Description
We consider the dynamic pricing problem of a monopolist ¯rm in a market with repeated interactions, where
demand is sensitive to the ¯rm's pricing history. Consumers have memory and are prone to human decision
making biases and cognitive limitations. As the ¯rm manipulates prices, consumers form a reference price
that adjusts as an anchoring standard based on price perceptions. Purchase decisions are made by assessing
prices as discounts or surcharges relative to the reference price, in the spirit of prospect theory.
We prove that optimal pricing policies induce a perception of monotonic prices, whereby consumers always
perceive a discount, respectively surcharge, relative to their expectations. The e®ect is that of a skimming
or penetration strategy. The ¯rm's optimal pricing path is monotonic on the long run, but not necessarily
at the introductory stage. If consumers are loss averse, we show that optimal prices converge to a constant
steady state price, characterized by a simple implicit equation; otherwise the optimal policy cycles. The
range of steady states is wider the more loss averse consumers are. Steady state prices decrease with the
strength of the reference e®ect, and with customers' memory, all else equal. O®ering lower prices to frequent
customers may be suboptimal, however, if these are less sensitive to price changes than occasional buyers.
If managers ignore such long term implications of their pricing strategy, the model indicates that they
will systematically price too low and lose revenue. Our results hold under very general reference dependent
demand models.
OPERATIONS RESEARCH
Vol. 00, No. 0, Xxxxx 0000, pp. 000–000
issn0030-364X| eissn1526-5463| 00| 0000| 0001
INFORMS
doi 10.1287/xxxx.0000.0000
c 0000 INFORMS
Dynamic Pricing Strategies with Reference E?ects
Ioana Popescu
INSEAD, Decision Sciences Area, Blvd. de Constance, 77300 Fontainebleau, France. [email protected],http://faculty.insead.edu/popescu/ioana
Yaozhong Wu
INSEAD, Technology and Operations Management Area, Blvd. de Constance, 77300 Fontainebleau, France.
[email protected]
We consider the dynamic pricing problem of a monopolist ?rm in a market with repeated interactions, where
demand is sensitive to the ?rm’s pricing history. Consumers have memory and are prone to human decision
making biases and cognitive limitations. As the ?rm manipulates prices, consumers form a reference price
that adjusts as an anchoring standard based on price perceptions. Purchase decisions are made by assessing
prices as discounts or surcharges relative to the reference price, in the spirit of prospect theory.
We prove that optimal pricing policies induce a perception of monotonic prices, whereby consumers always
perceive a discount, respectively surcharge, relative to their expectations. The e?ect is that of a skimming
or penetration strategy. The ?rm’s optimal pricing path is monotonic on the long run, but not necessarily
at the introductory stage. If consumers are loss averse, we show that optimal prices converge to a constant
steady state price, characterized by a simple implicit equation; otherwise the optimal policy cycles. The
range of steady states is wider the more loss averse consumers are. Steady state prices decrease with the
strength of the reference e?ect, and with customers’ memory, all else equal. O?ering lower prices to frequent
customers may be suboptimal, however, if these are less sensitive to price changes than occasional buyers.
If managers ignore such long term implications of their pricing strategy, the model indicates that they
will systematically price too low and lose revenue. Our results hold under very general reference dependent
demand models.
Subject classi?cations : dynamic programming: deterministic; marketing: pricing, promotion, buyer
behavior; inventory policies: marketing/pricing.
Area of review: Manufacturing, Service and Supply Chain Operations
History : Received February 2005; revised October 2005, January 2006; accepted February 2006
1. Introduction
Traditional economic, marketing and operational models view the consumer as a rational agent who
makes decisions based on current prices, income and market conditions. In a market with repeated
interactions, such as frequently purchased consumer goods (e.g. gasoline), services (e.g. resort
hotels, individual insurance) and B2B settings (e.g. media broadcasting, industrial maintenance),
customers’ purchase decisions are also determined by past observed prices.
As customers revisit the ?rm, they develop price expectations, or reference prices, which become
1
Popescu and Wu: Dynamic Pricing with Reference E?ects
2 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
the benchmark against which current prices are compared. Prices above the reference price appear
to be “high”, whereas prices below the reference price are perceived as “low”. The latter e?ect
stimulates short term demand and provides incentives for retailers to run price promotions as a
mechanism to increase short-term pro?ts. On the other hand, price promotions decrease consumers’
price expectations, and hence their willingness to buy the product at higher prices in the future.
(We ignore stock-piling e?ects and assume that consumers are fully informed about product quality,
and do not judge quality level by price.) For the ?rm, this means that high pro?ts today may come
at the expense of a loss in future demand, and hence less pro?t in the future. Therefore, a pro?t
maximizing ?rm must consider the long term implications of its pricing strategy.
Our goal is to characterize what types of pricing strategies are optimal in such repeated inter-
action markets. We investigate when the ?rm should use traditional skimming or penetration
strategies, and whether a constant price versus a cycling policy is optimal in the long run. A dis-
tinctive feature of this work, compared to traditional microeconomic and operational frameworks,
is that it relies on descriptive models of consumer behavior to derive pricing prescriptions under
complex dynamics. Ultimately, we aim to provide normative insights on how behavioral demand
parameters, such as customer loyalty and loss aversion, should reshape and re?ect in managerial
pricing decisions when customers have repetitive, long term relationships with the ?rm.
The marketing literature provides compelling empirical evidence for the dependence of demand
on past prices. Adaptation level theory (Helson 1964) predicts that customers respond to the
current price of a product by comparing it to an internal standard formed based on past price
exposures, called the reference price. While other reference price models exist in the literature, an
empirical comparison conducted by Briesch et al. (1997) shows that “the best [...] model is [...] one
that is based on the brand’s own price history”, i.e. an internal reference price mechanism.
The impact of the reference price on demand, called reference e?ect, is behaviorally explained at
the individual level by the postulates of prospect theory (Kahneman and Tversky 1979). Accord-
ingly, consumers perceive prices as gains (discounts) or losses (surcharges) relatively to a reference
price, and there is an inherent asymmetry in perception, in that losses loom larger than gains
of the same magnitude (loss aversion). In addition, there is diminishing sensitivity to both gains
and losses. The vast empirical validation of prospect theory in the context of reference prices is
best synthesized by Kalyanaram and Winer (1995). Their “key empirical generalizations” validate
the kinked S-shaped reference e?ect at the aggregate demand level. The value of this theory to
managers and researchers alike is that “it predicts how consumers actually behave, rather than
how they ought to behave” (Nagle and Holden 1995).
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 3
With few exceptions, the dynamic pricing literature is oblivious of such behavioral aspects
underlying demand. Recent surveys on dynamic pricing con?rm that the state of the art models
unrealistically assume demand to be given exogenously, and customers’ purchase decisions to be
based solely on the current price posted by the seller (Bitran and Caldentey 2003, Elmaghraby and
Keskino¸cak 2003). Similarly, in industry practice, prices are typically based on empirically esti-
mated demand models that re?ect consumer response conditional on current prices only, indicating
that ?rms typically follow myopic pricing policies.
Recent research trends in dynamic pricing and revenue management investigate the issue of
strategic buyer behavior, whereby consumers optimally time their purchase in anticipation of
future prices (Ovchinnikov and Milner 2005, Liu and van Ryzin 2005). In parallel work, Heidhues
and K¨oszegi (2005) propose an economic model where consumers’ stochastic reference points (for
consumption and price) are determined as a rational expectation “personal” equilibrium. They
investigate conditions for price stickiness in a static monopolistic framework with random cost.
In contrast, we are interested in incorporating the impact of past prices on demand in a dynamic
setting. Our consumers are more realistic than the traditional “homo economicus”, yet boundedly
rational. They have memory and are prone to human decision making biases (such as anchoring
e?ects and loss aversion) and cognitive limitations. In particular, they are unaware of their biases
and do not act strategically.
There are few demand models similar to ours in the dynamic pricing literature. Sorger (1988)
studies local stability of joint dynamic pricing and advertising policies under reference price e?ects,
but his results are not comparable to ours due to the compound advertising e?ect. Also, his
Assumption 2 c. is inconsistent with our model.
Kopalle et al. (1996) and Fibich et al. (2003) – henceforth KRA, respectively FGL – investigate
a problem similar to ours under a much simpler linear demand model with (kinked) linear refer-
ence e?ects. KRA provide computational insights from a discrete time formulation (adapted from
Greenleaf 1995), but their analytical results are limited. They conjecture and verify numerically
(on p.66) the monotonic convergence of optimal prices under asymmetric demand e?ects; some
special cases are proved in their Proposition 2. An elegant proof of this conjecture is due to FGL
in a continuous time control framework. Their approach, contingent on a linear demand model,
reduces the problem to solving a linear Euler equation. This cannot be solved explicitly for non-
linear models, hence “it is natural to ask whether the results of [their] study would remain valid for
more general demand functions” (p.729), and precisely what general structural properties, if not
Popescu and Wu: Dynamic Pricing with Reference E?ects
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just linearity, drive these results. We extend their insights, and provide stronger analytical results,
under very general modeling assumptions on demand.
We provide structural, as opposed to parametric results, that allow to solve the monopolist’s
dynamic pricing problem under a most general reference dependent demand model, that captures
non-linearities and dynamics in response to changes in the reference price. The model is based
on behavioral claims supported by empirical theories; it extends and generalizes existing models
used in the literature (Sorger 1988, Greenleaf 1995, KRA, FGL). Our main technical contributions,
detailed below, include a simple characterization of the steady state price, as well as convergence
and monotonicity properties of the optimal pricing strategy. From a methodological standpoint,
our approach suggests new ways of proving structural properties for dynamic programs with kinked
rewards, which are the minimum of two smooth functions.
We characterize the existence and uniqueness of an optimal constant pricing strategy on the long
run (steady state). If consumers are loss seeking, i.e. more responsive to discounts than surcharges,
the optimal policy cycles. If consumers are loss averse there is a range of steady states, reduced
to a unique one in the loss neutral case. We provide a simple closed form characterization of the
steady state, which allows for insightful sensitivity analysis with respect to behavioral parameters.
The value of the steady state price decreases with customers’ loyalty (memory e?ects) and with
their sensitivity to past prices (reference e?ects), all else equal. Also, the range of steady state
prices is wider the more loss averse consumers are. If customers are heterogeneous in terms of
their shopping frequency (and nothing else), retailers should o?er discounts to loyal customers,
and higher prices to occasional buyers (relatively to the unique price charged in a non-segmented
market). While such strategies are consistent with current practices by retailers in consumer goods
industries, our results indicate that they are not necessarily optimal if occasional buyers are more
responsive to price changes than loyal ones.
The steady state price is bounded between the pro?t maximizing price charged to consumers
who don’t form reference e?ects, and the steady state price charged by a myopic ?rm to reference
sensitive consumers. We show that ?rms who ignore the long term impact of reference price e?ects,
will myopically but systematically underprice.
Our results also characterize the transient monotonicity and convergence of price and reference
price strategies. In general, we show that the strategic monopolist seeks a monotonic reference price
strategy. If consumers’ reference price is initially high, they will be lead to perceive a gain in each
period, as the ?rm will consistently price below the reference price. This perception of monotonic
prices has the e?ect of a skimming strategy. Similarly, a low reference price leads to a penetration
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 5
type strategy. While high-low prices may be observed at the introductory stage, on the long run
prices are eventually monotonic in the same direction as reference prices. We provide conditions
for a monotonic pricing policy to be optimal (i.e. traditional skimming or penetration). This is true
for example if consumers’ memory is short (i.e. the reference price equals the last price observed),
or if reference prices have increasing marginal impact on demand. In particular, this includes the
linear models studied by KRA and FGL.
In summary, our results for general reference dependent demand models con?rm the robust-
ness of FGL and KRA insights concerning stability and steady state analysis, but not necessarily
monotonicity of the transient pricing policy. The latter appears to be a result of their linear refer-
ence e?ect assumption. Our approach is similar in spirit to the two-stage method in FGL, but our
analysis relies on structural arguments, as opposed to explicit solutions of a parametric model.
The remaining of the paper is structured as follows. Section 2 describes a very general model of
reference dependent demand based on behavioral ?ndings, in particular prospect theory. We sum-
marize behavioral claims that form the base of our various, parsimonious assumptions in the rest
of the paper. Section 3 presents the dynamic pricing model. The next two sections are structured
around consumer’s loss aversion. Section 4 focuses on models where consumers are loss neutral,
with the loss seeking case brie?y addressed at the end. We investigate the existence of the steady
state, and stability and monotonicity properties of the optimal price and reference price strategies;
these are benchmarked against myopic policies. Section 5 studies the same issues when consumers
are loss averse, i.e. the reference e?ect is consistent with prospect theory. Section 6 concludes our
?ndings.
2. General Reference Dependent Demand Model
This section describes how consumers make purchase decisions based on prices and reference prices.
The most prominent theory of reference dependent preferences is prospect theory (Kahneman and
Tversky 1979); the deterministic versions in Thaler (1985) and Tversky and Kahneman (1991) are
most relevant to our context.
The mental accounting framework (Thaler 1985) proposes that the total utility from purchasing
a product consists of two components: acquisition utility and transaction utility. The former cor-
responds to the monetary value of the deal, determined by the discrepancy between price and the
value of the product to the consumer. Transaction utility corresponds to the psychological value of
the deal, determined by the discrepancy between price and reference price. Thus, reference prices
a?ect customer behavior via the magnitude of the perceived “gain” or “loss” x =r ?p relative to
Popescu and Wu: Dynamic Pricing with Reference E?ects
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the reference point r. The microeconomic demand model induced by such individual preferences is
derived by Putler (1992), who validates it empirically with egg sales data. Empirical validation of
prospect theory at the demand level is remarkably synthesized by Kalyanaram and Winer (1995).
De?ne the reference e?ect on demand R(r ?p, r) =D(p, r) ?D(p, p) as the di?erential demand
at price p due to the reference price being r instead of the actual price p. Alternatively, R(x, r)
measures the impact on demand of a perceived discount/surcharge x =r?p relative to the reference
price r. Empirical evidence suggests that the larger magnitude of the perceived gain/loss, the
larger the corresponding absolute impact on demand; this is known as Reference Dependence (RD).
Furthermore, it has been observed that for normal goods purchase experience with higher past
prices increases the likelihood of buying the product for a given observed price. This leads to the
following standing assumption on the demand model:
Assumption 1 (Reference Dependent Demand). Demand D(p, r) is non-negative bounded
and continuous, decreasing in price p and increasing in the reference price r. The corresponding
reference e?ect R(x, r) is increasing in x.
In particular, the de?nition of the reference e?ect together with monotonicity of demand implies:
Remark 1. R(x, r) ?0 for x >0, R(x, r) ?0 for x 1; Loss Neutrality (LN): ? ?1; Loss Seeking (LS): ?(·) x
2
. If f is di?erentiable, this amounts to positiv-
ity of the cross-partial derivative f
xy
(x, y) ? 0 (partial derivatives are denoted by corresponding
subscripts). Submodularity is de?ned by the opposite inequality.
Claim (DC) states that R(x, r) is submodular on the gain, respectively loss domain; this is
irrespective of the gain-loss asymmetries discussed previously. An illustration of this claim under
prospect theory is provided in Figure 1.
2.3. Absolute and Relative Di?erence Models.
We next provide simple examples of reference e?ects used in the literature. The most common is
the absolute di?erence (AD) model R
AD
(x, r) =h(x), with h increasing and h(0) = 0. The shape
Popescu and Wu: Dynamic Pricing with Reference E?ects
8 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
Figure 1 Reference e?ect R(x, r) satisfying (PT) and (DC).
1
1.2
1.4
1.6
1.8
2
?1
?0.5
0
0.5
1
?2
?1.5
?1
?0.5
0
0.5
1
x
r
R
(
x
,
r
)
of the evaluation function h further controls for loss aversion, diminishing sensitivity etc. KRA
and FGL use AD models with h (piecewise) linear. An important criticism to AD models is their
failure to capture dynamics of the reference e?ect in response to changes in the reference price. A
natural generalization that satis?es (DC) is R(x, r) =h(x)s(r), with s a positive decreasing scaling
function.
Another model used in the literature is the relative di?erence (RD) model R
RD
(x, r) = h(
x
r
)
(with h increasing and h(0) =0), motivated by the Weber-Fechner law in psychophysics. This law
suggests that consumers perceive price di?erences in proportional, rather than absolute terms, a
theory that is experimentally supported by mental accounting studies, such as the small example
in the previous section (that can be explained by consumers not responding to the magnitude of
the price di?erential r ?p =50 Euro, but to the percentage di?erence relative to the base price:
r?p
r
= 50% vs. 4%). RD models have been used in conjunction with linear and logarithmic forms
for h (Winer 1988), as well as exponentials (Anderson and Rasmussen 2004). A generalization of
RD models is R(x, r) =h(xs(r)), with s a positive decreasing scaling function. This satis?es (DC)
if h is convex increasing, or h
is inelastic to changes in x.
While our analysis focuses on the most general reference dependence demand model, we system-
atically indicate how our general results specialize to the absolute and relative di?erence models.
3. Dynamic Pricing Model
This section sets up our dynamic pricing model, together with general assumptions on how con-
sumers’ memory works in adjusting reference prices based on price perceptions.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 9
We consider a pro?t maximizing monopolist who faces a homogeneous stream of repeated cus-
tomers over an in?nite time horizon. The monopolist decides in every period what price p to charge
to customers, from a bounded positive interval P = [0, ¯ p] (our results allow for a non-zero lower
bound on price; see the Remark of Section B in the Appendix). The resulting pro?t is ?(p, r) =
(p ?c)D(p, r), where r is the current reference price and D(p, r) the corresponding demand satis-
fying Assumption 1. We assume for simplicity and without loss of generality the variable cost to
be c =0; all our results hold for a non-zero constant cost c.
In each period, customers update their current reference price r
t
as they observe the new price
p
t
charged by the ?rm. The adaptive expectation framework Nerlove (1958) motivates a reference
price that is a weighted average of past prices, where more recent prices are given more weight.
Throughout of the paper, we assume that the reference price formation mechanism is given by:
r
t
=?r
t?1
+(1 ??) p
t?1
, ? ?[0, 1). (1)
Exponential smoothing is the most commonly used and empirically validated reference price
mechanism in the literature (e.g. Winer 1986, Sorger 1988, Greenleaf 1995, KRA, FGL). Some of
our results hold for more general mechanisms, including in?ation e?ects and price trends in general
(in this case our insights and results remain valid for in?ation-adjusted prices). The memory model
described by (1) allows for additional insights by quantifying the speed of the memory adjustment
process. The memory parameter ? captures how strongly the reference price depends on past prices.
Lower values of ? represent a shorter-term memory; in particular, if ? =0 then the reference price
equals the past price. Memory is an indicator for shopping frequency, hence can be a proxy for
customer loyalty. Actual estimated parameters range from ? = 0 (Krishnamurthi et al. 1992) to
? =0.925 (Greenleaf 1995), much depending on the type of product or service involved.
For an initial reference price r
0
, we model the ?rm’s long term pro?t optimization problem as:
V (r
0
) = sup
p
t
?P
?
t=0
?
t
?(p
t
, r
t
)
s.t. r
t
=?r
t?1
+(1 ??)p
t?1
, t ?1,
where ? ?(0, 1) is a discount factor. Pro?t-per-stage is bounded, so the value function is the unique
bounded solution of the following Bellman equation (Stokey et al. 1989):
V (r) = sup
p?P
W(p, r), where W(p, r) =?(p, r) +?V (?r +(1 ??)p). (2)
A consequence of Assumption 1 is that short term pro?t ?(p, r), and the value function are
increasing (Stokey et al. 1989, Theorem 4.7), i.e. larger reference prices are preferable for the ?rm.
Popescu and Wu: Dynamic Pricing with Reference E?ects
10 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
Remark 2. The value function V (r) is an increasing function of the reference price r.
Because the action set P is compact and all functions are continuous, there exists an optimal
stationary pricing policy that solves Problem (2), and associates with any given reference price r,
one or more optimal prices. We de?ne the optimal pricing policy as p
?
(r) = arg max
p
W(p, r). To
avoid the ambiguity of multiple solutions, arg max
x
f(x) refers without loss of generality to the
largest maximizer of the function f. The optimal reference price policy q
?
(r) =?r +(1 ??)p
?
(r)
provides the next reference price associated with the optimal pricing policy.
The pricing path {p
?
t
} starting with an initial state r
0
is computed sequentially as p
?
t
=p
?
(r
?
t
),
where the state variable is updated in each period via the reference price formation mechanism
r
?
t
= q
?
(r
?
t?1
). The sequence {r
?
t
} of optimal reference prices is referred to as the state path; for
the consumer, this is the sequence of perceived prices or price expectations. A steady state p
??
for Problem (2) is one from which it is suboptimal to deviate, i.e. p
?
(p
??
) =q
?
(p
??
) = p
??
, so the
optimal price path starting at r
0
=p
??
is constant.
Our goal is to investigate when such a steady state exists and to characterize stability and
monotonicity properties of the price and reference price paths. These results critically depend on
consumer’s reaction to discounts and surcharges, captured by loss aversion.
4. Model without Loss Aversion
This section focuses on the case when consumers are loss neutral (LN), i.e. the reference e?ect is
smooth; loss seeking behavior is investigated at the end of this section. We ?rst characterize the
unique steady state, and then investigate stability and monotonicity properties of the price and
reference price strategies, under very general modeling assumptions. Two additional assumptions
are made throughout Sections 4.1-4.3.
Assumption 2 (Smooth Reference E?ect). R(x, r) is twice di?erentiable in x and r.
The slope of the reference e?ect at x =0 is denoted ?(r) =R
x
(0, r). Our results require some typ-
ical technical assumptions on the short term pro?t ?(p, r) =?(p) +?
R
(p, r). Here ?(p) =?(p, p) =
pD(p, p) is the pro?t in a market where consumers don’t form reference e?ects (R?0), referred to
as base pro?t, and ?
R
(p, r) =pR(r ?p, r) is the pro?t from the reference e?ect, or reference pro?t.
Assumption 3 (Pro?t). (a) ?(p) is non-monotonic and concave in p. (b) ?
p
(r, r) =?
(r) ?r?(r)
is strictly decreasing in r. (c) ?
R
(p, r) is concave in p and supermodular in (p, r).
Some intuition and motivation for these assumptions is provided next. Non-monotonicity of ?(p)
states that the ?rm is not constrained to charge sub-optimal prices (in a market without reference
e?ects). This will allow to rule out pathological boundary steady states.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 11
Part (b) states that, if the ?rm charges the reference price, a small change in price has higher
impact on pro?t at lower reference prices. That is, marginal pro?t diminishes as price is set to
the reference price. Strict monotonicity is necessary to insure uniqueness of the steady state. A
su?cient condition for Assumption 3 (b), given (a), is r?(r) strictly increasing in r.
Diminishing marginal pro?t (concavity) is a typical economic assumption. Concavity of ? and
?
R
implies concavity of short term pro?t ?. The supermodularity (see p.7) assumption (c) states
that marginal short term pro?t increases with the reference price. This is consistent with modeling
assumptions used in the literature, and summarized by Winer (1988) (those demand functions are
typically additively separable, in particular linear, in price and reference price). Various sets of
su?cient conditions are provided in Section 5.3. One is supermodularity of demand D(p, r), which
in particular implies that demand is more elastic at higher reference prices. In support of this
assumption, Kamen and Toman (1970) found evidence that with higher reference prices, consumers
were more likely to switch brands of gasoline, for a given price di?erential.
4.1. Myopic Strategies for Loss Neutral Buyers
A ?rm that ignores the e?ect of current prices on future demand will focus on maximizing short
term pro?t. Intuitively, such a myopic ?rm should charge higher prices as consumers’ price expec-
tations are higher. This is insured by pro?t supermodularity (Assumption 3 (c)), via Topkis’
Theorem (Topkis 1998, Theorem 2.8.2 implies x
?
=arg maxf(x, y) increasing in y whenever f
is supermodular).
Lemma 1. The single period pro?t maximizing price ˜ p(r) =arg max
p?P
?(p, r) is increasing in r.
Formally, the problem solved by a myopic ?rm is equivalent to setting ? = 0 (full discounting)
in Problem (2). Consider two ?rms, one myopic, the other strategic (optimizer), starting with the
same initial reference price r
0
, and applying sequentially their respective pricing policies, while con-
sumers’ memory evolves according to the same ?-smoothing mechanism. The state of the system
at time t, for the myopic and optimizing ?rms are given by r
M
t
and r
?
t
, respectively. The corre-
sponding myopic, respectively optimal prices charged at time t are p
M
t
= ˜ p(r
M
t
) and p
?
t
= p
?
(r
?
t
).
We show that the myopic ?rm systematically underprices, following a monotonic pricing strategy.
The results are driven by supermodularity of short term pro?ts (Assumption 3 (c)).
Proposition 1 (Myopic vs. Strategic Firm). For any reference price r, the myopic ?rm
underprices the product, i.e. p
?
(r) ? ˜ p(r). Furthermore, for any initial reference price r
0
, the price
charged by the myopic ?rm at any point in time is less than the optimal price, i.e. p
M
t
?p
?
t
, for all
t. Moreover, all myopic price paths monotonically converges to a unique constant price p
M?
.
Popescu and Wu: Dynamic Pricing with Reference E?ects
12 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
The prices charged by the myopic ?rm are oblivious of their eroding e?ect on future demand,
hence future pro?ts. By charging higher prices, the optimizing ?rm trades o? current pro?ts for
future long-term pro?tability from higher reference prices. Figure 2 illustrates these results; in
this case, the observed di?erence between optimal and myopic pro?ts is about 10%. Throughout
the paper, numerical examples are calculated using the Compecon Matlab toolbox developed by
Miranda and Fackler (2002).
Figure 2 Myopic vs. optimal pricing policies; D(p, r)=1?2p +
r?p
5r
, P =[0.1, 0.3], ? =0.80, ? =.95.
5 10 15 20 25 30 35 40
0.15
0.2
0.25
t
P
r
i
c
e
Optimal (r
0
=0.3)
Myopic (r
0
=0.3)
Optimal (r=0.1)
Myopic (r
0
=0.1)
Figure 2 also suggests that the optimal pricing paths may converge monotonically to a single
steady state price. These issues are investigated next.
4.2. Steady State for Loss Neutral Buyers
De?ne the pro?t optimizing price in absence of reference e?ects ˜ p = arg max
p?P
?(p). The Bellman
equation (2) implies that, if a steady state p
??
exists, then V (p
??
) =
?(p
??
)
1 ??
, whereas generally
V (r) ?
?(r)
1 ??
for all r ?P. Assumption 3 (a,b) insures that such a steady state is unique and can
be computed as the solution of a simple implicit equation:
Theorem 1 (Steady State). If Problem (2) admits a steady state p
??
, then this is the unique
solution of:
?
(p)
1 ??
=
p?(p)
1 ???
. (3)
Furthermore, p
??
is continuously decreasing in ?, increasing in ?, and satis?es p
M?
?p
??
? ˜ p.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 13
At steady state, a small increase in price ?p leads to a marginal increase in the long-term base
pro?t equal to
?
(p)
1??
?p. However, on the transition, customers perceive a loss and reduce their
demand. The resulting pro?t loss is p?(p)?p in the ?rst period, ?p?(p)?p in the second period and
so on. Hence the implicit equation (3) re?ects the balance between maximizing long-term pro?ts
(without reference e?ects) and the discounted transitory costs due to the reference e?ect.
Equation (3) con?rms that myopic ?rms (? =0) set lower steady state prices p
M?
. On the long
run, strategic ?rms should charge lower prices when consumers form reference e?ects, than when
they do not, i.e. p
??
? ˜ p, approaching it as ? ?1.
In particular, Theorem 1 obtains the following closed form expressions for the corresponding
steady states for absolute and relative di?erence models discussed in Section 2.3. This generalizes
the FGL result for linear reference e?ects.
Corollary 1 (Steady State for AD and RD Models). Consider the problem with linear
base demand D(p, p) = b ? ap, and reference e?ect R
AD
(x, r) = h(x), respectively R
RD
(x, r) =
h(x/r). The corresponding steady states, if they exist, are p
??
AD
=
b
2a+?
and respectively p
??
RD
=
b??
2a
,
where ?=h
(0)
1??
1???
.
Consumer Heterogeneity and Sensitivity Analysis Insights. Consider a heterogeneous
market, where consumer types di?er in shopping frequency, captured by the memory parameter
?
i
, and sensitivity to past prices, i.e. the reference e?ect R
i
. Demands without reference e?ects are
assumed identical across segments (i.e. D
i
(p, p) =D(p, p)).
If the ?rm can segment the market, Theorem 1 indicates that on the long run, each segment
should be charged a steady state price p
??
i
that solves
?
(p)
p(1??)
=
?
i
(p)
1??
i
?
. The steady state price only
depends on the reference e?ect via its slope at zero, speci?cally, it decreases with the magnitude
of ?(·). That is, if consumers share the same memory parameter ?, those who are more sensitive
to past prices will be charged lower prices on the long run.
Remark 3. If ?
1
=?
2
and R
2
is a “stronger” reference e?ect than R
1
, i.e. if ?
1
(r) ??
2
(r) for all
r, then the corresponding steady state prices, if they exist, satisfy p
??
1
?p
??
2
.
On the other hand, if consumers only di?er in their memory parameter ?
i
, then segments with
higher ?
i
are charged lower steady state prices, all else equal. While charging lower prices to loyal
customers (proxied by higher ?
i
) is in line with current practices in consumer goods industries, our
results show that it may not be the optimal strategy if these shoppers are less sensitive to price
changes, as measured by ?
i
. In general, the ?rm should charge lower prices to segments with higher
Popescu and Wu: Dynamic Pricing with Reference E?ects
14 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
long run marginal sensitivity to the reference e?ect, as measured by ?
i
(·)/(1 ??
i
?). Speci?cally,
the same argument as Remark 3 shows that p
??
1
?p
??
2
, if
?
1
(r)
1??
1
?
?
?
2
(r)
1??
2
?
holds for all r.
Finally, for comparison purposes, consider the case where the ?rm cannot segment the market and
o?ers the same price to all segments. We show in Section D of the Appendix that the common steady
state price (if one exists) solves
?
(p)
1??
=p
N
i=1
?
i
(p)
1??
i
?
, and is bounded between the lowest and highest
prices charged to each consumer type in a perfectly segmented market p
??
?[min
i
p
??
i
, max
i
p
??
i
].
4.3. Stability and Monotonicity with Loss Neutral Buyers
This section investigates the existence and stability of the unique steady state price characterized
in Theorem 1. We show global convergence of the optimal prices to steady state, and study local
and global monotonicity properties of the state and optimal price paths.
We start with the most general results that require no further assumptions. Recall that the
reference price policy was de?ned by q
?
(r) =?r +(1 ??)p
?
(r).
Lemma 2 (Monotone Reference Price Policy).The reference price policy q
?
(r) increases in r.
The result follows from Assumption 3(c), and implies the following:
Theorem 2 (Global Stability and State Path Monotonicity). Problem (2) admits a unique
steady state p
??
, characterized by Theorem 1, to which all optimal price paths converge. All corre-
sponding reference price paths converge to p
??
monotonically.
Under the conditions of Theorem 2, the monopolist’s strategy involves decreasing customer’s
price expectations by always pricing below the current reference price, or vice versa. If the refer-
ence price is high, the optimal pricing policy is a skimming-like strategy: the ?rm starts with a
relatively high introduction price, then always prices below the reference price (r
t
> p
t
). In this
case, consumers perceive in each period a gain relative to their expectations, which stimulates
demand, and hence pro?ts in the current period. Low reference prices lead to a penetration-type
strategy, whereby the ?rm always prices above the customers’ reference price (r
t
< p
t
). The loss
in current pro?t from a negative reference e?ect, is outweighed (1) by a higher short-term pro?t
without reference e?ect, ?(p
?
t
), and (2) by the increase in future pro?ts due to higher reference
prices in the next period.
Theorem 2 does not guarantee monotonicity of the corresponding pricing paths. This holds
whenever the pricing policy is monotonic. Some su?cient conditions are provided next. Recall that
a real-valued C
2
function f(x, y) is strongly concave if there exist positive constants a and b such
that f(x, y) +ax
2
+by
2
is concave.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 15
Lemma 3 (Monotone Price Policy). Each of the following alternative conditions is su?cient
for the optimal pricing policy p
?
(r) in Problem (2) to be increasing in r :
(a) Consumers have short memory ? =0, i.e. r
t+1
=p
t
for all t;
(b) Demand D(p, r), or equivalently, the reference e?ect R(r ?p, r), is convex in r;
(c) Short term pro?t ?(p, r) is strongly concave on the interior of its domain, and there exists
a global constant K >0 such that ?
rr
(p, r) ??K and ?
pr
(p, r) ?K/2.
The result follows from supermodularity of W(p, r), the function on the right hand side of the
Bellman equation (2). This reduces to supermodularity of short term pro?t (Assumption 3 (c))
under condition (a). Condition (b) achieves this by additionally insuring convexity of short term
pro?t in r, and hence of the value function. Condition (c) insures a uniform lower bound on the
degree of concavity of the value function V, that is compensated by supermodularity of short term
pro?ts. At the limit, for K =0 we recover the conditions of part (b). Strong concavity is a technical
condition insuring second order di?erentiability of the value function.
Theorem 3 (Global Price Monotonicity). Under the conditions of Lemma 3, all optimal pric-
ing paths for Problem (2) converge monotonically to the unique steady state price p
??
, characterized
by Theorem 1.
Figure 3 Optimal pricing paths for D(p, r) =e
?p
+0.5(e
0.5(r?p)
?1), ? =0.8, ? =0.9.
5 10 15 20 25 30
0.74
0.76
0.78
0.8
0.82
0.84
0.86
0.88
t
P
r
i
c
e
In particular, we obtain the price monotonicity results of KRA and FGL for linear AD models
R
AD
(x, r) =?x; these satisfy part (b) of Lemma 3. Part (c) of Lemma 3 (but not (b)) obtains price
Popescu and Wu: Dynamic Pricing with Reference E?ects
16 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
monotonicity for linear RD models R
RD
(x, r) = ?x/r, on a price range P = [p, ¯ p] with ¯ p/p ?
4
?
2,
i.e. price dispersion not exceeding 18%.
Theorem 3 insures that only three possible pricing strategies are optimal: skimming (if initial
price expectations are high, start with a high price and then keep decreasing it), penetration (if
initial price expectations are low, start with a low price, then increase it), or constant pricing.
Moreover, after a transient regime, the optimal prices stabilize to a steady state price. The results
follow from the monotonicity of the policy function (Lemma 3). This also insures that the price
paths do not cross, i.e. any increase in the initial reference price shifts the entire pricing sequence
upwards. These results are illustrated with an example in Figure 3.
Furthermore, in a neighborhood of the steady state, prices are eventually monotonic, i.e. local
price monotonicity holds. Strong concavity of short term pro?ts is a technical condition that insures
di?erentiability of the value function and pricing policy.
Proposition 2 (Local Price Monotonicity). If ?(p, r) is strongly concave on the interior of
its domain, then all optimal pricing paths are locally monotonic around the steady state.
Local price monotonicity implies that, on the long run, the slope of the reference price policy
q
?
(r) =?r +(1 ??)p
?
(r) is steeper than ?. That is, reference prices converge to steady state at a
rate that is ultimately faster than the memory parameter ?.
Transient High-Low Pricing. At the introductory stage, however, the monopolist may have
an incentive to increase reference prices at a slower rate, achieved by high-low pricing. Such a
strategy, depicted in Figure 4, contrasts the insights obtained by KRA and FGL under linear
demand models. The optimal pricing policy p
?
(r) and price path {p
?
t
} are non-monotonic under the
conditions of Theorem 2 and Proposition 2, but violating Lemma 3. The pricing strategy illustrated
in Figure 4 is consistent for example with catastrophe insurance pricing in the U.S. before major
hurricanes, when consumers reference prices are much lower than pro?t optimizing prices (we thank
Paul Kleindorfer for suggesting this example; see endnote 1. for intuition).
This example illustrates that high-low pricing is technically possible in the loss neutral case,
contrary to previous insights in the literature; similar behavior can be observed for RD models.
Nevertheless, extensive numerical simulations revealed that the optimal pricing path is typically
monotonic; at most, it displays a relatively small direction swing from decreasing to increasing
prices, as in Figure 4 (simulated models for R(x, r) include xe
?ar
, x/r
k
, 1?e
?ax
, e
ax/r
k
?1, a, k >0
for a variety of parameters satisfying Assumptions 1-3). So it is fair to say that, if consumers are
loss neutral, we do not ?nd strong numerical support for transient high-low pricing, but cannot
rule it out either.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 17
Figure 4 High-low pricing policy p
?
(r) and optimal price path p
?
t
; D(p, p) = 2 ?0.5p, R(x, r) =0.5(e
1.5x/r
3
?1),
P =[1, 2], ? =0.950, ? =0.999, r
0
=1.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
1.9
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2
r
p
*
(r)
10 20 30 40 50 60 70 80 90 100
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2
t
P
r
ic
e
4.4. Loss Seeking Buyers
High-low pricing is provably optimal if consumers are focused on gains, i.e. loss seeking (LS). In this
case the optimal pricing policy for Problem (2) cycles. Consumers are loss seeking if they respond
more to discounts than to surcharges, i.e. if the reference e?ect has a convex kink at zero. This
type of asymmetry is inconsistent with prospect theory, but has found some empirical validation
in the marketing literature (e.g. Krishnamurthi et al. 1992, Greenleaf 1995). We address it for
completeness; only Assumption 1 stands in place for the next result:
Proposition 3 (Cyclic Policy). If buyers are loss seeking, i.e. the reference e?ect R(x, r) sat-
is?es (LS), then Problem (2) admits no steady state.
Proposition 1 in KRA obtains a similar result when R(x, r) =h(x) is piecewise linear in x.
Proposition 3 suggests that in order for a steady state price to exist, the reference e?ect R must
be smooth, or kinked in the direction predicted by prospect theory. This case is analyzed next.
5. Model with Loss Aversion
This section investigates the transient and long term behavior of the optimal dynamic pricing
policy when reference e?ects are kinked in the spirit of prospect theory. We ?rst set up the model;
two additional assumptions on the reference e?ect, respectively pro?t are made throughout this
section (replacing Assumptions 2 and 3 for the loss neutral case).
5.1. Model and Assumptions
At the aggregate level, prospect theory predicts that the reference e?ect is captured by an S-shaped
function of the reference gap x =r ?p, with a concave (loss averse) kink at x =0 (Kahneman and
Popescu and Wu: Dynamic Pricing with Reference E?ects
18 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
Tversky 1979, Tversky and Kahneman 1991).
The following assumption, made throughout this section, formalizes the technical aspects relevant
for our purposes. An illustration is provided in Figure 5. Recall that a function f(x) is single-
crossing if there exists a unique x
0
in its domain such that f(x
0
) =0. We denote 1
X
the indicator
function of the set X.
Figure 5 An example of kinked reference e?ect R
K
Assumption 4 (Kinked Reference E?ect). The kinked reference e?ect is given by
R
K
(x, r) =1
(x?0)
R
G
(x, r) +1
(x?0)
R
L
(x, r), (4)
where R
L
(x, r) and R
G
(x, r) are twice di?erentiable, increasing in x with R
G
(x, r)?R
L
(x, r) single-
crossing in x, and satisfying ?
L
(r) >?
G
(r) for all r.
Assumption 4 is a weaker formalization of prospect theory (PT), as detailed in Section 2.1,
including reference dependence (RD), and the weaker form of loss aversion (LA) in terms of the
index ?(r) =
?
L
(r)
?
G
(r)
, proposed by K¨obberling and Wakker (2005). We do not speci?cally assume
diminishing sensitivity (DS) (i.e. concavity/convexity of the gain/loss components), as it is imma-
terial for our analysis. The single crossing condition allows to write the kinked reference e?ect as
the minimum of the loss and gain counterparts on the relevant domain:
R
K
(x, r) =min(R
G
(x, r), R
L
(x, r)). (5)
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 19
This condition is unrestrictive in the context of Assumption 4 alone, because the loss and gain
parts of R
K
can always be smoothly extended so as to satisfy Eq. (5).
Based on (5), the pro?t per-period can be written as
?
K
(p, r) =min(?
G
(p, r), ?
L
(p, r)), (6)
where ?
G
(p, r) =?(p) +pR
G
(r ?p, r) and ?
L
(p, r) =?(p) +pR
L
(r ?p, r). In line with our previous
assumptions on pro?t, the following technical assumption is made in this section:
Assumption 5 (Gain/Loss Pro?t). The short term pro?ts ?
G
(p, r) and ?
L
(p, r) associated with
the smooth reference e?ects R
G
, respectively R
L
, satisfy Assumption 3.
Section 5.3 provides su?cient modeling conditions on the reference e?ect for Assumptions 4 and
5 to hold, in particular when R
K
is convex-concave, as predicted by prospect theory. In particular,
it provides concrete ways to construct R
L
, R
G
that satisfy Assumption 3, and so that Eq. (5) holds.
The corresponding Bellman equation for the problem with kinked demand e?ect is:
(K) V
K
(r) =sup
p
{?
K
(r, p) +?V
K
(?r +(1 ??)p)}. (7)
5.2. Steady State, Stability and Monotonicity
We extend the results obtained in Section 4 for the loss neutral case, to infer existence and stability
properties of the steady state for loss averse buyers. A key role in our analysis here is played by
the problem with smooth reference e?ect:
R
?
(x, r) =?R
G
(x, r) +(1 ??)R
L
(x, r). (8)
Denoting the corresponding short term pro?t ?
?
(p, r) =?(p) +pR
?
(r ?p, r), this problem is
(P
?
) V
?
(r) =sup
p
{?
?
(p, r) +?V
?
(?r +(1 ??)p)}. (9)
For any ? ?[0, 1], Problem (P
?
) satis?es our assumptions for the loss neutral case, so all the results
of Section 4 apply. Moreover, Eq. (5) implies
R
K
(x, r) =min(R
G
(x, r), R
L
(x, r)) ?R
?
(x, r), ? ?[0, 1]. (10)
The same relationship between short term pro?ts, and the corresponding value functions is implied.
The proof details of the following result are in the Appendix:
Lemma 4. For any ? ? [0, 1], Problem (P
?
) satis?es Assumptions 1-3. Its value function satis?es
V
?
(r) ?V
K
(r) for all r, and its steady state p
??
?
is also a steady state for Problem (K).
Popescu and Wu: Dynamic Pricing with Reference E?ects
20 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
By loss aversion and construction (8), the slopes of the reference e?ects R
L
, R
?
and R
G
satisfy
?
L
(r) ??
?
(r) ??
G
(r). Remark 3 further implies that the steady state prices of the corresponding
problems satisfy p
??
L
? p
??
?
? p
??
G
. Moreover, as ? ? [0, 1], the steady states p
??
?
span the interval
[p
??
L
, p
??
G
].
We are now ready to characterize the set of steady states for general non-linear loss-averse
demand e?ects, and their stability properties, as follows:
Theorem 4 (Steady State, Stability and Monotonicity under Loss Aversion). The set
of steady states for Problem (K) equals [p
??
L
, p
??
G
], so for an initial reference price r
0
?[p
??
L
, p
??
G
], the
optimal price path is constant p
?
t
=r
0
. For an initial reference price r
0
p
?
G
,
the reference price path {r
?
t
} converges monotonically to p
??
L
, respectively p
??
G
, and the corresponding
optimal price path {p
?
t
} also converges to the same steady state.
Our analysis, based on bounding the kinked reference e?ect, and hence corresponding value
function, by the full range of convex combinations of the smooth loss and gain counterparts,
suggests a general approach for obtaining structural properties and global stability in dynamic
programs with kinked reward structure. A similar approach is used by FGL in their analysis of
kinked linear demand models; their results are driven by the explicit solution of the linear demand
model, while ours rely on structural convexity arguments.
An illustration of the steady states, pricing strategies and corresponding value function under
loss aversion is provided in Figure 6. While technically, high-low pricing is possible within the
assumptions of Theorem 4 (e.g. using the model of Figure 4 for R
L
), numerical results suggest that
with most common demand models, prices are typically monotonic. The price monotonicity results
obtained in the loss neutral case extend as follows:
Proposition 4 (Price Monotonicity under Loss Aversion). If the problems with reference
e?ects R
G
and R
L
also satisfy the conditions of Lemma 3, respectively Proposition 2, then the price
paths for Problem (K) are globally, respectively locally monotonic. In particular, if consumers have
short term memory (? =0), then prices are globally monotonic.
Furthermore, the strategy of a myopic ?rm under loss aversion resembles that for the loss neutral
case, except that there is a range of myopic steady states. The proof of the next result follows the
combined spirit of Lemma 4, Theorem 4 and Proposition 1.
Proposition 5 (Myopic Prices under Loss Aversion). Given any initial reference price r
0
,
the myopic price path {p
M
t
} lies below the optimal price path {p
?
t
} and converges monotonically to
a myopic steady state. Furthermore, the set of myopic steady states equals [p
M?
L
, p
M?
G
], where p
M?
L
and p
M?
G
are the steady states for a myopic ?rm under reference e?ect R
L
, respectively R
G
.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 21
Figure 6 Pricing strategies and value function under prospect theory; D(p, p) = 2 ? 0.6p, R
G
(x, r) = .25(1 ?
e
?x/r
), R
L
(x, r) =0.8(e
x/r
?1), P =[1, 2], ? =0.80, ? =0.95, r
0
=1, 2.
5 10 15 20 25 30 35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
t
P
r
i
c
e
p
G
**
p
L
**
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
31
31.5
32
32.5
33
33.5
34
r
V
(
r
)
p
L
**
p
G
**
V
G
V
?
V
L
While the myopic ?rm systematically underprices, under loss aversion, its policy may be opti-
mal for a range of initial reference prices. For example, consider a piecewise linear AD model
with D(p, p) = 1 ?p, ?
L
= 0.5, ?
G
= 0.2, ¯ p = 0.6 and ? = 0.95, ? = 0.90. We obtain [p
M?
L
, p
M?
G
] =
[0.400, 0.455] and [p
??
L
, p
??
G
] =[0.426, 0.468]. In this case, if r
0
?[0.426, 0.455], the myopic (constant)
pricing strategy and the optimal one coincide.
5.3. Special Demand Models.
The assumptions underlying Theorem 4 impose additional, implicit conditions on the demand
model. These involve smoothly extending the loss and gain component of a given kinked reference
e?ect over the entire domain so as to satisfy Eq. (5), and verifying the technical pro?t Assumption
3 on each component. This section provides explicit modeling conditions on the reference e?ect that
are su?cient to validate these assumptions. We focus on reference e?ects satisfying the prospect
theory claims motivated in Section 2, including diminishing sensitivity (DS) and decreasing curva-
ture (DC). We further infer speci?c implications for AD and RD models. Finally, we explain how
our results extend the ones in the literature, obtained for linear AD models.
5.3.1. Prospect Theory Models. Given a kinked S-shaped reference e?ect R
K
(x, r), as pre-
dicted by prospect theory, the challenge in satisfying Assumption 4 is to smoothly “extend” the
convex (loss) and concave (gain) parts over the entire domain so as to satisfy Eq. (5). One such
construction involves a linear extension: R
L
(x, r) =x?
?
(r), x ?0, R
G
(x, r) =x?
+
(r), x ?0, where
?
?
(r), ?
+
(r) denote the left and right x-derivatives of R
K
(x, r) at x =0. This construction satis?es
Eq. (5) on P as long as (r ?¯ p)?
+
(r) ?R
L
(r ? ¯ p, r), where ¯ p is the upper bound on price. An alter-
native extension of the concave part that always satis?es Eq. (5) is R
G
(x, r) =R
K
(x, r)/?(r), x ?0,
Popescu and Wu: Dynamic Pricing with Reference E?ects
22 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
where ?(r) =
?
?
(r)
?
+
(r)
>1. The smooth extensions R
L
, R
G
are further conditioned to satisfy Assump-
tion 3 (intuitively motivated in Section 4). Su?cient conditions for pro?t concavity and supermod-
ularity are provided in the next lemma.
Lemma 5. Assumption 3 (c) holds if the reference e?ect R satis?es (LN) and (RD), and either:
1. R(r ?p, r) is supermodular in (p, r) and p
R
xx
(r ?p, r)
R
x
(r ?p, r)
?2; or
2. R(x, r) is concave in x and satis?es (DC); or
3. R(x, r) satis?es (DC) and the following two additional conditions:
(a) The marginal e?ect of discounts/surcharges R
x
is price inelastic, i.e. p
R
xx
(r ?p, r)
R
x
(r ?p, r)
?1;
(b) The marginal reference e?ect shift R
r
is price elastic, p
R
xr
(r ?p, r)
R
r
(r ?p, r)
?1, when R
r
(r?p, r)1 is increasing in r;
(c) r?
+
(r) and r?
?
(r) are strictly increasing in r.
The additional speci?cations of Proposition 6 are also supported by behavioral evidence. A
signi?cant loss aversion index (? 2, Ho and Zhang 2005) justi?es (b) on a limited price domain;
(b) implies (LA). Alternatively, (b’) states that the index of loss aversion is increasing in r, in
particular constant. Empirical evidence suggests risk neutrality for (small) losses, implying that
the convex part is actually linear in x, R
K
(x, r) =x?
?
(r), x ?0. In this case, condition (a) holds
and condition (b) is precisely (LA): ?
+
(r) ? ?
?
(r). Empirical support for (DC) is discussed in
Section 2. Kahneman and Tversky (1979, p.277) suggest a slow variation of the reference e?ect
with the reference level r, which supports (c) (i.e. ?(r) decreases slower than 1/r).
5.3.2. Absolute and Relative Di?erence Models. The conditions of Theorem 4 hold for
AD and RD reference e?ects R
AD
(x, r) =h(x) and R
RD
(x, r) =h(x/r), discussed in Section 2.3, as
long as the corresponding assumptions on pro?t are satis?ed. For both models, the only condition
that e?ectively needs to be veri?ed is pro?t supermodularity, which implies concavity of reference
pro?ts in price (joint concavity for RD models). For both models, supermodularity always holds
for the concave part (by part 2. of Lemma 5), so additional technical conditions concern only the
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 23
convex part. In particular, the results of Theorem 4 hold if the reference e?ect is linear in the
loss domain, e.g. for concave-kinked linear AD and RD models. The following result is a direct
consequence of Proposition 6, with Proposition 4 implying price monotonicity in the AD case.
Corollary 2 (Kinked Linear AD and RD Models). Assume that ?(p) is concave and non-
monotonic. The results of Theorem 4 and Proposition 2 hold for linear AD and RD-models with
a loss-averse kink; the corresponding steady states are characterized in Corollary 1. In addition,
global price monotonicity obtains for the AD case, or for ? =0.
5.3.3. Linear models. Comparison with the literature. FGL and KRA provide special
cases of our results for AD models with linear base demand D(p, p) =b ?ap and (kinked) linear
reference e?ects R
G
(x, r) =?
G
x, R
L
(x, r) =?
L
x, R
K
= min(R
L
, R
G
). FGL characterize the steady
state and prove monotonicity of the pricing path in the loss neutral case (?
G
=?
L
), by solving the
Euler equation explicitly in continuous time (their Proposition 1). They further use these results
in a two stage approach to characterize (in their Proposition 2) the pricing policies under loss
aversion (?
G
We consider the dynamic pricing problem of a monopolist ¯rm in a market with repeated interactions, where
demand is sensitive to the ¯rm's pricing history. Consumers have memory and are prone to human decision
making biases and cognitive limitations. As the ¯rm manipulates prices, consumers form a reference price
that adjusts as an anchoring standard based on price perceptions. Purchase decisions are made by assessing
prices as discounts or surcharges relative to the reference price, in the spirit of prospect theory.
We prove that optimal pricing policies induce a perception of monotonic prices, whereby consumers always
perceive a discount, respectively surcharge, relative to their expectations. The e®ect is that of a skimming
or penetration strategy. The ¯rm's optimal pricing path is monotonic on the long run, but not necessarily
at the introductory stage. If consumers are loss averse, we show that optimal prices converge to a constant
steady state price, characterized by a simple implicit equation; otherwise the optimal policy cycles. The
range of steady states is wider the more loss averse consumers are. Steady state prices decrease with the
strength of the reference e®ect, and with customers' memory, all else equal. O®ering lower prices to frequent
customers may be suboptimal, however, if these are less sensitive to price changes than occasional buyers.
If managers ignore such long term implications of their pricing strategy, the model indicates that they
will systematically price too low and lose revenue. Our results hold under very general reference dependent
demand models.
OPERATIONS RESEARCH
Vol. 00, No. 0, Xxxxx 0000, pp. 000–000
issn0030-364X| eissn1526-5463| 00| 0000| 0001
INFORMS
doi 10.1287/xxxx.0000.0000
c 0000 INFORMS
Dynamic Pricing Strategies with Reference E?ects
Ioana Popescu
INSEAD, Decision Sciences Area, Blvd. de Constance, 77300 Fontainebleau, France. [email protected],http://faculty.insead.edu/popescu/ioana
Yaozhong Wu
INSEAD, Technology and Operations Management Area, Blvd. de Constance, 77300 Fontainebleau, France.
[email protected]
We consider the dynamic pricing problem of a monopolist ?rm in a market with repeated interactions, where
demand is sensitive to the ?rm’s pricing history. Consumers have memory and are prone to human decision
making biases and cognitive limitations. As the ?rm manipulates prices, consumers form a reference price
that adjusts as an anchoring standard based on price perceptions. Purchase decisions are made by assessing
prices as discounts or surcharges relative to the reference price, in the spirit of prospect theory.
We prove that optimal pricing policies induce a perception of monotonic prices, whereby consumers always
perceive a discount, respectively surcharge, relative to their expectations. The e?ect is that of a skimming
or penetration strategy. The ?rm’s optimal pricing path is monotonic on the long run, but not necessarily
at the introductory stage. If consumers are loss averse, we show that optimal prices converge to a constant
steady state price, characterized by a simple implicit equation; otherwise the optimal policy cycles. The
range of steady states is wider the more loss averse consumers are. Steady state prices decrease with the
strength of the reference e?ect, and with customers’ memory, all else equal. O?ering lower prices to frequent
customers may be suboptimal, however, if these are less sensitive to price changes than occasional buyers.
If managers ignore such long term implications of their pricing strategy, the model indicates that they
will systematically price too low and lose revenue. Our results hold under very general reference dependent
demand models.
Subject classi?cations : dynamic programming: deterministic; marketing: pricing, promotion, buyer
behavior; inventory policies: marketing/pricing.
Area of review: Manufacturing, Service and Supply Chain Operations
History : Received February 2005; revised October 2005, January 2006; accepted February 2006
1. Introduction
Traditional economic, marketing and operational models view the consumer as a rational agent who
makes decisions based on current prices, income and market conditions. In a market with repeated
interactions, such as frequently purchased consumer goods (e.g. gasoline), services (e.g. resort
hotels, individual insurance) and B2B settings (e.g. media broadcasting, industrial maintenance),
customers’ purchase decisions are also determined by past observed prices.
As customers revisit the ?rm, they develop price expectations, or reference prices, which become
1
Popescu and Wu: Dynamic Pricing with Reference E?ects
2 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
the benchmark against which current prices are compared. Prices above the reference price appear
to be “high”, whereas prices below the reference price are perceived as “low”. The latter e?ect
stimulates short term demand and provides incentives for retailers to run price promotions as a
mechanism to increase short-term pro?ts. On the other hand, price promotions decrease consumers’
price expectations, and hence their willingness to buy the product at higher prices in the future.
(We ignore stock-piling e?ects and assume that consumers are fully informed about product quality,
and do not judge quality level by price.) For the ?rm, this means that high pro?ts today may come
at the expense of a loss in future demand, and hence less pro?t in the future. Therefore, a pro?t
maximizing ?rm must consider the long term implications of its pricing strategy.
Our goal is to characterize what types of pricing strategies are optimal in such repeated inter-
action markets. We investigate when the ?rm should use traditional skimming or penetration
strategies, and whether a constant price versus a cycling policy is optimal in the long run. A dis-
tinctive feature of this work, compared to traditional microeconomic and operational frameworks,
is that it relies on descriptive models of consumer behavior to derive pricing prescriptions under
complex dynamics. Ultimately, we aim to provide normative insights on how behavioral demand
parameters, such as customer loyalty and loss aversion, should reshape and re?ect in managerial
pricing decisions when customers have repetitive, long term relationships with the ?rm.
The marketing literature provides compelling empirical evidence for the dependence of demand
on past prices. Adaptation level theory (Helson 1964) predicts that customers respond to the
current price of a product by comparing it to an internal standard formed based on past price
exposures, called the reference price. While other reference price models exist in the literature, an
empirical comparison conducted by Briesch et al. (1997) shows that “the best [...] model is [...] one
that is based on the brand’s own price history”, i.e. an internal reference price mechanism.
The impact of the reference price on demand, called reference e?ect, is behaviorally explained at
the individual level by the postulates of prospect theory (Kahneman and Tversky 1979). Accord-
ingly, consumers perceive prices as gains (discounts) or losses (surcharges) relatively to a reference
price, and there is an inherent asymmetry in perception, in that losses loom larger than gains
of the same magnitude (loss aversion). In addition, there is diminishing sensitivity to both gains
and losses. The vast empirical validation of prospect theory in the context of reference prices is
best synthesized by Kalyanaram and Winer (1995). Their “key empirical generalizations” validate
the kinked S-shaped reference e?ect at the aggregate demand level. The value of this theory to
managers and researchers alike is that “it predicts how consumers actually behave, rather than
how they ought to behave” (Nagle and Holden 1995).
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 3
With few exceptions, the dynamic pricing literature is oblivious of such behavioral aspects
underlying demand. Recent surveys on dynamic pricing con?rm that the state of the art models
unrealistically assume demand to be given exogenously, and customers’ purchase decisions to be
based solely on the current price posted by the seller (Bitran and Caldentey 2003, Elmaghraby and
Keskino¸cak 2003). Similarly, in industry practice, prices are typically based on empirically esti-
mated demand models that re?ect consumer response conditional on current prices only, indicating
that ?rms typically follow myopic pricing policies.
Recent research trends in dynamic pricing and revenue management investigate the issue of
strategic buyer behavior, whereby consumers optimally time their purchase in anticipation of
future prices (Ovchinnikov and Milner 2005, Liu and van Ryzin 2005). In parallel work, Heidhues
and K¨oszegi (2005) propose an economic model where consumers’ stochastic reference points (for
consumption and price) are determined as a rational expectation “personal” equilibrium. They
investigate conditions for price stickiness in a static monopolistic framework with random cost.
In contrast, we are interested in incorporating the impact of past prices on demand in a dynamic
setting. Our consumers are more realistic than the traditional “homo economicus”, yet boundedly
rational. They have memory and are prone to human decision making biases (such as anchoring
e?ects and loss aversion) and cognitive limitations. In particular, they are unaware of their biases
and do not act strategically.
There are few demand models similar to ours in the dynamic pricing literature. Sorger (1988)
studies local stability of joint dynamic pricing and advertising policies under reference price e?ects,
but his results are not comparable to ours due to the compound advertising e?ect. Also, his
Assumption 2 c. is inconsistent with our model.
Kopalle et al. (1996) and Fibich et al. (2003) – henceforth KRA, respectively FGL – investigate
a problem similar to ours under a much simpler linear demand model with (kinked) linear refer-
ence e?ects. KRA provide computational insights from a discrete time formulation (adapted from
Greenleaf 1995), but their analytical results are limited. They conjecture and verify numerically
(on p.66) the monotonic convergence of optimal prices under asymmetric demand e?ects; some
special cases are proved in their Proposition 2. An elegant proof of this conjecture is due to FGL
in a continuous time control framework. Their approach, contingent on a linear demand model,
reduces the problem to solving a linear Euler equation. This cannot be solved explicitly for non-
linear models, hence “it is natural to ask whether the results of [their] study would remain valid for
more general demand functions” (p.729), and precisely what general structural properties, if not
Popescu and Wu: Dynamic Pricing with Reference E?ects
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just linearity, drive these results. We extend their insights, and provide stronger analytical results,
under very general modeling assumptions on demand.
We provide structural, as opposed to parametric results, that allow to solve the monopolist’s
dynamic pricing problem under a most general reference dependent demand model, that captures
non-linearities and dynamics in response to changes in the reference price. The model is based
on behavioral claims supported by empirical theories; it extends and generalizes existing models
used in the literature (Sorger 1988, Greenleaf 1995, KRA, FGL). Our main technical contributions,
detailed below, include a simple characterization of the steady state price, as well as convergence
and monotonicity properties of the optimal pricing strategy. From a methodological standpoint,
our approach suggests new ways of proving structural properties for dynamic programs with kinked
rewards, which are the minimum of two smooth functions.
We characterize the existence and uniqueness of an optimal constant pricing strategy on the long
run (steady state). If consumers are loss seeking, i.e. more responsive to discounts than surcharges,
the optimal policy cycles. If consumers are loss averse there is a range of steady states, reduced
to a unique one in the loss neutral case. We provide a simple closed form characterization of the
steady state, which allows for insightful sensitivity analysis with respect to behavioral parameters.
The value of the steady state price decreases with customers’ loyalty (memory e?ects) and with
their sensitivity to past prices (reference e?ects), all else equal. Also, the range of steady state
prices is wider the more loss averse consumers are. If customers are heterogeneous in terms of
their shopping frequency (and nothing else), retailers should o?er discounts to loyal customers,
and higher prices to occasional buyers (relatively to the unique price charged in a non-segmented
market). While such strategies are consistent with current practices by retailers in consumer goods
industries, our results indicate that they are not necessarily optimal if occasional buyers are more
responsive to price changes than loyal ones.
The steady state price is bounded between the pro?t maximizing price charged to consumers
who don’t form reference e?ects, and the steady state price charged by a myopic ?rm to reference
sensitive consumers. We show that ?rms who ignore the long term impact of reference price e?ects,
will myopically but systematically underprice.
Our results also characterize the transient monotonicity and convergence of price and reference
price strategies. In general, we show that the strategic monopolist seeks a monotonic reference price
strategy. If consumers’ reference price is initially high, they will be lead to perceive a gain in each
period, as the ?rm will consistently price below the reference price. This perception of monotonic
prices has the e?ect of a skimming strategy. Similarly, a low reference price leads to a penetration
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 5
type strategy. While high-low prices may be observed at the introductory stage, on the long run
prices are eventually monotonic in the same direction as reference prices. We provide conditions
for a monotonic pricing policy to be optimal (i.e. traditional skimming or penetration). This is true
for example if consumers’ memory is short (i.e. the reference price equals the last price observed),
or if reference prices have increasing marginal impact on demand. In particular, this includes the
linear models studied by KRA and FGL.
In summary, our results for general reference dependent demand models con?rm the robust-
ness of FGL and KRA insights concerning stability and steady state analysis, but not necessarily
monotonicity of the transient pricing policy. The latter appears to be a result of their linear refer-
ence e?ect assumption. Our approach is similar in spirit to the two-stage method in FGL, but our
analysis relies on structural arguments, as opposed to explicit solutions of a parametric model.
The remaining of the paper is structured as follows. Section 2 describes a very general model of
reference dependent demand based on behavioral ?ndings, in particular prospect theory. We sum-
marize behavioral claims that form the base of our various, parsimonious assumptions in the rest
of the paper. Section 3 presents the dynamic pricing model. The next two sections are structured
around consumer’s loss aversion. Section 4 focuses on models where consumers are loss neutral,
with the loss seeking case brie?y addressed at the end. We investigate the existence of the steady
state, and stability and monotonicity properties of the optimal price and reference price strategies;
these are benchmarked against myopic policies. Section 5 studies the same issues when consumers
are loss averse, i.e. the reference e?ect is consistent with prospect theory. Section 6 concludes our
?ndings.
2. General Reference Dependent Demand Model
This section describes how consumers make purchase decisions based on prices and reference prices.
The most prominent theory of reference dependent preferences is prospect theory (Kahneman and
Tversky 1979); the deterministic versions in Thaler (1985) and Tversky and Kahneman (1991) are
most relevant to our context.
The mental accounting framework (Thaler 1985) proposes that the total utility from purchasing
a product consists of two components: acquisition utility and transaction utility. The former cor-
responds to the monetary value of the deal, determined by the discrepancy between price and the
value of the product to the consumer. Transaction utility corresponds to the psychological value of
the deal, determined by the discrepancy between price and reference price. Thus, reference prices
a?ect customer behavior via the magnitude of the perceived “gain” or “loss” x =r ?p relative to
Popescu and Wu: Dynamic Pricing with Reference E?ects
6 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
the reference point r. The microeconomic demand model induced by such individual preferences is
derived by Putler (1992), who validates it empirically with egg sales data. Empirical validation of
prospect theory at the demand level is remarkably synthesized by Kalyanaram and Winer (1995).
De?ne the reference e?ect on demand R(r ?p, r) =D(p, r) ?D(p, p) as the di?erential demand
at price p due to the reference price being r instead of the actual price p. Alternatively, R(x, r)
measures the impact on demand of a perceived discount/surcharge x =r?p relative to the reference
price r. Empirical evidence suggests that the larger magnitude of the perceived gain/loss, the
larger the corresponding absolute impact on demand; this is known as Reference Dependence (RD).
Furthermore, it has been observed that for normal goods purchase experience with higher past
prices increases the likelihood of buying the product for a given observed price. This leads to the
following standing assumption on the demand model:
Assumption 1 (Reference Dependent Demand). Demand D(p, r) is non-negative bounded
and continuous, decreasing in price p and increasing in the reference price r. The corresponding
reference e?ect R(x, r) is increasing in x.
In particular, the de?nition of the reference e?ect together with monotonicity of demand implies:
Remark 1. R(x, r) ?0 for x >0, R(x, r) ?0 for x 1; Loss Neutrality (LN): ? ?1; Loss Seeking (LS): ?(·) x
2
. If f is di?erentiable, this amounts to positiv-
ity of the cross-partial derivative f
xy
(x, y) ? 0 (partial derivatives are denoted by corresponding
subscripts). Submodularity is de?ned by the opposite inequality.
Claim (DC) states that R(x, r) is submodular on the gain, respectively loss domain; this is
irrespective of the gain-loss asymmetries discussed previously. An illustration of this claim under
prospect theory is provided in Figure 1.
2.3. Absolute and Relative Di?erence Models.
We next provide simple examples of reference e?ects used in the literature. The most common is
the absolute di?erence (AD) model R
AD
(x, r) =h(x), with h increasing and h(0) = 0. The shape
Popescu and Wu: Dynamic Pricing with Reference E?ects
8 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
Figure 1 Reference e?ect R(x, r) satisfying (PT) and (DC).
1
1.2
1.4
1.6
1.8
2
?1
?0.5
0
0.5
1
?2
?1.5
?1
?0.5
0
0.5
1
x
r
R
(
x
,
r
)
of the evaluation function h further controls for loss aversion, diminishing sensitivity etc. KRA
and FGL use AD models with h (piecewise) linear. An important criticism to AD models is their
failure to capture dynamics of the reference e?ect in response to changes in the reference price. A
natural generalization that satis?es (DC) is R(x, r) =h(x)s(r), with s a positive decreasing scaling
function.
Another model used in the literature is the relative di?erence (RD) model R
RD
(x, r) = h(
x
r
)
(with h increasing and h(0) =0), motivated by the Weber-Fechner law in psychophysics. This law
suggests that consumers perceive price di?erences in proportional, rather than absolute terms, a
theory that is experimentally supported by mental accounting studies, such as the small example
in the previous section (that can be explained by consumers not responding to the magnitude of
the price di?erential r ?p =50 Euro, but to the percentage di?erence relative to the base price:
r?p
r
= 50% vs. 4%). RD models have been used in conjunction with linear and logarithmic forms
for h (Winer 1988), as well as exponentials (Anderson and Rasmussen 2004). A generalization of
RD models is R(x, r) =h(xs(r)), with s a positive decreasing scaling function. This satis?es (DC)
if h is convex increasing, or h
is inelastic to changes in x.
While our analysis focuses on the most general reference dependence demand model, we system-
atically indicate how our general results specialize to the absolute and relative di?erence models.
3. Dynamic Pricing Model
This section sets up our dynamic pricing model, together with general assumptions on how con-
sumers’ memory works in adjusting reference prices based on price perceptions.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 9
We consider a pro?t maximizing monopolist who faces a homogeneous stream of repeated cus-
tomers over an in?nite time horizon. The monopolist decides in every period what price p to charge
to customers, from a bounded positive interval P = [0, ¯ p] (our results allow for a non-zero lower
bound on price; see the Remark of Section B in the Appendix). The resulting pro?t is ?(p, r) =
(p ?c)D(p, r), where r is the current reference price and D(p, r) the corresponding demand satis-
fying Assumption 1. We assume for simplicity and without loss of generality the variable cost to
be c =0; all our results hold for a non-zero constant cost c.
In each period, customers update their current reference price r
t
as they observe the new price
p
t
charged by the ?rm. The adaptive expectation framework Nerlove (1958) motivates a reference
price that is a weighted average of past prices, where more recent prices are given more weight.
Throughout of the paper, we assume that the reference price formation mechanism is given by:
r
t
=?r
t?1
+(1 ??) p
t?1
, ? ?[0, 1). (1)
Exponential smoothing is the most commonly used and empirically validated reference price
mechanism in the literature (e.g. Winer 1986, Sorger 1988, Greenleaf 1995, KRA, FGL). Some of
our results hold for more general mechanisms, including in?ation e?ects and price trends in general
(in this case our insights and results remain valid for in?ation-adjusted prices). The memory model
described by (1) allows for additional insights by quantifying the speed of the memory adjustment
process. The memory parameter ? captures how strongly the reference price depends on past prices.
Lower values of ? represent a shorter-term memory; in particular, if ? =0 then the reference price
equals the past price. Memory is an indicator for shopping frequency, hence can be a proxy for
customer loyalty. Actual estimated parameters range from ? = 0 (Krishnamurthi et al. 1992) to
? =0.925 (Greenleaf 1995), much depending on the type of product or service involved.
For an initial reference price r
0
, we model the ?rm’s long term pro?t optimization problem as:
V (r
0
) = sup
p
t
?P
?
t=0
?
t
?(p
t
, r
t
)
s.t. r
t
=?r
t?1
+(1 ??)p
t?1
, t ?1,
where ? ?(0, 1) is a discount factor. Pro?t-per-stage is bounded, so the value function is the unique
bounded solution of the following Bellman equation (Stokey et al. 1989):
V (r) = sup
p?P
W(p, r), where W(p, r) =?(p, r) +?V (?r +(1 ??)p). (2)
A consequence of Assumption 1 is that short term pro?t ?(p, r), and the value function are
increasing (Stokey et al. 1989, Theorem 4.7), i.e. larger reference prices are preferable for the ?rm.
Popescu and Wu: Dynamic Pricing with Reference E?ects
10 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
Remark 2. The value function V (r) is an increasing function of the reference price r.
Because the action set P is compact and all functions are continuous, there exists an optimal
stationary pricing policy that solves Problem (2), and associates with any given reference price r,
one or more optimal prices. We de?ne the optimal pricing policy as p
?
(r) = arg max
p
W(p, r). To
avoid the ambiguity of multiple solutions, arg max
x
f(x) refers without loss of generality to the
largest maximizer of the function f. The optimal reference price policy q
?
(r) =?r +(1 ??)p
?
(r)
provides the next reference price associated with the optimal pricing policy.
The pricing path {p
?
t
} starting with an initial state r
0
is computed sequentially as p
?
t
=p
?
(r
?
t
),
where the state variable is updated in each period via the reference price formation mechanism
r
?
t
= q
?
(r
?
t?1
). The sequence {r
?
t
} of optimal reference prices is referred to as the state path; for
the consumer, this is the sequence of perceived prices or price expectations. A steady state p
??
for Problem (2) is one from which it is suboptimal to deviate, i.e. p
?
(p
??
) =q
?
(p
??
) = p
??
, so the
optimal price path starting at r
0
=p
??
is constant.
Our goal is to investigate when such a steady state exists and to characterize stability and
monotonicity properties of the price and reference price paths. These results critically depend on
consumer’s reaction to discounts and surcharges, captured by loss aversion.
4. Model without Loss Aversion
This section focuses on the case when consumers are loss neutral (LN), i.e. the reference e?ect is
smooth; loss seeking behavior is investigated at the end of this section. We ?rst characterize the
unique steady state, and then investigate stability and monotonicity properties of the price and
reference price strategies, under very general modeling assumptions. Two additional assumptions
are made throughout Sections 4.1-4.3.
Assumption 2 (Smooth Reference E?ect). R(x, r) is twice di?erentiable in x and r.
The slope of the reference e?ect at x =0 is denoted ?(r) =R
x
(0, r). Our results require some typ-
ical technical assumptions on the short term pro?t ?(p, r) =?(p) +?
R
(p, r). Here ?(p) =?(p, p) =
pD(p, p) is the pro?t in a market where consumers don’t form reference e?ects (R?0), referred to
as base pro?t, and ?
R
(p, r) =pR(r ?p, r) is the pro?t from the reference e?ect, or reference pro?t.
Assumption 3 (Pro?t). (a) ?(p) is non-monotonic and concave in p. (b) ?
p
(r, r) =?
(r) ?r?(r)
is strictly decreasing in r. (c) ?
R
(p, r) is concave in p and supermodular in (p, r).
Some intuition and motivation for these assumptions is provided next. Non-monotonicity of ?(p)
states that the ?rm is not constrained to charge sub-optimal prices (in a market without reference
e?ects). This will allow to rule out pathological boundary steady states.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 11
Part (b) states that, if the ?rm charges the reference price, a small change in price has higher
impact on pro?t at lower reference prices. That is, marginal pro?t diminishes as price is set to
the reference price. Strict monotonicity is necessary to insure uniqueness of the steady state. A
su?cient condition for Assumption 3 (b), given (a), is r?(r) strictly increasing in r.
Diminishing marginal pro?t (concavity) is a typical economic assumption. Concavity of ? and
?
R
implies concavity of short term pro?t ?. The supermodularity (see p.7) assumption (c) states
that marginal short term pro?t increases with the reference price. This is consistent with modeling
assumptions used in the literature, and summarized by Winer (1988) (those demand functions are
typically additively separable, in particular linear, in price and reference price). Various sets of
su?cient conditions are provided in Section 5.3. One is supermodularity of demand D(p, r), which
in particular implies that demand is more elastic at higher reference prices. In support of this
assumption, Kamen and Toman (1970) found evidence that with higher reference prices, consumers
were more likely to switch brands of gasoline, for a given price di?erential.
4.1. Myopic Strategies for Loss Neutral Buyers
A ?rm that ignores the e?ect of current prices on future demand will focus on maximizing short
term pro?t. Intuitively, such a myopic ?rm should charge higher prices as consumers’ price expec-
tations are higher. This is insured by pro?t supermodularity (Assumption 3 (c)), via Topkis’
Theorem (Topkis 1998, Theorem 2.8.2 implies x
?
=arg maxf(x, y) increasing in y whenever fis supermodular).
Lemma 1. The single period pro?t maximizing price ˜ p(r) =arg max
p?P
?(p, r) is increasing in r.
Formally, the problem solved by a myopic ?rm is equivalent to setting ? = 0 (full discounting)
in Problem (2). Consider two ?rms, one myopic, the other strategic (optimizer), starting with the
same initial reference price r
0
, and applying sequentially their respective pricing policies, while con-
sumers’ memory evolves according to the same ?-smoothing mechanism. The state of the system
at time t, for the myopic and optimizing ?rms are given by r
M
t
and r
?
t
, respectively. The corre-
sponding myopic, respectively optimal prices charged at time t are p
M
t
= ˜ p(r
M
t
) and p
?
t
= p
?
(r
?
t
).
We show that the myopic ?rm systematically underprices, following a monotonic pricing strategy.
The results are driven by supermodularity of short term pro?ts (Assumption 3 (c)).
Proposition 1 (Myopic vs. Strategic Firm). For any reference price r, the myopic ?rm
underprices the product, i.e. p
?
(r) ? ˜ p(r). Furthermore, for any initial reference price r
0
, the price
charged by the myopic ?rm at any point in time is less than the optimal price, i.e. p
M
t
?p
?
t
, for all
t. Moreover, all myopic price paths monotonically converges to a unique constant price p
M?
.
Popescu and Wu: Dynamic Pricing with Reference E?ects
12 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
The prices charged by the myopic ?rm are oblivious of their eroding e?ect on future demand,
hence future pro?ts. By charging higher prices, the optimizing ?rm trades o? current pro?ts for
future long-term pro?tability from higher reference prices. Figure 2 illustrates these results; in
this case, the observed di?erence between optimal and myopic pro?ts is about 10%. Throughout
the paper, numerical examples are calculated using the Compecon Matlab toolbox developed by
Miranda and Fackler (2002).
Figure 2 Myopic vs. optimal pricing policies; D(p, r)=1?2p +
r?p
5r
, P =[0.1, 0.3], ? =0.80, ? =.95.
5 10 15 20 25 30 35 40
0.15
0.2
0.25
t
P
r
i
c
e
Optimal (r
0
=0.3)
Myopic (r
0
=0.3)
Optimal (r=0.1)
Myopic (r
0
=0.1)
Figure 2 also suggests that the optimal pricing paths may converge monotonically to a single
steady state price. These issues are investigated next.
4.2. Steady State for Loss Neutral Buyers
De?ne the pro?t optimizing price in absence of reference e?ects ˜ p = arg max
p?P
?(p). The Bellman
equation (2) implies that, if a steady state p
??
exists, then V (p
??
) =
?(p
??
)
1 ??
, whereas generally
V (r) ?
?(r)
1 ??
for all r ?P. Assumption 3 (a,b) insures that such a steady state is unique and can
be computed as the solution of a simple implicit equation:
Theorem 1 (Steady State). If Problem (2) admits a steady state p
??
, then this is the unique
solution of:
?
(p)
1 ??
=
p?(p)
1 ???
. (3)
Furthermore, p
??
is continuously decreasing in ?, increasing in ?, and satis?es p
M?
?p
??
? ˜ p.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 13
At steady state, a small increase in price ?p leads to a marginal increase in the long-term base
pro?t equal to
?
(p)
1??
?p. However, on the transition, customers perceive a loss and reduce their
demand. The resulting pro?t loss is p?(p)?p in the ?rst period, ?p?(p)?p in the second period and
so on. Hence the implicit equation (3) re?ects the balance between maximizing long-term pro?ts
(without reference e?ects) and the discounted transitory costs due to the reference e?ect.
Equation (3) con?rms that myopic ?rms (? =0) set lower steady state prices p
M?
. On the long
run, strategic ?rms should charge lower prices when consumers form reference e?ects, than when
they do not, i.e. p
??
? ˜ p, approaching it as ? ?1.
In particular, Theorem 1 obtains the following closed form expressions for the corresponding
steady states for absolute and relative di?erence models discussed in Section 2.3. This generalizes
the FGL result for linear reference e?ects.
Corollary 1 (Steady State for AD and RD Models). Consider the problem with linear
base demand D(p, p) = b ? ap, and reference e?ect R
AD
(x, r) = h(x), respectively R
RD
(x, r) =
h(x/r). The corresponding steady states, if they exist, are p
??
AD
=
b
2a+?
and respectively p
??
RD
=
b??
2a
,
where ?=h
(0)
1??
1???
.
Consumer Heterogeneity and Sensitivity Analysis Insights. Consider a heterogeneous
market, where consumer types di?er in shopping frequency, captured by the memory parameter
?
i
, and sensitivity to past prices, i.e. the reference e?ect R
i
. Demands without reference e?ects are
assumed identical across segments (i.e. D
i
(p, p) =D(p, p)).
If the ?rm can segment the market, Theorem 1 indicates that on the long run, each segment
should be charged a steady state price p
??
i
that solves
?
(p)
p(1??)
=
?
i
(p)
1??
i
?
. The steady state price only
depends on the reference e?ect via its slope at zero, speci?cally, it decreases with the magnitude
of ?(·). That is, if consumers share the same memory parameter ?, those who are more sensitive
to past prices will be charged lower prices on the long run.
Remark 3. If ?
1
=?
2
and R
2
is a “stronger” reference e?ect than R
1
, i.e. if ?
1
(r) ??
2
(r) for all
r, then the corresponding steady state prices, if they exist, satisfy p
??
1
?p
??
2
.
On the other hand, if consumers only di?er in their memory parameter ?
i
, then segments with
higher ?
i
are charged lower steady state prices, all else equal. While charging lower prices to loyal
customers (proxied by higher ?
i
) is in line with current practices in consumer goods industries, our
results show that it may not be the optimal strategy if these shoppers are less sensitive to price
changes, as measured by ?
i
. In general, the ?rm should charge lower prices to segments with higher
Popescu and Wu: Dynamic Pricing with Reference E?ects
14 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
long run marginal sensitivity to the reference e?ect, as measured by ?
i
(·)/(1 ??
i
?). Speci?cally,
the same argument as Remark 3 shows that p
??
1
?p
??
2
, if
?
1
(r)
1??
1
?
?
?
2
(r)
1??
2
?
holds for all r.
Finally, for comparison purposes, consider the case where the ?rm cannot segment the market and
o?ers the same price to all segments. We show in Section D of the Appendix that the common steady
state price (if one exists) solves
?
(p)
1??
=p
N
i=1
?
i
(p)
1??
i
?
, and is bounded between the lowest and highest
prices charged to each consumer type in a perfectly segmented market p
??
?[min
i
p
??
i
, max
i
p
??
i
].
4.3. Stability and Monotonicity with Loss Neutral Buyers
This section investigates the existence and stability of the unique steady state price characterized
in Theorem 1. We show global convergence of the optimal prices to steady state, and study local
and global monotonicity properties of the state and optimal price paths.
We start with the most general results that require no further assumptions. Recall that the
reference price policy was de?ned by q
?
(r) =?r +(1 ??)p
?
(r).
Lemma 2 (Monotone Reference Price Policy).The reference price policy q
?
(r) increases in r.
The result follows from Assumption 3(c), and implies the following:
Theorem 2 (Global Stability and State Path Monotonicity). Problem (2) admits a unique
steady state p
??
, characterized by Theorem 1, to which all optimal price paths converge. All corre-
sponding reference price paths converge to p
??
monotonically.
Under the conditions of Theorem 2, the monopolist’s strategy involves decreasing customer’s
price expectations by always pricing below the current reference price, or vice versa. If the refer-
ence price is high, the optimal pricing policy is a skimming-like strategy: the ?rm starts with a
relatively high introduction price, then always prices below the reference price (r
t
> p
t
). In this
case, consumers perceive in each period a gain relative to their expectations, which stimulates
demand, and hence pro?ts in the current period. Low reference prices lead to a penetration-type
strategy, whereby the ?rm always prices above the customers’ reference price (r
t
< p
t
). The loss
in current pro?t from a negative reference e?ect, is outweighed (1) by a higher short-term pro?t
without reference e?ect, ?(p
?
t
), and (2) by the increase in future pro?ts due to higher reference
prices in the next period.
Theorem 2 does not guarantee monotonicity of the corresponding pricing paths. This holds
whenever the pricing policy is monotonic. Some su?cient conditions are provided next. Recall that
a real-valued C
2
function f(x, y) is strongly concave if there exist positive constants a and b such
that f(x, y) +ax
2
+by
2
is concave.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 15
Lemma 3 (Monotone Price Policy). Each of the following alternative conditions is su?cient
for the optimal pricing policy p
?
(r) in Problem (2) to be increasing in r :
(a) Consumers have short memory ? =0, i.e. r
t+1
=p
t
for all t;
(b) Demand D(p, r), or equivalently, the reference e?ect R(r ?p, r), is convex in r;
(c) Short term pro?t ?(p, r) is strongly concave on the interior of its domain, and there exists
a global constant K >0 such that ?
rr
(p, r) ??K and ?
pr
(p, r) ?K/2.
The result follows from supermodularity of W(p, r), the function on the right hand side of the
Bellman equation (2). This reduces to supermodularity of short term pro?t (Assumption 3 (c))
under condition (a). Condition (b) achieves this by additionally insuring convexity of short term
pro?t in r, and hence of the value function. Condition (c) insures a uniform lower bound on the
degree of concavity of the value function V, that is compensated by supermodularity of short term
pro?ts. At the limit, for K =0 we recover the conditions of part (b). Strong concavity is a technical
condition insuring second order di?erentiability of the value function.
Theorem 3 (Global Price Monotonicity). Under the conditions of Lemma 3, all optimal pric-
ing paths for Problem (2) converge monotonically to the unique steady state price p
??
, characterized
by Theorem 1.
Figure 3 Optimal pricing paths for D(p, r) =e
?p
+0.5(e
0.5(r?p)
?1), ? =0.8, ? =0.9.
5 10 15 20 25 30
0.74
0.76
0.78
0.8
0.82
0.84
0.86
0.88
t
P
r
i
c
e
In particular, we obtain the price monotonicity results of KRA and FGL for linear AD models
R
AD
(x, r) =?x; these satisfy part (b) of Lemma 3. Part (c) of Lemma 3 (but not (b)) obtains price
Popescu and Wu: Dynamic Pricing with Reference E?ects
16 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
monotonicity for linear RD models R
RD
(x, r) = ?x/r, on a price range P = [p, ¯ p] with ¯ p/p ?
4
?
2,
i.e. price dispersion not exceeding 18%.
Theorem 3 insures that only three possible pricing strategies are optimal: skimming (if initial
price expectations are high, start with a high price and then keep decreasing it), penetration (if
initial price expectations are low, start with a low price, then increase it), or constant pricing.
Moreover, after a transient regime, the optimal prices stabilize to a steady state price. The results
follow from the monotonicity of the policy function (Lemma 3). This also insures that the price
paths do not cross, i.e. any increase in the initial reference price shifts the entire pricing sequence
upwards. These results are illustrated with an example in Figure 3.
Furthermore, in a neighborhood of the steady state, prices are eventually monotonic, i.e. local
price monotonicity holds. Strong concavity of short term pro?ts is a technical condition that insures
di?erentiability of the value function and pricing policy.
Proposition 2 (Local Price Monotonicity). If ?(p, r) is strongly concave on the interior of
its domain, then all optimal pricing paths are locally monotonic around the steady state.
Local price monotonicity implies that, on the long run, the slope of the reference price policy
q
?
(r) =?r +(1 ??)p
?
(r) is steeper than ?. That is, reference prices converge to steady state at a
rate that is ultimately faster than the memory parameter ?.
Transient High-Low Pricing. At the introductory stage, however, the monopolist may have
an incentive to increase reference prices at a slower rate, achieved by high-low pricing. Such a
strategy, depicted in Figure 4, contrasts the insights obtained by KRA and FGL under linear
demand models. The optimal pricing policy p
?
(r) and price path {p
?
t
} are non-monotonic under the
conditions of Theorem 2 and Proposition 2, but violating Lemma 3. The pricing strategy illustrated
in Figure 4 is consistent for example with catastrophe insurance pricing in the U.S. before major
hurricanes, when consumers reference prices are much lower than pro?t optimizing prices (we thank
Paul Kleindorfer for suggesting this example; see endnote 1. for intuition).
This example illustrates that high-low pricing is technically possible in the loss neutral case,
contrary to previous insights in the literature; similar behavior can be observed for RD models.
Nevertheless, extensive numerical simulations revealed that the optimal pricing path is typically
monotonic; at most, it displays a relatively small direction swing from decreasing to increasing
prices, as in Figure 4 (simulated models for R(x, r) include xe
?ar
, x/r
k
, 1?e
?ax
, e
ax/r
k
?1, a, k >0
for a variety of parameters satisfying Assumptions 1-3). So it is fair to say that, if consumers are
loss neutral, we do not ?nd strong numerical support for transient high-low pricing, but cannot
rule it out either.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 17
Figure 4 High-low pricing policy p
?
(r) and optimal price path p
?
t
; D(p, p) = 2 ?0.5p, R(x, r) =0.5(e
1.5x/r
3
?1),
P =[1, 2], ? =0.950, ? =0.999, r
0
=1.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
1.9
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2
r
p
*
(r)
10 20 30 40 50 60 70 80 90 100
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2
t
P
r
ic
e
4.4. Loss Seeking Buyers
High-low pricing is provably optimal if consumers are focused on gains, i.e. loss seeking (LS). In this
case the optimal pricing policy for Problem (2) cycles. Consumers are loss seeking if they respond
more to discounts than to surcharges, i.e. if the reference e?ect has a convex kink at zero. This
type of asymmetry is inconsistent with prospect theory, but has found some empirical validation
in the marketing literature (e.g. Krishnamurthi et al. 1992, Greenleaf 1995). We address it for
completeness; only Assumption 1 stands in place for the next result:
Proposition 3 (Cyclic Policy). If buyers are loss seeking, i.e. the reference e?ect R(x, r) sat-
is?es (LS), then Problem (2) admits no steady state.
Proposition 1 in KRA obtains a similar result when R(x, r) =h(x) is piecewise linear in x.
Proposition 3 suggests that in order for a steady state price to exist, the reference e?ect R must
be smooth, or kinked in the direction predicted by prospect theory. This case is analyzed next.
5. Model with Loss Aversion
This section investigates the transient and long term behavior of the optimal dynamic pricing
policy when reference e?ects are kinked in the spirit of prospect theory. We ?rst set up the model;
two additional assumptions on the reference e?ect, respectively pro?t are made throughout this
section (replacing Assumptions 2 and 3 for the loss neutral case).
5.1. Model and Assumptions
At the aggregate level, prospect theory predicts that the reference e?ect is captured by an S-shaped
function of the reference gap x =r ?p, with a concave (loss averse) kink at x =0 (Kahneman and
Popescu and Wu: Dynamic Pricing with Reference E?ects
18 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
Tversky 1979, Tversky and Kahneman 1991).
The following assumption, made throughout this section, formalizes the technical aspects relevant
for our purposes. An illustration is provided in Figure 5. Recall that a function f(x) is single-
crossing if there exists a unique x
0
in its domain such that f(x
0
) =0. We denote 1
X
the indicator
function of the set X.
Figure 5 An example of kinked reference e?ect R
K
Assumption 4 (Kinked Reference E?ect). The kinked reference e?ect is given by
R
K
(x, r) =1
(x?0)
R
G
(x, r) +1
(x?0)
R
L
(x, r), (4)
where R
L
(x, r) and R
G
(x, r) are twice di?erentiable, increasing in x with R
G
(x, r)?R
L
(x, r) single-
crossing in x, and satisfying ?
L
(r) >?
G
(r) for all r.
Assumption 4 is a weaker formalization of prospect theory (PT), as detailed in Section 2.1,
including reference dependence (RD), and the weaker form of loss aversion (LA) in terms of the
index ?(r) =
?
L
(r)
?
G
(r)
, proposed by K¨obberling and Wakker (2005). We do not speci?cally assume
diminishing sensitivity (DS) (i.e. concavity/convexity of the gain/loss components), as it is imma-
terial for our analysis. The single crossing condition allows to write the kinked reference e?ect as
the minimum of the loss and gain counterparts on the relevant domain:
R
K
(x, r) =min(R
G
(x, r), R
L
(x, r)). (5)
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 19
This condition is unrestrictive in the context of Assumption 4 alone, because the loss and gain
parts of R
K
can always be smoothly extended so as to satisfy Eq. (5).
Based on (5), the pro?t per-period can be written as
?
K
(p, r) =min(?
G
(p, r), ?
L
(p, r)), (6)
where ?
G
(p, r) =?(p) +pR
G
(r ?p, r) and ?
L
(p, r) =?(p) +pR
L
(r ?p, r). In line with our previous
assumptions on pro?t, the following technical assumption is made in this section:
Assumption 5 (Gain/Loss Pro?t). The short term pro?ts ?
G
(p, r) and ?
L
(p, r) associated with
the smooth reference e?ects R
G
, respectively R
L
, satisfy Assumption 3.
Section 5.3 provides su?cient modeling conditions on the reference e?ect for Assumptions 4 and
5 to hold, in particular when R
K
is convex-concave, as predicted by prospect theory. In particular,
it provides concrete ways to construct R
L
, R
G
that satisfy Assumption 3, and so that Eq. (5) holds.
The corresponding Bellman equation for the problem with kinked demand e?ect is:
(K) V
K
(r) =sup
p
{?
K
(r, p) +?V
K
(?r +(1 ??)p)}. (7)
5.2. Steady State, Stability and Monotonicity
We extend the results obtained in Section 4 for the loss neutral case, to infer existence and stability
properties of the steady state for loss averse buyers. A key role in our analysis here is played by
the problem with smooth reference e?ect:
R
?
(x, r) =?R
G
(x, r) +(1 ??)R
L
(x, r). (8)
Denoting the corresponding short term pro?t ?
?
(p, r) =?(p) +pR
?
(r ?p, r), this problem is
(P
?
) V
?
(r) =sup
p
{?
?
(p, r) +?V
?
(?r +(1 ??)p)}. (9)
For any ? ?[0, 1], Problem (P
?
) satis?es our assumptions for the loss neutral case, so all the results
of Section 4 apply. Moreover, Eq. (5) implies
R
K
(x, r) =min(R
G
(x, r), R
L
(x, r)) ?R
?
(x, r), ? ?[0, 1]. (10)
The same relationship between short term pro?ts, and the corresponding value functions is implied.
The proof details of the following result are in the Appendix:
Lemma 4. For any ? ? [0, 1], Problem (P
?
) satis?es Assumptions 1-3. Its value function satis?es
V
?
(r) ?V
K
(r) for all r, and its steady state p
??
?
is also a steady state for Problem (K).
Popescu and Wu: Dynamic Pricing with Reference E?ects
20 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
By loss aversion and construction (8), the slopes of the reference e?ects R
L
, R
?
and R
G
satisfy
?
L
(r) ??
?
(r) ??
G
(r). Remark 3 further implies that the steady state prices of the corresponding
problems satisfy p
??
L
? p
??
?
? p
??
G
. Moreover, as ? ? [0, 1], the steady states p
??
?
span the interval
[p
??
L
, p
??
G
].
We are now ready to characterize the set of steady states for general non-linear loss-averse
demand e?ects, and their stability properties, as follows:
Theorem 4 (Steady State, Stability and Monotonicity under Loss Aversion). The set
of steady states for Problem (K) equals [p
??
L
, p
??
G
], so for an initial reference price r
0
?[p
??
L
, p
??
G
], the
optimal price path is constant p
?
t
=r
0
. For an initial reference price r
0
p
?
G
,
the reference price path {r
?
t
} converges monotonically to p
??
L
, respectively p
??
G
, and the corresponding
optimal price path {p
?
t
} also converges to the same steady state.
Our analysis, based on bounding the kinked reference e?ect, and hence corresponding value
function, by the full range of convex combinations of the smooth loss and gain counterparts,
suggests a general approach for obtaining structural properties and global stability in dynamic
programs with kinked reward structure. A similar approach is used by FGL in their analysis of
kinked linear demand models; their results are driven by the explicit solution of the linear demand
model, while ours rely on structural convexity arguments.
An illustration of the steady states, pricing strategies and corresponding value function under
loss aversion is provided in Figure 6. While technically, high-low pricing is possible within the
assumptions of Theorem 4 (e.g. using the model of Figure 4 for R
L
), numerical results suggest that
with most common demand models, prices are typically monotonic. The price monotonicity results
obtained in the loss neutral case extend as follows:
Proposition 4 (Price Monotonicity under Loss Aversion). If the problems with reference
e?ects R
G
and R
L
also satisfy the conditions of Lemma 3, respectively Proposition 2, then the price
paths for Problem (K) are globally, respectively locally monotonic. In particular, if consumers have
short term memory (? =0), then prices are globally monotonic.
Furthermore, the strategy of a myopic ?rm under loss aversion resembles that for the loss neutral
case, except that there is a range of myopic steady states. The proof of the next result follows the
combined spirit of Lemma 4, Theorem 4 and Proposition 1.
Proposition 5 (Myopic Prices under Loss Aversion). Given any initial reference price r
0
,
the myopic price path {p
M
t
} lies below the optimal price path {p
?
t
} and converges monotonically to
a myopic steady state. Furthermore, the set of myopic steady states equals [p
M?
L
, p
M?
G
], where p
M?
L
and p
M?
G
are the steady states for a myopic ?rm under reference e?ect R
L
, respectively R
G
.
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 21
Figure 6 Pricing strategies and value function under prospect theory; D(p, p) = 2 ? 0.6p, R
G
(x, r) = .25(1 ?
e
?x/r
), R
L
(x, r) =0.8(e
x/r
?1), P =[1, 2], ? =0.80, ? =0.95, r
0
=1, 2.
5 10 15 20 25 30 35
1.4
1.45
1.5
1.55
1.6
1.65
1.7
t
P
r
i
c
e
p
G
**
p
L
**
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
31
31.5
32
32.5
33
33.5
34
r
V
(
r
)
p
L
**
p
G
**
V
G
V
?
V
L
While the myopic ?rm systematically underprices, under loss aversion, its policy may be opti-
mal for a range of initial reference prices. For example, consider a piecewise linear AD model
with D(p, p) = 1 ?p, ?
L
= 0.5, ?
G
= 0.2, ¯ p = 0.6 and ? = 0.95, ? = 0.90. We obtain [p
M?
L
, p
M?
G
] =
[0.400, 0.455] and [p
??
L
, p
??
G
] =[0.426, 0.468]. In this case, if r
0
?[0.426, 0.455], the myopic (constant)
pricing strategy and the optimal one coincide.
5.3. Special Demand Models.
The assumptions underlying Theorem 4 impose additional, implicit conditions on the demand
model. These involve smoothly extending the loss and gain component of a given kinked reference
e?ect over the entire domain so as to satisfy Eq. (5), and verifying the technical pro?t Assumption
3 on each component. This section provides explicit modeling conditions on the reference e?ect that
are su?cient to validate these assumptions. We focus on reference e?ects satisfying the prospect
theory claims motivated in Section 2, including diminishing sensitivity (DS) and decreasing curva-
ture (DC). We further infer speci?c implications for AD and RD models. Finally, we explain how
our results extend the ones in the literature, obtained for linear AD models.
5.3.1. Prospect Theory Models. Given a kinked S-shaped reference e?ect R
K
(x, r), as pre-
dicted by prospect theory, the challenge in satisfying Assumption 4 is to smoothly “extend” the
convex (loss) and concave (gain) parts over the entire domain so as to satisfy Eq. (5). One such
construction involves a linear extension: R
L
(x, r) =x?
?
(r), x ?0, R
G
(x, r) =x?
+
(r), x ?0, where
?
?
(r), ?
+
(r) denote the left and right x-derivatives of R
K
(x, r) at x =0. This construction satis?es
Eq. (5) on P as long as (r ?¯ p)?
+
(r) ?R
L
(r ? ¯ p, r), where ¯ p is the upper bound on price. An alter-
native extension of the concave part that always satis?es Eq. (5) is R
G
(x, r) =R
K
(x, r)/?(r), x ?0,
Popescu and Wu: Dynamic Pricing with Reference E?ects
22 Operations Research 00(0), pp. 000–000, c 0000 INFORMS
where ?(r) =
?
?
(r)
?
+
(r)
>1. The smooth extensions R
L
, R
G
are further conditioned to satisfy Assump-
tion 3 (intuitively motivated in Section 4). Su?cient conditions for pro?t concavity and supermod-
ularity are provided in the next lemma.
Lemma 5. Assumption 3 (c) holds if the reference e?ect R satis?es (LN) and (RD), and either:
1. R(r ?p, r) is supermodular in (p, r) and p
R
xx
(r ?p, r)
R
x
(r ?p, r)
?2; or
2. R(x, r) is concave in x and satis?es (DC); or
3. R(x, r) satis?es (DC) and the following two additional conditions:
(a) The marginal e?ect of discounts/surcharges R
x
is price inelastic, i.e. p
R
xx
(r ?p, r)
R
x
(r ?p, r)
?1;
(b) The marginal reference e?ect shift R
r
is price elastic, p
R
xr
(r ?p, r)
R
r
(r ?p, r)
?1, when R
r
(r?p, r)1 is increasing in r;
(c) r?
+
(r) and r?
?
(r) are strictly increasing in r.
The additional speci?cations of Proposition 6 are also supported by behavioral evidence. A
signi?cant loss aversion index (? 2, Ho and Zhang 2005) justi?es (b) on a limited price domain;
(b) implies (LA). Alternatively, (b’) states that the index of loss aversion is increasing in r, in
particular constant. Empirical evidence suggests risk neutrality for (small) losses, implying that
the convex part is actually linear in x, R
K
(x, r) =x?
?
(r), x ?0. In this case, condition (a) holds
and condition (b) is precisely (LA): ?
+
(r) ? ?
?
(r). Empirical support for (DC) is discussed in
Section 2. Kahneman and Tversky (1979, p.277) suggest a slow variation of the reference e?ect
with the reference level r, which supports (c) (i.e. ?(r) decreases slower than 1/r).
5.3.2. Absolute and Relative Di?erence Models. The conditions of Theorem 4 hold for
AD and RD reference e?ects R
AD
(x, r) =h(x) and R
RD
(x, r) =h(x/r), discussed in Section 2.3, as
long as the corresponding assumptions on pro?t are satis?ed. For both models, the only condition
that e?ectively needs to be veri?ed is pro?t supermodularity, which implies concavity of reference
pro?ts in price (joint concavity for RD models). For both models, supermodularity always holds
for the concave part (by part 2. of Lemma 5), so additional technical conditions concern only the
Popescu and Wu: Dynamic Pricing with Reference E?ects
Operations Research 00(0), pp. 000–000, c 0000 INFORMS 23
convex part. In particular, the results of Theorem 4 hold if the reference e?ect is linear in the
loss domain, e.g. for concave-kinked linear AD and RD models. The following result is a direct
consequence of Proposition 6, with Proposition 4 implying price monotonicity in the AD case.
Corollary 2 (Kinked Linear AD and RD Models). Assume that ?(p) is concave and non-
monotonic. The results of Theorem 4 and Proposition 2 hold for linear AD and RD-models with
a loss-averse kink; the corresponding steady states are characterized in Corollary 1. In addition,
global price monotonicity obtains for the AD case, or for ? =0.
5.3.3. Linear models. Comparison with the literature. FGL and KRA provide special
cases of our results for AD models with linear base demand D(p, p) =b ?ap and (kinked) linear
reference e?ects R
G
(x, r) =?
G
x, R
L
(x, r) =?
L
x, R
K
= min(R
L
, R
G
). FGL characterize the steady
state and prove monotonicity of the pricing path in the loss neutral case (?
G
=?
L
), by solving the
Euler equation explicitly in continuous time (their Proposition 1). They further use these results
in a two stage approach to characterize (in their Proposition 2) the pricing policies under loss
aversion (?
G