Impulse Buying and Equilibrium Price
Dispersion¤
Simon P. Andersonyand André de Palmaz
June 2002
ABSTRACT
Many purchases are unplanned and so do not …t comfortably within standard
neoclassical consumer theory. We model such ”impulse” purchases by a simple
reservation price rule used by impatient consumers. The equilibrium exhibits
price dispersion in pure strategies, with lower price …rms earning higher pro…ts.
The range of price dispersion increases with the number of …rms: the highest price
is the monopoly one, while the lowest price tends to marginal cost. The average
transaction price remains substantially above marginal cost even in the limit.
Reducing the fraction of impulse buyers may increase prices, so bargain-hunters
may hurt other consumers.
KEY WORDS: Impulse goods, price dispersion, monopolistic competition.
¤ We
should like to thank Roman Kotiers, Richard Ruble, and Yutaka Yoshino for research
assistance and Regis Renault for suggesting alternative interpretations. Comments from conference participants at EARIE in Turin and at SETIT in Georgetown were helpful. The …rst
author gratefully acknowledges funding assistance from the NSF under Grant SBR-9617784
and from the Bankard Fund at the University of Virginia.
y
114 Rouss Hall, P.O. Box 4004182, Department of Economics, University of Virginia, Charlottesville, VA 22904-4182, USA.
z
Senior Member, Institut Universitaire de France and THEMA, 33 Bd. du Port, 95100
Cergy-Pontoise, FRANCE, CESifo member.
1
1
Introduction
Many goods bought by shoppers are unplanned purchases in the sense that the
consumers did not intend to buy the object when s/he left home. The evidence on the extent of the phenomenon varies according to the de…nition used
of ”unplanned purchase” and the fraction of unplanned purchases varies from
25% to 70% of total purchases, according to the commodity category.1 The term
”impulse” goods runs the gamut in the marketing, psychology and business literatures from unplanned purchases to compulsive shopping behavior by those with
psychological problems.
We de…ne impulse goods as those that were not intentionally sought out, nor
is the consumer reminded that s/he needs the good. Such goods are not on
shopping lists, nor are they forgotten from shopping lists. They are not search
goods.2 . Impusle goods do they …t comfortably into standard neo-classical
consumer theory. Yet, as is argued in the marketing literature, they may form a
very important part of shopping purchases in some categories.
We model unplanned purchases by recourse to a simple reservation price
rule. The consumer is reactive rather than proactive. S/he does not search,
but buys (or not) when s/he encounters an opportunity. Even if the purchase
is not planned, the consumer will not buy if it is too expensive. If the good is
passed over for being too expensive the …rst time it is encountered, the consumer
may nonetheless buy it when he comes across it later at a lower price, if that
price is below his reservation price for unplanned good. This simple model (for
which linear demand is a special case) leads to price equilibria that are always
dispersed when the reservation prices are continuously distributed.
Price dispersion has lately received a lot of attention. Most recent studies have
looked at web data since this is readily accessible (books, dvds, etc.). Another
interesting study is Lach (2002). He looks at 4 categories of goods in stores
in Israel, and …nds evidence of random behavior of prices after he controls for
location and product di¤erentiation di¤erences. This he interprets as supporting
Varian’s (1980) model of sales. However, 3 of the goods that Lach considers are
everyday shopping list goods (frozen chicken, ‡our, and instant co¤ee) while the
other (refrigerators) are search goods. Likewise, the studies that use web data also
suggest large price dispersion. Several attempts have been made to explain price
dispersion. Most such studies involve search goods (with homogeneous products)
where some consumers have zero search costs, and models generate mixed strategy
1 To
quote from one web-site on Point-of-Purchase sales: ”the research indicates that approximately 70 percent of brand purchase decisions were made at the point of purchase for
supermarket shoppers (60 percent unplanned; 4 percent substitute purchases; 6 percent generally
planned) .... a large 74 percent of mass merchandiser shoppers made in-store purchase decisions
(53 percent unplanned; 18 percent generally planned; 3 percent substitute).”
2
That is, the consumer does not actively search out the good but instead comes across it on
shopping trips with other purposes.
2
equilibria. As in Varian (1980), these equilibria are then interpreted as dispersed
prices.
In this paper we derive a demand system for impulse goods. The demand
system, its properties and the monopoly solution are presented in Section 2. The
oligopoly case is described in Section 3, while its implications are described in
Section 4. We show that the equilibria are necessarily dispersed and we characterize the price and pro…t ranking. The analysis suggests that even in the limiting
case where the number of …rms gets large, perfect competition is not achieved. In
Section 5, we consider the linear demand function case and discusses the explicit
solutions. Section 6 treats three extensions. Analysis of the multi-outlet monopolist sheds light on the property that oligopoly pro…ts can rise with the number of
…rms. Second, in a situation where low-price seekers coexist with impulse buyers,
we show the latter impose a negative externality on the other consumers. Third,
our basic model assumes that when the consumer encounters an impulse good,
s/he makes a snap decision whether to buy the good or not, without considering
the possibility of waiting to get the good elsewhere at a potentially lower price. In
that sense, impulse buyers are impatient buyers, who do not wish to wait and enter into a potentially complex decision process. Our last extension treats the issue
of the degree of patience, and shows that the fundamental dispersion properties of
the model still hold when consumers are su¢ciently impatient. However, patient
consumers induce a single-price equilibrium. Concluding remarks are presented
in Section 7.
2
Demand for impulse purchases
There are n …rms and production costs are zero. Each …rm sets the price for the
good it sells to maximize expected pro…t. There is a population of consumers with
mass normalized to unity. Consumers encounter the goods sequentially and in
random order. A consumer buys one unit as soon as she is faced with price below
her reservation price, and then exits the market. Each consumer has a speci…c
reservation price ± 2 [0; 1] and will not buy at all if the lowest price in the market
is above her reservation price. The distribution of reservation prices is given by
F (±), with a continuously di¤erentiable density f(±) for all prices ± 2 [0; 1]. Note
that any market in which marginal costs are constant and reservation prices are
bounded can always be reduced to this form by appropriate choice of units and
where the relevant support of consumer reservation prices is from marginal cost
to the highest reservation price (consumers whose reservation prices are below
marginal cost are simply dropped from the relevant population, since prices are
never below marginal costs in equilibrium).
2.1
The demand system
We now derive the demand system when consumers have disparate reservation
prices. Note that the random matching protocol implies that each consumer buys
3
with equal probability any good whose price is below her personal reservation
price.
We label the impulse goods such that 0 · p1 · p2 · ::: · pn · 1. Only
consumers with reservation prices exceeding pn will ever buy good n, and will
only do so when it is the …rst good encountered. Since the probability of having a
reservation price below pn is F (pn ), the mass of consumers who might potentially
buy good n is 1¡ F (pn ). Because this good carries the highest price, these
consumers are split equally among all goods. Hence, the demand for the most
expensive impulse good, n, is:
1
[1 ¡ F (pn )] .
(1)
n
We can de…ne demand recursively. The demand of the second lowest price
good is composed of two pieces. First, those consumers with a reservation price
above pn have a probability of 1 /n of purchasing this good. Second, the consumers who have a reservation price between pn¡1 and pn are equally likely to
purchase any of the n ¡ 1 goods below their reservation price. We can determine the demand for the good with the ith highest price in an analogous manner.
This demand comprises the demand addressed to the good with the next highest price (this follows from the way consumers are shared equally among goods
whose prices are below reservation levels) plus good i’s share of the consumers
whose reservation prices lie between pi and pi+1, which share is 1 /i . Using this
recursion, we can write the demand system as :
Dn =
Dn
Dn¡1
..
.
Di
..
.
D1
=
=
1
n
[1 ¡ F (pn )]
[F (pn ) ¡ F (pn¡1 )] + Dn
1
n¡1
= 1i [F (pi+1 ) ¡ F (pi )] + Di+1
(2)
= F (p2) ¡ F (p1) + D2 :
In this demand system, a good with a higher price attracts fewer consumers, as
expected (so Di = Dj whenever pj = pi, and Di < Dj whenever pj < pi . Firms
with prices above the lowest one are not obliterated when pn < 1. The overall
structure can be seen quite clearly by writing out in full the demand for the
lowest-priced good, which gives
D1 = F (p2) ¡ F (p1) +
1
1
[F (p3) ¡ F (p2)] + ¢ ¢ ¢ + [1 ¡ F (pn )] :
2
n
The demand system is illustrated in Figure 1. Figure 1a illustrates which
goods are purchased by which consumers (as a function of their reservation prices)
and the numbers in an area indicate the goods bought (with equal probability).
4
Figure 1b shows the same information with reference to the monopoly demand
curve. The fractions along the quantity axis denote the number of …rms sharing
a consumer segment.
Figure 1. The impulse good demand system
The demand system above has some interesting properties. Although the
demand for the good with the lowest price depends on all other prices, the demand
for the good with the next lowest price is independent of the level of the lowest
price. Similarly, demand for the good with the ith lowest price is independent of
the prices of all goods with lower prices than it since good i gets no demand from
consumers whose reservation prices are less than pi . Indeed, the prices of goods
1 through i ¡ 1 only determine how the demand from the F (pi ) consumers whose
reservation prices are less than pi is split up.
In summary, the demand for good i is independent of all lower prices and is a
continuous function of all higher prices. This is very di¤erent from the standard
(homogenous goods) framework in which the good with the lowest price is the
only one consumers buy. In our framework, when a …rm reduces its price (locally)
it picks up demand continuously from consumers who previously viewed it as too
expensive.
2.2
Monopoly preliminaries
Although we are primarily interested in price dispersion in a competitive setting,
the properties of the monopoly solution are key to describing the situation with
several …rms. For this reason, we take some time in elaborating the monopoly
solution.
We shall use the following technical assumptions, later referred to as A1:
Assumption 1A: F (±) is twice continuously di¤erentiable with F (0) = 0,
F (1) = 1, and f(±) > 0 for ± 2 (0; 1).
Assumption 1B: 1 /[1 ¡ F (±)] is strictly convex for ± 2 (0; 1).3
A stronger property is the log-concavity of f(±) which is veri…ed by most of
the densities commonly used in economics, such as the uniform, the truncated
normal, beta, exponential, and any concave function. However, any density that
is not quasi-concave violates Assumption 1B.
Assumption 1A introduces su¢cient continuity for simplicity and also embodies the normalization of the demand curve to have unit price and quantity
intercepts. Note that any kink down in the demand curve can be approximated
3 Equivalently,
1 ¡ F (±) is strictly (¡1)-concave. This condition ensures that the demand
function is well behaved (in a sense made precise below). It is implied by log-concavity of
1 ¡ F (±), which is in turn implied by log-concavity of f (±) (see Caplin and Nalebu¤, 1991).
5
arbitrarily closely with a twice continuously di¤erentiable function without violating Assumption 1B. Assumption 1B implies that the monopoly problem is
well behaved in a sense made precise below.
We shall use the subscript m to denote monopoly values. The pro…t function
facing the single …rm selling a single product is
¼ m(p) = pDm
where Dm = [1 ¡ F (p)]. Since ¼m (0) = ¼m (1) = 0, the monopoly price pm is
interior and given by the implicit solution to the …rst-order condition:
p=
1 ¡ F (p)
:
f(p)
(3)
The right-hand side of this expression is positive, decreasing and continuous in
p by A1. Therefore, there exists a unique solution pm 2 (0; 1) to (3), which
maximizes pro…t. We prove the remainder of the following result in the Appendix:
Lemma 1 (Monopoly) Under A1, there is a unique solution pm 2 (0; 1) to
the monopolist’s …rst-order condition, which is the unique maximizer of ¼ m(p).
Moreover, the monopoly pro…t function satis…es ¼00m (p) < 0 for all p for which
¼0m (p) ¸ 0. Equivalently, the condition 2f(p) + pf 0(p) > 0 holds for p · pm .
At an intuitive level, Assumption 1B implies that demand is not ”too” convex.
This ensures that the corresponding marginal revenue curve (with respect to
price) is decreasing whenever marginal revenue is non-negative. Equivalently,
this lemma establishes that the monopoly pro…t function is strictly concave up
through its maximum and thereafter it is decreasing.
3
Competitive price dispersion
One might expect that …rms selling impulse goods would behave like monopolists,
since consumers do not compare prices nor search. As we show below, there is
some truth in this statement, since one …rm charges the monopoly price regardless
of the number of competitors. However, the key property of an impulse good
market is that the prices are necessarily dispersed in equilibrium. All other …rms
will charge lower prices. To see this suppose that instead all …rms charged the
monopoly price. Then if one …rm cuts its price slightly, its demand rises by the
number of consumers whose reservation prices lie between the monopoly price
and its new price. However, if all …rms had reduced their prices in concert, then
each …rm would have received only 1 /n th of such consumers, and this is the
calculus of the monopoly problem.4
4
Since the monopoly price is such that the loss in revenue from price reduction is just
compensated by the increased revenue from extra customers (for a very small price change),
then a single …rm must gain when it cuts its price from the monopoly level since the lost revenue
on existing customers is much smaller because it has 1 /n th of the customer base of the group.
6
The pro…t of Firm i is ¼i = pi Di, where Di is given by equation (2). The
following lemma, proved in the Appendix, enables us to proceed henceforth solely
from interior solutions to the …rst-order conditions.
Lemma 2 (No bunching) Each …rm chooses a distinct price at any pure strategy equilibrium, i.e., p1 < p2 < ::: < pn.
The intuition is as follows. If two …rms were bunched at the same price,
then demand facing either of them is kinked at that price and demand is more
elastic for lower prices since the …rms splits the marginal consumer with fewer
rivals. Hence marginal revenue is greater to the right than to the left so that the
marginal revenue curve jumps up at the corresponding output. Therefore there
cannot be a pro…t maximum at such a point.
This means that any pure strategy price equilibrium involves price dispersion
(i.e. interior solutions to …rst-order conditions) for all …rms. The highest priced
…rm, n, charges the monopoly price pm (see (3)). This price is independent of
the number of …rms, because Firm n’s demand is independent of all other (lower)
prices.5 We can then solve for the candidate equilibrium prices from the top down.
The …rst-order condition is: Di + pi@Di /@pi = 0. Since @Di /@pi = ¡f (pi) /i , we
have the following simple relation between prices and demands at the candidate
equilibrium:
f(pi)
;
i = 1; :::; n:
i
The corresponding second-order conditions for local maxima are:
Di = pi
(4)
@ 2 ¼i
2f (pi) + pi f 0 (pi)
=
¡
< 0;
@p2i
i
where the inequality follows from Lemma 1 since pi < pm . Thus, the properties
of F (:) ensure that the …rst-order conditions do characterize local maxima.
The price expressions (4) can be used to construct a recursive relation for the
equilibrium prices. Rewriting equation (2) gives:
F (pi+1) ¡ F (pi)
;
i=1,...,n-1.
(5)
i
Since Di+1 = pi+1f(pi+1) /(i + 1) by (4) applied to Firm i + 1, we can substitute
for the demands in (5) to give:
Di = Di+1 +
(i + 1) [F (pi) + pi f(pi)] = (i + 1)F (pi+1) + ipi+1f(pi+1); i=1,...,n-1.
(6)
The next proposition crystallizes the no-bunching property of equilibrium with
impulse goods:
5
Although the corresponding pro…t level is 1 /n of the monopoly pro…t.
7
Proposition 1 (Price dispersion) Under A1, there is a unique candidate pure
strategy price equilibrium. It solves (3) and (6) and entails p1 < p2 < ::: < pn =
pm. Furthermore, the solution is a local equilibrium.
Proof. Given that 2f (p) + pf 0 (p) > 0 (by Lemma 1), expression [F (p) + pf (p)]
in (6) is increasing in p. Therefore, if pi > pi+1 we have a contradiction. Thus the
…rst-order conditions yield prices that are consistent with the ranking p1 < p2 <
::: < pn . It remains to show that there is a unique solution to (3) and (6). This
is true since pn = pm is uniquely determined and since pi is uniquely determined
from (6) given pi+1 under the condition that [F (p) + pf(p)]0 > 0 by Lemma 1.
The structure of the equilibrium prices is characterized in the next section,
and illustrated in the one after for a uniform density.
4
4.1
Dispersion properties
Price dispersion
First, as expected, more …rms provoke more competition in the following sense:
Proposition 2 (Falling prices) The ith lowest price in equilibrium (1 · i < n)
is strictly decreasing in the number of …rms, n. The price range pn ¡ p1 rises with
n.
Proof. Recall …rst equation (6):
(i + 1) [F (pi) + pi f(pi )] = (i + 1)F (pi+1 ) + ipi+1f (pi+1):
Lemma 1 established that F (p) + pf(p) is decreasing in p for p · pm and Proposition 2 showed that pi < pm for all i < n. Hence, the left-hand side of (6) is
increasing in pi while the right-hand side is increasing in pi+1. Any decrease in
pi+1 therefore elicits a decrease in pi . Adding a new …rm at the top price pm
causes the next highest price to fall, so prices fall all down the line. This means
that the lowest price falls with n. Together with the property that the highest
price, pn, is independent of the number of …rms, this implies that pn ¡ p1 is
increasing in n.
Hence, price dispersion increases with the number of …rms in the sense that
the price range in equilibrium broadens. Next we show that the k th highest price
rises with the number of …rms.
Proposition 3 (Rising prices) The k th highest price in equilibrium (1 < k ·
n ¡ 1) is strictly increasing in n.
8
Proof. The proof of Proposition 2 established that neighboring prices move
in the same direction. Since the highest price is una¤ected by entry, it su¢ces
to show that the second highest price rises with a new …rm. With n …rms,
the second highest price, pn¡1, is implicitly given by: F (pn¡1) + pn¡1f(pn¡1) =
F (pm ) + (n ¡ 1) /n pmf (pm). The right-hand side increases with n, so that pn¡1
must also increase with n (by Lemma 1 and Proposition 2, since then F (p)+pf(p)
is increasing in p).
The above results show that prices are always dispersed and fan out as the
number of …rms increases. The top price stays at the monopoly price and the
bottom price decreases. As we argue below, in the limiting case n ! 1, the
lowest price goes to the competitive price of zero. The asymmetry of equilibrium
prices is also re‡ected in asymmetric pro…ts.
4.2
Pro…t dispersion
Price dispersion is associated in our model with pro…t dispersion. Contrast for
example Butters’ (1977) model of advertising in which all …rms earn zero pro…t by
the equilibrium condition. In an oligopoly version of the Butters model, Robert
and Stahl (1993) …nd equilibrium price dispersion in that the equilibrium entails
non-degenerate mixed strategies. However, since the equilibrium mixture is the
same for all …rms, pro…ts are still equalized. Another model with price dispersion
is the Bargains and Rip-o¤s (or Tourists and Natives) set-up of Salop and Stiglitz
(1977). Since they also close the model with a zero pro…t condition for both the
Bargain and the Rip-o¤ …rms, pro…t asymmetries cannot arise. In our model,
we can allow for free entry by introducing positive entry costs. This will lead to
a …nite number of …rms in equilibrium, with all but the marginal …rm making
positive pro…ts. We now consider how pro…ts vary across …rms.
Proposition 4 (Pro…t ranking) Lower price …rms earn strictly higher pro…ts:
¼¤1 > ¼ ¤2 > ::: > ¼ ¤i > ::: > ¼¤n :
The total pro…t earned in the market place,
Pn
¤
i=1 ¼i ,
exceeds the monopoly pro…t.
Proof. Suppose not. Then there exists some …rm i for which ¼ ¤i · ¼ ¤i+1. But
i can always guarantee to earn the same pro…t as i + 1 by setting a price p¤i+1
(since the demand addressed to each …rm is independent of all lower prices). Since
though p¤i is the strict local maximizer of ¼i on [p¤i¡1; p¤i+1 ], pro…t is strictly higher
at p¤i than at p¤i+1. The last line follows because the pro…t of the highest-price
…rm is 1 /n th of the pro…t of the single monopolist, and all other …rms earn more
than the highest-price …rm.
The reason that the market pro…t exceeds the single product monopoly pro…t
is that there are multiple products o¤ered at di¤erent prices. This is a rather
9
unusual result with constant marginal cost. 6 It will not hold under Cournot
competition with a homogeneous product because there the Law of One Price
holds. 7 Interestingly, Stahl (1989) shows that more …rms (recall he has a mass of
consumers with zero search costs, and a …nite number of …rms, and so a mixed
strategy equilibrium) lead to an increase in prices (which result he terms ”more
monopolistic”) in the symmetric equilibrium density. However, Stahl (1989) does
not calculate the e¤ect on total pro…t of further entry. Our result holds because it
allows some price discrimination across consumer types with di¤erent reservation
prices.8 We pick up on this theme below in the Section 6.1 on the behavior of a
multi-outlet monopolist.
The pro…t ranking found in Proposition 4 is somewhat unusual in oligopoly
theory. In our setting, low price/high volume …rms earn the highest pro…ts. In
many other models, such as those of vertical di¤erentiation, and in asymmetric
discrete choice oligopoly models, high prices are associated with higher demands
(through the …rst-order conditions) and hence high-price …rms are predicted to
earn high pro…ts.9
4.3
Market prices
With impulse goods, not all consumers buy at the lowest price in the market.
Thus, even though we show below that the lowest price goes to zero (marginal
cost), this does not necessarily mean that the market solution e¤ectively attains
the competitive limit. It might also be that equilibrium prices pile up close to
marginal cost, so almost all consumers would buy at competitive prices. We now
show that this is not the case, and instead the average transaction price paid by
consumers is bounded away from marginal cost.
Proposition 5 (Margins) The demand-weighted market price strictly exceeds
marginal cost and is bounded below by pm Dm.
P
P
Proof. Let pa = ( ni=1 pi Di) = ni=1 Di denote the demand-weighted market price
and note that the denominator is bounded above by 1. The numerator exceeds
the monopoly pro…t by Proposition 4 and hence pa > pmDm , where pm is the
monopoly price given by (3).
6
With increasing marginal cost, clearly an oligopoly has an e¢ciency advantage in production, and so it is possible that total pro…ts are higher (in, say, a Cournot oligopoly). A similar
result can hold for a competitive industry.
7
The result can also hold under product di¤erentiation due to a market expansion e¤ect. To
illustrate, suppose half the consumers care only about product 1, while the other half are only
interested in product 2. Then two …rms in this ”industry” earn twice as much as one alone.
8 This is reminiscent of Salop’s (1977) noisy monopolist result, although Salop assumes that
consumers observes the prices set before searching – otherwise the monopolist will not be noisy,
and faces the Diamond (1971) paradox.
9
Although note that …rms with high mark-ups produce larger volumes in Cournot competition with homogenous goods.
10
It does perhaps seem unusual to compare prices against the yardstick of profits. This, though, is just a normalization issue since the total potential demand
(the quantity intercept on demand) has been set to unity. The generalization is
the pro…t per potential consumer.
Market forces do not drive prices to marginal cost for impulse goods. It is
not product di¤erentiation that underlies this result, since we have shown it with
a homogeneous good, and so it is distinct from Chamberlinian monopolistically
competitive mark-ups. It is also distinctive from the symmetric Chamberlinian
(1933) set-up because equilibria involve price dispersion, with distinct prices for
all …rms in a pure strategy equilibrium. Perhaps the closest result is that of
Butters (1977) who shows equilibrium price dispersion in a model of advertising,
although Butters uses a zero pro…t condition to close his model while we have
pro…t asymmetries.
The question of market performance in the face of imperfections is an old one.
Chamberlin (1933) was interested in the welfare economics of product diversity,
and subsequent authors (e.g. Hart, 1985 and Wolinsky, 1983) have re‡ected upon
the meaning of ”true” monopolistic competition. One recurrent issue in this
literature is whether the market price will converge to the competitive one as the
number of …rms gets large enough and when there are market frictions or product
di¤erentiation (see for example Wolinsky, 1983 and Perlo¤ and Salop, 1985).
Proposition 5 shows that the market solution stays well above the competitive
outcome even in the limit since the average (quantity-weighted) transaction price
is bounded below by the monopoly pro…t. On the other hand, we next show that
the lowest price in the market does converge to marginal cost. Our approach
provides an intriguing mix in this respect.
4.4
Limiting cases
There are two dimensions in which the market outcome ressembles the standard
competitive one as the number of …rms gets large.
Proposition 6 (Low price) The lowest price in the market tends to zero when
n goes to in…nity. Pro…ts for each …rm go to zero.
Proof. Recall …rst that the …rst-order condition for the lowest price …rm is
p1 = D1 /f(p1n) . Suppose that
p1 does not go to 0 and thus has a lower bound, p.
o
Let f = min f(p); f (pm ) be the lower bound of f (p) on (p; pm ), so that f (p1)
¸ f > 0 because f (p) is quasi-concave. Then D1 is also bounded below by p f .
Since Firm 2’s …rst-order condition is p2f (p2) /2 = D2, and since p2 > p1 > p,
D2 is bounded below by p f /2 . Following the same reasoning, Di is bounded
P
below by p f /i . Therefore, market demand is bounded below by p f ni=1 1 /i ,
which diverges as n ! 1. Then total demand is unbounded, a contradiction.
Consequently, Firm 1 charges a price which converges to 0 as n ! 1.
11
Since ¼ 1 = p1D1 , with D1 < 1, Firm 1’s pro…t clearly converges to 0 as
n ! 1. Given that ¼ 1 > ¼2 > :::¼n from Proposition 4, all pro…ts go to zero
with n.
Other properties are illustrated with the uniform density below, for which
we show that the di¤erence between consecutive prices falls as we climb the
price ladder. The implication is that more …rms price above the midpoint of
the equilibrium price range than below, and the average is also higher than the
midpoint.
4.5
Equilibrium existence
The model above is interesting for its asymmetric candidate equilibria. However,
the model is also intricate to deal with because the pro…t function is only piecewise
quasi-concave. The pro…t function may switch from sloping down to sloping up
at a price equal to a rival’s price. This feature means that a candidate pro…t
maximum must be carefully veri…ed by checking deviations into price ranges
de…ned by intervals between rivals’ prices. The problem stems from a demand
function that kinks out as one …rm’s price passes through that of a rival (and
hence a marginal revenue curve that jumps up at such a point – indeed, we used
this argument in showing Lemma 2). The demand kink in turn arises when a
…rm competes with fewer …rms at lower prices.
We can prove analytically two further properties that are useful in determining global equilibrium. First, the prices found constitute a local equilibrium
whereby each …rm’s pro…t is maximized provided it prices between its two neighbors. Indeed, we showed above that the unique candidate solution to the …rst
order conditions satis…es the ranking condition. Furthermore, each …rm’s pro…t
is maximized on the interval between its two neighbors’ prices since pro…ts on
these intervals are concave functions. A local equilibrium in this sense is still a
useful result because it ensures the solution is robust at least to price changes by
…rms that do not change the order of prices.
To prove the existence of a (global) equilibrium we must look at what happens
under all possible deviations. The class of such deviations we need to consider is
reduced because we can show that no …rm can earn more charging a higher price.
Indeed, by the fact we have proved the solution is a local equilibrium, it su¢ces
to show that no …rm i wishes to set a price strictly above pi+1 . If Firm i; i ¸ 1;
chose a p0i 2 (pj ; pj+1) ; j > i; (or indeed, p0i > pn) it would become the “new”
j th …rm, in the sense of setting the j th lowest price. But then its pro…t could not
exceed ¼j since the original pj was set to maximize ¼ j for p² (pj¡1; pj+1 ), Firm
i would now be choosing p0i in a smaller interval, [pj; pj+1], and the pro…t of the
…rm in the j th position is independent of p2 ; :::; pj¡1, the prices of all lower-price
…rms. Hence (using Proposition 4 above), ¼0i · ¼j < ¼ i.
In the next section we consider a uniform distribution and we verify numerically that there are no pro…table deviation from the candidate equilibrium. As
12
will be seen, the local equilibrium is also global, but the pro…t functions are
not quasi-concave, which would suggest that analytical proofs are unlikely to be
forthcoming.
5
Uniform distribution of reservation prices
The structure of the model can be easily comprehended for the uniform distribution that gives rise to linear demand. We can also get more precise characterization results for this case.
5.1
Price dispersion
For a uniform density, the highest price is given by (3) as pn = pm = 1 /2 . The
other prices are given by (6) as
2i + 1
pi+1 ;
i = 1; :::; n ¡ 1:
(7)
2 (i + 1)
This recurrent structure tells us several properties about the structure of
equilibrium price dispersion. Relative prices, pi+1 /pi , fall with i. Moreover, as
we show below using the closed form solution for prices, absolute prices di¤erences
also fall with i. This means that the density of equilibrium prices is thicker at
the top and tails o¤ for lower prices.
We can also study how price dispersion changes with n. First, it is readily
veri…ed that the i th lowest price (1 · i < n) is strictly decreasing in the number
of …rms (see Proposition 2). Second, the di¤erence between any pair of prices
decreases with the number of …rms. This follows from (7) since pi+1 ¡ pi =
pi /(2i + 1) and that pi decreases with n. The interpretation of this result is that
price coverage gets thicker with more …rms despite the broader range of prices.
The equilibrium can be readily computed for various values of n:
pi =
p¤1 =
p¤1 =
p¤1 =
31 ¤ 1
; p = ; for n = 2;
42 2 2
351
51
1
= 0:313; p¤2 =
= 0:417; p¤3 = for n = 3;
462
62
2
3571
= 0:273; p¤2 = 0:365; p¤3 = 0:438; p¤4 = 0:5 for n = 4, etc.
4682
The equilibrium prices are depicted in Figure 2.
Figure 2. Pattern of equilibrium prices as the number of …rms varies from 1 to
10.
13
For example, under duopoly, the high price …rm sells to 1 /4 of the consumer
population at a price of 1 /2 , while the low price …rm sells to 3 /8 of the population
at a price of 3 /8 . Total pro…ts under duopoly are thus 17 /64, this exceeds the
monopoly pro…t of 1 /4, which is consistent with Proposition 4. However, total
pro…ts do not monotonically increase with the number of …rms: we show below
that they fall to the monopoly level as the number of …rms gets large.
The explicit expression for the equilibrium prices is given by recursion by
(using the notation k!! ´ k ¢ (k ¡ 2) ¢ (k ¡ 4) :::):
p¤i
1
=
2
Ã
i!(2n ¡ 1)!!
2n¡in!(2i ¡ 1)!!
!
i = 1; :::; n:
This series veri…es the property p¤1 < p¤2 < ::: < p¤n : Writing out the double
factorial expressions yields:
p¤i
1
=
2
Ã
i!(i ¡ 1)!(2n ¡ 1)!
22(n¡i)(2i ¡ 1)!n!(n ¡ 1)!
!
i = 1; :::; n:
(8)
This equation can be used to verify the property noted above that absolute price
di¤erences contract toward the highest price. 10 The limit case for prices and
pro…ts as the number of …rms gets arbitrarily large is determined in the next
section.
5.2
Limit results for the uniform density case
p p
We can use Stirling’s approximation11 (which is i! ¼ 2¼ iii e¡i, for integer i)
on expression (8) as both i and n go to in…nity (with i /n …nite) to write:
p¤i
Ã
!
1³ p p ´
1
¼
2 ¼ i
p p
:
2
2 ¼ n
Fix x = i /n 2 (0; 1) to write the limiting price as
p¤(x) =
1p
x;
2
x 2 (0; 1):
Clearly this price is independent of n and it yields the monopoly price of 1 /2at
the upper end. This expression can be readily inverted to yield the cumulative
distribution of prices as G(p) = (2p)2 , which means a linear density of g(p) = 8p,
for p 2 (0;.1 /2
p ). The average limiting price across …rms is 1 /3 , while the median
price is 1 2 2 ¼ 0:353.
10
! 2i
From (7), we get ¢i+1; i = p i+1 ¡ pi = K i!(i¡1)
2 , with K > 0 a constant which only
(2i¡1)!
2i
depends on n. Then, ¢i+ 1;i ¡ ¢i; i¡ 1 = K (i¡1)!(i¡1)!2
>0
2(2i¡1)!
11
Note that the relative error using Stirling’s approximation is 1:7 % for n = 5, 0:8% for
n = 10 and 0:4% for n = 20.
14
The demand weighted price is the average price actually paid by consumers in
the market place. Half the consumers have reservation prices above the monopoly
level and so will buy the …rst good encountered. Given the distribution above,
the price they pay is the average price across …rms, which is 1 /3. A consumer
with a lower reservation price ± will buy as soon as she encounters a price below
that reservation level. The expected price paid is then 2± /3. Since the density of
reservation prices is uniform, we can apply the average value of ± (which is 1 /4
for these consumers) and so the average price they pay is 1 /6. Combining these
two averages, the average price paid in the market is pa = 1 /4. 12 Therefore, the
average price decreases from 1 /2 in the monopoly case to 1 /4 in the limiting
case n ! 1. The value of pa = 1 /4 is also the value of total pro…ts earned from
the market because all consumers buy. This means that the monopoly pro…t level
is attained in the market when the number of …rms is very large even though the
average transaction price is half of the monopoly price!
We now look at the limit properties of the pro…t ratios. By Proposition 4,
all other …rms earn less than Firm 1, and the greatest disparity between …rms’
pro…ts is between Firms 1 and n. Hence, we consider the limit:
à p
!2
µ
¶
¼1
n (p¤1)2
n(2n)!
lim
=n!1
lim
=n!1
lim
n!1
¼n
(p¤n )2
22n¡1 (n!)2
We can evaluate this expression using Stirling’s approximation to give:
lim
n!1
µ
¶
¼1
2
¼ p ¼ 1:27:
¼n
¼
Therefore, the ratio of pro…ts for any pair of …rms is no more than 1:27.
5.3
Existence: numerical examples
In order to check equilibrium existence, we calculated the pro…t of each …rm
when it deviates from its candidate equilibrium position, to any price in [0; 1].
Note that deviations may change the labelling of the …rms. This exercise is easily
done analytically for small numbers of …rms (two or three), and with the help
of a computer program (Matlab) for larger values of n. A representative plot of
pro…ts for Firm 5 of 10 is provided in Figure 3.
Figure 3 Pro…t function for Firm 5 when n = 10
We also veri…ed that any local equilibrium is also global for the other …rms
and for other values of n.
12
This limit
Pn can be veri…ed directly by using the formula p a =
Di = p i /i , i=1 Di ! 1, and with p i given by (8).
15
Pn
i=1
p i Di /
Pn
i=1
Di , with
6
Further directions
6.1
Comparison to multi-outlet monopoly
One of the results of the competitive analysis is that aggregate pro…ts increase
with the number of goods (at least initially, although eventually they may fall,
as seen in Section 5.2). The factors at play here are price discrimination and
competition. In this subsection, we hold the competitive e¤ect …xed by analyzing
the behavior of a monopolist selling multiple goods. For concreteness, we shall
refer to this as the multi-outlet monopoly in keeping with the impulse good idea
of a consumer who encounters opportunities randomly at di¤erent geographical
points.
We therefore derive the prices chosen by a single …rm that sells n products,
given the reservation price demand system. The multi-outlet monopolist sets n
prices, p1 ; :::; pn to maximize
¼=
n
X
i=1
pi Di =
n
X
i=1
pi
n+1
X
F (pj ) ¡ F (pj ¡1 )
;
j¡1
j=i+1
where we have de…ned F (pn+1) = 1.
We …rst derive the formula that determines the highest price, pn . Rearranging
the …rst-order condition yields
P
n¡1
1 ¡ F (pn )
pi
pn =
+ i=1 :
f (pn)
n¡1
(9)
The solution is readily compared to the competitive solution as given by (3),
which di¤ers only by the inclusion of the second term in the current incarnation.
This term denotes the extra pro…t gained on the other products sold, and constitutes a positive externality that is not internalized at the competitive solution.
Since this term is clearly positive, the right-hand side of (9) is clearly higher under
multi-outlet monopoly than under competition. Recalling that our assumption
n)
of (¡1)-concavity of 1 ¡ F implies that 1¡F(p
f(pn ) is a decreasing function, then
the solution to (9) is clearly higher than the solution to (3). The highest price
therefore exceeds the price that would be set by a single product monopolist.
We now derive the relevant expression for product k and proceed by recursion.
Indeed, the …rst-order condition for product k can be written as
X @Di
@¼
@Dk k¡1
= Dk + pk
+
pi
;
@pk
@pk
@pk
i=1
(10)
which is simply the extra revenue on product k plus the extra revenue on all
lower-priced products. Clearly the last term on the right-hand side is positive13
13
Recall that
@ Di
@p k
=
h
1
k¡ 1
i
¡ k1 f (pk ) for i < k, since an increase in p k transfers the marginal
f (p k ) consumers from being shared by k lower price …rms to being shared by k ¡ 1 lower price
…rms.
16
and has no counterpart in the corresponding equation for pk in the competitive
solution. The …rst two terms, the marginal revenue terms, are decreasing in pk
k
under the assumption of (¡1)-concavity of 1 ¡ F . The term @D
= ¡ f (pkk ) is
@pk
the same at any pk as in the competitive solution, so it remains to consider the
behavior of Dk . We have already shown that pn is higher under multi-outlet
monopoly. This though implies that Dn¡1 is higher at any value of pn¡1 < pn .
Hence the right-hand side of (10) is higher, and, since it is a decreasing function
of pk, leads to a higher solution. But then the same argument applies to the price
of product n ¡ 2 and so on back down all the product line. This establishes:
Proposition 7 (Multi-outlet monopoly) All prices are higher under multioutlet monopoly.
It is helpful to look at the direction in which pro…ts increase starting from
the duopoly equilibrium. First, the lower price has no e¤ect on pro…ts earned on
the higher-priced product, since the latter only caters to those consumers with
high reservation prices. However, the pro…t earned on the lower-priced product
are increasing in the higher price since a higher price increases the number of
consumers who buy the low-price good. Internalizing this externality means that
the monopolist will set a higher price (above pm ) on the top. Given this higher
price, it is also optimal to set a higher price on the other product, so that the
two-product monopolist sets both prices higher than under duopoly competition.
For example, consider a monopolist with two outlets and a uniform distribution
of reservation prices. The two-outlet monopoly sets higher prices (p1 = 37 and
p2 = 57 ) than the duopolists (who set prices p1 = 38 and p2 = 12 ). Moreover, the
price range is more than twice the size for the two-outlet monopoly.
6.2
An economy with economists
Studies of impulse buying suggest that only some consumers (often synonymous
with compulsive shoppers) are impulsive. Others plan their shopping (often using
shopping lists).
Here we investigate how the equilibrium prices change when we introduce a
fraction of consumers who always buy from the cheapest …rm, while the market
size remains …xed at 1. These consumers are assumed to have the same distribution of reservation prices as the others. For them, though, the reservation price
only serves to determine whether a consumers buys or not. If she buys, she always buys from the cheapest …rm (and so will not do so when the cheapest price
exceeds the reservation price).
Part of the interest here is to determine whether such consumers (economists)
”police” the market by causing …rms to charge lower prices. Clearly if there were
only such (classical) consumers present in the market, the outcome would be
the standard Bertrand equilibrium at which price equals marginal cost. The
surprising result here is that introducing economists serves actually to increase
17
prices. In other words, people who search out the lowest price can exert a negative
externality on those who are not so careful in their shopping.
To see this, consider the calculus of the lowest-priced …rm. This is the …rm
that will attract all the economists in the market. Recall the …rst-order condition
1
is D1 + p1 @D
@p1 = 0. Replacing some consumers with economists does not a¤ect the
1
derivative @D
since the economists always buy from the cheapest …rm. However,
@p1
increasing the fraction of economists does increase D1 because none of them
buy from the other …rms, whereas some of the population they are replacing
did buy from those …rms. Thus (holding other prices constant) the lowest price
rises. However, since the other prices are independent of the lowest price, the
assumption that the other prices are unchanged holds true. The result that price
can increase is dependent on the pure strategy equilibrium still holding true. If
there are too many economists, the only equilibria are in mixed strategies. 14
The uniform distribution provides an illustration. Let the fraction of economists be ¢, and suppose there are two …rms. For p1 < p2, the demand facing
the second …rm is (1 ¡ ¢) (1 ¡ p2 ) =2, which is half the non-economist consumers
whose reservation prices exceed p2 . As expected, the candidate equilibrium higher
price is 1=2 (the monopoly price). The demand facing the lower price …rm is
(1 ¡ ¢) (1 ¡ p2) =2 + (1 ¡ ¢) (p2 ¡ p1 ) + ¢ (1 ¡ p1)
This yields a candidate equilibrium lower price of pb1 = (3 + ¢) =8, which increases with ¢ as claimed. We now check that this is an equilibrium by …nding ¢ small enough that Firm 2 does not wish to deviate to just undercutting
the lower price (clearly this is the only deviation to check). Deviating yields a
pro…t pb1 [(1 ¡ ¢) (1 ¡ pb1 ) =2 + ¢ (1 ¡ pb1)] which is to be compared to the status quo pro…t of (1 ¡ ¢) =4. Substituting, deviation is unpro…table as long as
(3 + ¢) (5 ¡ ¢) (1 + ¢) < 32 (1 ¡ ¢), meaning that equilibrium exists as long as
the fraction of economists, ¢, is below its critical value of around 34:5 %.
Varian (1980, 1981) claimed that in his model of sales, it is possible that
increasing the number of uninformed consumers will decrease the price paid by
the informed. However, Morgan and Sefton (2001) have shown that this result
is not possible in Varian’s model. We have shown that such an e¤ect can arise
with impulse goods.15
6.3
Impulse goods with patient consumers
We have modeled the essence of impulse good purchases by supposing that consumers buy as soon as they encounter a price below their reservation price. This
1 4 Results
for that region indicate that increasing the number of economists beyond the initial
threshold does serve to decrease average prices and hence to improve economic e¢ciency.
15
Increasing the fraction of impulse purchasers has the same e¤ect on prices as decreasing
the fraction of economists in our model. Note though that we have considered a …xed number
of …rms while Varian uses a free entry mechanism.
18
makes sense to have exogenous reservation prices when consumers make a snap
decision about whether or not to buy. But this rule can be criticized as overly
simplistic: why does the consumer buy if her current surplus is low, rather than
holding-o¤ and waiting for a better opportunity? Notice that this behavior is
indeed optimal when consumers are su¢ciently impatient.
This leads us to consider equilibrium when consumers are patient. We shall
illustrate with a duopoly and the uniform density of the preceding section. We
shall show that the fundamental dispersion property remains when consumers are
not too patient, as does the fact that the low-price …rm earns more and market
revenue exceeds the monopoly level.
Assume, as before, that there is a uniform distribution of reservation prices
that determines the consumer willingness to pay. If a consumer with reservation
price ± pays price p today, her surplus is ± ¡ p, if she makes the transaction
b̄
b̄ is the discount factor. Let ¯ =
tomorrow,
.³
´her surplus is (± ¡ p), etc., where
b̄
2 ¡ b̄ be the e¤ective discount factor when there are two choices presented
randomly (and each encounter is random with probability one half). 16 Suppose
that the consumer has just encountered the high-price …rm, 2, and so observed
its actual price p2. If she decides to wait until she encounters the other …rm, 1,
her expected surplus is ¯ (± ¡ pe1), where pe1 is the expected price of Firm 1, which
is correct in equilibrium. Therefore she will buy from Firm 2 there and then if
and only if:
(± ¡ p2) > ¯ (± ¡ pe1) :
The reservation price of the consumer indi¤erent between buying and waiting
is then
1
e
±=
(p2 ¡ ¯pe1) > p2
1¡¯
so that all consumers with higher ± buy immediately from Firm 2.
a
17
As long as the actual price of Firm
h 1, p1,³ is´inot higher than p2, then the
demand addressed to Firm 2 is D2 = 12 1 ¡ F e± and so its candidate best-reply
price is given by
1
1¡
(2p2 ¡ ¯pe1 ) = 0
1 ¡¯
or
1 ¡ ¯ + ¯pe1
p2 =
:
2
16
The next encounter
is the
…rm with probability one half and this is discounted at
µ
¶ other
.³
´
2
b̄
b̄
rate b̄ , so ¯ = 2 + 4 + ::: = b̄
2 ¡ b̄ . If instead the …rst …rm encountered is random
but the other one is always encountered next, then the relevant discount factor is b̄.
17
It will be veri…ed below that neiher …rm wants to deviate from the purported equilibrium.
19
As long as pa1 · p2 , Firm 1’s demand is D1 =
and its candidate equilibrium price satis…es
1
2
h
³ ´i
h
³ ´
i
1 ¡ F e± + F e± ¡ F (pa1 )
i h
i
1h
1 ¡ e± + e± ¡ 2pa1 = 0:
2
Using the equilibrium condition that pa1 = pe1 = p1 along with Firm 2’s best-reply
price gives
"
Ã
!#
1
1
1 ¡ ¯ + ¯p1
1+
¡ ¯p1 ¡ 2p1 = 0;
2
1¡¯
2
or
3 (1 ¡ ¯)
:
8 ¡ 7¯
(11)
2 (1 ¡ ¯) (2 ¡ ¯)
:
8 ¡ 7¯
(12)
p¤1 =
Hence
p¤2 =
(so p¤2 = 23 (2 ¡ ¯) p¤1). 18 Now note that a necessary condition for p¤2 > p¤1 is that
¯ < 1=2, so there can be no pure strategy equilibrium with price dispersion unless
this holds. We show in the Appendix that no …rm wishes to deviate unilaterally
from the candidate price equilibrium.
Equilibrium pro…ts are given by:19
¼¤1
and
¼ ¤2 =
=
(p¤1 )2
9 (1 ¡ ¯)2
=
;
(8 ¡ 7¯)2
(13)
p¤2 h
¤i
(1 ¡ ¯) (2 ¡ ¯)2
1 ¡ e± = 2
;
2
(8 ¡ 7¯)2
(14)
so that the ranking remains unchanged, i.e. ¼ ¤1 > ¼ ¤2, for ¯ 2 [0; 1=2].
For higher ¯ there are single-price equilibria instead. To …nd them, consider
a symmetric candidate equilibrium, where each …rm charges p. The pro…t of
each …rm is ¼ = p (1 ¡ p) /2. If a …rm deviates and charges a lower price, pd ,
its demand is still half its previous constituency, plus the set of consumers with
reservation prices between pd and p. These latter consumers, when they come
across a price cheaper than expected, pd < p, now want to buy it (and they will
not wait for another opportunity since they still expect the other price to be p).
The pro…t of the deviant …rm is concave in pd:
¤
relation describing the indi¤erent consumer at equilibrium is e± = (4 ¡ 5¯) /(8 ¡ 7¯ ).
2
19
Using the …rst-order condition for Firm 1, p ¤1 = D1, so ¼ ¤1 = (p ¤1 ) . Similarly, ¼ ¤2 =
2
(p ¤2 ) /2 (1 ¡ ¯) .
1 8 The
20
¼ d = pd
µ
¶
1¡p
+ p ¡ pd ;
2
pd · p;
which is maximized for p¤d = minf(1 + p) /4 ; pg. If p > 1 /3 , then p¤d =
(1 + p) /4 < p and pro…t is higher with a lower price.20 Otherwise (i.e., if
p · 1 /3), p¤d = p and the …rm has no incentive to deviate by charging a lower
price. That is, only single-price candidate equilibria with prices no larger than a
third can survive deviation to a lower price.
So consider now a single-price candidate with p < 1 /3, and
suppose
h
³ ´i a …rm
1
deviates upwards by charging pu > p. Its demand is Du = 2 1 ¡ F ± , where
± = (pu ¡ ¯p) /(1 ¡ ¯). Note that the pro…t function, ¼u = puDu, is concave in
pu, and the derivative of the pro…t function at pu = p is
d¼u
(1 ¡ ¯) ¡ p (2 ¡ ¯)
=
;
dpu jpu =p
2 (1 ¡ ¯)
which is non-positive if p ¸ (1 ¡ ¯) /(2 ¡ ¯). In this case the …rm has no incentive
to deviate from the candidate price equilibrium, p (locally and globally given the
pro…t function is concave). Therefore, there exists an equilibrium price p¤ with
bunching if:
(1 ¡ ¯) /(2 ¡ ¯) · p¤ · 1 /3 .
Note that (1 ¡ ¯) /(2 ¡ ¯) · 1 /3, implies that ¯ ¸ 0:5. Pulling all this together,
we have:
Proposition 8 (Impatience) In the duopoly case with uniformly distributed
reservation prices and patient consumers, there exists a unique dispersed equilibrium if and only if ¯ < 1 /2. If ¯ ¸ 1 /2 , there exists a continuum of single-price
equilibria. Any such equilibrium involves a common price p¤ 2 [(1 ¡ ¯) /(2 ¡ ¯) ; 1 /3].
As long as the dispersed equilibrium exists, equilibrium …rm pro…ts decrease
in ¯ (see equations (13) and (14)). This is because consumers are willing to
wait for a better deal and this creates more competitive tension. With even
larger propensity to wait, the only equilibrium involves both …rms charging the
same price. This clearly shows that price dispersion in this model is a direct
result of the impulse good property: because consumer are unwillling to wait,
some …rms can charge higher prices than others in e equilibrium. The present
extension constitutes an alternative means of modeling more choosy consumers.
Instead of a fraction of (in…nitely patient) ”economists,” we have assumed that
all consumers are soemwhat patient.
20
Indeed, it is readily veri…ed that ¼ ¤d =
(1+ p)2
16
21
>
p(1¡ p)
2
for p > 1=3.
7
Concluding remarks
One notable feature of the impulse good model is that it generates asymmetric price equilibria in pure strategies. Whilst it is straightforward to generate
asymmetric price equilibria in standard Bertrand oligopoly models when …rms
di¤er according to exogenous di¤erences in costs or qualities (see Anderson and
de Palma, 2001, for example) the result here holds for ex-ante symmetric …rms
in a simple price game.21 There has been considerable interest in the literature
in generating equilibrium price dispersion with homogenous products - this has
been one of the major objectives of equilibrium models with consumer search (see
for example Carlson and McAfee, 1983). Price dispersion has also been generated in the literature through variations of Varian’s (1980) model of sales and the
consumer search models that follow a similar vein (e.g. Rob, 1985). However,
such dispersion arises as the realization of symmetric mixed strategy equilibrium
and there remains some uneasiness about the applicability of mixed strategies in
pricing games (although see Lach, 2002, for some empirical evidence).
We have assumed that the distribution of reservation prices has no mass
points. In practice, this is unlikely to be the case if individuals use rough rules of
thumb that round o¤ to the nearest dollar (say). Indeed, if the reservation price
rule for a mass of consumers is of the form ”buy if less than $10” then we would
expect prices of $ 9:99 if there are several …rms. Indeed, we would expect several
…rms to set the same price if there are enough of them.
Thus the general version of the model with mass points and round number
reservation prices can be expected to generate both clumping of …rms on certain
prices and the phenomenon of ”nines” in pricing (see also Basu, 1997, for an
alternative treatment of this problem that relies on the costs for consumers to
mentally process digits in prices).
Finally, one interpretation of our model is that consumers do not know the
parameters of the model (number of …rms, distribution of consumers, marginal
costs, etc) and so cannot rationally calculate expected equilibrium prices. They
therefore use a simple rule of thumb. We have though argued that our results
are robust in that they extend to su¢ciently impatient consumers who rationally
anticipate the equilibrium prices from the precise model. One way to relax the
assumption that consumers know the price distribution is to suppose that they
(rationally) anticipate or know the lowest price in the market and then buy if
the price encountered is close enough to that lowest ”anchor” price. This line of
enquiry might help further link the economic theory of consumer behaviour and
…rm pricing to studies in marketing.
21
In two-stage games, asymmetric price equilibria often arise in the second (price) stage when
di¤erent choices are made in the …rst stage (as for example in vertical di¤erentiation models see Anderson, de Palma, and Thisse, 1992, Ch.8).
22
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24
A
Appendices
Proof of Lemma 1 (monopoly)
Since ¼ 0m(p) = [1 ¡ F (p)]¡pf(p), then p · [1 ¡ F (p)] /f(p) whenever ¼0m (p) ¸
0. Moreover, ¼00m (p) = ¡2f(p) ¡ pf 0 (p). If f 0 (p) ¸ 0, then ¼00m (p) < 0, so assume that f 0 (p) < 0. Then ¼00m (p) · ¡2f(p) ¡ [1 ¡ F (p)] f 0 (p) /f(p) whenever
¼0m (p) ¸ 0. Hence ¼00m (p) < 0 for ¼0m (p) ¸ 0 if 2f 2 (p) + [1 ¡ F (p)] f 0 (p) > 0.
This latter condition is guaranteed by the condition that 1 /[1 ¡ F (p)] is strictly
convex (as is readily veri…ed by di¤erentiation).
Proof of Lemma 2 (no bunching).
We show that if more than one …rm chose some price pe 2 (0; 1), then any
e
such …rm would wish to deviate. Let the corresponding demand be D(p).
It
is helpful to recall that a necessary condition for a local maximum is that the
left pro…t derivative is non negative while the right one is non positive. Suppose
l ¸ 2 …rms set the same price pe and that k ¸ 0 …rms charge a lower price. The
left derivative of the demand addressed to Firm k + 1 at pe is (@ Dk+1 /@pk+1 )¡ =
¡f (pe) /(k + 1), while its right derivative is (@Dk+1 /@pk+1 )+ = ¡f (pe) /(k + l) .
The corresponding pro…t derivatives are:
Ã
@¼k+1
@pk+1
!
e (p
e)
pf
e ¡
= D(p)
<
(k + 1)
Ã
@¼k+1
@pk+1
!
+
= D( pe) ¡
e (p
e)
pf
;
(k + l)
which contradicts the necessary condition for a local maximum. Consequently,
the …rms charging pe will always have incentive to deviate and charge a price either
lower or larger than pe, so that bunching of 2 or more …rms is never a candidate
equilibrium.
Proof that no …rm wishes to deviate from dispersed equilibrium
with patient consumers.
We have to consider deviations which change the rankings of the …rms. First
consider Firm 2 deviating to charge a price pl < p¤1. Its demand is half the
consumers with reservation prices between p¤1 and 1 plus the consumers with
reservation prices between pl and p¤1, so Dl = 12 (1 ¡ p¤1) + (p¤1 ¡ pl ). The optimal
price of the deviating …rm is (1 + p¤1) /4 (when interior), so that the maximum
pro…t of the deviating …rm is ¼¤1l = (1 + p¤1)2 /16. Firm 2 will not deviate if:
¼¤1l
<
¼ ¤2
Ã
(p¤2)2
=
2 (1 ¡ ¯)
!
Using the expressions for p¤1 and p¤2 (see (11) and (12)), we get the condition:
p q
11 ¡ 10¯ ¡ 4 2 1 ¡ ¯ (2 ¡ ¯) · 0;
25
which is always satis…ed for ¯ · 0:5, and therefore the Firm 2 never wishes to
deviate.
Next, consider a deviation where Firm 1 charges a price ph > p¤2. The demand
¡¯p¤2
for …rm 1 is Dh = 12 (1 ¡ ± h) where ± h = ph1¡¯
. The optimized pro…t is ¼ ¤h =
2
(p¤h )
1¡¯+¯p¤2
¤
. Firm 1 will not deviate if ¼¤1 ¸ ¼ ¤h; using expression
2(1¡¯) where ph =
2
(12) and (13), this amounts to showing that
p q
8 ¡ 7¯ + 2¯ (2 ¡ ¯) ¡ 6 2 1 ¡ ¯ · 0;
which is satis…ed for ¯ · 0:5.
26