endogenous quality differentiation in congested markets

Endogenous Quality Differentiation in Congested MarketsAuthor(s): David ReitmanSource: The Journal of Industrial Economics, Vol. 39, No. 6 (Dec., 1991), pp. 621-647Published by: WileyStable URL: http://www.jstor.org/stable/2098666 .

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THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821 $2.00 Volume XXXIX December 1991 No. 6

ENDOGENOUS QUALITY DIFFERENTIATION IN CONGESTED MARKETS*

DAVID REITMAN

Firms selling a product with congestion externalities to a heterogeneous population of customers have an incentive to offer differentiated levels of quality. In a price competitive market differentiation arises endogenously through the prices chosen by firms. In equilibrium firms offer a range of prices, which induce an efficiency improving range of quality levels and allow customers to self-select their preferred price-quality combinations. Additional results suggest that, with many firms in the market, the choice of prices dominates the choice of service capacities in determining congestion levels.

I. INTRODUCTION

A COMMON feature of many forms of quality is that the level of quality depends on the level of demand. This holds for waiting times, reliability of service, crowded facilities, and, more generally, any product that requires that customers in effect share a limited amount of capacity available for service. Whenever any of these forms of "congestion-induced" quality affect the value of a product, the level of quality will be determined endogenously from the demand and capacity available at each producer. This is true whether the number and size of firms is determined by a planner, as is studied in much of the clubs literature (Buchanan [1965], Sandler and Tschirhart [1980]), or by a market mechanism (De Vany and Saving [1983], Scotchmer [1985a]).

As with any other form of quality, an efficient allocation generally assigns the same level of congestion-induced quality to every customer only if all customers have identical preferences. Otherwise, quality should be differentiated; for efficiency, those for whom not receiving a product is most costly should get the highest priority in the event of a shortage, those who do not mind waiting should wait the longest, and those who put the highest value on elbow room should get it. This can be accomplished by an explicit mechanism for assigning priority service. In the context of supply shortages, for example, firms can offer priority pricing schemes, as shown by Harris and Raviv [1981], Oren, Smith, and Wilson [1985], and Chao and Wilson [1987]. In the context of waiting lines, Lui [1985] has shown that an optimal priority scheme can be induced by requiring bribery payments to join the queue.

* This work is based on Chapter 2 of my doctoral dissertation at the Stanford University Graduate School of Business. I would like to thank Anat Admati, Jeremy Bulow, Suzanne Scotchmer, two referees, and especially Bob Wilson for helpful comments.

621



622 DAVID REITMAN

In the absence of an explicit priority scheme for assigning different quality levels, other mechanisms may arise that will move the outcome towards an efficient allocation. Rationing by waiting, as discussed by Barzel [1974] and Holt and Sherman [1982], can be understood in this framework. Customers placing the highest value on receiving a rationed product will arrive earliest, and other customers sort themselves according to their valuation of the product and cost of waiting. The waiting, though socially wasteful, induces a more efficient allocation of supply.

In a congested market, which is defined as a market for a product whose value is affected by congestion, there is a third, more fundamental way that differentiated quality levels are provided to a population of heterogeneous customers. It is clear that, if firms were to charge different prices, they would end up with different levels of congestion as customers sort among firms. The issue is whether this behavior can be sustained in equilibrium. In fact, firms serving a congested market will endogenously differentiate their quality levels by charging different prices and attracting different numbers of customers in equilibrium. Firms would earn positive profits if they all charged the same price, but profits increase if firms differentiate quality levels by offering different prices. In an oligopoly with a finite number of firms, a pure strategy equilibrium may not exist, but if it does, it is always asymmetric. This was demonstrated by Luski [1976] and Levhari and Luski [1978] for a duopoly model in which the firms competitively price Poisson queues.

With an infinite number of firms, however, a pure strategy equilibrium always exists. Each firm charges a different price, if the distribution of congestion costs is atomless, and as a result each offers a different quality level. Customers sort themselves according to their congestion costs. The equilibrium allocation is efficient, in the sense that no reallocation of customers among firms would lower aggregate costs of congestion. On the other hand, aggregate congestion costs are generally higher than they would be if all firms were to institute priority service. For example, in the context of waiting times, an efficient allocation would require that all customers with high waiting costs be served before those with lower waiting costs, which could be accomplished if all firms used a priority service mechanism. But with endogenously differentiated qualities, some low waiting cost customers who patronize a firm that on average has long delays may be served before some unlikely high waiting cost customers who buy from a firm with short delays on average.

It is always true in equilibrium that the quality among firms will differ endogenously because firms choose different prices. But quality also is increasing in the capacity level of firms. It might be expected that, for example, firms that serve a high quality market niche will choose a relatively large capacity in addition to a high price. In general, with a finite number of firms, firms will choose different capacity levels. However, at least for linear capacity costs when there are an infinte number of firms, the only equilibrium



ENDOGENOUS QUALITY DIFFERENTIATION IN CONGESTED MARKETS 623

has all firms choosing the same capacity level and differentiating their quality levels solely through prices. This suggests that, with many firms in the market, differentiation is done primarily through prices in equilibrium, rather than through other choices that may affect quality indirectly.

The next section introduces the model of congested markets that will be used. The third section contains a proof that the general oligopoly equilibrium is asymmetric, if it exists, and several examples with three firms. The results for the limit economy as the number of firms becomes infinite are derived in the fourth section. A discussion and comparison to related models follows.

II. THE MODEL

The actual dynamics that lead to congestion in markets are potentially quite complex, as customers arrive and are served according to some stochastic process, and as demand for service at each firm fluctuates. In order to incorporate within the model the minimal amount of dynamics necessary to retain the effects of congestion, assume that time is divided into periods, and consider only the number of customers who arrive each period. Each customer chooses one firm, and receives service from that firm along with all the other customers choosing the same firm. No subsequent attrition from a firm or jockeying among firms is permitted.

The level of quality received by every customer at any one firm in a given period is identical. This is the natural assumption for applications to clubs and related situations in which customers simultaneously utilize the resources of the firm. In a waiting time or service reliability context, the simplest interpretation is that customers base their choice on expected quality.' The price and expected level of quality at each firm are known by all customers at the time they make their choices. Thus we implicitly assume that each customer knows the aggregate characteristics of other customers in the market and can draw inferences about their choices. In equilibrium, the expectations about other customers' choices are fulfilled.

Each firm chooses its price, p, and its service capacity, y. The service capacity chosen by the firm affects the quality level received by customers and is assumed to be distinct from the capacity level that determines the number of customers that can be served per period. The two measures of capacity are of course related, but reflect decisions made in different time frames. For example, a supermarket has some maximum capacity for customers checked

'Deacon and Sonstelie [1985] study the lines at adjacent gas stations with a price discrepancy forced by differential ceiling price regulations. They find that their results are virtually identical if average waiting times are substituted for actual queue lengths, and speculate that customers either base their decisions on past experience, cannot observe queue lengths quickly enough to make a choice, or perhaps commit to a choice (by allocating time in their schedule, for example) before observing queue lengths. They also note a similar effect in other studies.



624 DAVID REITMAN

out per period that is determined by the size of the store and the number of checkout lanes. But it also makes a short term decision about how many of those lanes to operate at any given time when it schedules how many checkers will be working from shift to shift. For simplicity, the model focuses on the short run choice of price and service capacity, rather than the Edgeworth competition that would result if capacity played the dual role of determining the number of customers that can be served as well as the level of quality each receives.2 In keeping with the short-run nature of service capacity decisions, price and capacity decisions are assumed to be made simultaneously, in contrast to models of capacity choice that assume firms first choose capacity levels and subsequently engage in product market competition (e.g. Kreps and Scheinkman [1983]). Service capacity will be referred to simply as capacity for the rest of the paper. The unit of capacity is chosen so that a firm with a capacity of one can serve one customer in one unit of time. The cost function for capacity is K(y), with K'(y) > 0.

The production technology at each firm is assumed to be quite simple. Firms sell identical products and have identical technologies and cost functions. Production entails constant marginal costs, which can be assumed to be zero without loss of generality. Each customer imposes the same demands on the resources of the firm, and the quality of service is directly proportional to the capacity of the firm. This represents, for example, the technology of a service industry in which each worker serves one customer at a time, so that the size of the labor force is the fundamental measure of capacity. Thus the relative proportion of the number of customers, n, and capacity, y, determines the level of congestion at a firm. The congestion level is given by the function T(n/y), with T'( ) > 0.

On the demand side of the market, assume that D customers arrive in the market each period and that each customer buys one unit of product. Customers are heterogeneous, with types v distributed according to F(v) among the customers in the market, where F(v) is a cumulative distribution function. Each customer is infinitesimal, and their types form a continuum on [v, ve]. The distribution function is assumed to be twice continuously differentiable with 0 < F'(-) < oo, although some of the examples make use of type distributions that are discrete valued. The density function is f(v). The indirect utility function is assumed to be linearly separable in money. Thus a customer of type v served by the ith firm, with price pi and expected congestion level ti, has utility U(v, ti)-pi, with U, < 0 and Uv, < 0. The mixed partial assumption, which is common in self-selection models, ensures that customers sort among firms according to their type.

A more specific formulation of the model, which is adapted to a model of delay times and linear waiting costs, is used in all of the examples and in the last proposition of the paper, and will be referred to as the delay model. In the delay model, the relevant form of congestion is the expected delay time faced

2This dual role of capacity is studied in Reitman [1986].




by a customer arriving at a firm that serves customers at a constant rate. Thus T(n/y) = n/2y. Each customer's type is denoted by c, which is his unit cost of waiting. The waiting cost is assumed to be linear, so U(c, t) = R - ct, where R is a reservation price. Assume that R is sufficiently large so that it never binds in equilibrium.

The general model, which is assumed in all but the last proposition, covers other formulations of a waiting cost model as well as other forms of congestion-induced quality. These include:

* System Saturation. In queuing models with stochastic arrivals and service, the expected waiting time generally becomes infinite as the demand rate approaches the service rate. This effect can be captured in the general model by letting T(n/y) -s oo as n/y -+ 1.

* Discounting. Rather than having linear costs of waiting, customers may discount the value of a product by the amount of time elapsed until they receive service. In this case, even if customers use the same discount factors, their different valuations of the product induce different costs. Here v is interpreted as the individual customer's valuation, and the utility function has the form U(v, t) = ve -

, with v > 0. * Processor Sharing. Numerous forms of congestion can be characterized as

customers sharing a limited amount of capacity. Examples include computer loads, hiring an agent or consultant, and the attention of sales- people in a retail store. All of these examples can be characterized by a congestion function of the form T(n/y) = n/y.

* Reliability. If customers are rationed when demand exceeds supply, there is a quality attribute related to the probability that customers will not receive service. On the assumption that rationing is random, the congestion function in this example would be T(n/y) = 0 if n < y, and T(n/y) = 1- y/n if n > y.3

* Clubs. While the model used here is not as general as that typically used in the clubs literature, it is also relevant for clubs in which congestion has constant returns to scale in capacity, and customers have separable preferences. The congestion function T(n/y) establishes a common scale for measuring the level of crowding, which may be weighed by different customers differently (subject to the restrictions on the utility function).

The market can be formalized as a two-stage game. In the first stage firms choose prices and capacities simultaneously. Then in the second stage, customers, knowing the strategies picked by firms, choose which firm to be served by.4 In equilibrium, the quality offered by each firm depends on the

'The assumptions must be changed here to allow T'(-) = 0. 'The game can be described as a Nash-Stackelberg hybrid; firms play a Nash equilibrium

among themselves, and act as Stackelberg leaders with respect to customers. This terminology is used by Rob [1985]. The alternative formulation with customers and firms all choosing their strategies simultaneously never has a Nash equilibrium: given the choices made by customers about which firm to patronize, each firm prefers a higher price.



626 DAVID REITMAN

number of customers at the firm and its capacity, and the number of customers in turn depends on the price and anticipated quality at each firm. Customers distribute themselves among firms so that each customer cannot get a more favorable combination of price and quality at any other firm. The "mixed partial derivative" assumption, U,, < 0, is sufficient to guarantee that the most patient customers, with the lowest congestion costs, will choose to patronize the lowest priced, most congested firm. Similarly all other customers sort according to their type along the spectrum of price-congestion pairs offered in the market.5

Define v(i), for i = 1,... , I to be a permutation that maps the index of each firm into its order in the range of prices charged by all the firms; thus PC-'(1) < PC-'(2) < < PC-,) (the order assigned to firms with equal prices does not matter). Using the expected sorting behavior of customers in the second stage, the demand at each firm of an I firm oligopoly, in which firms set prices p = {P, ..., p} and capacities y = {Yi, ... , Yi} is

nflq-(l)(p,j) = F(v1)D

(1) nC,-1 (i)(J, y) = (F(vi)-F(vij 1))D

n- 1 (I)(-, y) = F(v _ ))D

where vi is the type of the customer that is indifferent between the firms with the ith and (i + 1)th lowest prices. These marginal customers are determined by the solution of a system of simultaneous equations, which are called the balance equations: for i < I,

(2) U (vi, T n( i(, Y)))-P? ti) = U (vi, T (ni + 1 (,y) _ i+t

The profits per period for firm i are I7I = pin(O) - K(yi). An equilibrium is a set of prices {p*,... ., p*. } and a set of capacities {Y,... ., y. } that solve

(3) Pi*, Yi*e arg max ypin,O) (pi5p-*ji5yi,-)-K(yi)}, Vi = 1, ... . I Pi, Yi

where n( ) solves (1) and (2). A solution to (3) actually produces an equivalence class of equilibria, with one equilibrium for each permutation of the I firms.

5 Suppose that the minimum type among customers selecting service at one firm is v and the maximum type among those selecting cheaper service at a competing firm is vJ, with v > v. Since in equilibrium the marginal customer is indifferent between service at the two firms, it must be that the expected congestion level is higher at the cheaper firm. If the customer of type v weakly prefers remaining at the more expensive firm, the customer of type v would prefer service there and would switch, since extra congestion is more costly to the higher type due to the mixed partial derivative assumption. It follows that the original prices and allocation of customers could not have been in equilibrium. For the solution to the customer sorting problem under more general assumptions, see Lee and Cohen [1985].




The problem of determining which firm ends up in which niche of the market is potentially an interesting one, since in an asymmetric equilibrium, firms generally earn different profits. But this problem will not be considered here.6 All equilibria will be characterized by assuming that a() is the identity mapping, so that the indices of firms are ordered by the prices they choose in equilibrium.

III. EQUILIBRIUM WITH A FINITE NUMBER OF FIRMS

The main results of the paper characterize the market equilibria as the number of firms in the market becomes large. But it is suggestive to look first at the market with a finite number of suppliers. As has been shown by Luski [1976] for duopoly queues, and by De Vany and Saving [1983] and Scotchmer [1985] in the homogeneous customer case, firms already in the market earn positive profits, despite selling an identical product under conditions of price competition. The reason is that endogenous quality determination adds continuity to the profit function. If a firm cuts its price, it is not able to attract all customers from its competitors because, as its demand increases, its quality level decreases. Thus the number of customers who switch to a firm that cuts its price is limited by the accompanying deterioration in its quality level. In equilibrium, the gain in profits from new customers if the price is dropped equals the revenues lost from existing customers.

If all customers have identical costs of congestion, the equilibrium will exist, and will be symmetric. But neither of these conditions necessarily hold if customers are heterogeneous. In a duopoly, the results derived by Luski [1976] and Levhari and Luski [1978] for a queuing model can be extended to our more general model of congestion and will only be summarized here, in order to focus instead on the equilibrium with more than two firms. As long as the demand curve is vertical, the equilibrium will be symmetric. However, if there are customers who are just indifferent between being served and remaining out of the market, the equilibrium, if it exists, will be asymmetric, with one firm offering a higher price and in consequence offering faster service to customers with higher waiting costs. In either case, the equilibrium may not exist; firms may choose to deviate from the solution to the first order conditions in order to serve the other firm's customers. This potential for nonexistence of an equilibrium will be demonstrated later in this section.

With three or more firms, the results are similar, except that the asymmetry of the equilibrium no longer hinges on whether the demand curve is vertical.

6 For one approach to this question, involving sequential entry, see Prescott and Visscher [1977] and Lane [1980]. These models assume that firms choose their quality level. However, when quality is endogenous, the presumption that early entrants can stake out preferred quality niches is less compelling.



628 DAVID REITMAN

The following proposition gives a sufficient condition for the equilibrium to be asymmetric:

Proposition 1. With I firms in the market, I > 3, the pure strategy equilibrium of the general model, if it exists and if F - '(I/I) < F- '(I - 1/I), is asymmetric.

The proof is given in the appendix. The sufficient condition requires that the customers on the margin of the highest and lowest Ith fractiles of the distribution have different types. In other words, the equilibrium will be asymmetric as long as the heterogeneity of types does not occur only in the tails of the distribution (when there are only two firms, all customers are in the lowest or highest fractile, and thus, with a vertical demand curve, the equilibrium is symmetric if it exists). The critical factor is whether, starting from symmetric strategies, a firm that changes its price faces the same type of marginal customer (who is indifferent between being served by the deviating firm and all others) regardless of whether it raises or lowers its price. If the type of the marginal customer differs depending on the direction of the price change, then the only possible equilibrium is asymmetric.

The following two examples with three firms illustrate the asymmetry of the equilibrium and the possibility of non-existence. The two examples differ in the strategies available to firms. In Example 1, firms are assumed to have symmetric, exogenously specified service capacities, and differentiate their quality levels solely through prices. In the following example, firms choose both their prices and their capacity levels.

Example 1: Differentiation through Prices Alone

Using the delay model, assume that there are only two type of customers: "impatient," with waiting cost CH, and ".patient," with waiting cost CL, where CH > CL. The number of customers of each type is assumed to satisfy the ex post condition that the marginal customer between the two lower priced firms has waiting cost CL, and the marginal customer between the two higher priced firms has waiting cost CH ( a 50-50 distribution works for all but extremely different values of CH and CL).7 Assume without loss of generality that the three firms are ordered such that Pi < P2 < p3. Thus firm 1 serves only patient customers, firm 3 serves only impatient customers, and firm 2 serves a mixture of both. Assume that each firm has capacity y. The balance equations representing the choice of the marginal customer between each pair of firms are

CLnl CLn2 Pl + L = P2 + 2

Alternatively, treat the distribution of customer waiting costs as arbitrary and continuous, and assume that CH and CL are the endogenously determined waiting costs of the two marginal customers after the market has obtained its equilibrium.




CHf2 CHfl3 P2 + =2 P3 +

2y ~2y

where n1 + n2 + n3 = D. Solving for the number of customers at each firm as a function of prices gives

D+ 4y ___

l1-3+ +3(P2-P1)+ 3(P3-P2) 3 CL 3CH

(4) n2 =D3C (P2-P1)+2 (P33-P2)

D 2y 4Y n3 = 3- (P2-P1)- (P3 -P2)

3 3CL 3cH

Differentiating these equations gives dni/dpi for each firm. Profits are II' = pini for i = 1, 2, 3; and the first order conditions are dHildpi =O ni + pidni/dpi = 0 for each i. The prices that solve the first-order conditions are

3CLnf1 3CLCHn2 3CHln3 Pi= ~ P =P3

-. 4y 2Y(CL + C) 4Y

Substituting these prices into the balance equations gives the only possible pure strategy distribution of customers and prices. The unique solution is

n, = - -, n~~~~~~~~~~~~ ~~~~~~~~~~~~~23 2(_____- 5 ( CH+CL 3 =3 K _ 2 H L) 3

D (CL(4CH + CL)) D ( CHCL

toy CH +CL ,J 22y C? +CL

D (CH(CH + 4CL)N P3

Oy CH+CL J

D2CL (4CH+CL 2 D6 ( CHCL

DCH (CH+ 4CL ) 2 73

75YV CH + CLJ

Prices, demands, and profits for a range of different type distributions are given in Table I. Note that, as long as customers are heterogeneous (CH # CL),

the only possible equilibrium is asymmetric. As CH increases, with CL held fixed, 7r3 becomes infinite as CH goes to infinity, while 7r1 and 72 remain finite. This implies that a Nash equilibrium cannot exist for all values of CH and CL:

as the ratio of waiting costs between patient and impatient customers increases, the profits obtained by the least congested firm will eventually induce all firms to attempt to have the highest price.

In order for this solution to be a Nash equilibrium, each firm must earn at



630 DAVID REITMAN

TABLE I

PRICES, NUMBERS OF CUSTOMERS, AND PROFITS WITH EXOGENOUS

CAPACITIES, Y = 1, D = 3, CL = 1, AND VARIOUS VALUES OF CH.

CH= 1 1.5 2 3 5 10

Pi 0.75 0.84 0.90 0.98 1.05 1.12

P2 0.75 0.90 1.00 1.13 1.25 1.36 p3 0.75 0.99 1.20 1.58 2.25 3.81

n, 1.00 1.12 1.20 1.30 1.40 1.49 n2 1.00 1.00 1.00 1.00 1.00 1.00

n3 1.00 0.88 0.80 0.70 0.60 0.51

[1 0.75 0.94 1.08 1.28 1.47 1.67 T12 0.75 0.90 1.00 1.13 1.25 1.36

r3 0.75 0.87 0.96 1.10 1.35 1.94

least as much profit as it would if it charged a different price. Under the solution to the necessary conditions, no firm would want to make a "small" change in its strategy that preserves the order of the firms' prices. But the price charged by each firm must be weakly preferred to all others, including those that switch the ordering of firms. Thus, each of six possible deviations-two for each of the three firms, since each firm can choose to be either the low, middle, or high priced firm, given the prices charged by the other two firms- must be checked. As it turns out, the middle firm will never (for any combination of CH and CL) want to undercut the low price firm, nor will the low price firm ever want to deviate to the middle. Both the low and the middle firms will deviate and charge the highest price if CH/CL is sufficiently large; the middle firm will deviate if the ratio exceeds 13.2, while the low firm will deviate if the ratio exceeds 13.4. Note that both of these imply an extremely heterogeneous population. Thus, for a moderate range of waiting costs, the asymmetric equilibrium exists.

The waiting time distribution attained in this example is less skewed than the efficient distribution. For example, the efficient allocation of customers to queues when half the customers have three times the unit waiting cost of the other half is to let one firm serve all the patient customers, and the other two firms split the impatient customers. In the equilibrium with CH = 3 and CL = 1, the lowest price firm gets only 43% of the customers. On the other hand, the equilibrium allocates waiting times better than if all firms charged a price equal to marginal cost and partially compensates for the loss of consumer surplus resulting from firms' power to set prices.8

IThat the low priced (and thus more desirable) firm is underutilized from an efficiency perspective appears at first to contradict the discussion of congestion externalities found in Pigou [1920] and Knight [1924]. It is true that the aggregate waiting time would be reduced if fewer customers were served by the low priced firm. However, for efficiency, we are concerned with aggregate waiting costs, rather than waiting times. In equilibrium, customers with lower waiting costs choose the low price firm, which therefore should be more crowded.




Example 2: Differentiation through Prices and Capacities

Consider the same market as in Example 1, but now suppose that firms choose their capacities. The cost of a unit of capacity is constant, and is equal to k. Customers again come in two types, with unit waiting costs CL and cH,

and firms again are ordered by their prices, Pl < P2 < p3. The balance equations and the first order conditions for prices are the same (except that yi replaces y) as in the previous example. There are also first order conditions for capacity, which are dfli/dyi = 0 pidni/dyi = k for each i. The first order conditions for each firm depend only on its own price and number of customers, as well as the capacities of all the firms. Thus the first order conditions can be solved to find the price and demand at each firm in terms of the capacities of all firms. The solution is pi= Y[k/2ai(Y-yi)]1/2 and ni= y[2kai/(Y-yi)]'12, where Y= Y1+Y2+Y3 and al = (Y-yl)/cL a2 = Y11CL+Y31CH, and a3 = (Y-Y3)/CH. Substituting these into the remaining equations, and letting r = cH/cL, the problem becomes

y1+2Y2+2Y3 =r 1/2 (ry1 + Y3 \Y3 ) I J ~~~~(Yl +Y2 +Y3) ?1 Y2+Y3 (ry1 + y3) (Y1 + Y3) ry,ry3

2y, + 2Y2 + Y3 7 1 \\1/2 (ryl +y3 1/2 = (Yl +Y2 + Y3) +

Y1 +Y2 (ryh +Y3)(Y1 +Y3) Y1 +Y3

r1/2y +rYr +Y3) 1/2 y +y1/2 r 3)1y+y33/ ~Y2+Y3 =Dyj2k)

The unique solution to the necessary conditions for various values of CH and CL is given in Table II. Once again, it is necessary to test whether these solutions are Nash equilibria. As it turns out, all but the last column are equilibria. When CH/CL = 10, the middle firm makes higher profits if it charges a higher price than the high priced firm, which upsets the equilibrium. As in the previous example, the solution is symmetric when all customers are identical. But for any other distribution of types, the firms spread out in price and congestion levels. As CH increases, holding CL constant, firm 1 captures an increasing share of the customers, while firm 3 increases its price dramatically relative to the other two firms. As is familiar from the product differentiation literature, firm 2's profits suffer from being caught in the middle. It is interesting to note that capacities in the solution do not differ greatly among the three firms. Essentially, the firms look alike in terms of size, but by setting different prices they attract widely varying congestion levels.

Under the assumptions of the model, the solution to the necessary conditions for equilibrium is unique (up to a permutation of the order of the firms). Thus, if any firm prefers to deviate from that solution, then no



632 DAVID REITMAN

TABLE II CAPACITIES, PRICES, NUMBERS OF CUSTOMERS, AND PROFITS FOR D = 3,

k = 0.5, CL = 1, AND VARIOUS VALUES OF CH.

CH 1 1.5 2 3 5 10

Yi 1.00 1.16 1.26 1.37 1.47 1.62 Y2 1.00 1.09 1.13 1.16 1.15 1.03 Y3 1.00 1.03 1.06 1.15 1.42 2.10

Pi 0.75 0.77 0.79 0.80 0.79 0.76 P2 0.75 0.66 0.85 0.88 0.90 0.91 p3 0.75 0.89 1.02 1.26 1.73 2.83

n1 1.00 1.16 1.26 1.37 1.47 1.62 n2 1.00 1.00 0.99 0.96 0.89 0.72 n3 1.00 0.84 0.75 0.67 0.64 0.66

H1 0.75 0.90 0.99 1.09 1.15 1.23 n2 0.75 0.81 0.84 0.85 0.80 0.66 rI3 0.75 0.75 0.77 0.84 1.10 1.88

equilibrium in pure strategies exists.9 In any case, the prices charged in equilibrium will never be symmetric, as long as the condition in Proposition 1 is met, or if there are customers on the margin of the market. If the asymmetric pure strategy equilibrium does not exist, the only alternative is that firms use a mixed strategy equilibrium.

There have been numerous papers explaining price variations in price competitive markets as mixed strategy equilibria (Shilony [1977], Varian [1980], and Sobel [1984] are examples). The possible existence of pure strategy equilibria with asymmetric prices is an appealing feature of congested markets, as it suggests that firms can maintain the same price/ quality niche from period to period. Furthermore, it is hard to justify the informational requirements of customer sorting if prices are fluctuating randomly each period. Nevertheless, there will always also be mixed strategy equilibria.'0 In fact, there is the potential for a much richer set of equilibria than is found in most of these dynamic pricing papers, because atoms in the mixed pricing strategy can not be ruled out, due to the continuity of profits with respect to prices. Some of the mixed strategy equilibria are symmetric, though of course the realization of prices from any mixed strategy equilibrium in any period will be asymmetric with positive probability. One symmetric,

9 The problem is the non quasi-concavity of profits; each firm would prefer to pick a price- quality niche on either side of a competitor than to imitate that competitor's strategy. This is discussed extensively in Dasgupta and Maskin [1986a, b].

" Theorem 6 in Dasgupta and Maskin [1986a] is sufficient for this result.




mixed strategy equilibrium with non-atomic strategies for example 1 is derived in the appendix.

IV. EQUILIBRIUM WITH A CONTINUUM OF FIRMS

In the previous section, we assumed that the number of firms in the market is fixed. But congestion induces positive profits that provide an incentive for entry into the market. In the long run, the number of firms is endogenous and is determined by the cost structure of production. In this section we consider one particular cost structure, which is that firms have vanishingly small fixed costs and constant marginal costs of production, leading to an infinite number of firms in the market." This cost structure is not too unreasonable an approximation for the kinds of markets (made-to-order products and some services) in which congestion is more likely to be a significant determininant of quality. The main result of this section is that with a continuum of firms, a pure strategy equilibrium exists, is asymmetric, and is efficient. In Proposition 2, this result is proved for firms that have symmetric, exogenously specified capacities. Then in Proposition 3, the same result is proved for endogenous capacities, but using the specific assumptions of the delay model.

Assume first that there is an infinite number of firms that each possess an equal share of the aggregate industry capacity. This aggregate capacity, L is given exogenously. Let N(i) be the cumulative number of customers served by the first (lowest priced) i firms. Throughout this section, N(i) will typically be written as Ni for ease of reading. Thus, firm i serves (Ni - -1) customers. Finally, define G(Ni) = F-'(Ni/D) to be the type of the customer on the margin between firms i and i + 1.

The equilibrium is efficient if no re-allocation of customers to firms increases their aggregate gross utility (excluding the price of service). To find the efficient allocation, we take the limit as the number of firms in the market increases. Let I be the number of firms in the market, and assume that, as the number of firms becomes infinite, the number of customers allocated to each firm becomes infinitesimal, and the cumulative number of customers and prices as a function of i become differentiable and strictly increasing in the limit. When Ni - -1 is small, the aggregate gross utility for all customers at firm i is approximately equal to the product of the utility of the most impatient customer at firm i and the total number of customers there, which is

(5) U T(i), ( Ll ))(Ni -i 1) L/II

1 Scotchmer [1985a] and, more generally, Mankiw and Whinston [1986] have shown in similar models that there will be a tendency for too many firms (relative to the efficient number) to enter the industry. On the other hand, stochastic effects may create additional economies of scale. For example, when queues are fluctuating randomly, the expected delay before service is reduced if capacity from several firms is combined in one firm with one large queue for service.



634 DAVID REITMAN

As the number of customers served by each firm goes to zero, this expression will be exact. Let s = i/I. Multiplying and dividing (5) by 1/I, then taking the limit as I - oo gives

U(G(Ns), T(Ns /L))N' ds

where s e (0, 1]. Aggregating over all firms gives the total gross utility in the market,

(6) |U(G(Ns), T(Ns'IL))Ns'ds

The optimal allocation of customers to firms is given by the function N(s) that maximizes (6), subject to the boundary conditions N(O) = 0 and N(1) = D. Let 4 = U(G(Ns), T(N8/L))Ns'. The Euler necessary condition (omitting arguments of functions to increase legibility) is

O= ) d _00

ONd s O LN')

(7) =--{ Uvt T'[N'] 2G' + Utt[T'] 2N'N"/L + Ut T"N'N" + 2Ut T'N"} L

This differential equation can be integrated once, becoming

Ut(G(Ns), T(N /L)) T'(N /L) [Nj ] 2 -

A (8)L L L

where A is an arbitrary positive constant. The optimal allocation of customers is found by integrating (8) again and solving with the two boundary conditions, provided that, in the solution, N > 0 and N' > 0.

Having found the efficient distribution of customers to firms, we can now compare it to the competitive solution. As the following proposition shows, the distribution that results in the competitive equilibrium is the efficient distribution. Moreover, the competitive equilibrium always exists. The proof is given in the appendix.

Proposition 2. The general model with a continuum of firms that have identical, exogenous capacities has a pure strategy equilibrium. The equilibrium induces the efficient allocation of customers to firms.

Example 3: Differentiation with a Continuum of Firms

As an example, suppose F(c) = c/b, for c E [0, b], and the utility and congestion functions are as given in the delay model. The solution to (8), using the boundary conditions, is N(s) = Ds213, which gives p(s) = (bD/3L)s'13 from




(A4) and n(s) = (2bD2/9L)ds from (A2). Aggregate industry profits are 2bD/9L > 0. Positive profits in the limit derive from the assumption that firms split a limited amount of capacity. If L increases as the number of firms increases, then industry profits disappear in the limit.

The model thus far is somewhat artificial because firms are forced to shrink as the number of firms grows. However, when capacity is chosen endogenously, the equilibrium turns out to be quite similar-as the number of firms in the industry grows, with a constant level of demand, each firm serves fewer customers, and therefore chooses to buy less capacity. But this is now a result of the model, rather than an assumption. The modeling cost of reintroducing capacity choice into the asymptotic model is that these results are derived only for the more specific delay model. The cost function for capacity is assumed to be linear, with marginal cost k. The results are unchanged if capacity costs are convex. However, the linear case is perhaps more significant, because there is no bias, as far as cost minimization goes, for firms to choose the same capacity level. Nevertheless, in equilibrium, all firms do have equal capacities. Let N(i) be defined as before, and let Y(i) be the cumulative capacity of the first (lowest priced) i firms in the market. Once again let s = i/I be the index of firms in the limit, s E (0, 1].

It is no longer possible to derive the equilibrium at the limit, as was done in the previous proposition. Profits for firm s are n, = p,N'ds-kY'ds, so the first order condition for capacity is 0 = psdN/d Y - k. With a continuum of firms, as a firm increases its capacity while holding its price constant, it picks up just enough customers to maintain the same level of congestion. That is, the change in capacity has no effect on congestion levels in the economy, due to the negligible size of each firm. Thus dNS'/d Y' = NS'/ YS, and the first order condition for capacity becomes pSN' = k Y'. This implies that firms earn zero profits in equilibrium, leading to an indeterminancy when analyzing the equilibrium at the limit: If firms earn zero profits, they are indifferent about their capacity level. Therefore it is necessary to look at the limit of the market equilibria, as the number of firms increases, rather than the equilibrium of the limit.

Lemma 1. In the delay model with a finite number of firms, the market clearing number of customers at firm i, given price and capacity choices at all firms, is given by

(9) ni = Y + 2yij=+ G(Nj ,) Y -2 G(N,1

The proof is given in the appendix. Using (9), the first-order condition for capacity at firm i, pidni/dyi - k = 0, becomes

(10) (YT-yi) ni k

Yi YT Pi



636 DAVID REITMAN

or, after rearranging,

(11l) yi(nEip) = YT(nip -kyi) In words, each firm sets its capacity so that its capacity times its revenues is proportional to its profits. To be an equilibrium in the limit, profits at each firm must be constant. If firms have different capacities in equilibrium, then those firms with larger capacities must have less revenues, if equation (11) is to be satisfied. Consequently, firms with larger capacities would have smaller profits and would prefer to decrease their capacity. Thus a necessary condition for equilibrium is that all firms choose equal capacities.

To verify that this solution is an equilibrium, first note that, when firms are infinitesimal, deviating from the equilibrium price is like assuming the role of another firm in the market, unless the price chosen is outside the range of those offered in equilibrium. But choosing a price other than those offered in equilibrium cannot improve profits since it induces an inefficiently low or high quality level. Thus to check for sufficiency, all that is necessary is to show that each firm prefers the equilibrium level of capacity. Associated with each firm s and price ps in equilibrium is a congestion level, ts, with ts = n,/2y, where y is the common equilibrium capacity level. When firm s chooses the equilibrium capacity level, it earns profits of fIS = psn- ky = (2pst, - k)y. Profits for each firm are zero, which implies that (2p,ts - k) = 0, since y > 0. Now if firm s changes its capacity strategy to A, it attracts the level of demand n that will maintain its quality level. Thus n' = 2yts, and profits from the new strategy are fl = 2pS9t S-k9 = (2psts-k)9 = 0. No combination of price and capacity will earn positive profits, and therefore the equilibrium exists. All that remains is to determine what the equilibrium level of capacity will be. Using (9) again, the first-order condition for price, ni + pidni/dpi = 0, becomes

(12) 2y+ [YT 1 i]ni YT L G(Ni) G(Ni- 1)j pi

Dividing (10) by (12) and solving for yi gives

I' (YT-yi)G(Ni- 1)G(Ni) 1/2

(13) y V= In 13 YI =2k [( YT- Yj) G(Ni) + Yi - 1 G(Ni - 1 )]

ni

In the limit as the number of firms becomes large, G(Ni 1) G(Ni), so (13) becomes

(14) Ys = ( kN)) s

A 01/2 = 2k( ds

in the limit, where A is the constant (determined by boundary conditions) from (8). Thus this analysis is consistent with the previous proposition; the




necessary condition on the distribution of customers among firms that is derived here in order for capacities to be equal is the same as that obtained when capacities are set equal exogenously. Solving (10) and (14) for prices in the limit gives

IG(Ni)k\ 1/2

Pi = 2 2

From Proposition 2, the distribution of customers is clearly optimal given the choice of capacities. Furthermore, it turns out that the choice of capacity is also optimal. More precisely, the efficient solution depends only on the aggregate level of capacity in the market and the requirement that there is an infinite number of firms with positive (density of) capacity. Any allocation of capacity among those firms will achieve the waiting cost minimizing solution with the appropriate allocation of customers to firms.

These results are proved by once again deriving the cost minimizing distribution of customers using the calculus of variations. Total waiting costs in the market (analogous to (6) with endogenous capacity) are

I1 [l G(Ns) [N ]2 (15) 2 - ds

This time let / = G(Nj) [Nfl 2/Y.'. The Euler condition is

&/ d OW EONds ( ON',!

G_(N) [N'_]2 (G(N)N" G'(N) [N']2 G(N)N'Y"] (16) = ~, -2 {l , + -y [y,]2}'

Simplifying (16) and rewriting gives

G'(N)N' 2N" 2Y" T+ =

G(N) N' Y'

The solution of this differential equation is

(17) G(N)=[N']2 BY'

where B is a constant of integration. Integrating the left hand side of (17) over the range of s gives twice the total waiting costs in the market. Integrating the right side of (17) over s gives a constant times Y(1), the total industry capacity. Thus, with the optimal choice of N, aggregate waiting costs are proportional to the total capacity, regardless of the distribution of capacity (as long as the density of capacity is positive for an infinite number of firms). Finding the optimal level of capacity now becomes a simple optimization problem. Let L = Y(1) be the aggregate amount of capacity. Since the allocation of total



638 DAVID REITMAN

capacity among firms does not affect the optimum, suppose that firms have equal capacity, with Y'(s) = L. Then, substituting in (13), aggregate waiting costs equal A/2L, where A is once again the constant of integration from (8). Total costs from waiting and capacity are A/2L + kL, which is minimized at L = (A/2k)"12. This is exactly the level of total capacity derived in (14) for the competitive analysis. Thus the asymptotic equilibrium not only generates prices that lead to the efficient allocation of customers to firms, but also generates the efficient level of industry capacity. Of course, this second sense of efficiency hinges on the assumption of a linear cost of capacity.

These results are summarized in the following proposition:

Proposition 3. In the delay model with a continuum of firms and simultaneous choice of price and capacity, the equilibrium exists and is unique. Firms choose equal capacity levels, aggregate capacity is finite, and each firm earns zero profits. The equilibrium level of capacity and the allocation of customers to firms is efficient.

There is a third sense in which the equilibrium is efficient. Consider the (non-atomic) set of customers with unit waiting costs c. In the limit, they are served by one firm, say s, charging price ps = (ck/2) 1/2. This price is efficient in the sense that it equals the marginal cost of serving another customer of type c, taking into account the congestion externality imposed by an extra customer on those already being served. To see this, suppose there is a demand curve for customers of type c, so that total demand is determined by the price and congestion level at firm s. If n, customers are served at firm s, with capacity y,, the total waiting plus capacity cost is cn/212y, + kys, which is minimized at ys = (c/2k)"12ns. At the optimal level of capacity, the marginal cost of serving another customer is cn,y, = (2kc)'12. Meanwhile the cost of service for the marginal customer of type c is the price plus the waiting cost, ps + cnsl2ys = (2kc) 12, which equals the marginal cost of service. Thus the price charged for service is exactly the efficient price, which insures that customers will choose to be served if and only if their benefit exceeds the social marginal cost.

Finally, although the distribution of customers to firms is efficient, in the sense that no other allocation would lead to lower aggregate waiting costs, there is still a degree of inefficiency since customers with high waiting costs are being served by high price firms while customers with low waiting costs are being served by low price firms. Full efficiency requires devoting all the market resources to serving impatient customers before any patient customers are served. For example, the aggregate waiting costs resulting after endogenous quality differentiation in Example 3 are 2bD/9L, where L is the total level of capacity in the industry. It is easy to compute what the aggregate waiting costs would be if customers are served by a single monopolist with capacity L, instead of by an infinite number of small, price competitive firms. This will generate the same aggregate waiting costs as if each infinitesimal




firm served the same number of customers. If the monopolist serves customers in random order, the aggregate waiting costs are bD/4L, while if they are served in the efficient order (which corresponds to the allocation that results if each firm offers complete priority service), aggregate waiting costs are bD/6L. Endogenous quality differentiation is an improvement over serving customers randomly, but there are still considerable improvements to be made from serving customers in an efficient order. Thus, there are still losses from the absence of a spot market for places in line, which can only be recovered through the use of priority service.'2

V. DISCUSSION

This model of congested markets, defined as markets for products with congestion-induced quality attributes, has been used to show that, when customers have heterogeneous preferences for quality, the competitive equilibrium itself will induce quality differentiation between firms through the prices and capacities selected by firms. Thus the losses in allocative efficiency from the absence of a spot market for priority service are to some extent recaptured by endogenous quality differentiation in the competitive equilibrium. This implies that the market for any product whose sale involves some aspect of congestion will be characterized by a range of prices. In some cases, there will be a pure strategy, asymmetric equilibrium; otherwise the only equilibria will involve mixed strategies. With an infinite number of firms in the market, under the conditions described in the previous section, a pure strategy equilibrium always exists and efficiently allocates customers to firms. Furthermore, in the specific model used to study capacity choice with an infinite number of firms, the aggregate level of capacity chosen is optimal, and prices for service are equal to the difference between the social and private cost of receiving service.

With many firms in the market, the only equilibrium has firms choosing equal service capacity levels and differentiating solely through prices. Clearly the role of capacity becomes more prominent when constraints on the number of customers served are incorporated. But the implication remains, for example, that convenience stores do not charge high prices because they are convenient. Rather, they are convenient because they charge high prices. Moreover, the high priced, high quality niche will surface regardless of

12Judging solely from this example, the gains from differentiation seem rather trivial- aggregate waiting costs are reduced by only about eleven percent. However, differentiation becomes increasingly important as the congestion costs of customers with high types, relative to the rest of the distribution, increase. If, in Example 3, the type distribution is F(c) = (c/b)', then as a approaches zero, which implies that there are a few very impatient customers among lots of patient customers, the ratio of aggregate waiting costs for the symmetric solution to those for the competitive solution becomes infinite, while in the limit, the competitive solution has twice the aggregate waiting costs of the perfect priority allocation.



640 DAVID REITMAN

additional features involving location, store size, and other factors. Asymmetric prices are critical in determining the quality niches of competing firms.

It is interesting to compare these results to those in related kinds of markets. The outcome appears similar to markets for products with directly chosen product quality (see Prescott and Visscher [1977], Shaked and Sutton [1982], and Moorthy [1982]). Firms all choose different quality levels, and firms offering higher quality also charge higher prices. However, there are several important differences. First, with directly chosen quality, a pure strategy equilibrium exists only if firms precommit to their quality level before engaging in price competition. If firms are free to vary both their quality level and their price, then, as Novshek [1980] has shown, no pure strategy equilibrium exists. In contrast, a pure strategy equilibrium may exist in congested markets, even though firms are free to vary both their price and their level of service capacity.

The reason firms differentiate their directly chosen quality levels is that, if two firms sold identical products, price competition would drive prices down to marginal cost-firms differentiate to avoid cutthroat price competition. In a product differentiation model, De Palma, Ginsburgh, Papageorgiou, and Thisse [1985] have shown that, if customers have idiosyncratic preferences among brands, then firms may in equilibrium choose identical brands. Idiosyncratic preferences make the response in demand to changes in price continuous, even if the brands are identical. The same phenomenon occurs in congested markets; profits are continuous in prices, even when firms have identical quality levels. Firms could choose to charge the same price, offer the same quality level, and still make positive profits. Nevertheless, profits increase if firms charge different prices, inducing quality differentiation and taking advantage of the resulting gains in efficiency.

As more firms enter the market for a product with directly chosen product quality, they fill the range of possible quality levels, and price competition drives price down to marginal cost. In one sense, this differs from congested markets. With an infinite number of firms in the market, prices are above the marginal cost of production; as modeled here, all firms have the same marginal costs of production, and all charge different prices in equilibrium. However, prices are equal to social marginal cost, which is the cost of production plus the cost of preserving the same quality level when serving additional customers.

The products received by customers are differentiated; customers buying at different firms receive different quality levels. Nevertheless, there is a real sense in which the products sold are homogeneous. One way to express the homogeneity is that any firm could replicate the product of a competitor just by changing its price. But the market for any homogeneous product will become differentiated if congestion matters at all to customers, in any form it may take, and customers differ in their distaste for congestion. Thus this study




of congested markets offers another reason why the market for a seemingly homogeneous product may be characterized by price dispersion.

Perhaps the most frequent explanation for price dispersion in markets for homogeneous products in the literature is search costs. There are two interesting differences between search cost models and congested markets. First, with a finite number of firms, the search cost equilibrium generally requires mixed strategies, while in congested markets, with some distributions of customer preferences, firms may use pure strategies. Thus firms in congested markets can potentially fill the same price-quality niche from period to period (and earn distinct levels of profit), if the market equilibrium used preserves the assignment of firms to niches each period. Furthermore, in search models, any price dispersion is socially costly-if all firms charged the same price, customers would buy from the first store, and would not waste time searching for the lowest price. In congested markets, however, the price dispersion is socially beneficial, as it induces customers to sort efficiently according to their costs from congestion.

DAVID REITMAN ACCEPTED APRIL 1991

Department of Economics, Ohio State University, Columbus, Ohio 43210-1172, USA.

APPENDIX

Proof of Proposition 1:

Suppose, by way of contradiction, that there is a symmetric equilibrium. The goal will be to show that, whatever the common level of price and capacity is, each firm would have an incentive to either raise or lower its price from the equilibrium level. When firms use a symmetric solution, customers are indifferent about which firm they choose, and so they are no longer necessarily partitioned according to their waiting cost among the various firms. However, if one firm did lower its price from the equilibrium level, it would attract only the most patient customers, and similarly, if that firm raised its price, it would attract only the most impatient customers. Thus, to simplify the argument, it is convenient to assume that customers do sort among the firms in the symmetric equilibrium, with firm 1 serving the most patient Ith fractile of the distribution and firm I serving the most impatient Ith fractile, and only check if either firm 1 would lower its price or firm I would raise its price.

Let p and y be the equilibrium levels of price and capacity. If firm 1 lowers its price, it attracts customers until the new balance equation is satisfied:

U (v, T (F(v1))) Pi = U (v T (( -F(vl))D ))



642 DAVID REITMAN

Differentiating the balance equation with respect to Pi gives

dvl [u ( T ( F(vY )) - Uv (I -F(vl)D))

+ Df(vl) Ut(v T(( ,))) T' ((v1))

+11 vl T'

When Pi = p, firm 1 gets the first Ith fractile of customers, or F(c1) = 1/I. Thus, evaluating this derivative at Pi = P,

dv1l (I-l)y

dp1 P1 = p IDf(vl) Ut(vl, T(D/Iy)) T'(D/Iy)

Profits are given by 1j = p1F(v1)D-K(y1), and so d1ll/dp1 = F(v1)D+ p1Df(v1)dv1/dp1. A necessary condition for the symmetric equilibrium is that firm 1 does not prefer to lower its price; that is, d1ll/dpl evaluated at Pi = p is non-negative. Solving for this derivative,

dP I D= -+pDf(v1)dvl dpl pi P I -dpl P = P

D (I-1)py

I 1Ut(v1, T(D/Iy)) T'(D/Iy) gives a necessary condition for the existence of the symmetric equilibrium:

dP|1 = -

Ut(v1, T(D/Iy)) >D (T-(DIpy dpl pi= pDTDIy

Similarly, if firm I raises its price, it loses customers until the following balance equation is satisfied:

U(VI-l,T( (-PIi)D))-P = U(V T ( F(v i)D))

Differentiating the balance equation by PI and solving at p, = p gives

dv__l -(I-l)y

dp1 Pi = P IDf (v 1 ) Ut(v1 - 1, T(D/Iy)) T'(D/Iy)

Profits for firm I are HI1 = p,(I - F(vI - 1))D, and the derivative of profits with respect to price is d111/dp1 = (1 -F(v,_ 1))D -pDf (vI 1)dv,_ 1/dp1. A necessary condition for the symmetric equilibrium is that firm I does not prefer to raise its price, or d11I/dpI evaluated at p, = p is non-positive. The derivative is

df, D =_-pDf(v,1 -) -

dpI PI=P I dpI PI=P D (I-1)py I 1Ut(v,_ 1, T(D/Iy)) T'(D/Iy)

so a second necessary condition for the existence of the symmetric equilibrium is

d < ? -Ut(v - 1, T(D/Iy)) < DT(D /IY)

dp1 PI=pDT(ly




Invoking the mixed partial derivative assumption U,t < 0, these two necessary conditions can only both be true if v1 = v- 1, which means that the customers in all but the lowest and highest Ith fractiles have the same type. If this middle group of customers is heterogeneous, the necessary conditions are violated, and therefore the symmetric equilibrium does not exist. I

A Symmetric, Mixed Strategy Equilibrium for Example 1:

Since firms can no longer be ordered by the price they charge in equilibrium, let pi be the price charged by firm i, and define Il"(pi, pi, Pk) to be the profits of the firm with the nth lowest price when the lowest price is pi, the next lowest is pj, and the highest is Pk-

Since the firms are symmetrical, I'1(P1,P2,P3) = n1(P1,P3,P2), and a similar expression holds for 1f2 and I13. Assume that each firm uses the mixed strategy determined by the distribution function F(p), which is conjectured to be differentiable. Let p be the largest price such that F(p) = 0, and let p be the smallest price such that F(pm-= ln Finally, let Hi(pi) be the pro-its of firm i using price pi, when the other two firms use the equilibrium strategy. Then

fll(pl) =2J P1 II'(p1p2,p3)dI(p2)dF(p3)

+ 2 J P 113(P2, p3, p1)dF(p2)dI(P3)

Jp P

fPP3 (D 4y 2y =2 i P Pi {-+ 3c (P2-P1)+ 3c (P3-P2)} dF(p2)dF(p3)

+21 J Pi - -(Pi-P2) - (P3 Pp)}dF(P2)dr(P3)

= P P3 Dc2y 4y + 2 Pi- (P3 - P2)- (Pl -P3) dF(P2)dF(P3).

3 3L3CH

P3 CL 4y P

+ - y pl(p)dF(p) \-3CH 3CL/ J,

(8y 4y Is yP 4y (Pi

A 3CH JP PdF(p) + 3c PdF(p)

The mixed strategy F(-) must be chosen so that profits are constant for all Pt in the range p < Pt - p. Taking the derivative of the profit expression, it can be shown that F must be of the form F(p) = B - A/p2. The constants A and B in general will depend on p and p-; however, p and p can be written in terms of A and B using the boundary conditions

A F(p) = B-- = 0 _ p2



644 DAVID REITMAN

and

A F(p)= B-- = 1

-p2

Solving these equations gives

CH9D 2C 2C2(CH- F(p) = CH D-H-CL) r CH-CL 64y2(C 3/2 -c 32)2p2

for 3D(cH/2 + C 1/2)'1/2c 3D(cA'2 + C 1'2)CHC 1/2

8y(cH +c 2cH 12 + CL) 8y(CH + CA'2c 1/2 + CL)

Thus firms tend to use lower prices more than higher prices, and the range increases as CH/CL increases. Expected profits are given by

3D2(C 1/2 + c 1/2)2cHC

1yc+ /2c 1/2 + )2- 16y(CH + C11 CL + CL)2

As might be expected, these profits are smaller than those of the lowest priced firm in the asymmetric pure strategy equilibrium, larger than those of the middle firm, larger than the profits of the highest price firm when CH/CL is relatively small (less than about 4.2) and smaller than those of the highest price firm when CH/CL is relatively large.

Proof of Proposition 2:

When there are an infinite number of firms, the set of balance equations is replaced by a differential equation involving prices and cumulative demand. The differential equation is found by taking the limit of a representative balance equation as the number of firms becomes infinite. With a finite number of firms, the balance equation between firm i and firm i + 1 is

U (G(N,), T (Ni - N - 1) )-pi = U (G(N ), T N(iV

1 l- Ni) pi+

Dividing both sides by 1/I gives

No

+ 1 _ Pi UG(Ni), T (Ni +

N

i) ))-U (G(N ), T

(Ni-j- ))) 1/I 1/I

Now defining s = i/I and taking the limit as I - oo gives

li U(G(N,),

T(N'+ 1/I/L)) - U(G(N,), T(Ns'/L))

(Al) = Ut(G(Ns), T(NS'/L)) T'(NS'/L)N "/L

Meanwhile, profits for firm i are

Ji= pi(Ni-N-=1) (A2) = PSN'ds

after taking limits. The first-order condition for profit maximization is

drI dN' S = N S

dps S + S dps




N"t

PS

Setting this equal to zero and solving gives

-pN, (A3) Ps=N'

Now, solving (Al) and (A3) gives

(A4) Ps =-Ut(G(N,), T(NS'/L)) T'(N /L)N'I/L

Differentiating,

(A5) PS, =- f U,tT'[N'] 2G' + Utt[T'] 2N'N"/L + Ut T"N'N" + UtT'N"} L

Finally, solving (Al) and (A5) gives

- 2Ut T'N" = Uvt T'[N'] 2G' + Utt[T'] 2N'N"/L + Ut T"N'N"

which is the same as (7). Thus the allocation of customers to firms resulting from competition is the same as the optimal distribution. Substituting (A4) into (A2) gives Js =-Ut(G(Ns), T(N5')) T'(N ) [N ] 2ds/L, which from (8) is constant for all s.

Each firm is infinitesimal and has no impact on the distribution of customers if it alters its price. Furthermore, if a firm deviates from the equilibrium and charges the same price as another firm, it receives the same demand and profits as that firm. Since all firms make the same profits, no firm has any incentive to deviate. Therefore, the asymmetric equilibrium exists. U

Proof of Lemma 1:

Let I, N(-), Y(-), and G( ) be defined as before, and define nj = N(j)-N(j-1) and yj = Y(j) - Y(j - 1). The balance equation between firm j-I and firm j is

G(Nj_1 ) nj -1 G(Nj _ 1 ) nj PJ1 i+ =Pj+ 2y

2yj- 1 2yj

Rearranging,

(A6) 2y=(pj--P) nj- nj-_ G(Nj 1 ' yj-1i

Define xj = (YI - Yj -)/yj. Multiplying both sides of (A6) by xj, then summing over j > 1 gives

' 2(YI-Yj_)(pj1-pj) I

n_ njFi

- Z nY-(YI-Y1) --

=D-YI n1 Y1

Solving for n1/y, gives

n, D I (pj-pj-1) 2 I Yj l(Pj-Pj-1)

Y1 YI i =2 G(Nj_- 1) YI j=2 G(Nj_- 1)



646 DAVID REITMAN

Now dividing (A7) by yj and summing over 2 j < i gives

(A8) ? (2 - ) Yi Yi j=2 G(Nj 1)

Therefore, substituting (A7) into (A8)

(A9) ni = Dyi +2Yi E )2y, 'j-2 (N ,)

which proves the lemma.

REFERENCES

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