorded to
water rights (98-99)
Cubic meters 6633 3000 4933 1193
Water accorded to
water rights (99-00)
Cubic meters 6633 3000 4969 1227
Education (98-99)
Years of
schooling
17 1 8.6 4.7
Experience (98-99) Years 50 1 28.6 14.3
Household size
(98-99)
Number of
people leaving in
the same house
18 1 4.9 3.6
Education (99-00)
Years of
schooling
17 2 8.7 4.6
105
Cont. Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Experience (99-00) Years 51 10 30.5 13.6
Household size
(99-00)
Number of people
leaving in the same
house
15 1 4.9 3.0
Land (98-99)
Total cultivated
land
116 0.2 12 21.3
Labor (98-99)
Total number of
hours on the season
per cultivated
hectare
7400 15.36 1099 1437
Fertilizers (98-99)
Kg. of nitrogen per
cultivated hectare
2818 0.00 320 591
Water used as input
by the farmer (98-99)
Cubic meters per
cultivated hectare
18720 1485 6958 5129
Land (99-00)
Total cultivated
Land
90 0.12 11.4 18.5
Labor (99-00)
Total number of
hours on the season
per cultivated
hectare
4533.3 8.5 987 1201
Fertilizers (99-00)
Kg. of nitrogen per
cultivated hectare
816.7 0.0 125.7 171
Water used as input
by the farmer (99-00)
Cubic meters per
cultivated hectare
93750 1782 8674 16081
106
Cont. Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Labor price (98-99) Chilean pesos 7800 2750 4238.8 919.7
Fertilizer price (98-
99 )
Chilean pesos 2717 51.9 985.5 686.7
Water price in the
spot market (98-99)
Chilean pesos 12.52 6.77 8.93 2.34
Labor price (99-00) Chilean pesos 5500 3250 4107.2 664.4
Fertilizer price (99-
00 )
Chilean pesos 6512.9
157.7 1133.6 1141.3
Water price in the
spot market (99-00)
Chilean pesos 9.72 2.11 6.61 2.65
Water right prices
(97-98)
Chilean pesos 456876 176775 286788 106160
Water right prices
(98-99)
Chilean pesos 554981 172496 361797 155830
Water right prices
(99-00)
Chilean pesos 588887 197404 332714 131889
Output price index
(98-99)
Chilean pesos 3.72 0.29 1 0.87
Output price index
(99-00)
Chilean pesos 5.14 0.35 1.16 1.28
Labor cost (98-99) Chilean pesos 31256007 15726 2010805 5434032
107
Cont. Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Fertilizer cost (98-
99)
Chilean pesos 10641458 16163 665187 1860068
Labor cost (99-00) Chilean pesos 42203503 12555 2314106 7486319
Fertilizer cost (99-
00)
Chilean pesos 3132494 0.0 325774 645616
Gross output
revenue per hectare
(98-99)
Chilean pesos 13255140 47324 1759103 2675375
Gross output
revenue per hectare
(99-00)
Chilean pesos 8923973 84547 1579336 2115119
108
I specify the variance-covariance matrix allowing different variances for the
disturbances of each equation and contemporaneous correlation among the
disturbances of the input first order conditions corresponding to the same farmer.
Heteroskedasticity for the errors of the Euler Equation is expected because they are
defined as
1 1 1 1 1 it it it t
? ? ? ? ? + +
+ + + +
, and it is reasonable to expect that the
[ ]
1
Var
t it
?
+
, i.e. the variance of the conditional volatility of the discounted water
right price for each farmer, differs among farmers. Nevertheless, the correlogram of
the square residuals, the conditional heteroskedasticity (ARCH) test and the Cusum
squares test indicate that the null hypothesis of homokedastic error terms of the Euler
Equation cannot be rejected. The same result is obtained for the error terms of the
input equations for water and fertilizer. In the case of the errors of the labor equation,
only the Cusum squares test does reject that hypothesis.
To test for correlation among errors corresponding to different farmers within
the same equation, I use the Ljung-Box Q-statistic which is commonly used to test
whether the disturbances are white noise. Based on this test I do not reject the
hypothesis that the errors are not correlated.
The resulting variance-covariance matrix can be summarized by:
( ) E
t
??? ? = =
0 0 0
11 * * * *
0
* 22 *
23 * 24 *
0
* 32 * 33 * 34 *
0
* 42 * 43 * 44 *
I I I I
N T N T N T N T
I I
N T N T I I
N T N T
I I I
N T N T N T N T
I I
N T N T N T N T
?
?
? ?
? ? ?
? ? ?
(
(
(
(
(
(
¸ ¸
Estimates of the error covariances between input equations are obtained from:
1
ˆ ˆ ˆ
1
N
gh gi hi
N
i
? ? ? = ¿
=
, where g=2,3,4 and h=2,3,4.
109
Residuals ˆ
gi
? and ˆ
hi
? are the maximum likelihood estimated residuals in each
iteration.
Efficient estimation of the Euler Equation is required for the Hausman test in
that equation (Wooldridge, 2002, Chapter 6.2.1). Because I ensure an efficient
estimation of that Equation with the above mentioned specification of the variance-
covariance matrix, I can use the Hausman test for possible endogeneity of some of the
regressors due to an omitted variables problem in the Euler Equation. Result of the
test indicates that I can not reject the null hypothesis that those variables are
individually and jointly exogenous. This provides evidence that the parameters of the
model are estimated consistently and that they are unbiased. In fact a maximum
likelihood ratio test for the null hypothesis that the variables are jointly exogenous
gives a value equal to 1.70 which is lower than the critical value of 12.8 for a Chi-
Squared with 5 degrees of freedom and a significance level of 5%. Moreover, the test
for the conditional mean of the error terms in each equation indicates that I can not
reject the hypothesis that the residuals have conditional mean equal to zero. This
result for the Euler Equation is fully consistent with the above result of the Hausman
test, because 0
1 2
E Z
hit
? ( =
+
¸ ¸
implies [ ] 0
1 2
E Z
hit
? =
+
.
A Jarque-Bera test for normality of the errors shows that I cannot reject the
null of normality for the residuals of each of the equations within the system but the
equation that describes first order condition for labor.
Parameter estimates are shown in Table 6.2. Results indicate that all the
parameters are significant at 1% of significance level, but the parameters for
110
household size and the dummy variable that indicate whether the farmer has livestock
are not statistically significant.
Table 6.2: Estimates of the parameters of the equation system using FIML
Coefficients
(Standard
Errors)
Estimates of the deterministic part of the production function
( vector? )
Water inputs per hectare 0.86
(0.10)
Labor per hectare 2.43
(0.08)
Fertilizer use per hectare 0.44
(0.03)
Estimates of the stochastic part of the production function (vector ? )
Parameters
Water inputs per hectare 2.28
(0.13)
Labor per hectare 0.04
(0.01)
Fertilizer use per hectare 0.15
(0.02)
Estimates of the parameters for the equation for farmers preferences
(vector ? )
Constant (
0
? )
5.2
(0.31)
Education -1.92
(0.08)
Experience -2.10
(0.14)
Family Size -0.12
(0.08)
Estimates of the parameters for the variance of
it
? (vector? )
Constant (
0
?
)
-5.1
(0.34)
111
Cont. Table 6.2: Estimates of the parameters of the equation system using FIML
Livestock ( D
l
)
-0.44
(7.44)
Credit ( D
c
)
7.77
(1.22)
Estimates of other parameters
Discount factor ( ? )
42
0.93
(0.05)
Estimates of other parameters
Values
Estimated variance of the Euler Equation
0.08
Estimated variance of the water equation
0.88
Estimated variance of the labor equation
1.06
Estimated variance of the fertilizer equation 0.88
Estimated correlation between water and fertilizer errors 0.02
Estimated correlation between labor and fertilizer errors 0.33
Estimated correlation between labor and water errors -0.12
Jarque-Bera Test for normality of the errors (p-value in
parenthesis)
Euler 0.31
(0.86)
Water 4.86
(0.09)
Labor 67.38
(0.0)
Fertilizer 5.74
(0.06)
Values
Number of observations 32
Maximum Likelihood 82.5
42
In the Euler equation I estimate the natural log of the discount factor ( ln ? ) and I
use the Deltha Method to obtain the standard deviation of the discounted factor, ? .
112
In order to check the accuracy of the estimation, I compute Theil’s Inequality
coefficient over gross output income per hectare
43
and water rights investments
44
.
That coefficient always falls between 0 and 1. If it takes the value of 0 there is a
perfect fit in the model; if it takes the value of 1, the predictive performance of the
model is bad. Moreover, Theil’s Inequality coefficient can be decomposed in the bias
(variance) proportion that indicates how far the mean (variance) of the predicted
values is from the mean (variance) of the actual data (Pindyck and Rubinfeld, 1998,
Chapter 8). For good forecasts, the bias and variance proportions are small. Because
gross output income per hectare and water rights investments have been normalized
by their standard deviation respectively, the comparison between the variance of the
predicted value and the variance of the actual data is meaningless. Thus, comparison
is limited to the bias proportion. For gross output income per hectare Theil’s
Inequality coefficient is 74% and the bias proportion is 1.7E-5. For water rights
investments Theil’s Inequality coefficient is 76% and the bias proportion is 1.01E-5.
Those low values of the bias in the means allow me to disregard the possibility of
systematic bias in the prediction of those variables with my model. Figures 6.1 and
6.2 report the actual value of output income per hectare and water right investments
against the imputed values for those variables, based on the estimates in Table 6.2.
43
On my model the expected value of gross output income per hectare corresponds
to:
0 1 2
P T w L F
it it
it it it
? ? ?
.
44
Investment in water rights corresponds to: ( )
1
N N
jt it it
? ?
?
113
Figure 6.1: Actual value of standardized gross output income per hectare
against the imputed values for that variable.
0 5 10 15 20 25 30
1
2
3
4
5
S
t
a
n
d
a
r
d
i
z
e
d
G
r
o
s
s
O
u
t
p
u
t
I
n
c
o
m
e
fitted actual
Figure 6.2: Actual value of standardized investment in water rights
against the imputed values for that variable
0 5 10 15 20 25 30
-6
-4
-2
0
2
4
6
S
t
a
n
d
a
r
d
i
z
e
d
N
e
t
I
n
v
e
s
t
m
e
n
t
i
n
W
a
t
e
r
R
i
g
h
t
s
fitted actual
114
Thus, at this time, I can ask whether heterogeneity of farmers’ preferences is a
valid hypothesis. That is tested by defining the null hypothesis that the parameters
, ,
1 2 3
? ? ? are jointly zero. If the null hypothesis is rejected, I infer that farmers
have heterogeneous preferences. A maximum likelihood ratio test is used to verify
the null hypothesis that 0
1 2 3
? ? ? = = = against the alternative hypothesis that at
least one of those parameters is different from zero. I obtain a value for the
maximum likelihood ratio test of 112.23 that leads me to strongly reject the null of
homogeneity on preferences.
Proper implementation of that test for heterogeneous preferences also requires
controlling for incomplete markets (Dubois, 2001). The above specification for
i
?
and equation (4.2.11) make clear that I am testing for differences in the farmers’
stochastic discount factors by allowing them to be a function of farmers’
characteristics. Those characteristics enter in the specification of the discount factor
in equation (4.2.12) through their possible relationship with farmers’ risk aversion.
Nevertheless, as I discussed on Chapter 4, Section 2, differences in reservation values
for a water right among farmers in the same water association may also differ due to
incomplete asset markets. Because it is well known that farmers in the Limarí Valley
face incomplete asset markets, testing for the effect of farmers characteristics on the
stochastic discount factor requires controlling for incomplete asset markets. One way
to do this is by allowing consumption to be a function of current income, which is a
clear consequence of incomplete asset markets. That approach is used in this
dissertation to control for incomplete asset markets. .
115
Estimates for the deterministic component of the production function
summarized in Table 6.2, show that the three inputs under analysis have a positive
effect on mean output. This result is as expected. In terms of output variance, the
three inputs have a positive marginal effect on yield variability. The result that
fertilizer has a positive marginal effect on production variance corroborates similar
findings by Love and Buccola (1991) and Just and Pope (1979). The positive effect
of labor on yield variability coincides with the result obtained by Di Falco, Chavas
and Smale (2006) for a sample of farmers from highlands of Ethiopia.
The finding of water as an increasing risk input is an unexpected result.
Usually, irrigation is considered as a risk-reducing input. As an example, irrigation
reduces the effect of frosts on some type of crops and so their yield variance. It may
be possible that the positive sign for water in the stochastic part of the production is
being caused because crops with higher water requirement are also the ones with
higher variance. Then, estimating the effect of water in output risk will require
controlling for the latter relationship. That can be done by estimating different
production functions for each specific crop type.
The signs of the estimates of the equation for risk aversion indicate that better
educated farmers and with more experience are less risk-averse. The result that more
educated farmers are less risk averse is consistent with the results obtained by Knight
et al. (2003) with household data from rural Ethiopia. A possible explication for that
relationship between risk aversion and education is that more educated farmers are
better able to manage risk and so they are willing to take more risk. Nevertheless,
that result contradicts the works of Bar-Shira et al. (1997) and Ajetomobi et al.
116
(2006), which find that higher levels of education are associated with greater risk
aversion.
The finding that farmers with higher level of experience exhibit a lower
degree of risk aversion confirms the result obtained by Z Bar-Shira et al. (1997).
Those authors explain that result by pointing out that risk is a complicated factor that
less-experienced farmers try to avoid.
Regarding the household size it can be hypothesized that the larger the size of
the family, the higher the subsistence consumption needs and given a fixed amount of
land, the lower the willingness of the farmer to assume risks. On the other hand,
family size might affect the labor capacity of the peasant household in which case a
larger family size implies greater capacity to assume risks. Furthermore, larger
households may diversify their activities and better insure themselves efficiently
reducing risk. Thus, those farmers will be lees reluctant to accept a bargain with an
uncertain payoff rather than another bargain with more certain but possibly lower
expected payoff. That makes them less risk-averse or more risk tolerant
45
. The result
that household size does not affect risk aversion suggests the convenience of
separating those two mentioned effects of that variable over risk aversion by
including a variable that represents farmer’s diversification and another one for the
farmer’s number of children in the specification for risk aversion of equation (6.2).
For the variance of the error term,
it
? , in the linear equation for “other net
incomes”, I
it
, I found that whether or not farmers invest in cattle does not affect the
45
The inverse of a person's risk aversion is sometimes called his risk tolerance
(Wikipedia, the free encyclopedia)
117
variance of I
it
. Furthermore, results indicated as it was expected that farmers with
access to credit for consumption or production have higher variance in I
it
.
Nevertheless, these results are not robust to different specifications of the variance of
it
? neither to variations in the number of observations in the sample.
The estimated value for the discount factor, ? =0.93, belongs to the expected
range for this parameter, 0 < ? <1. Nevertheless a value of 0.93 for the discount
factor seems to be too high to be credible and indicates that the model is not suitable
for the estimation of that discount factor. This occurs because I have an identification
problem with the discount factor: ? appears in the Euler Equation as the constant
term, which is capturing not only the value of the discount factor but also a possible
non zero mean of the error term as well as other constant terms of that equation.
Finally, the residuals for each equation based on the estimates in Table 6.2 are
plotted in Figure 6.3.
118
-4
-3
-2
-1
0
1
Euler
Water Labor Fertilizer
Figure 6.3: Plot of the residuals from the FIML regression based on estimates in
Table 6.2.
119
Appendix: A Hausman test for endogeneity
The error term in the Euler Equation (5.1.14),
1 1 it
?
+
, is a composite error:
1 1 1 1 1 it it it jt
? ? ? ? ? + +
+ + + +
. Thus for each t+1 period,
1 1 it
?
+
is the sum of an
unobserved effect,
1 it
?
+
, and two random errors
1 it
?
+
and
1 jt
?
+
. As I discussed in
Chapter 5,
1 it
?
+
represents a measure of the volatility of the discounted water right
prices. Because that unobserved effect, maximum likelihood estimation of equation
(5.1.14) may not be consistent due to endogeneity issues. This would occur if some
of the exogenous variables in equation (5.1.14) are correlated with
1 it
?
+
and hence
with the error term
1 1 it
?
+
. For example, if water right prices in t+1 were correlated
with the volatility of the discounted water right price in t+1, the FIML estimate of the
parameters in the model would be biased due to endogeneity. A similar situation may
also arise with respect to input quantities that affect output variance which may affect
the volatility of the stochastic discount factor. Thus, the potential presence of
endogeneity must be tested.
As in Di Falco, Chavas and Smale (2006) I use a residual-based form of the
Hausman test that turns to be asymptotically equivalent to the original form of the
Hausman test (Wooldridge, 2002, Chapter 6.2). The test involves estimating
auxiliary reduced-form regressions for the regressors suspected to be endogenous.
Those are linear regressions for each regressor suspected to be endogenous on a
constant, all the exogenous variables of the model and regressor specific instruments.
Then the Euler Equation is estimated including the reduced-form residuals as
additional explanatory variables. The joint statistical significance of the coefficients
120
associated with the residuals is then evaluated. If those parameters are jointly not
significant then the Hausman test does not reject the hypothesis of exogeneity of the
regressors. As Wooldridge (2002, Chapter 6.2.1) point outs, valid test for the
individual and the joint significance of those parameters associated with the residuals
requires an efficient estimation of the Euler Equation.
This test was implemented for those exogenous variables that are suspected of
being correlated with the volatility of the discounted water right prices. Table 6.3
shows the list of instruments that I use for each possible endogenous regressor.
Among those instruments I include all the exogenous variables in the system of
equations that are not correlated with the error term. Column 1 of that table contains
the list of variables that were tested to determine if they are statistically correlated
with the error term
1 1 it
?
+
.
Table 6.3: List of instruments to test for possible endogenous regressors.
Possible endogenous
regressors Instruments
1 1 1
s N v
jt it jt
? ?
+ + +
1 1 1
s N v
jt it jt
? ?
? ? ?
, ( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
, , , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
( )
1 1
N N
jt it it
? ?
+ +
( )
1 2 1
N N
jt it it
? ?
? ? ?
, ( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
, , , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
1
L
it +
( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
,
, , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
1
w
it +
( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
,
, , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
1
F
it +
( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
,
, , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
121
A maximum likelihood ratio test indicates that the coefficients on the reduced-
form residuals of the equations for the variables in column 1 of the above table were
jointly not statistically different from zero at a 5% of significance level.
122
Chapter 7: Conclusions and Suggestions for Future Research
In this dissertation I have shown that heterogeneous risk preferences among
farmers is a sufficient condition for water rights transfers when farmers can
simultaneously exchange water in a spot market with lower transaction costs. To
reach that goal I developed a model for water right reservation values in which water
rights are investment assets and where the link between the spot market and market
for water rights is explicitly considered. The model encompasses several aspects
related to water market activity and a farmer’s behavior. I described and measured
transaction activity in markets for water rights and in a spot market for water
volumes, in an existing market since 1981 in the Limarí Basin, Chile. That analysis
allowed me to conclude that both markets are reallocating a significant amount of
water among farmers, although the spot market has, by far, the highest amount of
traded water. I also showed that, contrary to what other researchers believe, the spot
market is active not only during drought years but also in years with average water
availability. I characterized optimal decision making by farmers over the number of
water rights to be held in each season. The model assumes incomplete asset markets,
output uncertainty, as well as uncertainty about future water availability and water
prices. Because investment decisions affect future levels of consumption and farmers
face uncertainty, the theoretical model for farmer decisions was modeled as a
stochastic dynamic problem. This results in a consumption-based capital asset
pricing model (CCAPM) which is described by an Euler Equation that ties asset
returns (water right returns in this case) to marginal rates of substitution for
consumption at different points in time. This model implies that the current period
123
reservation value of a water right is a function of the current value of the amount of
water accorded to water rights in the spot market, the expected future water rights
prices and the stochastic discount factor. Nevertheless, since most transfers of water
rights take place among farmers that belong to the same WUA and such farmers are
likely to have identical expectations, the primary basis for differences in reservation
values and for water right transactions are the differences in their stochastic discount
factors. Incomplete asset markets as well as heterogeneous risk preferences cause
differences in the stochastic discount factors among farmers.
Because asset markets are not complete, farmers value water rights not only as
a source of water for production but also as a means to insure themselves against bad
shocks. As a consequence, the future value of a water right is given by the expected
discounted marginal rate of return to holding a water right from time t to time t+1
plus a risk premium that is greater than zero. The latter implies that the reservation
value of a water right for the more risk-averse farmer is greater than that for the less
risk-averse farmer. This produces transfers of water rights from those farmers who
are least risk averse to the most risk-averse farmers. This approach also emphasizes
that water right transactions solve differences in attitudes towards risk among
farmers, whereas differences in water marginal return among farmers are solved in
the spot market.
The theoretical analysis provides the foundation for a case study of water
transfers for irrigation in the Limarí Basin, an important agricultural region in the
northern part of Chile, which has one of the most active Chilean irrigation water
markets. With micro level data from that basin I estimate a system of equations that
124
describes farmers’ optimal decisions over the number of water right to be held and
input quantities. The estimation of that system assumes that a farmer’s utility from
consumption is represented by a negative exponential utility function, and that the
production technology is described by a Just-Pope Cobb-Douglas production
function. The use of a negative exponential utility function imposes severe
restrictions in the model and the results are conditional to that specific functional
form. Nevertheless, that utility function allows me to characterize the absolute risk
aversion coefficient for each farmer as a function of his observable characteristics and
to develop a promising approach to jointly estimate the parameters that describe
farmers´ preferences and production technology considering farmers investment
decisions. This approach can be extended to more general utility functions although
that will require more advance methods of estimation.
The results of the estimation procedure indicate that the hypothesis of
heterogeneous risk preferences among farmers can not be rejected. Moreover, better
educated farmers and with more experience are less risk-averse. On the production
side, water, labor and fertilizers have positive impact on output mean per hectare.
The analysis of inputs on yield variability showed that those inputs are risk
increasing.
Up to now, research on agriculture finance has been characterized by the
dominance of real estate among the farmers’ assets. But now, due to the increasing
interest in establishing transferable water rights not married to land rights, research on
agriculture finance should move from considering land as the main asset to water
rights as a dominant asset in dry areas. Due to the special characteristics of water
125
resources, this new challenge offers a significant opportunity for future research.
That future research may include the analysis of the robustness of the conclusions
regarding farmers’ heterogeneous risk preferences to alternative functional forms of
the utility function. A suitable candidate is a linex utility function (Bell and Fishburn,
2001), consisting of a utility function that is the sum of an exponential function and a
linear function. This function has the important property of decreasing absolute risk
aversion while retaining some of the convenience of the exponential form. Moreover,
that functional form allows a closed form solution to the farmer decision problem that
I analyze in this dissertation.
Another possible extension would be to develop alternative ways of dealing
with the missing data problems that might allow me to use more of the data that I
have collected among farmers in the Limarí Valley.
The analysis of the role of risk differences, due to different types of crops or
distance from the reservoirs, on the reservation values for a water right is another
interesting extension of this work. The presence of speculative bubbles in water right
market prices, as suggested by Person and Michelsen (1994), is an appealing area for
future applied research on water price models. Other future research deals with
improving the mechanism by which prices are formed in the permanent market. This
could be done by the design of the right incentives to motivate farmers to reveal their
private information on the reservation values
46
. The existence of this non-disclosed
private information can reduce the number of transactions even when the reservation
value of the buyer is greater than that of the seller, and this may impede the efficient
46
Private information in reservation values for water rights includes information
about the farmer’s inter-seasonal discount rate and his attitude towards risk.
126
allocation of water. The formation of an options market in which farmers may obtain
options to purchase water during a dry year could be quite useful in addressing this
problem because an options market reveals the differences in attitude towards risk.
One of the most interesting problems in the formation of an options market is the
creation of an appropriate incentive framework for farmers who participate in such a
market.
127
Annex: Survey instrument
Date of Survey _________________Name of interviewee ________________________________
Association that provides you with water _________________
1. Interviewee’s home (either the tenant’s or the landlord’s)
Information about the tenant or landlord:
1.1 Age ______ Marital Status _______________
1.2 How many people live in your home? ________
1.3 Please give detailed information about each person living in your home
N° Kinship with
interviewee
¿Does
he/she work
in the farm?
Current
Age
Experience
in
agriculture
Level of
Education
achieved
Interviewee yes
Wife
Son/Daughter
Son/Daughter
Son/Daughter
Son/Daughter
2. Property and management of the land and water
2.1 How many lots do you own in the valley? _________ (No)
128
2.2 Please describe your lots?
Lot N° Location Area No. of water
shares
Name of the
Canal
Total
2.3 Quality and use of the farms
Lot
N°
Fertility* Slope* Erosion* Niter* Sown in
97/98
Sown in
98/99
Fertility* (1) high fertility (2) low fertility (3) poor quality of land
Slope* (1) flat (2) hillside (3) hill
Erosion* (1) no problem (2) some problem (3) serious problem
2.4. Do you rent either part or all of your property to other persons? Yes ___ No____
Since when/for how long? _____
How many hectares? _________
How much do you charge per year? _________
With how many water shares? _______
2.5 Do you rent any land properties from other persons? Yes ____ No______
Since when/for how long? ______
How many hectares? _________
How much do you pay per year? _________
With how many water shares? _______
129
2.6 Do you work part or the whole of the land with any partners? Yes ____ No______.
How many partners? __________ Are they next of kin (relatives)? Yes ___ No____
How many hectares?_________ Since when/for how long?_________
Please describe the type of contract made with your partner(s) (i.e. crops grown, land distribution, water, labor and
machinery each party supplies, etc.) and how you finance costs and production. ________________________
2.7 How did you purchase the farm?
Kind of Purchase Area
(hectares)
When did you
purchase it?
How much was a
hectare?
First purchase
Second purchase
Third purchase
Inherited
Due to agrarian
reform
Other
Total (verify)
2.8 Have you sold or divided your property so far? Yes___ No___
(If so, please fill in the chart below)
Type of operation Area
(Hectares)
When? How much did
you ask for each
hectare?
First sale
Second sale
Third sale
First partition
Second partition
Third partition
Other
Total
130
2.9 How did you purchase your water shares and how much did you pay for them?
Type of
acquisition
No. of
water
shares
When did
you buy
them?
With the
land?
How much did you
pay for each water
share?
Whom did
you buy it
from?
First buy
Second buy
Third buy
Inherited
Due to agrarian
reform
Other
2.10 Have you sold or distributed/divided part of you shares so far? Yes___ No___
(If so, please fill in the chart below)
Type of sale No. of water
shares
When did
you sell
them?
With the
land?
What were
you paid for
each water
share?
Whom did
you sell it to?
First sale
Second sale
Partitions
Other
Total
3 Agricultural Production over the Past Two Seasons
3.1. How much land did you sow in the 98/99 season?_________
3.1.1. How much land did you sow with partners? _________
3.2. How much land did you sow in the 97/98 season? _________
3.2.1. How much land did you sow with partners? ___________
131
3.3 What did you sow in the 97/98 and 98/99 seasons?
Sowing Hectares Were there any
losses due to
drought?
97/98 98/99 97/98 98/99
3.4 What was your harvest production in the two seasons?
Crop Total Production Harvested
Hectares
Unit of measure
(sacks, boxes, kilos,
etc.)
97/98 98/99 97/98 98/99 97/98 98/99
3.5 What were you paid for your products in the two seasons?
Name of Product Price per unit in
Pesos $
Weight unit
97/98 98/99 97/98 98/99
132
3.6 What sort of irrigation systems did you use for each crop in the 98/99 season? (Flooding, furrows, drip, etc)
Crop Irrigated
area
Type of irrigation system
used in the crop
3.7 What sort of irrigation system did you use for each crop in the 97/98 season? (Flooding, furrows, drip, etc)
Crop Irrigated
area
Type of irrigation system
used in the crop
3.8 Do you have grapevines? Yes___ No____
If so, please describe each of them
Grapevine
N°
N° de
hectares
N° of
shrubs
Years Type of grape Yield in
98/99
Regular
Yield
133
3.9 Do you have any other permanent crops? Yes____, No_____
(If so, please describe them)
Crop N° de
hectares
N° of
shrubs
Years Type of
product
Yield in
98/99
Regular
Yield
4. Use of Labor and Water
4.1 Do you have permanent workers in your farm? Yes____ No____
If so, how many? _______
4.2 How much labor did you use for each crop in the following activities over the past two seasons?
Grape 1 type) ________________No. hectares_______ No. shrubs________
Activity Season 97/98 Season 98/99
Time
span
Days No. of
people.
Salary Time
span
Days No. of
people.
Salary
Pruning/tying
Applications
Watering
Clearing
Harvest
Other
134
Grape 2 type) _____________No. hectares_______ No. shrubs___________
Activity Season 97/98 Season 98/99
Time
span
Days No. of
people
.
Salary Time
span
Days No. of
people.
Salary
Pruning/tying
Applications
Watering
Clearing
Harvest
Other
Crop No. 1_________________
Activity Season 97/98 Season 98/99
Time span Days No. of people. Salary Time span Days No. of people. Salary
Preparation of the land
Seedbed
Transplant
Seed Sowing
Applications
Watering/irrigation
Clearing
Harvest
Other
135
Crop N° 2_________________
Activity Season 97/98 Season 98/99
Time span Days No. of people. Salary Time span Days No. of people. Salary
Preparation of the land
Seedbed
Transplant
Seed Sowing
Applications
Watering/irrigation
Clearing
Harvest
Other
Crop N° 3________________
Activity Season 97/98 Season 98/99
Time span Days No. of
people.
Salary Time span Days No. of people. Salary
Preparation of the land
Seedbed
Transplant
Seed Sowing
Applications
Watering/irrigation
Clearing
Harvest
Other
136
4.3 Did you and your family take part in the pruning? Yes___ No____
4.4 Did you and your family take part in the sowings? Yes ___ No____
4.5 Did you and your family take part in the harvest? Yes ___ No____
4.6 Did your permanent workers take part in the pruning? Yes___ No____
4.7 Did your permanent workers take part in the sowings? Yes ___ No___
4.8 Did your permanent workers take part in the harvest? Yes ___ No___
4.9 How much do you pay your permanent workers? _______________
137
5 Use of Inputs and Other Expenses
5.1 Please state the amount/number and cost of inputs in the agricultural production in the last two seasons.
Grape Harvest 1_____________ hectares______ N° shrubs________
Input Amount.
97/98
Unit Total cost Amount
98/99
Unit Total cost
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or days
Freight
Grape Harvest 2_____________ hectares______ N° shrubs________
Input Amount.
97/98
Unit Total
cost
Amount
98/99
Unit Total
cost
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or
Freight
138
Crop 1_____________ hectares______
Input Amount
97/98
Unit Total cost Amount
98/99
Unit Total cost
Seeds
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or days
Freight
Crop 2_____________hectares______________
Input Amount
97/98
Unit Total cost Amount
98/99
Unit Total cost
Seeds
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or days
Freight
139
5.2 Did you rent in a tractor in the last two seasons? Yes ___ No ___
If so, answer the questions below
Grape Harvest
97/98 98/99
M/Hr Price M/Hr Price
Preparation of land ______ ______ _____ _____
Application ______ ______ _____ _____
Crop ______ ______ _____ _____
Other Harvests
97/98 98/99
M/Hr Price M/Hr Price
Preparation of land ______ ______ _____ _____
Application ______ ______ _____ _____
Crop ______ ______ _____ _____
If not, please answer
Did you use your own tractor? Yes ____ No _____
5.3 How did you use your own tractor (or tractors) in your own land in the last to seasons?
Grape Harvest
97/98 98/99
M/Hr M/Hr
Preparation of land ______ _____
Application ______ _____
Crop ______ _____
140
Other Harvests
97/98 98/99
M/Hr M/Hr
Preparation of land ______ _____
Application ______ _____
Crop ______ _____
5.4 Did you rent your tractor(s) to other farmers? Yes___ No____
If so, for how many hours _______ At what price? _______
5.5 Did you hire an accountant in the last two seasons? Yes_____, No_____
If so, how much did you pay him/her ______________
5.6 Did you buy water in the last two seasons? Yes ____ No _____
If so, please answer:
Amount purchased 97/98 _________ Price _______ ($ per m3)
Date of purchase __________
Name and address of salesman__________
How many salesmen did you deal with? __________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
Why did you buy? _____________
Amount purchased 98/99 _________ Price _______ ($ per m3)
Date of purchase __________
Name and address of salesman __________
How many salesmen did you deal with? _____________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
Why did you buy? _____________
141
5.7 Did you sell water in the last two seasons? Yes ____ No _____
If so, please answer:
Amount sold 97/98 _________ Price _______ ($ per m3)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer?_______________
Why did you sell?_____________
Amount sold 98/99 _________ Price _______ ($ per m3)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer? _______________
Why did you sell? _____________
5.8 Did you buy water shares in the last two seasons? Yes ____ No _____
If so, please answer the questions below
No. of shares bought 97/98 _________Price _______ ($ per share)
Date of purchase __________
Name and address of salesman __________
How many salesmen did you deal with?______________________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
How much time passed since you decided to buy shares until you got hold of them?
Why did you buy?_____________
142
No. of shares bought 98/99 _________ Price _______ ($ per share)
Date of purchase __________
Name and address of salesman __________
How many salesmen did you deal with? ______________________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
How much time passed since you decided to buy shares until you got hold of them? ___________
Why did you buy? _________________
5.9 Did you sell water shares in the last two seasons? Yes ____ No ________
No. Of shares sold in 97/98 _________Price _______ ($ per share)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer? _______________
How much time passed since you decided to sell your shares until you transferred them?______________
Why did you sell? _____________
No. Of shares sold in 98/99 _________ Price _______ ($ per share)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer? _______________
How much time passed since you decided to sell your shares until you transferred them? ______________
Why did you sell? _____________
5.10 Did you rent in water shares in the last two seasons? Yes ____ No _____
If so, please answer the questions below
Number of water shares leased in 97/98 _________Price _______ ($ per share)
143
Date of leasing __________
Duration of the leasing contract___________
Name and address of the renter__________
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the renter _________________
Why did you rent in water shares? _____________
Number of water shares leased in 98/99 _________Price _______ ($ per share)
Date of leasing __________
Duration of the leasing contract___________
Name and address of the renter__________
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the renter _________________
Why did you rent in water shares? _____________
5.11 Did you rent out water shares in the last two seasons? Yes ____ No _____
If so, please answer these questions below
Number of water shares leased in 97/98 _________ Price _______ ($ per share)
Date of leasing__________
Duration of the leasing contract___________
Name and address of the lessee__________
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the lessee__________?
Why did you leasing water shares? _________________
Number of water shares leased in 98/99_________ Price _______ ($ per share)
Date of leasing__________
Duration of the leasing contract___________
Name and address of the lessee__________
144
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the lessee__________?
Why did you rent out water shares?
6. Husbandry production in the last two seasons
6.1 Do you have livestock? Yes ____ , No_____ (If not, skip to the next section, if so, please describe your current stock
of cattle)
Kind Number Breed Approx. Total value
in Pesos $
6.1.1 How much livestock have you had each year?
1999_______________
1998_______________
1997_______________
6.2 Have you sold livestock in the last two years? Yes ___, No____
(If so, please describe your sales)
Sales Units When? Price per unit
1
st
sale
2
nd
sale
3
rd
sale
145
6.3 Have you bought livestock in the last two years? Yes ___, No____
(If so, please describe your purchases)
Buys Units When? Price per unit
1
st
buy
2
nd
buy
3
rd
buy
6.4 Dou you produce milk or cheese for sale? Yes ____, No_____
(If so, what was your average productivity per animal over the last three years?)
Productivity in 1999 _____________ Price of milk in 1999 __________
Productivity in 1998 _____________ Price of milk in 1998 __________
Productivity in 1997 _____________ Price of milk in 1997 __________
6.5 What are the main expenses (in pesos) per animal monthly or yearly for the husbandry production of your farm?
Expense per animal Monthly or Yearly
Feeding costs _____________ ________
Healthcare costs _____________ ________
Labor _____________ ________
Other expenses _____________ ________
6.6 Have you lost animals because of drought? Yes ___, No____
How many? ___
When? ___________
146
7. Productive tools
7.1 What equipment of your own do you have for the agricultural production of your farm?
Type of equipment
Units Years Old Equipment description
Tractors
Animals for labor
Fumigation and application
Equipment of the harvest
Production Transportation
Other equipment
7.2 Do you have wells and pumping equipment? Yes ____ No _____(if so, what are their characteristics ?)
Well No. Depth
(meters)
Age Pump
No.
Pumping
capacity
Age
7.3 How many hours have you used your pump (or pumps) to drain water from your well during each season?
97/98 __________ (Hours) 98/99 ___________ (Hours)
7.4 Do you have any storing system for surface water? (docks, tanks, etc.)?
Yes____ No _____ Type _______________
Age ___________.Covered area (m2) ________
Depth (meters) ___________.Capacity (m3) ___________
147
7.5 Do you have mechanized irrigation in your farm? Yes ____ No____
If so, please fill in the chart below
Irrigation
Implements
Years Detailed description
7.6 Do you have a contract with an export company or are you a member of a Pisco association? Yes ____, No ____
If so, please describe the contract:
7.6.1 Name of the company ____________________
7.6.2 Length of contract to date ____________________
7.6.3 Type of payment (monthly, bi-monthly, etc) _____________
7.7 Does the company provide you with one of these services?
7.7.1 Technical support (describe) _____________________________________________
7.7.2 Credit assistance (describe) _______________________________________________
7.7.3 Commercialization (describe) _____________________________________________
7.8 Did you purchase harvest insurance in the 97/98 season?
Whom did you purchase it from? ___________________ How much did you pay for it? _________________
7.9 Did you hire purchase harvest insurance in the 98/99 season?
Whom did you purchase it from? __________________ How much did you pay for it? __________________
148
8 Participation in Lending Markets and Subsidies
8.1 Have you applied for any production loans in the last two seasons?
Yes ___ No____ (If not, skip to question 8.3)
8.2 Have you received any production loans in the last two seasons?
Yes ____No___ (If so, please fill in the following chart)
Source of the loan Loan amount in Pesos $ Interest rate Time span
97/98 98/99 97/98 98/99 97/98 98/99
Commercial Bank
INDAP
Other sources
8.3 Have you applied to be granted a bonus or subsidy bonus to improve the irrigation system in your farm?
Yes ____ No_____ (If not, go to question 8.5)
8.4 Have you obtained any bonus from the government? Yes ___ No ____ (If not, go to question 8.5)
Did you already use the bonus? Yes ____ No____
Have you received or do you plan to receive any bank loans to complement the bonus or subsidy bonus? Yes ___ No____
What type of bonus did you apply for? For entrepreneurs ____, for farmers ____ both _____
Account for the money received in bonus or subsidy bonus ______ (Pesos)
8.5 If you have not applied for a bonus from the government, why is it so?
____________________________________________________________
THANK YOU VERY MUCH.
149
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doc_575368127.pdf
Social influence occurs when one's emotions, opinions, or behaviors are affected by others.[1] Social influence takes many forms and can be seen in conformity, socialization, peer pressure, obedience, leadership, persuasion, sales, and marketing. In 1958, Harvard psychologist, Herbert Kelman identified three broad varieties of social influence
ABSTRACT
Title of Dissertation:
THE INFLUENCE OF HETEROGENEOUS RISK
PREFERENCES ON WATER MARKET
ACTIVITY: AN APPLICATION TO THE PALOMA
SYSTEM OF THE LIMARÍ WATER BASIN,
CHILE
Oscar E Cristi, Doctor of Philosophy, 2007
Dissertation directed by:
Professor Lars Olson,
Department of Agricultural and Resource Economics
This dissertation contributes to our knowledge about water markets by
analyzing the factors that explain market transactions of water rights when there is
also a spot market for water volumes. I hypothesize that risk heterogeneity among
farmers can explain those transactions. To test the aforementioned hypothesis I
model farmers’ decisions on investment in water rights each season under the
assumptions that they face output risk and that uncertainty is generated by future
water availability and price. The first order condition to this problem, which is
represented by the Euler Equation, indicates that the current period reservation value
of a water right depends on the current value of the amount of water accorded to
water rights in the spot market, the stochastic discount factor, and the expected future
prices of water rights. Using the relationship between the reservation value of a water
right and the stochastic discount factor I show analytically how heterogeneous
preferences are a sufficient condition for an active market for water rights. Then, I
test for heterogeneous preferences by allowing them to be a function of specific
characteristics of farmers. That requires the estimation of a system of equations that
includes a parametric specification of the Euler Equation and the first order
conditions for optimal input quantities. For that, I use an exponential utility function
and a production function of the Just-Pope type. I jointly estimate the parameters that
describe a farmer’s utility function along with production function parameters. The
empirical application uses farmer micro-level data from a two-round survey that I
conducted on a sample of Limarí Basin farmers. That Basin is located in the northern
part of Chile and is characterized by an active water market that has existed since
1981. Evidence rejects the hypothesis of homogeneity among farmers and suggests
that those better educated and more experienced Limarí Basin farmers are less risk-
averse. Results also show that water, labor and fertilizers have a positive impact on
mean output per hectare but their effect on yield variability implies that those inputs
are risk increasing.
THE INFLUENCE OF HETEROGENEOUS RISK PREFERENCES ON WATER
MARKET ACTIVITY: AN APPLICATION TO THE PALOMA SYSTEM OF THE
LIMARÍ WATER BASIN, CHILE.
By
Oscar E. Cristi
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2007
Advisory Committee:
Professor Lars Olson
Professor Maureen Cropper
Professor Bruce Gardner
Professor Richard Just
Professor Erik Lichtenberg
© Copyright by
Oscar E. Cristi
2007
ii
Dedication
To my mother and Gislaine, Yuviza and Francois
iii
Acknowledgements
This dissertation would not have seen the light of day without the help of a
number of individuals and institutions. I thank the chair of my committee, Professor
Lars Olson, for his permanent encouragement and the valuable insights and advice
that he provided. I thank Professor Richard Just for the detailed comments and
valuable guidelines on previous versions which led to substantial improvements. I
am also appreciative of the input from committee members, especially Professor Erik
Lichtenberg whose careful reading of the manuscript yielded many suggestions for
clarifications and improvements.
I express my deepest gratitude to all the members of AREC for their
permanent support and enormous kindness and professionalism.
I thank the Universidad de los Andes, Chile, and the World Bank for the
financial support. Their generosity allowed me to gather the data for this dissertation.
I have had many fruitful discussions about the dissertation process with
Professor Ivar Strand and my colleagues. Especially important have been
conversations with Fernando Díaz, William Foster, Jean Sepulveda, Eugenio
Bobenrieth and Ricardo Bórquez.
Finally, I would like to thank Abel Mejía, from the World Bank, who not only
suggested me to study the economic aspects of water markets, but also provided
permanent support for this dissertation.
iv
Table of Contents
Dedication..................................................................................................................... ii
Acknowledgements...................................................................................................... iii
Table of Contents......................................................................................................... iv
List of Tables ............................................................................................................... vi
List of Figures ............................................................................................................. vii
Chapter 1: Introduction................................................................................................. 1
Chapter 2: Literature Review........................................................................................ 7
2.1 Water management ....................................................................................... 7
2.1.1 Water markets in Chile and the Chilean Limarí Basin ....................... 13
2.2 Asset pricing models................................................................................... 21
2.3 Identifying risk preferences ........................................................................ 25
Chapter 3: The Limarí River Basin’s Water Market .................................................. 29
3.1 Mechanism for the Allocation of Waters in the Paloma System................ 31
3.2 Market Activity........................................................................................... 34
3.2.1 Spot Market......................................................................................... 35
3.2.2 Permanent Market ............................................................................... 38
3.3 Price Behavior in the Water Market ........................................................... 41
3.3.1 Spot Market......................................................................................... 41
3.3.2 Permanent Market ............................................................................... 42
3.4 Market Expectations and Access to Information........................................ 44
3.5 Summary..................................................................................................... 46
Chapter 4: Theoretical Model ..................................................................................... 48
4.1 A Simple Model.......................................................................................... 48
4.2 The Farmer’s Decision Rule under Water and Output Uncertainty ........... 52
4.3 Market Frictions and Consumption-Based Asset Pricing for Water Rights66
4.4 Summary..................................................................................................... 72
Chapter 5: The Econometric Estimation..................................................................... 74
5.1 Parametric specification of the system of equations................................... 74
5.2 Data............................................................................................................. 89
Chapter 6: Estimation and results ............................................................................... 98
Appendix: A Hausman test for endogeneity........................................................ 119
v
Chapter 7: Conclusions and Suggestions for Future Research ................................. 122
Annex: Survey instrument ........................................................................................ 127
References................................................................................................................. 149
vi
List of Tables
Table 3.1 Average water accorded to water rights between 1980 and
2000…………………………………………………..…..
33
Table 5.1: Basic statistics of the main variables…………………….. 93
Table 6.1: Descriptive statistics for variables used in the estimation
of the system of equations…………………………......….
104
Table 6.2: Estimates of the parameters of the equation system using
FIML………………………………..…………………….
110
Table 6.3: List of instruments to test for possible endogenous
regressors…………………………………………………
120
vii
List of Figures
Figure 3.1: Limarí Basin and the Paloma System……............................. 31
Figure 3.2: Water accorded to water rights between 1980 and 2000…… 33
Figure 3.3: Transfers of water in the spot market…………………......... 38
Figure 3.4: Number of water rights transferred by each association
during the period 1980-2000………………………………..
39
Figure 3.5: Real average prices of water rights in each WUA…….......... 42
Figure 3.6: Reservation value for the right to one cubic meter of water
in ACEC……………………………………………………..
43
Figure 3.7: Reservation value for the right to one cubic meter of water
in ACER……………………………………………………..
44
Figure 3.8: Transfers of water through the spot market……………….... 45
Figure 4.1: Water right transactions…………………………………….. 70
Figure 6.1: Actual value of standardized output income per hectare
against the imputed values for that variable………………...
113
Figure 6.2: Actual value of standardized investment in water rights
against the imputed values for that variable………………...
113
Figure 6.3: Plot of the residuals from the FIML regression based on
estimates in Table 6.2………………………………….........
118
1
Chapter 1: Introduction
The study of water resources is a passionate task as their increasing value and
unique characteristics are appealing for economists working on natural resources and
have important implications for peoples’ well-being.
Almost any paper or document that tackles water management will begin by
pointing out that worldwide, even countries and regions where water is abundant face
increasing water scarcity (Rosegrant et al., 1997, Tsur, 2004, Saleth and Dinar, 2004).
Demands for water from all sectors – agriculture, industry, households, and even
environmental conservation – combined with increasing difficulty in developing new
structural solutions to increase water supply, explain growing water scarcity.
Projections of water withdrawals by sectors show the dramatic increase in pressure on
water resources over the next three decades (Rosegrant et al., 1997).
Water has special characteristics because it is not a resource like others that
can be easily appropriated, traded and used without affecting others. Water is a
mobile resource that generates multiple levels of physical inter-dependence among
users, while farmers’ water use and transfer decisions create externalities. Water
diversion requires expensive devices for volumetric or flow measurements and costly
conveyance systems. Finally, water for irrigation is characterized by the randomness
of supply and the high cost of reducing variability through storage capacity.
Since the 1950s, the literature has debated the merits of water allocation
institutions, and now wise water management policies are among the most crucial
challenges of nearly all countries. If Julian Simon – whom I met for the first time in
1986 and who encouraged me to come to study at the University of Maryland – were
2
still alive, surely he would have seen this problem as a challenge for the people.
Likely he would have said that this new challenge could best be resolved by allowing
people to deal freely with this problem, as he extensively documented in his works on
decreasing scarcity of several natural resources. The public-good approach with
public water ownership and state involvement in its development and distribution
does not work well in the present context of water scarcity. Thus, as Saleth and Dinar
(2004) point out, “The current trend is toward an alternative system that can allow
private decision-making in water resource development, allocation, and management.
For the alternative system to function effectively and equitably, legal changes are
needed to facilitate private and transferable water-rights system that ensures full legal,
physical and tenure certainty of water rights.”
Worldwide, irrigation continues to be, by far, the largest sector of water
consumption accounting for nearly 70% of water withdrawals worldwide and over
90% in low-income developing countries. Another 23 percent of water is used in
industry and the remainder is consumed by households. These numbers indicate that
the increasing demand for water will need to be met from water savings in irrigated
agriculture by improving efficiency. Traditionally, economists have argued that
efficiency can be achieved through a water pricing system that reflects water scarcity.
This approach has some problems. First, water users have been able to use their
political power to prevent major increases in water prices, especially for irrigation.
Second, the water authority needs to define a mechanism to value water, which may
differ from water users’ willingness to pay for water.
3
Given those problems, water markets are receiving increasing attention from
policymakers in an attempt to improve the efficiency of water allocation. Although,
in spite this and the fact that numerous informal water markets have evolved around
the world, very few countries have implemented formal and legal free markets for
water. This is a result in part from the difficulty that researchers face in studying
rights that are attached to a mobile resource like water. Therefore, relevant policy
questions about water markets have not been addressed with empirical evidence.
Moreover, policymakers have tended to claim state “ownership” of water and have
been reluctant to develop tradable water rights by separating them from land rights,
which would allow the transfer of the former. They recognize the theoretical value of
water market institutions, but some think that the number of recorded instances where
water is reallocated by market transactions is far too limited due to physical
constraints and third-party effects associated with water exchanges. There are also
doubts about whether the market can reallocate water to its optimal social use.
Hence, still the main question about water markets, as Saliba (1987) pointed out is
“Do water markets ‘work’?”
Chile has formal water markets that have been operating for more than 24
years, and has become, along with Australia, an example of how institutional reforms
that treat water as an economic good improve water-use efficiency and water
allocation. Nonetheless, more empirical research based on extended data is required
to support those hypotheses (Bauer, 2004). Thus, Chilean water markets offer an
excellent opportunity for researchers who seek to answer relevant policy questions
about water markets.
4
The purpose of this dissertation is to contribute to the understanding of water
right transactions. I attempt to answer a basic question: What explains water rights
trading when farmers can exchange water in the spot market, which has lower
transaction costs? In answering this question, I emphasize the link between the spot
and water rights markets where differences in marginal returns to water among
farmers are solved in the spot market while the water rights market address
differences in the stochastic discount factor among farmers.
In this dissertation, I develop a theoretical model in which I characterize
optimal decision making by farmers faced with the decision of investment in water
rights. This model assumes that farmers face water and output uncertainty, and that
they may trade water in the market for water rights and/or the spot market. Because
investment decisions affect future levels of consumption and farmers face
uncertainty, the theoretical model for farmer decisions is modeled as a stochastic
dynamic problem. This results in a Consumption Capital Asset Price Model
(CCAPM) whose solution is described by an Euler Equation that ties asset returns
(water right returns in this case) to marginal rates of substitution of consumption at
different points in time. This model provides insights into how farmers determine
their reservation value for water rights when it is considered an asset, often the main
asset for farmers, and it allows me to emphasize the role of farmers’ heterogeneous
risk preferences on their reservation values and on marketing activity.
This theoretical analysis provides the foundation for a case study of water
transfers for irrigation in the Limarí Basin, an important agricultural region in the
northern part of Chile, which has one of the most active Chilean irrigation water
5
markets. With farmer-level data obtained from a survey among farmers for two
different agricultural seasons, the case study allows me to estimate jointly the
parameters that describe a farmer’s utility function and production function. The
estimation is based on the Euler Equation and the first order conditions for input
quantities. Using observed economic behavior I test for heterogeneous preferences
among farmers. The use of an asset price model to jointly determine farmers’
preferences and production technology in developing countries is a contribution to the
literature on agricultural finance.
The present analysis of a Chilean water market and its empirical application
provide insights into how water markets work in a developing country. It is hoped
that the results of this dissertation will supplement our knowledge of the outcomes
and experiences of developing countries with active, but undocumented, water
markets, and help move the debate beyond principles to empirical results on the
operation of water markets. This analysis will serve countries that are contemplating
adopting market-based reallocation systems where policymakers wish to inform
themselves of other experiences in water markets in different cultures and geographic
regions before they make a decision.
This dissertation proceeds in 7 sections. In Chapter 2, I present a brief survey
of the literature on water markets. I then proceed to provide a detailed description of
the operation of the water market in the Limarí Basin. In Chapter 4, I develop a
theoretical model for optimal decision making by farmers who must decide on
investment in water rights and input quantities in every season. Data availability and
the econometric model are discussed in Chapter 5. In Chapter 6, I describe the
6
estimation procedures and I report estimation results. Finally, Chapter 7 concludes
with a summary of the main results and suggestions for future research.
7
Chapter 2: Literature Review
In this chapter I review the relevant portions of the extensive literature on
water management and water markets. I also review the part of the literature on asset
pricing models and the estimation of heterogeneous risk preferences that is related
with the analysis and methodology of this dissertation.
2.1 Water management
There is a broad consensus in the literature on water management that
increasing water scarcity requires a shift from supply-oriented approaches focused on
technical and hydrological solutions towards allocation-oriented approaches centered
on economic and institutional solutions that provide the right incentives for water
savings. These efforts focus primarily on agricultural irrigation, the main use of
water all around the world
1
(Saleth, 2004). This is the starting point for a wide
literature on alternative institutional arrangements to promote efficient use (Bruns and
Meinzen-Dick, 2000, Saleth and Dinar, 2004)
Since at least the 1950s, there has been a debate in the literature over socially
appropriate water allocation institutions (for an excellent summary of that debate, see
Lynne and Saarinen, 1993). Traditionally, economists and policymakers have argued
that what is needed is a centralized water pricing system
2
that reflects the opportunity
cost of water. Centralized water pricing systems have two main problems. First,
water users have been able to use their political power to prevent major increases in
1
Irrigation is the largest sector of water consumption; accounting for nearly 70% of
total water withdrawals worldwide and over 90% in low income developing countries.
2
In such a system the water authority determines the value of water in different uses
and fixes a price for the use of that resource.
8
water prices, especially for irrigation (Easter et al., 1998). Second, under a
centralized system of water pricing, the water authority needs to define a mechanism
to value water, which may differ from water users’ willingness to pay for water.
Non-market techniques to value water rights include the farm-budget residual
valuation for water
3
as in Hearne and Easter (1995 and 1997). Another approach is to
estimate the value of marginal productivity of water using a crop-water production
function. Marginal values of water in municipal uses or in instream and recreational
uses are usually derived by estimating consumer willingness-to-pay through
contingent valuation and travel cost methods. Another procedure that has been used
to evaluate willingness to pay for water is the least cost alternative technique as in
Hearne and Easter (1995 and 1997)
4
. Person and Michelsen (1994) offer a good
review of different methods for estimating water values and they summarize the
willingness to pay estimates for different water uses in studies until 1994.
The current trend of water allocation institutions is toward an alternative
system that allows private decision-making in water resource development,
allocation, and management (Saleth and Dinar, 2004). Thus, market-type allocation
institutions such as water markets and water banks, that recognize water as an
economic good rather than a social good, are receiving increasing policy attention in
attempts to improve the efficiency of water allocation (Rosengrant and Binswanger,
1994, Vermillion, 2000, Brookshire and Ganderton, 2004). In that framework, the
3
In the residual method, subtracting all non water and land input cost from the total
revenue yields a residual value, which can be viewed as the maximum price that the operator
could pay for land and water and still break even. The researcher then allocates the residual
value between the two components, land and water.
4
In this case, the technique is used to compare the present value of the cost of buying
water rights against the cost of building a new storage capacity to increase water availability
for municipal uses.
9
market provides the mechanism by which water is valued. This new paradigm has
been promoted by the United Nations at the Rio Convention on Environment and
Development in 1992 and by the World Bank (1993).
The literature on the design of water markets within the variety of legal
settings that exist around the world has focused on the distinction between formal and
informal markets. In the formal market a variety of transactions take place, such as
the rental of water rights, water volume sales for a specific time period, and water
entitlement transferences, whereas in the informal market only short term transactions
are observed (Bjornlund, 2004).
Formal water markets have been implemented in the western United States in
Colorado (early 1960’s) and California (since 1982), and in Chile (since 1981),
Australia (since 1983), South Africa (since 1998), New Zealand (since 1991) and
Mexico (since 1994). Peru, Bolivia, Argentina, Nicaragua are among the countries
that are discussing policy reforms oriented towards water markets (Bauer, 2004).
Saleth (2004) and Bruns and Meinzen-Dick (2000), advocate that for formal
water markets to function effectively and equitably, legal changes are needed to
ensure full legal, physical, and tenure certainty of water rights separated from rights
to land. The costs associated with these institutional changes necessary to move to
market mechanisms explain in part the reduced number of countries that have formal
markets (Coward, 2000). Those costs have been addressed by McCann and Easter
(2004), but as Saleth and Dinar (2004) point out, “A study of the full transaction cost
associated with the change to an alternative water allocation mechanism has not been
attempted to our knowledge.”
10
With regard to informal markets, those markets have been widely
implemented in a number of countries such as India (Saleth 1998), Pakistan
(Meinzen-Dick, 1998) and Jordan (Shatanawi and Orabi, 1994).
Most critics of water markets argue that they do not work at all or that
transactions are too few due to market failures, and that they are not compatible with
integrated water resource management because they do not jointly solve critical
economic, environmental and social issues (Bauer, 1995 and 2004, Crase et al., 2000
and 2003, Gleeson, 2003). Supporters emphasize the benefits of water markets and
how these markets are actively working in various parts of the world (Rosengrant and
Binswanger ,1994; Holden and Tobani ,1995; Briscoe, 1996)
The emergence of water markets as allocation institutions has led economic
analysis to focus on what type of market is likely to appear, how water prices may be
formed, and whether water markets improve efficiency of use by reallocating water to
its highest use value.
Up to now, the description of how water markets function has focused
primarily on developed regions including the Western States of the USA (Michelsen
and Young, 1993, Israel and Lund, 1995, Susan M Burke et al., 2004) and, more
recently, Australia (Bjornlund and McKay, 2002, Bjornlund, 2004, Crase et al.,
2004). Attention is now shifting toward the developing regions of Africa, Asia, and
Latin America. Examples of empirical research on Latin American countries are the
works done for Chile by Rios and Quiroz (1995) and Bauer (1995 and 2004).
Rosengrant and Binswanger (1994) describe markets in tradable water rights in Chile
and Mexico. A major contribution to the literature on the outcomes of water
11
negotiation is the book edited by Bruns and Meinzen-Dick (2000), which documents
cases primarily from South Asia and Indonesia. These cases show how negotiation is
frequently used by water users, and the successful outcomes that have resulted from
the process.
Understanding the factors that determine water right prices and their variation
has become important for establishing whether water markets reallocate water to its
most efficient use. Initial studies explain price formation as a result of a bargaining
process between farmers for which the value of marginal product of water differs.
Several studies use hedonic price function to estimate the value of water rights and
the factors that influence prices of water rights and its fluctuations, such as water
right characteristics, institutional constraints, physical transferability of water,
bargaining power of sellers and buyers, and speculative behavior over water right
prices (Colby et al., 1993, Person and Michelsen, 1994, Bjornlund and Mckay, 1998).
Numerous authors have constructed the theoretical arguments that some type of
market based trading mechanism would greatly increase the efficient use of water (for
a good summary of the state of art on water markets see Brookshire and Ganderton,
2004). The basic argument is well established: water is a natural resource with
varying value in different uses and with clearly defined social and political
constraints. In terms of applied research, qualitative analyses as well as increasingly
sophisticated empirical studies have been published to verify whether water markets
allocate water to its highest valued use. To make this determination several studies
simulate market performance (Saleth et al., 1991, Dinar and Latey, 1991, Tisdell et
al., 2004, Dinar et al., 1998, Murphy et al., 2000). Also, an increasing number of
12
studies provide empirical analysis for assessing water right market efficiency with
data on existing water markets, mainly in some states in the USA (Brown et al., 1982,
Saliva, 1987, Crouter, 1987, Michelsen, 1994, Rosegrant and Binswanger, 1994,
Brookshire at al., 2004) and Australia (Crase et al, 2000, Bjornlund ,2004). For the
case of developing countries such as Chile, Rios and Quiroz (1995) and Bauer (1995
and 2004) provide qualitative analysis of water market performance whereas Hearne
and Easter (1995 and 1997) provide quantitative analysis of water market efficiency.
A good number of these studies analyze whether water allocated through the market
moves from its lower to its highest value by empirically identifying who are the
buyers and the sellers. This is the case of Nieuwoudt and Armitage (2004), Bjornlund
(2004) and Crase et al. (2004) for developed countries as South Africa and Australia,
and Hadjigeorgalis (2000) and Zegarra (2002) for the water market in the Limarí
Valley, Chile. In general, studies that analyze water market efficiency conclude that
the major benefits of the formal market are associated with a reallocation of water to
1) more productive soils, 2) more efficient water users, 3) higher-value uses, and 4)
new developments and the consolidation of water into larger more viable units.
The work by Bjornlund (2004) is quite interesting because he measures and
compares temporary trade with permanent trade in the Goulburn System and Murray
System, Australia. Bjornlund (2004) finds that the temporary market has by far the
highest amount of traded water, and that the practice of using both markets has been
widely adopted to shift an irrigator’s risk position and to manage increased supply
uncertainty. He also indicates that trade in water rights surged after farmers became
familiar with water trading and aware of the potential benefits. This took around 7
13
years in the area he studied. Finally, he presents evidence that shows how liquidity
constraints cause farmers to participate as buyers in the temporary market because
they cannot afford to buy water rights. Crase et al. (2004) found similar results for the
water markets in the Murray Darling Basin of Australia. In New South Wales
permanent and temporary transactions took 10 years to be significant, with temporary
transactions always much more important. In Victoria’s water market trade in water
rights surged after 7 or 8 years. These results of Bjornlund and Crase et al. on the
relative size of the temporary market with respect to the permanent market and on the
time that the permanent market takes to become established are very similar to the
ones that I obtain for the Limarí Basin, and which are reported in Chapter 3 of this
dissertation.
Nevertheless, there still exists a real need for more applied work on water
markets. Brookshire and Ganderton (2004) point out that “it is necessary to
understand beyond theoretical considerations how well these alternative institutions
perform from an empirical standpoint and what are some of the institutional design
issues that remain” and that “It is also needed to move beyond the simple description
of markets to identify the forces operating within those markets.”
2.1.1 Water markets in Chile and the Chilean Limarí Basin
Chile, together with Australia, has become one of the world’s leading
examples of how institutional reforms that treat water as an economic good improve
water use efficiency and water allocation. Yet more empirical research based on
extended data needs to be done to support those hypotheses. Bauer (2004) points out
14
that “much of the discussion about Chilean water markets has been long on
theoretical or ideological argument and short on reliable information”.
The first real empirical study of water markets in Chile was done by Hearne
and Easter (1995 and 1997), followed by Hadjigeorgalis (2000) and Zegarra (2002).
All these studies examine the water market in the Limarí Basin. Hearne and Easter
(1995 and 1997) also analyze water markets in the adjacent Elqui Basin, and Cristi et
al. (2003) also use that basin in their case study.
The Limarí River Basin, in north central Chile, has attracted national and
international attention through the 1990s and first half of the 2000s. The Limarí
River Basin is the one example that is widely agreed to have an active and successful
agricultural water market, including both temporary and permanent sales, and even
local real estate agents broker and facilitate water rights trading. Hearne and Easter
(1995 and 1997) estimate economic gains (net return to society) and financial gains
(individual net benefits) from trade in that basin. Economic gains correspond to the
difference between the value of water to the buyer after a purchase and the value of
water to seller before a sale minus transaction costs of the transfer. Financial gains
for a seller equal the sale price less both the value of water to the seller and the
seller’s transaction cost, whereas for a buyer it is the difference between the value of
water to the buyer and the sum of the buyer’s purchase price and transaction costs.
They conclude that market transfers of water rights produce substantial economic and
financial gains from trade in the Limarí Basin.
While their results are interesting they do not consider key features of water
used in irrigation that affect individual values for water rights, such as uncertainty
15
about water availability and future water prices, output uncertainty, and farmers’
attitudes towards risk. They also do not consider that in the Limarí Basin there are
two markets for water that coexist and interact: the market for water rights sales
(permanent transactions or permanent water rights sales) and the spot market for
water (short-term or temporary transactions). As a consequence, they fail to
recognize the relationship between the value of a water right and the price of water in
the spot market.
Hadjigeorgalis (2000) provides the first empirical analysis of actual trading
outcomes in both spot and water rights markets in the Limarí Basin. For the spot
market, she measures the number of transactions, volumes sold, and the number of
participants - separated by buyers and sellers - for the period 1994-1997. She also
analyzes price behavior for the 95/96 and 96/97 seasons, using field data for around
332 farmers. She concludes that there exists an active spot water market with prices
highly sensitive to water scarcity, and that the facility to transfer water volumes
between sectors has resulted in an equalization of water prices for water volumes
between geographically segmented sectors. With respect to market activity in the
market for water rights she identifies the existence of physical constraints that prevent
transferring rights between different reservoirs and institutional constraints that
prevent trading rights that are stored within the same reservoir, but that have different
legal locations (i.e. farmers with water rights in different Water User Associations)
5
.
These constraints produce segmentation into local market sectors below the dams and
this segmentation allows for water right price differences between local markets. She
5
Cortés, M (1997) offers a lucid explanation of the legal constraints to water right
trades in the Paloma System.
16
also presents the first formal, theoretical analyses of the impact of risk and
uncertainty on water market trading and water decisions on the amount of water to be
used in the production process
6
. In her theoretical approach she allows for output
price and spot market price uncertainty, water endowment uncertainty and capital
production risk for farmers that produce perennial crops. She presents formal
expressions for reservation spot market prices and reservation values for water rights.
The former are a function of the net value of the marginal product of water in
irrigation, irrigation efficiency, risk aversion and uncertainty cost associated with
selling and buying water volumes. Reservation values for water rights – which she
derives by emphasizing that water rights are an asset – are a function of the sum over
time of the discounted per period net values of the marginal product of water in
irrigation (benefits from water use in irrigation less the cost of holding water rights),
irrigation efficiency, risk aversion, and uncertainty cost associated with stochastic
water supplies. For perennial crop producers, reservation value is also a function of
the risk of future loss of their stock of perennial crops from a water supply shortfall.
In the empirical application she analyzes market participation and the probability that
a farmer participates in either the spot market or the water right market, and whether
the farmer will buy or sell water volumes and/or water rights. Among the
explanatory variables, she includes risk aversion proxied by farmer’s wealth. She
shows that trades occur from farms with low irrigation efficiency to farms with high
irrigation efficiency and that transaction costs in the spot market as well as in the
6
Howitt (1998) provides a first theoretical analysis of the impact of risk and
uncertainty on water markets and water decisions in a case study for the existing water
market in California.
17
market for water rights are minimal. The main limitation of Hadjigeorgalis’s work is
that, although in the theoretical model she clearly addresses the effect of risk and risk
preferences in the farmers’ reservation value for a water right, in the empirical
application that relationship vanishes and is replaced by a set of prior assumptions
regarding what type of farmers are more or less risk averse. Thus she cannot clearly
show how risk affects farmers’ water trading decisions. An empirical test of the
relationship among farmers’ characteristics and risk aversion would have helped her
to explain what she called unexpected results. One of these unexpected results is that
perennial crop producers are not exclusively buyers of water rights but appear
consistently on both sides of the market. An empirical estimation of heterogeneous
risk preferences may show how differences in risk preferences explain differences in
reservation values for water rights among perennial crop producers. If such
differences exist then it would explain why water right trades occur as more risk-
averse perennial crop producers would buy water rights from those perennial crop
producers with lower risk aversion.
Zegarra (2002) focuses his research on the operation of the spot market in the
Limarí Valley in the face of an extremely negative shock: the severe drought of
96/97. He models farmers’ decisions about the amount of water to be used in
production and the amount of water to be sold in the spot market. Farmers reach
equilibrium when the water’s marginal of value product equals the spot market price
for water. Thus, farmers decide not to grow crops in those seasons in which the water
return for selling water in the spot market is greater than their expected income from
production. Farmers are risk neutral and with production functions characterized by a
18
minimum water requirement constraint, which results in a non-convexity of the
production technology. With production non-convexity one of the main assumptions
for Pareto efficient allocation through market transactions is broken. Heterogeneity
among farmers is given by their crop mix and this heterogeneity makes spot markets
work. By simulating expected income in different scenarios he tests the hypotheses
that the increasing presence of permanent crops creates demand rigidities that reduce
the effectiveness of spot water markets. He finds that as crops become more
concentrated in permanent crops the spot market water prices exhibit a higher average
value and a greater dispersion. He also analyzes a farmer’s participation in the spot
market, i.e. if a farmer trades in the spot market and if so whether he is a buyer a
seller or both. The main results from Zegarra are that the spot market for water
solves differences in the marginal return of water among farmers, promoting the
allocation of water from low value annual crops to high value permanent crops. He
finds that in the context of severe drought, the water market starts to be less effective
in allocating the resource, with greater water price dispersion. Unlike Hadjigeorgalis
(2000), he concludes that water rights are heterogeneous with statistically significant
differences in both the mean water per share and the standard deviation. He suggests
that there are low transaction costs in the spot market. The main limitation of
Zegarra’s work is that it does not take into account farmers’ risk aversion. This
omission weakens one of his main results: that the spot market price at which the
supply of water volumes starts to be greater than zero is $30 pesos. If farmers are risk
averse his model overestimates that value because a risk-averse farmer will be willing
19
to sell his seasonal amount of water – and obtain a sure income – at a price lesser than
his expected marginal return of water use in irrigation.
From the above literature review, it is possible to infer some guidelines for
economic water research. 1) There is a need to understand how well water markets
perform and the forces operating within those markets from an empirical standpoint,
especially in non developed countries. 2) There is a need to empirically estimate the
impact of risk and uncertainty on water market trading and water decisions on the
amount of water to be used in production. In the process it is advisable to infer
reservation values for water rights from a model that recognizes that water rights are
one of the farmer’s main assets. 3) When a market for water rights and a spot market
for water coexist there is a need to account for the link between the two markets in
order to understand how spot prices affect water right prices over time. It also needs
to be emphasized that the spot market resolves differences in the marginal return of
water while the market for water rights resolves differences in farmers’ reservation
values for a water right. The latter are due to farmers’ differences on risk of future
loss on their stock of perennial crops from a water supply shortfall as well as
heterogeneous risk preferences. 4) Finally, there is a need to move from institutional
constraints that explain price differences in the market for water rights across sectors,
to factors that explain differences in reservation values between farmers within the
same Water Users Association.
The present dissertation contributes to the literature on water markets by
providing new insights on several issues. It measures market activity and provides
the first estimation of the size of the temporary water trades (spot market) in relation
20
to the permanent markets (water right trades) in the most active Chilean water market.
The analysis shows that the volume of water traded on the spot market is several
times greater than on the permanent market. Contrary to what other researchers such
as Hadjigeorgalis (2000) believe, it illustrates how the spot market is active not only
during drought years but also in years with average water availability. The theoretical
model that I develop infers reservation values for a water right from an asset pricing
model that assumes heterogeneous risk preferences among farmers, incomplete asset
markets and uncertainty about output, future water availability and future water
prices. It also incorporates the interaction between the spot market and the market for
water rights where the spot market mainly resolves differences in the marginal return
to water and the market for water rights mainly resolves differences in the stochastic
discount factor among farmers. The model explains differences in reservation values
between farmers within the same Water Users Association as a function of farmers’
risk preferences. The empirical application estimates an asset price model for water
rights and input demands that allows testing for heterogeneous risk preferences
among farmers. In addition, the effect of water on the mean and variance of yields is
estimated using detailed farm data on output and input quantities for each crop. This
is an improvement from previous studies, such as Hearne and Easter (1995 and 1997)
and Hadjigeorgalis (2000), that rely on standard crop budgets to proxy the marginal
revenue of water use in irrigation. The role of risk differences due to different types
of crops or distance from the reservoirs is not included in this dissertation and it
represents an important future extension of this work.
21
2.2 Asset pricing models
In this dissertation I model water right reservation values using a capital asset
pricing model (CAPM). The empirical use of a CAPM requires choosing between the
conventional consumption-based capital asset-pricing model (CCAPM) and a
production-based capital asset-pricing model (PCAPM). Next I briefly review some
of the literature related to these approaches and some of the literature related to the
different issues embedded in the use of an asset pricing model to value water rights.
The CCAPM ties asset returns to marginal rates of substitution for
consumption at different points in time and so must use a utility function defined on
consumption over time. Alternatively, the PCAPM emphasizes the linkages between
asset returns and investment and production variables. In it, production is used
instead of consumption and so the production function is modeled instead of the
utility function. A production model is proposed by Cochrane (1991 and 1996),
where asset returns are tied to marginal rates of transformation (the rate at which the
firm can transform goods from date t to date t+1, i.e. the rate of return on
investment). Then he empirically tests the relationship between stock and investment
returns in which the investment/capital ratio is a key variable. Arroyo (1996)
explains asset returns as a function of capital productivity and the adjustment cost of
capital proxied by the investment/capital ratio. Those who propose the use of
PCAPM usually mention the mounting evidence against standard consumption-based
models of asset returns. The empirical evidence indicates that returns on equity seem
to be too high to be consistent with observed consumption behavior unless investors
are extremely risk averse: a risk aversion often too large to be credible (Arroyo, 1996,
22
Campbell et al., 1997). Cochrane (1996) and Campbell et al. (1977) point out that the
poor empirical performance of CCAPM in explaining asset returns may be due,
among other reasons, to measurement error in aggregate consumption and/or because
growth of aggregate consumption is very smooth. Another source of criticism of the
CCAPM arises from transactions costs, borrowing constraints and other market
frictions that may invalidate the condition that discounted expected marginal utilities
should be equilibrated across time, which is the heart of the consumption-based
capital asset pricing model. In spite of the potential advantages of PCAPM over
CCAPM, as I explain in Chapter 4, Section 4.2., I have chosen a consumption-based
model because it emphasizes the role that preferences over consumption have in the
determination of the reservation value for water rights, and as Moschini and
Hennessy (2001) point out “…one should keep in mind that farmers ultimately likely
care about their consumption, itself the result of an intertemporal decision”. In that
same line, Cochrane (2005, Chapter 9.1: 157) points out that “…good economists are
unhappy about a utility function that has wealth in it. Few of us are like Disney’s
Uncle Scrooge, who got pure enjoyment out of a daily swim in the coins in his vault.
Wealth is only valuable because it gives us access to more consumption. Utility
functions should always be written over consumption. One of the few real rules in
economics to keep our theories from being vacuous is that ad “hoc utility functions”
over other objects like wealth should eventually be defended as arising from a more
fundamental desire for consumption or leisure.”
In this dissertation the CCAPM is derived from farmers’ optimal decisions
about investment in water rights in each season. The optimality conditions of the
23
model are described by Euler Equations. Empirically, the Euler Equations are
estimated together with the first order conditions for input quantities, using farm-level
data.
A number of requirements for high quality empirical production research in
agriculture, or what Just (2000) calls guiding principles, are addressed by the way in
which farmers’ decisions are modeled in this dissertation. First, it deals with the need
to focus on long run considerations of investment and cost adjustments, and the need
to consider the role of serial correlation of farm income and the intertemporal
dependence of farmers’ marginal utilities. I model intertemporal decisions on
investment which emphasizes the long-run nature of farmers’ decisions. Although I
model farmers’ decisions assuming non-serial correlation of farmer’s consumption
and a time-additively separable utility function over consumption, the model
indirectly links consumption and utilities over different periods. This link arises
because in this model the optimal consumption path depends on the stock of water
rights which is related both to present and past investment in water rights (Bossaerts,
2002). Second, I identify risk preferences using an asset pricing model for water
rights, which arises from farmers’ investment decisions that reflect the greatest
consequences of risk on farmers’ decisions (Just, 2000, Just and Pope, 2003).
Usually, the problem is that data on asset choices are very limited, thus the data on
water right choices gathered for this dissertation provides an important piece of
information to be able to build an asset pricing model for these rights and from there
to analyze the effect of risk and risk preferences. Third, I use data at the individual
farmer level and I incorporate farmers’ heterogeneity, which helps to improve the
24
empirical quality of the CCAPM (Campbell et al., 1997, Heaton and Lucas, 1996,
Constantinides and Duffie, 1996). Studies that estimate Euler Equations or more
general first-order conditions with data at individual level include, among others,
Zeldes ( 1989), Langemeier and Patrick (1993) and Phimister (1995), all of them in
the context of testing for liquidity constraints in the permanent income/life cycle
model for consumption. Blundell et al. (1994) estimate an Euler Equation using
micro data in order to estimate the parameters of household preferences that
determine the allocation of goods within the period and over the life cycle. Most
models of asset pricing assume homogeneous preferences among individuals or,
equivalently, the existence of a representative agent. Allowing farmers to differ in
their utility functions is a contribution to the empirical literature on asset pricing. The
assumption of heterogeneous preferences is also a sufficient condition for the
occurrence of asset trading among individuals. Niehaus, (2001) considers a simple
economy, where only a riskless bond, shares of a stock and an option written on the
stock are available in the financial market, and shows that differences in investors’
preferences have an impact on asset prices and the amount of trading in the market.
He finds that the amount of trading and the price of the option grow with increasing
divergence in risk aversion, and the agents with a higher degree of risk aversion sell
shares and options and buy the riskless bond. The agents with a lower degree of risk
aversion take the opposite position: they buy shares and options and sell bonds. This
is the same approach that I use in my dissertation, where each farmer has one asset –
a water right – and the following options: to sell the water right and buy a riskless
asset (or just put the money in bank at a riskless interest rate), to take more risk by
25
selling the water right and buying water in the spot market for use in farming
activities, or to keep the water right and use it in a risky farming activity. These are
the relevant contributions of my dissertation to the literature on the analysis of
farmers’ behavior along time, subject to limitations associated with the extent of the
data which is at the farmer level for two agricultural seasons.
2.3 Identifying risk preferences
The inclusion of heterogeneous risk preferences in the CCAPM requires
attention to the literature on identifying risk preferences for agricultural producers. A
comprehensive review of the large literature on this issue exceeds the scope of this
dissertation. Thus I limit discussion to the main issues identified in the review by
Moschini and Hennessy (2001). I then review some of the studies that specify risk
aversion as a function of socioeconomic characteristics such as age, education level
and family size.
Moschini and Hennessy (2001) show how early empirical studies of
agricultural decision making under risk elicited risk preferences from choices
between hypothetical lotteries. Later, using an econometric approach, studies
imputed a measure of risk aversion from the divergence between actual farmers’
production decisions and optimal decisions under risk neutrality. Due to the
limitations of inferring risk from observed production decisions and because
hypothetical payout surveys can give unstable results, Binswanger (1980) made real
payments to peasants farmers in India to elicit risk preferences. Antle (1987)
described the optimality conditions of expected utility maximizing choices in terms of
a given individual’s absolute risk aversion and downside risk aversion, and as an
26
econometric procedure he used the generalized method of moments (GMM). Later
on, Antle (1989) developed a method to estimate risk preference structures separately
from the production technology. Myers (1989) assumed constant relative risk
aversion (CRRA) and joint lognormality of the distributions of output prices and
producer consumption, and developed a reduced-form rational expectations approach
to test for the aggregate level of relative risk aversion for US producers who store
crops. Exploiting technical attributes of CRRA and of constant partial relative risk
aversion (CPRRA), Pope (1988) developed implications for optimal choices by
individuals expressing such preferences. Several studies have followed Pope’s
approach or variations of it. Another characteristic of research that attempts to
determine farmers’ risk preference structures is that most of them are based on
aggregate data (Just, 2000). Exceptions to this are the already mentioned
Binswanger’s lottery experiment and the Bar-Shira, Just and Zilberman (1997) study.
For this dissertation it is relevant to make a brief review of the literature that
identifies risk preferences by assuming that farmer’s risk aversion is a function of
socioeconomic characteristics such as age, education level and family size, among
others. This is necessary because in my dissertation farmers’ risk preference
heterogeneity is tested by estimating risk aversion for each farmer as a function of his
socioeconomic characteristics. Moscardi and Janvry (1977) analyze the relationship
between risk aversion and a number of socioeconomic variables that characterize
Mexican peasant households, their access to income-generating opportunities, and
their relation to public institutions. Binswanger (1980) analyzes the effect of wealth,
education level, more progressive farmers, and off-farm salaries on farmer’s risk
27
aversion. Zeldes (1989) allows a household’s utility function to be influenced
linearly by tastes that may differ across families and shift across time. Tastes differ
due to observable (for the econometrician) and unobservable factors. The observable
factors, which vary across families and time, are family size, age and age squared.
This linear specification for family utility function is included in the Euler Equation
which is estimated with data at the family-level. Blundell et al. (1994), who also
estimate an Euler Equation, allow the parameters that describe individual preferences
over consumption to be a linear function of variables such as the number and age of
children and labor market status: whether the head of the house and/or the wife are in
paid employments, and the level of consumption itself. Dubois (2001) also
parameterizes agent preferences by specifying a linear function for the absolute risk
aversion coefficient as a function of observable individual characteristics (age,
household size, number of children, etc.).
At the end of this section it is worthwhile to mention that the approach
followed in this dissertation, where water rights are assets and farmers’ optimal
decisions about water rights are treated as an investment problem that affect present
and future income and consumption, is closely related to the so called literature on
Agriculture Finance. Barry and Robinson (2001) offer a good review of the main
issues in Agriculture Finance. One of those issues is intertemporal farm-level
analysis in the context of life cycle planning and performance models of farm
business, where production and consumption are linked. Intertemporal analysis is
expressed as the maximization of the utility of multiperiod consumption, constrained
by the present value of wealth and the available investment alternatives, including
28
both productive investments and lending and borrowing in a perfect or imperfect
financial market. A second issue is the effect of farmers’ risk attitudes on their
portfolio decisions. In this dissertation those two elements: intertemporal farm-level
analysis where production and consumption are linked and the effect of farmers’ risk
attitudes on their decisions are carefully considered.
Up to now, research on Agriculture Finance has focused on real estate as the
dominant asset for farmers. But now, due to the increasing interest on establishing
transferable water rights not married to land rights, research on Agriculture Finance
should also consider water rights as a primary asset in dry areas. Due to the special
characteristics of water resources, this new challenge offers a significant opportunity
for future research. This dissertation is an effort to contribute toward this goal.
29
Chapter 3: The Limarí River Basin’s Water Market
The coexistence of a market for water rights and a spot market for water
volumes is analyzed for four water user associations in the Limarí Basin in Chile’s
IVth Region. The existence of a legal framework that permits the transfer of water
rights independent of land rights has contributed to the development of a very active
water market with a variety of exchange mechanisms over the last 20 years. The
Limarí River basin is a semi-arid zone with approximately 65,000 hectares of
irrigated land used mainly in traditional crops such as maize, beans or potatoes,
horticultural production (artichokes, peppers and tomatoes), grains, grasses and other
valuable perennial crops such as avocados, export grapes and grapes used for pisco
7
.
The farmer base is diverse and consists of orchard owners, medium-sized farms
established by past land reform programs, and a few large multinational fruit
exporters. Each irrigation district possesses distinct climatic characteristics that favor
certain types of crops. The hydrologic system of the Limarí basin is characterized as
being primarily niveous, that is to say that it is fed from the snow-covered Andes
Mountains. The basin has an average annual precipitation of 140 ml. One essential
characteristic of this basin is the existence of three interconnected dams: Cogotí,
Recoleta, and the Paloma Dams. Together, these dams form the subbasin called the
Paloma System, which has a storage capacity of one billion cubic meters and
possesses a flexible physical system for the distribution of water based on floodgates
and a network of siphons and canals that allow interconnection to different irrigation
districts within this subbasin. The current Paloma System has six Water User
7
Grapes used in making local liquor.
30
Associations (WUAs), four of which are analyzed in this study: i) Junta de Vigilancia
del Río Limarí and its tributaries (JVRL); ii) Asociación de Canalistas del Canal
Camarico (ACCC); iii) Asociación de Canalistas del Embalse Recoleta (ACER); and
iv) Asociación de Canalistas del Embalse Cogotí (ACEC).
The data used in this section comes from a variety of sources. The series of
prices
8
and transferred water rights for the period 1981-1992 were reported by
Zegarra (2002) and, for the period 1992-2000, by Cristi et al. (2002) and Vicuña
(2000). These authors obtained this information through Conservador de Bienes
Raíces of Ovalle and the records of the WUAs. The series of prices and volumes of
water exchanged in the spot market were constructed using the records of the WUAs,
information obtained from the Direccion de Riego, and a farmer survey. This survey
was applied to a sample of farmers in the Limarí Basin on three occasions, and
information was collected for each of the five growing seasons between 1995 and
2000
9
. The surveyed sample was designed by Zegarra (2002)
10
who conducted the
first round survey. I conducted the second and third rounds (a detailed description of
the data is included in Chapter 5).
8
Unless otherwise noted, all prices are expressed in 1990 Chilean pesos. The
average peso-dollar exchange rate in 1990 was $304.9 pesos to the American dollar.
9
This survey was applied to 195 farmers of the region and contains production data
and figures on land and water use, among other information.
10
He did not develop a list of farmers to interview based on a random sample due to
the expense of finding each sampled individual; instead, he simulated random sampling for
farmers who were present at their farm when he conducted the survey. He began at some
point inside the irrigated area (stratum), interviewing farmers using a systematic round
skipping for close neighbors. This results in a sample, which is geographically representative
for each irrigation organization. The main limitation of this sampling procedure is that
farmers who were not present at the moment of the survey had zero probability of being
selected. The procedure also excludes farmers who, at the moment of the survey, had
abandoned production.
31
The following diagram illustrates the Limarí Basin and the Paloma System.
Figure 3.1: Limarí Basin and the Paloma System
3.1 Mechanism for the Allocation of Waters in the Paloma System
The allocation process determines the amount of water to be received by each
user. Conceptually, allocation is a distinct task from that of distribution. The latter is
defined as delivering water in accordance with allocations (Bruns and Meinzen,
2000). Legally, the Paloma system divides the Limarí basin into two districts: the
irrigation district that is located above the dams, and that below them, known as the
Paloma System. The Paloma System is the subject of this study. In this system,
water rights are defined in terms of cubic meters of stored water, and water is
COGOTÍ
DAM
Huatulame River
Grande River
Hurtado River
Limarí River
Grande River
Hurtado River
RECOLETA
DAM
LA PALOMA
DAM
OVALLE
Cogotí
Canal
Recoleta
main canal
OCEAN PACIFIC
Cogotí
distribution
canal
Camarico
Canal
Recoleta
distribution
Canal
Feeder canal
for Recoleta
Cogotí River
DAM
Artificial
canal
Natural
channel
32
distributed simultaneously to the farmers’ plots directly from the dams through its
associated canal network except during severe droughts when a rotating system of
distribution (also known as shifts) is implemented.
In the Paloma System, the responsibility of water allocation lies with the
Junta de Vigilancia del Sistema Paloma, which reports to the Dirección de Riego
(Irrigation Administration). Every year, the board adds up the amount of stored water
in the system and establishes a quantity of water for each irrigated area (and therefore
for each WUA) based on holder’s historical shares. The total volume of water to be
allocated in the system depends on existing levels. When the volume stored in the
system is less than 500 million m
3
, no more than half the stored water volume may be
assigned. When the system contains more than 500 million m
3
but less than 1 billion
m
3
, the maximum global assignment is 320 million m
3
. Lastly, when the volume
exceeds 1 billion m
3
, free use is granted to all WUAs.
Once water has been allocated among the WUAs, they assign the water
volume to their members. To do this, every season each WUA determines the
amount of water accorded to water rights. Then, by multiplying this amount by the
number of water rights owned by a farmer, it determines each farmer’s endowment of
water (expressed in cubic meters). During most seasons, the amount of water
accorded to water rights is determined by dividing the total allocated water to the
WUA by the total water rights that exist within it. Nevertheless, the expectation of
water availability for the next season can motivate the adoption of different criteria.
This distribution of water by the WUAs is at a farm or user level, and the WUA is
33
charged with the task of billing its associates for water use and administrative
expenses.
Table 3.1 and Figure 3.2 illustrate the variation in water accorded to water
rights for the associations under study between 1980 and 2000.
Table 3.1: Average water accorded to water rights between 1980 and 2000
WUA
AVERAGE
(m
3
)
STANDARD
DEVIATION
(m3)
ACCC 4,800 941
ACEC 5,193 1,182
ACER 3,955 1,168
JVRL 6,450 1,172
Source: Water User Associations
Figure 3.2: Water accorded to water rights between 1980 and 2000
0
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
8
0
-
8
1
8
2
-
8
3
8
4
-
8
5
8
6
-
8
7
8
8
-
8
9
9
0
-
9
1
9
2
-
9
3
9
4
-
9
5
9
6
-
9
7
9
8
-
9
9
Season
m3
Camarico Limarí Cogoti Recoleta
Source: Water user associations
11
11
In those seasons of free endowment, the highest amount of water accorded to water
rights for the period was recorded as that for those seasons.
34
3.2 Market Activity
The Limarí Basin beneath the dams is characterized as having an active water
market in which a spot market and a permanent transaction market coexist. In the
former, volumes of water are exchanged. In the latter, the purchase and sale of water
rights take place over time. After the annual allocation of water among water users
associations based on historical criteria and the posterior distribution of it among
farmers according to their number of water rights, the spot market reallocates this
resource to equalize differences in the marginal return to water among farmers
12
.
This process is facilitated by the existence of a significant number of farmers with
non-perennial crops that can, with relative ease, modify their water consumption by
varying the percentage of land used or the type of crops according to their water use
intensity. Moreover, water volume transactions are relatively easy due to the
existence of significant storage capacity. The use of flexible floodgates and the
proper operation of the water users associations also facilitate short term transactions.
Together, these factors support the existence of an active spot market. This
dissertation examines reasons for simultaneous water rights market and spot market
activity.
12
Differences in the marginal return to water are measured by Hearne and Easter
(1995). They estimated an average value for the marginal return to water rights in the case of
table grapes of US$ 856.7 and US$ 865.7 for the case of grapes used in pisco. This compares
to US$ 33.5 for potatoes and of US$317.5 for peppers, two of the main non-perennial crops
of the basin.
35
3.2.1 Spot Market
Because the allocation of water in each season among the WUAs in the
Paloma System does not necessarily coincide with the water demands of each WUA,
significant volumes of water are transferred among associations. As such, the
volumes of transfers received
13
by the four associations studied reached 24,189,000
m
3
in the 99/00 growing season representing 7% of the total amount of water assigned
to these associations. This number reached 16% for the 95/96 growing season, a year
of drought. As I analyze each association individually, I observe that, except for
ACER, which owns a volume of entries very similar to its actual outflows or debits,
the rest are net claimants of water (entries greater than debits) or net sellers (debits
greater than entries). ACCC and ACEC are examples of net claimants, and JVRL
stands out as an example of a net supplier. Thus, in the 99/00 growing season, the
irrigators of the ACCC and the ACEC obtained additional water rights from another
district equivalent to 34% and 7% of their water consumption, respectively. Such
figures reached 35% and 12%, respectively, in the 95/96 season. On the other hand,
during the 99/00 season the JVRL transferred 20% of its water assignment to other
associations while in the 95/96 season, transfers from that same association reached
40%.
That some associations are net water claimants or suppliers is explained by
differences in the marginal return to water and its availability. The fact that the
13
I can distinguish two types of water transfers: inter-association transfers and intra-
association transfers. The first group includes those transfers among WUAs, which can be
both entrances (when water is received from another association) and exits (when water is
transferred from one association to another). The second group includes those transfers
among irrigators within the same association.
36
ACCC and the ACEC are net claimants is explained by the presence of highly
profitable perennial crops such as grapes, and a significant share of irrigators who
belong to the ACEC develop crops on the land located above the Paloma Dam and
below the Cogotí Dam with excellent weather conditions and, thus, higher marginal
returns to water. In the case of JVRL, the practice of river water recovery by farmers
that increases their supply of water beyond that accorded to their water rights, partly
explains its condition as a net water seller.
Inside each association, irrigators produce different crops resulting in varying
marginal returns. This generates a significant level of internal water transfers
between those farmers with lower marginal returns to water and those with higher
marginal returns. During the 99/00 growing season, the total volume of internal
transfers for the WUAs under study reached 26,633,000 m
3
, which represents 8% of
the total amount of water assigned to these associations during this season. During
the period 1995-2000, the highest level of internal transfers occurred inside the
ACCC and ACEC associations accounting for 24% and 13%, respectively, of the total
amount of water assigned to each.
As the focus of this research is the water market, it is necessary to be precise
about the definition of water transfer in the spot market. First, it is necessary to
distinguish between the volumes of water transferred in the spot market from those
outside the spot market. In the former, transactions may occur between farmers in
different WUAs associations or among farmers within an association. Transfers that
the associations register as entries and debits have two different sources. The first
consists of transfers of water volumes made among different farmers and is counted
37
as water transferred in the spot market. The second are the so-called intra-farmer
water transfers. This is the case for farmers with two plots of land that are irrigated
with water from different WUAs, and the transfer reallocates water from one plot to
another. This is registered as a water transfer by the WUAs, but it occurs outside the
market, and is not a spot market activity. Transfers of water volumes among farmers
in different WUAs and transfers of water volumes among farmers within a same
WUA are considered spot market activities.
Unfortunately, it is not possible from the analysis of the WUAs records to
determine the exact magnitude of transfers between one farmer who has different
plots of land. Nevertheless, based on their experience, the executive directors of the
WUAs estimate that approximately 75% of the total transfers that imply an entry or
debit from one water association to another are made among different farmers, and
should be considered spot market transfers. For those transfers between an individual
WUA, records are clear on the type of transfer that occurs. Figure 3.3 illustrates spot
market activity obtained by adding together the total amount of transfers among
farmers within each WUA with 75% of the transfers that occur between different
WUAs
14
.
14
The evolution of spot market activity shows that the volume of water exchanged
reaches a low in the 97/98 growing season, an occurrence that is explained by a drought
during the first half of the season, and by heavy rains during winter that led the WUAs to
declare free river. This is to say, during that same season, there was a lack of supply at the
beginning of the season and a surplus later decreasing market activity for the season.
38
Figure 3.3: Transfers of water in the spot market
0
10
20
30
40
50
60
95-96 96-97 97-98 98-99 99-00
Season
Millions of m
3
0%
5%
10%
15%
20%
25%
Total transfers
Percentage of total transfers over the total amount of water
Source: Direccion de Riego, the Paloma System Administration.
3.2.2 Permanent Market
The spot market for water coexists with a separate market for water rights
(henceforth referred to as the permanent market). Beginning from an initial
assignment of water rights by the government, these rights are reallocated to farmers
through the market. When analyzing the activity of this market, I observe that the
total percentage of reallocated water rights, independent of land transactions during
the period 1980-2000, varies from 20% to 50% by WUA. Since the approval of the
water law in 1981, and until 2000, more than 27% of the total water rights have been
transferred through the permanent transactions market
15
.
15
Adding together the water rights of the different associations is imprecise because
the volumetric specification of these water rights is different for each association.
39
The analysis of the behavior of the permanent transactions market over time
shows how the activity in this market has grown. Figure 3.4 shows how the earliest
transactions had lower volumes of trade than later periods reflecting a market that
matured in its first decade.
Figure 3.4: Number of water rights transferred by each association during the
period 1980-2000.
0
200
400
600
800
1
9
8
0
-
8
1
1
9
8
2
-
8
3
1
9
8
4
-
8
5
1
9
8
6
-
8
7
1
9
8
8
-
8
9
1
9
9
0
-
9
1
1
9
9
2
-
9
3
1
9
9
4
-
9
5
1
9
9
6
-
9
7
1
9
9
8
-
9
9
Season
Water Rights
ACCC
ACEC
ACER
JVRL
Comparing the size of the spot market with the permanent market is difficult
because the former trades in volumes of water during a specific growing season while
a water right transaction implies the transfer of variable volumes of water over time.
In spite of these restrictions, I have compared the relative size of both markets
through the following steps: i) expressing the trade in rights in volumetric terms for
each season using the average amount of water accorded to water rights (see Table
3.1); ii) assuming that the sale (purchase) of a water right in any season is equivalent
to the sale (purchase) of the amount of water accorded to that right in the
40
spot market in all following seasons
16
; iii) assuming that only a portion of the
transfers among WUAs are conducted through the spot market due to the accounting
conventions discussed above in Section 3.2.1. These conventions require that we
consider the following two scenarios. In one, only transfers among different farmers
within a WUA are counted as spot market transactions (scenario 1); and in the other
transfers among different farmers within a WUA plus 75% of the transfers among
WUAs are counted as spot market transactions (scenario 2)
17
. A possible drawback
of this method is that it could lead to an over estimation of the relative size of the
permanent market because rights may be transferred several times in the period under
study resulting in the volume of water associated with those rights being double
counted. However, the history of water right transfers independent of land
transactions shows that few farmers who buy water rights sell them later. The
exceptions are a few water right holders who do not own land, but buy water rights to
sell them when prices rise. The detailed operation of the market indicates that
overestimation of the permanent market is not significant. As a result, I conclude that
16
One possible scenario is as follows:
Association XX 95-96 96-97 97-98 98-99 99-00
Exchanged water
rights (in cubic
meters)
100 200 100 50 100
Total cubic
meters
100 300 400 450 550
17
As I already mentioned, 75% of the total transfers that involve a change of water
association are assumed to be transfers among different farmers; therefore, they are
considered to spot market transactions. This assumption is based on interviews of the
executive directors of the four water user associations under study. Moreover, the content in
this chapter was presented in the in the city of Ovalle (Limarí Valley) in a seminar with more
than 20 attendees that included the executive directors and the water engineers of the WUAs.
During the presentation, I emphasized the assumption that few farmers that buy water rights
sell them later independent of land sales, and that approximately 75% of the transfers among
WUAs are done through the spot market. However, many non-attendees disagreed with those
assumptions.
41
for the period 1995-2000, the water transferred in the spot market was 3.8 times
greater than transfers in the permanent market under scenario 1 and nearly 7 times
that in scenario 2.
3.3 Price Behavior in the Water Market
To verify if a market behaves in an efficient manner, I test to determine
whether prices reflect the relative scarcity of water. In the following sections, I will
show that water prices in the spot market reflect its relative seasonal scarcity. In
regards to the permanent market, the systematic rise in the real prices of water rights
reflects a sustained expansion of the demand for water over time.
3.3.1 Spot Market
Evidence exists that the behavior of prices in the spot market reflects the
relative scarcity of the resource. Analysis of the period 1995-2000 illustrates that the
maximum real price per cubic meter of water is reached in all the WUAs in the 96/97
growing season, which suffered a drought. When comparing the real prices between
the 96/97 season and a normal season such as 99/00, prices during the drought season
are between 3 and 12 times greater than those of a normal season, varying by
association. The capacity of the spot market to reflect the relative scarcity of water
can also be observed through the correlation coefficient of water availability and the
average real price per cubic meter, which is –0.921 for the five growing seasons
between 1995 and 2000
18
.
18
This coefficient correlates the average real price of water in the spot market with
the sum of the water accorded to water rights in the WUAs between 1995 and 2000. The
price in the spot market is estimated as a weighted average of the prices as reported by
surveyed farmers that exchanged water. Overall, 123 observations of temporal transfers in
five growing seasons were used.
42
3.3.2 Permanent Market
The increasing scarcity of water due to a sustained expansion of the demand
over time has produced a systematic rise in the real prices of water rights. Thus,
during the period 1986-2000, the price of water rights experienced real increases
ranging from 41% to 240% depending upon the association. Figure 3.5 illustrates the
evolution of these prices over time.
Figure 3.5: Real average prices of water rights in each WUA
0
100.000
200.000
300.000
400.000
500.000
600.000
700.000
800.000
900.000
8
0
-
8
1
8
2
-
8
3
8
4
-
8
5
8
6
-
8
7
8
8
-
8
9
9
0
-
9
1
9
2
-
9
3
9
4
-
9
5
9
6
-
9
7
9
8
-
9
9
Season
$
1
9
9
0
Camarico
Cogot i
Recolet a
Junt a
Heterogeneity in the prices of water rights among associations tends to
diminish over time with exception of ACEC. The higher water rights prices in some
associations are largely explained by differences in the alternative mechanisms to
purchase water rights. More specifically, I can distinguish between two markets: The
first is restricted to the area below the Cogotí Dam and above the Paloma Dam, which
is characterized by limited suppliers of water rights due to the relatively small number
of farmers. In addition, several farmers in this area grow export grapes and have a
43
higher marginal return on water than farmers located in other zones in the basin. The
second one includes the area below the Paloma Dam, and is characterized by the
existence of a relatively large number of irrigators. Increasing the number of farmers
increases the dispersion of reservation values improving the probability of water right
sales. Figures 3.6 and 3.7 show the reservation values of a permanent right to one
cubic meter of water as reported by farmers from the ACEC and ACER WUAs. As
expected, water right reservation values are higher in the former market.
Figure 3.6: Reservation value for the right to one cubic meter of water in ACEC
0
1
2
3
4
5
D
o
e
s
n
'
t
k
n
o
w
0
-
1
2
5
1
2
5
-
1
7
5
1
7
5
-
2
2
5
2
2
5
-
2
7
5
2
7
5
-
3
2
5
3
2
5
-
4
5
0
4
5
0
-
5
5
0
5
5
0
-
6
5
0
g
r
e
a
t
e
r
t
h
a
n
$ (pesos)
F
r
e
q
u
e
n
c
y
0%
20%
40%
60%
80%
100%
120%
A
c
c
u
m
u
l
a
t
e
d
P
e
r
c
e
n
t
a
g
e
44
Figure 3.7: Reservation value for the right to one cubic meter of water in ACER
0
2
4
6
8
10
12
D
o
e
s
n
'
t
k
n
o
w
1
0
0
-
1
2
5
1
2
5
-
1
5
0
1
5
0
-
1
7
5
1
7
5
-
2
0
0
2
0
0
-
2
2
5
2
2
5
-
2
7
5
2
7
5
-
3
0
0
3
0
0
-
5
2
5
$ (pesos)
F
r
e
q
u
e
n
c
y
0%
20%
40%
60%
80%
100%
120%
A
c
c
u
m
u
l
a
t
e
d
P
e
r
c
e
n
t
a
g
e
3.4 Market Expectations and Access to Information
By observing water transactions in the spot market in each season, it is
possible to see that in most seasons the market does not become active until after the
passage of winter when the volume of rainfall is known and the need for water
increases due to the hot, dry summer weather. This seasonality provides some
support to the statement that the farmers have homogeneous information with respect
to the future; therefore, no additional benefits are associated with advance purchases
and sales of water volumes. The exception to this behavior is observed in the 97/98
growing season, which was preceded by a severe drought. The experience of three
previous years of drought and the relatively high prices in the spot market at the
beginning of that season could have motivated advance water purchases by farmers
45
who expected water prices to increase and advance water sales from sellers who
expected water prices to decrease. This seasonal behavior of the spot water market
and the advance sales of water in the 97/98 growing season are illustrated in Figure
3.8 for ACEC.
Figure 3.8: Transfers of water through the spot market
0
1.000.000
2.000.000
3.000.000
4.000.000
5.000.000
6.000.000
M
a
y
J
u
n
J
u
l
A
g
o
S
e
p
O
c
t
N
o
v
D
i
c
E
n
e
F
e
b
M
a
r
A
b
r
Month
m3:
96-97, 98-99, 99-00
seasons
0
50.000
100.000
150.000
200.000
250.000
300.000
m3: 97-98 season
96-97 season 98-99 season 99-00 season 97-98 season
With respect to rain expectations and the expected amount of water accorded
to water rights, I found agreement among surveyed farmers. When asked during the
third round survey about rain expectations for the following season, 70% responded
that it would be a normal year. In addition, farmers were asked in that same survey to
estimate the amount of water accorded to water rights in the following year as a share
of what they currently receive. Almost 50% of surveyed farmers estimated that the
water accorded to their water rights in the following period would be between 90%
and 120% of their current level. The majority of the remaining farmers (47%) admit
that they do not have an estimate of the future level of water accorded to water rights.
46
With regard to information access, I observe that the majority of the farmers are well
informed about water availability in the dams. When asked about the level of the
dams, 85% of them correctly answered that the dam was half-full or full
19
.
3.5 Summary
The current water law in Chile has been flexible enough to allow the Paloma
System, in the Limarí Basin, to develop not only an active market for water rights, but
also an active market for volumes of water, i.e. a spot market. The volume of water
exchanged in this market indicates that this mechanism of water allocation is highly
important. In the permanent market, more than 27% of existing water rights were
exchanged independently of land transfers between 1981 and 2000.
A flexible water market such as the spot market in the Paloma System allows
for the reallocation of water to those areas in which the water acquires its greatest
value. This market -contrary to what other researches believe- is active not only in
drought years but also in years with average water availability. Thus, in the 99/00
season characterized by the average water availability, approximately 14% of the
water allocated to the four main WUAs of the basin was reallocated through
exchange in the spot market. During the severe drought of the 95/96 season that
figure reached a value of 21%.
The Water User Associations are a primary factor in determining the correct
functioning of the market. Thanks to good organization and efficient management of
their records of water allocation, it has been possible to develop an active spot
market. Together with the Water Use Associations, it is also clear that the existence
19
At the time of the third round survey the amount of stored water stood at 70% of its
total capacity.
47
of a safe supply source, such as the three dams that form the Paloma System, is a
necessary condition for the existence of a spot market.
The correlation between the growth in the relative scarcity of water over time
and the prices of water rights, as well the correlation between the scarcity of water per
season and the price of water in the spot market, indicates that water markets operate
correctly. At the same time, for the market for water rights there is some
heterogeneity in prices that tends to diminish over time with exception of ACEC.
This heterogeneity is also captured when farmers are asked about their reservation
price for a water right. As expected, water right reservation values are higher for the
farmers that belong to ACEC.
Finally, survey questions regarding access to relevant information and water
market expectations supports the hypothesis that farmers’ access to information is
homogeneous and farmers form similar expectations with regard to future water
prices and the availability of water supplies. From this, I assume in the coming
chapters that farmers have homogeneous expectations with respect to the main
stochastic variables in the water markets.
48
Chapter 4: Theoretical Model
In this chapter I pursue an explanation for water rights trading, which arises
whenever there are differences in the reservation value of the water right asset. I use
this reservation value as a proxy for private valuation, and hypothesize that difference
in reservation values arise from heterogeneity in farmers’ preferences. To establish
the importance of heterogeneity in farmer preferences, I first consider a simple model
where investors are risk neutral. Under this assumption, the price of a water right is
equal to the expected discount sum over time of the spot market values of the amount
of water accorded to that water right.
4.1 A Simple Model
Let us define R
t
as the return on a water right in period t. It is a function of
the change in the water right price between t+1 and t, where
t
? denotes the water
right price in period t, the spot market price per cubic meter, s
t
, and the total water
(in cubic meters) accorded to each water right in season t, denoted by v
t
. I assume
that ? , s and v are stochastic variables that are revealed during each growing season.
Thus, at t the values of
t
? , s
t
and v
t
are known. Hence, the return on a water right is
what can be earned by buying a water right in period t, selling the assigned volume of
water on the spot market in period t+1, and then re-selling the water right in period
t+1:
1 1 1
.
1
s v
t t t t
R
t
t
? ?
?
? +
+ + +
=
+
(4.1.1)
49
In order to simplify the analysis I can assume that the expected water right
return is equal to a constant, R, and:
[ ] E R R
t t l
=
+
, for l= 0,1,2,.. (4.1.2)
where E
t
stands for the expectation operator conditional on the information set
available at time t, which is assumed to be known by all farmers.
The assumption that the expected return remains constant is sometimes known
as the Martingale Model of stock prices. Although that assumption of constant stock
returns contradicts the empirical evidence of returns behavior over time, it is
analytically convenient for the goal of this section, which is to show the importance
of heterogeneity in farmer preferences in the reservation values of water rights.
Developing the model with time-varying expected returns will lead to the same
conclusion, but the analysis is cumbersome because the relationship between prices
and returns becomes nonlinear (Campbell, et al., 1997, Chapter 7.1).
By taking expectations in (4.1.1) over
1 t
?
+
and
1 1
s v
t t + +
, imposing (4.1.2),
and rearranging terms, I obtain an equation that links the current water right price
with the next period’s expected water right price and the water price in the spot
market:
1 1 1
1
s v
t t t
E
t t
R
?
?
+ (
+ + +
=
(
+
¸ ¸
(4.1.3)
Recursively iterating forward the future prices of water rights and using the
Law of Iterated Expectations, the following equation is obtained:
1
1
1
l
L
E s v
t t t l t l
R
l
?
(
| |
(
= ¿
+ + |
(
+
\ .
=
¸ ¸
+
1
1
L
E
t t L
R
?
(
| |
(
+ |
+ ( \ .
¸ ¸
(4.1.4)
50
Ruling out the possibility of a rational bubble in the market, i.e. the water
right price
t
? is not expected to grow forever at a rate R or faster, the second term on
the right-hand side of this equation shrinks to zero as the time horizon, L, increases.
As such, equation (4.1.4) may be written as:
1
1
1
l
E s v
t t t l t l
R
l
?
(
?
| |
(
= ¿
+ + |
(
+
\ .
=
¸ ¸
(4.1.5)
If all farmers have identical expectations about water right returns, R , water
prices in the spot market, s
t l +
, and water quantities accorded to each water
right, v
t l +
, then they would have identical private water right valuations (or
reservation values), and no trading would take place.
To explain why water rights are exchanged in the Limarí Valley water market,
some type of heterogeneity among farmers must be assumed. As I mentioned in
Chapter 3, farmers’ market expectations and access to information indicate that
farmers have homogeneous information regarding the future of water prices and
water quantities accorded to water rights and, thus, homogeneous expectations. The
analysis of rain expectations and the expected amount of water accorded to water
rights for the coming season revealed a high coincidence in answers among surveyed
farmers who were also well informed about water availability in the dams.
Consequently, I focus on other types of heterogeneity among farmers to explain
differences in reservation values for a water right. One source for farmer
heterogeneity arises from differences in the marginal productivity of water due to
differences in soil quality, equipment (for example tractor use), and irrigation
systems, among others. These differences in marginal productivity are resolved in the
51
spot market for water, where farmers with higher marginal productivity will buy
water from farmers with lower marginal productivity until the differences vanish in
each agricultural season. Hadjigeorgalis, (2000) provides a theoretical and empirical
proof for this statement as well as empirical evidence of a unique price of water in the
spot market for the whole area below dams in the Limarí Valley. Accordingly,
differences in marginal productivity are not sufficient to explain trading in the market
for water rights.
Another potential source of heterogeneity is a variety of idiosyncratic random
shocks that affect farmers’ income. Constantinides (1996) and Heaton and Lucas
(1996), among others, consider the effect of idiosyncratic random shocks on asset
prices. Farmers in the Limarí Valley face independent shocks to their incomes as
revealed by the second survey when some farmers reported important frost damage to
their crops in just one night. Pest infestations are another potential source of shock,
although the farmers under study use pesticides. If farmers face incomplete asset
markets these shocks affect consumption and hence the reservation values for a water
right if farmers are not risk neutral. In practice, that is the case because most farmers
do not insure against transitory idiosyncratic shocks. The spot market for water helps
farmers to smooth household consumption, but does not totally preclude them from
the need to modify consumption due to idiosyncratic shocks.
Finally, heterogeneity of farmers’ preferences is a sufficient condition for
differences in the reservation value of water rights among farmers. In the next
section, I develop a model of a farmer’s decision for optimal consumption and
52
investment in water rights and use it to illustrate the role of incomplete asset
markets
20
and farmers’ preferences upon reservation values for water rights
4.2 The Farmer’s Decision Rule under Water and Output
Uncertainty
The goal of this section is to analyze the role of farmers’ preferences in
determining reservation values for water rights. I consider the case of irrigated
agricultural land in a region with stochastic water availability. The farmer has
income that can either be consumed or invested in water rights. In this problem
uncertainty comes from different sources: output uncertainty in season t, future spot
market and water right prices, and future amounts of water accorded to water rights.
The particular model I develop for farmer consumption and investment in
water rights is based on the conventional consumption-based capital asset-pricing
model (CCAPM) where the asset under study is a water right. In this model, water
right returns are linked to marginal rates of substitution for consumption at different
points in time. Alternatively, a production-based asset-pricing model (PCAPM) may
be developed emphasizing the linkages between asset returns and investment and
production variables. Data availability is key when it comes to deciding between the
CCAPM and the PCAPM. The first requires consumption data and the second
investment and capital data. Unfortunately, my data is not ideally suited to either of
these. First, while it reports farm equipment, it differs broadly in regards to numbers,
quality and age, and any effort to value farmer capital is unreliable. Second, it does
not contain information on consumption. I do have reliable data on current net
20
Incomplete asset markets occurs when some insurance markets are absent so that
farmers can not insure against each state of nature.
53
income which, in a framework with incomplete asset markets, is correlated with
consumption
21
. For this reason I have chosen the CCAPM.
I assume that the farmer’s optimal decision rule regarding consumption and
investment in each season is the solution to the maximization of the expectation of his
intertemporal utility of consumption subject to a budget constraint. I consider the
case of a farmer for whom the intertemporal utility of consumption takes the form of
a time-additively separable function, such that the expected present value of his utility
is given by:
( )
1
T
t
E U C
t it
t
? ¿
=
(4.2.1)
with i=1…,n and t=1…,T. Here n is the total number of farmers, T is the total
number of seasons, C
it
denotes farmer i’s consumption in season t and ? is a
constant discount factor measuring the rate of time preferences. Given that 0< ? <1,
one unit of utility tomorrow is valued less than one unit of utility today. The utility
function ( ) U is assumed to be twice continuously differentiable, with
( ) dU C
t
U
t
dC
t
? = >0 and
( )
2
2
d U C
t
U
t
dC
t
?? = <0, i.e. the marginal utility of consumption is
21
Under the permanent-income hypothesis, proposed by Milton Friedman in 1957,
increases and decreases in income that farmers see as temporary have little effect on their
consumption spending. Nevertheless, that independence between consumption and current
income requires the capacity of farmers to counteract specific random shocks on their current
income. They are able to do that if they have access to a complete asset market or have assets
that deliver wealth when they face unexpected reductions on their current income. If none of
those conditions is fulfilled then consumption is highly correlated with current income.
54
positive, but it decreases with consumption. Expectations in (4.2.1) are taken
conditional on information available at t.
The assumption of a time-additively separable utility is standard in the
literature, but it is not without restriction. It implies that farmers’ current utility is not
affected by the timing of the resolution of uncertainty (Duffie et al., 1997). In other
words, it rules out preferences for early or for late resolution of uncertainty. A
second drawback of this utility function is that the elasticity of intertemporal
substitution is inversely linked with the coefficient of relative risk aversion (Epstein
and Zin, 1981 and Just, 2000). Finally, farmers’ current utility is assumed to be
unaffected by past consumption. This rules out habit formation in consumption
where the marginal utility of future consumption is increasing with the level of past
consumption. Just (2000), demonstrates that assuming additive separability of utility
may be inadequate for studying longer-term agricultural production problems. This is
so because the expected utility over a time horizon with an additive separability
function does not take into account possible correlations across time of the variable
over which utility is specified. In my setting, the role of habit formation is mitigated
for two reasons: i) survey field experience indicates that durable goods are only a
small share of farmers’ consumption bundles in the Limarí Valley; and ii) optimal
consumption depends on state variables that are related to the past history of
consumption and investment. This indirectly links utilities of different periods
(Bossaerts, 2002). A primary advantage of time-additive utility is that preferences are
recursive and this allows dynamic programming methods to be used to analyze
optimal decisions. Chavas (2004) points out that nonadditive preferences are more
55
difficult to specify and evaluate, and he concludes by stating that “The reader should
keep in mind that the time-additive model remains a popular framework for the
analysis of dynamic behavior”.
Before developing a model of farmers’ consumption and investment
decisions, two main features of the water market under analysis require discussion.
First, the aggregate demand curve for water volumes in the spot market is assumed to
be deterministic. Consequently, s
jt
which is the price of water in the spot market of
WUA j in season t, varies from one season to another as water supply changes. Thus,
I have:
( )
s s v
jt jt
= . (4.2.2)
Because the number of water rights is fixed in every WUA, changes in water
supply are explained only by changes in v
jt
.
The second important feature of this case study is that water supply depends
on the level of water in the dams, which is known at the beginning of the agricultural
season in April. At this time each WUA determines the amount of water accorded to
each water right, v
jt
. Hence, uncertainty about water availability and the price of
water in the spot market is eliminated at the beginning of each season although it
remains unresolved for future seasons. Nevertheless, the WUAs may change the
amount of water accorded to water rights after April. This can be due to high levels
of rain in the winter or to the extraordinary size of snow packs, the main source of
water for the dams formed in the surrounding mountains during the winter months of
May, June and July, which could generate a water flow that overwhelms the storage
56
capacity of the dams during the summer. In these cases, the WUAs declare “free
river” so each farmer can take freely whatever amount of water he requires
22
. The
analysis of the time series of v for ACEC over 46 seasons shows that this WUA
declared free river in only 8 seasons and there were no seasons for which it modified
the amount of v announced at the beginning of the season. The JVRL announced a
modification on v at the beginning of a season in 5 out of the 22 observed seasons,
with an average change of 17%, and declared free river in four seasons. For the other
WUAs, either this information is not available or they did not modify v except when
declaring free river
23
. For the purposes of my analysis I assume that the normal case
is when water accorded to water rights is known at the beginning of the season and
does not change. This means that v
jt
and
( )
s s v
jt jt =
are known by the farmers at
that time.
The farmer’s stochastic problem is to choose a sequence of consumption over
time that maximizes the farmer’s expected utility in equation (4.2.1) subject to: an
22
In theory, the amount of water accorded to water rights announced at the beginning
of the season may be reduced although this never occurred in the years under study. In
addition, the WUA of the Cogotí dam permitted the dam to run dry in seven seasons between
1954 and 2000.
23
Number of seasons in which the WUAs declared free river.
WUA Number of growing seasons in the
available series for v
jt
Number of years declared
free river
ACCC 20 7
ACEC 46 8
JVRL 22 4
ACER 23 5
57
initial stock of water rights
0
N , a budget constraint, and a water availability
constraint.
Due to the fact that most of the farmers in the Limarí Valley face incomplete
markets
24
, it is reasonable to assume that farmers’ decisions over consumption
depend on net revenue from current production and current income from water
transactions. Thus, the budget constraint for farmer i in WUA j is defined by:
( ) ( ) C , ,
1
N N s z P T f w X r X I
it jt it it jt it it it it it it it it it
? ? ? = ? + + ? +
?
(4.2.3)
The first two terms of the right hand side of the budgets constraint represent
net income from water transactions, where
jt
? denotes the price of a water right in
WUAj at period t and is the same for all farmers in a WUAj; N
it
is the number of
water rights held by farmer i during season t; ( )
1
N N
jt it it
? ?
?
is the net cost of
investment in water rights at time t; s
jt
denotes the spot market price per cubic meter
in WUAj in season t, and z
it
is the volume of water exchanged in the spot market by
farmer i in season t, where z
it
>0 implies sale of water volumes and z
it
<0 implies
the purchase of water volumes. The third and fourth terms in the budget constraint
represent net revenue from production, where T
it
is the total amount of cultivated
land; ( ) , , f X w
it it it
? denotes a per hectare production function of an aggregate
output; w
it
denotes the volume of water used as an input in production and X
it
is a
vector of inputs different from water (all inputs are expressed in terms of per hectare);
it
? represents the value of a vector of shocks that are partially determined by nature;
24
Farmers face incomplete markets because only few of them have crop insurance
and access to credit is limited.
58
P
it
is the price of the aggregate output faced by farmer i in season t; and r
it
is the
vector of input prices for X
it
. Output and input prices are assumed to be known.
Finally, I
it
represents farmer’s “other net income” sources including consumption or
production credit, land transactions, livestock exchanges and work off-farm.
The water availability constraint is:
N v z T w
it jt it it it
? +
25
. (4.2.4)
It limits the amount of water that a farmer can use for irrigation, T w
it it
, and the total
volume that he can exchange in the spot market, z
it
.
Without violating the rule that utility is function of consumption
26
, I can
substitute the budget and the water constraints inside the utility function and express
the farmer’s expected present value of utility as:
( )
( )
( )
( ) ( )
, ,
1
1
T
t
E U N N s N v T w P T f w X r X I
jt it it jt it jt it it it it it it it it it it
t
? ? ? ? ? + ? + ? + ¿
?
=
(4.2.5)
The farmer’s problem now involves choosing investment in water rights and
input quantities to maximize his expected discounted sum of utility in equation
(4.2.5), subject to an initial stock of water rights. Hence, the consumption decision
problem has been transformed to an investment and production decision problem.
25
This assumption is an oversimplification because farmers in some WUAs may
occasionally use more water than the volume they obtain for the season by asking for an
advance on water from the next season. Farmers also may save water from one season to the
next, but there is a penalty of 15 to 20% of the endowment due to projected evaporation
losses making this practice rare (Zegarra 2002).
26
“..good economists are unhappy about a utility function that has wealth in it. Few
of us are like Disney’s Uncle Scrooge, who got pure enjoyment out of a daily swim in the
coins in his vault. Wealth is only valuable because it gives us access to more consumption.
Utility functions should always be written over consumption” (Cochrane, 2005, pg. 157).
59
The overall problem of finding the optimal sequence of water rights
transactions can be solved by using a standard dynamic programming approach, i.e.
by finding the optimal amount of water rights,
it
N for t=1,… T, and input quantities
( , w X
it it
), that satisfy the recursive functional equation:
( )
( )
( )
[ ]
1
( , )
, , 1 , ,
( , )
1
N N s N v T w
jt it it jt it jt it it
E U
t
V N Max
P T f w X r X I it it w X N
it it it it it it it it it it it
E V N
t it it
?
µ
?
? µ
¦ ¹
( | |
? + ? +
?
¦ ¦
( |
¦ ¦
( |
=
? ´ ` ? + ?
\ . ¸ ¸
¦ ¦
¦ ¦ +
+
¹ )
(4.2.6)
where
it
µ is the vector
it
µ =
( )
, , , , , v s T P r
jt jt jt it it it
? ? . For notational simplicity, the
dependence of the value function on the term
it
µ will be suppressed in what follows
unless it is needed to avoid confusion. The value function, ( )
1
V N
it ?
, denotes the
value of the stock of water rights held from t-1 to t,
1
N
it ?
, after the optimal sequence
of
it
N , t=1,…T has been determined and is the indirect utility function of that same
stock of water rights. In the first term the expectation is over
it
? (its value is resolved
at harvest time) and the random variable I
it
. The values of , , s v
jt jt jt
? are known
at t but their future values are unknown. The values of T
it
and
1
T
it +
are assumed to
be known. Thus, the expectation in the second term is over
1 it
?
+
, as well
as , ,
1 1 1
s v
jt j jt
?
+ + +
, and
1
I
it +
. Output prices, P
it
, and input prices, r
it
, are
certain. The control variables are N
it
, w
it
and X
it
. The state variables are
1
N
it ?
and
it
µ . I assume that ( ) U and ( ) f are continuously differentiable, the value
60
function is differentiable in N
it
, and that the optimization problem has an interior
solution.
The first order condition with respect to N
t
(omitting the indices i and j for
convenience) is:
( ) [ ]
1
E U s v E V
t t t t t t t
? ? ? ? ( ? ? =
+
¸ ¸
(4.2.7)
where ( )
1
V V N N
t t t
? = ? ?
+
is the marginal value of a water right held at the
beginning of period t+1.
Using the envelope theorem, the right hand term in (4.2.7) can be expressed in
terms of the discount factor, marginal utility, water right prices and the spot market
value of water accorded to each water right. I begin with the value function for
period t+1 evaluated at the optimal level of each one of the variables that are included
in it:
( ) ( )
( )
[ ]
1 1 1 1 1 1
1
1
( ) , ,
1 1 1 1 1 1
( )
1 1
N N s N v T w
t t t t t it t
t
E U
t
V N P f w X r X
t t t t t t t
E V N
t t
?
?
?
¦ ¹ (
| | ? + ? +
+ + + + + +
+
¦ ¦ ( |
+ ¦ ¦
|
( = ? ?
´ `
+ + + + + +
\ . ¸ ¸
¦ ¦
+
¦ ¦
+ +
¹ )
(4.2.8)
Then I differentiate equation (4.2.8) with respect to N
t
, to obtain:
[ ]
1 1 1 1
V E U
t t t t
? ? ? =
+ + + +
. (4.2.9)
Next I substitute equation (4.2.9) into equation (4.2.7), and use the Law of
Iterated Expectations, to obtain the first order condition with respect to N
t
:
( ) [ ] s v E U
t t t t t
? ? ? = [ ]
1 1
E U
t t t
? ? ?
+ +
(4.2.10)
Equation (4.2.10) is the so-called Euler Equation. The left hand side term
indicates how much the farmer’s utility increases if the available income for
61
consumption rises due to a sale of a water right at the price of
t
? . The deduction of
s v
t t
from
t
? , to get the net income from selling a water right, is due to the fact that
whenever a farmer sells a water right he will not be able to sell the amount of water
accorded to that water right, v
t
, on the spot market at a value of s v
t t
. The right hand
side term is the farmer’s expected discounted utility of keeping the water right from
season t to t+1, which may also be interpreted as the farmer’s expected forgone
discounted utility if he sells a water right in season t. In other words, this term can be
understood as the marginal cost of selling a water right in period t. Thus, (4.2.10)
reflects the optimality condition that the marginal benefit of selling a water right (the
left hand side term) is equal to its marginal cost (the right hand side term). This
allows me to interpret equation (4.2.10) as giving the marginal value or willingness to
pay for a water right (Cochrane, 2005, Chapter 2.1). More specifically, I can rewrite
the Euler Equation as:
( ) ( )
( )
( ) ( ) ( ) ( )
1 1 1 1 1 1 1
1 1
, ,
1 1 1 1 1 1 1 1
, ,
1
N N s N v T w
t t t t t t t t
U
t t
P T f w X r X I
t t t t t t t t
s v E
t t t t
E U N N s N v T w PT f w X r X I
t t t t t t t t t t t t t t t t t t
?
?
?
? ?
? ?
( | | ? + ? +
+ + + + + + +
? ( |
+ +
|
? ? +
(
+ + + + + + + +
\ .
= +
(
? ? ? + ? + ? +
?
(
(
¸ ¸
(4.2.11)
where the term on the right hand side of (4.2.11) is the willingness to pay or the
reservation value for a water right.
Furthermore, the stochastic discount factor in this problem is defined by the
random variable
1
M
t +
, where:
62
( ) ( )
( )
( ) ( ) ( ) ( )
1 1 1 1 1 1 1
1
, ,
1 1 1 1 1 1 1 1
1
, ,
1
N N s N v T w
t t t t t t t t
U
t
P T f w X r X I
t t t t t t t t
M
t
E U N N s N v T w PT f w X r X I
t t t t t t t t t t t t t t t t t t
?
?
?
? ?
| | ? + ? +
+ + + + + + +
?
|
+
|
? ? +
+ + + + + + + +
\ .
=
+
? ? ? + ? + ? +
?
(4.2.12)
Thus, the Euler Equation becomes:
[ ]
1 1
s v E M
t t t t t t
? ? = +
+ +
(4.2.13)
Following the tradition of the literature on asset pricing, the Euler Equation
can be written as:
1
1
1
s v
t t t
E M
t t
t t
?
? ?
(
+
= +
( +
¸ ¸
. (4.2.14)
The right hand side term in (4.2.14) is the discounted expected value of the
marginal rate of return to holding a water right from time t to time t+1. This marginal
return is comprised of two terms: the first term is s v
t t t
? , and it represents the rate of
return to a water right when the water accorded to it is sold in the spot market; and the
second term is
1 t t
? ?
+
, which represents the rate of change in the water right price
from season t to season t+1. Finally, expression (4.2.14) indicates that the optimal
investment in a water right occurs when the discounted expected value of the
marginal return to holding an extra water right is equal to 1. This formula is the
foundation of the consumption-based capital asset pricing model (CCAPM).
The Euler Equation can also be written in terms of the value of water accorded
to each water right in the spot market,
1 1 t t
s v
+ +
, for t = 0,1,2…T. To do this I solve
equation (4.2.13) forward by repeatedly substituting out the future price of water
rights,
1 t
?
+
, for l = 1,2…T, and I obtain:
63
[ ]
1 1
E M
t t t t t t
v s ? ? = +
+ +
[ ]
1 1 1 1 2 2
E M
t t t t t t
v s ? ? = +
+ + + + + +
[ ]
2 2 2 2 3 3
E M
t t t t t t
v s ? ? = +
+ + + + + +
[ ] [ ]
2 2 1 1 1 1 1 2
E M E E M M
t t t t t t t t t t t t t
v v v s s s ? ( = + +
+ + + + + + + +
¸ ¸
which, by the Law of Iterated Expectations, becomes:
[ ]
2 2 2 1 1 1 1
E M v M M v
t t t t t t t t t t t
v s s s ? = + +
+ + + + + + +
(4.2.15)
This equation states that the reservation value of a water right in period t is
determined by adding together the present and expected future values of the amount
of water accorded to a water right in the spot market, discounted by the stochastic
discount factor. In order to explain the economics behind this statement, consider a
representative farmer who lives only two periods, t and t+1, and has only one water
right, which is accorded one m
3
of water. If that farmer decides to use his cubic
meter of water as an input in crop production, he does so because he obtains a greater
return than what is expected from selling that water in the spot market. In
equilibrium, he would then buy water in the spot market until the marginal benefit of
water used as an input equals the expected present benefit of selling water in the spot
market
27
.
27
This is the case for a costless spot market for water such as that of the Limarí
Valley. Up to now, nobody has measured the real transaction cost on the spot market.
Zegarra (2002) indicates costs should be significant because when he conducted his survey in
the Limarí Valley, farmers were facing a severe drought and complaining about the difficulty
in purchasing water in the spot market. However, that is not in itself a proof of high
transaction costs. What happened is that, due to the severe drought, prices were quite high at
that point, and several farmers were not willing to pay those prices. Indeed an active water
market existed at that time in Ovalle’s plaza with water being sold to the highest bidder.
64
In order to examine the role of risk preferences and incomplete asset markets
in the reservation value for water rights as well as in water rights trading, it is
convenient to write the Euler Equation in terms of the covariance between the
stochastic discount factor and the price of a water right. Using the definition of
covariance [ ] [ ] [ ] [ ] ,
1 1 1 1 1 1
E M Cov M E M E
t t t t t t t t t t
? ? ? = +
+ + + + + +
, I can write
(4.2.13) with the sub index i and j as:
[ ]
,
1 1 1 1
s v E M E Cov M
jt jt jt t it t jt t it jt
? ? ?
( (
= + +
+ + + +
¸ ¸ ¸ ¸
(4.2.16)
Equation (4.2.16) splits the farmer’s reservation value of a water right into the
expected discounted marginal rate of return to holding a water right from time t to
time t+1,
[ ]
1 1
s v E M E
jt jt t it t jt
?
(
+
+ +
¸ ¸
, and a risk premium, ,
1 1
Cov M
t it jt
?
(
+ +
¸ ¸
.
Thus, differences in reservation values for a water right, among farmers in a same
water association and under the assumption of farmers’ identical expectations on
1 jt
?
+
, require differences in the expected value of the random variables
1
M
it +
and/or in the covariance between
1
M
it +
and
1 jt
?
+
.
If farmers are risk neutral then the expected value of
1
M
it +
will be the same
for all farmers and risk premium is zero. Risk neutral farmers utility function over
consumption can be represented by: ( )
0 1
U C C
it it
? ? = + . Since ( )
1
U C
it
? ? = , the
stochastic discount factor becomes
( )
( )
1
1
U C
it
M
it
E U C
t it
? ?
+
=
+
? (
¸ ¸
=
[ ]
1
1
E
t
??
?
?
= (Chavas,
2004, Chapter 10). In this case the covariance term, [ ] ,
1 1
Cov M
t it it
?
+ +
, would
65
vanish and the optimality condition reduces to
1
s v E
jt jt jt t jt
? ?? = +
+
, which is the
same as derived from the simple model in equation (4.1.5).
If farmers face complete asset markets that provide full insurance, then a
farmer’s specific random shocks to current income will not affect his consumption.
In this case the stochastic discount factor
1
M
it +
depends on aggregate temporal
shocks. If it is also assumed that farmers have homogenous preferences, then they are
all characterized by the same, unique stochastic discount factor, i.e.
1
M
it +
=
1
M
t +
(Campbell, 1997, Chapter 8.1). In this case the expected value of the random
variables
1
M
it +
and the covariance between
1
M
it +
and
1 jt
?
+
will not differ among
farmers and water right reservation values will be the same for all farmers.
To summarize, under the assumption that farmers have identical expectations
regarding
1 jt
?
+
, differences in reservation values for a water right among farmers in
the same water association can only arise if there are incomplete markets or if farmers
have heterogeneous preferences that are not risk neutral. In that case there are
multiple stochastic discount factors that satisfy equation (4.2.16).
How risk aversion affects water rights transactions depends on the sign of the
risk premium, ,
1 1
Cov M
t it jt
?
(
+ +
¸ ¸
. The risk premium increases with risk aversion,
so if the covariance between
1
M
it +
and
1 jt
?
+
is positive then more risk averse
farmers will place a higher value on water rights than less risk averse farmers, and the
former will buy water rights from the latter. In the case under study, with incomplete
asset markets, water rights are used not only to obtain water but also to smooth
consumption. This is because water rights allow farmers to insure themselves against
66
bad shocks to production income and hence consumption. Thus, water right prices
covary negatively with consumption and because a farmer’s marginal utility of
consumption is decreasing in consumption, the ,
1 1
Cov M
t it jt
?
(
+ +
¸ ¸
is positive.
Finally, optimal variable input quantities are obtained from the following first
order conditions:
( )
( )
* , , 0 E U P T f w X s
t t it it w it it it jt
?
(
? ? =
¸ ¸
(for water) (4.2.17)
( ) ( )
* , , 0 E U P T f w X r
t t it it x it it it it
?
( ? ? =
¸ ¸
(for other inputs) (4.2.18)
The first order condition for water in (4.2.17) can be written as:
( )
( ) ( )
[ ]
, , ,
, ,
Cov U P T f w X
t t it it w it it it
E P T f w X s
t it it w it it it jt
E U
t t
?
?
?
( = ?
¸ ¸
?
. This provides
an intuitive interpretation of the first order condition for water: at the optimal input
use, the expected marginal value product,
( )
, , E P T f w X
t it it w it it it
? (
¸ ¸
, is equal to the
input price, plus the marginal risk premium,
( ) ( )
[ ]
, , , Cov U P T f w X
t t it it w it it it
E U
t t
? ?
?
?
. An
identical interpretation applies to the first order conditions for the other inputs
(Chavas, 2004, Chapter 8).
4.3 Market Frictions and Consumption-Based Asset Pricing for
Water Rights
Before concluding this chapter, I discuss market frictions that may affect the
consumption-based capital asset pricing model upon which the expressions for the
farmer’s private reservation value of a water right are based. At the heart of the
consumption-based capital asset pricing model is the condition that discounted
expected marginal utilities should be equilibrated across time, as implied by equation
67
(4.2.14). In the presence of market frictions this equality may become an inequality
such as
1
1
1
t
E M
t t
s v
t t t
?
?
(
+
?
( +
?
¸ ¸
or as
1
1
1
t
E M
t t
s v
t t t
?
?
(
+
?
( +
?
¸ ¸
. He and Modest (1995)
review some of the market frictions that may produce such inequalities including: i) a
non-short sales constraint, which prevents the short selling of some assets; ii)
solvency constraints, which restrict the wealth process at some future date from
falling below some predetermined level; iii) transaction costs that include bid-ask
spreads and commissions; and iv) borrowing constraints, which preclude investor’s
current consumption from exceeding their current wealth.
The two-round survey I conducted among farmers in the Limarí Valley shows
that they have low levels of education, and most own less than 6 hectares. These
figures suggest that sophisticated market activities, such as short sales or solvency
constraints, are not relevant for this study. Excess liquidity and an intermediary
institution are necessary for short selling to work properly and the development of
more sophisticated markets where speculators can take short positions. Such is not
the case for water markets in Chile where the size of the market greatly reduces the
possibility that a market maker has a counterpart. The difficulty in hedging against
production shocks and the lack of crop insurance creates uncertainty with respect to
future income. The spot market for water plays a role in smoothing income, but not
nearly enough to avoid distress land sales by insolvent farmers. This suggests that for
the case under study there are two constraints that may be important: transaction costs
and borrowing constraints.
A large literature on the impact of transaction costs on asset prices has
developed. Some authors emphasize the cost in terms of bid-ask spreads (Luttmer,
68
1996), the existence of illiquid assets as a consequence of transaction costs (Vayanos
and Jean-Luc Vila, 1999), or the presence of transaction costs that are endogenously
determined because investors adjust trading frequencies (Constantinides, 1986).
Heaton and Lucas (1996) interpret transaction costs not as a trading cost, but rather as
a wedge between the borrowing and lending rates due to monitoring and other costs
incurred in each period. Gollier (2001) emphasizes that risk reduces long-term
savings on assets if the trading cost of the assets is high. In spite of differences in the
definition of transaction costs among authors, they do agree that for short periods,
transaction costs are important, but over time, they become insignificant. Thus, a
one-period transaction cost will be less significant if traders that buy an asset hold it
for many periods.
To show that transaction costs do not have much effect in the present study, I
start with Gollier’s (2001) findings on illiquid assets. I also assume that the existence
of a costless spot market helps to smooth income variations and minimize the
liquidity problem for water rights. For other types of transaction costs, both theory
and empirical evidence support the hypothesis that transaction costs are not relevant
for this study. In general, water right buyers want to keep them for long periods
minimizing the impact of transaction costs on farmer decisions, and on the price of
water rights or the degree of market activity. As mentioned in the previous chapter,
Hearne and Easter (1995) have estimated that the transaction costs associated with the
purchase and sale of water rights in the Limarí Valley range from 5% to 2% of the
price for buyers and sellers, respectively. Although these costs do not appear to be
high, problems arise when trade implies a change in the source of water because the
69
Direccion General de Aguas, a central governmental office, must approve the change
of source, a process which may take several months because it must ensure that the
petitioners meet certain requirements and evaluate possible negative externalities to
other farmers. The length of this process explains why most of the water rights
transactions are among farmers who obtain their water from the same source.
Secondly, the size (measured by the number of water rights exchanged) of each water
right transaction
28
is relatively small with little dispersion indicating that transaction
costs are insignificant. The following figure presents the frequency of the number of
water rights trades in each one of the 778 transactions included in the dataset.
28
The database contains information for 778 transactions of water rights
(independent of land transactions), in the four WUAs under study representing trade of
11,910 water rights.
70
Figure 4.1: Water Rights Transactions
0
20
40
60
80
100
120
0
,
0
9 3 6 9
1
2
1
5
1
8
2
5
4
0
5
5
7
0
8
5
2
0
0
Number of Traded Water Rights in each Transaction
F
r
e
q
u
e
n
c
y
As the Figure 4.1 illustrates, most transactions involve small amounts of water
rights, indicative of low transaction costs. In fact, 50% of all transactions involve 6
or fewer water rights, and 80% involve less than 20 while the mean value is 15 and
the mode is equal to 2. In dollar terms, a water right in 1999 had a minimum average
price of US$ 927.40 (in ACER) and a maximum average price of US$ 2,766.4 (in
ACEC). If I use the mode (2 water rights per transaction) and the upper bound of the
transaction cost estimated by Hearne and Easter (1995), i.e. 5% of the transaction
price, I arrive at an estimate for transaction cost that ranges from a minimum of $46.4
to a maximum of $138.32 U.S. dollars.
Borrowing or liquidity constraints may explain outcomes in
which
1
1
1
t
E M
t t
s v
t t t
?
?
(
+
?
( +
?
¸ ¸
. This implies that [ ]
1 1
s v E M
t t t t t t
? ? + ?
+ +
, i.e. the
71
marginal expected benefit of buying a water right (the left hand side of the
inequality) is greater than the marginal cost of buying a water right (right hand side
term of the inequality). In such a case the farmer has an incentive to buy water rights
and reduce his present consumption. With decreasing marginal utility of
consumption ( ( ) U C
t
?? <0), that will reduce the stochastic discount factor,
1
M
t +
moving the inequality toward an equality. But if the farmer faces liquidity or credit
constraints then he may not be able to buy all the water rights he desires and the
inequality may not vanish. Accordingly to the Chilean Water Code, water rights are
divorced from land. As a consequence water rights are accepted as collateral by most
banking institutions. This allows farmers to obtain loans to buy water rights. Thus,
credit constraints are mainly for consumption. The effect of this latter constraint is
included in my model through the budget constraint for consumption.
I also consider the impact of liquidity constraints on farmer savings decisions
and risk aversion. Deaton (1991), Carroll (1997), and Gollier (2001), among others,
show that the risk of facing a liquidity constraint in the future introduces an important
motive to save, since savings act as a buffer stock that reduces the probability that the
liquidity constraint will be binding in the future. Agents accumulate assets to insulate
themselves from a temporary drop in income that cannot be compensated by short-
term debts. Nevertheless, in the case under study farmers can smooth revenues from
production by trading in the spot market for water.
Now, if the inequality is such that [ ]
1 1
s v E M
t t t t t t
? ? + ?
+ +
, then the
marginal expected cost of selling a water right (the left hand side of the inequality) is
less than the marginal benefit of selling a water right in period t (the right hand side
72
term of the inequality). In this case the farmer wishes to sell water rights and increase
his consumption. A farmer without water rights is not able to do this leaving him at a
corner solution where the inequality is strict. From a theoretical point of view, such
farmers may exist, so I review the data to see how many farmers in the sample have
no water rights. I find that only 5% of farmers had no water rights.
4.4 Summary
In this chapter, I have developed a framework that models a farmer’s optimal
decision making in regards to investment in water rights and input quantities for each
growing season. This model is a consumption-based asset price model, and shows
that a private farmer’s reservation value for a water right depends on the value of his
stochastic discount factor. Further analysis of that discount factor allows me to relate
reservation values for water rights to the underlying preferences of farmers and risk
aversion. I have shown how heterogeneous preferences generate differences in the
stochastic discount factor, which creates differences in farmers’ private reservation
values for water rights and helps to explain water rights transactions among farmers.
Moreover, I show how more risk averse farmers have incentives to buy water rights
from farmers with lesser risk aversion. Lastly, I have discussed the effect of market
frictions on the water rights market and on the validity of my model. Among the
frictions that the literature addresses, I have focused on transaction costs and
borrowing constrains. The analysis of the data for the case under study indicates that
transaction costs do not affect the validity of the model and that the likelihood of a
corner solution with a farmer holding zero water rights and with a reservation value
less than water right market price is quite low. Moreover, borrowing constraints
73
affect consumption decisions but not water right purchases because the rights can be
used as collateral.
74
Chapter 5: The Econometric Estimation
In this section, I develop an approach to jointly estimate the parameters that
describe a farmer’s utility and production functions, based on observed economic
behavior. Joint estimation preserves estimation consistency and allows exploiting
cross-equations error correlations that might improve efficiency (Love and Buccola,
1991). I also summarize the data sources and the main characteristics of the data that
will be used in later estimations.
5.1 Parametric specification of the system of equations
The joint estimation of the parameters that describe a farmer’s utility and
production functions is performed through the simultaneous estimation of the
equation system described by the Euler Equation (4.2.10) and the first order
conditions that solve for the optimal variable input quantities (4.2.17) and (4.2.18).
That procedure requires a parametric specification of the instantaneous utility of
consumption, the stochastic production technology and the variable that represents
other net income, I
it
.
For the utility function, I have chosen an exponential function:
( ) ( ) exp U C C
i i i
? = ? ? (5.1.1)
where
i
? is restricted to be non-negative. The specification (5.1.1) has three main
features. First, the marginal utility of consumption is positive,
( ) ( ) exp U C C
i i i i
? ? ? = ? >0. Second, the utility function is concave,
( ) ( )
2
exp U C C
i i i
i
? ? ?? = ? ? <0, which implies aversion to risk. Third, it assumes that
75
the absolute Arrow-Pratt risk aversion coefficient for each farmer, denoted by
i
R
A
, is
constant (CARA) and equal to
i
? , i.e.
i
R
A
=
i
?
The CARA preferences embodied in the use of the negative exponential utility
function is a drawback of this specification because few decision-makers have the
implied characteristic that their attitude towards risk remains the same regardless their
wealth or asset position (Chavas, 2004, Chapter 4; Saha et al., 1994). Nevertheless,
risk preferences displaying constant risk aversion are extremely easy to deal with
analytically (Hammond, 1974). Moreover, the exponential function it is very suitable
as a local approximation to anyone’s utility function for evaluating small to moderate
gambles (Pratt, 1964). For these reasons the CARA preferences have been widely
used in applied decision analysis (Hammond, 1974, Keeney and Raffia, 1976,
Gregory, 1978, Love and Buccola, 1991).
To describe the stochastic production technology I use a Just-Pope Cobb-
Douglas (Just and Pope, 1978) production function that facilitates the estimation of
production risk endogenous to inputs. Moreover, the Just-Pope form holds the best
potential for mutual inference of preferences and technology (Love and Buccola,
1991). This function is:
( ) ( )
0 0 1 2 1 2
, , ; , , ; y f w L F h w L F w L F w L F
it it it
it it it it it it it it it it it it
? ? ? ? ? ?
? ? ? ? = + = +
(5.1.2)
where y is output per-hectare, w is water, L is labor, F is fertilizers (all inputs are
divided by the number of cultivated hectares), ? represents output uncertainty. The
vector of parameters of the non-stochastic component of the production function is
76
represented by ( ) , ,
1 2 3
? ? ? ? = , and ( ) , ,
1 2 3
? ? ? ? = is the vector of parameters of
the stochastic component of the production function. Output uncertainty,
it
? , is
assumed to be independent and identically distributed (i.i.d.) normal with
[ ] 0 E
it
? = and [ ] 1 V
it
? =
29
. Pope and Just (1977) point out that no generality is lost
in assuming [ ] 1 V
it
? = , since if
[ ]
2
V
it
? ?
?
= then the ( ) , , ; h w L F
it it it
? could simply
be modified by a multiplicative factor
2
?
?
. Moreover, as in Just and Pope (1977 and
1979) and Love and Buccola (1991) I assume that
0 1 2
0 E w L F
t it
it it it
? ? ?
?
(
=
(
¸ ¸
. That
assumption implies that input quantities are either not stochastic or stochastic, but
independent of ? .
Finally contemporaneous correlation among errors of the production function
is not excluded i.e. , 0 E
it jt
? ?
(
?
¸ ¸
From this production function specification, I obtain
[ ]
3 1 2
E y L w F
it
it it it
? ? ?
= and
[ ]
2
3 1 2
V y L w F
it
it it it
? ? ? | |
=
|
\ .
, which implies that the mean
and variance of output are endogenous to input decisions.
For the “other net income” variable, I
it
, I assume the following linear model:
29
The equation system described by the Euler equation and the first order conditions
that solve for the optimal variable input quantities are expressed in terms of conditional
moments to the set of information in t. However, because
it
? is assumed to be i.i.d. its
conditional and unconditional moments are the same (Cochrane, 2005, Chapter 1). Thus, if I
have that [ ] [ ] 0 E E
t it it
? ? = = I also can say that [ ] [ ] 0
1 1
E E
t it it
? ? = =
+ +
.
77
( )
( )
3 1 2
1
*
3 1 2
N N s N v T w P T L w F
jt it it jt it jt it it it it
it it it
I a
it it
P T L w F r X c
it it it it it i
it it it
? ? ?
?
? ? ?
?
| |
? + ? + +
?
|
= +?
|
|
? ? ?
\ .
(5.1.3)
where a is an unknown parameter, c
i
is “normal” consumption, and
it
? is a random
error that represents random aggregate shocks plus idiosyncratic shocks different
from production uncertainty that affect I
it
.
The variable I
it
represents farmer’s “other net income” sources as
consumption and production credit, land transactions, livestock exchanges and work
off-farm that can be used to afford consumption. Equation (5.1.3) assumes that part
of I
it
is endogenous and depends on the differences between net income from
production and water trading and some “normal” consumption level, c
i
, that is
farmer specific. For instance, a farmer that owns livestock and who faces a situation
in which her net production revenues plus water trading income is lesser than c
i
may
sell cattle.
Equation (5.1.3) also states that I
it
is a function of the deterministic part of
the net income, ( )
( )
3 1 2
1
N N s N v T w P T L w F r X
jt it it jt it jt it it it it it it
it it it
? ? ?
?
? ? + ? + ?
?
,
and two stochastic components: (1) farmer’s output specific or idiosyncratic random
shocks,
3 1 2
P T L w F
it it it
it it it
? ? ?
? , and (2) the random term
it
? .
I assume that output specific random shocks,
it
? , and
it
? are distributed
independently. That assumption follows from the fact that
it
? represents output
uncertainty, for instance pests, while
it
? represents uncertainty over “other net
78
income sources” such as random variations in bank interest rates. If asset markets
were complete then farmers’ specific random shocks would be aggregate temporal
shocks and the assumption of independent distributions may not be valid any more.
Nevertheless, because in the case under study asset markets are incomplete the above
independence assumption should hold.
Incomplete asset markets may also imply heterogeneous variances for I
it
among farmers. Different investment decisions of a surplus between “normal”
consumption level, c
i
, and production and water trading incomes may lead to
different variances in I
it
. Whether or not a farmer faces credit constraints may also
affect the variance of I
it
. These differences in the variance of I
it
among farmers
lead me to assume that
it
? is heteroskedastic. Thus I assume that
it
? distributes
normal with conditional expected mean zero and non constant conditional
variance
2
i
?
?
.
Replacing the Euler Equation (4.2.10) with these structural forms, I get:
79
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i jt
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
s v E
jt jt jt t
?
? ? ?
? ?
? ? ?
?
? ?
| | | | ? +
+ +
| |
|
|
? +
+ + + + +
|
|
+
| |
?
? ? + +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
|
|
? +?
+ \ .
= +
( )
( )
( )
1
exp 1
3 1 2
3 1 2
N N
jt it it
s N v T w
jt it jt it it
E a ac
t i i it
P T L W F r X
it it it it
it it it
P T L W F
it it it
it it it
?
?
? ? ?
? ? ?
?
(
(
(
(
(
(
(
(
(
(
(
( | | ( | | ? +
?
( | | (
( | | (
? +
( |
| (
? + ? +?
( |
| (
? ? +
( | | (
( | | (
( | |
(
\ .
\ . ¸ ¸
(
¸ ¸
(
(5.1.4)
Further simplification of equation (5.1.4) can be achieved by taking logs on both
sides
( )
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
ln ln ln
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
s v E
jt jt jt t
?
? ? ?
?
? ? ?
?
? ?
| | | | ? +
+ +
| |
| |
? +
+ + + + +
| |
+
| |
?
? ? +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
? +?
+ \ .
? = +
( )
( )
( )
1
1
exp 1
3 1 2
3 1 2
jt
N N
jt it it
s N v T w
jt it jt it it
E a ac
t i i it
P T L W F r X
it it it it
it it it
P T L W F
it it it
it it it
?
?
?
? ? ?
? ? ?
?
(
(
(
(
(
+ (
(
(
( |
|
(
(
(
| | ( | | ? +
?
( |
| (
( |
|
? +
( |
|
? + ? +?
( |
|
? ? +
( | |
( | |
( | |
\ . \ .
¸ ¸
¸ ¸
(
(
(
(
(
(
(
(
(5.1.5)
80
A convenient expression for the second term in the right hand of (5.1.5) can be obtained with the Jensen’s
inequality that states that:
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
ln
exp 1
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i jt
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
E
t
E
t i
?
? ? ?
? ?
? ? ?
?
?
| | | | ? +
+ +
| |
|
|
? +
+ + + + +
|
|
+
| |
?
? ? + +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
|
|
? +?
+ \ .
? + ( )
( )
( )
( )
1
1
exp
ln
1
3 1 2
3 1 2
N N
jt it it
a
i
E
t
N N
jt it it
s N v T w
jt it jt it it
a ac
i it
PT L W F r X
it it it it
it it it
PT L W F
it it it
it it it
?
?
?
? ? ?
? ? ?
?
(
?
+ +
(
(
(
+
(
?
(
(
(
(
(
( =
( | | ? + (
?
( |
(
( |
(
? +
( |
(
? +?
( |
(
? ? +
( |
(
( |
(
| (
(
\ . ¸ ¸
(
(
¸ ¸
( )
( )
( )
( )
1
1 1 1 1 1
3 1 2
1
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
1
exp 1
s N v T w
jt it jt it it
P T L W F r X jt
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
N N
jt it it
s N v T
jt it jt i
E a
t i
? ? ?
?
? ? ?
?
?
?
| | | | +
| |
|
|
? +
+ + + + +
|
|
| |
? ? + +
| | + + + +
+ + +
| |
| |
+ + +
+ + + \ .
|
|
? +?
+ \ .
? +
?
?
? +
( )
1
3 1 2
3 1 2
it
w
t it
ac
i it
PT L W F r X
it it it it
it it it
PT L W F
it it it
it it it
?
? ? ?
? ? ?
?
| |
|
|
|
|
|
|
|
|
|
|+
+
( | | | | |
( | | |
( | | |
+
( | | |
? +?
( | | |
? ? +
( | | |
( | | |
( | | |
\ .
\ . ¸ ¸
|
|
\ .
(5.1.6)
81
where
1 + it
? is a positive expression that corresponds to a Jensen’s inequality
adjustment. The economic intuition for the adjustment can be seen by noticing that
(using notation of Chapter 4)
1 + it
? =ln ln
1 1 1 1
E M E M
t it jt t it jt
? ?
( (
?
+ + + +
¸ ¸ ¸ ¸
,
where the difference on the right hand side of the expression is a measure of
conditional volatility of the discounted water right price for each farmer (Alvarez and
Jermann, 2005). As a special case, if the discounted price is distributed lognormal,
then the volatility measure
1 + it
? =
1
2
( )
ln 1
1
Var M jt
t it
? ( +
+
¸ ¸
(Campbell et al.
(1997, Chapter 8).
Substituting the right hand term of equation (5.1.6) in the Euler Equation
(5.1.5) I get:
( )
( )
( )
( )
1 1
1 1 1 1 1
1
3 1 2 exp
1 1 1 1
1 1 1
3 1 2
1 1 1
1 1 1
1
ln ln ln
N N
jt it it
s N v T w
jt it jt it it
a
P T L W F r X i
it it it it
it it it
P T L W F
it it it
it it it
ac
i it
s v E
jt jt jt t
?
? ? ?
?
? ? ?
?
? ?
| | | | ? +
+ +
|
|
| |
? +
+ + + + +
| |
+
|
|
?
? ? +
|
| + + + +
+ + +
| |
|
|
+ + +
+ + + \ .
? +?
+ \ .
? = +
( )
( )
( )
1
exp 1
3 1 2
3 1 2
N N
jt it it
s N v T w
jt it jt it it
E a ac
t i i it
P T L W F r X
it it it it
it it it
P T L W F
it it it
it it it
?
?
? ? ?
? ? ?
?
| |
|
|
|
|
|
|
|
| |
| |
|
(
| | | | | ? +
?
( | | |
( | | |
? +
( | | |
? + ? +?
( | | |
? ? +
( | | |
( | | |
( | |
\ .
\ . ¸ ¸
\ .
ln
1 1
E
t jt it
? ?
+
|
|
|
(
+
+ +
¸ ¸
(5.1.7)
82
Because
3 1 2
0
1 1 1
1 1 1
E P T L W F
t it it it
it it it
? ? ?
?
(
=
+ + +
( + + +
¸ ¸
and [ ] 0
1
E
t it
? =
+
, equation
(5.1.7) simplifies to:
30
( )
( )
( )
( )
( )
( )
( )
1 1
ln ln 1
1 1 1 1 1
3 1 2
1 1 1 1
1 1 1
1
ln exp 1
1
N N
jt it it
s v E a s N v T w ac
jt jt jt i t jt it jt it it i
P T L W F r X
it it it it
it it it
N N
jt it it
s N v T w
jt it jt it it
E a
t i
P T L W
it it
it it
?
? ? ?
? ? ?
?
?
? ?
( | |
? +
+ +
( |
( |
? = ? + ? + ? ?
( | + + + + +
( |
( |
? ?
+ + + +
+ + + \ . ¸ ¸
? +
?
? +
? + ln
1 1
3 2
3 1 2
ac E
i it t jt it
F r X
it it
it
P T L W F
it it it
it it it
? ?
?
? ? ?
?
( | | | |
( | |
( | |
( | |
(
? +? + +
( + + | |
¸ ¸
? ? +
( | |
( | |
( | |
\ . \ . ¸ ¸
(5.1.8)
Equations (4.2.3), (5.1.2) and (5.1.3) imply that the second term of the right
hand side of equation (5.1.8) represents the expectation in t of the expected
consumption in t+1 conditional to the information in that same period, i.e.
30
I have assumed that input quantities are not stochastic in the current period or
stochastically independent from? , i.e.
3 1 2
0 E P T L W F
t it it it
it it it
? ? ?
?
(
=
(
¸ ¸
. Now, to explore the
value of
3 1 2
1 1 1
1 1 1
E P T L W F
t it it it
it it it
? ? ?
?
(
+ + +
( + + +
¸ ¸
I use that assumption and the law of
iterated expectations to obtain:
3 3 1 2 1 2
1 1 1 1 1 1 1
1 1 1 1 1 1
E P T L W F E E P T L W F
t it it it t t it it it
it it it it it it
? ? ? ? ? ?
? ?
(
( (
=
+ + + + + + +
( ( ( + + + + + +
¸ ¸ ¸ ¸ ¸ ¸
=
[ ]
3 1 2
1 1 1 1 1
1 1 1
E E P T L W F E
t t it it t it
it it it
? ? ?
?
(
(
=
+ + + + +
( ( + + +
¸ ¸ ¸ ¸
3 1 2
0 0
1 1 1
1 1 1
E E P T L W F
t t it it
it it it
? ? ? (
(
=
+ + +
( ( + + +
¸ ¸ ¸ ¸
.
The result [ ] 0
1
E
t it
? =
+
follows from the law of iterated expectations and the
assumption that [ ] 0
1 1
E
t it
? =
+ +
. In fact the law of iterated expectations implies that
[ ] [ ]
1 1 1
E E E
t it t t it
( ? = ?
+ + +
¸ ¸
, and [ ] 0
1 1
E
t it
? =
+ +
implies
[ ]
0
1 1
E E
t t it
( ? =
+ +
¸ ¸
.
83
[ ]
1 1
E E C
t t it
(
+ +
¸ ¸
31
. I then write that expression in a more conventional time-series
notation as
[ ]
1 1 1 1
E E C C
t t it it it
? ( = +
+ + + +
¸ ¸
, where
1 it
?
+
is random error with
conditional expectation equal to 0 (Cochrane, 2005, Chapter 9).
Assuming [ ] 0
1
E
t it
? =
+
implies that the unconditional mean [ ] 0
1
E
it
? =
+
and that
1 it
?
+
is not correlated with the information set at time t (Wooldridge, 2002,
Chapter 2.2.3). Thus, the proposed transformation presumes that, based on the set of
information in period t, farmers can predict part of random consumption in t+1, but
there is another part of future consumption represented by
1 it
?
+
that can not be
predicted because is not related with the set of information at time t.
An identical argument can be used to transform the expectation of the natural
log of the price of a water right in the right hand side of equation (5.1.8) as:
ln ln
1 1 1 1
E E
t t jt jt jt
? ? ?
( (
= +
+ + + +
¸ ¸ ¸ ¸
. As in the case of the transformation for
consumption and for the same reasons that I have indicated above I assume
that 0
1
E
jt
t
?
(
=
+
¸ ¸
.
Substituting these transformations,
[ ]
1 1 1 1
E E C C
t t it it it
? ( = +
+ + + +
¸ ¸
and
ln ln
1 1 1 1
E E
t t jt jt jt
? ? ?
( (
= +
+ + + +
¸ ¸ ¸ ¸
, in equation (5.1.8) gives:
31
Because production uncertainty realized its value at the end of the season,
consumption in t+1 is not known until the end of that season. Thus, in t+1 the farmer only
has an expectation of his consumption level for that season.
84
( )
( )
( )
( )
( )
( )
( )
1 1
ln ln 1
1 1 1 1 1 1
3 1 2
1 1 1 1
1 1 1
1
ln exp 1
1
N N
jt it it
s v a s N v T w ac
jt jt jt i jt it jt it it i it
P T L W F r X
it it it it
it it it
N N
jt it it
s N v T w
jt it jt it it
E a
t i
P T L
it it
it
?
? ? ? ?
? ? ?
?
?
?
( | |
? +
+ +
( |
( |
? = ? + ? + ? + ?
( | + + + + + +
( |
( |
? ?
+ + + +
+ + + \ .
¸ ¸
? +
?
? +
? + ln
1 1 1
3 2
3 1 2
ac
i it jt jt it
W F r X
it it
it it
P T L W F
it it it
it it it
? ? ?
? ?
? ? ?
?
(
| | | |
( |
|
( |
|
( |
|
? +? + + +
( + + + |
|
? ? +
( |
|
( |
|
( | |
\ .
\ .
¸ ¸
(5.1.9)
In equation (5.1.9) the term ac
i
cancels out. Then, reordering terms, equation
(5.1.9) can be written as:
( )
( )
( )
( )
( )
( )
( )
1 1
ln ln 1 ln
1 1 1 1 1 1
3 1 2
1 1 1 1
1 1 1
1
ln exp 1
3 1 2
N N
jt it it
s v a s N v T w
jt jt jt i jt it jt it it jt
P T L W F r X
it it it it
it it it
N N
jt it it
s N v T w
jt it jt it it
E a
t i
P T L W F r X
it it it
it it it
?
? ? ? ?
? ? ?
?
?
? ? ?
| |
? +
+ +
|
|
? = ? + ? + + ?
| + + + + + +
|
|
? ?
+ + + +
+ + + \ .
? +
?
? +
? +
? ?
1 1
3 1 2
it it
it
P T L W F
it it it
it it it
?
? ? ?
?
(
| | | |
( |
|
( |
|
( |
|
+? +
( + |
|
+
( |
|
( | |
( | |
\ .
\ .
¸ ¸
(5.1.10)
where
1 1 1 1 1 it it it jt
? ? ? ? ? + +
+ + + +
is a composite error with
[ ] [ ]
1 1 1
E E
t it t it
? ? =
+ +
.
85
The structural forms proposed in equations (5.1.1), (5.1.2) and (5.1.3) implies
that the first order conditions for optimal quantities of water, labor and fertilizer,
represented by equations (4.2.17) and (4.2.18) are, respectively:
( )
( )
( )
1
exp 1
3 1 2
3 1 2
3 3 1 2 1 2
1 1
N N
jt it it
s N v T w
jt it jt it it
a ac
i i it
P T L W F r X
it it it it
it it it
E
t
P T L W F
it it it
it it it
P T w L F P T w L F
it it it it it
it it it it it it
s
jt
w
it
?
?
? ? ?
? ? ?
?
? ? ? ? ? ?
? ? ?
(
| | ? +
?
( |
( |
? +
( |
? + ? + ?
( |
? ? +
( |
( |
| (
\ . ¸ ¸
+
?
¸
0
(
(
(
(
(
(
=
(
(
(
(
(
(
¸
(5.1.11)
( )
( )
( )
1
exp 1
3 1 2
3 1 2
3 3 1 2 1 2
2 2
1
N N
jt it it
s N v T w
jt it jt it it
a ac
i i it
P T L W F r X
it it it it
it it it
E
t
P T L W F
it it it
it it it
P T w L F P T w L F
it it it it it
it it it it it it
r
it
L
it
?
?
? ? ?
? ? ?
?
? ? ? ? ? ?
? ? ?
( | | ? +
?
( |
( |
? +
( |
? + ? + ?
( |
? ? +
( |
( |
| (
\ .
¸ ¸
|
+
?
\
0
(
(
(
(
(
(
(
=
(
(
(
|
(
|
(
|
|
(
.
¸ ¸
(5.1.12)
( )
( )
( )
1
exp 1
3 1 2
3 1 2
3 3 1 2 1 2
3 3
2
N N
jt it it
s N v T w
jt it jt it it
a ac
i i it
P T L W F r X
it it it it
it it it
E
t
P T L W F
it it it
it it it
P T w L F P T w L F
it it it it it
it it it it it it
r
it
F
it
?
?
? ? ?
? ? ?
?
? ? ? ? ? ?
? ? ?
( | | ? +
?
( |
( |
? +
( |
? + ? + ?
( |
? ? +
( |
( |
| (
\ .
¸ ¸
|
+
?
\
0
(
(
(
(
(
(
(
=
(
(
(
|
(
|
(
|
|
(
.
¸ ¸
(5.1.13)
In the above equations the price for labor is denoted by
1
r , and for fertilizer by
2
r .
86
Using the already mentioned distributional assumptions for
it
? and
it
? , and
the assumption of distributional independence between
3 1 2
P T L W F
it it it
it it it
? ? ?
?
(
(
¸ ¸
and
it
? (discussed above), plus the properties of the moment generating functions of the
normal distributed variables
it
? and
it
?
32
, the system of equation (5.1.10) to (5.1.13)
simplifies to:
The Euler Equation:
( )
( )
( )
( )
( )
( )
( )
ln ln ln
1
1 1 1 1 1 1 1
1
3 1 2
1 1 1 1
1 1 1
1
1
3 1 2
s v
jt jt jt jt
N N s N v T w
jt it it jt it jt it it
a
i
P T L W F r X
it it it it
it it it
N N
jt it it
a s N v T w r X
i jt it jt it it it it
P T L W F
it it
it it it
? ? ?
?
?
? ? ?
?
?
? ? ?
? = + ?
+
| |
| |
? + ? +
+ + + + + + +
|
|
+ +
|
|
| ? | ?
+ + + +
+ + + \ .
\ .
? +
?
? + ? ? +
( )
2
2
1
2 2 2
3 1 2
1
1 1
2 2
i
a P T L W F
i it it i it
it it it
?
? ? ?
? ? ?
| |
|
|
?
|
|
|
\ .
| |
| |
?
|
+ ? +
| +
|
\ .
\ .
(5.1.14)
32
For a random variable X distributed normal with mean µ and variance
2
? the
generation moment function is: ( )
2 2
exp exp
2
t
E tx t
?
µ
| |
|
= +
|
\ .
. That implies
( )
( )
( )
2 2
exp
2
exp exp
2
E tx
t
E x tx t t
t
?
µ ? µ
| |
?
|
= = + +
| ?
\ .
or
( )
( )
2 2
2
exp exp
2
t
E x tx t t
?
µ ? µ
| |
|
? = ? ? +
|
\ .
87
The equations for the optimal variable input quantities:
( )
2
0 1 2
1
0 1 2 0
0
0
2it
a P T w L F
i it it
it it it P T w L F
it it
it it it
s
jt
w w
it it
? ? ?
? ? ? ? ?
?
?
| |
| |
| +
|
\ . |
? ? + =
|
|
|
\ .
(5.1.15)
( )
2
0 1 2
1
0 1 2 1
1
0
1 3it
a P T w L F
i it it
it it it P T w L F
it it
it it it
r
it
L L
it it
? ? ?
? ? ? ? ?
?
?
| |
| |
| +
|
\ . |
? ? + =
|
|
|
\ .
(5.1.16)
2
0 1 2
(1 )
0 1 2 2
2
0
2 4it
a P T w L F
i it it
it it it P T w L F
it it
it it it
r
it
F F
it it
? ? ?
? ? ? ? ?
?
?
| |
| |
| +
|
\ . |
? ? + =
|
|
|
\ .
(5.1.17)
Those first order conditions indicate that input decisions depend on the
marginal productivity of inputs, input prices, risk aversion and the effect of each input
on the variance of output. The last two effects on optimal input quantities are
captured by the second term within brackets in each of those equations. Given that
i
? is positive, that term is negative (positive) for risk decreasing (increasing) inputs.
Thus for a risk-averse farmer, when an input is risk decreasing (increasing), he has an
incentive to increase (decrease) the demand for this input.
It is important to notice that in equations (5.1.14) to (5.1.17) if the parameter a
is equal to -1, then risk aversion does not affect farmer decisions on either water
rights or input quantities. As it can be seen from equations (4.2.3) and (5.1.13) a
value of -1 for a implies that the farmer’s expected consumption in any season is
equal to his “normal” consumption level. Moreover, with a = -1 farmer’s
88
consumption is not affected by random shocks in production, i.e. C c
it i it
= + ? . This
implies that the farmer has production full insurance or unlimited access to “other
income sources” such as credits. Nevertheless, because neither production full
neither insurance nor unlimited access to “other income sources are feasible I rule out
the possibility of a = -1.
As in Love and Buccola (1991), Saha et al. (1994), Chavas an Holt (1996) and
Kumbhakar (2002), among others, I added to equations (5.1.15), (5.1.16) and (5.1.17)
the additive disturbances, , ,
2 3 4 it it it
? ? ? , associated with errors in optimization.
Pope and Just (2003) find credible evidence in U.S. agricultural production of errors
in optimization. I assume that these optimization mistakes occur in form of random
failures, which support a stochastic structure to the equation system. That stochastic
structure is needed to achieve an econometric estimation of the parameters of interest.
In addition, I assume that these
it
?
?
s have conditional expected value equal to zero. I
do not restrict the error terms of the first order condition for input quantities to be
independent among equations for each farmer. Only the error term of the Euler
Equation,
1 1 it
?
+
, is assumed to be independent of the error terms , ,
2 3 4 it it it
? ? ? .
This assumption about the independence of the error term of the Euler Equation with
respect to the error terms of the input equations is based on the structure of
1 1 it
?
+
.
That structure indicates that
1 1 it
?
+
is a function of the random variable
1 it
?
+
that
realizes its value in t+1 and which I expect not to be correlated with errors in
optimization for input quantities in period t.
89
I also assume that disturbances are correlated across farmers within equations.
Therefore, [ ] , 0 Cov
t hit hmt
? ? ? for i m ? and for h=1,2,3,4. I only exclude
correlations between errors associated with different farmers and across equations.
Given these assumptions regarding the error terms, ( ) , ,
1 2 3
? ? ? ? = , the
variance- covariance matrix of ? , which is denoted by ? is specified as follows:
?= ( ) E
t
??? =
0 0 0
11 * * *
0
* 22 23 * 24 *
0
* 32 * 33 34 *
0
* 42 * 43 * 44
I I I
N N N N N N
I I I
N N N N N N
I I I
N N N N N N
I I I
N N N N N N
? ?
? ?
? ?
? (
(
?
(
( ?
(
?
¸ ¸
Here ? is a matrix of order N*H x N*H with N the number of farmers, H the number
of equations, [ ] E
hh t h h
? ?? ? = and
( )
,
gh Cov git hit
t
? ? ?
=
, for g=2,3,4 and h=2,3,4.
5.2 Data
The data set has two parts and contains data for the four main WUAs
associations in the Limarí Valley
33
. The first part contains a cross-section time series
sample on farmers over two agricultural seasons (98/99 and 99/00). It includes micro
level data for farming activity. The second part of the data contains time series for: i)
water right prices and water right transactions for the period 1981 to 2001; ii) spot
market water prices and water transactions for the period 1995 to 2000; and iii) water
accorded to water rights for the period 1980 to 2000.
The farmer micro-level data is obtained from a two-round survey that I
conducted. This survey was performed in the Limarí Valley (see the survey
33
Those WUAs are: Asociación de Canalistas del Canal Camarico (ACCC);
Asociación de analistas del Embalse Cogotí (ACEC); Junta de Vigilancia del Río Limarí
(JVRL); and Asociación de Canalistas del Embalse Recoleta (ACER).
90
instrument in the Annex). Farmers from the main irrigated areas within this key
agricultural region were interviewed. The sample was designed by Zegarra (2002)
34
who conducted a previous survey on that same valley. In the first round (SI),
surveyed in 1999, I collected information for the 98/99 season from 161 farmers. The
second round (SII) was conducted in 2000 and I collected information for the 99/00
season from 151 farmers.
The farm level data set includes seasonal information on crop production,
input use for each crop, output and input prices, irrigation methods, water right
transactions, volume of water bought or sold in the spot market, land transactions,
livestock inventory, renting in or out of machinery, asset ownership, farmer’s
liabilities, household characteristics, family labor, well access and water storage
capacity, marketing, governmental subsidies to improve irrigation systems and water
expectations for the coming season.
As is usual in surveys that collect data, I have gaps in the data due to attrition
and survey non-response. Balgati (2001) presents the rate of attrition for a sample of
studies that use panel data. In my sample, attrition from the first to the second round
surveys that I conducted is comparable to that obtained in other empirical works.
Non-response is caused by farmers that have sold or have decided to abandon their
land, farmers previously interviewed that were not subsequently located and farmers
34
Zegarra did not develop a list of farmers to interview based on a random sample
due to the expense of finding each sampled individual; instead, he simulated random
sampling for farmers who were present at their farm when he conducted the survey. He
began at some point inside the irrigated area (stratum), interviewing farmers using a
systematic round skipping for close neighbors. This results in a sample, which is
geographically representative for each irrigation organization. The main limitation of this
sampling procedure is that farmers who were not present at the moment of the survey had
zero probability of being selected. The procedure also excludes farmers who, at the moment
of the survey, had abandoned production.
91
who refused to answer (7 and 4 farmers in the first and second rounds refused to
answer, respectively). The causes of non-response suggest few if any behavioral
reasons behind this problem; hence, the consequences of attrition appear to be
minimal.
There is missing data arising from partial response to survey questions. This
is mainly due to the fact that most farmers do not keep written records of the
information requested in the survey. In fact, only five among all surveyed farmers
had written records on most of the surveyed data. Thus, in most cases, a partial
response occurs when the respondent fails to answer a question because he has
difficulty recalling events that occurred in the past.
One way to handle the missing data problem is by imputing missing
observations. Nevertheless, as Cameron and Trivedi (2005) point out “there is a cost
of imputing missing data that comes from having to make assumptions to support a
procedure for generating proxies for the missing observations, and from the
approximation error inherent in such a procedure.”
Alternatively, it is possible to handle missing data by deleting them and
analyze only the reduced sample of “complete” observations. That procedure is
called listwise deletion. Its consequences for the econometric estimation depend on
the missing data mechanism. If the probability of missing data of the variables in the
data set depends neither on its own values nor on the values of other variables in the
data set, then missing data process is completely at random. In that case the
remaining set after listwise deletion remains a random sample from the original
population and the estimates based on it are consistent (Cameron and Trivedi, 2005).
92
If the probability of missing data on a variable does not depend on its value but may
depend on the values of other variables in the data set, then data is missing at random.
If the data set has gaps due to data missing at random and the parameters for the
missing data-generation process are unrelated to the parameters that one wants to
estimate, then the missing data problem is ignorable and the complete data set after
listwise deletion allows for consistent estimation of the parameters of interest
(Cameron and Trivedi, 2005). Nonetheless, under either the missing data complete at
random or just missing data at random assumptions, listwise deletion still reduces
efficiency in the estimation.
For the case of the missing data problem in my survey, the causes of non-
response suggest few if any behavioral reasons behind this problem. Furthermore,
missing data due to a partial response to survey questions is mainly related to the
difficulty of some farmers to recall past data. Thus, it seems reasonable to assume
my data set is characterized by missing data complete at random or at least missing at
random. Therefore I handle the missing data problem using listwise deletion.
The Euler Equation (5.1.14) links farmers’ decisions in two seasons. This
requires full data for each farmer in the two seasons under study. That requirement
causes substantial sample loss and after the listwise deletion process the number of
farmers that fulfill that requirement is 32.
93
The following table provides some statistics on the most important variables in the model for the whole sample as
well as for sub-sample of 32 farmers used in this dissertation (referred to in what follows as the sub-sample).
Table 5.1: Basic statistics of the main variables
Whole sample Sub-sample (3 2 farmers)
Variable SI
SII SI SII
Number of
observations.
mean Standard
deviation
Number of
observations
mean Standard
deviation
mean Standard
deviation
mean Standard
deviation
Cultivated land (hectare) 161 12.1 25.2 148 10.3 17.9 12 21.3 11.4 18.5
Labor (hours per hectare) 148 1242 2115 137 1021 1674 1099 1437 987 1201
Nitrogen (kilograms per
hectare) 117 258.2
458.6
137 109.8 151.5 320 591 125.7 170.9
Input water (cubic meters
per hectare)
144 15766 20763 132 9145 9997 6958 5129 8674 16081
Number of water rights 155 20.7 37 141 20.2 36 15.6 14.7 17.3 19.0
Education (years of
schooling)
151 8.3 4.7 139 8.0 4.7 8.6 4.7 8.7 4.6
Experience (years) 134 30.4 14.3 118 25.6 13.0 28.6 14.3 30.5 13.6
Household size 131 4.7 3.3 102 4.8 3.2 4.9 3.6 4.9 3.0
Percentage of multioutput
producers
53.7% 52.11% 31.3% 34.3%
94
Table 5.1 shows that sample mean values from the whole samples are close to
the sample mean values from the sub-samples for all the reported variables but input
water per hectare in survey I. The percentages of multioutput producers in the sub-
samples are lower than in the whole sample. If farmers’ decisions on the number of
crops they grow are related to their risk aversion, then differences in the percentage of
multioutput producers may cause a bias problem. That may be the case if an
estimation of risk aversion for a “representative farmer” is intended with the sub-
samples. In this dissertation I test for differences in risk aversion among farmers. As
I explain in next chapter, that test is based on the effect of farmer’s specific
characteristics upon his risk aversion and, as a consequence, it is not subject to the
above mentioned bias problem.
Annual average price for water rights and water right transaction time series
for the period 1981 to 2001 were obtained through the Conservador de Bienes Raíces
of Ovalle and the records kept by the WUAs
35
. Water prices and water transactions
in the spot market time series for the period 1995 to 2000 were constructed using the
records of the WUAs, information obtained from the Direccion de Riego and the two-
round survey. For water accorded to water rights I use data from the WUAs.
In the equation system (5.1.14) to (5.1.17), the land input is measured as the
total hectares of cultivated land. No distinction is made as to whether land is owned,
rented or sharecropped
36
.
35
The series for water right prices and transactions between 1981 and 1992 were
collected by Zegarra (2002) and between 1992 and 2000 by Cristi et al. (2002) and Vicuña
(2000).
36
Sharecroppers were considered as single producers.
95
Production input water is measured as the farmer’s total number of water
rights times the amount of water accorded to each right plus net sales of water
volumes
37
. The amount of water obtained through that formula was weighted by
farmer’s average irrigation efficiency
38
.
Labor is total hours per growing season and is obtained by grouping together
three different types of labor: family workers, permanent hired workers, and hired
workers for specific activities (temporary workers). The survey data show that some
farmers only use family work, others have permanent hired workers and others hire
workers for specific activities and time periods (temporary workers). For those
farmers with a mix of workers it is not possible to infer from the available data how
many hours worked correspond to each type of worker. Thus I restrict my
econometric results by making no distinction among type of workers.
Fertilizer is measured in kilograms of nitrogen.
37
Farmers may decide not to use all the water accorded to their water rights, but
saving water from one season to the next has a penalty of 15% to 20% of the endowment
which makes this practice rare (Zegarra 2002). Moreover, the existence of a price greater
than zero for water in the spot market implies that rational farmers will not resign to any
amount of water accorded to their water rights.
38
Because irrigation systems vary among land plots within the same farm I use
farmer’s average irrigation efficiency. This was calculated as the arithmetic mean of the
farmer’s irrigation efficiency in his different plots within the same farm:
( )
1
irrigation eficiency in plot q
q
total cultivated land
¿
=
,
where irrigation efficiency varies accordingly to the following table:
Irrigation
system/
Efficiency
Drip Sprinkler Furrow Flood
90% 75% 65% 45%
Source: Comisión Nacional de Riego, Gobierno de Chile
96
Some farmers in the sample produce more than one crop. For those farmers I
represent output price by a farmer-specific weighted average of all the farmgate
prices of the farmer’s crops. As in Saha et al. (1994) I use the product specific
income shares as weights for each crop price. Furthermore, because prices depend on
the arbitrary output units they value
39
I have divided each farmgate price by is sample
mean
40
. This scaling procedure does not affect the relations that I intend to estimate
econometrically with my model.
For wages paid to labor,
1
r , I use the per hour payments to temporary workers.
Those payments are a good proxy of the labor price that each farmer faces on the
labor market. I also estimate total labor cost by multiplying wages by total hours
worked. That procedure values the work of permanent hired workers and family
workers as equal with the work done by temporary workers. Three farmers in the
sample do not hire temporal workers. For them, I use the daily payment to their
permanent workers as the wage rate. Finally, three other farmers use only unpaid
family workers so I use the sample average of the per hour payments to temporary
39
As an example, potatoes can be measured in kilograms and there is a price for the
kilogram of potatoes. Nevertheless, potatoes are also exchanged in sack units of 50
kilograms and the price of a sack is higher than the price for a kilogram. In the estimation of
an average price the price of potatoes will be given more weight if I arbitrarily use the price
of a sack of potatoes. This problem can be eliminated by dividing potato prices by their
sample mean. The sample mean for the price of each crop has to be calculated over farmgate
prices that value identical units of that crop.
40
The sample average price of crop k in season t, p
kt
, is calculated as:
n
k
p p n
kt kit k
i
= ¿ where p
kit
is the farmgate price of crop k for farmer i in period t, and
n
k
is the total number of surveyed farmers that produce crop k in season t.
97
workers as proxy of their labor cost. Here I am assuming that family workers can at
least get the average wages for temporary workers by working off-farm.
The nitrogen price aggregate,
2
r , is computed as an arithmetic mean using
expenditure shares as the weights. Different nitrogen prices are obtained by dividing
unit fertilizer prices by kilograms of nitrogen per unit of fertilizer:
PF
fit
UN
f
(5.2.1)
where PF
fit
is the unit price of fertilizer f paid by farmer i in period t and UN
f
equals kilograms of nitrogen contained in one unit of fertilizer f. In the sample data
there is one farmer that reports a value of zero for nitrogen. Because the underlying
assumption of this dissertation is that observed farmers decisions are optimal, I take
that amount of nitrogen as the farmer’s optimal decision for the quantity of that input.
The nitrogen price for this observation is set at the arithmetic sample average of
nitrogen prices. The underlying assumption is that the farmer can do as well as the
average farmer in buying fertilizer input.
98
Chapter 6: Estimation and results
The estimation of the parameters , , a ? ? and the parameter vectors , ? ? , ?
is based on Full Information Maximum Likelihood (FIML) assuming a multivariate
normal distribution in the residuals of the system of equations. The likelihood of the
sample is:
( ) ( )
( )
( ) ( )
1
1
1
, , , , , 2 exp , , , , , , , , ) 2 2
2
NT
L a Z F Z F Z J
t
? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ?
? = ? ? ?
(6.1)
where Z=(Z
1
,Z
2
) is the vector of variables in the model. Z
1
is the set of endogenous
variables in the model, i.e. Z
1
=( ) , , N w X . Z
2
represent the set of exogenous
variables, i.e.
, , , , , , , , , , , ,
1, 1 1 1 1 1 1 1 2 , 2 1 1,
2
, ,
1 1 1, 1
s s v v P P T T r r r r w
jt jt jt jt jt jt it it it it i t i t i t i t it
Z
L F N N
it it it it
? ? | |
+ + + + + + + +
= |
|
+ + ? +
\ .
.
( ) , , , ,
det
1
F Z
J
Z
? ? ? ? ( ?
?
(
?
¸ ¸
is the Jacobian of the transformation from ? to Z
1
.
Consistent estimation of the system of equations (5.1.14) to (5.1.18) requires
that the exogenous variables in the model are not correlated with the error term of the
Euler Equation,
1 1 1 1 1 it it it jt
? ? ? ? ? + +
+ + + +
. Nevertheless, because that error term
contains an omitted variable,
1 it
?
+
, some of the regressors may be correlated with
1 1 it
?
+
. If so, the explanatory variables that are correlated with
1 1 it
?
+
are
endogenous (Kapetanios, 2004). Hence, I need to test for endogeneity of the
regressors in the Euler Equation. For that purpose I use a test proposed by Hausman
(1978), which I describe in detail in the Appendix at the end of this chapter. Also I
99
test whether the conditional mean of the error term in the Euler Equation is zero, i.e.
0
1 1 2
E Z
it
? ( =
+
¸ ¸
. The latter test procedure is as follows. First I obtain the FIML
estimated residuals for the Euler Equation. Then I estimate a linear regression of
those errors on a constant. If the constant is not statistically significant I do no reject
the hypothesis of a conditional mean value of zero for the error terms in the Euler
Equation.
Due to the lack of a longitudinal data, the estimation of the parameter for
preferences,
i
? , for each farmer is addressed by assuming that his utility function is
based on known farmer characteristics (Zeldes, 1989, Blundell et al., 1994, Dubois,
2001). Thus, the parameter that represents a farmer’s preferences is parameterized as
an exponential function of that farmer’s education (ED), experience (EXP) and
household size (HS)
i
? = ( ) exp
0 1 2 3
ED EXP HS
i i i
? ? ? ? + + + (6.2)
where , , ,
0 1 2 3
? ? ? ? are unknown parameters. The exponential form ensures that
i
? is positive. Hence, the right hand side of (6.2) replaces
i
? in the system of
equations (5.1.14) to (5.1.18).
For the variance of the random error
it
? in the equation for the variable I
it
,
equation (5.1.3), I propose the following specification:
( )
2
1 2
exp
0
D D
ic i il
? ?
? ? =
?
(6.3)
where , ,
0 1 2
? ? ? are unknown parameters, and D
il
and D
ic
are dummy variables
that indicate whether the farmer has livestock and whether the farmer has access to
100
consumption or production credit, respectively
41
. The exponential form for the
constant
0
? ensures that the variance of
it
? is positive.
Sample data indicates that a good number of farmers invest in livestock and
that cattle are sold and bought quiet often by the farmers, probably to mitigate
consumption volatility (Rosenzweig and Wolpin, 1993). The investment decision in
livestock should affect the variance of I
it
because asset returns do not have the same
variances in a real situation. Moreover, farmers without credit restrictions can
experience higher changes in I
it
because when it is needed they can substitute for
present net income from production and water transactions with market loans.
Furthermore, for those farmers, when income is greater than their “normal”
consumption levels they probably devote an important part of that difference to pay
their debts. Additionally, farmers with access to credit can face riskier activities and
therefore higher expected incomes precisely because they can solve consumption
smoothing through indebtedness. As a consequence, it is expected that farmers
without credit restrictions exhibit a higher variance in I
it
.
That structure for the variance of
it
? also allows me to control for the effect
of two main other income sources (credits and sales of cattle) when choosing optimal
decisions for consumption smoothing. In the present context of incomplete markets,
farmers maintain water rights not only for production but also for consumption
smoothing and this justifies the necessity for controlling from other income sources.
41
The variable D
il
takes the value of 2 if farmers have livestock and the value of 1
other wise. In the case of D
ic
it takes the value of 2 if farmers have access to credit and the
value of one other wise. I have used the value of 1 and 2 instead of 0 and 1 to allow
convergence in the estimation procedure.
101
This allows me to better explain within my model, decisions upon the number of
water rights held by a farmer.
In the construction of the dummy variable for credit, those farmers that have
received a credit in any of the agricultural seasons that go from the 95-96 to 99-00
seasons where classified as farmers with access to credit, otherwise they were labeled
as farmers without access to credit.
The parameter a in equation (5.1.3), the parameter
0
? in equation (6.3), as
well as the constants terms of the Just-Pope production function are not separately
identifiable from the parameter
0
? in the equation system (5.1.14) to (5.1.17). The
latter is the constant term in the specification for risk aversion in equation (6.2). Thus
the equation (5.1.14) should be written as:
( )
( )
( )
( )
( ) ( )
( )
ln ln ln
1
1 1 1 1 1 1 1
exp
0 1 2 3
3 1 2
1 1 1 1
1 1 1
1
exp
0 1 2 3
s v
jt jt jt jt
N N s N v T w
jt it it jt it jt it it
a ED EXP HS
i i i
P T L W F r X
it it it it
it it it
N N
jt it it
a ED EXP HS s
i i i j
? ? ?
?
? ? ? ?
? ? ?
?
? ? ? ?
? = + ?
+
| |
| |
? + ? +
+ + + + + + +
| |
+ + + + +
|
|
| ? | ?
+ + + +
+ + + \ . \ .
? +
?
+ + + +
( )
( )
( )
3 1 2
2
1
3 1 2
exp 2 2 2 2 2
0 1 2 3
2
1 2
exp 2 2 2 2
0 0 1 2 3 1 1
2
N v T w r X
t it jt it it it it
P T L W F
it it
it it it
a ED EXP HS P T L W F
it it
it it it
D D
ic il
ED EXP HS
it
? ? ?
? ? ?
? ? ? ?
? ?
? ? ? ? ? ?
| |
|
|
? ? ? + ?
|
|
|
\ .
| |
| |
|
+ + + + ?
|
|
\ .
\ .
+ + + + +
+
(6.4)
where ( ) ( ) exp 1 a a = + . Thus I estimate the parameters ln , , , , ,
0 1 2 3 0
? ? ? ? ? ?
plus the parameter vectors ? and ? . With
0
? =
0
a ? + and 2
0 0 0
? ? ? = +
. In the
102
input equations I also estimate
0
? instead of
0
? . As a consequence, I can not
estimate a specific value for the coefficient of risk aversion. Nevertheless, in spite of
the identification problem of
0
? , I can still test whether risk aversion is
heterogeneous among farmers.
FIML procedure requires an initial value for each of the parameters of the
system. Love and Buccola (1991) and Saha et al. (1994) estimate farmers’
preferences and production technology jointly in the presence of risk. They estimate
parameters for preferences and production technology from the first order conditions
of the maximization of the expect utility with respect to inputs. In both studies,
starting values for the production function parameters are provided by a prior
estimation of the Just-Pope production function. Nevertheless, if inputs are
endogenous, they should be correlated with the error term in the production function.
The Just-Pope parameter estimates are then inconsistent (Love and Buccola, 1991).
Due to that problem with the estimation of the Just-Pope production function, I have
chosen to obtain starting values for the production function parameters from a prior
estimation of the input demand system in my model. The estimates from those
equations provide the set of starting values for a new estimation of the whole system,
i.e. the input equations together with the Euler Equation.
For the constant discount factor, ? , I use as a starting value the reciprocal of
one plus the market annual interest rate on year 1998. For the constant in the
equation for the variance of
it
? (equation (6.3)) I use a value of zero as starting value
and a value of one for the parameters of livestock and credit.
103
Data used in the estimation presents large differences in the scale of the
variables. Therefore, I have re-scaled the values of the variables dividing each
variable by its sample standard deviation (with the only exception of dummy
variables of the equation (6.3)). Scaling the data facilitates convergence of the
estimation and does not affect the measurement of the underlying relationship among
the variables in the model neither the t-statistics, but it does affect the interpretation
of the parameter estimates (Carter et al., 1997, Chapter 6). The latter is not a problem
for the analysis developed in this dissertation that focus on the significance and the
sign of the parameter estimates rather than on their numerical values. Finally, a
constant term was added to each input equation in order to ensure a zero mean value
for the error terms.
The descriptive statistics for the variables used in the estimation of the system
of equations are reported in Table 6.1.
104
Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations.
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Number of water
shares (96-97)
Units 65.5 0.25 18.2 17.5
Number of water
shares (97-98)
Units 65 0.0 17.8 17.9
Number of water
shares (98-99)
Units 65 0.25 15.6 14.7
Number of water
shares (99-00)
Units 80 0.0 17.3 19
Water accorded to
water rights (97-98)
Cubic meters 6633 5000 6039 526.0
Water accorded to
water rights (98-99)
Cubic meters 6633 3000 4933 1193
Water accorded to
water rights (99-00)
Cubic meters 6633 3000 4969 1227
Education (98-99)
Years of
schooling
17 1 8.6 4.7
Experience (98-99) Years 50 1 28.6 14.3
Household size
(98-99)
Number of
people leaving in
the same house
18 1 4.9 3.6
Education (99-00)
Years of
schooling
17 2 8.7 4.6
105
Cont. Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Experience (99-00) Years 51 10 30.5 13.6
Household size
(99-00)
Number of people
leaving in the same
house
15 1 4.9 3.0
Land (98-99)
Total cultivated
land
116 0.2 12 21.3
Labor (98-99)
Total number of
hours on the season
per cultivated
hectare
7400 15.36 1099 1437
Fertilizers (98-99)
Kg. of nitrogen per
cultivated hectare
2818 0.00 320 591
Water used as input
by the farmer (98-99)
Cubic meters per
cultivated hectare
18720 1485 6958 5129
Land (99-00)
Total cultivated
Land
90 0.12 11.4 18.5
Labor (99-00)
Total number of
hours on the season
per cultivated
hectare
4533.3 8.5 987 1201
Fertilizers (99-00)
Kg. of nitrogen per
cultivated hectare
816.7 0.0 125.7 171
Water used as input
by the farmer (99-00)
Cubic meters per
cultivated hectare
93750 1782 8674 16081
106
Cont. Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Labor price (98-99) Chilean pesos 7800 2750 4238.8 919.7
Fertilizer price (98-
99 )
Chilean pesos 2717 51.9 985.5 686.7
Water price in the
spot market (98-99)
Chilean pesos 12.52 6.77 8.93 2.34
Labor price (99-00) Chilean pesos 5500 3250 4107.2 664.4
Fertilizer price (99-
00 )
Chilean pesos 6512.9
157.7 1133.6 1141.3
Water price in the
spot market (99-00)
Chilean pesos 9.72 2.11 6.61 2.65
Water right prices
(97-98)
Chilean pesos 456876 176775 286788 106160
Water right prices
(98-99)
Chilean pesos 554981 172496 361797 155830
Water right prices
(99-00)
Chilean pesos 588887 197404 332714 131889
Output price index
(98-99)
Chilean pesos 3.72 0.29 1 0.87
Output price index
(99-00)
Chilean pesos 5.14 0.35 1.16 1.28
Labor cost (98-99) Chilean pesos 31256007 15726 2010805 5434032
107
Cont. Table 6.1: Descriptive statistics for variables used in the estimation of the system of equations
Variable name
(season)
(all nominal
variables are in
Chilean pesos of
year 1990)
Variable
description Max Min Average
Standard
deviation
Fertilizer cost (98-
99)
Chilean pesos 10641458 16163 665187 1860068
Labor cost (99-00) Chilean pesos 42203503 12555 2314106 7486319
Fertilizer cost (99-
00)
Chilean pesos 3132494 0.0 325774 645616
Gross output
revenue per hectare
(98-99)
Chilean pesos 13255140 47324 1759103 2675375
Gross output
revenue per hectare
(99-00)
Chilean pesos 8923973 84547 1579336 2115119
108
I specify the variance-covariance matrix allowing different variances for the
disturbances of each equation and contemporaneous correlation among the
disturbances of the input first order conditions corresponding to the same farmer.
Heteroskedasticity for the errors of the Euler Equation is expected because they are
defined as
1 1 1 1 1 it it it t
? ? ? ? ? + +
+ + + +
, and it is reasonable to expect that the
[ ]
1
Var
t it
?
+
, i.e. the variance of the conditional volatility of the discounted water
right price for each farmer, differs among farmers. Nevertheless, the correlogram of
the square residuals, the conditional heteroskedasticity (ARCH) test and the Cusum
squares test indicate that the null hypothesis of homokedastic error terms of the Euler
Equation cannot be rejected. The same result is obtained for the error terms of the
input equations for water and fertilizer. In the case of the errors of the labor equation,
only the Cusum squares test does reject that hypothesis.
To test for correlation among errors corresponding to different farmers within
the same equation, I use the Ljung-Box Q-statistic which is commonly used to test
whether the disturbances are white noise. Based on this test I do not reject the
hypothesis that the errors are not correlated.
The resulting variance-covariance matrix can be summarized by:
( ) E
t
??? ? = =
0 0 0
11 * * * *
0
* 22 *
23 * 24 *
0
* 32 * 33 * 34 *
0
* 42 * 43 * 44 *
I I I I
N T N T N T N T
I I
N T N T I I
N T N T
I I I
N T N T N T N T
I I
N T N T N T N T
?
?
? ?
? ? ?
? ? ?
(
(
(
(
(
(
¸ ¸
Estimates of the error covariances between input equations are obtained from:
1
ˆ ˆ ˆ
1
N
gh gi hi
N
i
? ? ? = ¿
=
, where g=2,3,4 and h=2,3,4.
109
Residuals ˆ
gi
? and ˆ
hi
? are the maximum likelihood estimated residuals in each
iteration.
Efficient estimation of the Euler Equation is required for the Hausman test in
that equation (Wooldridge, 2002, Chapter 6.2.1). Because I ensure an efficient
estimation of that Equation with the above mentioned specification of the variance-
covariance matrix, I can use the Hausman test for possible endogeneity of some of the
regressors due to an omitted variables problem in the Euler Equation. Result of the
test indicates that I can not reject the null hypothesis that those variables are
individually and jointly exogenous. This provides evidence that the parameters of the
model are estimated consistently and that they are unbiased. In fact a maximum
likelihood ratio test for the null hypothesis that the variables are jointly exogenous
gives a value equal to 1.70 which is lower than the critical value of 12.8 for a Chi-
Squared with 5 degrees of freedom and a significance level of 5%. Moreover, the test
for the conditional mean of the error terms in each equation indicates that I can not
reject the hypothesis that the residuals have conditional mean equal to zero. This
result for the Euler Equation is fully consistent with the above result of the Hausman
test, because 0
1 2
E Z
hit
? ( =
+
¸ ¸
implies [ ] 0
1 2
E Z
hit
? =
+
.
A Jarque-Bera test for normality of the errors shows that I cannot reject the
null of normality for the residuals of each of the equations within the system but the
equation that describes first order condition for labor.
Parameter estimates are shown in Table 6.2. Results indicate that all the
parameters are significant at 1% of significance level, but the parameters for
110
household size and the dummy variable that indicate whether the farmer has livestock
are not statistically significant.
Table 6.2: Estimates of the parameters of the equation system using FIML
Coefficients
(Standard
Errors)
Estimates of the deterministic part of the production function
( vector? )
Water inputs per hectare 0.86
(0.10)
Labor per hectare 2.43
(0.08)
Fertilizer use per hectare 0.44
(0.03)
Estimates of the stochastic part of the production function (vector ? )
Parameters
Water inputs per hectare 2.28
(0.13)
Labor per hectare 0.04
(0.01)
Fertilizer use per hectare 0.15
(0.02)
Estimates of the parameters for the equation for farmers preferences
(vector ? )
Constant (
0
? )
5.2
(0.31)
Education -1.92
(0.08)
Experience -2.10
(0.14)
Family Size -0.12
(0.08)
Estimates of the parameters for the variance of
it
? (vector? )
Constant (
0
?
)
-5.1
(0.34)
111
Cont. Table 6.2: Estimates of the parameters of the equation system using FIML
Livestock ( D
l
)
-0.44
(7.44)
Credit ( D
c
)
7.77
(1.22)
Estimates of other parameters
Discount factor ( ? )
42
0.93
(0.05)
Estimates of other parameters
Values
Estimated variance of the Euler Equation
0.08
Estimated variance of the water equation
0.88
Estimated variance of the labor equation
1.06
Estimated variance of the fertilizer equation 0.88
Estimated correlation between water and fertilizer errors 0.02
Estimated correlation between labor and fertilizer errors 0.33
Estimated correlation between labor and water errors -0.12
Jarque-Bera Test for normality of the errors (p-value in
parenthesis)
Euler 0.31
(0.86)
Water 4.86
(0.09)
Labor 67.38
(0.0)
Fertilizer 5.74
(0.06)
Values
Number of observations 32
Maximum Likelihood 82.5
42
In the Euler equation I estimate the natural log of the discount factor ( ln ? ) and I
use the Deltha Method to obtain the standard deviation of the discounted factor, ? .
112
In order to check the accuracy of the estimation, I compute Theil’s Inequality
coefficient over gross output income per hectare
43
and water rights investments
44
.
That coefficient always falls between 0 and 1. If it takes the value of 0 there is a
perfect fit in the model; if it takes the value of 1, the predictive performance of the
model is bad. Moreover, Theil’s Inequality coefficient can be decomposed in the bias
(variance) proportion that indicates how far the mean (variance) of the predicted
values is from the mean (variance) of the actual data (Pindyck and Rubinfeld, 1998,
Chapter 8). For good forecasts, the bias and variance proportions are small. Because
gross output income per hectare and water rights investments have been normalized
by their standard deviation respectively, the comparison between the variance of the
predicted value and the variance of the actual data is meaningless. Thus, comparison
is limited to the bias proportion. For gross output income per hectare Theil’s
Inequality coefficient is 74% and the bias proportion is 1.7E-5. For water rights
investments Theil’s Inequality coefficient is 76% and the bias proportion is 1.01E-5.
Those low values of the bias in the means allow me to disregard the possibility of
systematic bias in the prediction of those variables with my model. Figures 6.1 and
6.2 report the actual value of output income per hectare and water right investments
against the imputed values for those variables, based on the estimates in Table 6.2.
43
On my model the expected value of gross output income per hectare corresponds
to:
0 1 2
P T w L F
it it
it it it
? ? ?
.
44
Investment in water rights corresponds to: ( )
1
N N
jt it it
? ?
?
113
Figure 6.1: Actual value of standardized gross output income per hectare
against the imputed values for that variable.
0 5 10 15 20 25 30
1
2
3
4
5
S
t
a
n
d
a
r
d
i
z
e
d
G
r
o
s
s
O
u
t
p
u
t
I
n
c
o
m
e
fitted actual
Figure 6.2: Actual value of standardized investment in water rights
against the imputed values for that variable
0 5 10 15 20 25 30
-6
-4
-2
0
2
4
6
S
t
a
n
d
a
r
d
i
z
e
d
N
e
t
I
n
v
e
s
t
m
e
n
t
i
n
W
a
t
e
r
R
i
g
h
t
s
fitted actual
114
Thus, at this time, I can ask whether heterogeneity of farmers’ preferences is a
valid hypothesis. That is tested by defining the null hypothesis that the parameters
, ,
1 2 3
? ? ? are jointly zero. If the null hypothesis is rejected, I infer that farmers
have heterogeneous preferences. A maximum likelihood ratio test is used to verify
the null hypothesis that 0
1 2 3
? ? ? = = = against the alternative hypothesis that at
least one of those parameters is different from zero. I obtain a value for the
maximum likelihood ratio test of 112.23 that leads me to strongly reject the null of
homogeneity on preferences.
Proper implementation of that test for heterogeneous preferences also requires
controlling for incomplete markets (Dubois, 2001). The above specification for
i
?
and equation (4.2.11) make clear that I am testing for differences in the farmers’
stochastic discount factors by allowing them to be a function of farmers’
characteristics. Those characteristics enter in the specification of the discount factor
in equation (4.2.12) through their possible relationship with farmers’ risk aversion.
Nevertheless, as I discussed on Chapter 4, Section 2, differences in reservation values
for a water right among farmers in the same water association may also differ due to
incomplete asset markets. Because it is well known that farmers in the Limarí Valley
face incomplete asset markets, testing for the effect of farmers characteristics on the
stochastic discount factor requires controlling for incomplete asset markets. One way
to do this is by allowing consumption to be a function of current income, which is a
clear consequence of incomplete asset markets. That approach is used in this
dissertation to control for incomplete asset markets. .
115
Estimates for the deterministic component of the production function
summarized in Table 6.2, show that the three inputs under analysis have a positive
effect on mean output. This result is as expected. In terms of output variance, the
three inputs have a positive marginal effect on yield variability. The result that
fertilizer has a positive marginal effect on production variance corroborates similar
findings by Love and Buccola (1991) and Just and Pope (1979). The positive effect
of labor on yield variability coincides with the result obtained by Di Falco, Chavas
and Smale (2006) for a sample of farmers from highlands of Ethiopia.
The finding of water as an increasing risk input is an unexpected result.
Usually, irrigation is considered as a risk-reducing input. As an example, irrigation
reduces the effect of frosts on some type of crops and so their yield variance. It may
be possible that the positive sign for water in the stochastic part of the production is
being caused because crops with higher water requirement are also the ones with
higher variance. Then, estimating the effect of water in output risk will require
controlling for the latter relationship. That can be done by estimating different
production functions for each specific crop type.
The signs of the estimates of the equation for risk aversion indicate that better
educated farmers and with more experience are less risk-averse. The result that more
educated farmers are less risk averse is consistent with the results obtained by Knight
et al. (2003) with household data from rural Ethiopia. A possible explication for that
relationship between risk aversion and education is that more educated farmers are
better able to manage risk and so they are willing to take more risk. Nevertheless,
that result contradicts the works of Bar-Shira et al. (1997) and Ajetomobi et al.
116
(2006), which find that higher levels of education are associated with greater risk
aversion.
The finding that farmers with higher level of experience exhibit a lower
degree of risk aversion confirms the result obtained by Z Bar-Shira et al. (1997).
Those authors explain that result by pointing out that risk is a complicated factor that
less-experienced farmers try to avoid.
Regarding the household size it can be hypothesized that the larger the size of
the family, the higher the subsistence consumption needs and given a fixed amount of
land, the lower the willingness of the farmer to assume risks. On the other hand,
family size might affect the labor capacity of the peasant household in which case a
larger family size implies greater capacity to assume risks. Furthermore, larger
households may diversify their activities and better insure themselves efficiently
reducing risk. Thus, those farmers will be lees reluctant to accept a bargain with an
uncertain payoff rather than another bargain with more certain but possibly lower
expected payoff. That makes them less risk-averse or more risk tolerant
45
. The result
that household size does not affect risk aversion suggests the convenience of
separating those two mentioned effects of that variable over risk aversion by
including a variable that represents farmer’s diversification and another one for the
farmer’s number of children in the specification for risk aversion of equation (6.2).
For the variance of the error term,
it
? , in the linear equation for “other net
incomes”, I
it
, I found that whether or not farmers invest in cattle does not affect the
45
The inverse of a person's risk aversion is sometimes called his risk tolerance
(Wikipedia, the free encyclopedia)
117
variance of I
it
. Furthermore, results indicated as it was expected that farmers with
access to credit for consumption or production have higher variance in I
it
.
Nevertheless, these results are not robust to different specifications of the variance of
it
? neither to variations in the number of observations in the sample.
The estimated value for the discount factor, ? =0.93, belongs to the expected
range for this parameter, 0 < ? <1. Nevertheless a value of 0.93 for the discount
factor seems to be too high to be credible and indicates that the model is not suitable
for the estimation of that discount factor. This occurs because I have an identification
problem with the discount factor: ? appears in the Euler Equation as the constant
term, which is capturing not only the value of the discount factor but also a possible
non zero mean of the error term as well as other constant terms of that equation.
Finally, the residuals for each equation based on the estimates in Table 6.2 are
plotted in Figure 6.3.
118
-4
-3
-2
-1
0
1
Euler
Water Labor Fertilizer
Figure 6.3: Plot of the residuals from the FIML regression based on estimates in
Table 6.2.
119
Appendix: A Hausman test for endogeneity
The error term in the Euler Equation (5.1.14),
1 1 it
?
+
, is a composite error:
1 1 1 1 1 it it it jt
? ? ? ? ? + +
+ + + +
. Thus for each t+1 period,
1 1 it
?
+
is the sum of an
unobserved effect,
1 it
?
+
, and two random errors
1 it
?
+
and
1 jt
?
+
. As I discussed in
Chapter 5,
1 it
?
+
represents a measure of the volatility of the discounted water right
prices. Because that unobserved effect, maximum likelihood estimation of equation
(5.1.14) may not be consistent due to endogeneity issues. This would occur if some
of the exogenous variables in equation (5.1.14) are correlated with
1 it
?
+
and hence
with the error term
1 1 it
?
+
. For example, if water right prices in t+1 were correlated
with the volatility of the discounted water right price in t+1, the FIML estimate of the
parameters in the model would be biased due to endogeneity. A similar situation may
also arise with respect to input quantities that affect output variance which may affect
the volatility of the stochastic discount factor. Thus, the potential presence of
endogeneity must be tested.
As in Di Falco, Chavas and Smale (2006) I use a residual-based form of the
Hausman test that turns to be asymptotically equivalent to the original form of the
Hausman test (Wooldridge, 2002, Chapter 6.2). The test involves estimating
auxiliary reduced-form regressions for the regressors suspected to be endogenous.
Those are linear regressions for each regressor suspected to be endogenous on a
constant, all the exogenous variables of the model and regressor specific instruments.
Then the Euler Equation is estimated including the reduced-form residuals as
additional explanatory variables. The joint statistical significance of the coefficients
120
associated with the residuals is then evaluated. If those parameters are jointly not
significant then the Hausman test does not reject the hypothesis of exogeneity of the
regressors. As Wooldridge (2002, Chapter 6.2.1) point outs, valid test for the
individual and the joint significance of those parameters associated with the residuals
requires an efficient estimation of the Euler Equation.
This test was implemented for those exogenous variables that are suspected of
being correlated with the volatility of the discounted water right prices. Table 6.3
shows the list of instruments that I use for each possible endogenous regressor.
Among those instruments I include all the exogenous variables in the system of
equations that are not correlated with the error term. Column 1 of that table contains
the list of variables that were tested to determine if they are statistically correlated
with the error term
1 1 it
?
+
.
Table 6.3: List of instruments to test for possible endogenous regressors.
Possible endogenous
regressors Instruments
1 1 1
s N v
jt it jt
? ?
+ + +
1 1 1
s N v
jt it jt
? ?
? ? ?
, ( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
, , , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
( )
1 1
N N
jt it it
? ?
+ +
( )
1 2 1
N N
jt it it
? ?
? ? ?
, ( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
, , , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
1
L
it +
( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
,
, , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
1
w
it +
( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
,
, , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
1
F
it +
( )
1
N N
it it
?
?
, T
it
,
1
T
it +
, P
it
,
1
P
it +
,
1
s
jt +
, s
jt
,
, , ,
1 1 1 2 2 1
r r r r
it it it it + +
, v
jt
, Educ.
,
Exp., Household size.
121
A maximum likelihood ratio test indicates that the coefficients on the reduced-
form residuals of the equations for the variables in column 1 of the above table were
jointly not statistically different from zero at a 5% of significance level.
122
Chapter 7: Conclusions and Suggestions for Future Research
In this dissertation I have shown that heterogeneous risk preferences among
farmers is a sufficient condition for water rights transfers when farmers can
simultaneously exchange water in a spot market with lower transaction costs. To
reach that goal I developed a model for water right reservation values in which water
rights are investment assets and where the link between the spot market and market
for water rights is explicitly considered. The model encompasses several aspects
related to water market activity and a farmer’s behavior. I described and measured
transaction activity in markets for water rights and in a spot market for water
volumes, in an existing market since 1981 in the Limarí Basin, Chile. That analysis
allowed me to conclude that both markets are reallocating a significant amount of
water among farmers, although the spot market has, by far, the highest amount of
traded water. I also showed that, contrary to what other researchers believe, the spot
market is active not only during drought years but also in years with average water
availability. I characterized optimal decision making by farmers over the number of
water rights to be held in each season. The model assumes incomplete asset markets,
output uncertainty, as well as uncertainty about future water availability and water
prices. Because investment decisions affect future levels of consumption and farmers
face uncertainty, the theoretical model for farmer decisions was modeled as a
stochastic dynamic problem. This results in a consumption-based capital asset
pricing model (CCAPM) which is described by an Euler Equation that ties asset
returns (water right returns in this case) to marginal rates of substitution for
consumption at different points in time. This model implies that the current period
123
reservation value of a water right is a function of the current value of the amount of
water accorded to water rights in the spot market, the expected future water rights
prices and the stochastic discount factor. Nevertheless, since most transfers of water
rights take place among farmers that belong to the same WUA and such farmers are
likely to have identical expectations, the primary basis for differences in reservation
values and for water right transactions are the differences in their stochastic discount
factors. Incomplete asset markets as well as heterogeneous risk preferences cause
differences in the stochastic discount factors among farmers.
Because asset markets are not complete, farmers value water rights not only as
a source of water for production but also as a means to insure themselves against bad
shocks. As a consequence, the future value of a water right is given by the expected
discounted marginal rate of return to holding a water right from time t to time t+1
plus a risk premium that is greater than zero. The latter implies that the reservation
value of a water right for the more risk-averse farmer is greater than that for the less
risk-averse farmer. This produces transfers of water rights from those farmers who
are least risk averse to the most risk-averse farmers. This approach also emphasizes
that water right transactions solve differences in attitudes towards risk among
farmers, whereas differences in water marginal return among farmers are solved in
the spot market.
The theoretical analysis provides the foundation for a case study of water
transfers for irrigation in the Limarí Basin, an important agricultural region in the
northern part of Chile, which has one of the most active Chilean irrigation water
markets. With micro level data from that basin I estimate a system of equations that
124
describes farmers’ optimal decisions over the number of water right to be held and
input quantities. The estimation of that system assumes that a farmer’s utility from
consumption is represented by a negative exponential utility function, and that the
production technology is described by a Just-Pope Cobb-Douglas production
function. The use of a negative exponential utility function imposes severe
restrictions in the model and the results are conditional to that specific functional
form. Nevertheless, that utility function allows me to characterize the absolute risk
aversion coefficient for each farmer as a function of his observable characteristics and
to develop a promising approach to jointly estimate the parameters that describe
farmers´ preferences and production technology considering farmers investment
decisions. This approach can be extended to more general utility functions although
that will require more advance methods of estimation.
The results of the estimation procedure indicate that the hypothesis of
heterogeneous risk preferences among farmers can not be rejected. Moreover, better
educated farmers and with more experience are less risk-averse. On the production
side, water, labor and fertilizers have positive impact on output mean per hectare.
The analysis of inputs on yield variability showed that those inputs are risk
increasing.
Up to now, research on agriculture finance has been characterized by the
dominance of real estate among the farmers’ assets. But now, due to the increasing
interest in establishing transferable water rights not married to land rights, research on
agriculture finance should move from considering land as the main asset to water
rights as a dominant asset in dry areas. Due to the special characteristics of water
125
resources, this new challenge offers a significant opportunity for future research.
That future research may include the analysis of the robustness of the conclusions
regarding farmers’ heterogeneous risk preferences to alternative functional forms of
the utility function. A suitable candidate is a linex utility function (Bell and Fishburn,
2001), consisting of a utility function that is the sum of an exponential function and a
linear function. This function has the important property of decreasing absolute risk
aversion while retaining some of the convenience of the exponential form. Moreover,
that functional form allows a closed form solution to the farmer decision problem that
I analyze in this dissertation.
Another possible extension would be to develop alternative ways of dealing
with the missing data problems that might allow me to use more of the data that I
have collected among farmers in the Limarí Valley.
The analysis of the role of risk differences, due to different types of crops or
distance from the reservoirs, on the reservation values for a water right is another
interesting extension of this work. The presence of speculative bubbles in water right
market prices, as suggested by Person and Michelsen (1994), is an appealing area for
future applied research on water price models. Other future research deals with
improving the mechanism by which prices are formed in the permanent market. This
could be done by the design of the right incentives to motivate farmers to reveal their
private information on the reservation values
46
. The existence of this non-disclosed
private information can reduce the number of transactions even when the reservation
value of the buyer is greater than that of the seller, and this may impede the efficient
46
Private information in reservation values for water rights includes information
about the farmer’s inter-seasonal discount rate and his attitude towards risk.
126
allocation of water. The formation of an options market in which farmers may obtain
options to purchase water during a dry year could be quite useful in addressing this
problem because an options market reveals the differences in attitude towards risk.
One of the most interesting problems in the formation of an options market is the
creation of an appropriate incentive framework for farmers who participate in such a
market.
127
Annex: Survey instrument
Date of Survey _________________Name of interviewee ________________________________
Association that provides you with water _________________
1. Interviewee’s home (either the tenant’s or the landlord’s)
Information about the tenant or landlord:
1.1 Age ______ Marital Status _______________
1.2 How many people live in your home? ________
1.3 Please give detailed information about each person living in your home
N° Kinship with
interviewee
¿Does
he/she work
in the farm?
Current
Age
Experience
in
agriculture
Level of
Education
achieved
Interviewee yes
Wife
Son/Daughter
Son/Daughter
Son/Daughter
Son/Daughter
2. Property and management of the land and water
2.1 How many lots do you own in the valley? _________ (No)
128
2.2 Please describe your lots?
Lot N° Location Area No. of water
shares
Name of the
Canal
Total
2.3 Quality and use of the farms
Lot
N°
Fertility* Slope* Erosion* Niter* Sown in
97/98
Sown in
98/99
Fertility* (1) high fertility (2) low fertility (3) poor quality of land
Slope* (1) flat (2) hillside (3) hill
Erosion* (1) no problem (2) some problem (3) serious problem
2.4. Do you rent either part or all of your property to other persons? Yes ___ No____
Since when/for how long? _____
How many hectares? _________
How much do you charge per year? _________
With how many water shares? _______
2.5 Do you rent any land properties from other persons? Yes ____ No______
Since when/for how long? ______
How many hectares? _________
How much do you pay per year? _________
With how many water shares? _______
129
2.6 Do you work part or the whole of the land with any partners? Yes ____ No______.
How many partners? __________ Are they next of kin (relatives)? Yes ___ No____
How many hectares?_________ Since when/for how long?_________
Please describe the type of contract made with your partner(s) (i.e. crops grown, land distribution, water, labor and
machinery each party supplies, etc.) and how you finance costs and production. ________________________
2.7 How did you purchase the farm?
Kind of Purchase Area
(hectares)
When did you
purchase it?
How much was a
hectare?
First purchase
Second purchase
Third purchase
Inherited
Due to agrarian
reform
Other
Total (verify)
2.8 Have you sold or divided your property so far? Yes___ No___
(If so, please fill in the chart below)
Type of operation Area
(Hectares)
When? How much did
you ask for each
hectare?
First sale
Second sale
Third sale
First partition
Second partition
Third partition
Other
Total
130
2.9 How did you purchase your water shares and how much did you pay for them?
Type of
acquisition
No. of
water
shares
When did
you buy
them?
With the
land?
How much did you
pay for each water
share?
Whom did
you buy it
from?
First buy
Second buy
Third buy
Inherited
Due to agrarian
reform
Other
2.10 Have you sold or distributed/divided part of you shares so far? Yes___ No___
(If so, please fill in the chart below)
Type of sale No. of water
shares
When did
you sell
them?
With the
land?
What were
you paid for
each water
share?
Whom did
you sell it to?
First sale
Second sale
Partitions
Other
Total
3 Agricultural Production over the Past Two Seasons
3.1. How much land did you sow in the 98/99 season?_________
3.1.1. How much land did you sow with partners? _________
3.2. How much land did you sow in the 97/98 season? _________
3.2.1. How much land did you sow with partners? ___________
131
3.3 What did you sow in the 97/98 and 98/99 seasons?
Sowing Hectares Were there any
losses due to
drought?
97/98 98/99 97/98 98/99
3.4 What was your harvest production in the two seasons?
Crop Total Production Harvested
Hectares
Unit of measure
(sacks, boxes, kilos,
etc.)
97/98 98/99 97/98 98/99 97/98 98/99
3.5 What were you paid for your products in the two seasons?
Name of Product Price per unit in
Pesos $
Weight unit
97/98 98/99 97/98 98/99
132
3.6 What sort of irrigation systems did you use for each crop in the 98/99 season? (Flooding, furrows, drip, etc)
Crop Irrigated
area
Type of irrigation system
used in the crop
3.7 What sort of irrigation system did you use for each crop in the 97/98 season? (Flooding, furrows, drip, etc)
Crop Irrigated
area
Type of irrigation system
used in the crop
3.8 Do you have grapevines? Yes___ No____
If so, please describe each of them
Grapevine
N°
N° de
hectares
N° of
shrubs
Years Type of grape Yield in
98/99
Regular
Yield
133
3.9 Do you have any other permanent crops? Yes____, No_____
(If so, please describe them)
Crop N° de
hectares
N° of
shrubs
Years Type of
product
Yield in
98/99
Regular
Yield
4. Use of Labor and Water
4.1 Do you have permanent workers in your farm? Yes____ No____
If so, how many? _______
4.2 How much labor did you use for each crop in the following activities over the past two seasons?
Grape 1 type) ________________No. hectares_______ No. shrubs________
Activity Season 97/98 Season 98/99
Time
span
Days No. of
people.
Salary Time
span
Days No. of
people.
Salary
Pruning/tying
Applications
Watering
Clearing
Harvest
Other
134
Grape 2 type) _____________No. hectares_______ No. shrubs___________
Activity Season 97/98 Season 98/99
Time
span
Days No. of
people
.
Salary Time
span
Days No. of
people.
Salary
Pruning/tying
Applications
Watering
Clearing
Harvest
Other
Crop No. 1_________________
Activity Season 97/98 Season 98/99
Time span Days No. of people. Salary Time span Days No. of people. Salary
Preparation of the land
Seedbed
Transplant
Seed Sowing
Applications
Watering/irrigation
Clearing
Harvest
Other
135
Crop N° 2_________________
Activity Season 97/98 Season 98/99
Time span Days No. of people. Salary Time span Days No. of people. Salary
Preparation of the land
Seedbed
Transplant
Seed Sowing
Applications
Watering/irrigation
Clearing
Harvest
Other
Crop N° 3________________
Activity Season 97/98 Season 98/99
Time span Days No. of
people.
Salary Time span Days No. of people. Salary
Preparation of the land
Seedbed
Transplant
Seed Sowing
Applications
Watering/irrigation
Clearing
Harvest
Other
136
4.3 Did you and your family take part in the pruning? Yes___ No____
4.4 Did you and your family take part in the sowings? Yes ___ No____
4.5 Did you and your family take part in the harvest? Yes ___ No____
4.6 Did your permanent workers take part in the pruning? Yes___ No____
4.7 Did your permanent workers take part in the sowings? Yes ___ No___
4.8 Did your permanent workers take part in the harvest? Yes ___ No___
4.9 How much do you pay your permanent workers? _______________
137
5 Use of Inputs and Other Expenses
5.1 Please state the amount/number and cost of inputs in the agricultural production in the last two seasons.
Grape Harvest 1_____________ hectares______ N° shrubs________
Input Amount.
97/98
Unit Total cost Amount
98/99
Unit Total cost
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or days
Freight
Grape Harvest 2_____________ hectares______ N° shrubs________
Input Amount.
97/98
Unit Total
cost
Amount
98/99
Unit Total
cost
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or
Freight
138
Crop 1_____________ hectares______
Input Amount
97/98
Unit Total cost Amount
98/99
Unit Total cost
Seeds
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or days
Freight
Crop 2_____________hectares______________
Input Amount
97/98
Unit Total cost Amount
98/99
Unit Total cost
Seeds
Fertilizers
Pesticide/insecticide/fungicide
Pumping M/hr or days
Freight
139
5.2 Did you rent in a tractor in the last two seasons? Yes ___ No ___
If so, answer the questions below
Grape Harvest
97/98 98/99
M/Hr Price M/Hr Price
Preparation of land ______ ______ _____ _____
Application ______ ______ _____ _____
Crop ______ ______ _____ _____
Other Harvests
97/98 98/99
M/Hr Price M/Hr Price
Preparation of land ______ ______ _____ _____
Application ______ ______ _____ _____
Crop ______ ______ _____ _____
If not, please answer
Did you use your own tractor? Yes ____ No _____
5.3 How did you use your own tractor (or tractors) in your own land in the last to seasons?
Grape Harvest
97/98 98/99
M/Hr M/Hr
Preparation of land ______ _____
Application ______ _____
Crop ______ _____
140
Other Harvests
97/98 98/99
M/Hr M/Hr
Preparation of land ______ _____
Application ______ _____
Crop ______ _____
5.4 Did you rent your tractor(s) to other farmers? Yes___ No____
If so, for how many hours _______ At what price? _______
5.5 Did you hire an accountant in the last two seasons? Yes_____, No_____
If so, how much did you pay him/her ______________
5.6 Did you buy water in the last two seasons? Yes ____ No _____
If so, please answer:
Amount purchased 97/98 _________ Price _______ ($ per m3)
Date of purchase __________
Name and address of salesman__________
How many salesmen did you deal with? __________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
Why did you buy? _____________
Amount purchased 98/99 _________ Price _______ ($ per m3)
Date of purchase __________
Name and address of salesman __________
How many salesmen did you deal with? _____________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
Why did you buy? _____________
141
5.7 Did you sell water in the last two seasons? Yes ____ No _____
If so, please answer:
Amount sold 97/98 _________ Price _______ ($ per m3)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer?_______________
Why did you sell?_____________
Amount sold 98/99 _________ Price _______ ($ per m3)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer? _______________
Why did you sell? _____________
5.8 Did you buy water shares in the last two seasons? Yes ____ No _____
If so, please answer the questions below
No. of shares bought 97/98 _________Price _______ ($ per share)
Date of purchase __________
Name and address of salesman __________
How many salesmen did you deal with?______________________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
How much time passed since you decided to buy shares until you got hold of them?
Why did you buy?_____________
142
No. of shares bought 98/99 _________ Price _______ ($ per share)
Date of purchase __________
Name and address of salesman __________
How many salesmen did you deal with? ______________________
Did you use a middleman to negotiate the purchase? ___________
How did you contact the salesman? _________________
How much time passed since you decided to buy shares until you got hold of them? ___________
Why did you buy? _________________
5.9 Did you sell water shares in the last two seasons? Yes ____ No ________
No. Of shares sold in 97/98 _________Price _______ ($ per share)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer? _______________
How much time passed since you decided to sell your shares until you transferred them?______________
Why did you sell? _____________
No. Of shares sold in 98/99 _________ Price _______ ($ per share)
Date of sale __________
Name and address of buyer __________
How many buyers did you deal with? _____________
Did you use a middleman to negotiate the sale? ___________
How did you contact the buyer? _______________
How much time passed since you decided to sell your shares until you transferred them? ______________
Why did you sell? _____________
5.10 Did you rent in water shares in the last two seasons? Yes ____ No _____
If so, please answer the questions below
Number of water shares leased in 97/98 _________Price _______ ($ per share)
143
Date of leasing __________
Duration of the leasing contract___________
Name and address of the renter__________
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the renter _________________
Why did you rent in water shares? _____________
Number of water shares leased in 98/99 _________Price _______ ($ per share)
Date of leasing __________
Duration of the leasing contract___________
Name and address of the renter__________
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the renter _________________
Why did you rent in water shares? _____________
5.11 Did you rent out water shares in the last two seasons? Yes ____ No _____
If so, please answer these questions below
Number of water shares leased in 97/98 _________ Price _______ ($ per share)
Date of leasing__________
Duration of the leasing contract___________
Name and address of the lessee__________
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the lessee__________?
Why did you leasing water shares? _________________
Number of water shares leased in 98/99_________ Price _______ ($ per share)
Date of leasing__________
Duration of the leasing contract___________
Name and address of the lessee__________
144
How many people did you negotiate the leasing with? __________
Did you use any middlemen to negotiate the leasing? __________
How did you contact the lessee__________?
Why did you rent out water shares?
6. Husbandry production in the last two seasons
6.1 Do you have livestock? Yes ____ , No_____ (If not, skip to the next section, if so, please describe your current stock
of cattle)
Kind Number Breed Approx. Total value
in Pesos $
6.1.1 How much livestock have you had each year?
1999_______________
1998_______________
1997_______________
6.2 Have you sold livestock in the last two years? Yes ___, No____
(If so, please describe your sales)
Sales Units When? Price per unit
1
st
sale
2
nd
sale
3
rd
sale
145
6.3 Have you bought livestock in the last two years? Yes ___, No____
(If so, please describe your purchases)
Buys Units When? Price per unit
1
st
buy
2
nd
buy
3
rd
buy
6.4 Dou you produce milk or cheese for sale? Yes ____, No_____
(If so, what was your average productivity per animal over the last three years?)
Productivity in 1999 _____________ Price of milk in 1999 __________
Productivity in 1998 _____________ Price of milk in 1998 __________
Productivity in 1997 _____________ Price of milk in 1997 __________
6.5 What are the main expenses (in pesos) per animal monthly or yearly for the husbandry production of your farm?
Expense per animal Monthly or Yearly
Feeding costs _____________ ________
Healthcare costs _____________ ________
Labor _____________ ________
Other expenses _____________ ________
6.6 Have you lost animals because of drought? Yes ___, No____
How many? ___
When? ___________
146
7. Productive tools
7.1 What equipment of your own do you have for the agricultural production of your farm?
Type of equipment
Units Years Old Equipment description
Tractors
Animals for labor
Fumigation and application
Equipment of the harvest
Production Transportation
Other equipment
7.2 Do you have wells and pumping equipment? Yes ____ No _____(if so, what are their characteristics ?)
Well No. Depth
(meters)
Age Pump
No.
Pumping
capacity
Age
7.3 How many hours have you used your pump (or pumps) to drain water from your well during each season?
97/98 __________ (Hours) 98/99 ___________ (Hours)
7.4 Do you have any storing system for surface water? (docks, tanks, etc.)?
Yes____ No _____ Type _______________
Age ___________.Covered area (m2) ________
Depth (meters) ___________.Capacity (m3) ___________
147
7.5 Do you have mechanized irrigation in your farm? Yes ____ No____
If so, please fill in the chart below
Irrigation
Implements
Years Detailed description
7.6 Do you have a contract with an export company or are you a member of a Pisco association? Yes ____, No ____
If so, please describe the contract:
7.6.1 Name of the company ____________________
7.6.2 Length of contract to date ____________________
7.6.3 Type of payment (monthly, bi-monthly, etc) _____________
7.7 Does the company provide you with one of these services?
7.7.1 Technical support (describe) _____________________________________________
7.7.2 Credit assistance (describe) _______________________________________________
7.7.3 Commercialization (describe) _____________________________________________
7.8 Did you purchase harvest insurance in the 97/98 season?
Whom did you purchase it from? ___________________ How much did you pay for it? _________________
7.9 Did you hire purchase harvest insurance in the 98/99 season?
Whom did you purchase it from? __________________ How much did you pay for it? __________________
148
8 Participation in Lending Markets and Subsidies
8.1 Have you applied for any production loans in the last two seasons?
Yes ___ No____ (If not, skip to question 8.3)
8.2 Have you received any production loans in the last two seasons?
Yes ____No___ (If so, please fill in the following chart)
Source of the loan Loan amount in Pesos $ Interest rate Time span
97/98 98/99 97/98 98/99 97/98 98/99
Commercial Bank
INDAP
Other sources
8.3 Have you applied to be granted a bonus or subsidy bonus to improve the irrigation system in your farm?
Yes ____ No_____ (If not, go to question 8.5)
8.4 Have you obtained any bonus from the government? Yes ___ No ____ (If not, go to question 8.5)
Did you already use the bonus? Yes ____ No____
Have you received or do you plan to receive any bank loans to complement the bonus or subsidy bonus? Yes ___ No____
What type of bonus did you apply for? For entrepreneurs ____, for farmers ____ both _____
Account for the money received in bonus or subsidy bonus ______ (Pesos)
8.5 If you have not applied for a bonus from the government, why is it so?
____________________________________________________________
THANK YOU VERY MUCH.
149
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