A Model of Sales
Hal R. Varian
The American Economic Review, Vol. 70, No. 4. (Sep., 1980), pp. 651-659.
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A Model of Sales
Economists have belatedly come to recog-
nize that the "law of one price" is no law at
all. Most retail markets are instead char-
acterized by a rather large degree of price
dispersion. The challenge to economic the-
ory is to describe how such price dispersion
can persist in markets where at least some
consumers behave in a rational manner.
Starting with the seminal paper of George
Stigler, a number of economic theorists have
proposed models to describe this phenome-
non of equilibrium price dispersion. See, for
example, Gerard Butters, John Pratt, David
Wise, and Richard Zeckhauser, Michael
Rothschild, Steven Salop, Salop and Joseph
Stiglitz (1977), Yuval Shilony, Stiglitz, and
Louis Wilde and Alan Schwartz.
Most of the models of price dispersion
referred to above are concerned with
analyzing "spatial" price dispersion; that is,
a situation where several stores contempora-
neously offer an identical item at different
prices. A nice example of such a model is
the "bargains and ripoffs" paper of Salop
and Stiglitz (1977). They consider a market
with two kinds of consumers; the "in-
formed" consumers know the entire distri-
bution of offered prices, while the "unin-
formed" consumers know nothing about the
distribution of prices. Hence the informed
consumers always go to a low-priced store,
while the uninformed consumers shop at
random. The stores have identical U-shaped
cost curves and behave as monopolistically
competitive price setters. Salop and Stiglitz
show that for some parameter configura-
tions, the market equilibrium takes a form
where some fraction of the stores sell at the
competitive price (minimum average cost)
and some fraction sell at a higher price. The
'University of Michigan. Research support by the
National Science Foundation and the Guggenheirn
Memorial Foundation is gratefully acknowledged.
Helpful comments were received from James Adams,
Paul Courant, Vincent Crawford, and John Pamr.
Research assistance was provided by Todd Lanski.
high-price stores' clientele consists only of
uninformed consumers, but there is a suffi-
ciently large number of them to keep the
stores in business.
In ,the Salop and Stiglitz model-as in all
the models of spatial price dispersion-some
stores are supposed to persistently sell their
product at a lower price than other stores. If
consumers can learn from experience, this
persistence of price dispersion seems rather
implausible.
An alternative type of price dispersion
might be called "temporal" price dispersion.
In a market exhibiting temporal price dis-
persion, we would see each store varying its
price over time. At any moment, a cross
section of the market would exhibit price
dispersion; but because of the intentional
fluctuations in price, consumers cannot learn
by experience about stores that consistently
have low prices, and hence price dispersion
may be expected to persist.
One does not have to look far to find the
real world analog of such behavior. It is
common to observe retail markets where
stores deliberately change their prices over
time-that is, where stores have sales. A
casual glance at the daily newspaper indi-
cates that such behavior is very common. A
high percentage of advertising seems to be
directed at informing people of limited
duration sales of food, clothing, and appli-
ances.
Given the prevalence of sales as a form of
retailing, it is surprising that so little atten-
tion has been paid to sales in the literature
of economic theory. In fact, I know of no
work in economic theory that explicitly ex-
amines the rationale of price dispersion by
means of sales.' However, the work of
Shilony does provide an implicit rationale
for the use of sales as a marketing device.
'salop and Stiglitz' 1976 paper is concerned with
"spatial" price dispersion rather than temporal price
dispersion.
652 THE AMERICAN ECONOMIC REVIE W SEPTEMBER 1980
Shilony examines an oligopolistic market
where consumers can purchase costlessly
from neighborhood stores, but incur a
"search cost" if they venture to more distant
stores in search of a lower price. He shows
that no Nash equilibrium exists in pure pric-
ing strategies. On the other hand, Shilony
does establish the existence of an equilibrium
mixed strategy-that is, a strategy where
firms randomize their prices. Such a strategy
could be interpreted as stores having ran-
domly chosen sales.
In this paper, I explicitly address the
question of sales equilibria. The model may
be regarded as a combination of the Salop-
Stiglitz and the Shilony models described
above. As in the Salop-Stiglitz model, it will
be assumed that there are informed and
uninformed consumers. As in the Shilony
model, I will allow for the possibility of
randomized pricing strategies by stores. I
will be interested in characterizing the
equilibrium behavior in such markets.
In the model to be described below, firms
engage in sales behavior in an attempt to
price discriminate between informed and
uninformed customers. This is of course only
one aspect of real world sales behavior.
Other reasons for sales behavior might in-
clude inventory costs, cyclical fluctuations
in costs or demand, loss leader behavior,
advertising behavior, and so on. The theo-
retical examination of these motives is left
for future work.
I. The Model
Let us suppose there is a large number of
consumers who each desire to purchase, at
most, one unit of some good. The maximum
price any consumer will pay for the good- a
consumer's reservation price-will be de-
noted by r. Consumers come in two types,
informed and ~n in fo rmed .~Uninformed
consumers shop for the item by choosing a
store at random; if the price of the item in
that store is less than r , the consumer
2 ~ o rnow, the uninformed-informed distinction is
exogenously given. The decision to become informed or
uninformed will be examined in Section 111.
purchases it. Informed consumers, on the
other hand, know the whole distribution of
prices, and in particular they know the
lowest available price at any time. Hence,
they go to the store with the lowest price
and ~urchase the item there.
0;e might think of a model where stores
advertise their sale prices in the weekly
newspaper. Informed consumers read the
newspaper and uninformed consumers do
not. Let 1 > 0 be the number of informed
consumers, and M >0 the number of unin-
formed consumers. Let n be the number of
stores, and let U= M/n be the number of
uninformed consumers per store.
Each store has a density function f ( p )
which indicates the probability with which it
charges each price p. In its choice of this
pricing strategy, each firm takes as given the
pricing strategies chosen by the other firms
and the demand behavior of the consumers.
Only the case of a symmetric equilibrium
will be examined. where each firm chooses
the same pricing strategy.3
Each week, each store randomly chooses
a price according to its density function
f (p ) . A store succeeds in its sale if it turns
out to have the lowest price of the n prices
being offered. In this case the store will get
I+ U customers. If a store fails to have the
lowest price, it will get only its share of
uninformed customers, namely U. If two or
more stores charge the lowest price, it will
be considered a tie, and the low-price stores
will each get an equal share of the informed
customers.
Finally the stores are characterized by
identical, strictly declining average cost
curve^.^ The cost curve of a representative
firm will be denoted by c(q). It will be
assumed that entry occurs until (expected)
profits are driven to zero. Thus we will be
examining a symmetric monopolistically
competitive equilibrium in pricing strategies.
3Some justification for this symmetry assumption is
given by Proposition 9 in the Appendix.
4The motivation for this assumption is the casual
observation that retail stores are characterized by fixed
costs of rent and sales force, plus constant variable
costs-the wholesale cost-of the item being sold.
653 VOL. 70 NO. 4 VARZAN: A MODEL OF SALES
11. The Analysis
The maximum number of customers a
store can get is I + U. Let p* = c ( I + U ) /
( I + U ) be the average cost associated with
this number of customers.
PROPOSITION 1: f ( p ) =0 for p >r or p <
P*.
PROOF:
No price above the reservation price will
be charged since there is zero demand at
any such price. No price less that p* will be
charged since only negative profits can re-
sult from such a price.
PROPOSITION 2: There is no ~ymmetric
equilibrium where all stores charge the same
price.
PROOF:
Suppose that all stores were charging a
single price p with r l p >p*. Then a slight
cut in price by one of the stores would
capture all of the informed market, and thus
make a positive profit. If all stores were
charging p*, each would get an equal share
of the market and thus be making negative
profits.
Proposition 2 is simply a variant of the
well-known argument that declining average
cost curves and "competitive" behavior are
incompatible. I therefore concentrate on
establishing the nature of a price-randomiz-
ing solution. Recall that p is a point mass
of a probability density function f if there
is positive probability concentrated at p .
PROPOSITION 3 . There are nopoint masses
in the equilibrium pricing strategies.5
PROOF:
The intuition of this argument is seen to
be quite straightforward. If some price p
were charged with positive probability, there
would be a positive probability of a tie at p .
5Proposition 9 in the Appendix provides a partial
converse to this assertion.
If a deviant store charged a slightly lower
price, p -e, with the same probability with
which the other stores charged p , it would
lose profits on order e, but gain a fixed
positive amount of profits when the other
stores tied. Thus for small e its profits would
be positive, contradicting the assumption of
equilibrium.
Let us proceed to a detailed formulation
of this argument. First note that p* can
never be charged with positive probability,
for when p* is the lowest price charged,
profits are zero, and if there is a tie at p*,
profits are negative. Suppose then that p >p*
is charged with positive probability.
The number of points of positive mass in
any probability distribution must be count-
able so we can find an arbitrarily small e
such that p -e is charged with probability 0.
Consider what happens if we charge p -e
with the probability with which we used to
charge p , and charge p with probability 0.
The increase in profits will be
-Pr(Pi>p all i ) ( p ( I + U ) - c ( I + U ) )
- P r (P i
=p-e all i , Pi=p for k stores)
k = 2
( ( p - e ) ( I + U ) - c ( I + U ) )
n
- x Pr (P ,Zp all i , Pi=p fork stores)
k - 2
( p ( U + I / k ) - c ( U + I l k ) )
As e approaches zero, the sum of the first
four terms approaches zero, while the sum
of the last two terms remains a positive
number. Hence for small e profits are posi-
tive, contradicting the assumption of an
equilibrium strategy.
Proposition 3 expresses the essential dif-
ference between models of spatial price
654 THE AMERICAN ECONOMIC REVIEW SEPTEMBER I980
dispersion and models of temporal price dis-
persion. Most models of spatial price disper-
sion, such as the Salop-Stiglitz model or the
Wilde-Schwartz model, have equilibria with
specific prices being charged with positive
probability mass. The above argument shows
that such strategies cannot be profit-max-
imizing Nash behavior in a temporal ran-
expected profit; for if some price yields a
greater profit than some other price it would
pay to increase the frequency with which
the more profitable price were charged. Since
we require zero profits due to free entry, this
common level of profit must be zero.6 This
argument yields
.2
domizing model.
Since there are no point masses in the
equilibrium density, the cumulative distribu-
tion function will be a continuous function
on (p*, r). Let F (p ) be the cumulative dis-
tribution function for f (p) ; thus f ( p )=
F ( p ) almost everywhere.
We can now construct the expected profit
function for a representative store. When a
store charges price p, exactly two events are
relevant. It may be that p is the smallest
price being charged, in which case, the given
store gets all of the informed customers.
This event happens only if all the other
stores charge prices higher than p, an event
which has probability (1 -F(p))"-'. On the
other hand, there may be some store with a
lower price, in which case the store in ques-
tion o d y gets its share of the uninformed
customers. This event happens with proba-
bility I - (1 -F(~))"- I . (By Proposition 3
we can neglect the probability of any ties.)
Hence the expected profit of a representa-
tive store is
where q ( p ) =p (U+ I ) -c(U+ I )
The maximization problem of the firm is
to choose the density function f ( p ) so as to
maximize expected profits subject to the
constraints:
It is clear that all prices that are charged
with positive density must yield the same
PROPOSITION 4: Iff ( p ) >0, then
(Of course, Proposition 4 also follows di-
rectly from the application of the Kuhn-
Tucker theorem to the specified maximiza-
tion problem.) Rearranging this equation,
we have a formula for the equilibrium
cumulative distribution function:
Note that the denominator of this fraction is
negative for any p between p* and r. Hence
the numerator must be negative so that
profits in the event of failure are definitely
negative. The construction of (1 -F(p))"-'
is illustrated in Figure 1. At each p where
f (p)>O we can construct 7~~ ( p ) and rS(p)
as illustrated and take the relevant ratio.
Proposition 4 gives us an explicit expression
for the equilibrium distribution function at
those values of p where f (p ) >0. If this is to
be a legitimate candidate for a cumulative
distribution function, it should be an in-
creasing function of p . This is easy to verify:
PROPOSITION 5: ?(p)/(7if ( p ) - ~ ( p ) )
is strictly decreasing in p.
PROOF:
Taking the derivative it suffices to show
that
60ne can also formulate the model with a fixed
number of firms. In this case, expected profits must be
equal to II , (r) .
655 VOL. 70 NO. 4 VARIAN: A MODEL OF SALES
1 1
U I t U quantity
Using the definitions of nf and T , this can
be rearranged to yield
which is obvious since average cost has been
assumed to strictly decrease.
Of course, Proposition 4 characterizes the
equilibrium density function only for those
prices where f ( p ) >0 . In order to fully char-
acterize the equilibrium behavior, we need
to establish which prices are charged with
positive density.
First, it is clear that prices close to p*
must be charged with positive density:
PROPOSITION 6: F(p* + E ) >0 for any
&>0.
PROOF:
If not, some store could charge p* + ~ / 2 ,
and thereby undercut the rest of the market
and make positive profits.
Similarly we can characterize the be-
havior of f ( p ) near its upper limit.
PROPOSITION 7: F(r-E ) < 1for any E >0 .
PROOF: '
Suppose not, and let p^ 0 , so charging r with probability 1
could make a positive profit.
Propositions 6 and 7 show that prices
near p* and r are charged with positive
density. It is now easy to show:
PROPOSITION 8: There is no gap ( p , , p2 )
where f ( p ) - 0 .
PROOF:
If not, let p , p ,, p^ will make larger prof-
its than p , . Since p , must make zero profits,
this shows that chargingj with probability 1
will make positive profits.
We now have a complete characterization
of the equilibrium density: f ( p ) >O for all p
in ( p * , r ) and f ( p ) = F' (p ) , where
We can also solve for the endogenous
variables n andp*. First, note that if a store
charges r , it only gets the uninformed
customers, and profits must therefore satisfy
?(r )=O. Similarly, if a store charges p* it
gets all the informed customers with proba-
bility 1 so rS ( p* )=0 .These two equations
can be used to determine n and p*.
'A heuristic proof is presented here and a more
rigorous proof in the Appendix. (The same holds true
for Proposition 8.)
656 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1980
As an example, let us compute the
equilibrium density when the cost function
has fixed cost k >0 and zero marginal cost.
Then
Since ~ , ( r ) =0, and U= M/n
(4) rM/n-k=O
or
( 5 ) n =rM/k
Thus
Since n,(p*) =0, we have
The equilibrium distribution function can
be found by substituting (2) and (3) into (1).
We have
Substituting from (6) and rearranging, we
find
The equilibrium density function is found
by differentiating (9):
(10) f ( p ) =F ' ( p )
1
--I 1
( k / I ) n - ' (I /p- l / r ) n - '
=-
n-1 P
Let
P* r price
FIGURE2. GRAPHOF f(p)= l/p(l -p/r)
Then f ( p ) can be written as
If n is reasonably large, m will be approxi-
mately 1, so f (p ) will be proportional to
This density is illustrated in Figure 2. Note
that stores tend to charge extreme prices
with higher probability than they charge
intermediate prices. This seems intuitively
plausible; a store would like to discriminate
in its pricing and charge informed customers
p* (to keep their business) and charge unin-
formed customers r (to exploit their surplus)