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Durable Services Monopolists Do Better than Durable Goods Monopolists Author(s): John Spicer and Dan Bernhardt Source: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 30, No. 4a (Nov., 1997), pp. 975-990 Published by: Wiley on behalf of the Canadian Economics Association Stable URL: http://www.jstor.org/stable/136282 . Accessed: 12/06/2014 18:39 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Wiley and Canadian Economics Association are collaborating with JSTOR to digitize, preserve and extend access to The Canadian Journal of Economics / Revue canadienne d'Economique. http://www.jstor.org This content downloaded from 185.44.78.76 on Thu, 12 Jun 2014 18:39:42 PM All use subject to JSTOR Terms and Conditions

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Durable Services Monopolists Do Better than Durable Goods MonopolistsAuthor(s): John Spicer and Dan BernhardtSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 30, No. 4a(Nov., 1997), pp. 975-990Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/136282 .

Accessed: 12/06/2014 18:39

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

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Wiley and Canadian Economics Association are collaborating with JSTOR to digitize, preserve and extendaccess to The Canadian Journal of Economics / Revue canadienne d'Economique.

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Page 2: Durable Services Monopolists Do Better than Durable Goods Monopolists

Durable services monopolists do better than durable goods monopolists J O H N S P I C E R Queen's University D A N B E R N H A R D T Queen's University and University of Illinois at Urbana Champaign

Abstract. We consider a monopolist who can commit to a price path when selling a good for which individual consumers demrand at most one unit. Only if the monopolist can also commit to destroying unpurchased stock and the timing of stock release, can she earn more than a static monopolist. This additional commitment power has empirical relevance for a wide range of goods that we term 'durable services.' The inability to store output facilitates price discrimination by a durable services monopolist by permitting the credible threat of rationing. We detail sufficient conditions for a durable services monopolist to produce more than a static or durable goods monopolist produces.

Les monopoleurs font mieux dans les services durables que dans les biens durables. Les auteurs considerent le cas d'un monopoleur qui peut s'engager 'a maintenir un sentier de prix quand il vend un bien pour lecluel la demande des consommateurs individuels est au plus une unit6. C'est seulement si le monopoleur s'engage 'a d6truire les inventaires non- achet6s et ce a un moment pr6cis, qu'il peut gagner plus que le monopoleur en statique. Ce pouvoir d'engagement additionnnel a de la pertinence sur le plan empirique pour tout un 6ventail de produits qu'on appelle 'services durables.' L'incapacit6 a stocker le produit facilite la discrimination par les prix pour le monopoleur qui produit des 'services durables' enlui permettant de brandir une menace cr6dible de rationnement. Les auteurs d6finissent les conditions suffisantes pour qu'un monopoleur de services durables produise plus qu'un monopoleur en statique ou un monopoleur de biens durables.

1. INTRODUCTION

Consider a WordPerfect tutorial and a computer. To the consumer, both are goods for which a single purchase will often suffice. The lessons one learns from the tutorial can be applied forever without having to take another course; and the com- puter, once purchased, can be 'consumed' for a long period of time. By appropriate discounting, it is possible to express the lifetime utility a consumer will get from purchasing either of these two goods in period t. Sellers of either good, therefore,

Canadian Journal of Economics Revue canadienne d'Economique, XXX, No. 4a November novembre 1997. Printed in Canada Imprime au Canada

0008-4085 / 97 / 975-90 $1.50 ? Canadian Economics Association

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find themselves providing their own intertemporal competition even in the absence of rival producers.

However, an important difference between these two goods exists on the pro- duction side. In the case of computers, what the seller fails to sell today will be added to inventory, and another attempt at selling it can be made tomorrow. In contrast, if the consumer does not take the tutorial today, the monopolist may not be able to offer an extra lesson on the following day; he cannot store the tutorial. Consequently, only the computer meets the seller's criteria for a durable good. This difference has implications for the ability of a monopolist to commit. In particular, for goods and services that cannot easily be stored, the seller can credibly commit to a one period 'take-it-or-leave-it' price. If consumers decide to 'leave-it,' the good will not be added to the inventory and the possibility of rationing appears.

There are many goods and services that have the feature that the seller can credibly commit to not intertemporally storing unsold units and the buyer demands only a single unit (or is satiated for some long period of time). Many services often require individual attention when serving the customer and are demanded only once. For example, a white-water raft ride will have a daily capacity constraint determined by the number of rafts and guides available; and the thrill of the first ride may well satiate a consumer's demand for rides.' Other examples include ear piercing and tatooing (Dennis Rodman, excepting), hair cuts, many medical procedures (hip replacements, dentures, etc.), and artistic endeavours, such as concerts and theatre.

In this paper we compare the pricing behaviour of a monopolist who sells goods that she can store with the pricing behaviour of a seller who cannot intertempo- rally store her product. We refer to the former as a 'durable goods monopolist' and the latter as a 'durable services monopolist.' This labelling is meant to be only suggestive, since, for example, the hip replacement would be considered a durable service. Also, if the monopolist can make a credible commitment that limits in- ventory, perhaps a binding pledge to denate unsold stock to a charity, then it, too, would be classified as a durable service.

We allow each type of monopolist credibly to limit total output. That is, the seller can somehow put in place a significant cost that will convince consumers that she really will keep her promise to restrict total output. For a service such as a computing tutorial, access to computers can limit how many (effective) tutorials can be given. In the durable goods case, the oft-cited example of how this commitment can be made credible is an artist publically destroying his lithograph. Nevertheless, we go out of our way to make it difficult for the monopolist to extract additional profits from her commitment powers. We assume that consumers are arbitrarily patient. As a consequence the monopolist can entice consumers to pay higher prices only by the threat of rationing.

Bagnoli, Salant, and Swierzbinski (1989) allow a durable goods monopolist to make prices conditional on the history of sales. They derive conditions when there are a finite number of consumers that permit a durable goods monopolist to do better

1 We thank Curtis Eaton for this example and more generally for focusing our attention on the distinction between what we term 'durable goods' and 'durable services.'

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than merely setting price equal to the valuation of the marginal consumer. They show that for high enough discount factors there is a subgame perfect equilibrium in which the firm does better by playing a 'Pacman' strategy. The monopolist makes a myopic 'take-it-or-leaveit' offer in each period equal to the valuation of the highest remaining consumer, and consumers adopt a 'get-it-while-you-can' strategy. For discount factors approaching one the profits tend to those of a perfectly discriminating monopolist.

Al-Najjar (1992) shows that the optimality of this 'Pacman' strategy is poten- tially undermined if the monopolist has any uncertainty about consumer valuations: it is no longer necessarily optimal for the monopolist to make a take-it-or-leave-it offer, since with positive probability along the equilibrium path, no sale will be made. Then, since the monopolist would anticipate the possibility that no con- sumer had the conjectured valuation, she will lower her price, and the equilibrium unravels. Such a problem is not encountered in our setting, although the durable services monopolist does adopt a 'Pacman' strategy, picking of the high-valuation consumers first.

Despite stacking the deck against the monopolist by assuming arbitrarily patient consumers, we show that a durable services monopolist able to commilt to output can earn greater profits than those she would obtain were she always to charge the static monopoly price. We show how the monopolist's profit maximization problem can be formalized as a simple programming problem, and then we characterize its solution. This solution features a declining sequence of prices and the monopolist's selling a single unit in each periodJ. The effect of the pricing commitment is to con- vince high-valuation consumers to purchase the service early at higher prices, rather than risk being rational if they try holding out for a lower price. If a high-valuation consumer fails to purchase at the high price, then there is an over-subscription for the service in the next period at the lower price, so the high-valuation customer is no longer certain to be served. Indeed, this over-subscription will carry on as he competes with other higher-valuation consumers at future dates.

We contrast this result with that obtained when the monopolist is selling a durable good and cannot destroy unsold stock from her inventory. This has a dra- matic effect: she is unable to extract greater profits than a 'static' monopolist. The threat to ration high valuation consumers loses all bite. This is in keeping with the findings of Stokey (1979). In her model Stokey assumes that the monopolist can commit to a price path,2 although pricing decisions occur in continuous rather than discrete time. The product is introduced at date To and sold for a finite length of time unit T1, an exogenously given date known to all agents at the commencement of the game. To focus on price discrimination as a cause for sales of a durable good occurring at different dates and at different prices, Stokey initially assumes

2 Ausubel and Deneckere (1989) justify the assumption of commitment to a pricing sequence in the 'no-gap case' where there is non-negligible set of consumers with valuations at or below marginal cost. They show the existence of equilibria in which the monopolist selling a durable good approximately earns static monopoly profits by lowering price over time at an arbitrarily slow rate.

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no production costs. For a wide class of utility functions, she finds that the op- timal pricing decision of the monpolist entails all sales being made instantaneously. This result continues to hold if production costs are allowed to decline over time, provided they decline slowly enough.

Two other authors that look at rationing as a means of increasing monopoly profits are Ferguson (1994) and van Cayseele (1991). Van Cayseels shows that if lower-valuation consumers can be identified from the pool of willing purchases at a given price and if the monopolist serves these low-valuation consumers first (threatening higher valuation would-be purchasers with rationing), then intertem- poral price discrimination may be feasible even though high-valuation consumers rationally foresee future price drops. In Ferguson's paper, the monopolist uses shortages as a means for inducing buyers to self-select. Those with high valua- tions, to ensure access, are willing to pay a higher price or buy the good as part of a bundle, while lower-valuation consumers are willing to risk being rationed while paying the base price. In a similar spirit, Donaldson and Eaton (1981) use time as a means of discriminating between consumers. A monopolist separates the market into two sub markets, offering a (low-money, high-time) price to the long-suffering, and a (high-money, low-time) price to the impatient. The monopo- list does not need to identify an individual's group membership, since consumers self-select their preferred pricing bundle.

Our paper is most in keeping with that of Ferguson, with consumers' self- selecting the date at which to purchase. In the case of a durable service, high- valuation consumers pay a high price early on to avoid the threat of being rationed that accompanies a decision to wait for lower subsequent prices.

In section II of the paper the model is outlined. The monopolist's actions when she produces a durable good are characterized in section III. In section IV we demonstrate that a durable services monopolist is able to earn greater profits; we characterize the monopolist's actions and derive conditions under which the mo- nopolist will serve at least as many customers as a static monopolist. Conclusions are offered in section V. All proofs are left to the appendix.

11. THE MODEL

There is a single profit-maximizing monopolist who sells a good or service that can be produced at constant marginal cost, c > 0. The monopolist sells the good in periods t = 1, 2, .... To simplify the notation we refer to prices, pt, as net of the monopolist's marginal cost. Hence, it is possible that the price may be negative, although the gross price is always positive.

Mindful of the results of Bagnoli, Salant, and Swierzbinski (1989) and Levine and Pesendorfer (1995),3 we assume that the monopolist faces demands from a

3 These papers show how results may differ significantly when there is a large but finite number of agents, and when there is an infinite number of agents. Our analysis holds only when there are a finite number of positive valuation consumers, because it is not possible in our model selectively to ration agents of measure zero.

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finite number of positive net reserve valuation consumers. Consumer l's valuation of the good, net of the marginal cost c > 0, is denoted vl. There are L consumers with positive reserve valuations, ordered without loss of generality such that v1 > V2 > ... > VI > *-- > VL > 0. It is assumed that a consumer's valuation of a second unit of the good is zero; no one demands more than one unit of the good.

Denote the initial distribution of net valuations by Q1. The monopolist knows this initial distribution of net valuations,4 but she is unable to determine which consumer has which valuation, preventing her from discriminating against a consumer who seeks to make a purchase. Therefore if, in any period t, n consumers seek a purchase when there are only q goods available, where q < n, the probability any given potential consumer makes a purchase in that period is given by q/n; the probability is independent of the consumer's valuation. One can think of this probability as reflecting a first-come, first-served approach to sales in any period, where arrival time within a period is randomly determined by nature. Let Qt be the distribution of net valuations of unserved consumers at the beginning of period t. To simplify arguments we assume that consumers know Qt, although equilibrium outcomes are unaffected if consumers only know Ql.

Let Pt denote the price charged by the monopolist at each date, t, and let qt > 0 represent the quantity of the good the monopolist makes available to the market at period t. For the durable goods monopolist we assume that once the stock is delivered, it remains available until sold. A strategy for the monopolist is a vector, S = {pt, qt, }l-1. In the equilibria we outline, after a number of periods the price is set arbitrarily high and no more sales occur. To motivate this, we note that the same results would accrue if instead we allowed the monopolist to credibly commit to cease operations at a certain date.

The seller and all buyers have lexicographic preferences with regard to time and expected pay-off; they always prefer a high expected pay-off to a low expected pay-off, but if expected pay-offs in two periods are the same, then an agent prefers receiving the pay-off sooner rather than later. So, too, if Pt > v Vt, then if Pt = v, an unfulfilled consumer with valuation v attempts to make a purchase at t. In this sense, consumers and the monopolist effectively act as arbitrarily patient, risk- neutral agents.

The monopolist chooses S to maximize profits

00

II- >Ptst, t=I

where st represents sales in periods t. Given the sequence of prices and stock availability to which the monopolist has

committed and the past purchases by other consumers, each consumer 1 who has yet to purchase must decide whether to attempt to purchase at each period t. A pure strategy for a consumer 1 is a vector of decision rules, hl(S, Et, Qt) -+ {0, 1}.

4 This is not crucial to the analysis.

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If an attempt to purchase is made, h' = 1; otherwise h' = 0. Et details whether consumer 1 has made a purchase by period t. Clearly, if Et = 1, then t+ - land hti = ?, i > 0. A consumer's period pay-off is U1 = v -Pt if a purchase is made in period t and is zero if the consumer fails to make a purchase. Off the equilibrium path analysis requires that we consider expected period pay-offs because for certain consumers there may be a positive probability of not being served in the period(s) in which they seek to make a purchase.

111. DURABLE GOODS MONOPOLIST

As a basis for comparison we first consider a durable goods monopolist, a seller unable to commit to destroying any stock once it has been produced. However, the monopolist can commit to permanently ceasing production. We verify that the monopolist cannot earn greater profits than those of a static monopolist by probabilistically rationing consumers.

Given the time preferences of consumers, rationing can occur only in one period, and once it has occurred there can be no more sales. Consequently, if the monopolist wants consumer 1 to purchase at a high price to avoid rationing, she has to set the same price for the 1 - 1 higher-valuation consumers even though they would have been willing to pay an even higher price. This is because if consumer 1 is going to be served with probability one in period t, then there is no incentive for any consumer to pay a price greater than Pt in any period before t. Therefore, only in period one can there be purchases at a price higher than VN, where N is the total number of goods the monopolist ships to the market.

In period two, the monopolist has to set P2 < vN+1 if she is to make the threat of rationing credible, the necessary condition for consumers to be persuaded to purchase at a high price in period one. The second-period price must be set sufficiently low that any potential consumer not purchasing in the first period faces the possibility of being rationed.

Let N* be the quantity sold and fI* be the profits earned by a 'static monopolist,' a seller who sells stock only in the first period. N* is the solution to maxN NVN.

PROPOSITION 1. In the subgame perfect equilibrium, the durable goods monopolist sells N* goods in the first period. There are no sales in subsequent periods.

This result is similar to Stokey's. An immediate corollary is that the durable goods monopolist earns the same profits as a static monopolist.

COROLLARY 1. A durable goods monopolist with the powers to set total quantity and commit to a price path makes the same profits as a static monopolist, serving the N* highest-valuation consumers in the first period at price vN*. At each future date, pt > VN*.

The monopolist is unable to do better than a static monopolist. To do so would require sales in two periods, with a price greater than vN* in period one. However,

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profits are a linear function of the proportion of consumers still seeking a purchase in period two. This linearity means that the proportion of those seeking a purchase in period two who are served should be either zero or one. If everyone is to be served in period two, then no one would buy in period one when p, > P2, but we know that the monopolist's preference for earlier sales ensure that she would prefer to sell in period one.

If pt > VN*, t > 1, the N* highest-valuation consumers will pay any price up to, and including, their valuation in period one. Thus, the problem for the monopolist is identical to the one faced by a static monopolist; maximize total net revenue in a single period, NVN.

It also follows immediately that if the monopolist can commit only to quantities in each period, but not to prices, she can do no better than a static monopolist. By committing to the static monopoly output level, she can ensure static monopoly profits. She can do no better, however, because of the time preferences of con- sumers, since they will wait unitil the monopolist reduces price to that which a static monopolist would charge. Thus, all equilibria are (outcome) equivalent to that where the monopolist releases N* goods in period one and charges VN* in all periods, and all the N* highest-valuation consumers buy at date one.

IV. A DURABLE SERVICES MONOPOLIST

We now contrast the results in section III with what obtains when the firm can credibly commit to destroying stock if it is not purchased in a certain period, as well as to delaying the release of stock. This commitment power has empirical relevance for goods, such as the white-water raft ride, that cannot be stored.

Clearly, the monopolist can at least obtain the static monopoly level of profits. We now show that the firm does best by staggering the release of goods and adopting the pricing strategy outlined in the following proposition.

PROPOSITION 2. In equilibrium, the durable services monopolist release one good in every period. Given the subgame perfect equilibrium output level, N, the prices charged in periods t = 1, 2, ... , N, are given by

j P(N_ j) = VN + 0.5(VN-i --- VN-i+1), j 1, .. ,N- 1.

i= 1

In period N the price is VN. For t > N, Pt - v1. The firm's profit maximization problem can be rewritten as

N-1

max NvN + 0.5 (vN-i - vN-i+1)(N - i)- iN1

Along the equilibrium path, consumer 1 seeks a purchase in all periods from period 1 on to period N. She is served in period 1. In equilibrium, the N highest-valuation consumers make a purchase: 61 - 1, 1 = 1, ...,N.

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This result reflects the firm's ability to exploit consumers' fears of being ra- tioned. By releasing just one good in every period the monopolist takes fullest ad- vantage of her ability to ratio consumers. The highest-valuation consumer is willing to pay more to avoid rationing than any other unserved consumer. Therefore, in every period the monopolist releases a good and targets the highest-valuation con- sumer remaining to be served, setting the greatest price she can such that the consumer does not anticipate a better expected period pay-off by delaying seeking a purchase.

Clearly this tactic results in a higher price's being charged than if more than one unit became available; with more than one good available the firm has to set a price attractive to the lowest-valuation consumer to whom she is selling in that period or allow some units to go to waste. Since the monopolist prefers an early sale to a later sale only if the price is at least as large today as could be charged at a later date, she prefers to sell a single good and receive the highest price she can from the highest-valuation consumer remaining. Therefore, she delays releasing goods for sale to other consumers until future periods. The seller finds a low capacity more profitable; that is, she prefers a one-person raft to a larger vessel.

The prices charged to each consumer 1 differ as a function of the number of units, N, that the monopolist sells. The more units the monopolist sells, the less she can get for each of them. This reflects the intertemporal competition that the monopolist provides against herself. The greater is N, the smaller is the probability that a high- valuation consumer who delays a purchase is rationed: she has more opportunities to purchase at a later date if she does not purchase during the period in which the monopolist intends to sell to her. This offsets the advantage, in terms of additional sales, that accrues from making more units available. It also underlines the incentive for the consumer to purchase during the period in which the monopolist intended to sell to them. If we altered the game so that, instead, the monopolist announced a non-binding price path but could commit to dates at which output was available, including a final date, then remaining prices would be revised upward if a consumer failed to purchase in the period he was supposed to.

The optimal number of units to be made available can be calculated by solving the maximization problem detailed in the proposition. The expression depends only on the valuations and N, because this is sufficient information to determine the price each consumer pays. This programming problem reflects the fact that the higher are prices at future dates, the more a higher-valuation consumer is willing to pay rather than defer and be stochastically rationed. As is true when the monopolist cannot commit to destroying stock, it never pays to ration stochastically the last consumer: the linearity of profits in the rationing probability implies a corner solution.

Hence, to calculate the price to charge each consumer if a total of N units are to be sold, the monopolist solves recursively, recognizing that she can charge the Nth valuation consumer VN but no more, and working backward from there, solving for the price to charge the (N - 1)th consumer and then the (N - 2)th, and so on, until the appropriate price to charge in each of the first N periods had been calculated.

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Example 1. Suppose there are two consumers with valuations 50 and 30. Then the durable services monopolist sells a single unit in each of the first two periods, at prices pi = 40 and P2 = 30. The monopolist sets an arbitrarily high price in periods t > 2. The high-valuation consumer receives an expected pay-off of 10 if he purchases in period one. If he delays purchasing, his expected pay-off is also 10, since his probability of being served in period two is only one-half. Clearly, the low-valuation consumer will seek a purchase in period two, provided the commitment to cease production in subsequent periods is credible.

The durable goods monopolist also sells two units, but both consumers pay a price of 30 in period one. Because the high-valuation consumer knows the durable goods monopolist will sell to the low-valuation consumer in her final period of operations, he cannot be enticed to pay a higher price than 30, since the good will be stored should he not purchase. If, instead, the second consumer had a valuation of 20 rather than 30, then the durable goods monopolist would sell but one unit, and a price of 50 is optimal. In contrast, the durable services monopolist would continue to sell two units, at prices pi = 35 and P2 = 20.

Example 2. Now suppose two consumers have valuations 60 and 40, respectively, and eighty consumers have valuation 1.5 Then the durable services monopolist sells two units at prices p, - 50 and P2 = 40. In contrast, the durable goods monopolist sells 82 units at a price of 1. The difference arises because, to price discriminate effectively, the durable services monopolist cannot give the highest- valuation consumers too many opportunities to purchase the good. If the durable services monopolist sold more than two units, she could not charge the highest- valuation consumer more than 30, so offering more than two units is less profitable.

Denote the optimal quantity sold by a durable services monopolist by Nd, and denote her profits by 11d. Together, examples 1 and 2 prove the following result.

PROPOSITION 3. For different dist-ributions of consumer valuations it is possible that a dynamic durable services monopolist able to commit to total output will sell to more than, to less than, or to the same number of consumers as she would were she a static monopolist; that is, Nd - N*.

For a static monopolist, selling to N + j consumers, instead of N consumers, yields extra revenue of jVN+j from the additional j consumers served, but the N highest-valuation consumers collectively pay N(VN - VN+j) less. Consequently, if the difference between VN and VN+j is large enough, a static monopolist would sell N goods instead of N + j goods. Conversely, if VN - VN+j is small, a static monopolist would sell more stock.

The durable services monopolist does not necessarily want to sell more stock when VN -VN+j is small. This is because the extra availability reduces the probability that higher-valuation consumers would not be served if they delayed attempting to

5 This is not strictly in keeping with the assumption that v1 > vl+], but the example holds if V3 = V82 + E, e sufficiently small.

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make a purchase, and this reduced rationing threat lowers the rent the monopolist is able to extract from the high-valuation consumers. Therefore, if there is a large group of consumers with similar moderate valuations, it is possible that a durable services monopolist with the ability to commit to output would choose not to sell to these consumers, preferring instead to extract greater rents from consumers with much higher valuations.

In contrast, because a static monopolist is unable to discriminate when selling to the highest-valuation consumers, she faces a lower opportunity cost, in terms of the revenue she has to forgo from the highest-valuation consumers if she does sell to the group of consumers with similar valuations.

Returning to the case when (VN - VN+j) is large, the static monopolist's inability to price discriminate is the reason for her to favour selling N. A firm that extracts rents by delaying the availability of some service, however, might decide to sell the extra j goods. The monopolist need not reduce the prices she would charge the top N-valuation consumers by the margin (VN - VN+j). Hence, she might prefer to sell the extra j goods.

Proposition 3 demonstrated that it is necessary to know something about the distribution of consumers' valuations before comparing N* and Nd. Proposition 4 details sufficient conditions for Nd ? N*.

PROPOSITION 4. There exists a a > 0 such that, if (vi - vi+1) - (vi_1 - vi) < a, i < N* then the durable services monopolist serves at least as many consumers as the static monopolist. A special case for which this is true is when valuations decline at a constant rate; that is, vi - v+1 = k Vi.

The quantity a static monopolist sells, N*, maximizes NVN, a term that also appears in the maximization problem of the durable services monopolist. The ad- ditional term in the durable-goods monopolist's pay-offs, ZN 0.5(vN-i - vNi+1)

(N-i), captures those profits that accrue because of her ability to price discriminate. As N increases, the price a consumer is willing to pay to purchase immediately falls, because the danger of being rationed is less. There are more consumers for the monopolist to price discriminate against, however, so the profits resulting from price discrimination may rise or fall as N increases.

The more positive net-valuation consumers that are served, the more efficient is the outcome in the sense that total surplus is increased. Therefore proposition 4 offers conditions that ensure that a dynamic durable services monopolist is more efficient than a static monopolist. Were we to link the consumer demands to form a 'continuous demand curve,' we would find the discriminating monopolist is more efficient so long as the curve is not too convex; concave and linear demand curves satisfy this requirement.

While the durable-services monopolist who can commit may sell either more or less than the static monopolist, there is an unconditional characterization of the relationship between fI* and nd.

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PROPOSITION 5. Whenever more than one consumer is served, the profits of the durable services monopolist strictly exceed those she would earn were she a static monopolist. When only one consumer is served, profits are the same.

This result simply reflects the fact that the durable services monopolist suc- cessfully price discriminates. Of course, when the optimal amount of stock to sell is one, no discrimination occurs: the price charged is v1 (and the dynamic seller credibly commits to cease selling at that date), the same as the price charged by a static monopolist. This is the only case where the two types of monopolists earn the same profit.

V. CONCLUSION

As Stokey noted, in a dynamic setting a number of factors may influence the price path charged by a monopolist selling a good for which consumers want one unit at most. In this paper we focus on the possibility that the price path is designed to permit price discrimination. Consequently we assume constant unit costs and suppose that consumers are perfectly patient. Qualitatively similar results would follow were we to relax these assumptions, but the results would not be as stark. A richer model might also be developed if the consumption benefits to the consumer were not perfectly durable or if we allowed for the possibility of new consumers' arriving. Our model could then be applied, for example, to haircuts and automobile repairs.

Since the service sector accounts for a large amount of economic activity, the distinction between what we term durable services and durable goods seems impor- tant. A dynamic monopolist who can commit both to release and withdrawal dates for stock and to prices at each date earns strictly greater profits than a static mo- nopolist, provided she sells more than one unit. If the monopolist can commit only to prices and total output, however, then she cannot successfully price discriminate and does no better than a static monopolist. Whether price discrimination allows for a more efficient outcome depends on the distribution of consumer demands. The assumption of a finite number of positive-valuation consumers is key: discrim- inating profits are possible only when there are a finite number of consumers, since it is impossible in our economy selectively to ration agents of measure zero.

If the monopolist could credibly commit to dates at which she will sell at vastly discounted prices to clear stock, the seller would be able to use the threat of low- valuation consumers purchasing to extract rent from higher valuation consumers, even if the good could be stored. In this case one would observe a roller-coastering of prices, with prices increasing and decreasing between periods.

Throughout the paper we assume that the monopolist is perfectly patient. How- ever, all the results continue to hold for a range of seller's discount factors close enough to unity. So, in contrast to the result of Wang (1994), the introduction of commitment makes it possible for a monopolist with a smaller discount factor than the buyer to earn profits greater than those of a static monopolist.

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APPENDIX

Proof of proposition 1 and corollary 1. We first argue that only prices in the first two periods matter.

Because the firm prefers revenue today to revenue tomorrow, in equilibrium prices Pt will be set so that there are no periods without sales preceding periods with sales; were there no sales in equilibrium in period t at price Pt and the next sales were at some future date t + k at price Pt+k, then the monopolist could set the revised price p' = Pt+k, receive the same revenues as before, and receive the revenues earlier. The consumer who, under the initial pricing plan, has ht+k = 1 but in all prior periods ht+j = 0 would, under the revised pricing plan, have ht 1; the periods in which a consumer chooses not to purchase have no influence on later purchase decisions. If the firm prices and releases goods so that she expects to make sales in period t + 1 but not period t, the firm would be better off because her time preferences bring all prices and release dates from period t + 1 on forward by one period, eliminating the original price she proposed for period t. Consequently, it must be that there is no 'gap' between sales dates.

An optimizing consumer, given the choice of two prices in two different periods, purchases in the period with the higher price only if there is a positive probability of being rationed in the lower-priced period. This implies that all stock must be sold in at most two periods - were there three periods in which stock were purchased along the equilibrium path, then consumers would not be rationed in the first two sale periods because there would be stock remaining for the third period. But then they all want to purchase in the lower-price period, and, if the prices are the same, they want to purchase at the earlier date. Since there can be no gaps of time between when stock is sold, it must be that stock is sold only at the first two dates.

Suppose the monopolist decides to sell N goods in period one and wants to determine Q, the number of goods to sell in period two, and prices P1 and P2. Let VN+J be the valuation of the marginal consumer who seeks a purchase in period two, so P2 = VN+J. The valuation of the marginal consumer buying in period one iS VN, SO

VN -P1 = Q

+ 1 (VN - VN+J).

Therefore, the problem for the monopolist is to maximize

Np1 + QP2,

where Q ? J. Rearranging, the objective function of the monopolist becomes

N [VN - Q (VN -VN+J) + QVN+J.

Since the function is linear in Q, the optimum number of second-period sales is either Q = 0 or Q = J. If Q = 0, however, then the monopolist's problem is to maximize NVN, while if Q = J, the problem reduces to maximize (N +J)vN+J. Both problems are identical to the profit-maximizing problem for the static monopolist.

The preferences of the monopolist then ensure that all sales occur in period one at VN*, with future prices merely set to force consumers to buy in period one. U

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Proof of proposition 2. The pre erences of the monopolist ensure that there will not be periods without sales prior to periods with sales. Hence, Pt < v1.

We first verify that, given the prices specified in the proposition, consumers optimize. We then show that the monopolist cannot eam greater profits with some other sequence of prices and shipments.

Along any path where no low-valuation consumer j with valuation vj > vN+1

purchases before period j, consumer j's strategy specifies that he attempt to buy in all periods j, j + 1, . .. , N, until successful.

Clearly, h' = 0 for all Pt > vl, so consumers N + 1, . . . ,L never seek a purchase, and the only period in which consumer N seeks a purchase is period N. The strategy outlined for consumer N - 1 is also optimal. The pay-off from seeking a purchase in N -1,

VN-1 PN-1 -(VNI - VV)7 2

the pay-off from not seeking a purchase until period N. The pay-off from seeking a purchase in a period prior to N - 1, given the equilibrium strategies of all other players, is no greater than

q(vN-l -PN-2) + (1 - q)(vN_1 -PN-1),

where q is the probability 3N-2 1. Since PN-2 > PN-1 the consumer should not seek a purchase in periods prior to N - 1. Working recursively, we can show that no consumer has an incentive to deviate in equilibrium from the strategies detailed in the proposition.

Now consider what the optimal response of a consumer should be if other consumer(s) deviate from their equilibrium strategy. Clearly, for consumers with valuation v, < VN, they optimally continue to not seek a purchase. Consumer N also has no incentive to deviate from her equilibrium strategy, since it is never optimal to purchase when v, < pt.

For consumer N - 1, suppose J higher-valuation consumers were not served in the period they were supposed to be in equilibrium. Then the pay-off to consumer N - 1 if she continues to attempt to purchase in N - 1 is

J I(VN-1 -pIN-1)0 J I + +I(VN-1 -PN)

J + 1 [2 N 1 JN + J 1 V P) i = - [-)(VN+ 7 1 PN) (VN-1 -PN)

1 [3J+ (V1 1 J + L 2J +2 PA/I] 1

which is the pay-off to waiting until N before seeking a purchase. Hence, the pay- off from seeking a purchase in period N - 1 is at least as great as the pay-off from waiting until N before first seeking a purchase.

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988 John Spicer and Dan Bernhardt

Consider now the expected pay-off to seeking a purchase in a period prior to N - 1. Let q denote the probability that consumer is served prior to N - 1. The lowest price prior to N - 1 iS PN-2- Therefore, the pay-off to seeking a purchase earlier than N - 1 can be no greater than

q(VN-1 -PN-2) + ( 1- 3) 1(VN-1 -PN)+- 2J+ 2(N1 P)

Since (VN-1 -PN-2) < (VN-1 -PN) and (3J + 1)/(2J + 2) > 1 it is not optimal for the (N - 1)th consumer to attempt purchases prior to N - 1 even when J higher-valuation consumers failed to get served in the periods they were supposed to in equilibrium. So too, one can recursively determine that if J higher-valuation consumers were not served in the period they were supposed to be in equilibrium, a consumer 1 < N should continue to attempt to purchase in all periods 1, 1 + 1 ... N until successful.

Now suppose 3k = 1 for 1 < k < N. This is a zero probability event off the path where a high-valuation consumer has purchased too soon. If it did occur, one subgame perfect equilibrium would be for all other consumers to pursue the same strategy as before, with the exception of consumer 1, who now does not attempt to buy prior to k.

Finally, suppose that there are M > 0 agents for whom 3k = 1 for 1 < k <N, and Z > 0 agents for whom & 1 for J > N. Then one subgame perfect equilibrium set of strategies would be for all consumers to pursue the same strategy as before: consumer 1 attempts to purchase in periods 1, 1 + 1, . . . ,N.

We have now specified consumer strategies off the equilibrium path and shown that the strategies are subgame perfect.

Suppose that N has been chosen optimally, but that the pricing strategy is not optimal. Then it must be that at least one consumer would be either willing to pay more or willing to purchase earlier. Clearly consumer N will not pay more than VN, so only in period N will he seek a purchase with positive probability of being served. Therefore, each N - 1 highest-valuation consumer has a probability no greater than 0.5 of being served in period N.

Consequently, consumer N - 1 is willing to pay no more than pN-1, since at this price she is just indifferent between buying or waiting a period; that is,

VN-1 -PN-1 = O.S(VN- - VN).

Rearranging,

PN-1 = O.S(VN-I + VN)

is the maximum price that the consumer N - 1 is willing to pay in period N - 1. Working back inductively, similar reasoning shows that the first N - 2 prices

stated in the proposition correspond to the highest prices the N -2 higher-valuation consumers are willing to pay, and that setting prices this high requires that goods not sold in the period they were supposed to be sold in are destroyed. Since, in equilibrium, a single sale is made in each period, summing over the prices from periods 1 to N gives the firm's profit maximization problem as a function of N and

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Qt. This assumes that the prices charged to later consumers should not be reduced: that is, in equilibrium none of the N highest-valuation consumers is probabilistically rationed.

For a given N, suppose the firm were to ration probabilistically at the final date with positive sales. Set price PN such that VN- +J > PN, some J > 0. Suppose that in period N, Q C {O,... ,J} units arrive for sale. Along the equilibrium path the N - 1 highest-valuation consumers have been sold to in the first N -1 dates. Given Q and PN, the prices generating the greatest revenues can be calculated as before:

PN-1 - - J 1

(VN_I -PN), and so on.

Given these prices as a function of Q and PN, the monopolist's problem is N-1

PNmVNaI+x,QE , E>Pisl + QPN.

i=1

Observe that fixing PN, profits are linear in Q, implying Q- 0 or J. Therefore, the assumption of no probabilistic rationing is correct.

Lastly, note that the monopolist's prices can be supported only by the threat of rationing of a targeted purchaser if he fails to purchase at the appropriate price, so that there must be destruction of any existing stock before new stock is introduced. .

Proof of proposition 4. Suppose not. Then there must be some j > 0 such that N*-I

N*VN* + 0 O.51(vN*-i - VN*i+)(N* - i) < (N* -j)vN*]j i= 1

N*-j-1

+ X, 0.51(VN*j-i - VN*_j-i+1)(N -j -i)- i=1

where N* is the quantity that maximizes NVN. By definition, N*VN* -(N* -j)VN* -j 0. Therefore, if the proposition is incorrect, it must be that

N*-j-1

E 0.51[(vN*-i - vN*-i+1)(* - i) - (vN*j-i - VN*-j-i+1)(N -j ^-i)]

i=1 N*-I

+ > 0.5'(vN*-i - VN*-i+1)(N- i) < 0.

i=N*-j

This rearranges to N*;j-1

0.5i(N* - i)[(vN*-i -VN*-i+) - (VN*-j-i -VN--i+)]

N*-j-1

+ E 0.5'(VN*j-i -VN*-j-i+)

N*-1

+ E 0.5'(vN*-i - VN*-i+)(N - i) < 0. i=N*-j

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990 John Spicer and Dan Bernhardt

By assumption, however, (vl - v1+1) > 0. Therefore, the only term that can be negative on the left-hand side of the inequality is

N*-j-1 > 0.5 (N - i)[(vN*-i - VN*-i+) - (VN*-j-i-VN*-j-i+)], i=1

but is must be that if [(vN*-i - vN*-i+1) - (VN*-j-i -vN*]j-i+1)] is small enough that the inequality can no longer hold. Hence, the supposition at the start of this proof does not hold. U

Proof of proposition 5. Immediate. For any N, including N*, the static monopolist's profits are NVN, which can never exceed NVN + ENI 0.5'(N - i)(vN-i - vN-i+1).

Since H*(N*) ? HId(N*), it must be true that H*(N*) ? ld(Nd).

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