ECONOMIC RESEARCH CENTER DISCUSSION PAPER
A u g u s t 2003

No. 145CYCLICAL PRICE MOVEMENTSIN

AN

ATOMISTIC

MARKETby
T a d a s h i M I N A G A W A*
Shin KAWAI**

ABSTRACTThis paper explains price movements countercyclical to business cycles in an
atomistic market. A recession makes a portion of low-income consumers leave
the market and the fraction of low-price stores decreases. Thus, the remaining
consumers' reservation prices for storage increase and consequently low-pricestores charge higher prices, and vice versa. It is also shown that the equilibrium
is characterized as a symmetric mixed strategy Nash equilibrium.
JEL classification: L11, L16
KEYWORDS
Price movements, Business cycles, Atomistic market, Mixed strategy

Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya, 464-8601, Japan. Tel.: +81-52-789-2384. Fax.: +81-52-789-4924.
** Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya, 464-8601, Japan. E-mail: d980103d@mbox.nagoya-u.ac.jp
This paper is received July23 2003.
*

1

Introduction

Empirical studies show that price markups over marginal cost behave countercyclically to business cycles (see, for example, Bils, 1987). This behavior
is observed particularly in unconcentrated industries (see Domowitz, et al.,1986a,b).1 If these observations are accurate, we cannot predict that pricemoves procyclically to business cycles. Classical theory, however, insiststhat price always moves procyclically to business cycles2 if the short-run
marginal cost curves are upward sloping. It seems that the classical theory
fails to explain some of the actual markets’ price phenomena.
The purpose of this paper is to explain theoretically the countercyclicalbehavior of price3 to business cycles in an atomistic market. By the atomistic market we mean the market which retains all the usual characteristics
of perfectly competitive markets except in that individual firms decide theprice they will charge, and the purchasers react passively to these prices (seeHey, 1974). Although the study of price movements has been devoted mainly
to the explanation of price rigidities, which are represented by the kinked demand curve theory (see, for example, Okun, 1981 and Stiglitz, 1987), several
papers make clear countercyclical price behavior (Phelps and Winter, 1970;
Stiglitz, 1984; Rotemberg and Saloner, 1986; Bils, 1989a,b; and Wilson andReynolds, 2002). They explain, for example within customer markets, that
customers with high elasticity of demands leave the market while customers
with inelastic demands remain in a period of recession. As a result, themarket demand curve becomes inelastic and firms can increase the pricein order to maximize their profits. These studies, however, focus mainlyon imperfectly competitive markets such as monopolies and oligopolies and1

For a survey of these empirical studies, see Carlton (1989).
We consider business cycles which are caused by movements in demand.
3
We are interested in price movements rather than markups.
2

1

thus only a few attempts have been made to analyze atomistic markets. In
the following, we exploit the Stiglitz’s model (1979)(1989), which studies
the relation between price distribution and search costs. In our model, adiﬀerence in income, instead of a distribution of search costs, is introduced.
A recession makes a portion of low-income consumers leave the market andthe fraction of low-price stores decreases.4 Thus, the remaining consumers’
reservation prices for the storage increase and consequently low-price stores
charge higher prices, and vice versa. We also find that equilibrium is attained by firms’ mixed strategic pricing behavior in the atomistic market.
The next section constructs a model for consumers and firms in partial equilibrium. Section 3 provides countercyclical movements in price. In
section 4, equilibrium is characterized as a symmetric mixed strategy Nashequilibrium. Section 5 provides concluding remarks.

2

The ModelThis economy is an overlapping generations model. Time is discrete, starts
at t = 1, and is infinite, thus t = 1, 2, . . . . There is one indivisible commodity
x. There are always L new-born consumers in each period. They live fortwo periods.5 They are identical in tastes but diﬀering in income; let 1 − π
be the percentage of consumers with high-income yh and π ∈ (0, 1) be thefraction with low-income yl . There are N identical firms (or stores) whichsell the commodity x. Let 1 − λ be the proportion of firms selling at ph andλ ∈ [0, 1] be the fraction selling at pl in each period. Each firm is engaged in
random pricing. Suppose that both L and N are a large number and bothconsumers and firms take π and λ as given.
Consider the market which retains all the usual characteristics of per4
5

We treat business cycles as exogenously given.
Since our model is partial equilibrium, we ignore the decision problems in labor market.

2fectly competitive markets except in that individual firms decide the pricethey will charge, and the purchasers react passively to these prices. Further, each individual goes shopping and chooses one store randomly among
N stores. They cannot shop at the other stores in that period.6 Supposethat the consumers know a priori the distributions of prices and incomesbut they cannot diﬀerentiate between stores of high and low price until one
is chosen. Assume that the firms know a priori both distributions but theycannot distinguish between the high- and low-income consumers at theirstore.7

2.1

Consumer’s problem

Each individual consumes at most one unit commodity x in each period.
They can purchase another unit for future consumption in the first period.8
Let X = {0, 1} be the consumption set of x, and x1 ∈ X denote the consumption of x in the first period. Let x21 ∈ X and x22 ∈ X denote thex’s second period consumption purchased in the first period and that purchased in the second period, respectively. They are mutually exclusive (i.e.
x21 x22 = 0). The common preferences of consumers are defined by the
expected utility function:u(x1 , x21 , x22 )= E[(1 + ax1 )(y − p1 (x1 + x21 )) + β(1 + ax21 + ax22 )(y − p2 x22 )], (1)x21 x22 = 0, 0 < β < 1, a > 0

6

For simplicity, it is assumed that the second search cost is prohibitively expensive.

For more detailed discussion about search costs, see Stiglitz (1989).
7
Hence, there is no price discrimination in this market.
8
The storage cost is not entailed directly in this model. Instead, discount factor β plays
the same role as the storage cost.

3

where β, y ∈ R+ and pi ∈ R+ are a discount factor, the income of eachperiod, and the price of the i-th period, respectively.9 Prices are randomvariable with a certain distribution λ. Note that the cases u(0, 1, 1) andu(1, 1, 1) do not exist by definition. Assume now the following condition:
ASSUMPTION 1.
β(1 + a) < 1.
The following lemma is established.
LEMMA 1. Let the utility function (1) satisfy Assumption 1. If a consumer chooses x21 = 1, then he has already chosen x1 = 1.
Proof. See Appendix A.
Lemma 1 means that if a consumer purchases only one unit of the commodity x in the first period, then he consumes it in that period, that is, theassumption excludes the case u(0,1,0). The utility function, for example,
takes the following forms:10
u(0, 0, 0) = y + βy,(2)u(1, 0, 1) = (1 + a)(y − p1 ) + β(1 + a)(y − p2 ),(3)

u(1, 1, 0) = (1 + a)(y − 2p1 ) + β(1 + a)y.

(4)

Although the discrete choice of consumption is originated in the early writings of A. Marshall,11 the above type of utility function is used by Gabszewicz and Thisse (1979)(1980) or Shaked and Sutton (1982). Becauseof this form, even with identical preferences, the consumers’ reservation
prices 12 are no longer identical since they will vary in accordance with theirincome.
9

The term (y − pi xi ) represents the residual income in each period.
There are basically five cases, that is, u(1, 0, 0) and u(0, 0, 1) are also possible.
11
See Whitaker (1975); Ch.II.
12
The term reservation price is defined as a price which makes the consumer indiﬀerent
104The method of dynamic programming
Here, we apply the method of dynamic programming. This method is usefulfor analysis of dynamical discrete choice with uncertainty. Figure 1 illustrates the decision tree of this program. There are five stages of decision

Figure 1: The decision tree: The bold line illustrates the optimal paths
when condition (5) is satisfied.

to purchasing the commodity or not. We assume that consumers purchase the commodity
when the price is equal to their reservation price: they purchase it when the price is equal
to or less than the reservation price.

5

making. The first period contains three of these stages. First, the consumer
decides not to enter, or else to enter the market and chooses one store randomly. If he decides to enter, he finds a high-price store with probability(1 − λ) or a low-price store with probability λ. Second, in the store, theconsumer decides to purchase x (i.e. x1 = 1) or not (i.e. x1 = 0), taking
the price charged by the store as given (i.e. ph or pl ). Third, he decides
to purchase another x for storage (i.e. x21 = 1) or not. If he purchases it
for storage, then the total life plan is finished and he merely consumes it in
the second period. The latter two stages in the second period are the sameas the former one in the first period. The bold line illustrates the optimal
paths when the reservation prices satisfy the following condition:p̂1 = p̂22 ≥ ph > p̂21 ≥ pl ,(5)

where p̂i denotes xi -reservation price. Since there is no participation cost (ortransactions cost), we can conclude that the individual enters the market as
long as there is a chance to purchase x; i.e., ph > pl > p̂i is not satisfied.13
In next three arguments, we derive the three reservation prices p̂1 , p̂21 , p̂22
at which the consumer decides purchase x in each stage and deduces thecorresponding individual demand functions.
The choice of x22
Suppose the consumer has been at the bottom nodes of the decision tree. In
this stage, the consumer is in a store in the second period. Hence, he doesn’t
have x for storage (i.e. x21 = 0). The x22 -reservation price is derived fromthe following condition:u(x1 , 0, 1) = u(x1 , 0, 0)
13(6)

We can skip the question of entering the market in each period. See Appendix B.

6or, from (1),(1 + ax1 )(y − p1 x1 ) + β(1 + a)(y − p2 ) = (1 + ax1 )(y − p1 x1 ) + βy(7)where (p1 , x1 ) is a history which has already been realized at this stage.
Solving for p2 , we obtain the x22 -reservation price:µ¶ap̂22 (y) =
y.
1+a

(8)

We find that p̂22 depends on y. Therefore, the demand function of period 2
can be written as
x22 =



1, p̂22 (y) ≥ p2 ,0, p2 ≥ p̂22 (y).

(9)

Since there is no participation cost, it is the suﬃcient condition of entering
the market that there is a store charging equal to or less than the x22 reservation price.14
The choice of x21
From Lemma 1, we can conclude that the consumer has already purchased x
for immediate consumption (i.e. x1 = 1) when he decides whether or not topurchase for storage or not. Here, we encounter an interperiod substitution
problem. The x21 -reservation price is deduced from the following equation:from (1),(1 + a)(y − 2p1 ) + β(1 + a)y(10)= (1 + a)(y − p1 ) + βE[(1 + ax22 )(y − p2 x22 )],where E[ · ] is an expected utility function of the second period when x21 = 0.
Note that, from (9), this function varies according to the level of p̂22 (y),hence income levels.
14

See Appendix B.

7

If p̂22 (y) ≥ ph > pl , then the consumer purchases x (i.e. x22 = 1)
irrespective of price charged by stores. Thus, in this case, the expectedutility function of the second period becomesE[(1 + ax22 )(y − p2 x22 )] = (1 − λ)(1 + a)(y − ph ) + λ(1 + a)(y − pl )(11)

= (1 + a)(y − ((1 − λ)ph + λpl )).
Substituting (11) into (10) and solving for p1 , we obtain the x21 -reservation
price when p̂22 (y) ≥ ph > pl :p̂21 (λ, ph , pl ) = β((1 − λ)ph + λpl ).

(12)

We find that p̂21 depends on the average price p̄ (= (1 − λ)ph + λpl ).
Next, if ph > p̂22 (y) ≥ pl , then the consumer does not purchase x (i.e.
x22 = 0) at a high price store while he purchases x (i.e. x22 = 1) at a low
price store. Therefore, the expected utility function of the second period
becomes
E[(1 + ax22 )(y − p2 x22 )] = (1 − λ)y + λ(1 + a)(y − pl )(13)

= (1 + λa)y − λ(1 + a)pl .
Substituting (13) into (10), we obtain the x21 -reservation price when ph >p̂22 (y) ≥ pl :p̂21 (λ, pl , y) = β((1 − λ)p̂22 (y) + λpl ).

(14)

Hence, these x21 -reservation prices (12), (14) can be written asp̂21 (λ, ph , pl , y) = min{β((1 − λ)ph + λpl ), β((1 − λ)p̂22 (y) + λpl )}.

(15)

The choice of x1
From Lemma 1, if p1 is equal to or less than p̂21 (λ, ph , pl , y), then x21 = 1;
hence, x1 = 1. Thus, we must deduce the x1 -reservation price when x21 = 0.
8

The x1 -reservation price is deduced from the following equation: from (1),(1 + a)(y − p1 ) + βE[(1 + ax22 )(y − p2 x22 )](16)= y + βE[(1 + ax22 )(y − p2 x22 )].
Solving for p1 , we obtain the x1 -reservation price:µ¶ay.
p̂1 (y) =
1+a

(17)

We find that p̂1 (y) = p̂22 (y). From (15), we also find thatp̂1 (y) = p̂22 (y) > p̂21 (λ, ph , pl , y).

(18)

From (18), we can confirm that, when x21 = 1 (i.e. p̂21 (λ, ph , pl , y) ≥ p1 ),the consumer has already purchased x for consumption in this period: i.e.,
x1 = 1.
Summing up the above three arguments, individual demand functions
can be written as
period 1
2, p̂1 (y) > p̂21 (λ, ph , pl , y) ≥ p1 ,

x1 + x21 = 1, p̂1 (y) ≥ p1 > p̂21 (λ, ph , pl , y),0, p > p̂ (y) > p̂ (λ, p , p , y),
1
1
21
h l

(19)

period 2 (when x21 = 0),

x22 =



1,


0,

p̂22 (y) ≥ p2 ,p2 ≥ p̂22 (y).

These demand functions are illustrated in Figure 2.

9

(20)

p1

p2

Period 1p̂1 (y)Period 2(x21 = 0)p̂22 (y)p̂21 (λ, ph , pl , y)

0

1

x

2

0

1

2

x

Figure 2: Indivisual demand curves.

2.2

Firm’s problem and market equilibrium

We now turn to a discussion of firm’s problem and market equilibrium. Since
consumers cannot change the store during the period, firms have some monopolistic power against them. However, they cannot discriminate between
high- and low-income consumers. There is thus no price discrimination in
our model. The store sets one of the reservation prices in each period inorder to maximize its profit. For simplicity, we assume that the marginal
cost of production is zero.15 Therefore, the profit maximization is equal tomaximization of its revenue.16
Suppose that the firms (or stores) know the income distribution and15
16

Each period has a fixed cost that determines the number of firms in the market.
The total profit of the representative firm is described as follows:∞Xδ t−1 (pt qt − c),

(21)

t=1

where δ and c denote discount factor and fixed costs respectively. Since we assume thatfirms and consumers take price distributions λ as constant, we are allowed to treat the
maximization problem as static. Furthermore, we can ignore the fixed cost when comparing profits.

10

the demand function which they face (the individual firm’s demand function). Since there are high-income and low-income consumers, the reservation prices can be divided into two values, respectively. The form of thedemand function, thus, varies in accordance with a diﬀerence in income andthe corresponding strategies used by firms. Consider a set of alternative
strategies: a high price strategy which sells one unit to each high-income
consumer at his reservation price (i.e. ph = p̂1 (yh ) = p̂22 (yh )), or a low pricestrategy which sells two units to each high-income consumer at a low price(i.e. pl = p̂21 (λ, ph , pl )). Figure 3 illustrates the demand function under
this strategy.17 Each firm attracts not only L/N young consumers but alsopHigh-income
Low-income

ph = p̂1 (yh ) = p̂22 (yh )Average cost curvep̂1 (yl ) = p̂22 (yl )pl = p̂21 (λ, ph , pl )p̂21 (λ, pl , yl )0AA : (2 − λ)(1 − π)(L/N )C : (2 + (1 − λ)(1 − π))(L/N )

B

C

D

x

B : (2 − λ(1 − π))(L/N )D : (3 − λ(1 − π))(L/N )Figure 3: An individual firm’s demand curve.
[(1 − λ)L]/N old consumers who selected a high-price store and (πλL)/N
low-income old consumers who selected a low-price store, when they were17

It is assumed that p̂1 (yl ) ≥ p̂21 (λ, ph , pl ) = pl .

p̂21 (λ, pl , yl ). See Appendix C.

11

In this case, p̂21 (λ, ph , pl ) >young.18 Furthermore, each high-income consumer purchases one unit of x
in high-price stores. Then, sales qh of each ph firm are given by (A in Fig.
3)qh =((1 − π) + (1 − π)(1 − λ))L
.
N

(22)

Sales of the low price firms are higher; each high-income young consumerpurchases two unit of x and each low-income one purchases one unit of x
(C in Fig. 3), hence,ql =(2(1 − π) + π + (1 − λ) + πλ)L
.
N

(23)

Since we are interested in this particular case,19 we make the following
assumption about a range of relative income.
ASSUMPTION 2.
(1 − π) > θy ≥ β,

θy ≡

yl
.
yh

The former inequality guarantees that the high price strategy whichsells at the reservation prices of high-income consumers (i.e. ph = p̂1 (yh ) =p̂22 (yh )) is more profitable than that of low-income consumers. The latter
inequality guarantees that p̂1 (yl ) ≥ p̂21 (λ, ph , pl ) = pl.
LEMMA 2. Let utility function (1) satisfy Assumption 1, and let θy satisfy
Assumption 2, then the profit maximizing conditions of high price and low
price strategies are determined by
µ
¶
a
yh ,
ph =
1+a
µ
¶
β(1 − λ)pl =ph .
1 − βλ
18
19

(24)
(25)

Each low-income consumer purchases at most one unit of x in each period.
Although it is possible to consider several strategies in general cases, some of them

are not irrelevant to the main subject and the others only complicate the discussion. See
Appendix C.

12

Proof. See Appendix C.
This lemma ensures that there are at most two prices in equilibrium.
If a two price equilibrium (TPE) exists, then the profit from each strategy
must be equal to each other: i.e. ph qh = pl ql . Therefore, profit maximizing
and equal profit conditions must be satisfied in TPE: from (22), (23) and(24), (25),plβ(1 − λ)=, profit maximizing condition,ph1 − βλpl(2 − λ)(1 − π)Pl ≡=, equal profit condition.
ph(2 + (1 − λ)(1 − π))

Pl ≡

(26)
(27)

This establishes the following proposition.
Pl
1βprofit maximizing2(1−π)
3−πequal profit
1−π
2

0

λ∗

1

λ

Figure 4: The existence and uniqueness of λ∗ and Pl .

PROPOSITION 1. Let utility function (1) satisfy Assumption 1, let θysatisfy Assumption 2, and if β ≥ 2/3, then there is unique distribution

λ∗ ∈ (0, 1) which satisfies profit maximizing and equal profit conditions:(2 − λ∗ )(1 − π)β(1 − λ∗ ).
=(2 + (1 − λ∗ )(1 − π))
1 − βλ∗Proof. See Appendix D.
13

(28)

3

Cyclical price movements

We are now ready to consider the firm’s reaction to changes in economic
situations. In order to focus on two-price equilibrium, we make the following
assumption regarding β.
ASSUMPTION 3.
2β≥ .
3This assumption seems reasonable. Further, together with Assumption
2, this implies that the proportion of high-income consumers is a large
enough and the diﬀerence in income is not large in the economy.
Figure 4 indicates that the proportion of low-price stores decreases withdecreasing the fraction of low-income consumers; as a result, low prices
increase. The following proposition formalizes this argument.
PROPOSITION 2. Let utility function (1) satisfy Assumption 1 and let
θy and β satisfy Assumptions 2 and 3, respectively, then
dpl
< 0.
dπ

(29)

Proof. See Appendix E.
This is our main proposition.20 The movements of prices are countercyclicalto business cycles. For simplicity, suppose that only a fraction of low-income
consumers lost their jobs in recessions.21 As a result, they reduce their reservation prices below the prevailing prices and leave the market; the fraction of
low-income π decreases. Since the low price strategy becomes less profitable,
the proportion of low-price stores λ∗ decreases with π. On the other hand,
high-income young consumers in a low-price store raise their reservation
20
21

Although we assume that β is constant, it can change with business cycles.
See Appendix F.

14

Pl
1

Pl∗

0

λ∗

1

λ

Figure 5: Eﬀect of the decrease in fraction of low-income consumers
price for storage with decreases in λ∗ because the probability of choosing a
low-price store diminishes in the next period. Therefore, firms can raise thelevel of low prices during recessions,22 and vice versa.(See Figure 5)

4

TPE as a mixed strategy Nash equilibrium

The equilibrium discussed above (TPE) is derived from profit maximizing
and equal profit conditions. The equal profit condition, however, is appliedexternally to the model. It does not indicate how firms determine the strategy in each period. We provide here that TPE can be derived merely from afirm’s profit maximizing behavior. We will find that the equal profit condition is obtained endogenously as a result of this behavior. The equilibrium
is characterized as a symmetric mixed strategy Nash equilibrium.23 In theequilibrium, we find that firms price ph with probability (1 − λ∗ ) and pl with
λ∗ . We shall discuss this in detail.

Consider a strategic game which consists of N identical firms, and two
22
23

The average price p̄ increases as λ∗ decreases, while ph remain constant.
See Osborne and Rubinstein (1994), Ch.3.

15

strategies which consists of high-price strategy and low-price strategy. Payoﬀ
functions are defined by their revenue: from (22), (23), (24), and (25),
µ
¶ ½
¾
a
((1 − π) + (1 − π)(1 − λ))L
yh
,
(30)
ph qh =
1+a
N
µ
¶ ½
¾
β(1 − λ)(2(1 − π) + π + (1 − λ) + πλ)Lpl q l =ph.
(31)
1 − βλN
We now focus on a symmetric Nash equilibrium. Let firm i denote i-th
individual firm and let firms −i denote the other firms which take the same
strategy. Since N is a large number, we can treat it as λ = 0 when firms −itake a high-price strategy and as λ = 1 when they take a low-price strategy.
Therefore, from (30) and (31), the payoﬀ matrix for this game is given by
Figure 6. If firms −i take a high-price strategy, then the low-price strategy
is more profitable for firm i, and vice versa. This implies that there is no
single price equilibrium. The following lemma formalizes this argument.

High
³

High

ph

Low

βph

Firm iFirms −iLow2(1−π)LN´³´(3−π)LN

ph³(1−π)LN´

0Figure 6: Payoﬀ matrix: Firm i’s payoﬀ is written in each cell.

LEMMA 3. Let utility function (1) satisfy Assumption 1 and let θy and βsatisfy Assumptions 2 and 3, respectively; then there is no symmetric purestrategy Nash equilibrium.
Proof. See Appendix G.

16

Next, consider each firm adopts a mixed strategy composed of both
high-price and low-price strategies. If firms −i take a low-price strategy
with probability λ (hence a high-price strategy with probability 1 − λ), thenit coincides with the price distribution λ in the market according to thelaw of large numbers. Corresponding firm i’s payoﬀ function of each purestrategy is shown by (30), (31). Hence, firm i’s expected payoﬀ function of
mixed strategy q is¶¾
½ µ(2 − λ)(1 − π)L(1 − q) ph
N
½µ
¶ µ
¶¾
β(1 − λ)(2 + (1 − λ)(1 − π))L+qph.
1 − βλN(32)

From this payoﬀ function (32) and the equal profit condition (28) in Proposition 1, we can find that symmetric mixed strategy Nash equilibrium coincides with TPE. The following proposition formalizes this argument.
q
1

0

λ∗

1

λ

Figure 7: Symmetric Mixed Strategy Nash Equilibrium
PROPOSITION 3. Let utility function (1) satisfy Assumption 1 and let
θy and β satisfy Assumptions 2 and 3, respectively; there is then a uniquesymmetric mixed strategy Nash equilibrium λ∗ ∈ (0, 1). This equilibriumcoincides with TPE.
17

Proof. See Appendix H.

5Concluding remarks

We demonstrated how market prices are determined and react to businesscycles in an atomistic market. Prices move countercyclically to businesscycles because a low-price strategy becomes less (more) profitable duringa recession (boom).24 We must, however, notice that our results depend
strongly on the type of commodity addressed. For example, low-income
consumers may not leave grocery markets during a recession, though they
will leave the durable goods markets. Our model seems more suitable forthe latter case because we consider the market from which a fraction of
low-income consumers exit during a recession.
Although we focused on the equilibrium of steady states, cyclical pricemovements should be analyzed from a dynamic point of view since they
have intrinsically dynamic structures. However, as Bils (1989) shows, suchan analysis will be extremely complicated.
Finally, we supposed here that firms are engaged in random pricing; thatis, the store which charges low prices does not necessarily charge the same
price in the next period. In some actual markets, however, we can observethat each firm continues to charge the same price even though prices are
distributed in the market. In this case, consumers’ information regarding
low price stores increases, and, as a result, uncertainty may diminish. It will
be necessary to extend our model along this line.

24

By the structure of overlapping generations, there are always potential consumers

having a new demand for storage in the market. This is important to show the benefits ofa low-price strategy, and hence our main propositions. It is not assumed clearly in Stiglitz(1979)(1989).

18

Appendix A. proof of Lemma 1
LEMMA 1. Let the utility function (1) satisfy Assumption 1. If a consumer chooses x21 = 1, then he has already chosen x1 = 1.
Proof. Suppose x1 = 0 when x21 = 1 , then , from x21 x22 = 0, the utility
is u(0, 1, 0). From (1), the conditions u(1, 0, 0) ≥ u(0, 0, 0) and u(0, 1, 0) ≥u(0, 0, 0) are the following, respectively:(1 + a)(y − p1 ) + βy ≥ y + βy,¶µay ≥ p1 ,
1+a

(33)

and
y − p1 + β(1 + a)y ≥ y + βy,
βay ≥ p1 .
From Assumption 1, (33) and (34),¶µay > βay ≥ p1 .
1+a

(34)

(35)

This inequality means that if p1 ≤ βay he can choose both u(1, 0, 0) andu(0, 1, 0) in the store. Here, the condition u(1, 0, 0) > u(0, 1, 0) is(1 + a)(y − p1 ) + βy > y − p1 + β(1 + a)y,(1 − β)y > p1 .

(36)

From Assumption 1, this condition is satisfied for all p1 ≤ βay; that is, hedoes not choose the case u(0, 1, 0). It is a contradiction. The desired result
is satisfied.

19

Appendix B. conditions of entering the markets
The condition of entering the second period market
case#1 If p̂22 (y) ≥ ph > pl , then, from (11), the expected utility function of
the second period isE[(1 + ax22 )(y − p2 x22 )] = (1 + a)(y − ((1 − λ)ph + λpl ))(37)= (1 + a)(y − p̄),
where p̄ = (1 − λ)ph + λpl . A consumer decides to enter the second
period market if(1 + ax1 )(y − p1 x1 ) + β[(1 + a)(y − p̄)](38)

≥ (1 + ax1 )(y − p1 x1 ) + βy.
Hence,(1 + a)(y − p̄) ≥ y,¶µay ≥ p̄,1+a(39)

p̂22 (y) ≥ p̄.
Since case#1 always satisfies this inequality, the consumer enters themarket.
case#2 If ph > p̂22 (y) ≥ pl , then, from (13), the expected utility function of
the second period isE[(1 + ax22 )(y − p2 x22 )] = (1 + λa)y − λ(1 + a)pl .

(40)

A consumer decides to enter the market if(1 + ax1 )(y − p1 x1 ) + β[(1 + λa)y − λ(1 + a)pl ]≥ (1 + ax1 )(y − p1 x1 ) + βy
20

(41)

Hence,(1 + λa)y − λ(1 + a)pl ≥ y,¶µay ≥ pl ,1+a(42)

p̂22 (y) ≥ pl .
Since case#2 satisfies this inequality, the consumer enters the market.
The condition of entering the first period market
case#1-1 If p̂1 = p̂22 (y) ≥ ph > p̂21 (λ, ph , pl ) ≥ pl , then the condition of entering
the market is(1 − λ) {(1 + a)(y − ph ) + β(1 + a)(y − p̄)}+ λ {(1 + a)(y − 2pl )) + β(1 + a)y}(43)

≥ y + β(1 + a)(y − p̄).
Hence,(p̂22 (y) − (1 − λ)(1 − βλ)ph ) ≥ (λ(2 − λβ))pl .

(44)

Since p̂1 (y) = p̂22 (y) ≥ ph , this condition is satisfied if(1 + β(1 − λ))ph ≥ (2 − βλ)pl ,ph ≥ 2pl − β p̄,ph ≥ 2pl − p̂21 (λ, ph , pl ).

(45)

and from p̂21 (λ, ph , pl ) ≥ pl , the suﬃcient condition of (45) isph ≥ p̂21 (λ, ph , pl ).

(46)

Since case#1-1 satisfies this inequality, the consumer enters the market.
21

case#1-2 If p̂1 (y) = p̂22 (y) ≥ ph > pl > p̂21 (λ, ph , pl ), then the condition ofentering the market is(1 − λ) {(1 + a)(y − ph ) + β(1 + a)(y − p̄)}+ λ {(1 + a)(y − pl )) + β(1 + a)(y − p̄)}(47)

≥ y + β(1 + a)(y − p̄).
Hence,(1 + a)(y − p̄) ≥ y,µ¶ay ≥ p̄,1+a(48)

p̂1 (y) = p̂22 (y) ≥ p̄.
Since case#1-2 satisfies this inequality, the consumer enters the market.
case#2-1 If ph > p̂1 (y) = p̂22 (y) > p̂21 (λ, pl , y) ≥ pl , then the condition ofentering the market is(1 − λ) {y + β[(1 + λa)y − λ(1 + a)pl ]}+ λ {(1 + a)(y − 2pl ) + β(1 + a)y}(49)

≥ y + β[(1 + λa)y − λ(1 + a)pl ].
Hence,(1 + β(1 − λ))p̂22 (y) ≥ (2 − βλ)pl ,p̂22 (y) ≥ 2pl − p̂21 (λ, pl , y).

(50)

Since p̂21 (λ, pl , y) ≥ pl , the suﬃcient condition of (50) isp̂22 (y) ≥ p̂21 (λ, pl , y).

(51)

Since case#2-1 satisfies this inequality, the consumer enters the market.
22

case#2-2 If ph > p̂1 (y) = p̂22 (y) ≥ pl > p̂21 (λ, pl , y), then the condition ofentering the market is[(1 + λa)y − λ(1 + a)pl ]+ β[(1 + λa)y − λ(1 + a)pl ](52)≥ y + β[(1 + λa)y − λ(1 + a)pl ].
Hence,[(1 + λa)y − λ(1 + a)pl ] ≥ y,¶µay ≥ pl ,1+ap̂22 (y) ≥ pl .

(53)

Since case#2-2 satisfies this inequality, the consumer enters the market.

Appendix C. proof of Lemma 2
LEMMA 2. Let utility function (1) satisfy Assumption 1, and let θy satisfy
Assumption 2, then the profit maximizing conditions of high price and low
price strategies are determined by
µ
¶
a
yh ,
ph =
1+a
µ
¶
β(1 − λ)pl =ph .
1 − βλ(54)(55)

Proof. There are three steps. First, consider the profit of a high-price strategy. We have already known the profit of setting price ph for high-income:ph = p̂1 (yh ) = p̂22 (yh ); that is,µ¶
aph qh =yh (1 − π)(2 − λ)(L/N ).
1+a

(56)

If the firm set a price ph for low-income consumers: i.e., ph = p̂1 (yl ) =p̂22 (yl ), then the x21 -reservation price of high-income consumers coincides
23with that of low-income consumers; i.e., p̂21 (λ, ph , pl ) = p̂21 (λ, pl , yl ). Hencep̂1 (yl ) > p̂21 (λ, ph , pl ). Therefore, each firm attracts L/N young consumers
and [(1 − λ)L]/N old consumers who selected a high price store when they
were young. The profit of this case isµ¶ayl (2 − λ)(L/N ).
ph qh =
1+a

(57)

Comparing (56) with (57), we find that setting (54) as a high-price strategy
is more profitable if(1 − π)yh > yl .

(58)This condition is satisfied by Assumption 2. Notice that, under this assumption, we can exclude the case ph = p̂1 (yl ) = p̂22 (yl ).
Next, consider the profit of a low-price strategy under ph = p̂1 (yh ).
Suppose that p̂1 (yl ) ≥ p̂21 (λ, ph , pl ). There are two reservation prices for a
low-price strategy in accordance with income levels (See Fig. 3):
p̂21 (λ, ph , pl ) = β((1 − λ)ph + λpl ),(59)

p̂21 (λ, pl , yl ) = β((1 − λ)p̂22 (yl ) + λpl ).

(60)

Note that p̂21 (λ, ph , pl ) > p̂21 (λ, pl , yl ). If the firm sets a low price for highincome consumers: i.e., pl = p̂21 (λ, ph , pl ), hence,µ¶β(1 − λ)ph ,pl =1 − βλ(61)

then, each firm attracts L/N young consumers, [(1 − λ)L]/N old consumers
who selected a high-price store when they were young and (πλL)/N lowincome old consumers who selected a low-price store when they were young.
Since each young high-income consumer purchases two units of x, the profitof this case can be written as (C in Fig. 3)µ¶µ¶β(1 − λ)ayh (2 + (1 − λ)(1 − π))(L/N ).
pl q l =1 − βλ
1+a
24

(62)

If the firm sets a low price for low-income consumers: i.e., pl = p̂21 (λ, pl , yl ),then each firm attracts L/N young consumers, [(1 − λ)L]/N old consumers
who selected a high-price store when they were young and and (πλL)/N
low-income old consumers who selected a low-price store when they wereyoung. Since each young consumer purchases two units of x, the profit ofthis case can be written as (D in Fig. 3)¶µ¶µaβ(1 − λ)yl (3 − λ(1 − π))(L/N ).
pl q l =1 − βλ
1+a(63)Comparing (62) with (63), we find that setting (61) as a low-price strategy
is more profitable if
yh (2 + (1 − λ)(1 − π)) > yl (3 − λ(1 − π)).

(64)

It follows that1−π
> (1 − π) > θy ,3 − λ(1 − π)∀ λ ∈ (0, 1).

This condition is satisfied by Assumption 2.
Last, from (61), the suﬃcient condition of p̂1 (yl ) ≥ p̂21 (λ, ph , pl ) for any
λ ∈ (0, 1) under ph = p̂1 (yh ) is¶µ¶µβ(1 − λ)
a
a
yl ≥
yh ,
1+a
1 − βλ1+aβ(1 − λ).
θy ≥1 − βλ(65)

This inequality is satisfied by Assumption 2 for all λ ∈ [0, 1].

Appendix D. proof of Proposition 1
PROPOSITION 1. Let utility function (1) satisfy Assumption 1, let θysatisfy Assumption 2, and if β ≥ 2/3, then there is unique distribution

λ∗ ∈ (0, 1) which satisfies profit maximizing and equal profit conditions:(2 − λ∗ )(1 − π)β(1 − λ∗ ).
=(2 + (1 − λ∗ )(1 − π))
1 − βλ∗25(66)

Proof. Consider LHS and RHS in (66) as functions of λ, that is,(2 − λ)(1 − π),
(2 + (1 − λ)(1 − π))β(1 − λ).
RHS(λ) =1 − βλLHS(λ) =

(67)
(68)

These functions are continuous and strictly decreasing with respect to λ:i.e.,∀ λ ∈ [0, 1],∂LHS(λ)< 0,∂λ∂RHS(λ)
< 0.
∂λ

(69)

Next, if λ = 1, then
LHS =

1−π
,
2

RHS = 0.

(70)This means that LHS > RHS when λ = 1.
If λ = 0, thenLHS =2(1 − π),3−πRHS = β.

(71)

From (71), if2(1 − π)
< β,
3−π

(72)

then, from (69), there is unique λ∗ ∈ (0, 1). This condition is satisfied forall π ∈ (0, 1) if β ≥ 2/3. (See Fig. 4)

Appendix E. proof of Proposition 2
PROPOSITION 2. Let utility function (1) satisfy Assumption 1 and let
θy and β satisfy Assumptions 2 and 3, respectively, then
dpl
< 0.
dπ

26

(73)

Proof. Since Proposition 1 is satisfied under Assumptions 1, 2, and 3, let
λ∗ ∈ (0, 1) denote the equilibrium in which profit maximizing and equal

profit conditions (66) are satisfied; i.e., LHS(λ∗ ) = RHS(λ∗ ). Taking theimplicit derivative of (66) with respect to π at λ∗ :
dλ∗
=³
dπ
=³

2(2−λ∗ )(2+(1−λ∗ )(1−π))2

´

β
1−π
∗
∗
1−βλ∗ (1 − RHS(λ )) − 2+(1−λ∗ )(1−π) (1 − LHS(λ ))2(2−λ∗ )(2+(1−λ∗ )(1−π))2β1−π1−βλ∗ − 2+(1−λ∗ )(1−π)

´

(74)

.

(1 − LHS(λ∗ ))The suﬃcient condition that the denominator of (74) remains positive is
β > 1/2. From Assumption 3, this condition is satisfied. Hence, from (74),
dλ∗
> 0,
dπ

(75)

dpl
< 0.
dλ

(76)

dpl dλ∗
dpl
= ∗·
< 0.
dπ
dλ
dπ

(77)

and from (61),Therefore, from (75) and (76),(See Fig. 5)

Appendix F. changes in π
Consider a phase of a recession. Let 1 − β and 1 − α be the proportionof high-income and low-income consumers, respectively, who remain in themarket and let β and α be the fraction of those who leave the market becauseof unemployment. The fraction π decreases if(1 − α)πL< π.
(1 − β)(1 − π)L + (1 − α)πL
27

(78)

Hence,α > β.

(79)

Therefore, we find that π decreases as long as α is larger than β.

Appendix G. proof of Lemma 3
LEMMA 3. Let utility function (1) satisfy Assumption 1 and let θy and βsatisfy Assumptions 2 and 3, respectively; then there is no symmetric purestrategy Nash equilibrium.
Proof. The payoﬀ matrix for this game is given by Figure 6. First, consider
that firms −i take a low-price strategy; i.e., λ = 1, thenµ¶(1 − π)L
> 0.
ph
N

(80)

Thus, firm i takes a high-price strategy. Under an assumption of symmetricity, this implies λ = 0. It is a contradiction. Next, consider firms −i take ahigh-price strategy; i.e., λ = 0. If the following inequality is satisfied,µµ¶¶(3 − π)L2(1 − π)L
> ph
.
(81)
βph
N
N
Hence,β>2(1 − π)
.
3−π

(82)

From π ∈ (0, 1), RHS in (82) takes the value of interval (0, 2/3). This condition is satisfied by Assumption 3. Thus, firm i takes a low-price strategy.
This implies λ = 1. It is a contradiction. Therefore, the desired result issatisfied.

28

Appendix H. proof of Proposition 3
PROPOSITION 3. Let utility function (1) satisfy Assumption 1 and let
θy and β satisfy Assumptions 2 and 3, respectively; there is then a uniquesymmetric mixed strategy Nash equilibrium λ∗ ∈ (0, 1). This equilibriumcoincides with TPE.
Proof. Let 1 − q and q denote firm i’s mixed strategy of high price and low
price, respectively. Similary, let 1 − λ and λ denote the firms −i’s mixed
strategy. Since there are a large number of firms, firms −i’s mixed strategy
can be regarded as an actually observed price distribution λ. Thus the profitof each pure strategy for firm i can be written as
µ
¶ ½
¾
a
((1 − π) + (1 − π)(1 − λ))Lyh,
ph qh =
1+a
N
µ
¶ ½
¾
β(1 − λ)(2(1 − π) + π + (1 − λ) + πλ)Lpl q l =ph.
1 − βλN
Hence, the expected payoﬀ function is¶¾½ µ(2 − λ)(1 − π)L(1 − q) ph
N
½µ
¶ µ
¶¾
β(1 − λ)(2 + (1 − λ)(1 − π))L+qph.
1 − βλ
N

(83)
(84)

(85)

From (85), if
µ
¶ µ
¶ µ
¶
(2 − λ)(1 − π)L(2 + (1 − λ)(1 − π))Lβ(1 − λ)
>
ph
.
ph
N
1 − βλNHence(2 − λ)(1 − π)β(1 − λ)>.
(2 + (1 − λ)(1 − π))
1 − βλ(86)

Firm i then chooses q = 0 to maximize its profit since the expected payoﬀ
function (85) is linear with respect to q. To the contrary, if LHS < RHS in(86), then firm i chooses q = 1. If LHS = RHS, then the payoﬀ is indiﬀerent
29for q ∈ [0, 1]. Similar to the arguments in the proof of Proposition 1, there

exists unique λ∗ ∈ (0, 1) such that(2 − λ∗ )(1 − π)β(1 − λ∗ ).
=(2 + (1 − λ∗ )(1 − π))
1 − βλ∗(87)

Then, LHS > RHS for all λ ∈ [0, λ∗ ) and LHS < RHS for all λ <∈ (λ∗ , 1].
Therefore, the best response function of firm i is given by Figure 7. Under
the symmetricity assumption, q = λ∗ is the symmetric mixed strategy Nashequilibrium. From (87) and (28) in Proposition 1, the value of λ∗ coincides
with TPE.

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