REBATES, INVENTORIES, AND INTERTEMPORAL PRICE DISCRIMINATION.

AULT, RICHARD W. ; BEARD, T. RANDOLPH ; LABAND, DAVID N. 等

RICHARD P. SABA [*]

We demonstrate that universally redeemed rebates can increase manufacturer profits by reducing the incentives of downstream retailers to hoard inventories when optimal wholesale prices vary predictably over time. By bypassing retailers and making direct contracts with buyers, the manufacturer can increase the variations in effective prices paid by consumers without concomitantly creating larger incentives for retailers to hold inventories. During profitable, high-demand periods, manufacturer revenues are ordinarily constrained by "competition" from retailer inventories, thus limiting profits. However, by selectively offering rebates to consumers while maintaining high wholesale prices, low-demand periods can be accommodated without inducing retailer hoarding. (JEL D4, L1)

General Motors Corp. is emptying its bag of marketing tricks in an all-out effort to force its U.S. market share back up to 30% of light-vehicle sales. That is good news for GM customers, because the manufacturer is offering financing at less than 1% or cash rebates of as much as $3,000 a vehicle.

--Wall Street Journal, Sept. 16, 1998

I. INTRODUCTION

We tackle a fundamental economic question: if General Motors or any manufacturer wants to increase unit sales, why does the necessary price cut occur indirectly in the form of a rebate? It is well known that in cases of perfect competition or monopoly a tax (or subsidy) on buyers is equivalent to the same tax (or subsidy) on sellers in that each leads to the same net prices and the same equilibrium quantities. A rebate is effectively a subsidy offered to the buyer, whereas a cut in the wholesale price is effectively a subsidy to the seller. Under conditions of perfect competition or monopoly, a manufacturer should be able to achieve identical outcomes in either of the two ways. The challenge is to explain why rebates or coupons are used. Redemption of coupons and rebates is costly to both consumers and issuers, while a price cut of equivalent size can be implemented at a much lower cost. One would expect the price reduction strategy to dominate the strategy of issuing coupons or rebates to all buyers, yet it apparently does not. Throughout the 1990s, for example, Chrysler, Ford, and General Motors consistently offered a more extensive array of consumer-oriented indirect price cuts (via rebates) than dealer incentive programs. [1]

The most common explanation for the use of rebates and coupons is that they enable either manufacturers or retailers who possess some monopoly power to engage in third-degree price discrimination. [2] Consumers whose reservation prices lie below the price charged by the manufacturer or retailer are induced to purchase the item by a sufficiently large rebate or coupon.

Yet certain aspects of the practice of offering coupons and rebates are inconsistent with the standard price discrimination explanation. Rebates often are given automatically to all customers at the time of purchase. Purchasers of automobiles and other big-ticket items are given the option of applying the rebate to their down payment or receiving the cash. As noted by Hanley [1987] and Higgins [1989], sellers of less costly items recenfly have started providing "instant" rebates or on-package coupons, which can easily be used by all buyers at the time of purchase. These innovations have dramatically lowered redemption costs for consumers and boosted redemption rates. [3] Moreover, Bowman [1980] demonstrates that, holding redemption costs constant, there is a positive relationship between the size of the coupon/rebate and the likelihood of consumer redemption. The implication of this empirical regularity is expressed succinctly by Gerstner and Hess [1991b,p. 5]: "While price discrimination is a reasonable exp lanation for cents-off coupons, for dollars-off rebates such as those given by manufacturers of household appliances it is not. Most consumers are likely to redeem these large rebates."

Gerstner and Hess [1991a, 1991b, 1995] have developed a model in which rebates enhance manufacturer profits even with 100% redemption. They assume two groups of buyers with different reservation prices and different redemption costs. This implies that a $1 reduction in the wholesale price is not equivalent to a $1 rebate offered to all buyers. A sufficient increase in the proportion of low-price buyers can make it profitable for the manufacturer to offer a coupon or rebate but not to offer an equal reduction in the wholesale price. [4] Moreover, when a product is distributed through a single, independent retailer, a trade deal may not be fully passed on to consumers, whereas a manufacturer-to-consumer rebate or coupon is fully passed.

In this article we present an economic theory of rebating that is similar to the Gerstner and Hess model in that it assumes 100% redemption by consumers. Unlike previous analyses, however, we suggest that rebates and coupons can be used to support intertemporal price discrimination by reducing inventory hoarding by retailers. When demand fluctuates so that optimal wholesale prices vary over time, retailers may have an incentive to accumulate inventory during low-price periods for disposal when business conditions (and prices) improve. [5] The strategic use of rebates and coupons, which are price cuts available only to final consumers, can increase manufacturer profits by reducing retailer incentives to undermine the effectiveness of wholesale price discrimination. Rebates and coupons thus have a dynamic function that cannot be duplicated by trade deals.

We introduce our formal model in section II. In section III we offer several complications and extensions of the model, none of which overturn our general results. In section IV we present preliminary empirical evidence on rebating activity that is inconsistent with the price discrimination explanation but fully consistent with our theory. Concluding comments round out our presentation.

II. MODELS OF REBATES

Universal rebates can increase manufacturer profits by reducing the incentives of downstream retailers to hoard inventories when optimal wholesale prices vary over time in a predictable manner. By bypassing the retailers and making direct contracts with buyers, the manufacturer is able to increase the variations in effective prices paid by consumers without concomitantly creating larger incentives for retailers to hold inventories. During profitable, high-demand periods, manufacturer revenues are ordinarily constrained by "competition" from retailer inventories, thus limiting profits. However, by selectively offering rebates to consumers while maintaining high wholesale prices, lowdemand periods can be accommodated without inducing retailer hoarding.

In this section we present a model that illustrates the rebate/inventory mechanism. In all of our analyses, we maintain the following:

* First, we assume that all retailers in our models are price takers in the wholesale market. This assumption rules out strategic stockpiling and/or monopsonistic behavior by retailers, which, though interesting, is not the focus of this paper.

* Second, our analysis is restricted to uniform pricing. The issue of nonuniform contracts between the manufacturer and retailers is a potentially important one, although antitrust laws and practical considerations may limit the ability of the manufacturer to engage in certain types of discriminatory, nonuniform pricing.

* Third, in each case we envision the following time line. In any period, t, the manufacturer observes the level of inventories among retailers and the product demand curve. The manufacturer then announces a uniform price for the good and perhaps a rebate payable only to consumers who buy in period t. The retailers accept these prices and select their most profitable retail market quantities and inventory accumulation. Retailers, while behaving atomistically in input markets, are strategically aware and make rational forecasts of future wholesale prices. Information is complete and symmetric. [6]

Current Period Competition and Payoffs

In any period t, the retailer(s) bring forward an inventory, denoted [I.sub.t-1], from the last period. The inventory level, the level of product market demand, and costs of all players are common knowledge. After observing [I.sub.t-1] and market demand, the manufacturer selects a uniform wholesale price, [w.sub.t] The retailer observes this price and market demand and selects its desired retail market quantity, [Q.sub.t], and level of inventory, [I.sub.t]. Payoffs to the manufacturer and retailer are made and the game enters the next period. Information is perfect.

The profit of the retailer in period t is

(1) [[[pi].sup.r].sub.t] = [P.sub.t][Q.sub.t] - [w.sub.t]([Q.sub.t] + [I.sub.t] - [I.sub.t-1])

- i/2[([I.sub.t-1]).sup.2]

where [P.sub.t] is retail price, [Q.sub.t] is quantity sold at retail in period t, [w.sub.t] is wholesale price, [I.sub.t] represents purchases for inventory by the retailer in period t intended for disposal in period t + 1, [I.sub.t-1] is inventory brought forward into t from t - 1 for disposal in t, i [greater than] 0 is an inventory storage cost parameter, and inventory storage costs are quadratic in the level of inventory brought forward. [7] This formulation utilizes several simplifications. First, inventories experience "one-hoss shay" depreciation, lasting one period. This formulation is necessary for the tractable analysis of what follows but is distinct from formulations in which inventory decays over time at, for example, an exponential rate. Second, our specification of the inventory cost function implies that at zero inventory the marginal cost of storing one unit of inventory is zero. Third, the formulation in equation (1) implies that the retailer experiences no costs of selling aside from the costs of goods obtained either from current wholesale sales or inventory.

Current retail price, [P.sub.t], is assumed to depend on market quantity, [Q.sub.t], and the level of consumer rebate, [R.sub.t], if any. Rebates are direct cash payments from the manufacturer to the customer, bypassing the retailer. Unlike a wholesale price cut, a rebate is always unavailable to the retailer and does not affect the costs of adding inventory as a trade deal does. A final consumer must purchase the item at retail in the rebate period for the rebate to be paid. This fact supports the strategic importance of rebates for intertemporal price discrimination.

Profits to the manufacturer in period t are

(2) [[[phi].sup.m].sub.t] = ([w.sub.t] - [R.sub.t])[Q.sub.t] + [w.sub.t]([I.sub.t] - [I.sub.t-1])

- C([Q.sub.t] + [I.sub.t] - [I.sub.t-1]),

where C(.) is the cost function of the manufacturer.

In any period t, the manufacturer observes [I.sub.t-1] and retail market demand before selecting [w.sub.t]. The retailer takes [w.sub.t] as given and selects [Q.sub.t] and [I.sub.t]. Our specification in equations (1) and (2) involves an additional simplification: For [I.sub.t], for example, to have the same meaning in both definitions, there must be a single retailer. If there were, say, N retailers, one would need to modify equations (1) and (2) so that "market" inventory [I.sub.t] was equal to the sum of the inventories of the various retailers and so on. We ignore this complication for now, but return to it presently. [8]

We begin our analysis by assuming there exists a single retail "monopoly," perhaps a franchisee, such as an automobile dealership. Although the retailer has power in the output market, we impose the condition that inventories not be a strategic tool of the retailer to influence future prices; the retailer is treated as a price-taker in the input market. Thus, the competitive instruments of the players are wholesale prices and retail quantities, with the optimal inventory condition (introduced below) as a "rule of the game." Furthermore, the prices used to calculate optimal inventories must be correct in equilibrium. All this is common knowledge.

Let [I.sub.o] be the initial inventory at time 0, [[P.sub.t]] the sequence of period retail demands, and note that the optimal inventory condition of the retailer is [[I.sup.*].sub.t] = [([delta]i).sup.-1]([delta][w.sub.t+1] - [w.sub.t]) when [[I.sup.*].sub.t] [greater than] 0, and [delta] is the one-period discount factor (0 [less than] [delta] [less than] 1). This condition merely establishes that the cost at the margin of a unit sold in period t should be equal regardless of whether that unit is bought wholesale in t or brought forward as inventory from t - 1. This model cannot be solved by the straightforward application of optimization techniques because we simultaneously require rational expectations about future prices and assume that the manufacturer is unable to credibly commit to a future pricing policy. Thus, should the manufacturer raise wholesale prices in t, it is necessary to determine the effect in equilibrium of this price increase on future prices and price expectations. In particular, one must show that a wholesale price increase will decrease inventory brought forward in equilibrium, despite the fact that decreased inventories today create an incentive for higher prices tomorrow, thus producing a countervailing incentive to hoard. To address this problem, we specify a Markov perfect equilibrium (MPE) for our model.

An MPE is a set of "dynamic reaction functions," [[w.sup.*].sub.t], ([I.sub.t-1]), [[Q.sup.*].sub.t]([[w.sup.*].sub.t], [I.sub.t-1]) such that

(1) [[w.sup.*].sub.t]([I.sub.t-1]) maximizes the period t present value of manufacturer profits for a given inventory level, [I.sub.t-1], and [[Q.sup.*].sub.t]([[w.sup.*].sub.t], [I.sub.t-1]);

(2) [[Q.sup.*].sub.t] ([[w.sup.*].sub.t], [I.sub.t-1] is the retailer's best response to the wholesale price announcement [[w.sup.*].sub.t], given inventory on hand, [I.sub.t-1]; and

(3) [I.sub.t] evolves according to the restriction

[[I.sup.*].sub.t] = max{0, [([delta]i).sup.-1] ([delta][w.sub.t+1]([[I.sup.*].sub.t]) - [[w.sup.*].sub.t]([I.sub.t-1]))}.

The "dynamic reaction functions," [[w.sup.*].sub.t]([I.sub.t-1]) and [[Q.sup.*].sub.t]([[w.sup.*].sub.t], [I.sub.t-1] depend only on the payoff-relevant states of the system rather than the whole history of the game, and are thus Markov strategies in the sense of Maskin and Tirole [1988]. Consequently, any Markov equilibrium of this game is simultaneously a perfect equilibrium: if one player ignores all but the payoff-relevant history, the other can do the same. The reaction functions [[w.sup.*].sub.t] and [[Q.sup.*].sub.t] specify optimal choices by the players. The optimal strategy of the retailer is to: (1) use up inherited inventory while (2) calculating the static, current-period, profit-maximizing price of output given [w.sub.t] and (3) purchasing the profit-maximizing current output less inherited inventory.

The Fudenberg and Tirole [1991] conditions for optimal Markov strategies allow us to prove that a wholesale price increase at t will not in equilibrium increase inventories at future dates. Letting [[[pi].sup.m].sub.t] denote the manufacturer's stage game (i.e., current) payoff, we have the conditions ([[partial].sup.2][[[pi].sup.m].sub.t]/[partial][w.sub.t][I.sub.t-1] = -1 [less than] 0, and [[partial].sup.2][[pi].sub.t]/[partial][Q.sub.t][partial[I.sub.t-1] = 0. This shows that the stage payoffs satisfy a negative sorting condition, implying that a higher-state variable, [I.sub.t-1], makes a lower action [w.sub.t] more desirable for the manufacturer. In other words, the greater the inventory sold by retailers in period t, the lower the optimal wholesale price the manufacturer selects in t. Because the game is separable and sequential, we can conclude that the optimal wholesale price, [[w.sup.*].sub.t]([I.sub.t-1]), is decreasing in [I.sub.t-1]. Payoffs to the manufacturer are decreasing in the inventory level, and, for any pattern of future demands, the present value of profits in period t is decreasing in [I.sub.t-1].

We now illustrate the effect of a rebate introduced in period t-k, k [epsilon] [1, 2,...], when the equilibrium initially implies that [[I.sup.*].sub.t-k] [greater than] 0, [[I.sup.*].sub.t-k+1] [greater than] 0,... [[I.sup.*].sub.t-1] [greater than] 0, [[I.sup.*].sub.t] = 0. Our analysis utilizes the relationship [[I.sup.*].sub.t] = [([delta]i).sup.-1]([delta][[w.sup.*].sub.t+1]([I.sub.t]) - [[w.sup.*].sub.t]([I.sub.t-1])), so that we have

(3) [delta][[I.sup.*].sub.t] = [([delta]i).sup.-1]([delta][[w.sup.*].sub.t+1]([delta][[I.sup.*].sub. t])

- [[w.sup.*].sub.t]([delta][I.sub.t-1])([I.sub.t-1])).

For any period, optimality requires

(4) [delta][[I.sup.*].sub.t](1 - [(i).sup.-1][[w.sup.*].sub.t])

= -[([delta]i).sup.-1][[w.sup.*].sub.t-1]([delta][I.sub.t-1]).

Since [[w.sup.*].sub.t] [less than or equal to] 0 for any t, we conclude that [delta][[I.sup.*].sub.t] and [delta][[I.sup.*].sub.t-1] have the same sign. Noting that at period t - 1 an increase in [w.sub.t-1] lowers [[I.sup.*].sub.t-1] (and [[I.sup.*].sub.t] = 0 by assumption) and working backward, it is apparent that an increase in [w.sub.t-k] lowers equilibrium inventories in periods t-k, t-k+1,..., t-1, and, since [[I.sup.*].sub.t] = 0 by assumption, [delta][[I.sup.*].sub.t] = 0 for a marginal change in [w.sub.t-1]. Summarizing, if periods t-k through t-1 exhibit positive inventories, and period t exhibits zero inventory accumulation, then a wholesale price increase in period t-k lowers inventories in periods t-k, t-k+1,..., t-1, has no effect on inventory in period t (which is zero), and raises optimal wholesale prices in periods t-k+1, t-k+2,..., t-1, and t. The wholesale price change also has a "direct" effect of discouraging retail sales in period t-k, thereby reducing wholesale sales for the current market in t-k.

We now consider whether a rebate matched to a wholesale price increase in period t-k is profitable to the manufacturer. Performing the experiment of raising [w.sub.t-k] and [R.sub.t-k] by a small, identical increment, the effect on profits of the manufacturer is

(5) [delta][[pi].sup.m] = [[[sigma].sup.k].sub.j=0][[delta].sup.j][([w.sub.t-k+j] - [C'.sub.t-k+j])

- [delta]([w.sub.t-k+j+1] - [C'.sub.t-k+j+1])][delta][I.sub.t-k+j]

because for a small change in [w.sub.t-k], the induced effects of tiny changes in future prices on profits are zero when such prices are optimal to begin with. The existence of positive inventories in periods t-k, t-k+1,..., t-1 presupposes that the relationship [delta][w.sub.t-k+1] - [w.sub.t-k] [greater than] 0 holds. The rebate cum wholesale price increase experiment is guaranteed to be profitable when [delta][m.sub.t-k+j+1] [greater than] [m.sub.t-k+j], where [m.sub.i] is the margin in period i, since [delta]I [less than or equal to] 0 in each period.

III. COMPLICATIONS AND EXTENSIONS

Several points deserve consideration here. First, downstream market structure changes do not undermine our conclusion. Suppose, for example, that the downstream retail sector exhibits competitive behavior, so that [[P.sup.*].sub.t] = [[w.sup.*].sub.t]. This benefits the manufacturer, but has no effect on inventory behavior by retailers. A small adjustment to the analysis then produces the conclusion that rebates are profitable under the condition that equation (5) is positive, although equilibrium prices are not the same.

The existence of multiple retailers, either franchise monopolies or competitors, is also an interesting complication. In these cases, total inventory in t, [I.sub.t], is just the sum of retailer inventories. When all retailers are identical, a symmetric equilibrium for N firms involves qualitatively similar behavior. Again, however, rebates with wholesale price increases are profitable when discounted margins are rising with time.

The comparative static properties of the equilibrium are potentially empirically useful. Unfortunately, the complexity of the game in its most general setting precludes such an analysis. However, some insight into both comparative statics properties and the intuition underlying the mechanism can be presented using a very simple example.

A Simple Example

Let there be two periods (t = 1, 2), no discounting, costless manufacturing, and competitive and costless retailing. [9] Suppose that period 1 is a "low" demand period, and t = 2 has "high" demand. To obtain closed-form solutions for all choices, assume that [Q.sub.1] = [A.sub.1] - [p.sub.1] - [R.sub.1], and [Q.sub.2] = [A.sub.2] - [p.sub.2] - [R.sub.2], where [A.sub.2] [greater than] [A.sub.1] Solving by backward induction without rebates yields the conditions:

(6) (i) [[[pi].sup.m].sub.2] = [[([A.sub.2] - [I.sub.1])/2].sup.2]

(ii) [[I.sup.*].sub.t] = [(i + .5).sup.-1] ([A.sub.2]/2 - [w.sub.1]).

Result (6)(ii) implies that [partial][[I.sup.*].sub.t]/[partial][w.sub.1] [less than] 0 for an arbitrary increase in [w.sub.1]: this illustrates the consequence of the negative sorting condition satisfied by the optimal strategies. Direct calculation illustrates that for introduction of an increase in [w.sub.1] with an equivalent increase in [R.sub.1]([delta][w.sub.1] = [delta][R.sub.1] [greater than] 0), one obtains [delta][[pi].sup.m] = [(i+.5).sup.-1]([[w.sup.*].sub.2] - [[w.sup.*].sub.1]), where [[w.sup.*].sub.1] = ([A.sub.2] + [A.sub.1](1 + 2i))/4(1+i) and [[w.sup.*].sub.2] = ([A.sub.1] + [A.sub.2](1 + 2i))/4(1+i). With rebates used optimally, one obtains [[w.sup.*].sub.1] = [A.sub.2]/2, [[w.sup.*].sub.2] = [A.sub.2]/2, [[R.sup.*].sub.1] = ([A.sub.2] - [A.sub.1])/2, [[R.sup.*].sub.2] = 0. Thus, rebates eliminate wholesale price differences and inventories completely in this simple example. We note also that demand shifts have interesting effects: If [A.sub.1] increases, [[w.sup.*].sub.1] and [[w.sup.*].s ub.2] are unaffected but [[R.sup.*].sub.1] rises. If [A.sub.2] increases, all prices rise (except [[R.sup.*].sub.2], which is, of course, zero). Neither [[w.sup.*].sub.1], [[w.sup.*].sub.2], nor [[R.sup.*].sub.1] depend on the carrying cost, i.

These results illustrate the basic intuition of the rebate mechanism: by bypassing retailers, the manufacturer can manipulate effective prices without suffering the consequences of rational inventory behavior by retailers. Inventories are competition for the manufacturer, and reductions in wholesale price differences between periods can lessen this competition. Rebates allow the manufacturer to do this while maintaining discriminatory retail prices. [10]

IV. SUPPORTING EVIDENCE

Our findings have several empirical implications. First, we expect universally applied rebates to be issued during low-demand periods. Second, rebates should reduce retailer "speculative" inventories below the level of inventories expected when rebates are not used. Finally, rebates should be associated with a convergence in wholesale prices across time. All of these conclusions are potentially testable, however, reliable information on rebate activities, wholesale prices, and demand conditions is difficult to obtain. Thus, we limit ourselves here to informal, suggestive evidence.

We focus our analysis on the U.S. automobile industry, in which rebating is a well-established marketing practice. Two sources of data permit us to shed admittedly casual empirical light on rebating activity: newspaper advertisements and Automotive News. Figure 1 reports the fraction of display ads for automobiles, published in the Atlanta Constitution and the Chicago Tribune from January 1975 through December 1997, in which rebates to buyers were identified explicitly. [11] This fraction is markedly higher in periods associated with economic downturns--1975, the early 1980s, and just prior to the recession of the early 1990s. This pattern seems to be consistent with our theory of rebating, which holds that automatic rebates should be used in low-demand periods. We note that rebating seems to have become an increasingly important component of automobile manufacturers' pricing strategy since the mid-1980s. This may help explain the otherwise inconsistent finding of relatively high fractions of advertisements w ith rebates in the strong economic years in the mid-to-late 1990s.

In Figures 2 and 3 we identify the number of active rebate programs offered by Chrysler, Ford, and General Motors per month, as reported in the Incentive Watch section of Automotive News, and the average dollar amount of the rebate (by month) for the period January 1990 through December 1995. [12] We observe considerable month-to-month variation in rebating activity, measured in terms of both numbers of rebate programs and average dollar amount of rebate offered. The average rebate amount clearly exhibits a seasonal pattern. The drop in October closely follows the introduction of new models each year. The summer and early fall rebates are designed to clear the lots of the "old" models prior to introduction of the "new" models. This appears to be consistent with our theory, as we would expect demand for the "old" model to fall at the end of the model year. However, we do not have data on manufacturer wholesale prices or dealer inventories; a more encompassing analysis should relate the observed variation in re bates to both.

V. CONCLUSION

The conventional wisdom about the economic function of rebates and coupons is that they facilitate third-degree price discrimination by manufacturers by serving as the mechanism whereby individuals self-sort by willingness to pay. Conventional wisdom does not, however, explain manufacturer use of rebates and coupons when redemption is complete. In this paper, we demonstrate that strategic use of rebates and coupons can increase manufacturers' profits by mitigating arbitrage by retailers across temporally separated markets.

Our theory differs from the recent contributions of Gerstner and Hess, who provide explanations for manufacturer-to-consumer rebates and coupons with 100% redemption. They argue that such a strategy may be used by a manufacturer to coordinate price promotions when distribution occurs through a channel. As argued by Gerstner and Hess, the direct-to-customer rebate strategy is more desirable from the manufacturer's perspective than a trade deal, in which the manufacturer reduces the wholesale price to the distributor, because the retailer may not pass the full price reduction on to customers. We demonstrate that even when trade deals are fully passed on, retailers could cut into a monopoly manufacturer's profits by accumulating inventory at the low price for resale once the manufacturer discontinued the trade deal. For this reason, the manufacturer would find a rebate/coupon strategy more profitable than a trade deal.

To frustrate the arbitrage behavior of consumers, the manufacturer of a commodity with relatively low storage costs will find it useful to limit the number of units that consumers can purchase with the rebate/coupon. The fact that we observe such limitations on consumer redemption of coupons is consistent with the notion that manufacturers may worry about the downstream inventory aspects of fluctuating prices.

(*.) We acknowledge with appreciation the assistance of Amy Smith of PROMO Magazine and especially Jack Teahen of Automotive News. Helpful comments were received from Preston McAfee, David Kamerschen, Nancy Lutz, Scott Masten, William Neilson, John Sophocleus, and two anonymous reviewers. We are, however, solely responsible for any errors.

Ault: Associate Professor of Economics, Auburn University, Auburn, Ala. 36849. Phone 1-334-844-2919, Fax 1-334-844-4615, E-mail auttric@mail.auburn.edu

Beard: Associate Professor of Economics, Auburn University, Auburn, Ala. 36849. Phone 1-334-844-2918, Fax 1-334-844-4615, E-mail beardtr@mail.auburn.edu

Laband: Professor, Forest Policy Center, 205 M. White Smith Bldg., Auburn University, Auburn, Ala. 36849. Phone 1-334-844-1074, Fax 1-334-844-1084, E-mail labandn@mail.auburn.edu

Saba: Associate Professor of Economics, Auburn University, Auburn, Ala, 36849. Phone 1-334-844-2922, Fax 1-334-844-4615, E-mail sabaric@mail.auburn.edu

(1.) Our claim in this regard comes after analyzing the customer and dealer incentive programs advertised weekly in Automotive News from January 1, 1990, through December 31, 1995. In 1995, for example, automobile rebate programs outnumbered dealer incentive programs by a 10-to-1 margin.

(2.) Interested readers can consult Conlisk et al. [1984], Jeuland and Narasimhan [1985], and Narasimban [1984].

(3.) See Manufacturers Coupon Control Center [1988], Miracle [1995].

(4.) Conlisk et al. [1984] employ a similar model to explain periodic sales by a durable goods monopolist.

(5.) Our argument has its genesis in the well-documented notion that manufacturer trade deals to distributors influence the distributors' inventory behavior (see, for example, Goodman and Moody [1970], Chevalier and Curhan [1976], and Blattberg and Levin [1987]).

(6.) We assume throughout that the manufacturer is unable to credibly commit to future prices. This issue (price precommitment) is reprised in note 7.

(7.) The quadratic form of inventory costs, familiar from the literature on inventories, is not necessary to our general results but greatly simplifies some of our analytical presentation.

(8.) We thank an anonymous referee for pointing this out to us.

(9.) One implication of these assumptions is that retail and wholesale prices will be identical.

(10.) As pointed out to us by an anonymous referee, with a monopolist retailer the optimal retail prices are [[p.sup.*].sub.1] = .25[A.sub.1] + .5[A.sub.2] and [[p.sup.*].sub.2] = .75[A.sub.2]. The monopolist retailer increases price in the second (high-demand) period relative to the first (low-demand) period, even though the wholesale prices are the same in both periods. Thus, the manufacturer is not able to capture all of the surplus when the retailer has monopoly power. Note also that in this analysis if [A.sub.2] increases, all wholesale and retail prices increase, as does the period 1 rebate. However, if [A.sub.1] increases, both wholesale prices and the retail price in period 2 remain unchanged, the retail price in period 1 increases, and the rebate in period 1 decreases. We extend special thanks to this reviewer for these contributions to our analysis. We note also that, if the manufacturer could commit to future prices, one obtains [partial][[I.sup.*].sub.1]/[partial][w.sub.1] = [(-i).sup.-1], whereas without price pre-commitment one obtains [partial][[I.sup.*].sub.1]/[partial][w.sub.1] = -[(i + 1/2).sup.-1].

(11.) We analyzed advertisements placed in the first Sunday paper of each month.

(12.) We analyzed this information for the first issue of each month (Automotive News is a weekly publication). Though our discussion and data are limited to the automobile industry, universal rebates are used in the sale of other big-ticket items, like home appliances, computers and other electronics, furniture, and mobile homes. A variety of low-cost items now come packaged with instant-redemption coupons.

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