Study on Impact of Social Interactions on Economic Outcomes

Description
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect.

ABSTRACT
Title of dissertation: ESSAYS ON THE IMPACT OF
SOCIAL INTERACTIONS ON
ECONOMIC OUTCOMES
Nathalia Perez Rojas
Doctor of Philosophy, 2007
Dissertation directed by: Professor Rachel Kranton
Department of Economics
This dissertation consists of two essays, which address the question of how
social interactions shape economic outcomes. The ?rst essay examines crime and
criminal networks. The second one studies immigration, assimilation, and ethnic
enclaves.
The ?rst essay o?ers a formal model of crime. Criminals often do not act
alone. Rather, they form networks of collaboration. How does law enforcement
a?ect criminal activity and structure of those networks? Using a network game, I
show that increased enforcement actually can lead to sparse networks and thereby
to an increase in criminal activity. When criminal activity requires a certain degree
of specialization, criminals will form sparse networks, which generate the highest
level of crime and are the hardest to disrupt. I also show that heavy surveillance
and large ?nes do not deter crime for these networks.
The second essay examines the impact that residential location decisions have
on economic outcomes of immigrants. About two thirds of the immigrants that
arrived to the United States between 1997 and 2006 settled in six States only. Using
a simultaneous-move game on residential choices I show that when all immigrants are
unskilled they cluster in an enclave and earn very low wages, although they would be
better o? assimilating. Hence the enclave is ‘trap’. Introducing skill heterogeneity
among immigrants reverses the result: the enclave equilibrium becomes socially
preferred to assimilation.
ESSAYS ON THE IMPACT OF SOCIAL INTERACTIONS
ON ECONOMIC OUTCOMES
by
Nathalia Perez Rojas
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial ful?llment
of the requirements for the degree of
Doctor of Philosophy
2007
Advisory Committee:
Professor Rachel Kranton, Chair/Advisor
Professor Larry Ausubel
Professor Peter Reuter
Professor John Shea
Professor Dan Vincent
c Copyright by
Nathalia Perez Rojas
2007
Acknowledgements
I would like to thank my main advisor, Rachel Kranton, for her encouragement
and dedication, and also, for the many productive discussions we had since the
inception of this project. I am indebted to Dan Vincent for his frank and challenging
comments, and for helping me to become more rigorous in my work. To Larry
Ausubel, thank you for your crucial feedback and for your generous advice. I am
very grateful to John Shea, who took the time to read my dissertation and provide
me with very useful (and incredibly detailed) comments. I would also like to thank
Peter Reuter for reading my dissertation with fresh eyes and making it better.
To my family, thanks for being so loving and supportive. Mom and Dad, words
are not enough, thanks for everything. Andres, I could not have asked for a better
companion in this journey. Thank you for your love, understanding and patience.
To my brother, Esteban, thank you for cheering me up in di?cult moments. Your
energy took out the best of me. My deepest gratitude to Carlos Alberto, Juan
Carlos and Luca. Helena, I miss you very much. Thanks to you too.
ii
Table of Contents
List of Figures iv
1 Crime Networks 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Setup of the Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 The Game: Strategic Criminal E?ort Choices . . . . . . . . . . . . . 12
1.4 Decentralized Link Formation . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Pairwise Stability . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Policy Interventions . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Larger Populations of Criminals . . . . . . . . . . . . . . . . . . . . . 25
1.5.1 Equilibrium Crime in Large Populations . . . . . . . . . . . . 26
1.5.2 Pairwise Stable Networks in Large Populations . . . . . . . . . 27
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Immigration, Assimilation and Ethnic Enclaves 32
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 A Game of Residential Clustering . . . . . . . . . . . . . . . . . . . . 35
2.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 The Game: Residential Location Choice . . . . . . . . . . . . 40
2.3 A Game with Residential and Entrepreneurial Clustering . . . . . . . 44
2.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.2 The Game: Residential and Workplace Decisions . . . . . . . 50
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
A Proofs of Propositions of Chapter 1 58
B Proofs of Propositions of Chapter 2 63
Bibliography 72
iii
List of Figures
1.1 Networks with three criminals . . . . . . . . . . . . . . . . . . . . . . 6
1.2 PWS networks for a ?xed ?ne f = 150 . . . . . . . . . . . . . . . . . 23
iv
Chapter 1
Crime Networks
1.1 Introduction
Criminals often do not act alone. Rather, they form networks of collaboration.
This chapter examines the impact of law enforcement on those networks and on the
resulting level of crime. I de?ne a network as a group of criminals and the pattern
of communication links between them. Moreover, I refer to a sparse or di?used
network as one that connects a given number of criminals with the fewest links. I
?nd that sparsely connected networks generate the most crime and are the hardest to
dismantle. Also, within a di?used network, criminals who establish communication
links with the fewest other agents exert the highest level of crime e?ort. Sparse
networks generate the most crime, and even heavy surveillance and large ?nes do
not a?ect their shape or their level of criminal activity.
Criminal networks participate in a wide range of illegal activities, such as
drug tra?cking, arms smuggling, and terrorism (Naim [22]). Many hierarchical
Ma?a-like organizations have shifted their structure towards networks of loosely
aligned criminals (Williams [33]). For example, the Colombian cocaine trade, long
dominated by the cartels of Medell´?n and Cali, is now run by independent and spe-
cialized tra?cking organizations in Colombia, Mexico, and the U.S.
1
Decentralized
1
For a description of the process of cocaine smuggling into the U.S., see INCSR reports of the
1
crime networks are typically sparsely connected and very ?exible. These features
allow them to be less visible and to change their structure constantly, which makes
it di?cult for law enforcement to dismantle them (Reuter [25], Sageman [28] and
Williams [32]).
This chapter shows that, for a given number of criminals, those networks that
connect (directly or indirectly) all criminals with the fewest number of links yield the
highest level of crime. Within such sparsely connected network, criminals that have
few links are less visible to the police through connections, and supply high criminal
e?ort. Well-connected criminals, in contrast, exert low e?ort, instead providing
connections among distant agents. In networks that are densely connected, all
criminals face large penalties because of their large number of links, and thus they
optimally supply low e?ort.
To demonstrate these results, I analyze a simultaneous-move game in which
criminals in a given network independently select the amount of criminal e?ort to
exert. The basic model has three criminals; I compare the levels of crime generated
by all possible con?gurations of the network.
2
I make the following assumptions:
First, no criminal can undertake illicit activity by himself: all agents need at least one
link to participate and to exert e?ort. Second, links allow criminals to coordinate on
their e?orts and thus to derive spillover bene?ts. These e?ort spillovers are stronger
US Department of State [9]. On the cost associated with each phase of the cocaine smuggling
process see Reuter [26]. Fuentes [14] has a very detailed description of the Colombian cocaine
‘distribution’ cells operating in the US during the 1990s.
2
In section 1.5 I extend the model to more than three criminals.
2
between pairs of criminals who are more closely connected (in terms of path length).
Finally, the penalties from engaging in criminal activity increase with both the links
a criminal has and the amount of e?ort he supplies.
I show that if the e?ort spillovers are not too strong, then the static game has a
unique Nash Equilibrium. Suppose that well-connected criminals face a signi?cantly
larger marginal cost of e?ort than do poorly-connected criminals. Then, within a
di?used network, well-connected criminals will supply less e?ort than criminals with
fewer links. When I compare crime levels across networks I ?nd two features that
lead to the most crime: connectedness and sparseness of the network. Networks
that connect all criminals with the fewest possible links lead to the most crime.
Given that sparse networks generate the most crime, under what conditions do
they form? To address this question, I extend the model and allow criminals to form
links, and then to select the level of e?ort to exert. I ?nd that di?used networks
are likely to emerge and can a?ord to pay large ?nes and face heavy surveillance by
the police. Further, when spillovers are su?ciently strong, large penalties do not
a?ect the structure of di?used networks or their level of crime.
This study contributes to the literature on the economic theory of networks,
and the economics of crime. Previous works by Calv´ o-Armengol and Zenou [6] and
Ballester, Calv´o-Armengol and Zenou [2] studied the e?ect of social networks on
criminal behavior. The ?rst paper shows that the decision among otherwise iden-
tical agents to get involved in crime depends on each agent’s position in the social
network. In the second paper, the authors develop a measure of centrality for
3
each player (Bonacich measure) and ask how this measure a?ects the individual
choice of criminal e?ort. These papers are in line with the extensive literature on
the economics of crime, which focuses both at the theoretical and empirical levels
on the incentives for agents to engage in criminal activities.
3
I depart from this
approach and assume that all agents are criminals, and study instead the decisions
that agents make on who to communicate with and how much crime e?ort to sup-
ply. This study is also related to the theoretical literature on drug markets, which
analyzes the e?ect that law enforcement policies have on agents situated at di?erent
levels of the drug production and distribution chain, and ultimately on the number
of consumers in the streets (see for example Poret [23] and Chiu et al. [7]).
Adding to the previous literature, my work assumes that agents in the network
are criminals, who must collaborate with each other in the illicit activity. In my
model criminal e?orts of all agents that are linked directly or indirectly are strategic
complements. Each criminal derives bene?ts from the collaboration through e?ort
spillovers. In Calv´o-Armengol and Zenou [6] and Ballester, Calv´ o-Armengol and
Zenou [2] there are both local (direct neighbor) complementarities and global sub-
stitutabilities in criminal e?ort. The complementarities re?ect peer e?ects, while
the substitutabilities re?ect the competition for the booty among criminals who are
not connected directly.
By expanding our understanding of crime, this study can inform law enforce-
ment policy. The resiliency of decentralized crime networks requires crime ?ghting
3
For a review of the literature on the economics of crime, see Freeman [12] and DiIulio [10].
4
policies that di?er from those targeted towards hierarchical organizations. In the
context of decentralized networks, changes in the penalties will a?ect the level of
crime and, more importantly, the structure. As the structure of a crime network
changes, law enforcement policies that were e?ective in the past may be useless. I
show that when e?ort complementarities are su?ciently strong, tougher penalties
do not discourage criminal behavior. Policies that target criminal activity and links
among criminals, rather than those links alone, are more e?ective in reducing crime.
The rest of the chapter is organized as follows. In Section 1.2, I describe
a static game in which, for a given network, criminals strategically select criminal
e?ort. In Section 1.3, I solve for the Nash Equilibrium of the game, and compare
equilibrium outcomes across all networks with three criminals. In Section 1.4, I
specify a two-stage game: ?rst criminals form links, and then they choose criminal
e?ort. I look for networks that are pairwise stable; i.e. ones in which no criminal
has the incentive to sever a link, and no unlinked pair wants to form a link (Jackson
and Wolinsky [17]). In Section 1.5, I extend the analysis to larger networks and
consider a dynamic process of network formation. Section 2.4 concludes.
1.2 Setup of the Game
There are three criminals. Denote the set of criminals by N = {1, 2, 3} and
index each agent by i = 1, 2, 3. A network g is the collection of communication
links between criminals (or nodes) that belong to N. Links allow criminals to
communicate and thereby coordinate on their crime e?orts. A communication link
5
2 2 2 2
1 1 1 1 3 3 3 3
a. b. c. d.
Figure 1.1: Networks with three criminals
between agents i and j where i, j ? N is represented by g
ij
= 1. If i and j are not
directly connected then g
ij
= 0. I normalize g
ii
= 0.
Given three criminals, there are four possible network con?gurations. The ?rst
network is the empty network, where all criminals are isolated (Figure 1.1a). The
second network is the Single-Link Network, which has only two criminals linked.
This network is described by g
I
12
= 1, and g
I
13
= g
I
23
= 0 (Figure 1.1b). The third
network is a Star represented by g
S
with g
S
12
= 1, g
S
23
= 1 and g
S
13
= 0. Here only one
criminal –the center of the Star– is directly linked to the other two nodes (Figure
1.1c). The last structure is the Complete network that has each agent connected
to every other agent (g
C
12
= 1, g
C
13
= 1 and g
C
23
= 1, Figure 1.1d).
I assume that agents are homogeneous and that the value of links only depends
on the network structure, not on the identity of agents. For example, a Star network
with g
S
12
= 1, g
S
23
= 1 and g
S
13
= 0 generates the same value as one with g
S
12
= 1,
g
S
13
= 1 and g
S
23
= 0.
The total number of links that agent i has in network g equals N
g
i
=

j?N
g
ij
.
Let the vector N
g
= [N
g
1
, N
g
2
, N
g
3
] represent the pro?le of the number of links each
6
criminal has.
To measure the distance between agents, let s
g
i
= (s
i1
, s
i2
, s
i3
) for i = 1, 2, 3.
Each element s
it
of the vector s
g
i
corresponds to the inverse of the shortest distance
in network g between agents i and t. As a convention s
ii
= 0. If i and t are
not connected (directly or indirectly) then s
it
= 0. The magnitude of s
it
depends
on the link pattern of the network and in particular, on the links that agent i has
within it. Let s
g
= [s
g
1
; s
g
2
; s
g
3
] be a symmetric matrix in which the it-th element
corresponds to s
it
.
Given a network g, criminals strategically select how much e?ort to exert.
Denote the criminal e?ort of agent i by e
g
i
.
4
Let e
g
= (e
g
1
, e
g
2
, e
g
3
), represent the
pro?le of e?orts of all criminals in network g. De?ne the level of criminal activity
of a network as the sum of e?orts of all of its members. Crime and crime e?ort are
used interchangeably throughout.
The payo? to a criminal depends on his level of e?ort (e
g
i
), his links (N
g
i
), other
criminals’ e?orts (e
g
j
), proximity to them (s
ij
), and two law enforcement parameters.
The law enforcement parameters are a ?ne (f), and an intensity of law enforcement
(µ). For example, µ can be the probability that law enforcement will put under
surveillance any criminal.
4
As a convention, subscripts refer to nodes or criminals (i), while superscripts refer to networks
(g).
7
Given a network g, the payo? to criminal i equals:
Y
g
i
= B(e
g
i
, N
g
i
) +

j=i,j?N
K(e
g
i
, e
g
j
, s
ij
; ?) ??(e
g
i
, N
g
i
; µ, f) (1.1)
Agents can participate in the criminal activity only if they are connected, i.e.
if N
g
i
= 0 then e
g
i
= 0, and Y
g
i
= 0. More precisely, I normalize to zero the payo?
of a criminal that has no links. Thus the e?ort that a linked criminal supplies is
interpreted as the additional crime e?ort driven by the gains from coordination and
communication with other criminals.
The ?rst term in (1.1) is the private bene?t derived from own links and own
e?ort. Even if all other criminals in the network exert minimal or no e?ort, agent
i still bene?ts from his connections (B(e
i
, N
g
i
) > 0 even if ?j ? N, e
g
j
= 0). B(.)
is increasing in all of its terms and is weakly concave in e?ort:
?B
?e
g
i
> 0 ,
?B
?N
g
i
> 0
and
?
2
B
?(e
g
i
)
2
? 0. And, having more links makes own e?ort more productive. Thus
criminals with more links derive a larger marginal bene?t of e?ort:
?
2
B
?N
g
i
?e
g
i
> 0.
The second term in (1.1) is the bene?t derived from e?ort spillovers.

j=i,j?N
K(e
g
i
, e
g
j
, s
ij
; ?) is increasing in own e?ort (e
g
i
), other criminals’ e?orts
(e
g
j
), proximity to them (s
ij
) and the strength of spillovers (?):
?K
?e
g
i
> 0,
?K
?e
g
j
> 0,
?K
?s
ij
> 0,
?K
??
> 0. Further spillovers are weakly concave in own e?ort:
?
2
K
?(e
g
i
)
2
? 0.
The most important assumption of the model is that e
i
and e
j
are strategic com-
plements. The strength of these complementarities is measured by the parameter
? > 0, i.e.
?
2
K(.)
?e
i
?e
j
= h(?, .) > 0 and
?h
??
> 0. If each node has a very particular
skill or knowledge that vastly enhances the value of the criminal e?ort put in by all
8
others in the network, then ? is large. When ? is small e?ort complementarities
are weak and the gains from collaboration are small. Moreover, stronger spillovers
increase the marginal bene?t of e?ort:
?
2
K(.)
???e
i
> 0.
The last term in (1.1), ?(e
g
i
, N
g
i
; µ, f), is the cost of engaging in a criminal
activity. The law enforcement parameter µ a?ects the likelihood that a criminal
will get caught. A criminal who gets caught by the police must pay an exogenous
?ne f. Thus I interpret the function ?(e
g
i
, N
g
i
; µ, f) as the ?ne payment. The cost
of engaging in criminal activity is increasing in e
g
i
, N
g
i
, and in the law enforcement
parameters (µ and f):
??
?e
g
i
> 0,
??
?N
g
i
> 0,
??
?µ
> 0 and
??
?f
> 0. Further the ?ne
payment is strictly convex in e
g
i
, i.e.
?
2
?
?(e
g
i
)
2
> 0. Larger penalties increase the cost
of additional e?ort:
?
2
?
?e
g
i
?f
> 0 and
?
2
?
?e
g
i
?µ
> 0. Even at the margin, increases in the
?ne ( f) and in the intensity of law enforcement (µ) act as criminal deterrents by
raising the marginal cost of criminal e?ort. And, heavier penalties are more costly
to criminals with more links:
?
2
?
?N
g
i
?f
> 0 and
?
2
?
?N
g
i
?µ
> 0.
I am interested in examining how law enforcement policies shape criminal
e?ort choices in a given network. The ?ne (f) and the probability of surveillance
(µ) a?ect Nash Equilibrium e?orts through ?(e
g
i
, N
g
i
; µ, f), the cost of being part
of a crime network. I assume that criminals with more links ?nd it more costly
to marginally increase their e?ort relative to poorly connected criminals:
?
2
?
?e
g
i
?N
g
i
>
0. For any given e?ort level, having more links increases the likelihood of being
captured. Within a network, well connected criminals are more visible to the police
because of these links. This visibility gives them the incentive to supply less e?ort
9
than sparsely connected agents. For example, if all criminals in the Star network
exert the same level of e?ort, then increasing it marginally is more costly for the
center of the Star than for the corners.
Each criminal in the network faces a tradeo? between the bene?t from e?ort
coordination and the cost associated to getting caught. A well-connected crimi-
nal coordinates on e?orts with several other criminals, and thereby derives larger
spillover bene?ts than a poorly connected agent does. However, holding the e?ort
level ?xed, a well-connected criminal is also more likely get caught through links
than a criminal with fewer connections.
To motivate the model, consider the following situation. Suppose that the
process of drug smuggling consists of three phases: processing the coca leaf into
cocaine, smuggling the cocaine into the foreign country (e.g. the U.S.) and ?nally,
distributing and retailing it. Now assume that each of these activities is undertaken
by a di?erent agent within the crime network. The ?rst agent is the producer
or Colombian drug-lord, the second agent is the smuggler, and the third is the
distributor or dealer. The criminal e?ort of the drug-lord includes such activities
as growing the coca and then re?ning it to produce the cocaine. Hence e
g
i
refers to
e?ort put into the criminal activity itself, and it excludes any action to avoid being
captured by the police.
The intuition behind the assumptions of the model are as follows for this ex-
ample: Criminals must have at least one link to participate in the criminal activity.
Therefore, the drug-lord needs connections either with the smuggler or the distrib-
10
utor or both in order to derive a non-zero bene?t from his economic activity. Recall
that the crime e?ort of an isolated criminal is normalized to zero. For example,
when the drug-lord has no connections he goes to the spot market and has an anony-
mous transaction with a smuggler. In such a transaction there are no bene?ts from
e?ort spillovers or from connections. Therefore the value of that transaction is
no larger than that of a transaction in which the drug-lord and the smuggler agree
on the packaging and the delivery time of the cocaine. Further, the more links a
criminal has, the larger his marginal bene?t from e?ort. Suppose that the smuggler
marginally increases the amount of cocaine brought illegally into the U.S. Then his
marginal bene?t is larger when he is connected to two dealers than one.
The key assumption of the model is e?ort spillovers. In this scenario one
example would be: The Colombian drug-lord makes an R&D investment that allows
him to process better quality/high-purity cocaine at a low cost. This improvement
in quality gives the distributor/dealer the incentive to search for customers who
are willing to pay a premium for the high-purity cocaine. The dealer responds
to the increased R&D e?ort of the drug-lord by supplying more e?ort. These
marginal increases in e?ort become more productive if the tra?ckers use electronic
communications that are encrypted and thus very secure (? increases). Then it
becomes harder for the police to tap into their communications. Such a change
gives criminals the incentive to collaborate more closely with each other (i.e. a
marginal increase in ? raises the marginal bene?t of e
g
i
).
11
1.3 The Game: Strategic Criminal E?ort Choices
I specify a simultaneous-move game as follows. For a given network g, each
criminal i selects e
g
i
to maximize his own payo?. For a given network, criminals
play a simultaneous move game in criminal e?orts. I compare Nash Equilibrium
(NE) e?orts across all network con?gurations shown in Figure 1.1.
Denote a pro?le of Nash Equilibrium e?orts by e
g?
= (e
g?
1
, e
g?
2
, e
g?
3
).
From (1.1) the payo? to criminal i in network g is:
Y
g
i
= B(e
i
, N
g
i
) +

j=i,j?N
K(e
i
, e
j
, s
ij
; ?) ??(e
i
, N
g
i
; µ, f) (1.2)
The pro?le of NE e?orts (e
?
) is determined by the set of ?rst order conditions given
by:
?Y
g
i
?e
i
=
?B(e
i
, N
g
i
)
?e
i
+

j=i,j?N
?K(e
i
, e
?
j
, s
ij
; ?)
?e
i
?
??(e
i
, N
g
i
; µ, f)
?e
i
(1.3)
= 0?i ? N
Given that the strategy sets are one-dimensional (e?ort levels), and that pay-
o?s are continuous and concave in e?ort, the following condition guarantees that a
unique Nash Equilibrium pro?le of e?orts (e
?
) exists:
¸
¸
¸
¸
¸
?
2
B
? (e
i
)
2
+

j=i,j?N
?
2
K
? (e
i
)
2
?
?
2
?
? (e
i
)
2
¸
¸
¸
¸
¸
>

j=i,j?N
?
2
K
?e
i
?e
j
?i ? N (1.4)
When this inequality is satis?ed, Best Response functions are contraction mappings,
and the system of ?rst-order conditions given by (1.3) has a unique solution (Fried-
man [13] and Vives [30]). A property of this Nash Equilibrium is symmetry: agents
12
in identical structural positions in the network adopt the same strategy. For ex-
ample, at the NE, players in the corners of the Star (Figure 1.1c) exert the same
amount of criminal e?ort. The inequalities in (1.4) suggest that the Best Response
of criminal i changes proportionately more with a marginal increase in own e?ort
(e
g
i
) than with a similar increase in other criminals’ e?ort (e
g
j
). Conditions in (1.4)
hold if the network e?ects (?) are not too strong, and if the cost of engaging in crim-
inal activity is su?ciently convex in own e?ort. When the e?ort complementarities
are very strong, then an individual might choose either to supply almost no e?ort
at all, given that any small and positive amount of e?ort is extremely productive or
to supply very high e?ort that feeds back through larger e?orts of all others in the
network. Thus if ? is too large, then the game can have multiple equilibria.
Let ¯ ? be the largest ? for which a unique Nash Equilibrium exists. The
following analysis applies for all ? ? ¯ ?.
The ?rst result of the model is that within a network criminals with few links
exert more e?ort than those with more links. Well-connected criminals are more
likely to be captured because of their links and they choose to supply low e?ort. In
contrast, criminals with few links are less visible through the links and can exert
more e?ort. Within a network, when the penalties for engaging in criminal activity
depend on both e
g
i
and N
g
i
criminals who have few links supply more e?ort than
well-connected agents. The next proposition formalizes the result.
Proposition 1.1. Let ? ? ¯ ?. Suppose that
?
2
?
?e
g
i
?N
g
i
> 0 is su?ciently large. Then
13
within a network, sparsely connected criminals exert more e?ort than those with
more links.
Proof. All proofs are in Appendix A.
More precisely the assumption of large
?
2
?
?e
g
i
?N
g
i
in Proposition 1.1 requires that
?
2
?
?e
g
i
?N
g
i
>
?
2
B
?e
g
i
?N
g
i
. This assumption suggests that a criminal who is well connected
faces a higher marginal cost of e?ort and a lower marginal bene?t relative to an agent
that is poorly connected. If two criminals who di?er in their number of links face the
same marginal e?ort cost (i.e.
?
2
?
?e
g
i
?N
g
i
= 0), then it would be more pro?table for the
well-connected criminal to increase his e?ort relative to the poorly connected agent
(the e?ort spillover bene?ts of the latter are lower). Furthermore, the well-connected
criminal, by exerting marginally more e?ort, leads to stronger e?ort spillovers that
feed back to everyone else in the network. This feed back translates into more crime
than that which would result from a poorly-connected agent increasing his e?ort.
If instead, the marginal cost of e?ort increases with the number of connections,
say because a ‘tax’ is imposed on each link, then a well-connected criminal has the
incentive to supply less e?ort than a poorly-connected individual (see the Star in
Figure 1.1c). The ‘tax’ on links leads to a decentralization of crime e?ort in the
network: criminals that are sparsely connected supply more e?ort than criminals
with more links. In my model the penalties play the role of the ‘tax’.
The system of ?rst order conditions given by (1.3) yield Nash Equilibrium
e?orts of the form e
g?
i
= e
i
(N
g
, s
g
; ?, µ, f) for all i ? N in g. These NE e?orts
14
are increasing in the strength of spillover e?ects (?) , and are decreasing in the
law enforcement parameters (µ and f).
5
Stronger spillovers increase the marginal
bene?t of e?ort and lead to higher e
g?
i
. In contrast, tougher ?nes (f) or better
surveillance technology (µ) make criminal e?ort more costly to exert and lead to
smaller e
g?
i
.
For example, in the Star network in Figure 1.1c players 1 and 3 have only one
link, while player 2 has two links (N
S
2
> N
S
1
= N
S
3
). Further, agents 1 and 3 are
symmetric: each is a step away from player 2 and two steps away from the other.
Then the NE e?orts of the Star network are e
?
center
< e
?
corner
for corner = 1, 3.
Criminals that are sparsely connected select higher e?ort levels than well connected
criminals.
Proposition 1.1 illustrates the role of asymmetries between spillovers and in-
dividual e?ort costs in shaping optimal e?ort choices. Links let criminals maximize
the bene?t of e?ort spillovers. But having more links increases the chance of get-
ting caught. Less-connected criminals have a lower probability of being captured
through links, and derive lower spillover bene?ts due to indirect connections. Given
the strong e?ect of links on the marginal cost of e?ort and the large penalties, less-
connected criminals supply more e?ort than their counterparts.
For a ?xed level of e?ort, the direct bene?ts (B(e
i
, N
g
i
)) from participating
in the network are larger for criminals with more links. Similarly, spillovers are
5
These properties are derived using the assumptions on the cross-partial derivatives and the
concavity of Y
g
i
on e
g
i
, and by totally di?erentiating the system of ?rst-order conditions (1.3).
15
greater for well-connected criminals: they tend to have higher s
ij
’s that make

j=i,j?N
K(e
i
, e
j
, s
ij
; ?) large. Meanwhile, the cost of engaging in criminal activity
is increasing in both e
i
and N
g
i
. For a given e?ort level, having more links increases
the probability of getting caught and paying a ?ne. Given the strong complemen-
tarities between e
g
i
and N
g
i
in ?(.) and the tough penalties for crime, well-connected
criminals will supply less criminal e?ort than their counterparts in equilibrium.
Large penalties linked to e
g
i
and N
g
i
drive well-connected criminals to supply
little e?ort, and instead channel spillovers among otherwise distant nodes. Those
that exert more e?ort communicate or have links with few criminals. Thus, better
connected nodes are not the most dangerous (in terms of crime e?ort level).
I now derive the central result: that sparse networks generate the most crime.
Sparse networks that (directly or indirectly) connect all criminals with the fewest
links yield the highest level of crime. In a sparse network, criminals with few
links can exert high e?ort. Their high e?ort will feed back to the well-connected
criminals through spillovers and will lead them to increase their own e?ort as well.
In contrast, in a densely-connected network all criminals face a high probability of
getting caught because of the links and optimally supply low e?ort.
Proposition 1.2. Let ? ? ¯ ?. Suppose that B(.) and ?(.) are homogenous of degree
one in links. Then sparse networks connecting all agents induce the highest (NE)
level of crime: e
S?
> e
C?
> e
I?
where e
g?
=

i?N
e
g?
i
. Further e
?
corner
> e
?
center
>
e
C?
i
? e
I?
i
.
16
Sparser networks motivate criminals to supply more e?ort than they would
in more densely connected networks with the same number of nodes.
6
In di?used
networks, criminals at the periphery exert the most e?ort; this feeds back through
spillovers to agents who are densely connected, and induces them to supply high
e?ort. Thus e
?
center
> e
C?
i
, and consequently, e
S?
> e
C?
.
The second result of proposition 1.2 (e
C?
i
? e
I?
i
) follows from the homogeneity
of B(.) and ?(.) in links and from the spillover bene?ts. The homogeneity assump-
tion implies that if criminal i has one link and criminal j has two links, then B(.)
and ?(.) are twice as large for criminal j as for criminal i. In both the single-link
and the Complete network, connected criminals are only one step away from each
other; thus s
ij
= s = 1 for all connected criminals. Given that all players in the
Complete network have identical positions, I can use this symmetry to calculate the
e?ort spillovers for criminal i as 2K(e
i
, e
j
, s, ?). Similarly, the spillover bene?t of
a connected player in the Single-link network is K(e
i
, e
j
, s, ?). Then the payo?s to
criminals in the Complete network are an increasing monotonic transformation of
the payo?s to connected players in the single-link network. Hence e
I?
i
= e
C?
i
for
N
I
i
> 0. The gains from spillovers and private bene?ts in the complete network
relative to the single-link network (B
C
+ K
C
= 2(B
I
+ K
I
)) fully o?set the higher
costs of links (?
C
= 2?
I
).
If, contrary to the assumptions above, the costs of participating in a crime
6
This result requires the connected components of both networks to have the same number of
nodes. A connected component is a set of nodes that are linked, either directly or indirectly.
17
network increase in own e?ort only while the bene?ts increase both in e?ort and
links, then well-connected criminals would choose to exert the most e?ort. Conse-
quently, more densely connected networks would turn out to be the most dangerous
(i.e. with the highest e
g?
=

i?N
e
g?
i
).
1.4 Decentralized Link Formation
If sparse networks are associated with the most crime, when should we expect
these networks to emerge? To answer this question, let us now suppose that pairs
of criminals must agree to form links. While two criminals agree to a link, either
one can sever it unilaterally. I extend the basic model and specify a two-stage game.
Fix (?, f, µ) and let ? ? ¯ ?. First criminals form links.
7
A network emerges and is
publicly observed. Then, given the network structure, criminals strategically select
levels of criminal e?ort . I solve the game using ‘backward induction.’ Given the
NE e?orts of the second stage of the game, criminal i in network g anticipates in
the ?rst stage a payo? equal to Y
g
i
(N
g
, s
g
; ?, µ, f) = Y
g
i
(e
g?
i
, N
g
i
; ?, µ, f). I look for
networks that are pairwise stable. This equilibrium concept is developed by Jackson
and Wolinsky [17]. A network is pairwise stable if no pair of unlinked agents agree
to a new link and if no agent wants to unilaterally sever a link.
7
I assume that the cost of forming a link is zero (i.e. c = 0). The results do not change
signi?cantly if I let c > 0.
18
1.4.1 Pairwise Stability
Start with network g. Suppose that previously unlinked criminals i and j add
a link to g. Denote the new structure as g +ij. Let g ?ij be the resulting network
when the existing link g
ij
is removed (i.e. g
ij
= 0.)
A network g is pairwise stable (PWS) if:
(1) ?g
ij
= 1, Y
g
i
(N
g
, s
g
; .) ? Y
g?ij
i
(N
g?ij
, s
g?ij
; .) and
Y
g
j
(N
g
, s
g
; .) ? Y
g?ij
j
(N
g?ij
, s
g?ij
; .); and
(2) ?g
ij
= 0, if Y
g+ij
i
(N
g+ij
, s
g+ij
; .) > Y
g
i
(N
g
, s
g
; .) then Y
g+ij
j
(N
g+ij
, s
g+ij
; .) <
Y
g
j
(N
g
, s
g
; .).
The ?rst condition says that no agent wants to sever a link in g. The second
condition says that no pair of agents gain by forming a new link. Criminals will
form or sever links only if they can earn a larger payo? with the deviation. Pairwise
stability allows at most two criminals to coordinate on forming a link. Thus, the
link formation process is decentralized. For example, the Star network is PWS
if: 1) criminals at the corners optimally do not form a link (i.e. Y
corner
(N
S
, s
S
; .) >
Y
C
i
(N
C
, s
C
; .)); 2) the player at the center optimally does not sever either of his links
(Y
center
(N
S
, s
S
; ?, µ, f) > Y
I
connected
(N
I
, s
I
; .)); and 3) no corner optimally severs his
link (Y
corner
(N
S
, s
S
; .) > 0).
I apply PWS to the link formation game.
Let the ?ne payment be ?(e
g?
i
, N
g
i
; µ, f) = ?(e
g?
i
, N
g
i
; f) +?(N
g
i
, µ, f). Under
this speci?cation criminals get caught either because they exert high criminal e?ort
19
and thereby increase their visibility to the police (?(e
g?
i
, N
g
i
; f)), or because they
are put under surveillance (?(N
g
i
, µ, f)). When a criminal supplies high e?ort, and
this e?ort gets him caught, he pays a ?ne equal to ?(e
g?
i
, N
g
i
; f) . To illustrate the
intuition of this penalty consider the following example: Suppose that the amount
of e?ort an agent puts into committing a crime is an increasing function of the
fraction of time he spends on criminal activity. The higher the e?ort, the more
time an agent spends on criminal activity, the more likely the police will observe
and capture him (i.e. ?(e
g?
i
, N
g
i
; f) is increasing in e
g?
i
). Moreover, for any given
e?ort level, the cost of exerting e?ort is larger for criminals with more links because
they are more visible to the police (i.e. ?(e
g?
i
, N
g
i
; f) is also increasing in N
g?
i
). A
criminal also can get caught if he is put under surveillance, in which case he pays a
?ne equal to ?(N
g
i
, µ, f). We can imagine that the probability of a criminal being
put under surveillance is increasing in the number of links he has (i.e. ?(N
g
i
, µ, f) is
increasing in N
g
i
). In this situation, the police need not observe a criminal engaging
in crime in order to ?ne him. Links are enough to punish a criminal who is under
surveillance. For example, the surveillance could consist of the police tapping into
the communications of a criminal. Once the police intercept the communications of
a criminal, he is captured and pays a ?ne accordingly.
8
8
A more intuitive speci?cation is ?(e
g
i
, N
g
i
; µ, f) = ?(e
g
i
)?(N
g
i
, µ)f . Here a criminal is captured
only if, conditional on being under surveillance, he is observed in criminal activity. Suppose that
the network is put under surveillance with some exogenous probability µ ? (0, 1). Then the
probability that criminal i is put under surveillance is ?(N
g
i
, µ) ? (0, 1), with ?(.) increasing in all
of its arguments. Let ?(e
g
i
) ? (0, 1) be the probability that the police observe a criminal while
committing a crime. Hence a criminal gets caught with probability ?(e
g
i
)?(N
g
i
, µ).
20
For a given network g and its corresponding NE crime e?orts, criminal i an-
ticipates a payo? in the ?rst period equal to :
Y
g
i
(N
g
, s
g
; ?, µ, f) = B(e
g?
i
, N
g
i
) +

j=i,j?N
K(e
g?
i
, e
g?
j
, s
ij
; ?) ??(e
g?
i
, N
g
i
; µ, f) (1.5)
= B(e
g?
i
, N
g
i
) +

j=i,j?N
K(e
g?
i
, e
g?
j
, s
ij
; ?) ? (1.6)
[?(e
g?
i
, N
g
i
; f) + ?(N
g
i
, µ, f)]
When ?(e
g?
i
, N
g
i
; µ, f) = ?(e
g?
i
, N
g
i
; f) + ?(N
g
i
, µ, f) NE crime e?orts in (1.5) are
independent of the intensity of law enforcement (µ): e
g?
i
= e
g
i
(N
g
, s
g
; ?, f).
9
High µ
discourages link formation without a?ecting e?ort choices. µ can be interpreted as
the ability of law enforcement to tap into the communications of the criminals.
For more precise results, I use a speci?c functional form. Consider the follow-
ing payo? function:
Y
g
i
= N
g
i
e
g
i
+ ?s
ij
e
g
i
e
g
j
+ ?s
ik
e
g
i
e
g
k
?
N
g
i
¯
N
(e
g
i
)
2
2
f ?
N
g
i
¯
N
µf (1.7)
for i = j = k and i, j, k = 1, 2, 3.
¯
N = 6 is the largest possible number of links
in a network with three players. Let e
g
i
? (0, 1). This function is a particular
representation of (1.5). The ?rst term (N
g
i
e
g
i
) is the private bene?t of own links
(N
g
i
) and e?ort (e
g
i
). The second and third terms are the bene?ts from spillovers.
The last two terms,
N
g
i
¯
N
(e
g
i
)
2
2
f +
N
g
i
¯
N
µf are the cost of engaging in criminal behavior.
Here, the probability of surveillance is independent of the amount of e?ort a criminal
9
It turns out that in the speci?cation of the ?ne payment in the previous footnote NE e?orts
are functions of both µ and f , which make the analysis of PWS networks less tractable than with
the speci?cation in (1.5).
21
exerts. The higher the e?ort of a criminal, the more likely he is to get caught.
Moreover for a given e?ort level, criminals with more links are more likely to be
captured (
N
g
i
¯
N
(e
g
i
)
2
2
f increases with N
g
i
and e
g
i
). Additionally, criminals with more
links are more likely to be put under surveillance and to get caught (
N
g
i
¯
N
µf increases
with N
g
i
).
With the functional form in (1.7) I can characterize all PWS networks using
an algorithm that I construct in Matlab.
10
I ?nd that if the penalties for engaging
in criminal activity (µ and f) are large enough to drive the payo?s of all networks
with at least one link negative, then the only PWS network is the empty network.
Let the range of ?nes (f) be such that there are some values of (?, µ) for which
the payo?s in networks with links are non-negative. I now derive the third main
result. Sparse networks with a high degree of specialization lead to the highest
level of crime and are the hardest to dismantle. They can sustain even very large
?nes and heavy surveillance. If ? is su?ciently large, then the Star network forms
regardless of the level of f and µ.
Proposition 1.3. Fix the ?ne f and let ? ? ¯ ?.
11
If spillovers are su?ciently strong
(? close to ¯ ?), then di?used networks can sustain heavy surveillance by the police
and large ?nes. These features make sparse networks hard to disrupt. Di?used
networks have the fewest possible number of links and connect (directly or indirectly)
all criminals. These networks decrease the probability and the cost of getting caught,
10
The Matlab code is available upon request.
11
I ?x f = 150.
22
EMPTY
NETWORK
STAR /
EMPTY
1
w

e
n
f
o
r
c
e
m
e
n
t

(
?
)

NETWORK
COMPLETE
STAR
?
low
?
high
Effort complementarities (?)
?
low
?
high
0
I
n
t
e
n
s
i
t
y

o
f

t
h
e

l
a
w
Figure 1.2: PWS networks for a ?xed ?ne f = 150
and achieve the highest level of criminal activity.
Under the assumptions of Proposition 1.2 the single-link network is never
PWS (Figure 1.2). At least one of the conditions required for PWS is never met.
12
Criminals either prefer to belong to a network with a larger connected component
or not to be connected at all. For some combinations of (?, µ), the Star and the
Empty networks are mutually PWS.
Regardless of the level of surveillance (µ) and for some ?xed ?ne f, if e?ort
12
The single-link network is PWS if: 1) no connected criminal wants to sever his link
(Y
I
connected
(N
I
, s
I
; ?, µ) > 0) and 2) the isolated criminal and a connected one don’t agree to
a link (Y
I
connected
(N
I
, s
I
; ?, µ) > Y
center
(N
S
, s
S
; ?, µ) and Y
corner
(N
S
, s
S
; ?, µ) < 0).
23
complementarities are strong (i.e. ? close to ¯ ?), then criminals form the Star net-
work. When ? is large and close to ¯ ?, all criminals have an incentive to connect
with the fewest links possible: a high ? leads to large NE crime e?orts, and high
levels of e?ort increase the cost of engaging in criminal activity. Anticipating high
e?ort in the second stage, criminals will choose to form a di?used network in the
?rst stage. This choice translates into the highest possible level of crime.
1.4.2 Policy Interventions
How do changes in law enforcement policies a?ect the network structure and
its level of crime?
Using the functional form in (1.7), I ?x the strength of the spillovers (?) and
analyze how changes in the intensity of law enforcement (µ) alters the network
structure.
13
Let µ
g
belong to the set of surveillance probabilities in which network
g is PWS. If spillovers are weak, then law enforcement policies targeted towards
very densely connected networks lead to more crime.
Set ? = ?
low
as in Figure 1.2. For such ?
low
, the Complete, the Star and
the Empty network are all PWS for some range of µ. Law enforcement intensities
for which each of these networks is PWS can be ranked as follows: µ
C
< µ
S
?
µ
empty
. From Proposition 1.2 e
C?
< e
S?
and e
S?
> 0. Thus, for low values of ?,
13
In the future I would like to look at how varying the ?ne (f) a?ects crime. This case is slightly
more complicated than that in which f is ?xed and µ varies. When f changes it a?ects not only
the network structure but also NE crime e?orts.
24
increases in the surveillance activity can be counterproductive. If the change in the
surveillance probability is not large enough to make some criminals drop out, then
the resulting network yields more crime. Increases in the penalties for engaging in
criminal activity can increase crime. As the complete network becomes sparser, the
probability of getting caught through links decreases for some criminals, and overall
criminal activity goes up.
1.5 Larger Populations of Criminals
Using the speci?c functional form of the payo?s in (1.7), I analyze in this
section the behavior of large populations of criminals. Now there are |N| criminals
for N = {1, 2, 3, ...}. The largest possible number of links in a network with |N|
criminals is
¯
N = |N| ?(|N| ?1). Following (1.7) the payo? to criminal i in network
g is:
Y
g
i
= N
g
i
e
g
i
+ ?

j=i,j?N
s
ij
e
g
i
e
g
j
?
N
g
i
¯
N
(e
g
i
)
2
2
f ?
N
g
i
¯
N
µf (1.8)
To obtain the following results I construct an algorithm for large populations
of criminals using Matlab. I solve the game as follows. I start at the second-stage:
given a network structure g, I solve for the NE criminal e?orts, which maximize
(1.8) for all i ? N. Then I turn to the ?rst-stage of the game and ?nd the networks
that are PWS.
25
1.5.1 Equilibrium Crime in Large Populations
Within a large network, criminals with fewer links exert more e?ort than those
with more links. Note that B = N
g
i
e
g
i
and ? =
N
g
i
¯
N
(e
g
i
)
2
2
f +
N
g
i
¯
N
µf are homogeneous
of degree one in links. Then from Proposition 1.2 it follows that the NE crime
e?orts of a Complete network with |N| criminals are described by e
C?
i
. And the
aggregate crime level of a Complete network is e
C?
= |N| e
C?
i
, which is increasing in
|N|.
When networks with more than three criminals are considered, there can be
several connected components. Intuitively, a connected component is a group of
nodes that are linked to each other either directly or indirectly. Whether i and j
belong to the same connected component can be seen by looking at s
ij
: if s
ij
> 0
then i and j are in the same connected component. If s
ij
= 0, then i and j are not
connected (directly or indirectly).
Networks sparser than the Complete, and such that ?i ? N, N
g
i
> 0, yield at
least the same level of aggregate crime as the Complete network. Suppose there
exists some large network g that has at least two connected components. If each of
the components is maximally connected, – i.e. if within a component each criminal
can reach every other criminal in just one step– then the level of criminal activity
of this network is identical to that of a Complete network with the same number of
agents. For example, suppose that g

is a Complete network with four criminals.
Let g

be such that it only has two links, g

12
= 1 and g

34
= 1. Then the NE crime
e?orts of g

and g

coincide.
26
Fix |N| > 3. Suppose that there are two networks g

and g

and that both
networks have a single connected component linking all criminals. Using the algo-
rithm in Matlab for larger populations of criminals, I ?nd that if g

can be obtained
by cutting links from g

and if g

and g

di?er signi?cantly on their link density,
then the sparser structure g

leads to more crime.
1.5.2 Pairwise Stable Networks in Large Populations
After calculating the NE e?orts for a given network structure, I turn to the
?rst stage and ask which network is likely to emerge. I ?x f and let (?, µ) vary.
14
Using the existence results of Jackson and Watts [16] I know that for any (? , µ)
there exists at least a PWS network or a closed cycle of networks. In the case of
three criminals, for any pair (? , µ) there always exists a PWS network and there
are no closed cycles. With larger populations of criminals, I can ?nd combinations
of (? , µ) for which no PWS network exists. The absence of a PWS network for
such pairs of (? , µ) raises the possibility of having cycling networks in these areas
of the parameter space.
Consider the following dynamic process of link formation proposed by Jackson
and Watts [16]: A set of N criminals form network g. In each period t a pair (i, j)
is selected with some positive probability p
ij
where

ij
p
ij
= 1. Criminals i and
j either can form a link, resulting in network g

= g + ij, or each can sever the
link g
ij
= 1 so that g

= g ? ij. In every period, a pair of criminals is randomly
14
Again, set f = 150 and z = 1.
27
selected and decides whether to form a link or to sever an existing link.
15
If a
dynamic process that starts from network g leads with strictly positive probability
to network g

, then an improving path exists from g to g

. A closed cycle C is a
set of networks such that for any g, g

? C there exists an improving path from g to
g

and all networks in the path also belong to C. Networks that are PWS in the
static game are always reached in this dynamic process.
Applying the existence results of Jackson and Watts [16] , for any combination
(?, µ) there is at least a PWS network or a closed cycle. We can ?nd ranges of (?, µ)
in which there are closed cycles and/or PWS networks. And in regions for which
no network is PWS, criminal activity is going on through cycling networks. For
example, let N = {1, 2, 3, 4}. Start at network g with g
12
= g
13
= 1 and g
i4
= 0
for i = 1, 2, 3. A cycle can exist over some range of (?, µ) as follows: player 1
severs link g
13
so that the new network g

has just one link g

12
= 1. Then criminals
3 and 4 connect and g

forms with g

12
= 1 and g

34
= 1. Next, players 2 and 3
form a link and a line results: g

12
= g

23
= g

34
= 1. Finally, players 3 and 4 are
selected and player 3 severs the link g

34
= 1. This leads back to the original network
g
12
= g
13
= 1. Similar examples can be constructed for larger networks.
15
This process is myopic because when pairs of agents are deciding on forming or a severing link,
they do not take into account future decisions of other agents to alter the resulting network g

.
28
1.6 Conclusion
In this chapter of the dissertation I construct a network model that captures
the strategic interactions among criminals who jointly engage in an illicit activity.
This theoretical framework is appropriate for understanding how decentralized crime
networks operate and how they react to changes in law enforcement policies.
I specify a static game in which for a given network, criminals select the level of
e?ort to exert. I solve the game for all possible network con?gurations and compare
the levels of crime generated. My ?rst result is that, within a di?used network,
sparsely connected criminals are the most dangerous. When I compare crime across
networks that di?er in their link density, I get my second result: networks that are
sparsely connected yield the highest level of crime. From a policy perspective,
densely connected networks thus should be preferred to di?used networks.
I then extend the model and allow criminals to form links endogenously. Using
a speci?c functional form, I ?nd that the degree of specialization of a network
determines its resiliency to law enforcement policies. My third result suggests
that sparse networks with a high degree of specialization, or strong spillovers, are
very hard to dismantle. Even large ?nes and heavy surveillance can be ine?ective
in altering their structure or their level of crime.
Finally, I derive results for larger populations of criminals. I observe that e?ort
choices within a larger network resemble those of networks with three criminals.
Extending the result that sparse networks yield more crime to larger populations of
29
criminals requires comparing networks that di?er signi?cantly in their link density.
The model provides intuition on how crime networks operate. In the drug
tra?cking example, once the kingpins of the Colombian cartels were killed or put
behind bars in the early 1990s, new drug-lords started to emerge. In contrast to
the kingpins, these new drug-lords chose to maintain a low pro?le in order to reduce
their visibility to law enforcement. They opted to stay small and to collaborate
instead with criminal organizations in Mexico and the U.S. During the second
half of the 1990s and the beginning of the twenty-?rst century this partnership was
coupled with a steady increase in the total amount of cocaine smuggled into the U.S.
Looser and more decentralized structures raised the volume of the cocaine smuggled.
The collaboration between criminals is not exclusive to the drug-smuggling business,
many illicit activities have led criminals to form decentralized networks (e.g. human
smuggling, arms smuggling, and terrorism).
The analysis o?ered is just a beginning in terms of our understanding of crime
networks. Two avenues are worth exploring in the future. The ?rst is letting
law enforcement be a strategic player in the game. We could imagine a (repeated)
three-stage game proceeding as follows. In the ?rst stage, the police announce the
penalties for engaging in criminal behavior and the intensity of law enforcement.
Then, criminals form links and the network structure is publicly observed. Finally,
in the third-stage, criminals choose e?ort given a network structure. If the degree of
specialization of the network is publicly known, then the police set penalties in the
?rst stage that will lead to the lowest level of crime, and the game will end. If the
30
degree of specialization of the network is not known to the police, then the police
and the criminals will interact repeatedly. Through repeated interaction, the police
will learn the degree of specialization of the network by observing its structure and
its level of crime, and respond by changing the penalties accordingly.
The second avenue to explore relates to the consequences of the strength of
e?ort spillovers within a crime network. I showed that when the e?ort spillovers
are su?ciently strong, sparse networks form and further, they are very hard to
dismantle. This result is relevant to the extent that crime networks are highly
specialized, so that e?ort spillovers are strong. To better understand the role of
specialization, for example, we could allow for heterogeneity in the value of the links
according to the identity (and skills) of each of the criminals. This exercise could
guide crime ?ghting policy.
31
Chapter 2
Immigration, Assimilation and Ethnic Enclaves
2.1 Introduction
Residential clustering by immigrants, i.e. the formation of ethnic enclaves,
is fairly common in the United States: about two thirds of the immigrants that
arrived between 1997 and 2006 settled in six States only (California, New York,
Florida, Texas, New Jersey and Illinois).
1
This chapter examines the impact that
residential location decisions have on economic outcomes of immigrants. I introduce
a simultaneous-move game in which immigrants decide whether to settle among
natives and assimilate or to cluster and form an ethnic enclave. The results of the
model show that the skill mix within the enclave (or the ‘quality’ of the enclave)
shapes the economic outcomes of immigrants. If all immigrants are unskilled and if
an equilibrium exists in which the enclave forms, then it is trap or a bad equilibrium.
In contrast, if both skilled and unskilled immigrants move to the enclave, I ?nd
that the enclave equilibrium is socially preferred to that in which all immigrants
assimilate. And regardless of where unskilled immigrants locate, their wages are
higher if a positive fraction of skilled co-ethnics settle in the enclave.
Previous literature in immigration suggests that the bene?ts and costs of liv-
1
According to the Immigration Statistics of the U.S. Department of Homeland Security. This
calculation takes into account legal residents only.
32
ing in an enclave depend on the quality of the enclave (see for example Borjas [5]
and Edin et al. [11]). In this chapter I propose a model with skill heterogeneity
among immigrants. In the model skilled immigrants who settle the enclave become
entrepreneurs and hire unskilled co-ethnics. This assumption allows me to assess
the quality of an enclave in terms both of the availability of jobs and the value
of the output produced in it. I make the following additional assumptions: ?rst,
there are language complementarities in production (both in the enclave and out of
it); second, immigrants that settle among natives are more likely to assimilate than
those who live in the enclave; third, immigrants that assimilate earn higher wages
in the general labor market (because of stronger language complementarities with
natives). Finally, unskilled immigrants who work in the enclave eventually become
self-employed, and thus gain upward mobility without assimilation.
2
I specify a simultaneous-move game in residential location decisions. First,
immigrants choose where to live. If some skilled people go to the enclave then a
labor market emerges in it; and in that case, immigrants decide whether to work
in the enclave or in the general labor market. I solve for the Nash Equilibria of
the game. I start by studying residential location decisions of a pool of identical
unskilled immigrants. I ?nd, ?rst, if the bene?ts from assimilation (other than
higher wages) are rather small, then unskilled immigrants do not assimilate and
earn very low wages. Second, if native employers cannot tell apart unskilled immi-
grants who assimilate, and if the wages for assimilated unskilled immigrants are not
2
In the enclave workers receive on-the-job training and informal advice from the entrepreneurs,
and eventually start their own businesses.
33
su?ciently large, then all unskilled immigrants cluster in the enclave, which emerges
as a poverty trap.
I then ask whether entrepreneurship in the enclave improves the economic
outcomes of unskilled immigrants who settle in it. The model yields the following
set of results. First, no enclave exists in which there is excess labor demand. Skilled
immigrants move to the enclave only if the supply of labor in it is abundant. One
plausible explanation for this outcome is that the value of assimilation is correlated
with skill. Second, if the enclave ever forms, then it is socially preferred to the
assimilation equilibrium. Once skilled immigrants settle in the enclave, it is no
longer a poverty trap. Hence the quality of the enclave matters when studying
the economic outcomes of immigrants who cluster. Third, ethnic enclaves and
ethnic enterprises improve the economic outcomes not only of immigrants that live
in the enclave, but also of those who live out of it. The demand for labor in the
enclave soaks up part or all of the unskilled (unassimilated) labor supply, and allows
unskilled immigrants that assimilate to earn wages comparable to those of natives in
the general labor market. Finally, better quality enclaves allow unskilled immigrants
to achieve upward mobility faster.
This study contributes to the literature on immigration. By accounting for
the ‘quality’ of an enclave I am able to explain the wide set of experiences of diverse
immigrant groups in the U.S.
3
Empirical studies in the economics of immigration
3
Sociologists have studied extensively the immigrant enclaves in the U.S. See for example Light
and Gold [20] and Portes and Rumbaut [27].
34
have shown that the quality of an enclave matters. For example Edin et al. [11]
examine the economic outcomes of refugee immigrants in Sweden and ?nd that those
who live in enclaves with high rates of self-employment have positive returns from
living there, while people who settle in enclaves with mostly unskilled individuals
experience lower earnings possibly due to clustering itself.
4
Finally Borjas [5] reports
that people who settle in the enclaves and do not acquire the social norms and skills
of the U.S. (i.e. assimilate), have wages growing at a slower pace than that of the
rest of the population.
The rest of the chapter is organized as follows. Section 2.2, describes a
simultaneous-move game in which unskilled immigrants choose where to live. I
solve for the Nash Equilibria in residential location decisions. In Section 2.3, I
incorporate skilled immigrants and introduce a labor market in the enclave. Then
I solve for the Nash equilibria and Pareto-rank them. Section 2.4 concludes.
2.2 A Game of Residential Clustering
2.2.1 Setup
Suppose there exists a continuum of unskilled immigrants with unit measure
that arrive to a large metropolitan area in the U.S. Each person decides indepen-
dently and noncooperatively whether to settle in a neighborhood with his co-ethnics
or in an area where the majority of the population is native. The location decision
4
In their study of the Cuban and Haitian enclaves in Miami Portes and Stepick [24] reach a
similar conclusion.
35
determines the likelihood that a person assimilates: I assume that an immigrant is
more likely to assimilate if he chooses to live among natives than if he settles with
his co-ethnics. The intuition for this assumption is as follows. As in Lazear [19]
we could imagine a situation in which individuals can trade only if they speak the
same language. Immigrants can assimilate in order to expand their pool of poten-
tial trading partners. The incentive to assimilate is stronger for people who live
and work among natives than for those who settle with co-ethnics. I also assume
that immigrants who acquire the skills and speak the language of the host country
earn higher wages in the general labor market. Thus in my model assimilation, or
equivalently living among natives, leads to upward economic mobility. In contrast,
residential clustering of (unskilled) immigrants hampers this process and can lead
to worse economic outcomes; in particular to lower wages (for empirical evidence
see Edin, et al. [11] and Borjas [5]).
De?ne the enclave as the residential neighborhood where unskilled immigrants
cluster, and let 0 ? n
u
? 1 represent the fraction of people who settle in the it.
Suppose that ¯ w
uc
(n
u
) is the wage earned by an immigrant who lives in the enclave.
Let J (n
u
) ? [0, K] be the cost of ?nding a job for an immigrant of the enclave.
Assume that individuals within the enclave share information about potential jobs,
and that an informal network of job contacts emerges. Further, suppose that the
larger the enclave, the ‘thicker’ the network and the lower the cost to an individual
of ?nding a job: J

(n
u
) < 0 and J

(n
u
) > 0 (thus J (0) = K and J (1) = 0 ).
5
I
5
For empirical evidence on the e?ciency of these ethnic networks in channeling job information
see Munshi [21] and Waldinger and Lichter ([31], p. 83, 104-105).
36
assume that immigrants derive utility from sharing common culture. This bene?t
is captured by the function h(n
u
) with h

(n
u
) > 0 and h

(n
u
) < 0. The sign of the
second derivative suggests some crowding e?ect as more people move to the enclave.
For example, if too many people settle in the enclave, it may be harder to get a
spot for the kids in the bilingual school of the neighborhood. Thus the utility of
an unskilled immigrant who settles in the enclave equals:
U
uc
(n
u
) = ¯ w
uc
(n
u
) ?J (n
u
) + h(n
u
) (2.1)
Suppose that an immigrant who goes to a neighborhood where natives are
majority assimilates. Let the costs/bene?ts of assimilation be given by b. b < 0
corresponds to the costs of acquiring the host country skills, or learning the language
and the social norms of natives. In contrast b > 0 represents the bene?ts of learning
the social norms of natives, which for example, might prevent the immigrant from
being discriminated against; b > 0 could also account for the gains derived from
having access to high quality public services (e.g. schools). Suppose that an
immigrant who assimilates faces a job ?nding cost equal to K and receives wage
¯ w
us
(n
u
). Therefore the utility received by an unskilled immigrant who assimilates
is:
U
us
(n
u
) = ¯ w
us
(n
u
) ?K + b (2.2)
I assume that the general labor market works as follows. All immigrants,
assimilated or not, compete for jobs. There are a large number of ?rms hiring
both immigrants and natives. Suppose that there are language complementarities
37
in production: If a worker speaks English his marginal product is MPh, and if he
does not then his marginal product is MPl < MPh.
6
If language ability is fully
observable, then a worker that speaks English earns w
us
= MPh and one that does
not receives w
uc
= MPl. Suppose that the only characteristic of a worker that
is observable to the employers is her ethnicity. Then all employers pay natives
w
us
= MPh. However, the employers cannot tell apart those immigrants who
assimilate from those who do not. The information available to ?rms is that a
fraction n
u
of immigrants lives in the enclave and therefore, do not speak English.
They also know that an immigrant who assimilates reveals to his boss that he speaks
English with some positive probability 0 < ? < 1.
7
The average productivity and therefore the expected wage of immigrants who
do not reveal is:
w = Pr [low|reveal = 0] ? MPl + Pr [high|reveal = 0] ? MPh (2.3)
=
n
u
n
u
+ (1 ?n
u
) (1 ??)
w
uc
+
(1 ?n
u
) (1 ??)
n
u
+ (1 ?n
u
) (1 ??)
w
us
= ¯ w
uc
(n
u
)
and
? wuc
?nu
< 0, the larger the enclave, the lower the fraction of the workers that are
6
In their study of the immigrant labor market in the area of Los Angeles, Waldinger and Lichter
([31], p. 69-72) conclude that the job assignment of an immigrant (and consequently the wage)
depends on her ?uency in the English language.
7
This setup could be thought of as a reduced form model of statistical discrimination (e.g.
Aigner and Cain [1]). All immigrants must take a test that measures imperfectly the likelihood that
a person speaks English. While immigrants who do not assimilate fail the exam with probability
1, those who assimilate pass the exam only with probability ?.
38
assimilated among the pool of individuals who do not reveal, the lower the wage for
the individual who does not assimilate. An immigrant who assimilates expects to
receive wage ¯ w
us
equal to:
¯ w
us
(n
u
) = ?w
us
+ (1 ??) ¯ w
uc
(n
u
) (2.4)
for ¯ w
uc
(n
u
) given by equation (2.3). Notice
? wus
?nu
= (1 ??)
? wuc
?nu
< 0. For a given ?,
the larger the enclave, the lower the pooling wage ¯ w
uc
(n
u
), the lower ¯ w
us
(n
u
). Using
(2.2) and (2.4) it is straightforward to show that
?Uus
?nu
< 0. Not always being able
to di?erentiate from the enclave immigrant in the labor market, the individual that
assimilates faces a negative externality from the enclave in the labor market, which
is larger for higher n
u
(enclave size). Because ¯ w
us
(n
u
) ? w
us
, ¯ w
us
(n
u
) ? ¯ w
uc
(n
u
)
and thus assimilated immigrants receive wages no lower than the wage of a person
living in the enclave.
The utility of immigrants who settle in the enclave is larger the higher is n
u
:
?U
uc
?n
u
=
? ¯ w
uc
?n
u
?J

(n
u
) + h

(n
u
) > 0
? h

(n
u
) ?J

(n
u
) >
¸
¸
¸
¸
? ¯ w
uc
?n
u
¸
¸
¸
¸
In contrast, the utility of someone who assimilates is lower for higher values of n
u
:
?Uus
?nu
=
? wus
?nu
= (1 ??)
? wuc
?nu
< 0. Additionally I assume that
¸
¸
¸
?J
?nu
¸
¸
¸ <
¸
¸
¸
? wuc
?nu
¸
¸
¸ <
¸
¸
¸
?J
?nu
¸
¸
¸ + h

(n
u
).
39
2.2.2 The Game: Residential Location Choice
I specify a simultaneous-move game. Immigrants decide independently and
noncooperatively whether to settle in the enclave or among natives. An enclave n
?
u
is a NE if for such n
?
u
an immigrant’s best response is to settle in the enclave (i.e.
U
uc
(n
?
u
) ? U
us
(n
?
u
)). Given that a fraction n
u
of immigrants go to the enclave, an
individual settles in the enclave if the utility he receives there is larger than the utility
he derives from assimilation (U
uc
(n
u
) ? U
us
(n
u
)), otherwise if U
us
(n
u
) > U
uc
(n
u
) his
best response is to assimilate.
Suppose that no one goes to the enclave (n
u
= 0) then the wages of all im-
migrants are ¯ w
us
(0) = ¯ w
uc
(0) = w
us
. If there are no bene?ts from assimilation
other than high wages, i.e. if b < 0, then all immigrants strictly prefer to settle in
the enclave (U
us
(0) < U
uc
(0)); and in that case living out of the enclave is never
a Nash Equilibrium. When n
u
= 0 and b < 0 an immigrant has the incentive to
unilaterally deviate and settle in the enclave. In doing so he free rides on the high
wages received by his co-ethnics and forgoes the assimilation cost b. In contrast
if assimilation translates not only into higher wages but also into being (socially)
less discriminated against (b > 0), then no enclave forming can be a NE: for n
u
= 0
if b > 0 it is a Best Response to assimilate (and thus n
?
u
= 0). The following
propositions formalize the results.
Proposition 2.1. If b < 0 then the unique NE is the enclave forming (n
?
u
= 1).
Proof. All proofs are in the appendix.
40
Immigrant groups that are likely to be discriminated against, or for whom
b < 0, do not to assimilate, and instead cluster in an ethnic neighborhood at the
expense of earning low wages. As an example consider the Haitian refugees that
arrived to Miami in 1980. They were black and unskilled and chose to cluster in an
ethnic neighborhood. They faced racial discrimination, and ultimately remained
unemployed or held jobs at very low wages (Portes and Stepick [24]).
Proposition 2.2. Let h(1) > b ? K. If b > 0 and w
us
? w
uc
?
h(1)?b+K
?
then the
unique Nash equilibrium is everyone assimilating (n
?
u
= 0).
Suppose everyone is going to the enclave and n
u
= 1. When all immigrants
go to the enclave if the wage di?erential net of search costs ( ¯ w
us
(1) ? ¯ w
uc
(1) ?K)
is su?ciently large to o?set the relative bene?ts of sharing common culture (i.e.
h(1) ? b), then it is individually optimal to assimilate. This incentive is stronger
when assimilated workers are more likely to reveal as such (i.e. when ? high). Larger
? leads to higher wages for assimilated people ( ¯ w
us
) and lower for those who live in
the enclave ( ¯ w
uc
). If at n
u
= 1 the opposite happens, i.e. if the wage di?erential
is no larger than the relative bene?ts of sharing common culture, then both the
enclave forming and everyone assimilating are NE of the game.
Proposition 2.3. Let h(1) > b ? K. If b > 0 and w
us
? w
uc
<
h(1)?b+K
?
then
multiple equilibria exist:
i. no enclave forming is a NE, n
?
u
= 0;
ii. the enclave forming is a NE, n
?
u
= 1 , and
iii. the enclave forming with n
?
u
? (0, 1) is a NE.
41
When the wage di?erential is no larger than the (highest) net bene?ts of culture
(h(1) ? b) then multiple equilibria emerge: If no one goes to the enclave, then an
individual prefers not to go to the enclave. In contrast, when she expects all others
to go to the enclave, then her best response is to settle in the enclave. And there
is an n
u
? (0, 1) where the person is indi?erent between settling in the enclave or
out of it. For such n
u
the wage di?erential is identical to the cultural gains in the
enclave. For given (w
us
, w
uc
, h(1) , b, K) higher ? make it more likely for a group
to assimilate.
According to the model then the existence of ethnic enclaves of unskilled peo-
ple (e.g. Mexicans) in the U.S. is partly driven by the inability of the employers
to tell apart the assimilated immigrants. Lower ? decreases the wages of assimi-
lated immigrants and makes assimilation less attractive.
8
Under what conditions
is assimilation socially preferred to clustering? The next proposition addresses this
question.
Proposition 2.4. Suppose that the conditions in proposition 2.3 are satis?ed. If
h(1) ? b + K < w
us
? w
uc
<
h(1)?b+K
?
then the enclave equilibrium is a ‘trap’.
The equilibrium in which all immigrants spread-out Pareto-dominates the enclave
8
One possible solution to ? being low is for the employers to hire a bilingual supervisor at a
low cost. Although not modeled directly, we could imagine that the interaction with the bilingual
supervisor raises the marginal productivity of all enclave workers. Indeed there is empirical
evidence of sweatshops in the area of Los Angeles, where bilingual supervisors are hired to interact
with Latino workers in order to improve their productivity through more e?ective communication
(Waldinger and Lichter [31] p. 69).
42
equilibrium. If w
us
? w
uc
< h(1) ? b + K, then the enclave equilibrium Pareto-
dominates the assimilation equilibrium.
If assimilation translates into su?ciently high wages, then the enclave equi-
librium is a trap. In contrast if the wage premium from assimilation is not too
large relative to the cultural bene?ts of the enclave, then the assimilation equilib-
rium is the ‘bad’ equilibrium. Thus if the wage gap is in an intermediate range,
immigrants are socially better o? assimilating. However, this equilibrium may fail
to be achieved if there is a lack of coordination among immigrants.
9
For given
assimilation bene?t (b) and job ?nding cost (K), when all immigrants assimilate,
the gains from assimilation are the largest possible because everyone earns the same
wage as natives (w
us
); in contrast, when all individuals go to the enclave wages
are very low (w
uc
), but the bene?ts from common culture are the largest possible
(U
uc
(1) = w
uc
+ h(1)).
From the conditions in propositions (2.2) to (2.4), if employers can readily
assess assimilation (i.e. ? is close to 1) and can reward it properly (i.e. if w
us
?
w
uc
is su?ciently large) then immigrants will assimilate: the region of w
us
? w
uc
for which multiple equilibria exist shrinks as ? goes to 1, and the assimilation
equilibrium is more likely to emerge as the unique equilibrium. If the government
could observe better the ability of immigrants to ‘speak’ English relative to the
employers, for example because it gives a more comprehensive set of exams, then
9
In the next section I introduce skill heterogeneity among immigrants and show that having a
labor market in the enclave improves the outcomes of unskilled immigrants, and furthermore, that
the enclave equilibrium is always the socially preferred equilibrium.
43
residential clustering would be less likely to occur.
2.3 A Game with Residential and Entrepreneurial Clustering
In this section I allow for skill heterogeneity in the immigrant pool. Speci?cally
I consider two types of immigrants: skilled (h) and unskilled (u). Let n
h
denote the
fraction of immigrants who move to the enclave in equilibrium. The presence of
skilled immigrants in the enclave increases the bene?ts of clustering for the unskilled
co-ethnics in two basic ways: ?rst, skilled immigrants who settle in the enclave
become entrepreneurs and create a demand for unskilled labor. Workers ?lling in
these jobs receive training and ?nancial advice from their employers and eventually
move on to start their own businesses.
10
Thus jobs in the enclave give unskilled
immigrants the opportunity to gain upward mobility without assimilation. Second,
skilled agents start up immigrant-oriented businesses including legal advice, credit
unions or healthcare services that further raise the bene?t of living in the enclave
_
h(n
u
, n
h
) > 0,
?h
?n
h
> 0,
?
2
h
?n
h
?nu
> 0
_
. These assumptions are based on studies by
sociologists on ethnic enclaves (e.g. Light and Gold [20]).
2.3.1 Setup
For d < 1 let n
h
? [0, d] be the fraction of skilled individuals who locate in
the enclave. The assumptions on the unskilled population are the same as those
of the previous section. I describe an enclave with a vector (n
u
, n
h
). Immigrants
10
There is extensive evidence on self-employment in immigrant communities (e.g. Koreans and
Cubans ), see Light and Gold [20].
44
can either live in the enclave (c) or out of it (s). Once they make their residential
choice (c or s), they decide where to work: each immigrant can work in the enclave
or in the general labor market (c or s). The place where an immigrant chooses to
live a?ects his labor market outcomes.
I specify a game as follows: ?rst immigrants decide independently and non-
cooperatively where to settle; and then they decide where to work. Once location
choices have been made an enclave labor market emerges.
The production technology in the enclave is as follows. If an entrepreneur hires
an unskilled worker they produce two units of output (q (unskilled, skilled) = 2),
which yield some revenue 2y > 0. If the entrepreneur decides to be self-employed,
then he produces one unit of output (q (skilled) = 1) for which he receives y. And
the unskilled person by herself produces no output q (unskilled) = 0. Hence the
production technology in the enclave is described by:
q (unskilled, skilled) = 2
q (skilled) = 1
q (unskilled) = 0
Let w
x
denote the clearing wage in the enclave labor market. I assume that the
cost of ?nding a job in the enclave is zero, because the ethnic network channels
information on these jobs more e?ectively than it does on jobs out of the enclave.
Language complementarities between an entrepreneur and a co-ethnic worker allow
for upward mobility without assimilation: workers initially earn w
x
, and after a
45
fraction d of time, they become self-employed and earn income y ? w
x
. Given
that an unskilled person spends a fraction of time d < 1 with the entrepreneur, the
entrepreneur requires of 1/d unskilled individuals to produce 2 units of output (and
thus n
h
? [0, d]).
The enclave wage w
x
follows a reduced-form bargaining model. De?ne ? =
dnu
n
h
as the ratio of labor supply and demand within the enclave. If there is excess
labor supply in the enclave (? > 1) entrepreneurs have more bargaining power
and pay workers a wage equal their outside option, which is the wage they would
receive in the general labor market net of the job ?nding cost. Hence for dn
u
> n
h
,
? > 1 the enclave wage is w
x
= ¯ w
uc
(n
u
, n
h
) ? J (n
u
, n
h
). When there is no excess
labor demand or supply (dn
u
= n
h
, ? = 1) parties have equal bargaining power and
the wage equals the wage an immigrant would receive in the general labor market
(w
x
= w
us
?K). If there is excess labor demand workers have more bargaining
power and get half of the production surplus w
x
= y. Summarizing,
= y if dn
u
< n
h
(2.5a)
w
x
= w
us
?K if dn
u
= n
h
(2.5b)
= ¯ w
uc
(n
u
, n
h
) ?J (n
u
, n
h
) if dn
u
> n
h
(2.5c)
Let b > 0. When there is no excess labor supply or demand (? = 1) unskilled
immigrants who locate in the enclave always ?nd jobs in it. They receive wage
w
x
= w
us
? K. Let U
uc
(n
u
, n
h
) represent the utility received by an unskilled
individual who lives and works in the enclave. This utility is equal to:
U
uc
(n
u
, n
h
) = d (w
us
?K) + (1 ?d)y + h(n
u
, n
h
)
46
For any 0 ? n
u
, n
h
? 1 the utility received by an unskilled individual who lives
and works in the enclave must be no larger than that of a self-employed immigrant
U
self
(n
u
, n
h
) where
U
self
(n
u
, n
h
) = y + h(n
u
, n
h
)
and thus y ? w
us
?K. An immigrant who settles in the enclave and works in the
general labor market earns utility:
U
uc,out
(n
u
, n
h
) = w
us
?K + h(n
u
, n
h
)
and all immigrants who settle in the enclave prefer to work in it so that U
uc
(.) ?
U
uc,out
(.). If only one immigrant chooses to work out of the enclave then he receives
wage ¯ w
uc
(0) = w
us
. If the immigrant assimilates and works in the general labor
market, he derives utility:
U
us
(n
u
, n
h
) = w
us
+ b ?K
where w
us
= ¯ w
us
(0). Finally if the assimilated immigrant works in the enclave he
receives utility equal to:
U
us,in
= d (w
us
?K) + (1 ?d)y + b ?K
When there is excess labor demand (? < 1) all unskilled immigrants ?nd jobs
in the enclave and w
x
= y. Then the utilities derived from each option become:
U
uc
(n
u
, n
h
) = y + h(n
u
, n
h
)
U
uc,out
(n
u
, n
h
) = w
us
?K + h(n
u
, n
h
)
U
us
(n
u
, n
h
) = w
us
+ b ?K
U
us,in
(n
u
, n
h
) = y + b ?K
47
When there is excess labor supply (? > 1) unskilled immigrants who settle in
the enclave ?nd jobs in it only with probability 0 <
1
?
< 1. A fraction n
u
?
n
h
d
of
unskilled persons search for jobs in the general labor market. Thus:
U
uc
(n
u
, n
h
) =
1
?
[dw
x
+ (1 ?d)y] +
? ?1
?
[ ¯ w
uc
(n
u
, n
h
) ?J (n
u
, n
h
)] + h(n
u
, n
h
)
U
uc,out
(n
u
, n
h
) = ¯ w
uc
(n
u
, n
h
) ?J (n
u
, n
h
) + h(n
u
, n
h
)
U
us
(n
u
, n
h
) = ¯ w
us
(n
u
, n
h
) + b ?K
U
us,in
(n
u
, n
h
) =
1
?
[dw
x
+ (1 ?d)y] +
? ?1
?
¯ w
us
(n
u
, n
h
) + b ?K
A skilled immigrant can live in the enclave and become an entrepreneur, live
in the enclave or work out of it, live out of the enclave and work in the general
labor market or live out and work in the enclave. When labor demand equals labor
supply (? = 1), all entrepreneurs hire workers and pay wage w
x
= w
us
?K. Given
such (n
u
, n
h
) an entrepreneur earns utility U
hc
equal to:
U
hc
(n
u
, n
h
) = 2y ?(w
us
?K) + h(n
u
, n
h
)
If the skilled individual decides to assimilate, then he earns utility:
U
hs
(n
u
, n
h
) = U
hs
= w
h
+ b
h
?Z
here b
h
> 0 is the bene?t of assimilation and Z > 0 is the cost of ?nding a job in
the general labor market. Let U
hs
(n
u
, n
h
) > U
us
(n
u
, n
h
) so that skilled immigrants
gain more from assimilating relative to unskilled individuals. This assumption
suggests that the monetary and non-monetary bene?ts from assimilation are larger
for a doctor (skilled) than for a janitor (unskilled person). If the skilled individual
48
lives in the enclave and decides to work out of it he receives utility:
U
hc,out
= w
us
?K + h(n
u
, n
h
)
= U
uc,out
Without assimilation, the skilled immigrant competes with unskilled co-ethnics
in the general labor market. Finally if the immigrant assimilates and decides to
become an entrepreneur in the enclave, he receives utility:
U
hs,in
(n
u
, n
h
) = 2y ?(w
us
?K) + b
h
?Z
When there is excess labor demand (? < 1) an entrepreneur is matched to
an unskilled worker with probability ? and pays him wage w
x
= y. For ? < 1 the
utilities become:
U
hc
(n
u
, n
h
) = y + h(n
u
, n
h
)
U
hs
(n
u
, n
h
) = U
hs
= w
h
+ b
h
?Z
U
hc,out
= w
us
?K + h(n
u
, n
h
)
U
hs,in
(n
u
, n
h
) = y + b
h
?Z
If there is excess labor supply in the enclave (? > 1) all entrepreneurs are
guaranteed to get workers. In that case the enclave wage is w
x
= ¯ w
uc
(n
u
, n
h
) ?
J (n
u
, n
h
) for ¯ w
uc
(n
u
, n
h
) ? J (n
u
, n
h
) ? y. When there is excess labor supply the
49
alternatives available to a skilled immigrant yield utilities equal to:
U
hc
(n
u
, n
h
) = 2y ?[ ¯ w
uc
(n
u
, n
h
) ?J (n
u
, n
h
)] + h(n
u
, n
h
)
U
hs
(n
u
, n
h
) = U
hs
= w
h
+ b
h
?Z
U
hc,out
= ¯ w
uc
(n
u
, n
h
) ?J (n
u
, n
h
) + h(n
u
, n
h
)
U
hs,in
(n
u
, n
h
) = 2y ?[ ¯ w
uc
(n
u
, n
h
) ?J (n
u
, n
h
)] + b
h
?Z
2.3.2 The Game: Residential and Workplace Decisions
I now solve for the Nash Equilibria of the game. For i = u, h and given
(n
u
, n
h
) an immigrant decides to live and work in the enclave only if:
U
ic
(n
u
, n
h
) ? U
is
(n
u
, n
h
) for i = u, h
? U
ic,out
(n
u
, n
h
)
? U
is,in
(n
u
, n
h
)
The full enclave (1, d) is a NE if for such (1, d) an immigrant’s best response
is to settle and work in the enclave. Similarly, full assimilation (0, 0) is a NE if
given (0, 0) an immigrant’s best response is to assimilate and work in the general
economy. Denote a NE enclave by (n
?
u
, n
?
h
). An enclave (0, 0) < (n
?
u
, n
?
h
) < (1, d)
is an interior NE if ?rst, for such (n
?
u
, n
?
h
) all immigrants are indi?erent between
settling and working in the enclave or living and working out of it; and second, all
immigrants (weakly) prefer either option to living and working in di?erent areas.
De?ne 0 ? n
ucrit
, n
hcrit
? 1 such that an immigrant is indi?erent between locating
in the enclave or out of it, i.e. for i = u, h U
ic
(n
ucrit
, n
hcrit
) = U
is
(n
ucrit
, n
hcrit
).
50
The ?rst result of the model is that there exist NE with no excess labor supply
or demand in the enclave, i.e. 0 ? n
?
u
=
n
?
h
d
? 1. These equilibria emerge when
the wage skilled immigrants receive out of the enclave (w
h
) is large, and when
the cultural bene?ts of the enclave are smaller than the non-monetary gains from
assimilation for unskilled immigrants (i.e. when b ? h(n
u
, n
h
)).
NE with no excess labor supply or demand emerge if the workers in the enclave
get upward mobility at a speed equal to
´
d =
U
hs
?Uus?(y?wus+K)
y?wus+K
, where U
us
= w
us
+
b ?K. If it takes longer for an individual to gain upward mobility in the enclave,
i.e. if d >
´
d, then the person prefers to assimilate and the enclave is no longer
an equilibrium. An unskilled immigrant gets upward mobility faster (i.e.
´
d is
smaller) when the entrepreneur is able to extract more rents from the him (i.e.
when y ?w
us
+K is large) and when the utility he receives if he assimilates is large
(U
us
). Therefore, holding all other variables ?xed, an increase in the value of output
, for example due to improvements in the production technology, increases the
rents received by the entrepreneurs and allows unskilled immigrants to become self-
employed faster (
´
d decreases). Finally, if a skilled immigrant assimilates, he receives
utility U
hs
. And the higher this utility is, the larger the share of the worker’s surplus
that he as an entrepreneur must receive in order for him to locate in the enclave
(i.e.
´
d is increasing in U
hs
). The following proposition formalizes these results.
Proposition 2.5. For b > 0 let w
us
?K ? y , w
h
? 2y ?w
us
+K and b
h
? Z. If
U
hs
> y + b and
d =
U
hs
?y ?b
y ?w
us
+ K
51
then i. no enclave forming is an equilibrium (n
?
u
= n
?
h
= 0);
ii. The enclave forming is an equilibrium (n
?
u
= 1, n
?
h
= d);
iii. An interior equilibrium exists with 0 < n
?
h
= dn
?
u
< d; and
iv. The enclave is Pareto-superior to the assimilation equilibrium.
Allowing for skill heterogeneity in the pool of immigrants leads to an improve-
ment in the economic outcomes of unskilled immigrants who live in the enclave in
comparison to a situation in which the all immigrants are unskilled. In the enclave
with skilled immigrants, unskilled individuals can get upward mobility without as-
similation. Consequently in this setup the enclave is no longer a trap or a ‘bad’
equilibrium, regardless of skill all immigrants are better o? moving to the enclave.
Although the wage in the enclave (w
x
= w
us
?K) is lower than that of the gen-
eral labor market ( ¯ w
uc
(0) = w
us
), unskilled immigrants are better o? settling in the
enclave because they still get upward mobility and additionally, they derive bene-
?t from sharing common culture. At the same time, having entrepreneurs in the
enclave reduces its negative externality on the wages of assimilated unskilled indi-
viduals; in fact for n
?
h
= dn
?
u
the negative externality completely disappears and
¯ w
us
(n
u
) = w
us
.
The second result of the model is that there exists an interior NE with excess
labor supply in the enclave, i.e. 0 <
n
?
h
d
< n
?
u
< 1. For such an equilibrium
? (n
?
u
, n
?
h
) =
(1?d)[y? wuc(.)+J(.)]?J(.)
2[y? wuc(.)+J(.)]?U
hs
+ wus(.)+b?K?J(.)
. This ? emerges if the wage of a
skilled and assimilated immigrant (w
h
) is su?ciently large, if the value of the enclave
output y is high and if unskilled immigrants spend a relatively large fraction of time
52
in the job (i.e. if d is high). Thus enclaves with excess labor supply may fail to
form if the value of the output produced in them is too low. When that occurs,
workers need stay even longer with an entrepreneur (d has to be very large) so that
the skilled person’s utility from living and working in the enclave is large enough
to discourage him from assimilating. Furthermore when d is too large unskilled
individuals could choose not to settle in the enclave. If the rents extracted by
the entrepreneur are large, then more skilled immigrants will have the incentive to
settle in the enclave and the equilibrium excess labor supply will be smaller (i.e. ? is
decreasing in (y ? ¯ w
uc
(.) + J (.))). Finally, as the utility received by an assimilated
skilled immigrant gets larger, the excess labor supply in the enclave increases. The
next proposition summarizes these results.
Proposition 2.6. For b > 0 let w
us
?K < y, w
h
>> y and
h(1, d) ? max {b ?(1 ?d) (y ?w
us
+ K) ; U
hs
?2y + w
us
?K}
then i. no enclave forming is an equilibrium (n
?
u
= n
?
h
= 0);
ii. The enclave forming is an equilibrium (n
?
u
= 1, n
?
h
= d);
iii. An interior equilibrium exists with 0 < n
?
h
< dn
?
u
< d only if ¯ w
us
(n
?
u
, n
?
h
) ?
y, w
h
? 2y ? ¯ w
uc
(.) + J (.); d ?
y? wus(.)
y? wuc(.)+J(.)
and
? (n
?
u
, n
?
h
) =
(1 ?d) [y ? ¯ w
uc
(.)] ?dJ (.)
2 [y ? ¯ w
uc
(.)] ?U
hs
+ ¯ w
us
(.) + b ?K ?J (.)
iv. The enclave is Pareto-superior to the assimilation equilibrium.
The presence of immigrant entrepreneurs in the enclave improves the economic
outcomes of all unskilled co-ethnics. The quality of the enclave a?ects the economic
53
outcomes of all immigrants. Even unskilled persons that assimilate bene?t from
the presence of the entrepreneurs because fewer co-ethnics who live in the enclave
end up working in the general labor market. For a given probability of a worker
revealing as assimilated (?), the fewer enclave people working in the general labor
market, the higher the average marginal productivity of workers who do not reveal,
the higher the wages for all immigrants in the general market.
The bene?ts of the enclaves with entrepreneurs are apparent: Portes and
Stepick [24] compare black Cubans and Haitians who arrived to Miami in 1980
and ?nd that the Cubans were able to ?nd jobs in the Cuban enclave, and even
comparable jobs in the general economy. In contrast the Haitians, who did not
have an ethnic economy, experienced high rates of unemployment and operated
mostly in the informal economy. An extreme case of proposition 2.6 is an equilib-
rium given by n
?
u
= 1 and n
?
h
= 0, in which sorting by skill occurs. All unskilled
immigrants cluster, while the skilled ones assimilate. This type of equilibrium is
consistent with the recent wave of Chinese migration into the U.S.: highly skilled
and educated individuals assimilate, while very low skilled people tend to cluster
(Karas [18]).
Finally no equilibrium exists in which there is excess labor demand in the
enclave. If there were excess labor demand, then the wage would be w
x
= y
and skilled and unskilled immigrants in the enclave would all earn the same (self-
employment) utility: U
uc
(.) = U
hc
(.) = U
self
(.) = y + h(n
u
, n
h
). For an enclave
with excess labor demand (dn
u
< n
h
) to be an equilibrium we require U
us
(.) =
54
U
uc
(.) = U
hc
(.) = U
hs
. But this equality can never hold because by assumption
U
us
(.) < U
hs
. Skilled immigrants do not cluster by themselves. They ?nd enclaves
attractive because they can hire co-ethnics fairly easily. The ethnic network that
emerges in the enclave seems to be a stronger magnet for unskilled people than for
skilled persons (Portes and Rumbaut [27]). For example, Filipino immigrants who
are highly skilled (typically doctors) have never formed ethnic enclaves (Karas [18]).
One possible explanation for why enclaves with excess labor demand never form is
that language complementarities in the general labor market are stronger for skilled
people than for unskilled persons. Hence skilled people are more likely to assimilate
relative to unskilled individuals. In fact, Lang et al. [3] ?nd that among Russian
immigrants in Israel, the value of learning Hebrew is large for skilled individuals and
close to zero for unskilled people.
2.4 Conclusion
In this chapter I construct a model to study the e?ect that residential location
choices have on the economic outcomes of immigrants. It is a game in which immi-
grants decide simultaneously and independently where to settle. I start by analyzing
the strategic decisions of an homogeneous group of unskilled immigrants. When the
bene?ts of assimilation come only through higher wages, immigrants decide to settle
in an enclave. In order for immigrants to be willing to assimilate, they must perceive
some positive non-monetary bene?t from assimilation (such as less discrimination in
their social endeavor). When there are monetary and non-monetary bene?ts from
55
assimilation, and when employers in the general labor market are unlikely to screen
out the assimilated immigrants, then multiple equilibria emerge. And in such case,
it is very likely that the enclave equilibrium is a poverty trap. Immigrants may end
up forming the enclave because of a lack of coordination in their decisions, although
they could all be better o? if they assimilated.
I then modify the game so that both skilled and unskilled immigrants decide
where to settle. I assume that skilled immigrants who settle in the enclave be-
come entrepreneurs and have a positive demand for unskilled labor. Thus adding
skilled immigrants to the model, opens the possibility for a labor market within
the enclave. I show that enclaves that emerge in equilibrium never have excess
labor demand. Skilled people have a stronger incentive to assimilate than unskilled
individuals. If the enclave emerges in equilibrium, the speed of upward mobility
of unskilled immigrants is increasing in the value of the output produced in the en-
clave. Furthermore, if the value of the output produced in the enclave is too small,
then the enclave may fail to form. For prevailing wages in the general economy, en-
clave entrepreneurs have the incentive to produce output that has more technology
embedded because they can then extract larger rents from the workers. Finally the
results of the model suggest that when immigrants with a mix of skills settle in the
enclave, the enclave equilibrium is the socially-preferred outcome. When skilled
immigrants locate in the enclave, the quality of the enclave improves and it is no
longer a trap (all individuals are better-o? clustering).
Throughout the analysis I assume that immigrants make decisions indepen-
56
dently. Although in the model an individual takes into account the social gains of
clustering when making his decision, this approach may con?ict with the empirical
evidence, which shows that the decision of one individual to migrate is conditioned
on the decisions of others in his social network (e.g. Portes and Rumbaut [27] and
Munshi [21]). One way to reconcile my approach with the evidence is to partition
the fraction of immigrants who settle in the enclave in smaller communities. Coor-
dination among members of a community could lead them to get out of the enclave
trap whenever it is likely to emerge. As the community gets larger, coordination
becomes harder, and in that case the results of my model would remain unchanged.
Two policy recommendations emerge from this analysis. The ?rst is that if
the pool of immigrants is uniformly unskilled, then the government can help native
employers in screening out unskilled immigrants who assimilate. For example the
government could provide immigrants with English lessons and then give a compre-
hensive examinations which would be required for employment. The second recom-
mendation is that the government could extend credits to immigrant entrepreneurs,
who could improve the technology of their production, allow unskilled co-ethnics
to achieve upward mobility sooner, and also lessen the negative externality that
the enclave imposes on the wages of assimilated unskilled immigrants. And better
technologies available to the enclave entrepreneurs guarantee that the enclave forms,
which prevents the unskilled people from clustering by themselves and earning very
low wages in the general labor market.
57
Chapter A
Proofs of Propositions of Chapter 1
Proof of proposition 1.1. The Nash equilibrium in e?orts of the second stage is
unique only if:
¸
¸
¸
¸
¸
?
2
B
? (e
i
)
2
+

j=i,j?N
?
2
K
? (e
i
)
2
?
?
2
?
? (e
i
)
2
¸
¸
¸
¸
¸
>

j=i, j?CC
i
?
2
K
?e
i
?e
j
for all i ? N (A.1)
In this game I consider Star networks (g
S
= {12, 23}) with symmetric Nash equilibria
in which e
?
1
= e
?
3
. The pro?le of NE e?orts in these networks is represented by
e
?
= (e
?
1
, e
?
2
, e
?
1
). All the analysis that follows looks at two players only, assuming
that players 1 and 3 behave identically.
Rewriting condition A.1 for two players and rearranging terms yields:
?
2
K
?e
i
?e
j
¸
¸
¸
?
2
B
?(e
i
)
2
+
?
2
K
?(e
i
)
2
?
?
2
?
?(e
i
)
2
¸
¸
¸
=
?e
i
?e
j
< 1 for i = j, i, j (A.2)
The condition A.2 has two implications: ?rst, because e?orts are strategic comple-
ments then any BR
i
is increasing in e
j
if s
ij
> 0. Second, the absolute value of the
slope of the reaction function of any player is less than one.
Under the assumptions made on the payo? functions the uniqueness condition
A.1 is always satis?ed. Let g
S
= {12, 23}. I will show that the pro?le of e?orts
_
e
S
1
, e
S
2
, e
S
3
_
= (ˆ e
1
, ˆ e
1
, ˆ e
1
) is not a NE and furthermore that at the NE e
?
corner
> e
?
center
.
Suppose that (ˆ e
1
, ˆ e
1
, ˆ e
1
) is the NE. The payo? to player 1 (corner) in the Star
58
network is:
Y
S
1
= B(e
1
; N
S
1
) + K(e
1
, e
2
, s
12
; ?) + K(e
1
, e
3
, s
13
; ?) ??(e
1
, N
S
1
; µ, f)
where s
13
=
1
2
s
12
. Taking the ?rst order condition and using the symmetry between
players 1 and 3 yields:
?Y
S
1
?e
1
|
e
1
=ˆ e
1
e
2
=e
3
=ˆ e
1
=
?B(ˆ e
1
; N
S
1
)
?ˆ e
1
+
?K(ˆ e
1
, ˆ e
1
, s
12
; ?)
?ˆ e
1
+
?K(ˆ e
1
, ˆ e
1
,
1
2
s
12
; ?)
?ˆ e
1
?
??(ˆ e
1
, N
S
1
; µ, f)
?ˆ e
1
= 0
or equivalently,
?K(ˆ e
1
, ˆ e
1
, ?, s
12
)
?ˆ e
1
=
??(ˆ e
1
, N
S
1
; µ, f)
?ˆ e
1
?
?B(ˆ e
1
; N
S
1
)
?ˆ e
1
?
?K(ˆ e
1
, ˆ e
1
,
1
2
s
12
; ?)
?ˆ e
1
(A.3)
The payo? to player 2 is:
Y
S
2
= B(e
2
; N
S
2
) + K(e
2
, e
1
, s
12
; ?) + K(e
2
, e
3
, s
12
; ?) ???(e
2
, N
S
2
; µ, f)
and the best reply function is given by:
?Y
S
2
?e
2
|
e
2
=ˆ e
1
e
1
=e
3
=ˆ e
1
=
?B(ˆ e
1
; N
S
2
)
?ˆ e
1
+
?K(ˆ e
1
, ˆ e
1
, s
12
; ?)
?ˆ e
1
+
?K(ˆ e
1
, ˆ e
1
, s
12
; ?)
?ˆ e
1
(A.4)
?
??(ˆ e
1
, N
S
2
; µ, f)
?ˆ e
1
=
_
?B(ˆ e
1
; N
S
2
)
?ˆ e
1
?
?B(ˆ e
1
; N
S
1
)
?ˆ e
1
_
+
_
?K(ˆ e
1
, ˆ e
1
, s
12
; ?)
?ˆ e
1
?
?K(ˆ e
1
, ˆ e
1
,
1
2
s
12
; ?)
?ˆ e
1
_
(A.5)
?
_
??(ˆ e
1
, N
S
2
; µ, f)
?ˆ e
1
?
??(ˆ e
1
, N
S
1
; µ, f)
?ˆ e
1
_
where the last equality follows from substituting A.3 in A.4.
Given that own e?ort and links are strategic complements in both B(.) and
C(.), and N
S
2
> N
S
1
, it follows that
?B(ˆ e
1
;N
S
2
)
?ˆ e
1
?
?B(ˆ e
1
;N
S
1
)
?ˆ e
1
> 0 and
??(ˆ e
1
,N
S
2
;µ,f)
?ˆ e
1
?
??(ˆ e
1
,N
S
1
;µ,f)
?ˆ e
1
> 0. Bene?ts from spillovers decrease with distance, i.e. K(ˆ e
1
, ˆ e
1
, ?; s
12
) >
K(ˆ e
1
, ˆ e
1
, ?;
1
2
s
12
). And spillovers are linearly increasing in own e?ort. Then
59
?K(ˆ e
1
,ˆ e
1
,s
12
,?)
?ˆ e
1
?
?K(ˆ e
1
,ˆ e
1
,
1
2
s
12
;?)
?ˆ e
1
> 0 . All terms in brackets in A.5 are strictly positive.
The di?erence
??(ˆ e
1
,N
S
2
;µ,f)
?ˆ e
1
?
??(ˆ e
1
,N
S
1
;µ,f)
?ˆ e
1
is an approximation to
?
2
?(e
1
,N
S
1
;µ,f)
?e
1
?N
1
. By
assumption these complementarities are large so that
??(ˆ e
1
,N
S
2
;µ,f)
?ˆ e
1
?
??(ˆ e
1
,N
S
1
;µ,f)
?ˆ e
1
is
strictly positive and large. But then
?Y
S
2
?e
2
|
e
2
=ˆ e
1
< 0, and
_
e
S
1
, e
S
2
, e
S
3
_
= (ˆ e
1
, ˆ e
1
, ˆ e
1
)
would not be a NE.
At e
2
= ˆ e
1
, the marginal cost of e?ort for the center of the Star is larger than
the marginal bene?ts, thus it is optimal for the agent to select some e?ort level
ˆ e
2
< ˆ e
1
. Since
?
2
K(e
1
,e
1
,s
12
;?)
?e
1
?e
2
> 0, player 1 would respond to ˆ e
2
by decreasing his
e?ort too. Given that A.2 holds, player 1 decreases his e?ort by less than the initial
decrease in e?ort of player 2. Then the NE of a Star network is characterized by
_
e
S
1
, e
S
2
, e
S
3
_
= (e
?
1
, e
?
2
, e
?
1
) with e
?
1
> e
?
2
.
Given the existence of a NE in which e
?
1
> e
?
2
, any equilibrium of the form e
?
2
>
e
?
1
is ruled out. The latter equilibrium would require that around its neighborhood
the (absolute value of the) slope of at least one of the reaction functions be greater
than 1, which would violate A.2.
Proof of proposition 1.2. I now compare crime e?orts across networks. Any crimi-
nal in the complete network has two links and is a step away from any other criminal.
Similarly, in the Star network, the center has two links and is only a step away from
the other two criminals ( N
center
= N
C
i
= 2, s(center, j) = s (i, j) = s
12
, j = 1, 3).
Let the pro?le of NE e?orts of the Star be
_
e
S
1
, e
S
2
, e
S
3
_
= (e
?
corner
, e
?
center
, e
?
corner
).
60
Evaluating the ?rst-order condition of the center of the Star at the NE:
?Y
S
center
?e
center
|
ecenter=e
?
center
=
=
?B(e
?
center
; N
center
)
?e
?
center
+ 2
?K(e
?
center
, e
?
corner
, s
12
; ?)
?e
?
center
?
??(e
?
center
, N
center
; µ, f)
?e
?
center
= 0
or equivalently,
??(e
?
center
, N
center
; µ, f)
?e
?
center
?
?B(e
?
center
; N
center
)
?e
?
center
= 2
?K(e
?
center
, e
?
corner
, s
12
; ?)
?e
?
center
(A.6)
Suppose that all nodes in the complete network select an e?ort level e
C
i
= e
?
center
:
?Y
C
i
?e
C
i
|
e
C
i
=e
?
center
=
?B(e
?
center
; N
center
)
?e
?
center
+2
?K(e
?
center
, e
?
center
, s
12
; ?)
?e
?
center
?
??(e
?
center
, N
center
; µ, f)
?e
?
center
(A.7)
Substituting A.6 in A.7 and for e
?
center
< e
?
corner
(from proposition 1.1),
?Y
C
i
?e
C
i
|
e
C
i
=e
?
center
= 2
_
?K(e
?
center
, e
?
center
, s
12
; ?)
?e
?
center
?
?K(e
?
center
, e
?
corner
, s
12
; ?)
?e
?
center
_
< 0
e
?
center
is not the NE e?ort level of criminals in the complete network. At e
?
center
the
marginal cost of e?ort is larger than the marginal bene?t. Thus the NE e?ort levels
are e
C?
i
< e
?
center
< e
?
corner
.
In the complete network all nodes have two links, and are one step away from
each other. Therefore payo?s are symmetric. Without loss of generality consider
player 1’s payo? for given e
2
:
Y
C
1
= B(e
C
1
, N
C
1
) + K(e
C
1
, e
C
2
, s
12
; ?) + K(e
C
1
, e
C
3
, s
12
; ?) ??(e
C
1
, N
C
1
; µ, f)
= 2
_
B(e
C
1
, 1) + K(e
C
1
, e
C
2
, ?, s
12
) ??(e
C
1
, 1; µ, f)
¸
with e
2
= e
3
, s
12
= s
13
(A.8)
61
using H.O.D.1 of B(.) and ?(.) in N
g
i
and symmetry.
In the single-link network, g
I
= {12}, only two agents are connected. The
payo? to player 1 in this network is:
Y
I
1
= B(e
I
1
, N
I
1
) + K(e
I
1
, e
I
2
, s
12
; ?) ??(e
C
1
, N
C
1
; µ, f)
= B(e
I
1
, 1) + K(e
I
1
, e
I
2
, ?, s
12
) ??(e
I
1
, 1; µ, f) (A.9)
From A.8 and A.9, Y
C
1
= 2Y
I
1
. The payo? of the complete network is an increasing
monotonic transformation of that of a connected node in the single-link network.
Thus for given (e
2
, e
3
), if e
?
1
maximizes A.8 then it also maximizes A.9. Hence
e
C
i
= e
I
connected
> e
I
isolated
= 0.
62
Chapter B
Proofs of Propositions of Chapter 2
Proof of proposition 2.1. For n
u
= 0: U
us
(0) = w
us
+b?K and U
uc
(0) = w
us
?K. If
b < 0 then U
uc
(0) > U
us
(0) and an immigrant unilaterally deviated and settles in the
enclave. Consequently everyone assimilating (n
?
u
= 0) is never a Nash Equilibrium
(NE). Given that
?Uus
?nu
< 0 and
?Uuc
?nu
> 0 then for n
u
> 0 U
uc
(n
u
) > U
us
(n
u
).
Therefore, if b < 0, for 0 ? n
u
? 1, it is always a best response to go to the enclave.
Thus n
?
u
= 1 is the unique Nash Equilibrium.
Proof of proposition 2.2. For b > 0 and n
u
= 0: U
us
(0) = w
us
+ b ? K > w
us
?
K = U
uc
(0), all immigrants assimilating is a Nash Equilibrium. Now I show that
if h(1) > b ? K and w
us
? w
uc
?
h(1)?b+K
?
, then full assimilation is the unique
equilibrium. Given that
?Uus
?nu
< 0 and
?Uuc
?nu
> 0, the lowest possible utility received
by an immigrant who assimilates is U
us
(1), and the highest possible if he settles
in the enclave is U
uc
(1). If at n
u
= 1 U
us
(1) ? U
uc
(1), then all immigrants
weakly prefer to assimilate for any 0 ? n
u
? 1. Notice U
us
(1) = w
us
+ b ? K ?
(1 ??) (w
us
?w
uc
) ? w
uc
+ h(1) = U
uc
(1) ? ? (w
us
?w
uc
) ? h(1) ? b + K > 0,
which holds given the assumptions above.
Proof of proposition 2.3. For b > 0 all immigrants assimilating is a Nash Equilib-
rium. By assumption h(1) > b ? K and w
us
? w
uc
<
h(1)?b+K
?
. Is the enclave
63
forming also a NE? U
uc
(1) = w
uc
+ h(1) ? w
us
+ b ? K ? (1 ??) (w
us
?w
uc
)
? ? (w
us
?w
uc
) ? h(1) ?b +K, hence the enclave is a NE if w
us
?w
uc
<
h(1)?b+K
?
,
which is always satis?ed. If U
us
(0) > U
uc
(0) and U
uc
(1) > U
us
(1), then there exists
0 < n
u
< 1 such that U
uc
(n
u
) = U
us
(n
u
). For given 0 < n
u
< 1 an immigrant is
indi?erent between going to the enclave or assimilating, and thus 0 < n
?
u
< 1 is an
interior Nash Equilibrium.
Proof of proposition 2.4. By assumption both the enclave and full assimilation are
NE of the game. All immigrants are better o? assimilating when U
us
(0) > U
uc
(1) .
Equivalently, U
us
(0) = w
us
+b?K > w
uc
+h(1) = U
uc
(1), which holds for w
us
?w
uc
>
h(1) ? b + K. The enclave equilibrium is Pareto superior to the assimilation
equilibrium when U
us
(0) < U
uc
(1), which requires w
us
? w
uc
< h(1) ? b + K. As
? gets closer to 1, if w
us
?w
uc
is not su?ciently large, the assimilation equilibrium
may emerge as the ‘bad’ equilibrium.
Proof of proposition 2.5. Using the conditions b > 0, w
us
?K ? y , 2y ?w
us
+K ?
w
h
? 2y , b ?K ? b
h
?Z ? b, U
hs
> y + b and
y ?w
us
+ K ?(b ?b
h
+ Z)
w
h
?y
? d ?
min
_
y ?w
us
+ K ?(b ?b
h
+ Z)
w
h
?y
,
U
hs
?y ?w
us
+ K ?b
y
_
I show that three equilibria exist: i. n
?
u
= n
?
h
= 0; ii. dn
?
u
= n
?
h
= d ; and iii.
0 < dn
?
u
= n
?
h
< d. The ?rst step is to show that (n
?
u
, n
?
h
) = 0 is a NE. Given
(n
?
u
, n
?
h
) = 0 is it a best response to assimilate? Start with the unskilled immigrants.
64
For (n
?
u
, n
?
h
) = 0: U
uc
(0, 0) = U
uc,out
(0, 0) = w
us
?K and U
us
(0, 0) = U
us,in
(0, 0) =
w
us
? K + b. These equations imply that U
us
(0, 0) > U
uc
(0, 0) = U
uc,out
(0, 0)
and U
us
(0, 0) ? U
us,in
(0, 0). For (n
?
u
, n
?
h
) = 0 the best response of an unskilled
immigrant is to assimilate and work out of the enclave. Now I show that it is also
a best response for a skilled immigrant to assimilated and work in the general labor
market when (n
?
u
, n
?
h
) = 0. When (n
?
u
, n
?
h
) = 0 U
hc
(0, 0) = y, U
hs
= w
h
+ b
h
? Z ,
U
hc,out
= U
uc,out
and U
hs,in
= y +b
h
?Z. By assumption U
hs
> y, and thus a skilled
immigrant prefers to assimilated and work out of the enclave. And he prefers this
alternative to either of the other two options: U
hc,out
(0, 0) = w
us
?K < y < U
hs
,
and U
hs,in
(0, 0) = y + b
h
?Z < U
hs
since w
h
> y. Thus (n
?
u
, n
?
h
) = 0 is a NE.
The second step is to show that 0 < dn
?
u
= n
?
h
< d is a NE. An unskilled
immigrant chooses among the four alternatives, which yield utilities: U
uc
(n
?
u
, n
?
h
) =
dw
x
(n
?
u
, n
?
h
) +(1 ?d) y +h(n
?
u
, n
?
h
), U
us
(n
?
u
, n
?
h
) = w
us
+b ?K, U
uc,out
= w
us
?K +
h(n
?
u
, n
?
h
) and U
us
(n
?
u
, n
?
h
) = dw
x
(n
?
u
, n
?
h
)+(1 ?d) y+b?K. Given (n
?
u
, n
?
h
) < (1, d)
an unskilled immigrant is indi?erent between living and working in the enclave
or living and working out of the enclave if U
us
(n
?
u
, n
?
h
) = U
uc
(n
?
u
, n
?
h
), which is
equivalent to w
x
(n
?
u
, n
?
h
) = y ?
U
hs
?wus?b+K
1+d
. This equilibrium wage is always
non-negative because by assumption
U
hs
?wus?b+K?y
y
? d. The person strictly
prefers to live and work in the enclave to live in and work out of the enclave:
U
uc
(n
?
u
, n
?
h
) = U
us
(n
?
u
, n
?
h
) ? U
uc,out
(n
?
u
, n
?
h
) ? b ? h(n
?
u
, n
?
h
). From U
hs
(n
?
u
, n
?
h
) =
U
hc
(n
?
u
, n
?
h
) we ?nd h(n
?
u
, n
?
h
) =
dU
hs
?wus?b+K
1+d
? y. Now b ?
dU
hs
?wus?b+K
1+d
? y
only if d ?
y?wus+K
U
hs
?y?b
? w
h
? 2y ? w
us
+ K , and this inequality holds by as-
sumption. Hence U
uc
(n
?
u
, n
?
h
) ? U
uc,out
(n
?
u
, n
?
h
). Finally, an unskilled immigrant
65
that assimilates prefers to work out of the enclave to working in the enclave if:
U
us
(n
?
u
, n
?
h
) ? U
us,in
(n
?
u
, n
?
h
) ? U
uc
(n
?
u
, n
?
h
) ? U
us,in
(n
?
u
, n
?
h
) ? h(n
?
u
, n
?
h
) ? b ? K.
This last condition holds whenever h(n
?
u
, n
?
h
) ? b
h
? Z ? b ? K. Below I prove
that this is the case. Now I show that for 0 < dn
?
u
= n
?
h
< d a skilled immigrant is
indi?erent between settling and working in the enclave or out of it, and prefers either
option to all others. Given (n
?
u
, n
?
h
) < (1, d) the alternatives available to a skilled in-
dividual yield utilities: U
hc
(n
?
u
, n
?
h
) = 2y?w
x
(n
?
u
, n
?
h
)+h(n
?
u
, n
?
h
), U
hs
= w
h
+b
h
?Z,
U
hc,out
(n
?
u
, n
?
h
) = U
uc,out
(n
?
u
, n
?
h
), and U
hs,in
(n
?
u
, n
?
h
) = 2y ?w
x
(n
?
u
, n
?
h
) +K +b
h
?Z.
Given the enclave (n
?
u
, n
?
h
) < (1, d) a skilled immigrant is indi?erent between living
and working in the enclave or out of it if U
hs
= U
hc
(n
?
u
, n
?
h
), which is equivalent
to h(n
?
u
, n
?
h
) =
dU
hs
?wus?b+K
1+d
? y > 0. A skilled immigrant prefers to settle and
work in the enclave than assimilate and become an entrepreneur in the enclave, i.e.
U
hc
(n
?
u
, n
?
h
) ? U
hs,in
(n
?
u
, n
?
h
) ? h(n
?
u
, n
?
h
) ? b
h
?Z ? d ?
y?wus+K?(b?b
h
+Z)
w
h
?y
. And he
never chooses to live in the enclave and work out of it: U
hc
(n
?
u
, n
?
h
) ? U
uc
(n
?
u
, n
?
h
) ?
U
uc,out
(n
?
u
, n
?
h
) = U
hc,out
(n
?
u
, n
?
h
). Thus 0 < dn
?
u
= n
?
h
< d is a NE.
The third step is to show that everyone locating and working in the en-
clave is also a NE (n
?
u
= 1, n
?
h
= d). Start with the unskilled immigrants:
U
uc
(1, d) = dw
x
(1, d) +(1 ?d) y +h(1, d) ; U
us
(1, d) = w
us
?K +b; U
uc,out
(1, d) =
w
us
? K + h(1, d) and U
us
(1, d) = dw
x
(1, d) + (1 ?d) y + b ? K. Notice that
U
uc
(1, d) ? U
us
(1, d) only if w
?
x
(1, d) ? y ?
1
d
[y ?w
us
+ K ?b + h(1, d)] , which
holds by assumption. Since h(n
?
u
, n
?
h
) =
dU
hs
?wus?b+K
1+d
? y and h(.) is increas-
ing in both of its arguments, then h(1, d) ? h(n
?
u
, n
?
h
) ? b ? b ? K. Hence
U
uc
(1, d) ? U
uc,out
(1, d) or equivalently, h(1, d) ? b ? K is always met. Similarly
66
U
uc
(n
?
u
, n
?
h
) ? U
us,in
(n
?
u
, n
?
h
) because h(n
?
u
, n
?
h
) ? b?K. Then h(1, d) > h(n
?
u
, n
?
h
) ?
b ?K. Now look at the skilled immigrants. The utilities from all alternatives are:
U
hc
(1, d) = 2y?w
x
(1, d)+h(1, d) ; U
hs
(1, d) = U
hs
; U
hc,out
(1, d) = w
us
?K+h(1, d)
and U
hs
(1, d) = U
hs
. Notice that for 0 < dn
?
u
= n
?
h
< d, U
hs
= U
hc
(n
?
u
, n
?
h
). Be-
cause U
hc
(n
u
, n
h
) is increasing in both of its arguments, then U
hc
(1, d) > U
hs
. Thus
given (n
?
u
= 1, n
?
h
= d), the best response of a skilled immigrant is to live and work
in the enclave. Furthermore, U
hc
(1, d) ? U
uc
(1, d) > U
uc,out
(1, d) = U
hc,out
(1, d),
and U
hc
(n
u
, n
h
) > U
hs,in
(1, d) ? h(1, d) ? h(n
?
u
, n
?
h
) > b
h
?Z.
Finally, the enclave equilibrium is Pareto-superior to the assimilation equilib-
rium: i. U
us
(0, 0) = w
us
? K + b < dw
x
(1, d) + (1 ?d) y + h(1, d) = U
uc
(1, d) ?
w
?
x
(1, d) ? y?
1
d
[y ?w
us
+ K ?b + h(1, d)] ; ii. U
hs
= U
hc
(n
?
u
, n
?
h
) < U
hc
(1, d).
Proof of proposition 2.6. Using the conditions b > 0, y > min [w
us
?K, ¯ w
us
(n
u
, n
h
)],
y << w
h
? 2y and b?K ? b
h
?Z, I show that three equilibria exist: i. n
?
u
= n
?
h
= 0;
ii. dn
?
u
= n
?
h
= 1 ; and iii. 0 < n
?
h
< dn
?
u
< d. From the proof of proposition 2.5,
the conditions for the existence of NE n
?
u
= n
?
h
= 0 and dn
?
u
= n
?
h
= 1 are:
w
h
? 2y (B.1)
h(1, d) ?
w
us
+ b ?K ?dU
hs
1 + d
?y (B.2)
h(1, d) ? b ?K (B.3)
h(1, d) ? b
h
?Z (B.4)
y ?
1
d
[y ?w
us
+ K ?b + h(1, d)] ? w
?
x
(1, d) ? 2y ?U
hs
+ h(1, d) (B.5)
67
By assumption equations (B.1), (B.2) and (B.5) are satis?ed. Two additional con-
ditions are d ? min
_
y? wus(.)
w
h
?y
,
y? wuc(.)+J(.)
U
hs
? wus(.)?b+K?(y? wuc(.)+J(.))
_
, and
y ? ¯ w
uc
(.) + J (.) ?d (w
h
?y)
¯ w
us
(.) ? ¯ w
uc
(.) + J (.) + b
h
?Z
? ?
?
(n
u
, n
h
) ?
d [U
hs
+ ¯ w
us
(.) ? ¯ w
uc
(.) + J (.)]
d [y ? ¯ w
uc
(.) + J (.)] + y ? ¯ w
us
(.) + d [ ¯ w
us
(.) + b ?K]
.
d ?
y? wus(.)
w
h
?y
guarantees that the lower bound of ? is smaller than its upper bound.
Further the upper bound of ? is greater than one if h(n
?
u
, n
?
h
) > b ? K. One
equilibrium condition is that h(n
?
u
, n
?
h
) ? b
h
? Z, since b
h
? Z ? b ? K, then the
conditions for ? > 1 and (B.3)-(B.4) will be met.
I ?rst show that the interior equilibrium exists, and will then show that the
remaining constraints above are met. I now prove that an interior Nash Equi-
librium exists (0 < n
?
h
< dn
?
u
< d and ?
?
(.) = ? =
dn
?
u
n
?
h
> 1). For given
(n
?
u
, n
?
h
), the utilities from each possible action taken by an unskilled immigrant
are: U
uc
(n
?
u
, n
?
h
) = U
uc
(.) =
1
?
[dw
x
(n
?
u
, n
?
h
) + (1 ?d) y] +
??1
?
( ¯ w
uc
(.) ?J (.)) +h(.);
U
us
(n
?
u
, n
?
h
) = ¯ w
us
(.) + b ?K; U
uc,out
(n
?
u
, n
?
h
) = ¯ w
uc
(.) ?J (.) + h(.); and
U
us,in
(n
?
u
, n
?
h
) =
1
?
[dw
x
(n
?
u
, n
?
h
) + (1 ?d) y] +
??1
?
¯ w
us
(.) +b ?K. The util-
ities for a skilled immigrant are:U
hc
(n
?
u
, n
?
h
) = 2y ? w
x
(n
?
u
, n
?
h
) + J (.) + h(n
?
u
, n
?
h
),
U
hs
= w
h
+ b
h
? Z, U
hc,out
(n
?
u
, n
?
h
) = U
uc,out
(n
?
u
, n
?
h
), and U
hs,in
(n
?
u
, n
?
h
) = 2y ?
w
x
(n
?
u
, n
?
h
) + J (.) + b
h
? Z. (n
?
u
, n
?
h
) is a NE only if an immigrant is indif-
ferent between living and working in the enclave, or assimilating and working
in the general economy. Thus U
uc
(n
?
u
, n
?
h
) = U
us
(n
?
u
, n
?
h
) ? w
x
(n
?
u
, n
?
h
) = y +
?
d
[ ¯ w
us
(.) + b ?K ? ¯ w
uc
(.) + J (.) ?h(.)]?
1
d
[y ? ¯ w
uc
(.) + J (.)]. And U
hc
(n
?
u
, n
?
h
) =
U
hs
(n
?
u
, n
?
h
) ? h(n
?
u
, n
?
h
) =
d(U
hs
?y)+?
?
(.)[ wus(.)? wuc(.)+b?K+J(.)]?[y? wuc(.)+J(.)]
?
?
(.)+d
.
68
Substituting h(.) in w
x
(.) yields
w
x
(n
?
u
, n
?
h
) = y+
1
?
?
(.)+d
{?
?
(.) [ ¯ w
us
(.) + b ?K] + (? ?1) [y ? ¯ w
uc
(.) + J (.)] ?U
hs
}.
An unskilled agent prefers settling and working in the enclave to assimilating and
working in the enclave only if U
uc
(n
?
u
, n
?
h
) ? U
us,in
(n
?
u
, n
?
h
) ? w
x
(n
?
u
, n
?
h
) ? y ?
1
d
[y ? ¯ w
us
(.)] ? ? ?
d[U
hs
+ wus(.)? wuc(.)+J(.)]
d[y? wuc(.)+J(.)]+y? wus(.)+d[ wus(.)+b?K]
. Similarly, he prefers to live
and work in the enclave to living in and working out of the enclave if U
uc
(n
?
u
, n
?
h
) ?
U
uc,out
(n
?
u
, n
?
h
) ? w
x
(n
?
u
, n
?
h
) ? y?
1
d
[y ? ¯ w
uc
(.) + J (.)] ? d ?
y? wuc(.)+J(.)
U
hs
? wus(.)?b+K?(y? wuc(.)+J(.))
.
Thus given interior (n
?
u
, n
?
h
) an unskilled person is indi?erent between living and
working in the enclave or living and working out of it, and both alternatives are
preferred to all others.
Next, I show that for given (n
?
u
, n
?
h
) skilled immigrants prefer both to lo-
cate and live in one place, than live and work in di?erent places. In equilib-
rium U
hs
= U
hc
(n
?
u
, n
?
h
) ? U
hs,in
(n
?
u
, n
?
h
) only if h(n
?
u
, n
?
h
) ? b
h
? Z, or equiva-
lently if ? ?
y? wuc(.)+J(.)?d(w
h
?y)
wus(.)? wuc(.)+J(.)+b
h
?Z
, which is satis?ed by assumption. But then
h(1, d) ? h(n
?
u
, n
?
h
) ? b
h
? Z, and so equation (B.4) above is also met and ? > 1.
Notice that U
hc,out
(n
?
u
, n
?
h
) = U
uc,out
(n
?
u
, n
?
h
) ? U
hc
(n
?
u
, n
?
h
) because y ? w
x
?
w
us
? K ? ¯ w
uc
(.) ? J (.). Therefore, given (n
?
u
, n
?
h
) a skilled immigrant is in-
di?erent between living and working in the enclave or out, and both alternatives
are preferred to all others. Thus an interior NE with excess labor supply exists.
Finally because both n
?
u
= n
?
h
= 0 and dn
?
u
= n
?
h
= 1 are NE, and utilities at these
equilibria do not change from those of the proof of proposition (2.5), the enclave
Pareto-dominates assimilation.
69
Proof of Proposition ??. Using the conditions b > 0, w
us
? y << w
h
? 2y , U
hs
>
y + b, b ?K ? b
h
?Z, and
d ?
(y ?w
us
) [U
hs
?w
us
?b + K]
(w
h
?y) [U
hs
?U
us
?2 (y ?w
us
)]
y ?
1
d
[y ?w
us
+ K ?b + h(1, d)] ? w
?
x
(1, d) ? 2y ?U
hs
+ h(1, d)
d [U
hs
?y ?b]
y ?w
us
+ K
? ?
?
(n
u
, n
h
) ?
d [U
hs
?y ?b + K]
y ?w
us
I show that three equilibria exist: i. n
?
u
= n
?
h
= 0; ii. dn
?
u
= n
?
h
= 1 ; and
iii. 0 < dn
?
u
< n
?
h
< d. By assumption equations (B.1), (B.2) and (B.5) are
satis?ed. I will show that equations (B.3) and (B.4) are met. First I prove that
an interior Nash Equilibrium exists (0 < dn
?
u
< n
?
h
< d and ?
?
(.) = ? =
dn
?
u
n
?
h
< 1).
For given (n
?
u
, n
?
h
), the utilities from each possible action taken by an unskilled
immigrant are U
uc
(n
?
u
, n
?
h
) = dw
x
(n
?
u
, n
?
h
) + (1 ?d) y + h(n
?
u
, n
?
h
), U
us
(n
?
u
, n
?
h
) =
w
us
+b?K, U
uc,out
= w
us
?K+h(n
?
u
, n
?
h
) and U
us
(n
?
u
, n
?
h
) = dw
x
(n
?
u
, n
?
h
)+(1 ?d) y+
b ? K. For a skilled immigrant they are: U
hc
(n
?
u
, n
?
h
) = ? (2y ?w
x
(n
?
u
, n
?
h
)) +
(1 ??) y +h(n
?
u
, n
?
h
), U
hs
= w
h
+b
h
?Z, U
hs,in
= ? (2y ?w
x
(n
?
u
, n
?
h
)) +(1 ??) y +
b
h
? Z and U
hc,out
= U
uc,out
= w
us
? K + h(n
?
u
, n
?
h
). An unskilled immigrant is
indi?erent between living and working in the enclave and living and working out of
it if U
uc
(n
?
u
, n
?
h
) = U
us
(n
?
u
, n
?
h
) ? w
x
(n
?
u
, n
?
h
) = y ?
1
d
[y ?w
us
+ b + h(n
?
u
, n
?
h
) ?K];
and a skilled immigrant is indi?erent between these two options if U
hc
(n
?
u
, n
?
h
) =
U
hs
(n
?
u
, n
?
h
), ? h(n
?
u
, n
?
h
) = U
hs
?y?? (y ?w
x
(n
?
u
, n
?
h
)). Using these two equations I
?nd w
x
(n
?
u
, n
?
h
) = y?
U
hs
?(wus+b?K)
?+d
and h(n
?
u
, n
?
h
) =
d
d+?
U
hs
+
?
?+d
(w
us
+ b ?K)?y.
An unskilled immigrant prefers to live and work in the same place rather than live in
the enclave and work out of it only if U
uc
(n
?
u
, n
?
h
) = U
us
(n
?
u
, n
?
h
) ? U
uc,out
(n
?
u
, n
?
h
) ?
70
b ? h(n
?
u
, n
?
h
) ? ? ?
d[U
hs
?y?b]
y?wus+K
. He also prefers the former option to assimilating and
working in the enclave if U
uc
(n
?
u
, n
?
h
) = U
us
(n
?
u
, n
?
h
) ? U
us,in
(n
?
u
, n
?
h
) ? h(n
?
u
, n
?
h
) ?
b ?K ? ? ?
d[U
hs
?y?b+K]
y?wus
.
A skilled immigrant prefers to live and work in the same place rather than live
in the enclave and work out of it if U
hc
(n
?
u
, n
?
h
) = U
hs
(n
?
u
, n
?
h
) ? U
hc,out
(n
?
u
, n
?
h
) =
U
uc,out
(n
?
u
, n
?
h
). Notice U
hc
(n
?
u
, n
?
h
) ? U
uc
(n
?
u
, n
?
h
) because (? + d) y ? (? + d) w
x
(n
?
u
, n
?
h
).
Since U
uc
(n
?
u
, n
?
h
) ? U
uc,out
(n
?
u
, n
?
h
) = U
hc,out
(n
?
u
, n
?
h
), then U
hc
(n
?
u
, n
?
h
) = U
hs
(n
?
u
, n
?
h
) ?
U
hc,out
(n
?
u
, n
?
h
). A skilled immigrant also prefers to live and work in the same place
instead of assimilating and working in the enclave if U
hc
(n
?
u
, n
?
h
) = U
hs
(n
?
u
, n
?
h
) ?
U
hs,in
(n
?
u
, n
?
h
) ? w
x
(n
?
u
, n
?
h
) ? 2y ?w
h
? ? ?
U
hs?(wus+b?K)
w
h
?y
?d. For
d ?
(y?wus)[U
hs
?wus?b+K]
(w
h
?y)[U
hs
?Uus?2(y?wus)]
,
U
hs?(wus+b?K)
w
h
?y
?d ?
d[U
hs
?y?b+K]
y?wus
? ?, hence
U
hs
(n
?
u
, n
?
h
) ? U
hs,in
(n
?
u
, n
?
h
). It remails to show that inequalities (B.3) and
(B.4) are met. From U
uc
(n
?
u
, n
?
h
) = U
us
(n
?
u
, n
?
h
) ? U
us,in
(n
?
u
, n
?
h
) I ?nd that
h(n
?
u
, n
?
h
) ? b ? K; but then h(1, d) ? h(n
?
u
, n
?
h
) ? b ? K. From U
hc
(n
?
u
, n
?
h
) ?
U
hs,in
(n
?
u
, n
?
h
) h(n
?
u
, n
?
h
) ? b
h
? Z + (1 ??) (w
h
?y). Since b ? K ? b
h
? Z and
(1 ??) (w
h
?y) > 0, then h(1, d) ? h(n
?
u
, n
?
h
) > b
h
?Z.
71
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