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A Model of Sales Hal R. Varian The American Economic Review, Vol. 70, No. 4. (Sep., 1980), pp. 651-659. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28198009%2970%3A4%3C651%3AAMOS%3E2.0.CO%3B2-A The American Economic Review is currently published by American Economic Association. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/aea.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. http://www.jstor.org Thu Mar 13 19:21:17 2008 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)