Most-Favored-Customer Protection versus Price Discrimination over Time

--------------------------

I. P. L. Png University of California, Los Angeles

How should a seller price capacity that has no salvage value to heterogeneous customers whose valuations are private information? There are two periods, and the seller cannot precommit to prices in the later period. One option is price discrimination: first price high, then discount later if excess capacity remains. By offering mostfavored-customer protection, the seller can charge more in advance but will leave capacity unsold with positive probability. She favors the MFC protection when capacity is large and leans toward price discrimination when customers are more uncertain about the degree of excess demand in the first period.

I.

Introduction

Any production capacity that cannot be carried over to another date is perishable. Some common examples are the services of a team of production workers in a factory, aircraft seats on a scheduled flight, cars in a rental agency, hotel rooms, and all forms of real property for lease, whether industrial, commercial, or residential. In many such settings, the seller is uncertain about customer demand, as frequently illustrated when U.S. domestic automobile manufacturers offer large cash rebates or cut-rate financing to clear excess inventory, or at airports when passengers with confirmed reservations cannot "Pricing by Likelihood of Availability without Commitment" (UCLA Business Economics Working Paper no. 87-4) reports a discrete version of this research_ I am grateful for support from a UCLA Faculty Career Development Award and for advice from Sushil Bikhchandani, Jeremy Bulow, Y_ N. Chen, Thomas Cooper, David Hirshleifer, John Marner, and the referee and Washington University, Saint Louis_ [journal of Pohttcal Economy, 1991, vol. 99, no 5J © 1991 by The University of Chicago_ All nghts reserved_ 0022-3808/91/9905-0006$01.50

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embark. ·Kut, unfortunately, as these examples indicate, the seller often must set production capacity before she receives full information about customer demand. This paper aims to analyze how a seller should price such capacity to heterogeneous customers with private information about their valuations. The seller ideally would wish to charge customers according to their willingness to pay. In studying this problem, Harris and Raviv (1981) assumed that the seller could commit in advance to the prices that she announces. They show that when the potential demand exceeds capacity, the seller will maximize profit through "priority pricing." Under such a scheme, the higher the price paid, the higher the buyer's priority of service; that is, those who pay less are less likely to obtain the service. In general, these prices will be set such that some portion of the population, with lower valuation of the product, will not buy at all (see also Chao and Wilson 1987). Harris and Raviv assume that the seller commits irrevocably to her price structure and hence cannot adjust prices according to new information about demand. In particular, if the total sales fall below the available capacity, the seller cannot cut the price to clear the remainder. This approach describes well the marketing of electricity capacity, in which the seller receives customer orders the instant before production takes place. Sales of airline and theater seats, however, take place over time: after a period of advance sales, the seller will have improved information about the remaining customer demand. She can then adjust the price for subsequent sales accordingly. Lazear (1986) focuses on this adjustment. In his setting, customers buy as soon as the quoted price falls below their valuation; in particular, they cannot wait to speculate on a lower price in a subsequent period. Lazear's analysis applies to situations in which the number of potential customers is so large relative to the quantity of the product available that the probability of obtaining the product at a later period is negligible. In the present paper, I allow customers to act strategically, but the seller may commit to a price in a later period only by a credible, legally enforceable contract. Like Lazear, I show that one option for the seller is to discriminate between the customers by offering prices that decline over time. As the price declines, so does the likelihood of obtaining a unit of capacity. The uncertainty of obtaining a service at a lower price compels customers with a higher valuation to buy in advance.' If the seller discriminates in this way, she will sell all available capacity. 1 For example, tickets to Broadway shows and other New York performing arts events are available in advance at the regular price or at a substantial discount on the

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The higher-valuation customers' willingness to gamble on a price cut in the following period, however, limits the seller's ability to extract their consumer surplus. The seller can mitigate this problem by offering most-favored-customer (MFC) protection, which guarantees customers who buy early that they will benefit from subsequent reductions in price. The MFC protection allows the seller to raise the price for advance sales. I show, however, that despite the MFC protection, if advance sales are sufficiently weak, the seller will cut the price over time because the revenue from additional sales outweighs the rebates to the early (high-valuation) buyers. If the advance sales are stronger, the seller will choose to leave excess capacity unsold, which means that social welfare will fall below that under price discrimination. How will the seller choose between price discrimination and MFC protection? When her capacity is large, she will prefer MFC protection because it supports a higher price for advance sales. An increase in uncertainty about the degree to which the number of highvaluation customers exceeds capacity allows the seller to charge more under discrimination but does not affect MFC pricing. Hence, such an increase will raise the seller's revenue from price discrimination relative to that from MFC protection. My analysis also applies to an employer who seeks to hire a fixed number of employees from a population of potential workers who differ in their reservation wage. Like the seller of a fixed capacity, the employer must decide whether to discriminate in wages. Indeed, the wide use of most-favored-treatment provisions in labor contracts suggests that workers are very sensitive to wage discrimination. This paper is structured as follows. I present the basic setting in Section II and then solve for the profit-maximizing path of prices under two scenarios: first, when the seller discriminates on price (Sec. III) and, second, when the seller offers MFC protection (Sec. IV). Next, I show that the seller will never sell in one period only (Sec. V) and then study how the seller will choose between price discrimination and MFC protection (Sec. VI). Section VII presents some empirical implications of this choice, and Section VIII considers how the seller should select capacity. The paper concludes (Sec. IX) by relating the present work to other recent pricing research and remarking on directions for future work.

day of the performance. Likewise. borrowers can secure funds from a lender by paying an advance commitment fee on top of the rate on the funds actually used. Alternatively, they could apply for a loan as and when they require funds, but then risk not obtaining the required money.

PROTECTION VERSUS DISCRIMINATION

II.

101 3

Setting

Let each customer have inelastic demand for one unit of service from the capacity, and attach a value of either VI or Vh > VI to the service. I shall refer to customers with valuations VI and Vh as being low and high types, respectively. Every individual's valuation is private information and is known to neither other customers nor the seller. A (random) total of x E [0, 1] customers are high types, while the remainder are low. Let the distribution of x on the condition that one customer is a high type be (x). The customers may not resell the serVice. Sellers in many businesses have a capacity that is almost fixed over some horizon. This is true of all forms of real property for lease in which construction of new capacity may take substantial time. It is also true of manufacturing capacity over shorter planning horizons. Accordingly, let the seller have some exogenous capacity k, and suppose that unsold capacity has no salvage value. 2 To prevent the problem from being trivial, I assume that the seller's capacity is less than the total customer population. ASSUMPTION. < k < 1. Let there be only two selling periods. Customers take the seller's price as given and do not haggle. If, at any time, there should be excess demand, the available units will be allocated by lottery. The seller aims to choose prices in each period to maximize total profit from the standpoint of the first period. The only commitments that she may make are those that buyers will have incentive to enforce in courts of law. For simplicity, I assume that the seller and customers are all risk neutral and do not discount profits and utility received in the second period (see fig. 1).

°

III.

Price Discrimination

In this section, I suppose that the seller does not commit in the first period to second-period prices. The seller has three alternatives: (i) sell in both periods, (ii) sell only in period 1, or (iii) sell only in period 2. I shall discuss two-period sales here and one-period sales in Section V. If the seller were to price to sell to the low-valuation customers in the first period, the high types would buy as well; hence sales would occur in the first period only. Accordingly, the only feasible strategy of two-period sales is the one in which all the high2 In such instances, the seller can adjust to information about demand only through her pricing policy.

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t= 1

t=2

seller chooses pricing strategy and sets firstperiod price

seller decides whether to sell and sets secondperiod price

buyer decides whether to buy

buyer decides whether to buy

FIG.

I.-Sequence of actions

valuation customers buy in advance and the low types wait until the second period. Since the seller cannot precommit, I begin by solving for the price P2 in period 2. At that time, all remaining customers have a low valuation, VI. Hence, if capacity remains, the seller will maximize profits from that point onward by setting P2 = VI' thereby extracting all the surplus of the low-type buyers. In the first period, each customer must weigh the net expected utility of buying immediately against waiting until period 2. Consider a customer who buys in period lor, more precisely, attempts to do so. If the total number of high-valuation customers does not exceed the seller's capacity, x ~ k, the customer will obtain the service with certainty. If, however, x > k, each customer attempting to buy must join a lottery and will obtain the service with probability k/x. Hence the probability that a customer will obtain the service if he buys in period 1 is (J"I

(k) clef = (k)

+

ilkx

-d(x).

(1)

k

Let PI be the seller's first-period price; then a high-valuation customer's net expected utility from buying in advance will be

(2) Consider a customer who does not buy in the first period and waits until the second. If the number of high-valuation customers falls short of capacity, x < k, the seller will have k - x units available in the second period. Then the seller will cut the price to P2 = VI' and all 1 - x low types will wish to buy as well; hence the probability of obtaining the service is (k - x)/(1 - x). If, however, x 2: k, all capacity will be sold in the first period. Thus the probability that a customer

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will obtain the service in period 2 is defJkk - X a2(k) = -1-d(x). o - x

(3)

Accordingly, a high-valuation customer's net expected utility if he waits to buy in period 2 will be (4)

To maximize profits, the seller should set first-period price PI so that each high-valuation customer will be just indifferent between buying immediately and waiting. 3 Then [Vh -

PI (k)]al (k)

= (Vh - vI)a2(k);4

(5)

hence a2(k) PI(k) = Vh - (Vh - VI) al(k)'

(6)

Notice from (1) and (3) that for k E (0,1), al(k) > a2(k) > 0, which implies that PI (k) E (VI' vh)' The lower likelihood of availability in the second period affects price discrimination by persuading the highvaluation customers to buy early. Since PI < Vh' the high types will retain some consumer surplus. Further, PI > VI confirms that the low types will not buy in the first period. If x :::; k, the seller will receive revenue of PI (k)x in the first period and vI(k - x) in the second. If, however, x > k, the seller will receive revenue of PI (k)k in the first period and nothing subsequently. Hence, from the standpoint of period 1, the seller's expected revenue from sales in both periods is R (k) = PI (k)

I:

xd(x)

= cg max[vI(k IV.

+ VI

- x), 0]

J: (k -

x)d(x)

+ PI (k)k

f

d(x)

(7)

+ PI (k)cg min (x, k).

Most-Favored-Customer Protection

Under the preceding strategy, the high-valuation customers' willingness to gamble on a lower price in the following period limits the seller's ability to extract their consumer surplus. The seller can mitig If the price PI were lower, then a high type would strictly prefer to buy in period I, and the seller could raise PI slightly and thereby increase revenue. 4 Thomas Cooper has observed that this implies Vh - PI (k) = (Vii - Vt)[u2(k)/ul (k)]; i.e., conditional on being able to obtain the service in advance, a high-valuation customer receives equal net expected utility from buying early and by waiting.

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gate this problem by promising customers who buy early that they will benefit from any subsequent price cut. In industrial marketing, such guarantees take the form of MFC provisions. s A retail example is Chrysler's December 1989 announcement of $1,000 rebates or cutrate 3.9 percent financing on new minivans and its promise to all purchasers that if it raised the rebate before October 1990, it would "make up the difference" to them ("Chrysler's New Incentives" [1989, p. BS]). If indeed the seller should reduce the price, buyers with MFC protection will have a private incentive to enforce the provisions in courts of law. As in the setting of price discrimination, the seller has three alternatives: (i) sell in both periods, (ii) sell only in period 1, or (iii) sell only in period 2. Again, I defer discussion of one-period sales to the next section. Further, the only feasible strategy of selling in both periods has all high-valuation customers buying in the first period and the low types waiting. 6 Let and P2 denote the prices in the first and second periods, respectively. In the first period, each customer will weigh the net expected utility from buying immediately against the net expected utility from waiting. The MFC provision assures every first-period customer that he will enjoy the lowest price offered by the seller, whenever it is offered; hence a customer loses nothing on price by buying early. By buying early, however, he can raise his chance of obtaining the product. Accordingly, a high-valuation customer will buy in period 1 at any price up to Vh' Thus the seller should set

PI

(8)

PI

Given Vh' the low types will wait to buy in the second period. If the number of high types (weakly) exceeds the seller's capacity, x 2: k, there will be no capacity to sell in period 2, and hence the seller's revenue will be simply (9)

Suppose instead that x < k. Then k x units will be available in period 2. If the seller were to cut the price to VI' she could sell an additional k - x units to the low-valuation customers but then must refund Vh - VI to each of the x high types who bought in period 1; hence revenue net of the rebate will be (10) 5 Butz (1987) notes that such provisions were widely adopted in contracts between natural gas pipelines and gas producers in the United States. 6 If the seller were to price to sell to the low-valuation customers in the first period, the high types would buy as well; hence there would be no sales in the second period.

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If the seller does not cut the price to VI' I would argue that she should set P2 = Vh' in case some high-valuation customer should mistakenly not buy in advance. In equilibrium, all high types buy in period 1; hence the seller's revenue from P2 = Vh will be x = VhX' Let

Pi

(11) then in the case in which x < k, the seller will set P2 VI if vlk > VhX (i.e., x < k) and will set P2 = Vh otherwise. Therefore, total revenue will be (12) The MFC provision inhibits the seller from reducing the price because any discount must be extended to all those who purchased earlier as well. 7 But if advance sales are sufficiently weak (because high-valuation customers are too few), the seller will cut the price. By (9) and (12), total revenue with an MFC provision is R(k) = ~{max[vlk, Vh min (x, k)]}.

V.

(13)

One- versus Two-Period Sales

I have analyzed price discrimination over two periods and sales with MFC protection over two periods. In this section, I show that the seller will always prefer to sell with protection over two periods rather than sell in one period only, with or without protection. Consider first a strategy of selling in the first period only. I claim that such a strategy is credible only if the seller sets price Pi = VI and sells all capacity in the first period. If she were to post Pi > VI' only high-valuation customers would buy; if capacity were to be available in period 2, the seller could not prevent herself from cutting the price to sell to the low-type customers. This is precisely the scenario of two-period sales analyzed in Section III above. Likewise, if the seller offered MFC protection and some first-period price > VI' she would fall into the two-period setting of Section IV. If the seller were to set first-period price Pi = VI' all customers would buy,8 and she would receive revenue of vI~[min(k, 1)] = vlk, which, by (13), falls short of R(k), the revenue from selling with an MFC provision over two periods.

Pi

This is one of the central arguments in Cooper (1986) and Salop (1986). Suppose that, off the equilibrium path, some capacity remained in period 2. Then the seller would set a price of at least Vi; hence all customers (weakly) prefer to buy in period I. 7

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Consider next a strategy of selling in the second period only. To accomplish this, the seller must post a first-period price above Vh so that no one will buy then. In the second period, the seller will make a larger profit from price P2 = Vh than any P2 E (v[, Vh); hence she should price at either P2 = Vh to obtain revenue of Vh ~ [min(x, k)] or P2 = VI' which yields revenue of v[k. But, by (13), revenue from MFC sales over two periods is R(k) :::::: vh~[min(x, k)] and R(k) :::::: v[k. The same arguments apply if the seller initially offers an MFC provision and sells in the second period only. Accordingly, the following proposition holds. PROPOSITION 1. The seller will always (weakly) prefer to guarantee MFC treatment to first-period buyers and sell over two periods than to sell in one period only. Selling over two periods allows the seller to collect and use information about customer demand. By offering an MFC guarantee in the first period and selling over two periods, the seller creates an option: to choose between (i) revenue of Vh min (x, k) by selling only to the high types at their valuation, v", and (ii) revenue of v[k by selling to both types at VI' Most crucially, the seller can choose between these alternatives after learning the relative numbers of high vis-a-vis low types. In contrast, if the seller sells in one period only, she must choose between the two alternatives before receiving this information and hence will obtain less revenue. 9 VI.

Price Discrimination versus Most-Favored-Customer Treatment

In light of proposition 1, the seller need only consider two alternatives: (i) price discriminating with first-period price PI (k) given by (6) and, if capacity remains in period 2, cutting the price to P2 = VI; and (ii) offering MFC commitment and price = Vh in the first period and, in the second, discounting to P2 = V[ if x < h. lO I now consider how the distribution, <1>, of x and capacity, k, influence the seller's choice between price discrimination and MFC protection. II The MFC protection provides the seller an option to choose be-

PI

9 I assumed implicitly that the MFC provision did not impose additional administrative costs. In reality, of course, the rebates required by MFC protection may be sufficiently troublesome that the seller may prefer to sell only in one period. For instance, most U.S. cinemas do not sell tickets in advance, thereby effectively committing to sales only in the final period. 10 Since k = kv/v", this condition implies that x < k, i.e., that there will be some capacity in the second period. II By contrast, in Lazear (1986), customers buy whenever the price falls below their valuation; hence the advance price is constant and MFC protection does not add to the seller's profits.

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tween revenues of v/k and Vh min (x, k) after she learns the number of 'high-valuation customers, x. Increased uncertainty does raise the value of the option inherent in the MFC strategy. I shall show, however, that additional uncertainty may also increase revenue from the price discrimination strategy by raising the pressure on highvaluation customers to buy in advance. Let increases in uncertainty be represented by the following concept. DEFINITION. A distribution <1>' is a mean-preserving spread of the distribution on the interval [XI' x2] if <1>' (x) = (x) , for all X in [0, xd and [x2' 1], and the conditional distribution <1>' /[<1> (x2) - (XI)] is a mean-preserving spread, in the sense of Rothschild and Stiglitz (1970), of the conditional distribution /[(x2) - (XI)].12 Now consider the effect of increased uncertainty on pricing under discrimination. If the distribution of x is spread on the lower range [0, k], then this will not affect the likelihood of obtaining the service for customers who buy in the first period while reducing the chances in the second. Accordingly, by adding pressure on customers to buy in advance, it will allow the seller to increase PI. A spread of the distribution on the upper range [k, 1] will raise the probability of obtaining the service in advance while not affecting the likelihood in the second period. Hence, this spread will also permit the seller to raise PI. PROPOSITION 2. A mean-preserving spread of the distribution on either [0, k] or [k, 1] will raise PI' the first-period price under price discrimination. 13 Under both price discrimination and MFC strategies, a change in will also affect the balance of sales in the first vis-a-vis second period. By (7), revenue from price discrimination depends on x and PI according to . PIX + v/(k - x) x E [0, k) R(x) =

{

x E [k, 1].

plk

By contrast, changes in will not affect MFC pricing and hence can affect MFC revenue only through sales. From (13), MFC revenue is k

x E [O,k)

v:x

x E [k, k)

Vhk

x E [k, 1].

V

R(x) =

{

See figure 2 for sketches of R(x) and R(x). 12 A mean-preserving spread on an interval will preserve the mean of the original distribution but need not lead to a distribution that is a mean-preserving spread of the original distribution (over its entire range). 13 I present the proofs of propositions 2 and 4 in the Appendix.

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-

-

-

-

-

- - ~ - - - - - -

,

-

-

-

-

- - - --,(-'-------

R(x)

- 01--,-------

R(x)

- - - - - - - J - - - - - - - - - - -

o

x k

R(x)-R(x)

FIG. 2.-Revenue and sales under price discrimination and MFC protection

For all x E [k, 1], sales under both price discrimination and MFC strategies will be k and hence do not vary with x. Now by proposition 2, a mean-preserving spread of the distribution on [k, 1] will raise the first-period discriminatory price, Pl. It will not affect prices under the MFC strategy. This proves that when the degree to which the number of high-valuation customers exceeds capacity becomes more uncertain, price discrimination revenue will rise relative to MFC revenue. PROPOSITION 3. A mean-preserving spread of on the interval [k, 1] will raise the seller's revenue from price discrimination relative to revenue from MFC treatment. By figure 2, the difference between price discrimination and MFC revenue, R(x) - R(x), is concave in x on the interval [0, k]. Hence, when any effects on price are ignored, a mean-preserving spread on the interval [0, k] would reduce price discrimination revenue relative to MFC revenue. By proposition 2, however, the spread causes a countervailing increase in the first-period discriminatory price, Pl. Indeed, examples may be constructed in which the price effect outweighs the effect on sales. Thus the direction of the effect of increased uncertainty on the choice between price discrimination and MFC treatment is very sensitive to the range of x over which the uncertainty rises. How will an increase in capacity, k, affect the seller's choice between price discrimination and MFC treatment? Consider the effect of an increase in capacity on pricing under price discrimination. An addi-

PROTECTION VERSUS DISCRIMINATION

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tion to capacity will raise the likelihood of obtaining the product in the second period relative to the probability of obtaining it in the first. Hence it forces the seller to reduce PI (k). PROPOSITION 4. Under price discrimination, the first-period price, PI (k), falls with capacity, k. Accordingly, under price discrimination, additional capacity has two countervailing effects: by (7), it raises sales, but it reduces the first-period price (the second-period price is constant at P2 = VI)' In the polar case of k = 1, the first-period price PI = vI; hence, by (7), revenue R = VI' By contrast, under MFC treatment, a capacity increase will raise sales but not affect prices and hence will unequivocally add to revenue. If k = 1, then by (13), revenue R = ~ max(v"x, VI) 2:: VI = R.14 Thus the seller will prefer the MFC treatment when capacity is large essentially because it allows her to maintain a high price for advance sales. PROPOSITION 5. If capacity is sufficiently large, the seller will prefer to offer MFC protection. The experience of the U.S. market for large electrical turbine generators during the 1960s lends some support to proposition 5. 15 Following the federal conviction of several electrical equipment manufacturers and individual executives for price fixing, a number of electric power utilities discovered that the manufacturers had engaged in widespread price discrimination over the period 1948-62. In May 1963, General Electric introduced a "price protection clause" that provided that if prices were reduced, General Electric would refund the difference to any customer who had bought in the preceding 6 months. It claimed that this provision responded to the power utilities' fear of price discrimination. 16 The case in which x takes a uniform distribution provides some additional insight into the choice between price discrimination and MFC protection. Suppose that x is uniformly distributed on [0, 1]. By (7) and (13), price discrimination revenue is

14 The inequality will be strict under the mild assumption that x > v/v" with positive probability. 15 The following account is based on Sultan (1974, pp. 84-124) and U.S. vs. General Electric Co. and Westinghouse Electric Corp. ("Plaintiff's Memorandum in Support of a Proposed Modification to the Final Judgment Entered on October I, 1962, against Each Defendant," Fed. Register 42 [March 30, 1977]). 16 Westinghouse, the other remaining domestic manufacturer, quickly followed General Electric's move. Cooper (1986) and Salop (1986) analyzed the anticompetitive effects of these provisions.

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1.00 , - - - - - - - - - - - - - - ,

0.02 0.00 +--=O'_ _ _ _-----"~----j

0.00

-0.07 - + - - - - - - - - - - - 1 0.00 100

+---~=---------j

-1 00 + - - - - - - - - - - - - - - j 0.00 1.00

FIG. 3.-Difference between price discrimination and MFC revenue as a function of capacity in the uniform case.

and MFC revenue is

Hence the difference in revenue between price discrimination and MFC protection is

and

=

-k(l - !:)2

(T2 (Tl

+ .3!.i. k2 2Vh

by (5). Thus, for sufficiently large Vh' the right-hand side is negative and R (k) < R (k), all k. Accordingly, if the dispersion of the two types' valuations is sufficiently large, the seller will prefer the MFC protection. With respect to the effect of capacity on the choice between price discrimination and MFC protection, figure 3 illustrates the difference R(k) - R(k) as a function of k for alternative values of VI and Vh' In the first example, VI = 1 and vh = 1.5, and the seller prefers price discrimination for k < 0.7 and MFC protection otherwise. In the second example, VI remains at one while Vh is higher at three. Then the cut-off capacity level falls below 0.1. These calculations project

PROTECTION VERSUS DISCRIMINATION

102 3

beyond proposition 5 to suggest, first, that the choice is monotone in capacity and, second, that the more heterogeneous the customers, the wider the range of capacity over which the seller will choose the MFC protection. 17

VII.

Empirical Differences

Under the strategy of price discrimination, the high types' willingness to gamble on a price cut limits the seller's first-period price to PI (k) < Vh (see eq. [6] above). By guaranteeing no discrimination, the MFC provision relaxes this constraint and thereby frees the seller to raise her price for advance sales to = Vh. 18 REMARK 1. The MFC treatment will lead the seller to raise the price for advance sales. Both price discrimination and MFC strategies allow as many highvaluation customers as possible to buy (in the first period). The two strategies, however, treat low types differently. Under price discrimination, the seller will always sell to the low types if capacity remains in the second period. Under an MFC strategy, however, she will cut the second-period price if and only if x < k. Hence, if there is excess capacity but x ~ k, the seller will prefer to leave it unsold. Thus, in the present setting, there will be idle capacity only if there is no price discrimination. 19 Since unused capacity has zero salvage value, each additional sale (to a low-valuation customer) will raise expected total surplus. The MFC provision will, however, inhibit the seller from clearing excess capacity through price cuts, and hence the following will be true. REMARK 2. In sales with an MFC provision, more capacity will be left idle and expected total surplus will be less than under price discrimination.

PI

VIII.

Choice of Capacity

So far, the analysis has taken service capacity, k, to be exogenous. I turn next to the seller's choice of k. After deciding on k, the seller 17 These two examples also show that neither price discrimination nor MFC protection dominates the other. 18 Cooper (1986) showed that if several sellers compete on price and one seller adopts the MFC strategy and raises the price, she will induce her competitors to raise their price as well. 19 In the model of Harris and Raviv (1981), in which the seller can precommit to prices, the seller discriminates through priority prices. The priority prices will be so high that some capacity will remain idle under certain realizations of customer demand. Hence, if the seller can precommit, price discrimination and idle capacity may arise together.

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will move to a pricing stage in which she must choose between price discrimination and MFC treatment. Accordingly, the seller's expected revenue from capacity k will be max[R(k), R(k)]. Assume, for simplicity, that capacity can be used only once, and let C(k) be the cost of building capacity. Then the seller should choose k to maximize expected profit: max[R(k), R(k)] - C(k).

(14)

Unfortunately, neither R(k) nor R(k) is concave in k, and even if both were, the maximum of the two, max[R(k), R(k)], need not be concave. Thus it seems very difficult to generally characterize the seller's choice of capacity. To illustrate the seller's capacity choice, let VI = 1, vh = l.5, x be uniformly distributed on [0, 1], and the marginal cost of capacity be a constant, c, so that C(k) = ck. From a social standpoint, the expected benefit to consumers from an additional unit of capacity is VI Pr(x :s; k) + Vh Pr(x > k) = k + l.5 (1 - k). Then if c = 0.7, the socially efficient capacity would be k = 1. The seller will instead choose k = 0.96 and MFC pricing, and she will leave capacity unsold with probability 0.96 - (0.96/l.5) = 0.32. By contrast, if c = l.3, the socially efficient capacity is given by 1.5 - 0.5k = l.3 (i.e., k = 0.4). The seller chooses k = 0.268 and price discrimination. Rationing will occur in the first period with probability .732, and with certainty in the second period. See table 1 for the detailed results. 20

IX.

Concluding Remarks

The analysis here is related closely to the work of Bulow (1982), Butz (1987), Moorthy (1988), Bagnoli, Salant, and Swierzbinski (1989), and others on the pricing of a durable good. These models generally assume that the manufacturer knows the customer demand with certainty. Having produced and sold some quantity of the good, the manufacturer always is tempted to make and sell more in the next period. Her customers will, however, look ahead, understand that the manufacturer will cut the price in the future, and be reluctant to buy at a high initial price. Under these circumstances, the manufacturer would like, if she can, to commit never to cut the price. Several ways of doing so include limiting her output, as the lithographer does 20 Since restricting capacity is one way to raise the price under price discrimination, it might be thought that capacity would be larger with MFC protection. The calculations reported in the table show that this need not be so. Under MFC protection, a larger capacity will be unattractive if the seller subsequently chooses to leave some capacity unsold.

PROTECTION VERSUS DISCRIMINATION

102 5

TABLE 1

PROFIT-MAXIMIZING CAPACITY UNDER PRICE DISCRIMINATION AND MFC PROTECTION WHEN VI = 1, Vh = 1.5, AND X HAS UNIFORM DISTRIBUTION PRICE DISCRIMINATION

c

.7 1.3

MFC PROTECTION

k

R(k) - ck

k

R(k) - ck

.79 .268

.36 .0282

.96 .24

.384 .024

by smashing her plates; converting her product into a nondurable by a policy of only renting the good; and offering MFC protection. What if the manufacturer is uncertain about customer demand? Then her best pricing strategy will depend on whether she must incur significant setup costs each time she manufactures the good. If production does not require any such costs, she should post a very high price initially together with MFC protection, wait to learn the demand, and then adjust the price and produce more accordingly.21 Many manufacturers, however, do incur substantial setup costS. 22 A manufacturer who incurs such setup costs faces a problem much like the one analyzed in this paper and must choose between price discrimination and MFC protection. Where a seller is uncertain about customer demand, MFC treatment does not imply that prices will remain fixed. If advance sales are sufficiently weak, she will cut the price even at the cost of rebates to customers who have purchased in advance. I implicitly assumed that the seller could not resolve her uncertainty about customer demand prior to offering the capacity for sale. In practice, the model applies to the extent that the seller cannot or chooses not to learn customer demand perfectly through market research. Note also that MFC provisions are effective only if buyers are well informed about the seller's prices, as, for instance, when all sales take place at advertised prices. These provisions are more difficult to implement when the buyer and seller negotiate in secret. The price discrimination analyzed here and MFC protection are two polar cases of a broader class of pricing mechanisms in which the seller contracts to provide partial rebates if she should cut the price. With partial rebates, if the seller does cut the price, early customers

21 If the initial price is too high, the seller must pay rebates to customers who buy early. 22 The background to Chrysler's MFC offer provides evidence of such costs: "Chrysler concluded the cost of 45 days of minivan incentives would be 'less than the cost of taking out the overtime that we've got scheduled in the first quarter'" ("Chrysler's New Incentives" [1989, p. B5]).

JOURNAL OF POLITICAL ECONOMY

will end up paying a higher (net) price than later customers. Potential customers, however, will probably find partial rebates considerably more difficult to comprehend. They must know the capacity and the distribution of x, calculate the critical level below which the seller will cut the price, and then decide whether to buy in advance. By contrast, their decision under MFC protection is simple: buy at any price below their valuation. Decision making under price discrimination requires only that they know the capacity and the distribution of x. Just as partial rebates will be more difficult for customers to understand, they require more computation on the seller's part. These two factors may explain why, in practice, price discrimination and MFC protection are much more widely used. 23 In the setting of this paper, lotteries occur because customer demand is uncertain and because the seller will not expand capacity to meet all potential demand. In related work by DeGraba and O'Hara (1990), the seller has fixed capacity and faces uncertain demand. She sets prices that decline over time in order to sell down each identical customer's individual demand curve. In Pratt and Zeckhauser (n.d.) and Wilson (1988), the seller chooses to allocate some items by lottery essentially because her revenue is not a concave function of sales. She maximizes profits through a two-price strategy: customers who buy at the higher price are guaranteed the product, whereas those who buy at the lower price must join a lottery. This scheme makes the revenue function concave. Randomization-of price rather than allocation-also arises in the analyses of Varian (1980), Sobel (1984), and Doyle (1986). Here, oligopolists compete by price in a market with heterogeneous customers. Each seller randomly cuts the price to capture customers with a lower reservation price (Varian and Sobel) or new customers with uncertainty about the quality of the product (Doyle). Note, however, that each customer who buys will obtain the item with certainty.24 Appendix Proof of Proposition 2 By (6), first-period price PI = Vh - (Vh - v/)«(T2/(TI)' Now, by (1), (TI = (k) + (k/x)d (x), so a mean-preserving spread on [0, k] will not affect (TI' Since

n

23 By analogy, if there are more than two customer types, the seller may choose to offer the various types different rebates. I am grateful to the reviewer for noting this possibility. 24 Varian (1989) reviews this work in detail.

PROTECTION VERSUS DISCRIMINATION

1027

the integrand k/x is convex in x, and a mean-preserving spread on [k, I] will raise (x). Since .f..(k - x) = ~[-(1 - x) - (k - X)(-I)] = -2(1- k)(1- x)-3<0, ax 2 I - x ax (l - x)2

the integrand (k - x)/(l - x) is concave in x. Thus a mean-preserving spread on [0, k] will reduce ' is a mean-preserving spread of on [0, k], then ' is a mean-preserving spread of on [k, I]. Q.E.D.

Proof of Proposition 4 By (6), PI falls with k if the conditional probability
2~

dk

[

d
[pr(x -

=

< k) +

fo k -k

X

f~

-I-d(x) x

d(X)]

fl I

f:

I

~ x d(x)

-d(x)

k X

fk_k_d(x) [-k1 Pr(x
fl

+

- fk k - x d(x) !d(x) ol-x kX

Thus
PI

2:

fl

!d(X)] kX

0.

falls with k. Q.E.D.

References

Bagnoli, Mark; Salant, Stephen W.; and Swierzbinski, Joseph E. "DurableGoods Monopoly with Discrete Demand." IP.E. 97 (December 1989): 1459-78. Bulow, Jeremy I. "Durable-Goods Monopolists." IP.E. 90 (April 1982): 314-32. Butz, David A. "Intertemporal Discrimination and Commitment in LongTerm Contracts: A 'New' Explanation for Most-Favored-Nation Provisions." Manuscript. Los Angeles: Univ. California, Dept. Econ., October 1987. Chao, Hung-po, and Wilson, Robert. "Priority Service: Pricing, Investment, and Market Organization." A..E.R. 77 (December 1987): 899-916. "Chrysler's New Incentives on Minivans Take Their Cue from Appliance Stores." Wall St. I (December 18, 1989). Cooper, Thomas E. "Most-Favored-Customer Pricing and Tacit Collusion." Rand I Econ. 17 (Autumn 1986): 377-88.

JOURNAL OF POLITICAL ECONOMY

DeGraba, Patrick, and O'Hara, Maureen. "Extracting Rents with Forward Contracts." Manuscript. Ithaca, N.Y.: Cornell Univ.,Johnson Grad. School Management, May 1990. Doyle, Chris. "Intertemporal Price Discrimination, Uncertainty and Introductory Offers." Econ. J. 96 (suppl., 1986): 71-82. Harris, Milton, and Raviv, Artur. "A Theory of Monopoly Pricing Schemes with Demand Uncertainty." A.E.R. 71 (June 1981): 347-65. Lazear, Edward P. "Retail Pricing and Clearance Sales." A.E.R. 76 (March 1986): 14-32. Moorthy, K. Sridhar. "Consumer Expectations and the Pricing of Durables." In Issues in Pricing: Theory and Research, edited by Timothy M. Devinney. Lexington, Mass.: Lexington, 1988. Pratt, John W., and Zeckhauser, Richard. "Smoothing the Bumps in Profit Maximization: Pro Rata Tenders, Monopolist's Roulette, and Other Rationed Transactions." Manuscript. Cambridge, Mass.: Harvard Univ., n.d. Rothschild, Michael, and Stiglitz, Joseph E. "Increasing Risk: I. A Definition." J. Econ. Theory 2 (September 1970): 225-43. Salop, Steven C. "Practices That (Credibly) Facilitate Oligopoly Co-ordination." In New Developments in the Analysis of Market Structure, edited by Joseph E. Stiglitz and G. Frank Mathewson. Cambridge, Mass.: MIT Press (for Internat. Econ. Assoc.), 1986. Sobel, Joel. "The Timing of Sales." Rev. Econ. Studies 51 (July 1984): 353-68. Sultan, Ralph G. M. Pricing in the Electrical Oligopoly. Vol. l. Competition or Collusion. Boston: Harvard Univ., Grad. School Bus. Admin., Div. Res., 1974. Varian, Hal R. "A Model of Sales." A.E.R. 70 (September 1980): 651-59. - - - . "Price Discrimination." In Handbook of Industrial Organization, vol. 1, edited by Richard Schmalensee and Robert Willig. New York: NorthHolland, 1989. Wilson, Charles A. "On the Optimal Pricing Policy of a Monopolist." J.P.E. 96 (February 1988): 164-76.