Dynamic diminishing marginal utility and tacit collusion
Ke Liu and Yuk-fai Fongy
Preliminary Draft (Please Do Not Cite)
Abstract
Diminishing marginal utility is usually treated as a static property, while in many situations it is more appropriately modeled as a dynamic phenomenon. This paper investigates
how dynamic diminishing marginal utility in‡uences …rms’ ability to tacitly collude, and
shows that it crucially depends on whether diminishing marginal utility is with or without
externality. When diminishing marginal utility is without externality, i.e., when consumption of one …rm’s product this period only lowers that …rm’s and no other …rm’s product’s
marginal utility next period, collusion is easier to sustain since consumers can switch among
…rms to avoid diminishing marginal utility on the equilibrium path, but at least some consumers cannot switch when a …rm deviates to capture all consumers. When diminishing
marginal utility is with externality, i.e., when consumption of one …rm’s product this period
lowers all …rms’products’marginal utility next period, collusion is (weakly) harder to sustain due to demand ‡uctuations endogenously induced by consumers’purchasing decisions
on the equilibrium path.
Key Words: Dynamic Diminishing Marginal Utility, Tacit Collusion, Consumer Sophistication, Demand Fluctuation
1
Introduction
Economics Department, Hong Kong University of Science & Technology Business School; email: kli-
[email protected].
y
Economics Department, Hong Kong University of Science & Technology Business School; email:
[email protected].
1
Diminishing marginal utility is well accepted textbook knowledge. It is usually treated as a
static property: in a static model or a period of a dynamic model, the more the consumer
consumes, the lower is her marginal utility in that period. However, in many situations, it is
more appropriately modeled as a dynamic phenomenon. For example, if you had Italian food
last weekend, you may prefer trying Japanese food this weekend. Or if you have been watching
a lot of superhero movies in the last three months, you will not want to watch another one for
at least some time.
Dynamic diminishing marginal utility has some potential implications on pricing and purchasing decisions. When repeat purchase of consumers is likely, a …rm may raise its price to
prevent the consumers from over-consuming now, so that their future marginal utilities (and
thus their demand for the …rm) will remain high. Instead, when consumers’repeat purchase
is unlikely, only the consumers want to preserve their future marginal utilities. In this case,
the …rm has to lower its price to encourage them to consume now, compensating them for the
loss of future marginal utilities.
While some papers studying dynamic diminishing marginal utility focus on consumers’
behavior or the interaction between consumers and …rms, no study considers the interaction between …rms, and thus the e¤ect of dynamic diminishing marginal utility on market
competition. The main purpose of this paper is to show how the assumption of dynamic
diminishing marginal utility a¤ects …rms’ ability to tacitly collude, and thus whether competition is strengthened or weekend under this assumption. We analyze a market in which
in…nitely-lived …rms serve a measure one of in…nitely-lived consumers through competition in
price. We compare the subgame perfect Nash equilibria (SPNE) in which consumers’preferences exhibit diminishing marginal utility with the SPNE in which consumers have constant
marginal utilities. We consider two di¤erent cases of diminishing marginal utility. When a
consumer’s consumption of a …rm’s product this period only lowers her marginal utility of the
…rm’s product next period and not other …rms’, we say that this consumer has diminishing
marginal utility without externality. For example, after listening to some music by Lincoln
2
Park, you may not want to buy their albums for some time, but you may still enjoy albums by
other musicians. When products of all …rms are completely homogeneous so that a consumer’s
consumption of a …rm’s product this period lowers her marginal utility of all …rms’products
next period, we say that this consumer has diminishing marginal utility with externality. For
example, if you have pizza for dinner today, you may not want to go to any pizza restaurant
tomorrow.
Take the case of constant marginal utility as a benchmark. We …rst prove that …rms earn
supranormal pro…ts for a wider range of discount factors when consumers have diminishing
marginal utility without externality. When consumers have constant marginal utility, a deviating …rm can capture the entire industry pro…t in one period before losing all future pro…ts
by undercutting the collusive price. When consumers have diminishing marginal utility without externality, we characterize an equilibrium in which consumers switch among …rms on the
equilibrium path and do not make repeat purchase from any …rm in any two consecutive periods, so that they avoid utility loss due to diminishing marginal utility. When a …rm deviates
to capture all consumers, it has to undercut the collusive price substantially so that those
who purchased from it last period, and by now have lowered marginal utilities, will buy from
it. If it only undercuts the collusive price by an in…nitesimal amount, then it will lose its old
customers, failing to attract all consumers. Therefore, there does not exist a deviation strategy
that allows the deviator to capture the entire industry pro…t in one period before losing all
future pro…ts.
Next we prove that compared to the case of constant marginal utility, …rms earn supranormal pro…ts for a narrower range of discount factors when consumers have diminishing marginal
utility with externality, if consumers are unsophisticated and consumers’consumption in the
previous period causes their marginal utilities to drop too much. We consider two types of
collusive equilibrium. The …rst one is the consecutive-purchase collusive equilibrium. In this
equilibrium, consumers consume every period, and from the second period on, consumers’
marginal utilities and the collusive price stay at a low level. Since each consumer has the
3
same marginal utility for all …rms’ product every period, a deviating …rm can capture the
entire industry pro…t in one period before losing all future pro…ts by undercutting the collusive price. Thus the critical discount factor to sustain this equilibrium is the same as the
critical discount factor to sustain collusion in the case of constant marginal utility. However,
when consumers’consumption in the previous period causes their marginal utilities to drop too
much, the consecutive-purchase collusive equilibrium fails to exist. This is because consumers
prefer not consuming when their marginal utilities are low even if the collusive price is zero.
The second type of equilibrium is the alternate-purchase collusive equilibrium in which
consumers only consume in odd periods. In even periods consumers abstain from consumption
to wait for their marginal utilities to be restored. Due to demand ‡uctuations endogenously
induced by consumers’purchasing decisions on the equilibrium path, this equilibrium is equivalent to the collusive equilibrium in the case of constant marginal utility with lower frequency
of …rms’interaction, making collusion harder to sustain.
We have two more important observations. One is that in the consecutive-purchase collusive
equilibrium, …rms are unable to fully capture consumers’ surplus. The reason is as follows.
Consumers will have lower marginal utilities for the products in the next period if they consume
in the current period. For consumers to consume consecutively on the equilibrium path, …rms
have to compensate consumers for such a cost. Therefore, the highest collusive price in any
period should let the consumers’ surplus be equal to the discounted cost of the lowering of
marginal utilities in the next period.
Another observation is that when consumers have diminishing marginal utility with externality and the extent to which consumers’marginal utilities drop due to consumption in the
previous period is moderate, there exists a “gap” in the range of discount factors in which
no collusion is sustainable. The reason is two-folds. First, the consecutive-purchase collusive
equilibrium does not exist, since consumers with low marginal utilities are su¢ ciently patient
so that they prefer abstaining from consumption and waiting for their marginal utilities to be
restored. Second, the alternate-purchase collusive equilibrium does not exist either, since …rms
4
are not patient enough to abstain from deviation from the collusive price.
For the main analysis (in section 4), we assume that consumers are unsophisticated in the
sense that when they observe any deviation from the equilibrium prices, they believe that in
the following period the prices will return to the equilibrium level. We discuss in section 5 the
alternative assumption that consumers are sophisticated. In other words, whey they observe
any deviation from the equilibrium prices, they believe that in the following period the price
war will begin. We …nd that diminishing marginal utility without externality still facilitates
collusion, while collusion is now as easy to sustain in the case of diminishing marginal utility
with externality as in the case of constant marginal utility.
The rest of the paper is organized as follows. The related literature is reviewed in section 2.
Section 3 provides the model and section 4 includes the analysis for both cases of diminishing
marginal utility. We discuss some alternative assumptions in section 5 and conclude in section
6.
2
Related Literature
There are several theoretical studies in the literature that are related to dynamic diminishing
marginal utility. For example, McAlister (1982) study why consumers prefer intertemporal
brand switching by using a dynamic attribute satiation model of variety-seeking behavior.
The paper assumes that each product can be represented by the values they take on for its
constituent attributes. A consumption history, which is a collection of products, can thus be
represented by the attribute "inventories" it generates. There exists a decreasing marginal
relationship between the attribute "inventories" that would result from the consumption of a
particular product and preference for that product. Under these assumptions, a consumer’s
preference for a product drops as her past consumption of the product increases. Becker and
Murphy (1988) analyze the consumption cycle by introducing stocks of consumption capital,
which are generated by past consumption. For example, to get cycles of overeating and dieting,
5
the eating capital stock must be compensatory with eating, while the weight stock must be
substitutable. Here the e¤ect of the weight stock is similar to diminishing marginal utility in
our paper.
Our paper is related to the literature on collusion and switching costs. Padilla (1995)
shows that collusion is harder to sustain with switching costs. Switching costs reduces the
severity of punishment when collusion breaks down by relaxing competition, and also reduces
the deviation pro…t since consumers are harder to switch to new suppliers. The …rst e¤ect
dominates. Our paper is not a corollary of Padilla (1995), since in our model diminishing
marginal utility does not a¤ect the severity of punishment when collusion breaks down. In
some sense, diminishing marginal utility without externality can be considered as a case of
negative switching costs, while diminishing marginal utility with externality can be considered
as a case of zero switching costs.
Bos, Peeters and Pot (2010) analyze the e¤ect of consumer inertia on tacit collusion. They
…nd that if consumers are inert enough, collusion can be achieved for all discount factors.
However, their result is based on the assumption that …rms’prices in the current period a¤ect
their sales in the next period directly. Under this assumption, collusion is harder to sustain for
higher discount factors (when …rms are more patient), since …rms are more likely to undercut
the collusive price so that they can get more sales in the future. We do not have this e¤ect in
our model, since …rms’prices do not a¤ect their sales in future periods directly.
The fact that consumers’ marginal utilities may ‡uctuate in our model links our paper
to the literature on demand ‡uctuations and collusion, for example, Rotemberg and Saloner
(1986), Haltiwanger and Harrington (1991) and Bagwell and Staiger (1995). While demand
‡uctuations are exogenous in their paper, they are endogenously induced by consumers’purchasing decisions in our model. In the case of diminishing marginal utility without externality,
there are no demand ‡uctuations on the equilibrium path since consumers consume every period. In the case of diminishing marginal utility with externality, there are demand ‡uctuations
in the alternate-purchase collusive equilibrium, which makes collusion harder to sustain.
6
Our paper also provides a reason why product di¤erentiation may facilitate collusion1 .
Although we do not explicitly model any ex-ante product di¤erentiation, we assume that there
exists some heterogeneity between products, which determines whether diminishing marginal
utility is with externality or not. When products are less di¤erentiated, diminishing marginal
utility is more likely to be with externality, and collusion may be harder to sustain. Instead,
when products are more di¤erentiated, diminishing marginal utility is more likely to be without
externality, and collusion is easier to sustain.
The discussion on the assumption of sophisticated consumers in section 5 links our paper
to the literature on collusion and intertemporal demand2 . A property commonly used in this
literature is that consumers’ anticipation of future price wars limits …rms’ ability to deviate
since consumers may delay consumption. The reason is that consumers can buy the same
products in the future with lower prices. However, in our model the reason why consumers
may delay consumption after …rms’ deviation is di¤erent: there is a cost for consumers to
consume now, in terms of lower marginal utilities in the future.
Finally, one closely related paper is Fong and Liu (2011). They show that loyalty programs
can facilitate tacit collusion. The reason is that a deviating …rm must lower its price substantially to attract all consumers, including those who are other …rms’ established consumers,
which means that it cannot capture the entire industry pro…t in one period before losing all
future pro…ts. Our paper di¤ers from theirs in the following aspects: in our model each …rm
o¤ers a uniform price to all consumers, while in their model each …rm can o¤er di¤erent prices
to new consumers and its established consumers; our model characterizes a stochastic game in
which the state every period is determined by consumers’purchasing decisions in the previous
period, while in their model …rms always use non-Markovian strategies; the reason why the
1
For the e¤ect of product di¤erentiation on tacit collusion, see Deneckere(1983), Chang(1991), Ross(1992)
and Rothschild(1992).
2
See for example Ausubel and Deneckere (1987) and Gul (1987) on durable goods pricing, Dutta, Matros
and Weibull (2011) on intertemporal demand, Liski and Montero (2006) and Aichele (2013) on forward contract
and Dana and Fong (2011) on intertemporal bundling.
7
deviant cannot capture the entire industry pro…t in one period is that some consumers have
lower marginal utilities for its products in our model, and that some consumers have access to
a lower "loyalty" price o¤ered by its competitors in their model.
3
The Model
There is a continuum of homogeneous consumers of measure one and n (n
3) …rms selling
identical perishable products.3 All players live forever. Firms compete in prices. Each …rm has
a marginal cost of zero. Each consumer demands at most one product and at most one unit of
that product every period. Each consumer gets an instantaneous utility of zero if she does not
consume in the current period and gets an instantaneous utility equal to her marginal utility of
the good if she consumes. In the case of constant marginal utility, consumers’marginal utilities
for all …rms’ products are V every period. Consider the following two cases of diminishing
marginal utility.
Diminishing marginal utility without externality: A consumer’s valuation for …rm
i’s product in period t is V if he did not consume i’s product in period t
1, and
V if
otherwise.
Diminishing marginal utility with externality: A consumer’s valuation for …rm i’s
product in period t is V if he did not consume any product in period t 1, and V if otherwise.
In the above de…nitions, (1
) 2 (0; 1) measures the extent to which marginal utility
diminishes dynamically. When a consumer’s marginal utility for a …rm’s product is
V , we
say that the consumer is “tired”of the …rm’s product. Thus when consumers have diminishing
marginal utility without externality, they are only "tired" of the speci…c product that they
consumed in the previous period; when consumers have diminishing marginal utility with
externality, they are "tired" of all products if they ever consumed in the previous period. We
assume that …rms cannot identify consumers who are "tired", so that each …rm is also unable
3
Throughout the paper we assume at least three …rms to simplify the analysis, which is true in many real-life
markets.
8
to discriminate between consumers who are "tired" of its own product and consumers who are
"tired" of other …rms’products.
Also note that in both de…nitions we assume that the e¤ect of diminishing marginal utility
only lasts for one period. For example, if a consumer purchases a product i in period t and does
not consume any products after period t, she is "tired" of product i (for the case of diminishing
marginal utility without externality) or all the products (for the case of diminishing marginal
utility with externality) only in period t + 1, but not in any period after period t + 1. We
discuss in section 5 the alternative assumption of more enduring diminishing marginal utility.
The common discount factor for consumers and …rms is
2 (0; 1).
Both …rms and consumers are forward-looking so that they maximize lifetime pro…ts and
utilities respectively. Firms are sophisticated so they correctly anticipate competitors’pricing
strategies in equilibrium. However, as in the IO literature, we consider two cases of consumer
sophistication.
Unsophisticated consumers: whey they observe any deviation from the equilibrium
prices, they believe that in the following period the prices will return to the equilibrium level.
Sophisticated consumers: their belief is fully consistent with …rms’s equilibrium strategies.
We …rst consider unsophisticated consumers in our main analysis, and will discuss the
alternative assumption in section 5.
Finally, we assume that …rms cannot o¤er negative prices both on and o¤ the equilibrium
path. Also, there is a public randomization device so that when the industry can sustain a
pro…t level
, the entire set of pro…ts [0; ] is sustainable by applying public randomization at
the beginning of the …rst period.
9
4
Analysis
For each feasible pro…t level, we compare the sets of discount factors for which tacit collusion is
sustainable between the case of constant marginal utility and the case of diminishing marginal
utility. In the latter case, the game is a stochastic game in which the state every period depends
on consumers’marginal utilities for all …rms’products. We restrict our attention to symmetric
equilibria in which …rms and consumers follow the same strategies for the same state. We also
assume that each consumer has zero measure so that any unilateral deviation by consumers
would not be detected, and the transition of the state would not be a¤ected by such deviation.
We assume unsophisticated consumers in this section.
4.1
Tacit collusion when consumers have constant marginal utility
When consumers have constant marginal utility, they have marginal utilities of V for all …rms’
products every period regardless of their purchasing decisions in the previous period. Consider
the following collusive equilibrium. On the equilibrium path, …rms price at p
V every period.
O¤ the equilibrium path, if any …rm deviates in price, then in all future periods, all …rms revert
to the SPNE of marginal cost pricing. Given that all …rms set price equal to zero and that
each consumer has the same marginal utility for all products, any …rm raising the price above
zero will not get any consumer. Therefore, it is a best response to set price equal to zero. This
constitutes the most severe punishment and hence characterizes the highest sustainable pro…t.
A deviating …rm can get a pro…t arbitrarily close to p by undercutting the collusive price
in any period, and zero pro…t thereafter. Thus …rms have no incentive to deviate in any period
if and only if
p
n(1
p
)
1
We summarize the result as follows.
10
1
n
Lemma 1 Assume that consumers have constant marginal utility. Denote
the discounted
value of the highest sustainable industry pro…t assessed from the …rst period on. Then
8
>
< V
if
1 n1
1
=
>
: 0
if < 1 n1
4.2
Tacit collusion when consumers have diminishing marginal utility with-
out externality
When consumers have diminishing marginal utility without externality, the game is a stochastic
game in which the state is characterized by (k1 ; k2 ; :::; kn ), where ki is the measure of consumers
that consumed product i in the previous period. In other words, ki of the consumers are
"tired" of product i in the current period. To get the highest sustainable industry pro…t for
any discount factor, consider the following collusive equilibrium.
U
On the equilibrium path, when the state is (0; 0; :::; 0), …rms price at pDM
1
1
n
V and
of the consumers consume each …rm’s product; when the state is ( n1 ; n1 ; :::; n1 ), …rms price
U
at pDM
2
V,
1
n
of the consumers consume each …rm’s product, and each consumer does
not consume the product that she is "tired" of; in all other states, …rms collude at whatever
sustainable price and consumers do whatever they want to do. In this equilibrium, each group
of consumers of measure
1
n
rotates among …rms so that consumers never consume products
that they are "tired" of.
O¤ the equilibrium path, if any …rm deviates in price, then in all future periods, all …rms
revert to the SPNE of marginal cost pricing in all states. Given that all …rms set price equal
to zero and that each consumer is "tired" of at most one product in any state, any …rm raising
the price above zero will not get any consumer4 . Therefore, it is a best response to set price
equal to zero. This constitutes the most severe punishment and hence characterizes the highest
sustainable pro…t.
4
This depends on the assumption of n
3, since for any consumer in any state, there are at least n
products for which her marginal utility is V .
11
1 …rms’
It is easy to prove that consumers have no incentive to unilaterally deviate in purchasing
decisions.
Now consider the …rms’ incentive to deviate in any period except the …rst period. The
state stays at ( n1 ; n1 ; :::; n1 ) on the equilibrium path from the second period on, which means
that there are
1
n
of the consumers who are "tired" of each …rm’s product every period. A
U to steal only
deviating …rm can either undercut pDM
2
U
"tired" of its product, or cut its price to pDM
2
(1
n 1
n
of the consumers who are not
) V to steal all consumers. Therefore,
the deviation pro…t is (arbitrarily close to)
max
U
1) pDM
U
2
; max pDM
2
n
(n
(1
) V; 0
Thus …rms have no incentive to deviate if and only if
U
1 pDM
2
n 1
max
U
1) pDM
U
2
; pDM
2
n
(n
(1
)V
Next consider the …rms’ incentive to deviate in the …rst period. Let
DM U
2
discounted value of the industry pro…t assessed from the second period on. Given
denote the
DM U ,
2
…rms
have no incentives to deviate in the …rst period if and only if
1 DM U
p
+
n 1
DM U
2
U
pDM
1
To summarize the result, the following proposition characterizes the highest sustainable
industry pro…t for any discount factor assessed from the …rst period on.
Proposition 1 Assume that consumers have diminishing marginal utility without externality.
Let
DM U
denote the discounted value of the highest sustainable industry pro…t assessed from
the …rst period on. Let ~ = 1
DM U
=
n 1
o .
(n 1)
max
; n
n
8
>
>
>
>
>
>
>
>
<
Then
V
1
V
1
>
V
>
>
1
>
>
>
>
>
: 0
if
if ~
n
n 1
1
1
1
n(1
)
n
n 1
if 1
if
12
1
n
1
<1
<~
1
n 1
<1
1
n
1
n 1
Proposition 1 shows that diminishing marginal utility without externality facilitates collusion by allowing …rms to sustain supranormal pro…ts for
2 [1
1
n 1; 1
1
n ),
which is not
possible in the case of constant marginal utility. There are two reasons. First, in the collusive equilibrium we characterize when consumers have diminishing marginal utility without
externality, consumers never consume products that they are "tired" of on the equilibrium
path because of consumers’switching among …rms, so that the collusive industry pro…t is as
high as the one in the case of constant marginal utility. Actually, when consumers have diminishing marginal utility without externality, the collusive industry pro…t in this equilibrium
is even higher than the monopoly pro…t when there is only one …rm. The reason is that the
consumers who are "tired" of the monopolist’s products have to abstain from consumption
for one period for their marginal utilities to be restored, while in this collusive equilibrium
consumers can consume products that they are not "tired" of every period. The second reason
is that consumers prefer switching to new suppliers when they are "tired", so that the one
period deviation pro…t decreases. A …rm that deviates in any period except the …rst period
has to undercut the collusive price substantially to attract all consumers so that it is unable
to capture the entire industry pro…t in one period before losing all future pro…ts.
4.3
Tacit collusion when consumers have diminishing marginal utility with
externality
When consumers have diminishing marginal utility with externality, the game is a stochastic
game in which the state is characterized by k, where k is the measure of consumers that
consumed any product in the previous period. In other words, consumers of measure k are
"tired" of all the products in the current period. To get the highest sustainable industry pro…t
for any discount factor, consider the following two types of collusive equilibrium.
The consecutive-purchase collusive equilibrium: on the equilibrium path, when the
state is k = 0, …rms price at pCON
1
V and
1
n
of the consumers consume each …rm’s product;
when the state is k = 1, …rms price at pCON
2
13
V and
1
n
of the consumers consume each
…rm’s product; in all other states, …rms collude at whatever sustainable price and consumers
do whatever they want to do. In this equilibrium, consumers consume every period, and from
the second period on, consumers always consume products that they are "tired" of.
The alternate-purchase collusive equilibrium: on the equilibrium path, when the
state is k = 0, …rms price at pALT
1
V and
1
n
of the consumers consume each …rm’s product;
when the state is k = 1, …rms price at pALT
and no consumer consumes; in all other states,
2
…rms collude at whatever sustainable price and consumers do whatever they want to do. In this
equilibrium, consumers only consume in odd periods so that they never consume any products
that they are "tired" of.
O¤ the equilibrium path, the punishment is the same for the two types of collusive equilibrium. If any …rm deviates in price, then in all future periods, all …rms revert to the SPNE
of marginal cost pricing in all states. Given that all …rms set price equal to zero and that each
consumer is either "tired" of all …rms’ products or "tired" of no products in any state, any
…rm raising the price above zero will not get any consumer. Therefore, it is a best response to
set price equal to zero. This constitutes the most severe punishment and hence characterizes
the highest sustainable pro…t.
First consider the consecutive-purchase collusive equilibrium. The following lemma shows
that …rms are unable to fully capture consumers’surplus in this equilibrium.
Lemma 2 In the consecutive-purchase collusive equilibrium, consumers have no incentive to
deviate unilaterally if and only if pCON
1
(1 +
)V and pCON
2
Intuitively, consumers lower their marginal utilities from V to
( +
)V .
V in period t + 1 once
they consume in period t. The lowering of marginal utilities, discounted to period t, is then
(1
V
)V . For consumers to consume when they are not "tired", …rms must keep their surplus
pCON
, at least as high as (1
1
)V . Similarly, for consumers to consume when they are
"tired", …rms must keep their surplus V
pCON
, at least as high as (1
2
)V . Therefore there
are upper bounds for the collusive prices so that …rms are unable to fully capture consumers’
14
surplus in the consecutive-purchase collusive equilibrium.
The following proposition characterizes the highest sustainable industry pro…t for any discount factor assessed from the …rst period on in the consecutive-purchase collusive equilibrium,
given that consumers have no incentive to deviate unilaterally.
Proposition 2 Assume that consumers have diminishing marginal utility with externality
and
1
. Let
CON
denote the discounted value of the highest sustainable industry
pro…t assessed from the …rst period on in the consecutive-purchase collusive equilibrium, where
pCON
1
(1 +
CON
)V and pCON
( +
2
8
h
>
n
< min
(n 1)(1 ) ( +
=
>
: 0
)V . Then
)V; 1+21
2
V
i
if
1
1
n
if
<1
1
n
Proposition 2 shows that it is as easy to sustain collusion when consumers have diminishing
marginal utility with externality and the consecutive-purchase collusive strategy is played as
when consumers have constant marginal utility. The reason is as follows. In the consecutivepurchase collusive equilibrium, which is stationary from the second period on, each consumer
has the same marginal utility for all …rms’ products every period, so that a …rm deviating
in any period except the …rst period can capture the entire industry pro…t in one period by
undercutting the collusive price before losing
1
n
of the same amount of industry pro…t in all
future periods. Thus the critical discount factor to sustain this equilibrium is 1
1
n,
which
is the same as the critical discount factor to sustain collusion when consumers have constant
marginal utility.
However, when consumers’consumption in the previous period causes their marginal utilities to drop too much (
>
1
), the consecutive-purchase collusive equilibrium fails to exist.
This is because consumers prefer not consuming when their marginal utilities are low even
if the collusive price is zero. In this case, we need to consider the alternate-purchase collusive equilibrium. The following lemma shows that consumers have no incentive to unilaterally
deviate in this equilibrium given that the collusive price in even periods is su¢ ciently high.
15
Lemma 3 In the alternate-purchase collusive equilibrium, consumers have no incentive to
deviate unilaterally if pALT
is su¢ ciently high.
2
Intuitively, if the collusive price in even periods is su¢ ciently high in this equilibrium,
consumers will abstain from consumption in even periods no matter they are "tired" or not.
Thus the lowering of marginal utilities in even periods that results from consumption in odd
periods is no longer a "cost" so that consumers have no incentive to deviate in odd periods,
which implies that …rms can fully capture consumers’surplus in odd periods in this equilibrium.
The following proposition characterizes the highest sustainable industry pro…t for any discount factor assessed from the …rst period on in the alternate-purchase collusive equilibrium,
given that consumers have no incentive to deviate unilaterally.
Proposition 3 Assume that consumers have diminishing marginal utility with externality.
ALT
Let
denote the discounted value of the highest sustainable industry pro…t assessed from
the …rst period on in the alternate-purchase collusive equilibrium, where pALT
is su¢ ciently
2
high so that consumers have no incentive to deviate unilaterally. Let
1
2
n(1
.Then
)
ALT
q
if
1
n
1
and
ALT
if
q
1
1
n
<
< 1.
=
=
8
>
<
8
>
<
V
if
2
1
>
: 0
if
V
1
>
: 0
2
if
if
<
q
1
n
1
1
n
q
b>
<b
1
q
1
b be the solution to
1
n
Proposition 3 shows that collusion is harder to sustain when consumers have diminishing
marginal utility with externality and the alternate-purchase collusive strategy is played than
when consumers have constant marginal utility. The reason is as follows. In the alternatepurchase collusive equilibrium, there are demand ‡uctuations endogenously induced by con16
sumers’purchasing decisions on the equilibrium path, which makes collusion harder to sustain.
For example, if a …rm deviates in an odd period in which consumers’marginal utilities are high,
it can capture the entire industry pro…t in one period by undercutting the collusive price by
an in…nitesimal amount, before losing future collusive pro…ts which come in every other period
from this period on. If a …rm deviates in an even period in which consumers’marginal utilities
are low, it can fully capture the surplus of all consumers in this period by undercutting the
collusive price substantially, before losing future collusive pro…ts which come in every other
period from the next period on.
Now we analyze the sustainability of collusion when consumers have diminishing marginal
utility with externality, assuming that …rms can freely choose from the two types of collusive
equilibrium. Let
DM E
denote the discounted value of the highest sustainable industry pro…t
assessed from the …rst period on. We present the results for di¤erent values of .
1) If
1
1
1
n,
DM E
=
8
>
<
V
1
>
: 0
2
if
if
<
q
1
1
n
1
1
n
q
The consecutive-purchase collusive equilibrium does not exist since consumers prefer abstaining from consumption when they are "tired". The alternate-purchase collusive equilibrium
q
1 n1 .
can be sustained for
q
2) If 1 n1 < 1
1 n1 ,
DM E
=
8
>
V
>
>
>
1
>
>
>
>
< 0
>
>
min
>
>
>
>
>
>
: 0
q
if
2
if
h
n
(n 1)(1
)(
+
)V; 1+21
2
V
i
1
if 1
if
1
n
1
<
1
n
<1
q
<
1
n
1
1
1
n
Only the consecutive-purchase collusive equilibrium can be sustained for 1
q
Only the alternate-purchase collusive equilibrium can be sustained for
1
1
n
1
n.
<
1
.
Note that
there is a “gap”in the range of discount factors, in which neither collusive equilibrium can be
17
sustained.
q
3) If 1
DM E
q
1
1
n<1
q
1 1
q n ,
1
1
1 n
8
>
V
>
>
>
1 2
>
>
h
h
>
>
n
< max V 2 ; min
(n 1)(1
1
=
h
>
n
>
min
>
>
(n 1)(1 ) ( +
>
>
>
>
: 0
if
)(
+
)V; 1+21
)V; 1+21
i
2
V
2
V
ii
if
1
q
if 1
if
1
n
1
1
n
1
n
<1
<1
q
< 1
1
n
Only the consecutive-purchase collusive equilibrium can be sustained for 1
1
n.
Only the alternate-purchase collusive equilibrium can be sustained for
q
<1 .
Both types of collusive equilibrium can be sustained for 1 n1
4) If
1
DM E
>
q
1 1
q n
1
1
1 n
1
n
<
1
.
> 1,
8
h
h
>
n
>
max 1 V 2 ; min (n 1)(1
>
>
<
h
n
=
min (n 1)(1
)( +
>
>
>
>
: 0
)( +
)V; 1+21
)V; 1+21
i
2
V
2
V
ii
if
if 1
if
b
1
n
<1
Only the consecutive-purchase collusive equilibrium can be sustained for 1
Both types of collusive equilibrium can be sustained for
b.
1
n
1
n
<b
< b.
To summarize, collusion is harder to sustain when consumers have diminishing marginal
utility with externality and is small enough. When 1
1 n1 , …rms can sustain supranorq
mal pro…ts for 2 [1 n1 ; 1 n1 ) in the case of constant marginal utility, which is not possible
in the case of diminishing marginal utility with externality. The reason is as follows. When
is small enough, consumers prefer abstaining from consumption when they are "tired", so that
the consecutive-purchase collusive equilibrium fails to exist. Firms can only choose to sustain
the alternate-purchase collusive equilibrium, in which there are demand ‡uctuations on the
equilibrium path so that …rms have more incentive to deviate.
18
5
Discussions
In section 4, we derive that compared to the benchmark case of constant marginal utility,
diminishing marginal utility without externality facilitates tacit collusion, while diminishing
marginal utility with externality makes collusion harder to sustain if
is small enough. In this
section, we check whether our main results still hold by considering the following alternative
assumptions.
5.1
Sophisticated consumers
Throughout the main analysis we assumed that consumers are not sophisticated enough to
anticipate a price war upon observing a price cut. In this section we adopt the alternative
assumption that consumers are sophisticated, to investigate how our main results may change.
We consider the same equilibria as we characterize in the main analysis.
We …nd that diminishing marginal utility without externality nevertheless facilitates tacit
collusion. Since consumers can always …nd some …rm’s product that they are not "tired" of in
all states, their purchasing decisions only depend on their utilities in the current period, which
means that the most pro…table deviation strategy for …rms does not change: in any period
U by an in…nitesimal amount
except the …rst period a deviating …rm can either undercut pDM
2
U
to steal the consumers who are not "tired" of its product or cut its price to pDM
2
to steal all consumers before losing all future pro…ts
to sustain collusion is still 1
U
pDM
2
1
n 1
(1
)V
. Thus the critical discount factor
1
n 1.
However, we show that collusion is now as easy to sustain in the case of diminishing
marginal utility with externality as in the case of constant marginal utility for all values of
. For
1
1
1
n.
the consecutive-purchase collusive equilibrium can be sustained if and only if
Assume
>
1
now so that the consecutive-purchase collusive equilibrium fails
to exist. Consider the alternate-purchase collusive equilibrium.
Following any deviation by some …rm in an even period, if a consumer purchases from the
19
deviator, her utility stream starting from the next period on is (0; V; 0; V; :::); if she abstains
from consumption, her utility stream starting from the next period on is (V; 0; V; 0; :::). The
cost of purchasing this period, in terms of future utility loss, is
V
V
2
2
1
=
2
1
V
1+
Since the forward-looking consumers can anticipate such a cost, the …rm’s deviation pro…t
must be less than or equal to
V
1+
V
, which is negative since we assume
>
1
. Thus
the incentive constraint for …rms not to deviate in even periods always holds.
Following any deviation by some …rm in an odd period, if a consumer purchases from the
deviator, her utility stream starting from the next period on is (0; V; 0; V; :::); if she abstains
from consumption, her utility stream starting from the next period on is (V; 0; V; 0; :::). The
cost of purchasing in the current period, in terms of future utility loss, is
V
1
V
2
2
1
=
2
V
1+
Since the forward-looking consumers can anticipate such a cost, the …rm’s deviation pro…t
must be less than or equal to
V
V
1+
=
V
1+
Thus the incentive constraint for …rms not to deviate in odd periods is
pALT
1
2
n(1
)
min
V
; pALT
(1 + ) 1
which is easiest to satisfy when pALT
=V.
1
Therefore the alternate-purchase collusive equilibrium is sustainable if and only if
1
1
n
Recall that when consumers are unsophisticated, diminishing marginal utility with externality makes collusion harder to sustain if
is small enough, because of demand ‡uctuations
endogenously induced by consumers’ purchasing decisions in the alternate-purchase collusive
20
equilibrium. When consumers are sophisticated, a deviating …rm has to undercut the collusive
price substantially to attract the consumers in the alternate-purchase collusive equilibrium,
since otherwise consumers would delay consumption so that their marginal utilities stay at a
high level. Therefore consumers’ forward-looking behavior upon observing a price cut facilitates collusion so that the "demand ‡uctuation e¤ect" is o¤set.
5.2
Enduring diminishing marginal utility
We have assumed that the e¤ect of diminishing marginal utility lasts for only one period in our
main analysis. In other words, consumption in period t only lowers the consumer’s marginal
utility in period t + 1. It is natural to ask whether our results continue to hold if the e¤ect of
diminishing marginal utility is more enduring. To answer this question, we now assume that
the e¤ect of diminishing marginal utility lasts for M (M
2) periods. We focus on the case of
M + 25 . The formal de…nitions are as follows.
n
Diminishing marginal utility without externality: A consumer’s valuation for …rm
i’s product in period t is V if he did not consume i’s product from period t
1 to period t
M,
and V if otherwise.
Diminishing marginal utility with externality: A consumer’s valuation for …rm i’s
product in period t is V if he did not consume any product from period t
1 to period t
M,
and V if otherwise.
We keep the assumption of unsophisticated consumers and focus on symmetric equilibrium
in which …rms and consumers follow the same strategies for the same state.
The result that diminishing marginal utility without externality facilitates collusion is
strengthened. There are enough …rms among which consumers can switch, which implies that
on the equilibrium path consumers never consume products that they are "tired" of. Moreover,
a deviating …rm in any period except the …rst period can either undercut the collusive price
5
Like the assumption of n
3 in the main analysis, this assumption is to ensure that marginal cost pricing
in all states is a SPNE so that we can construct the most severe punishment in which …rms get zero pro…ts.
21
by an in…nitesimal amount to attract the consumers who are not "tired" of its product or
undercut substantially to attract all consumers, so that it cannot capture the entire industry
pro…t in one period before losing all future pro…ts. Recall that the key incentive constraint in
the case of diminishing marginal utility without externality in the main analysis is
U
1 pDM
2
n 1
max
(n
U
1) pDM
U
2
; pDM
2
n
(1
)V
Under the alternative assumption, this incentive constraint is replaced by
U
1 pDM
2
n 1
max
(n
U
M ) pDM
U
2
; pDM
2
n
(1
)V
Thus …rms have less incentive to deviate when the e¤ect of diminishing marginal utility is
more enduring, since there are more consumers who are "tired" of some …rm’s product in any
period except the …rst period so that a deviating …rm has to undercut more substantially to
attract all consumers.
The result that diminishing marginal utility with externality makes collusion harder to
sustain if
is small enough is also strengthened under the alternative assumption. We can
construct a collusive equilibrium which is similar to the consecutive-purchase collusive equilibrium, which fails to exist when
is small enough. In this case, we have to consider a collusive
equilibrium that is similar to the alternate-purchase collusive equilibrium in which there are
demand ‡uctuations on the equilibrium path. Recall that the key incentive constraint in the
alternate-purchase collusive equilibrium in the main analysis is
pALT
1
2
n(1
)
pALT
1
Under the alternative assumption, this incentive constraint is replaced by
pALT
1
n(1
M +1
)
pALT
1
Thus the "demand ‡uctuation e¤ect" is strengthened when the e¤ect of diminishing marginal
utility is more enduring. Since consumers may be "tired" for more than one period, there is
a longer "delay" between two consecutive periods in which …rms make positive pro…ts in this
equilibrium so that …rms have more incentive to deviate.
22
5.3
Asymmetric collusive equilibrium
In the main analysis, we proved that diminishing marginal utility with externality makes
collusion harder to sustain if
is small enough, due to the demand ‡uctuations in the alternate-
purchase collusive equilibrium. Since we only focused on symmetric equilibrium, it is natural
to ask whether this result may change if this restriction is relaxed. In this section, we prove
that this result still holds even if we are able to construct an asymmetric collusive equilibrium
to avoid the demand ‡uctuations on the equilibrium path.
We keep the assumption of unsophisticated consumers and focus on equilibrium in which
…rms and consumers follow the same strategies for the same state. Consumers have diminishing
marginal utility with externality and
is su¢ ciently small so that the consecutive-purchase
collusive equilibrium does not exist.
To get rid of the "demand ‡uctuation e¤ect" in the alternate-purchase collusive equilibrium,
we focus on the following candidate equilibrium: on the equilibrium path, when the state is
k = 0, …rms price at pASY
1
the state is k =
1
2,
V and
1
2n
…rms price at pASY
2
of the consumers consume each …rm’s product; when
V and
1
2n
of the consumers consume each …rm’s
product; in all other states, …rms collude at whatever sustainable price and consumers do
whatever they want to do. In this equilibrium, half of the consumers consume every period
and each consumer consumes in every other period.
O¤ the equilibrium path, if any …rm deviates in price, all …rms revert to the SPNE of
marginal cost pricing in all states. Given that all …rms set price equal to zero and that each
consumer is either "tired" of all …rms’products or "tired" of no products in any period, any
…rm raising the price above zero will not get any consumer. Therefore, it is a best response to
set price equal to zero. This constitutes the most severe punishment and hence characterizes
the highest sustainable pro…t.
There are no demand ‡uctuations for the …rms in the candidate equilibrium. However,
we show that collusion is nevertheless harder to sustain when consumers have diminishing
23
marginal utility with externality and the candidate equilibrium is played than when consumers
have constant marginal utility.
For consumers to be indi¤erent between starting consuming from the …rst period and
starting consuming from the second period, we must have
(V
pASY
)+
1
2
pASY
)
2
(V
1
2
pASY
)
2
(V
=
1
2
or equivalently,
pASY
=
1
V + pASY
2
1+
When a …rm deviates in the …rst period, it can undercut pASY
by an in…nitesimal amount
1
to attract all consumers, since consumers prefer purchasing from the deviant in the …rst period
to starting consuming from the second period. Thus the incentive constraint for …rms not to
deviate in the …rst period is
pASY
1
+
2n
1
pASY
2
2n
pASY
1
or equivalently,
1
1
(2n
pASY
1) p1ASY + 1
2
Note that the right-hand-side is increasing in
pASY
1
pASY
2
. Thus the incentive constraint is easiest
= pASY
=V.
to satisfy for pASY
1
2
Therefore the candidate equilibrium is sustainable only if
1
6
1
>1
2n
1
n
Conclusion
In this paper, we illustrate how dynamic diminishing marginal utility can a¤ect …rms’ability to
tacitly collude. When diminishing marginal utility is without externality so that consumption
of some …rm’s product only lowers the consumer’s marginal utility of this …rm’s product in
24
the future, tacit collusion is easier to sustain. Consumers’ switching among …rms on the
equilibrium path avoids the lowering of collusive pro…ts due to consumers’ utility loss. Also
consumers’tendency to switch to new products prevents a deviating …rm from capturing the
entire industry pro…t in one period before losing all future pro…ts.
When diminishing marginal utility is with externality so that consumption of some product
lowers the consumer’s marginal utility of all …rms’ products in the future, tacit collusion is
harder to sustain if consumers’consumption in the previous period causes their marginal utilities to drop too much. We …rst characterize an equilibrium (the consecutive-purchase collusive
equilibrium) in which consumers consume every period. While collusion is as easy to sustain in
the consecutive-purchase collusive equilibrium as in the case of constant marginal utility, this
equilibrium fails to exist when
is small enough. We then characterize an equilibrium (the
alternate-purchase collusive equilibrium) in which consumers only consume in odd periods.
Collusion is harder to sustain in the alternate-purchase collusive equilibrium than in the case
of constant marginal utility, due to demand ‡uctuations on the equilibrium path.
Our results relied on several assumptions. We assumed that there are more than two …rms.
If there are only two …rms, we conjecture that the main results still hold even though the
characterization of the most severe punishment could be very complicated. We also assumed
that consumers are small so that any unilateral deviation by consumers would not be detected.
If consumers could act collectively, …rms’ability to tacitly collude would be diminished.
More importantly, we assumed in the main analysis that consumers are unsophisticated
so that they always expect prices to be the equilibrium prices, even after a price cut. When
consumers are sophisticated so that they anticipate a price war upon observing a price cut, we
show that tacit collusion is nevertheless easier to sustain in the case of diminishing marginal
utility without externality than in the case of constant marginal utility. However, tacit collusion is now as easy to sustain in the case of diminishing marginal utility with externality as in
the case of constant marginal utility. The reason is that a deviating …rm has to undercut the
collusive price more substantially in the alternate-purchase collusive equilibrium, since other25
wise consumers would delay consumption upon observing a price cut to keep their marginal
utilities high.
Note that diminishing marginal utility with externality nevertheless makes collusion harder
to sustain when
is small enough if we can make the following assumption: when consumers
observe any unilateral deviation by some …rm, they believe that with probability
from the next period on the forever price war will begin, and with probability 1
( < 1)
in the next
period the prices will return to the equilibrium level. In this sense, our results are still robust
to changes in the assumption of consumer sophistication.
One interesting extension for future research is to explicitly model ex-ante product differentiation. In our analysis, we only capture products’ heterogeneity by distinguishing the
case of diminishing marginal utility without externality between the case with externality. In
real markets, consumption of some product may lower the consumer’s marginal utility for not
all but certain types of products. This has some important empirical implications. We may
need to analyze product di¤erentiation in an intertemporal perspective, namely the e¤ect of
consumption of some product on future demand of other products.
Appendix
Lemma A1 Assume that consumers have diminishing marginal utility without externality.
Let
DM U
2
denote the discounted value of the highest sustainable industry pro…t assessed from
the second period on. Then
DM U
2
where [1
1 ~
n 1; )
= ? if
=
(n 1)
n
8
>
>
>
>
>
<
>
>
>
>
>
:
V
1
V
1
1
1
1
n(1
if
~
if 1
1
n 1
)
0
if
<~
1
n 1
<1
.
By rearranging the incentive constraint for any period except the …rst period we get
1 1
n1
max
n
1
n
;1
26
(1
)
V
U
pDM
2
Note that 1
V
DM U . For …rms to collude at pDM U = V (so
) pDM
U is increasing in p2
2
(1
2
that
DM U
2
=
V
1
), the incentive constraint becomes
1 1
n1
(n
max
1)
n
;
or equivalently,
1
max
1 ~
n 1; )
It is easy to verify that [1
2 [1
1 ~
n 1; )
n
= ? if
1
o
(n 1)
n ;
(n 1)
n
=~
n
(n 1)
n
. Now suppose
<
and consider
6= ?.
This means
1
(1
)>
1 1
n1
(n
1)
n
U has to be chosen low enough so that
To satisfy the incentive constraint, pDM
2
1 1
n1
1
(1
)
V
U
pDM
2
or equivalently,
U
pDM
2
which implies that
If
<1
1
n 1,
DM U
2
=
V
1
V
1
1
1
n(1
1
1
n(1 )
1
.
)
which means
1 1
n1
<
(n
1)
n
no supranormal pro…t can be sustained from the second period on.
Proof of Proposition 1.
By rearranging the incentive constraint for the …rst period we get
DM U
2
U
pDM
1
Note that
DM U
2
n 1
is increasing in
DM U .
2
n
To get
27
1
DM U ,
let
DM U
2
=
DM U .
2
If
~, then
DM U
2
V
1
=
, and …rms have no incentive to deviate in the …rst period if
and only if
V
U
pDM
1
n
U = V (so that
For …rms to collude at pDM
1
11
DM U
=
V
1
), the incentive constraint becomes
V
V
n
11
or equivalently,
1
n
1
If
2 [~; 1
1
n ),
DM U
2
V
1
=
U has to be chosen low enough so that the incentive
and pDM
1
constraint binds. This implies
U
pDM
=
1
or
DM U
If
2 [1
=
V
1
V
n
11
n
n 1.
1 ~
n 1 ; ),
DM U
2
=
V
1
1
1
1
n(1
)
U has to be chosen low enough so that
and pDM
1
the incentive constraint binds. This implies
U
pDM
=
1
or
If
DM U
<1
=
V
1
1
n 1,
1
1
1
n(1
DM U
2
)
V
n
1
11
1
1
n(1 )
n
n 1.
= 0. It must be
DM U
= 0.
Proof of Lemma 2.
It is su¢ cient to prove that one-shot deviations are not pro…table for each consumer in the
state k = 0 and k = 1.
When the state is k = 0, each consumer’most pro…table deviation is as follows: she does
not consume in the current period and consumes every period from the next period on. The
incentive constraint for such deviation is
(V
pCON
)+
1
1
( V
pCON
)
2
(V
28
pCON
)+
2
2
1
( V
pCON
)
2
or equivalently,
pCON
1
(1 +
)V
When the state is k = 1, each consumer’most pro…table deviation is as follows: she does
not consume in the current period and consumes every period from the next period on. The
incentive constraint for such deviation is
pCON
2
V
pCON
)+
2
(V
1
2
( V
1
pCON
)
2
or equivalently,
pCON
2
( +
)V
Thus
pCON
1
V
and
pCON
2
V
Proof of Proposition 2.
It is su¢ cient to prove that one-shot deviations are not pro…table for each …rm in the state
k = 0 and k = 1.
When the state is k = 0, a deviating …rm can attract all consumers by undercutting pCON
.
1
The incentive constraint for such deviation is
1 CON
(p
+
n 1
1
pCON
)
2
pCON
1
or equivalently,
pCON
1
(n
1)(1
)
pCON
2
When the state is k = 1, a deviating …rm can attract all consumers by undercutting pCON
.
2
The incentive constraint for such deviation is
pCON
2
pCON
2
n(1
)
29
or equivalently,
1
n
1
If
1
1
n,
pCON
has to be chosen low enough so that either the incentive constraint for
1
…rms’deviation or the incentive constraint for consumers’deviation binds in the state k = 0,
and pCON
has to be chosen low enough so that the incentive constraint for consumers’deviation
2
binds in the state k = 1. This implies
pCON
=( +
2
)V
and
pCON
= min
1
or
CON
If
<1
= min
1
n,
h
n
(n 1)(1
(n
1)(1
)
( +
)V; 1+21
)( +
pCON
= 0. It must be
2
CON
= 0.
)V; (1 +
2
)V
i
V .
Proof of Lemma 3.
It is su¢ cient to prove that one-shot deviations are not pro…table for each consumer in the
state k = 0 and k = 1.
When the state is k = 0, each consumer can deviate by not consuming now and choose
from in…nite deviation strategies from the next period on. The incentive constraints for all
possible deviation strategies are
+3
1
1
2
pALT
)
1
(V
+3
1
1
where
2
(V
(1
1
pALT
)
1
+1
2
)
(V
(1
1
pALT
)+
2
+1
2
)
(V
+1
( V
)
pALT
1
pALT
)
2
= 1; 3; 5:::.
When the state is k = 1, each consumer can deviate by consuming now and choose from
in…nite deviation strategies from the next period on. The incentive constraints for all possible
deviation strategies are
(V
pALT
)
1
( V
pALT
)+ ( V
2
30
pALT
)
1
pALT
)
1
(V
(1
1
+2
2
)
(V
pALT
)
1
+2
(1
1
where
2
)
(V
2
pALT
)+
2
( V
pALT
)
1
pALT
2
V
(1
1
)
2
2
pALT
)+
2
( V
pALT
)+
2
(V
(1
1
)
2
+1
( V
pALT
)
1
pALT
)
2
(V
= 2; 4; 6:::.
Thus the incentive constraints are all satis…ed if pALT
is su¢ ciently high.
2
Proof of Proposition 3.
It is su¢ cient to prove that one-shot deviations are not pro…table for each …rm in the state
k = 0 and k = 1.
When the state is k = 0, a deviating …rm can attract all consumers by undercutting pALT
.
1
The incentive constraint for such deviation is
pALT
1
2
n(1
)
pALT
1
When the state is k = 1, if a …rm deviates to a price (denote this price pd ), consumers
would purchase from it rather than not consume this period if and only if at least one of the
following conditions is satis…ed:
(V
pALT
)<( V
1
(V
(1
1
+2
2
)
pALT
)<( V
1
(V
+2
(1
1
where
2
)
(V
pd ) + ( V
pALT
)< V
1
2
pd ) +
(1
1
pALT
)<( V
1
pALT
)
1
pd
)
2
(V
2
pd ) +
(1
1
pALT
)+
2
)
2
(V
+1
( V
pALT
)
1
pALT
)
2
= 2; 4; 6:::.
Note that consumers are less likely to purchase from the deviating …rm in the state k = 1
when pALT
is higher. Let pALT
= 1. The incentive constraint for …rms’deviation in the state
2
2
k = 1 is
pALT
1
2
n(1
)
max ( +
)V; (
31
)V + pALT
1
Combine the incentive constraints in the state k = 0 and k = 1:
1
n(1
2
max ( +
)
V
;(
pALT
1
)
)
V
+ 1; 1
pALT
1
, collusion is easiest to sustain when pALT
= V , and the combined incentive
1
If
constraint becomes:
1
2
n(1
If
<
1
)
, collusion is easiest to sustain when pALT
= V , and the combined incentive
1
constraint becomes:
1
2
n(1
If
>
1
, for all pALT
1
)
V , the combined incentive constraint becomes:
1
2
n(1
Thus for
q
1
1
n,
1
)
1
collusion is easiest to sustain when pALT
= V , and the combined
1
incentive constraint becomes:
r
1
n
1
q
< 1 n1 , we must have pALT
= 0;
1
q
for 1 n1 < < 1, collusion is easiest to sustain when pALT
= V , and the combined
1
If
incentive constraint becomes:
If
< e, we must have pALT
= 0.
1
b>
r
1
1
n
References
[1] Aichele, M. (2013). Abuse of forward contracts to semi-collude in volatile markets.
[2] Ausubel, L. M., & Deneckere, R. J. (1987). One is almost enough for monopoly. The
RAND Journal of Economics, 255-274.
32
[3] Bagwell, K., & Staiger, R. W. (1995). Collusion over the business cycle (No. w5056).
National Bureau of Economic Research.
[4] Becker, G. S., & Murphy, K. M. (1988). A theory of rational addiction. The journal of
political economy, 675-700.
[5] Bos, A. M., Peeters, R. J., & Pot, E. A. (2010). Competition versus collusion: the impact
of consumer inertia. METEOR, Maastricht research school of Economics of Technology
and Organizations.
[6] Chang, M. H. (1991). The e¤ects of product di¤erentiation on collusive pricing. International Journal of Industrial Organization, 9(3), 453-469.
[7] Dana, J. D., & Fong, Y. F. (2011). Long-Lived Consumers, Intertemporal Bundling and
Collusion. The Journal of Industrial Economics, 59(4), 609-629.
[8] Deneckere, R. (1983). Duopoly supergames with product di¤erentiation. Economics Letters, 11(1), 37-42.
[9] Fong, Y. F., & Liu, Q. (2011). Loyalty rewards facilitate tacit collusion. Journal of Economics & Management Strategy, 20(3), 739-775.
[10] Gul, F. (1987). Noncooperative collusion in durable goods oligopoly. The RAND Journal
of Economics, 248-254.
[11] Haltiwanger, J., & Harrington Jr, J. E. (1991). The impact of cyclical demand movements
on collusive behavior. The RAND Journal of Economics, 89-106.
[12] Liski, M., & Montero, J. P. (2006). Forward trading and collusion in oligopoly. Journal of
Economic Theory, 131(1), 212-230.
[13] Matros, A., Dutta, P., & Weibull, J. W. (2011). Bertrand competition with forwardlooking consumers.
33
[14] McAlister, L. (1982). A dynamic attribute satiation model of variety-seeking behavior.
Journal of Consumer Research, 141-150.
[15] Padilla, A. J. (1995). Revisiting dynamic duopoly with consumer switching costs. Journal
of Economic Theory, 67(2), 520-530.
[16] Ross, T. W. (1992). Cartel stability and product di¤erentiation. International Journal of
Industrial Organization, 10(1), 1-13.
[17] Rotemberg, J. J., & Saloner, G. (1986). A supergame-theoretic model of price wars during
booms. The American Economic Review, 390-407.
[18] Rothschild, R. (1992). On the sustainability of collusion in di¤erentiated duopolies. Economics Letters, 40(1), 33-37.
34