Intertemporal Price-Quality Discrimination and the Coase Conjecture

Journal of Mathematical Economics 42 (2006) 896–940

Intertemporal price–quality discriminationand the Coase conjecture

Praveen Kumar∗C.T. Bauer College of Business, University of Houston, Houston, TX 77204-6021, USA

Received 8 March 2002; received in revised form 18 January 2006; accepted 20 April 2006Available online 17 July 2006

Abstract

We examine time-consistent intertemporal price–quality discrimination by a durable goods monopolist,when there are a continuum of buyer demand-intensities with respect to product quality, and it is profitablefor the monopolist to trade with the marginal buyer-type (i.e., the “gap” case). We show that along everysubgame perfect equilibrium path, with probability 1, prices and qualities decline over time, and the marketis completely and monotonically depleted according to buyer-type in a finite number of offers. But, unlike thefixed quality literature, the monopolist may randomize over price–quality offers along the equilibrium path.We also show that the Coase conjecture continues to be valid here, but in a form that is significantly differentfrom the usual formulation. In the limit, as the time between offers evaporates, the monopolist makes acontinuum of offers and perfectly screens the market. However, he effectively cannot price-discriminate,because the equilibrium profits converge to the complete “pooling” profits that would be made if the entiremarket had the marginal buyer-type’s valuation.© 2006 Elsevier B.V. All rights reserved.

JEL classification: D40; D42; C72

Keywords: Durable goods monopoly; Quality discrimination; Coase conjecture

1. Introduction

A durable goods monopolist’s ability to exploit market power to make supranormal profits,and the attendant implications for efficient resource allocation, have been a central focus ofeconomic theory. Coase (1972) has famously summed up the “time-inconsistency” problem ofthe monopolist by arguing that if there are no constraints on capacity and the rate of sales, then

∗ Tel.: +1 713 743 4770; fax: +1 713 743 4789.E-mail address: [email protected].

0304-4068/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2006.04.013

mailto:[email protected]

dx.doi.org/10.1016/j.jmateco.2006.04.013

P. Kumar / Journal of Mathematical Economics 42 (2006) 896–940 897

“the whole process would take place in a twinkle of an eye;” that is, with continuous tradingthe market would be saturated at every point in time, and the durable goods monopoly would beefficient.

However, in practice, sellers routinely attempt to intertemporally price-discriminate by vary-ing not only prices but also product qualities over time. This paper examines whether strategicintertemporal price and quality variation allows the monopolist to make supranormal profits evenunder the “Coase conditions,” when there are no constraints on capacity and the rate of sales, andthe restriction of time-consistency is imposed on the monopolist’s offers.

We study a model in which a durable goods monopolist faces a market with a continuum ofbuyer demand-intensities for quality (or a continuum of buyer-types in terms of quality valuation).The monopolist has constant unit costs of production, and it is strictly profitable for the monopolistto trade with the marginal consumer, i.e., we focus on the “gap” case where the Coase conjectureis valid when product quality is fixed.1 In contrast to this literature, we allow the monopolist tomake different price–quality offers over time.2

Strategic intertemporal quality variation is potentially very important for the monopolist’sprice discrimination ability. In the fixed quality model, intertemporal price discrimination canoccur only when buyers have differing marginal willingness to pay for consumption todayversus consumption tomorrow. But under the Coase conditions, the monopolist’s time consis-tent strategy in the “gap” case is to move down the demand curve infinitely fast, leading toan evaporation of intertemporal price-discrimination ability. However, if the monopolist canstrategically vary product quality over time, then he may be able to induce inframarginal buy-ers to accept the initial offers with higher prices and qualities, even if the time between of-fers evaporates, by threatening a rapid (product) quality decline to lower quality offers in thefuture.

It turns out that allowing the monopolist to vary both price and quality over time raises fun-damental issues (regarding intertemporal price discrimination) that go above and beyond theimplications for the Coase conjecture. In the fixed quality model, the pattern of intertemporalprice discrimination is apparent: the inframarginal buyers will purchase first and prices will de-cline over time. Fudenberg et al. (1985) and Gul et al. (1986) sharpen this intuition and showthat there is a generically unique equilibrium path along which prices decline deterministically.But with flexible quality it is possible that there exist equilibria where both prices and qual-ities increase over time (at least for some duration), and the high buyer-types delay purchasein anticipation of higher quality offers. An additional complication is the possibility of ran-domization by the monopolist along the equilibrium path, unlike the fixed quality case wherethe monopolist never randomizes along the equilibrium path (e.g., Ausubel and Deneckere,1989).

1 In the fixed-quality durable goods monopoly literature, the Coase conjecture has been verified for the “gap” caseby Gul et al. (1986). For the “no gap” case, the Coase conjecture is valid if the players in the market follow stationarystrategies (Stokey, 1981), but not if they are allowed to follow history-dependent strategies (Ausubel and Deneckere,1989). Kahn (1986) shows that the Coase conjecture will not hold if there are capacity constraints on production. TheCoase conjecture also does not apply when the monopolist can offer product lines (Kuhn and Padilla, 1996; Wang, 1998)or when there are only a finite number of buyer-types (Bagnolli et al., 1989).

2 But we do not allow the monopolist to offer product lines or quality schedules. However, our results would applywhenever the number of product versions offered at any given time is less than the diversity in buyer tastes. This is theeconomically reasonable assumption whenever there are fixed costs – such as fixed reconfiguration costs for manufacturingassembly lines and fixed selling and distribution costs – in offering additional qualities.

898 P. Kumar / Journal of Mathematical Economics 42 (2006) 896–940

However, and inspite of these potential complexities, the main result of the paper providesa strong characterization of all equilibrium paths. We show that in every subgame perfect equi-librium, the market is monotonically depleted according to buyer-type in a finite number ofoffers, and prices and qualities will decline monotonically over time. Furthermore, the equilib-rium quality in every offer is socially efficient for the marginal buyer-type accepting the offerand the equilibrium price makes this buyer-type just indifferent between accepting the currentoffer and deferring to accept the lower price–quality combination that is offered in the nextperiod.

The arguments required to establish these restrictions on equilibrium paths are consider-ably more involved than those used in the (fixed quality) literature. Consider a market for thedurable good with a continuum of infinitely lived non-atomic buyers indexed by z ∈ [0, 1], andsuppose that consumer demand-intensity for quality is strictly increasing in z. If the productquality is exogenously fixed, then it is well known that along the equilibrium path the remain-ing market is of the form [0, z′), 0 < z′ ≤ 1, and that the marginal consumer (z = 0) will beserved last. With discounting and in the “gap” case, trade occurs with the marginal consumerin a finite number of offers, since the maximal (buyer) valuation in the remaining market musteventually fall so low that profits from immediate dissipation exceed the present value of fu-ture perfect discrimination profits. This “finite dissipation” property implies that all equilib-rium paths satisfy the backwards recursion property, and one can then establish that pricesare deterministically declining along a generically unique equilibrium path. The restriction onthe set of remaining buyers in the market (i.e., the connected interval [0, z′)) is crucial in thisargument.

This sequence of arguments is no longer valid when quality is flexible and the monopolistcan strategically vary product quality. For, the set of buyers in the remaining market will bethe connected interval [0, z′) only under some very restrictive time-paths of product qualities; forexample, if qualities are always non-increasing in expectation over time. However, the equilibriumtime-path of product qualities itself depends on the nature of the remaining market. Hence, thereis an apparent circularity of argument, and the characterization of generic equilibrium paths issignificantly more difficult once the assumption of a fixed quality is relaxed.

We prove directly that the market will be dissipated in a finite number of offers along allequilibrium paths, irrespective of the history of prices and qualities. The key insight here is that,unlike the usual time-inconsistency problem of the durable goods monopolist, with flexible qualitythe monopolist faces the time-inconsistency problem of not raising prices against inframarginalbuyers who delay purchase in anticipation of higher product quality. We use this fact to show thatthe set of subgame perfect price–quality paths is significantly circumscribed, and that the finitedissipation property applies to flexible quality equilibrium paths as well. Consequently, all equi-librium paths satisfy the backwards recursion property, and we are therefore able to characterizethe evolution of prices, qualities, and the residual market simultaneously.

Apart from its intrinsic interest, this characterization is central to the examination of the Coaseconjecture when the monopolist can strategically and intertemporally vary product quality. We findthat as the time between offers evaporates, all equilibrium paths are characterized by a “vanishing-sales” (per offer) property: in an ultimately futile attempt to preserve his price-discriminationability, the monopolist reduces the rate of price decline by reducing the rate of market depletionaccording to buyer-types in a given offer. (Indeed, in the continuous time limit, the monopolistmakes a continuum of offers.) However, the infinitesimal rate of market depletion also reducesthe rate of quality decline, because the monopolist’s time-consistent policy ties the offered qualityto the marginal buyer-type in each offer.


Consequently, the monopolist’s attempt to make inframarginal buyers accept high qualityoffers at prices that generate higher profit margins by threatening them with a rapid rate of qualitydecline is not credible, when there is no restriction on the rate of offers. In the continuous-time limit, there are a continuum of trades with a uniform profit margin on every trade, namely,the margin obtained from trading with the marginal buyer-type in the market (z = 0). Thus,as the time between offers evaporates, the monopolist’s total profits shrink to the “completepooling” profits that would result from serving out the entire market in just one price–qualityoffer.

This result should be contrasted to the outcome in the one-shot game when the monopolistcan offer price–quality menus. In this game, the monopolist can effectively price discriminateby distorting the quality offers to almost every buyer-type (Mussa and Rosen, 1978). Similarly,and in contrast to our result, the monopolist retains intertemporal price discrimination abilityin the continuous time limit when there are only a finite number of buyer-types, because themonopolist’s threat of a non-infinitesimal drop in quality over time is time-consistent (Kumar,2002). An especially interesting implication of this is that increasing the diversity of buyer-typescan have a pernicious effect on a durable goods monopolist’s ability to intertemporally price–quality discriminate under the Coase assumptions.

Our results also complement other aspects of the durable goods monopoly literature. Liang(2000) examines a durable goods monopoly model when the product physically depreciates somestochastic rate.3 He finds that physical depreciation can sometimes allow the monopolist to ef-fectively price discriminate even under the Coase conditions. Essentially, physical depreciationforces the inframarginal buyers to “return” to the market, and the monopolist can strategicallyexploit this fact. By comparison, our model assumes infinite durability but allows variation inproduct performance due to other aspects (of the product) that provide utility to buyers.4 Thus,unlike the physical depreciation model, trading can permanently cease in our set-up. This is thecrucial difference that explains our relatively strong characterization of equilibrium price–qualitypaths and the monopolist’s price discrimination ability.

Deneckere and McAfee (1996), Fudenberg and Tirole (1998), and Ellison and Fudenberg(2000) consider models of durable goods monopoly where product quality is upgraded over time,and derive a number of implications for pricing and social welfare. However, these papers generallypresume that the monopolist has the ability to commit to a quality evolution path; i.e., the extent ofthe quality improvement is usually given exogenously. By contrast, this paper examines equilib-rium outcomes when there is no credible way for the monopolist to commit to future quality paths.

In the remaining paper, Section 2 specifies the game and the solution concept. Section 3records some useful benchmark outcomes. Section 4 provides the general characterization of

3 Following the seminal work of Swan (1970), other works that examine durable goods monopoly with physical depre-ciation include Bulow (1982), Rust (1985), Waldman (1996), and Hendel and Lizzeri (1999).

4 There are important differences between intertemporal quality variations that arise due to technological obsolescence(or retrogression) and those that arise due to physical depreciation alone. In the latter situation, product quality – measuredin terms of utility or service flow per unit time – must necessarily be higher for newer goods. Importantly, for products ofa given durability, the difference between the qualities of various product vintages is fixed by the given rate of physicaldepreciation. By contrast, the more general product performance interpretation of quality adopted in this paper allowsthe quality of the new goods to be non-monotonically related to existing product vintages. Moreover, in our model thedifference between the qualities of the various product vintages will generally be time-varying in an endogenous fashion,unlike the physical depreciation models. This difference is exogenously given in the physical depreciation models, sincethe monopolist is not allowed to optimally vary the rate of physical depreciation over time.


equilibrium paths. Section 5 proves the analog of the Coase conjecture in the intertemporalquality discrimination game, and Section 6 concludes. All proofs are collected in Appendix A.

2. The model

We consider the market for an infinitely durable commodity that can be produced at a variety ofquality levels, q ∈ Q ≡ [0, q]. Higher qualities are costlier to produce. Products of quality q ∈ Qare produced at a constant unit cost of bq2/2, b > 0. The market is served by a single producer.The demand for the good is represented by a continuum of non-atomic and infinitely lived buyersindexed by z ∈ Z = [0, 1]. Each consumer is in the market for only one unit of the good and exitsafter making the purchase.

The market opens at regularly spaced intervals, t ∈ I+ (the set of non-negative integers).Market equilibrium in each period t is characterized by a price–quality pair, θt ≡ (pt, qt), offeredby the producer, and the set of buyers accepting this offer. If buyer-type z accepts a price–qualityoffer θt , then his or her utility is

u(pt, qt, z) = δt[(γ + βz)qt − pt]. (1)

Here, 0 < δ < 1, is a discount factor, and γ and β are positive parameters representing the buyers’intensity of preference for quality relative to the numeraire good. γ is a demand-intensity factorthat is common across the various consumer types, while βz is a type-specific demand-intensityfactor.

Thus, there are a continuum of non-atomic (buyer) valuation types in the market, and exceptfor the common valuation parameter (γ), the preference distribution specified in (1) is similarto that adopted by the quality discrimination literature (Mussa and Rosen, 1978, and onwards).The assumption γ > 0 here is the analog of the “gap” assumption in the (fixed-quality) durablegoods monopoly literature. There, the role of the “gap” assumption is to ensure that the marginalconsumer (z = 0) receives a positive price offer along any equilibrium path. Similarly, here theassumption γ > 0 implies that the marginal consumer receives a positive price–quality offer alongany equilibrium path.

The monopolist also discounts future profits with the discount factor δ, and cannot ob-serve the consumer’s “type” (i.e., the index z ∈ Z). But since buyers have heterogeneous val-uations for product quality (cf. Eq. (1)), the monopolist can potentially intertemporally price–quality discriminate by allocating various buyer-types to different prices and qualities overtime.

Let At denote the set of buyers who accept the period-t offer, θt . We shall make appropriatemeasurability restrictions below that will ensure thatAt ∈ �[Z], where�[X] denotes the (Borel)sigma algebra onX. Market equilibrium at date t ∈ I+ is the pair (θt, At). For a given t, a historyup-to the date t is the profile: ht = (θ0, . . . , θt−1;A0, . . . , At−1), and the space of such historiesis inductively generated as the sets Ht = (P ×Q×�[Z])t , where P = R+ is the feasible spaceof prices. (We take H0 = ∅.) Players remaining in the market have a perfect recall (cf. Kuhn,1953).

A pure strategy for the monopolist is the sequence of mappings ω = {ωt}∞t=0, ωt : Ht →P ×Q. In the usual fashion (Kuhn, 1953; Aumann, 1964), a behavior strategy for the monopolistis denoted as the sequence of mappings λ = {λt}∞t=0, λt : Ht → (P ×Q), where (X) is thespace of probability measures on X. Next, consumer strategies are taken to be measurable withrespect to the current offer. So, letH ′

t = Ht ×�[P ×Q]. A strategy for the buyers is the sequenceof mappings α = {αt}∞t=0, αt : H ′

t × Z → {0, 1}. “1” signifies the decision to accept θt , and “0”


denotes the alternative. We note that for each t ∈ I+, and given anyh′t ∈ H ′

t ,αt(h′t , z) is measurable

for each buyer-type z ∈ Z.5 A decision never to purchase gives the buyer zero utility.6

We now specify a subgame perfect equilibrium (SPE) in the game at hand. Let X and Ydenote the space of strategies for the monopolist and the buyers, respectively. A pair of strategyprofiles φ = (ω,α) generates over time a path of price–quality offers, {θφτ = (pφτ , q

φτ )}∞τ=0, and

corresponding sales {m(Aφτ )}∞τ=0, where m(Aφτ ) is the Lebesgue measure of the set of acceptingbuyers a date τ. The pattern of sales can be recursively computed, and given the foregoingmeasurability restrictions, defines the payoffs for the players. We let (φ; δ) and V (φ, z; δ),respectively, denote the present value of the producer’s profits and the utility of buyer-type z,when the discount factor is δ and the chosen strategy pair is φ. That is,

V (φ, z; δ) = max{u(pφ0 , q

φ0 , z), u(pφ1 , q

φ1 , z), . . .

};

(φ; δ) =∞∑τ=0

δτ

[(pφτ − b(qφτ )2

2)m(Aφτ )

]. (2)

A strategy profile φ = (ω,α) is a Nash equilibrium (NE) if and only if

(ω,α; δ) ≥ (ω,α; δ), ∀ω ∈ X; V (ω,αz,α−z; δ) ≥ V (ω, αz,α−z; δ),

∀αz ∈ Yz, z ∈ Z. (3)

Here, αz is the projection of α on the zth component, and similarly for Yz. Similarly, α−z is theprojection of α on the set Z − {z}.

A given strategy profile (ω,α) itself induces strategy sub-profiles (ω|ht ,α|ht ) and (ω|h′t,α|h′

t)

after histories ht and h′t , respectively. A strategy profile φ is a SPE if (ω|ht ,α|ht ) is a Nash

equilibrium for the subgame ht for every ht ∈ Ht, t ∈ I+, and similarly for (ω|h′t,α|h′

t), for any

h′t ∈ H ′

t , for any t ∈ I+. A behavior strategy SPE is similarly defined, by replacing ω with λ.7

Finally, for any fixed δ, we will let �∗(δ) denote the set of SPE for this discount factor, with σdenoting a generic element of this set. And, the set continuation equilibria given any history ht(or h′

t) will be denoted by �∗|ht (or �∗|h′t).

We restrict attention to the subset �(δ) ⊆ �∗(δ) in which deviations by measure zero sets ofbuyers do not change the actions of the other buyers or the monopolist. We thus assume thatplayers’ equilibrium strategies are constant over consumer acceptance histories that differ on setsof measure zero. Let the acceptance sets A and A be called equivalent if m(A− A) = 0. The re-striction then is that for every pure strategy σ ∈ �(·), ωt(·;A0, . . . , At−1) = ωt(·; A0, . . . , At−1),for each t ∈ I+, and every αt(·;A0, . . . , At−1, z) = αt(·; A0, . . . , At−1, z), for each t ∈ I+, everyz ∈ Z. The restriction on the monopolist’s equilibrium behavior strategies are similarly formu-lated. This assumption serves to focus on equilibria where buyers act as price takers, and is the

5 Under the rules of the game, αt(h′t , z) ≡ 0 for every z ∈ ∪t−1

j=0Aj ; i.e., the set of buyers who have already acceptedbefore period t.

6 Consistent with the game-theoretic literature on durable goods monopoly (see, e.g., Gul et al., 1986; Ausubel andDeneckere, 1989), we only consider pure strategies for the buyers because buyers are assumed to accept an offer if theyare indifferent between accepting an offer and rejecting it.

7 The specification of consumer and monopolist payoffs when the monopolist randomizes is a notationally tedious butstraightforward extension of (2). (See Appendix A for details.)


maintained hypothesis in the literature on fixed-quality durable goods monopoly (e.g., Gul et al.,1986; Ausubel and Deneckere, 1989).8

3. Some benchmark outcomes

3.1. Equilibria with fixed quality

As a benchmark, consider the equilibrium outcomes in the model when product quality isexogenously fixed, and the monopolist can only vary price offers over time. Ideally, the monopolistwould like to make the profit-maximizing offer in the first-stage, and cease production. However,this policy is not time-consistent for the monopolist. Buyers will rationally expect the opportunisticmonopolist to intertemporally price discriminate, and continue to make offers till the marginalbuyer-type (z = 0) is served, given that γ > 0. The price will be expected to fall to at least themarginal consumer’s valuation, and hence the inframarginal buyers will not be willing to acceptthe initial (profit-maximizing) offer. The culmination of this logic under the Coase assumptionsis the Coase conjecture.

Gul et al. (1986) formalize the Coase conjecture and show that for the “gap” case there isgenerically a unique SPE where agents play weak-Markov strategies,9 prices decline determin-istically, and the market is dissipated in a finite number of offers. And, as the time betweenoffers evaporates, the market is dissipated immediately with an offer that is arbitrarily close tothe constant unit cost. Observe that if the quality was arbitrarily fixed in the current model, thenthe Coase conjecture would be valid here because there exists a well-defined profit maximizingprice–quality offer for the marginal consumer (z = 0); viz., θ ≡ (p = γ2/b, q = γ/b).

3.2. Flexible quality

Suppose now that the choice of product quality was flexible, as modeled above. Then, it isstraightforward to show that the socially efficient quality allocation for each z ∈ Z is qe(z) =(γ + βz)/b. In a competitive market, this allocation would be offered at cost; i.e., pe(z) = (γ +βz)2/2b. On the other hand, suppose that the monopolist could offer the perfectly discriminatingprice–quality menu that instantaneously screens all the buyer-types in the market. Interestingly,the quality allocation remains socially efficient, but (of course) the monopolist prices to extractthe entire consumer-surplus. Thus, the perfectly discriminating price–quality allocations is thefunction, (pm(z) = (γ + βz)2/b, qm(z) = (γ + βz)/b), yielding a profit of (γ + βz)2/2b, for eachz ∈ Z.

Suppose, however, that the monopolist could attempt intertemporal price–quality discrim-ination subject to time-consistency constraints, as in the model at hand. It is then not ob-vious that the powerful characterization of equilibrium paths available for the fixed-qualitymodel would extend straightforwardly, for the reasons mentioned in the introduction. More-over, the validity of some appropriate form of the Coase conjecture itself needs to beinvestigated.

8 Gul et al. (1986) show that this restriction on equilibrium strategies is not innocuous; i.e., �(δ) ⊂ �∗(δ), in general.9 In other words, players’ strategies along the equilibrium path are strong-Markov and depend only on the measure of

the remaining market along the equilibrium, but off-the-equilibrium path players’ use history-dependent strategies thatcorrelate with the previous period’s price.


4. Equilibria with flexible quality

We provide a strong characterization of all equilibrium paths, summed up in Theorem 1 below.This result asserts that prices and qualities must be monotonically declining along any equilibriumpath, and that the market will be monotonically depleted in terms of consumer demand-intensityin a finite number of offers. Furthermore, there are positive sales every period, as long as themarket is not completely depleted.

Theorem 1. In any σ ∈ �, with probability 1,

(i) the market is entirely served by some finite period T σ ;(ii) m(Aσt ) > 0, for each 0 ≤ t ≤ T σ ;

(iii) qσt > qσt+1 and pσt > pσt+1, for each 0 ≤ t ≤ T σ − 1;(iv) in every period 1 ≤ t ≤ T σ , the set of remaining buyers (if non-empty) is the connected set

[0, z′), for some 0 < z′ < 1.

For the reasons mentioned earlier, our proof strategy is to first establish that the market will bedissipated in a finite number of offers along all equilibrium paths, without making any assumptionon the nature of the remaining market set (or the nature of equilibrium quality paths). This proofis based on a series of facts, stated as Lemmas below.

Lemma 1 establishes that along any equilibrium path, the remaining market set has a Lebesguemeasure that is strictly bounded away from zero. Lemmas 2 and 3 use this property to show thatthe (extensive) marginal consumer in the market cannot expect positive surplus in any subgame.It follows from this property that at every stage, either the remaining market set is empty or itcontains an open neighborhood around buyer-type z = 0 (Lemma 4). Next, Lemma 5 shows thatat least a fixed fraction of any remaining market must be depleted in a specified number of dates,for any history, and along any equilibrium path. Further, this lower bound on the proportion of(the residual) market to be depleted is uniform in that it is independent of both the history and thediscount factor. Lemmas 6–11 then build on these facts to establish the finite dissipation property(Theorem 1(i)).

In turn, the finite dissipation property implies that all equilibrium paths satisfy the backwardsrecursion property, and this property is exploited in Lemma 12 to show that there are positivesales in every period (Theorem 1(ii)) and that qualities and prices are (strictly) monotonicallydeclining over time as long as the market is not fully served (Theorem 1(iii)). Finally, Lemma 13uses Theorem 1(iii) to establish that the set of buyers remaining in the market must be connectedalong the equilibrium path (Theorem 1(iv)).

Unlike the fixed quality model, we cannot rule out randomization when quality is flexible.It is well known that the monopolist never randomizes along the equilibrium path in the fixed-quality model (Gul et al., 1986; Ausubel and Deneckere, 1989). To see why, suppose that in somesubgame the remaining market is [0, z′), and the monopolist randomizes over price offers, withthe expected price Ep. Let p be the supremum of the support of the monopolist’s randomized(pricing) strategy. But, along the equilibrium path, buyer-type z′ must be indifferent between theprevious price, say p−1, and Ep. Normalizing, without loss of generality, the quality at q = 1,the equilibrium condition is: p−1(z′;Ep) = (γ + βz′)(1 − δ) + δEp. It is apparent that p−1 isstrictly increasing in Ep. Hence, it is the monopolist’s strictly dominant strategy to choose onlyp in the current stage, since this allows him to extract a higher surplus from those buyers thataccepted in the previous stage.


However, let quality be flexible, and suppose that buyer-type z′ is indifferent between the offer(p−1, q−1) in the pervious period and the randomized offer (Ep,Eq) in the current period. Then,the equilibrium condition is:p−1 = (γ + βz′)(q−1 − Eq) + δEp. But along the equilibrium path,prices and qualities must be positively correlated for offers to be acceptable to buyers. Hence, theprevious price p−1 is not necessarily increasing in both prices and qualities today, and we cannotrule out ex ante equilibrium price paths where the monopolist is indifferent between high and lowprice–quality offers.

To deal with randomization, we will use the following notation. Fix any strategy λ for the mo-nopolist, and at any ht ∈ Ht , t ∈ I+, let μλ

t : �[P ×Q] ×Ht → [0, 1] be a probability measuresuch that for anyB ∈ �[P ×Q], μλ

t (B, ht) is the probability that the monopolist will offer θt ∈ Bin the subgame ht . Furthermore, let ϒλ

t (ht) be the support of μλt (ht), and denote the projections

of ϒλt (ht) on P and Q as ϒλ

t,P (ht) and ϒλt,Q(ht), respectively. Thus, all equilibrium price–quality

offers in period t belong to ϒλt (ht). We will let Dσt denote the set of remaining buyers in period

t along any equilibrium path σ. The (extensive) marginal and the supremum buyer-types in themarket will be denoted by zt ≡ inf Dσt and zt ≡ supDσt , respectively.

Lemma 1 below uses the continuity of consumer preferences to show that along any equilibriumpath either the market is completely served or the remaining set of buyers has a measure that isstrictly bounded away from zero.10

Lemma 1. Fix any σ ∈ �. At every t ∈ I+, for any given ht ∈ Ht , eitherDσt = ∅ orm(Dσt ) ≥ ε,for some ε > 0.

Lemma 1 implies that every offer made by the monopolist along the equilibrium path mustbe unimprovable in terms of expected profits in the subgame. Lemma 2 uses this fact to put animportant restriction on the terminating offer (the offer that is accepted by all remaining buyers),if such an offer exists. We establish that the terminating offer must extract the surplus of the(extensive) marginal buyer-type. This result thus extends (Gul et al., 1986, Lemma 1) to the caseof flexible quality.

Lemma 2. Fix any σ ∈ �. Suppose that there exists some period T ∈ I+ and hT ∈ HT such thatDσT �= ∅ but DσT+1 = ∅. If θT ∈ ϒλ

T (hT ), then pT = (γ + βzT )qT .

Lemma 3 uses Lemma 2 to prove a “zero surplus” property for the (extensive) marginal buyer-type in any subgame. That is, we show that along the equilibrium path, in every subgame, the(extensive) marginal buyer-type must have zero expected utility.

Lemma 3. Fix anyσ ∈ �. For any t ∈ I+, and everyht ∈ Ht such thatDt �= ∅, αt(ht, (p, q), z) =1,∀z ∈ Dt , for any offer (p, q) such that p < (γ + βzt)q.

According to Lemma 3, any offer that gives the extensive marginal consumer a non-negativeutility must be the terminating offer, since it is accepted by all the remaining buyers. We show infact that any inframarginal buyer-type who rejects such an offer will regret ex post. It immediatelyfollows from Lemma 3 that the marginal buyer-type z = 0 must be in the market, as long as themarket is not completely served. Moreover, Lemmas 2 and 3 imply that the perfectly discriminatingprice–quality offer for the marginal buyer-type; i.e., θ (cf. Section 3.1) will be accepted by anyremaining set of buyers. Consequently, we will refer to θ as the “pooling” offer.

10 Due to the possibility of behavior strategy equilibria, all the results hold with probability 1.


Importantly, Lemma 3 does not necessarily imply thatDσt is the connected set [0, z′) (for t ≥ 1).But Lemmas 1 and 3 together do imply that along any equilibrium path the set of remaining buyerswill be a countable union of intervals, and will contain some neighborhood around the marginalconsumer.

Lemma 4. For any σ ∈ �, at any ht ∈ Ht, t ∈ I+, eitherDσt = ∅ orDσt = [0, zt(1)) ∪ �t , where�t is a countable union of intervals whose complement is closed.

We now build on Lemmas 1–4 to establish that the market must be dissipated in a finite numberof offers, along any equilibrium path. As noted above, Lemma 5 below first shows that along anyequilibrium path at least a positive fraction of the remaining market is depleted in a finite numberof offers. Moreover, this minimal depletion proportion is uniformly bounded below; i.e., it isindependent of the discount factor and the price–quality history.

Lemma 5. Fix any σ ∈ �(δ) and any ht ∈ Ht, t ∈ I+, such that m(Dσt ) > 0. Then, there existsan integer κ(δ) such that m(Dσt+κ(δ)) ≤ (1 − ν)m(Dσt ), where ν ≡ [γ2/2(γ + β)2] > 0.

Observe that ν is increasing in γ . Thus, the minimal market depletion between periods t andt + κ(δ); viz., νm(Dσt ), is an increasing function of γ . Intuitively, the higher is the valuation ofthe marginal consumer (γ), the greater are the profits from immediately serving out the entireremaining market, and higher is the monopolist’s opportunity cost of delaying market depletion.11

It can also be shown (cf. Appendix A) that κ(δ) is a non-decreasing function of δ, and thatlimδ↑1 κ(δ) = ∞.

An immediate consequence of Lemma 5 is that any given positive measure subset of the marketmust be depleted in a finite number of offers along any equilibrium path.

Lemma 6. Fix any 0 < m < 1. There exists some finite date T (m, δ) such thatm(DσT (m,δ)) ≤ m,in every σ ∈ �(δ).

Next, we also verify that if the remaining market set is connected and has a sufficiently smallmeasure, then the market will be completely dissipated in the next offer with probability 1. Becauseof discounting, the monopolist prefers the immediate profits from the pooling offer rather thanthe delayed profits from implementing intertemporal price–quality discrimination.

Lemma 7. Fix any σ ∈ �(δ). There exists some z(δ) > 0 such that for any ht ∈ Ht, t ∈ I+, ifDσt = [0, z′), with z′ ≤ z(δ), then Dσt+1 = ∅.

If Dσt was guaranteed to be connected, as in the fixed quality case, then Lemmas 5–7 wouldessentially yield the finite dissipation result, since Lemma 6 would ensure thatDσt = [0, z′), withz′ ≤ z(δ), in a finite number of offers along any equilibrium path. But with flexible quality theremaining market set can only be restricted in the fashion of Lemma 4. However, Lemmas 4 and5 do ensure that,

Lemma 8. For a given δ, there exists a finite date T (δ) such thatDστ = [0, zτ(1)) ∪ �τ, ∀τ ≥ T (δ),where zτ(1) ≤ z(δ) (cf. Lemma 7). Furthermore, if Dστ = [0, zτ(1)) ∪ �τ , then a consumer set

11 Note that ν = 0, trivially, if γ = 0; i.e., there is no meaningful lower bound on the minimal market served in a finitenumber of offers along any equilibrium path. This point verifies the intuition in the literature that the “gap” assumptionimposes considerable power (Fudenberg et al., 1985; Gul et al., 1986; Ausubel and Deneckere, 1989). If the valuation ofthe marginal buyer-type is not strictly bounded away from the constant cost of production of the marginal quality (i.e.,zero), then the finite dissipation result (Theorem 1(i)) will not generally obtain.


B ⊂ [0, zτ(1)), with m(B) > 0, must accept an offer from the seller in a finite number of periodsafter the date τ.

The first part of Lemma 8 follows from Lemma 6 which establishes that the measure of theresidual market must fall below z(δ) (a positive number) at a finite date. The last part of thisLemma also uses similar logic: if no consumer set of positive measure in [0, zτ(1)) accepts anoffer in a finite number of dates, then the measure of the residual market must remain abovezτ(1) > 0, contradicting Lemma 6.

The next two results (Lemmas 9 and 10 below) now complete the argument needed to establishthe finite dissipation property. Lemma 9 first shows that there is an essential “connectedness” inthe set of accepting buyers along the equilibrium path. That is, every buyer-type that lies betweenthe infemum and supremum of the acceptance set in any period accepts the same offer. Let, forany h′

t ∈ H ′t , t ∈ I+, z−t (h′

t) ≡ inf At(h′t), and z+t (h′

t) ≡ supAt(h′t). Then,

Lemma 9. For every t ∈ I+, for any givenht ∈ Ht and θt ∈ ϒλt (ht), if 0 < z−t (h′

t) < z+t (h′t) < zt ,

then, upto closure, Dσt+1 = Dσt − Aσt (h′t) = {z ∈ Dσt |z < z−t (h′

t)}⋃{

z ∈ Dσt |z > z+t (h′t)}

.

The next Lemma uses Lemma 9 to prove that if the neighborhood around the marginal consumerinDσt is sufficiently small so that zt(1) ≤ z(δ), and if a consumer set of positive measure in [0, zt(1))accepts θt , then the market is completely dissipated in at most two further offers.

Lemma 10. Fix any σ ∈ �(δ). Take any t ∈ I+, and any ht ∈ Ht such thatDσt = [0, zt(1)) ∪ �t(cf. Lemma 4). If zt(1) ≤ z(δ) (cf. Lemma 7), and there exists some θt ∈ ϒλ

t (ht) and some B ⊂[0, zt(1)), with m(B) > 0, such that αt(ht, θt, z) = 1, ∀z ∈ B, then Dσt+2 = ∅.

If the hypothesis of Lemma 10 is true, then Lemma 9 implies thatDσt+1 = [0, zt+1(1)) ∪ �t+1,where zt+1(1) < z1(t) ≤ z(δ) and �t+1 = {z ∈ Dσt |z > z+t (h′

t)}. However, Lemma 10 shows that�t+1 must then be empty. Observe that if the infemum buyer-type in �t+1 rejects θt , then it mustexpect strictly higher utility in σ|h′

t. But Lemma 10 establishes that this assumption is violated

in every continuation game. Arguing inductively, the acceptance set in period t must then coverall the inframarginal buyer-types; i.e.,Dσt+1 = [0, z−t (h′

t)). But, by hypothesis, z−t (h′t) < z(δ), and

hence Lemma 7 implies that Dσt+2 = ∅.We can now collect together the previous Lemmas to establish the finite dissipation property

(Theorem 1(i)).

Lemma 11. Fix any σ ∈ �(δ). There exists some T σ ∈ I+ such that DσTσ+1 = ∅.

The finite dissipation property ensures that all equilibrium paths satisfy the backward recursionproperty. We use backwards induction from the terminal offer (θ) to provide a strong characteri-zation of equilibrium price, quality, sales, and market depletion patterns.12

Lemma 12. Fix any σ ∈ �(δ). For any given ht ∈ Ht, 0 ≤ t ≤ T σ , if θt ∈ ϒλt (ht), then

(i) m(At(h′t)) > 0;

(ii) At(h′t) = {z ∈ Dt(ht)|z−t (h′

t) ≤ z < zt(h′t)};

(iii) pt = (γ + βz−t (h′t))qt − δVσt+1(h′

t , z−t (h′

t));(iv) qt = (γ + βz−t (h′

t))/b > qt+1.

12 We recall that, by the established notation, h′t = (ht, θt), z

−t (h′

t) ≡ inf At(h′t), z

+t (h′

t) ≡ supAt(h′t), zt = inf Dt(ht),

and zt = supDt(ht). Also, Vσt+1(h′

t , z) denotes the expected utility of buyer-type z along σ|h′t. See Appendix A for the

complete backwards induction argument.


Lemma 12 establishes properties (ii) and (iii) of Theorem 1. In fact, it provides substan-tial additional economic content. For instance, it shows that the equilibrium quality offered atevery date is efficient for the (extensive) marginal buyer-type accepting the offer at that date.Moreover, the price is set so as to make this buyer-type just indifferent between accepting orrejecting the offer (and waiting for the next period’s offer). We will return to these propertiesbelow.

It now remains to demonstrate connectedness of the remaining market set (cf. Theorem 1(iv)).The argument, given in the following Lemma, also completes the proof of Theorem 1.

Lemma 13. Fix any σ ∈ �, and suppose that for every t, qσt ≥ qσt+1. Then, for every period τ,and z, z′ such that z′ > z, if z �∈ Dστ , then z′ �∈ Dστ as well.

Theorem 1 (specifically, Lemma 12) provides a characterization of prices and qualities ex post;i.e., in terms of the realizations of the monopolist’s equilibrium behavioral strategy, λ. However, itis useful to derive the expected price, quality, and market depletion in equilibrium. To this end, weuse the fact that along any equilibrium path the measure of the remaining market set 0 < z′ ≤ 1 issufficient to determine all future prices, qualities, and sales. Moreover, the monopolist’s presentvalue of profits with the state z′ satisfies the dynamic programming (DP) equation:

(z′) = maxz∈[0,z′]

[(p(z) − b(q(z))2

2)(z′ − z) + δ (z)

], (4)

s.t., (i) q(z) = (γ + βz)

b(≡ q(z)); (ii)p(z) = (γ + βz)q− δV ∗(z)(≡ p(z)). (5)

Here, V ∗ : Z → R+, is the expected utility function for the state, z ∈ Z, and will be soon quan-tified. Constraints (i) and (ii) are derived from Lemma 12. Now, fix any state 0 < z′ ≤ 1, and letz(z′) be the argmax correspondence of (4) and (5). We define,

z∗(z′) ≡{z ∈ z(z′)|(γ + βz′)q(z) − p(z) = inf

x∈z(z′)[(γ + βz′)q(x) − p(x)

]}(6)

Then, given (6), one recursively defines, V ∗(z′) ≡ (γ + βz′)q(z) − p(z), ∀z ∈ z∗(z′).z∗(z′) is the subset of the optimal correspondence from the (DP) equation with the property that

the associated optimal price and quality yields the minimum expected utility to the inframarginalconsumer z′. Eq. (6) thus defines the support of the monopolist’s equilibrium market depletionstrategy: for any given state, z′, the equilibrium market served in the current period is (z′ − z), z ∈z∗(z′). However, if z∗(z′) is multi-valued, then the monopolist may randomize over its elements,and there can be multiple equilibrium paths. In particular, the randomization rule can be history-dependent.

To reformulate an earlier point, it is transparent from (6) why the monopolist will not randomizewhen quality is fixed: for, in this case, z∗(·) is trivially a singleton, because it picks the elementassociated with the lowest price. However, when both prices and qualities are flexible, then therecan exist multiple price–quality pairs that can satisfy (6). But notice that z(·) is a monotone non-decreasing correspondence, and hence it can be multivalued on at most a countable number ofstates.

We now state the evolution of prices, qualities, and sales along generic equilibrium paths. Atany t ∈ I+, with the state, zt , the relevant history is ht = (zt, θt), where θt = (θ0, θ1, . . . , θt−1), isthe history of price–quality offers made by the monopolist upto the period t. A behavior strategyfor the monopolist is the sequence of mappings, λ = {λσt : Z × (P ×Q)t−1 → ([0, 1])}Tt=0, so


that the image of λt(zt, θt) is a probability measure on the remaining market set [0, zt). The supportof λ is the recursively generated correspondence z(zt) (cf. Eq. (6)).

Theorem 2. In any σ ∈ �(δ), for every t ∈ I+, and given any ht = (zt, θt), there is a proba-bility measure μt(zt, θt ; δ) with a support zt(zt ; δ) ⊆ [0, zt), such that along the equilibrium paththe expected next stage state is zσt+1(zt, θt ; δ) = Eμ(z) < zt . Moreover, the equilibrium expectedprice–quality offer in period t is (pσt (zt, θt ; δ), qσt (zt, θt ; δ)), with

qσt (zt, θt ; δ) = (γ + βzσt+1(zt, θt ; δ))

b(7)

pσt (zt, θt ; δ) = Eμ

((γ + βz)2

b− δV ∗σ

t+1(z; δ)

), (8)

where V ∗σk (z; δ) is the indirect expected utility of the buyer-type z in period k ∈ I+, when the

state is z. Furthermore, all σ ∈ �(δ) are payoff-equivalent for the monopolist: (σ; δ) = (δ),for some (δ) > γ2/2b, ∀σ ∈ �(δ).

Thus, along any σ ∈ �, at any t ∈ I+, and given any history ht , the monopolist’s presentexpected value for profits ( σt ) and the equilibrium correspondence (zt) are functions of the state(zt) alone; that is, they are strong-Markov. But, as noted before, the equilibrium price, quality, andmarket dissipation policies may be non-stationary (or history dependent) since the monopolistmay use history-dependence randomization rules over z∗σt , whenever it is multivalued. However,all equilibrium paths must be payoff-equivalent for the monopolist since the present expectedvalue of profits in any subgame depend only on the state. But the expected utility of every buyer-type may depend on the price–quality history; that is, the various continuation equilibria need notbe payoff-equivalent for the remaining buyers.

5. Intertemporal quality discrimination and the Coase conjecture

The characterization of equilibria in Theorems 1 and 2 allows us to address another basicissue: if the monopolist has constant unit costs of production, and can make offers arbitrarily fast,can he still use intertemporal quality variation to price discriminate, subject to time-consistencyconstraints? In this section, we explicate equilibrium outcomes in the limit as δ ↑ 1. Our mainresult here is based on a “vanishing-sales” property of the monopolist’s optimal policy as thetime between offers evaporates: as δ approaches 1, the monopolist optimally shrinks the set ofbuyer-types served by each offer in an ultimately futile attempt to protect his price-discriminationability.

Indeed, in the limit as δ ↑ 1, the monopolist makes a continuum of price–quality offers, and thelimiting price–quality allocation of buyer-types are expressed through continuous functions onZ.These allocations retain the basic properties of the equilibrium allocations specified in Theorem2, subject to the provision that the difference between two adjacent states shrinks to 0 as thediscount factor approaches 1. Following the formal analysis, we provide further discussion of theeconomic basis of our result.

5.1. Equilibrium behavior as δ ↑ 1

For any given δ ∈ (0, 1) and σ ∈ �(δ), let ψσk (δ) represent the collection of states z′ ∈ Z withthe property that in any period j ∈ I+, if Dj = [0, z′), then Dσj+k = ∅ (a.s.). That is, the market


is dissipated in (at most) k more offers (including the period-j offer) almost surely. Also, letψj(δ) ≡ ∪σ∈�(δ)ψ

σj (δ) and zj(δ) ≡ sup ψj(δ). Hence, by the definition of zj(δ), if at any period t,

the state is z′ ≤ zj(δ), then Dσt+j+1 = ∅ (a.s.), for every σ ∈ �(δ).The content of the vanishing-sales property is then this:

Proposition 1. limδ↑1 zk(δ) = 0 for each k = 1, 2, . . ..

That is, for every k ∈ I+, and for any given ε > 0, there exists some 0 < δk < 1 such thatzk(δ) < ε for every δ > δk. Alternatively, let d(X) denote the diameter of the set X. Then, anotherway to state the vanishing-sales property is that, for every ε > 0, there exists some δ(ε) < 1 suchthat for every δ > δ(ε), d(ψk(δ)) < ε.

Because all equilibrium paths satisfy the backwards recursion property (cf. Theorem 1(i)), theproof of Proposition 1 starts by demonstrating that limδ↑1 z1(δ) = 0. Now, by definition, z1(δ) =sup{z ∈ (0, 1)|z∗(z′) = {0},∀z′ < z}, is the supremum of the (T σ − 1)-period states (cf. Theorem1(i)), along all equilibrium paths σ ∈ �. Hence, the interpretation of limδ↑1 z1(δ) = 0 is that, fordiscount factors sufficiently close to 1, the market is not dissipated unless the remaining marketsize is arbitrarily close to zero. This is an intuitively appealing property of equilibrium paths, andconsistent with results in the sequential bargaining and durable goods monopoly literatures (e.g.,Fudenberg et al., 1985; Gul et al., 1986).

Indeed, the linear-quadratic structure of our model allows one to compute z1(δ) explicitly andstudy its properties in the limit as δ ↑ 1.

Lemma 14.

z1(δ) =

⎧⎪⎨⎪⎩

min(

1, γ2β

)if 0 < δ ≤ 3

4

min

(1,

2γ[√

1−δ−(1−δ)]

β

)if 3

4 < δ < 1.

It is transparent from Lemma 14 that limδ↑1 z1(δ) = 0. But if the monopolist will optimallydissipate markets of infinitesimally small size as the discount factor approaches 1, then intuitionsuggests that z2(δ) cannot be strictly bounded away from zero for all δ. For, it is unlikely that themonopolist will optimally serve remaining markets of some fixed size with at most two offers,independent of the discount factor, but will only dissipate or serve-out a remaining market ifit is arbitrarily small, when the discount factors are sufficiently high. Indeed, using backwardrecursion, we find that, limδ↑1 zk(δ) = 0, for each k = 2, . . ..

To understand fully the implications of Proposition 1, we need some notation that is based onTheorem 2. There, it is shown that for any given σ ∈ �(δ), 0 < δ < 1, if the state is z′ and theprice–quality history is θ ∈ �∞, then there exists a mapping zσ : Z ×�∞ → Z, such that alongthe equilibrium path, zσ(z′, θ; δ) represents the expected next-period state, and the expected priceand quality offers can be written as (pσ(z′, θ; δ), qσ(z′, θ; δ)).

Proposition 1 then implies that for every ε > 0, and any 0 < z′ < 1, there exists δ(ε, z′) < 1such that (z′ − zσ(z′, θ; δ)) < ε for every δ > δ(ε, z′), whenever the state is (z′, θ), for any givenθ ∈ �∞. In effect, the equilibrium market depletion at any stage becomes infinitesimally smallas the discount factor approaches 1. But if there are vanishingly small sales with probability 1 atevery state, then even with possibility of multiple solutions to the (DP) equation (cf. (4) and (5)),one can bound the equilibrium price and quality offers for any given state with arbitrary precisionas the time between offers evaporates.


Then, we define for each z′ ∈ Z,

p∗(z′) ≡ γ2 + β(γz′ + β(z′)2/2)

b(9)

q∗(z′) ≡ (γ + βz′)b

. (10)

A comparison with the socially efficient price–quality allocation (pe(z), qe(z))z∈Z, given in Sec-tion 3.2, shows that q∗(z′) maximizes aggregate surplus, and p∗(z′) is the cost-plus fixed markuppricing rule for every quality sold. More precisely,

q∗ ∈ arg maxq(·)

∫ 1

0[(γ + βz)q(z) − b(q(z))2/2] dz. (11)

And, for each z′ ∈ Z,

p∗(z′) = b(q∗(z′))2

2+ γ2

2b. (12)

We now show that the equilibrium price and quality policies, be they stationary or non-stationary, must converge pointwise to the functions (p∗(·), q∗(·)) in the limit as δ ↑ 1.

Theorem 3. For every ε > 0, and given any z′ ∈ Z, there exists some δ∗(ε, z′) < 1 such that forall δ > δ∗(ε, z′), for any θ ∈ �∞,

qσ(z′, θ; δ) ≥ q∗(z′) − ε, (13)

pσ(z′, θ; δ) ≤ p∗(z′) + ε. (14)

Theorem 3 implies that as the time between offers evaporates, the monopolist makes an arbi-trarily large number of price–quality offers along the equilibrium path. However, the markup onany accepted offer shrinks toward γ2/2b (cf. (12)). The implications of Theorem 3 for the presentvalue of the monopolist’s equilibrium path profits as δ approaches 1 are therefore striking.

Corollary 1. For every ε > 0, there exists some δ(ε) < 1 such that for every δ > δ(ε), ( (δ) −(γ2/2b)) < ε.

Theorem 3 and Corollary 1 establish that a Coase conjecture type result applies even whenthe durable goods monopolist is allowed to intertemporally quality discriminate through strategicquality innovation or obsolescence, at least under the conditions annunciated in the model athand. In the limit as the time between offers evaporates, the monopolist’s profits are arbitrarilyclose to the profits from immediately dissipating the market from the pooling offer, because themonopolist can always obtain γ2/2b as profits by completely serving the market with the optimalpooling offer (cf. Theorem 1). Thus, and consistent with the spirit of the Coase conjecture, themonopolist is unable to effectively price-discriminate as the time between offers evaporates.

We reiterate that the form of the Coase conjecture derived here is significantly different fromthe usual formulation. In the fixed quality case, only one offer (the competitive one) is made inthe continuous time limit. By contrast, here the monopolist makes a continuum of offers.

5.2. Discussion

Theorem 3 and Corollary 1 establish that the monopolist cannot intertemporally price dis-criminate even when quality is flexible under these assumptions: there are a continuum of


non-atomic buyer-types, unit costs are constant, the rate of sales is unrestrained, the monop-olist is subject to time-consistency constraints, and buyers are rational. It turns out that thepresence of a continuum of buyer-types with respect to the demand-intensity for quality (asspecified in the consumer preferences in (1)) is important for this result. This point deservesexplication.

Without loss of generality, consider any pure-strategy equilibrium, σ ∈ �(δ), and at somedate t, let the remaining market be Dσt [0, zt). Then, the equilibrium specifies that the state in thesucceeding period is zt+1 = zσ(zt ; δ), and the market depletion at date t is (zt − zt+1). Furthermore,from Lemma 12(iii), we know that,pσt = (γ + βzt+1)qσt − δVσt+1(zt+1). Clearly, for any given qσt ,the period-t price, pσt , is positively related to the valuation of the marginal buyer-type in period-t,i.e., zt+1. Hence, pσt , is negatively related to the market depletion at this date: the lower is zt+1, thelower is pσt . But as long as δ � 1, the monopolist’s desire to reduce market depletion and increasethe current price is offset by the cost of waiting. However, as the time between offers evaporates,the monopolist’s cost of waiting also evaporates, and the optimal rate of market depletion at eachstage also falls, as formalized in Proposition 1.

The falling rate of market depletion has a perverse effect on the monopolist’s ability to in-tertemporally price–quality discriminate. Because qσt = (γ + βzσ(zt ; δ))/b (cf. Lemma 12(iv)),as zσ(zt ; δ) ↑ zt for every date t, qσt+1 ↑ qσt , for each t, as well. Thus, the rate of quality decline(or obsolescence) also vanishes locally as δ approaches 1, and the monopolist cannot credibly usea fast rate of quality decline to induce high demand-intensity buyers to accept high quality offersat supra-marginal prices.

We note that the rate of market depletion is unbounded below only if there are a continuumof buyer-types. The simplest illustration of this point is to examine the situation with only twobuyer-types, a case that is a polar opposite of the one considered in this paper. Let the buyerpreferences be:

u(pt, qt, z) ={δt[β�qt − pt

]if z ∈ [0, 1

2 )

δt[βhqt − pt

]if z ∈ ( 1

2 , 1].(15)

where βh > β� > 0. Now there are a continuum of non-atomistic buyers, but obviously only twobuyer (valuation) types—high (i.e., βh) and low (i.e., β�).

It is intuitively clear that along the equilibrium path the monopolist will make at most twooffers, and deplete at least one-half of the total market in each offer. Indeed, Kumar (2002)considers this market and shows that in the unique SPE, for (βh − β�) sufficiently large, themarket is depleted in two offers, {p∗

t , q∗t }2t=1, accepted by the high- and low-type buyers, re-

spectively, where p∗1 = [βh(βh − δβ�) + δβ�]/b and p∗

2 = (β�)2/b.13 Unlike the situation inTheorem 3, the initial period price does not converge to the low-type’s valuation even in thelimit as δ ↑ 1, as long as βh > β�. Hence, the monopolist is able to intertemporally price-discriminate in the continuous time limit: we can check that, limδ↑1

∗0 > (β�)2/2b (the pooling

profits).The reason why the monopolist is able to retain some price discriminating ability under the

Coase assumptions in the two buyer-type case is that the rate of quality decline between the twooffers is strictly bounded away from zero along the equilibrium path for all δ as long as βh > β�.

13 As in Theorem 2, the quality provision is efficient in each period, and the marginal buyer-type in each period, i.e., thehigh buyer-type in period 1 and the low buyer-type in period 2, are just indifferent between accepting and refusing themonopolist’s offer.


However, a finite quality decline is guaranteed only because the difference in the valuation ofthe two (trivially adjacent) buyer-types is strictly bounded away from zero by assumption. Bycontrast, with a continuum of buyer-types, the valuation difference between adjacent buyer-typesis unbounded below; and, therefore, so also is the rate of quality decline as the time betweenoffers evaporates. Note that the monopolist would like to threaten inframarginal buyer-types witha fast rate of quality depletion, but this threat is non-credible in the continuous time limit with acontinuum of buyer-types.

As we mentioned above, buyer preferences of the type specified in (1) are typically assumed inthe quality discrimination literature (Mussa and Rosen, 1978, and onwards). But the implicationsof the diversity of buyer demand-intensities for a durable goods monopolist’s inter-temporalprice–quality discrimination ability under the Coase assumptions have not been emphasized inthe literature.

6. Conclusions

Allowing a durable goods monopolist to strategically vary product quality over time isof substantial interest, but significantly complicates the analysis of equilibrium intertempo-ral price discrimination: equilibria where prices and qualities are non-monotone with re-spect to time are possible and randomization by the monopolist cannot be ruled out. Despitethese potential complications, we provide a strong characterization of all equilibrium pathsfor the “gap” case: along every (subgame perfect) equilibrium path, prices and qualities de-cline over time, and the market is completely and monotonically depleted according the buy-ers’ demand-intensity for quality in a finite number of offers. The key point is that unlikethe usual time-inconsistency problem of the durable goods monopolist, with flexible qual-ity the monopolist faces the time-inconsistency problem of not raising prices against infra-marginal buyers who delay purchase in anticipation of higher product quality offers in thefuture.

Moreover, if there are a continuum of buyer-types then the Coase conjecture continues to bevalid here, albeit in a significantly different form, with the monopolist making a continuum ofoffers and allocating each buyer-type to their socially efficient quality at a constant profit-margin.Thus, the power of time-consistency in restricting the monopolist’s ability to intertemporally pricediscriminate extends to the case where the monopolist is allowed the flexibility to vary productqualities strategically over time, as long as there is sufficient diversity of buyer demand-intensitieswith respect to quality.

Our analysis emphasizes the role of price and quality expectations in determining equilib-rium outcomes. But these expectations are determined by the institutional or market structureof the model. For example, it matters whether it is common knowledge that the monopolistwill not face an entry threat in the future or whether the set of feasible product qualities isfixed. Outcomes are quite different if the monopolist could credibly restrain his incentive toraise prices on the high-valuation that delay purchase in anticipation of high quality products(e.g., Kumar, 2002). Outcomes may also be very different if the feasible quality set is ini-tially uncertain due to endogenous technological progress through creative destruction (e.g.,Aghion and Howitt, 1992) or technological investment by the monopolist (e.g., Fishman andRob, 2000). Extensions of the model to incorporate institutional or market features that allowthe monopolist to credibly constrain his price gouging ability ex post and examine his in-centives for varying the feasible quality set through R&D are important directions for futureresearch.


Acknowledgements

I thank two anonymous referees for helpful comments. I also thank Dilip Abreu, LarryAusubel, Charles Kahn, Faruk Gul, Peter Hammond, Debraj Ray, Hugo Sonnenschein, JeanTirole, Michael Waldman, seminar participants at the University of British Columbia, Carnegie-Mellon, Queens, Rice, Stanford, the Indian Statistical Institute, and the Winter EconometricSociety Meetings for useful comments or discussions. All remaining shortcomings are my ownresponsibility.

Appendix A. Proofs

We start by establishing some notation. Fix a buyer strategy profile α, and let, for anyht ∈ Ht, t ∈ I+, Cα

t (z, ht) ≡ {θ ∈ P ×Q|αt(ht, θt, z) = 1}; i.e., Cαt is the set of price–quality

offers that will be accepted by buyer-type z at t with history ht . Cαt denotes the complement

of Cαt . The measurability restrictions on α ensure that Cα

t and Cαt are both measurable sets.

Since h′t = (ht, θt), a specification of α defines a sequence of non-stochastic transition map-

pings ξαt+1 : H ′t → Ht+1, t ∈ I+, so that ht+1 = ξαt+1(ht) ∈ Ht+1. For any two time-periods

τ and t, τ ≥ t, Hτ(ht) ⊆ Hτ denotes the set of period-τ histories that include ht . We willalso distinguish the set of pure strategy SPE, � ⊆ �, from the set of behavior strategy SPE,� ⊆ �.

Note that a given strategy profile φ = (λ,α) allows the recursive computation of the expectedutility of any buyer-type z ∈ Dt ⊆ Z at any period t ∈ I+, given any ht .

Vφt (z, ht) =

∫Cαt (z,ht )

[(γ + βz)qt − pt

]dμλ

t (ht) + δ

∫Cαt (z,ht )

Vφt+1(z, ξαt+1(ht, θt)) dμλ

t (ht).

(A.1)

(Vφt (z, ht) is set equal to +∞ if it is not finite.) A SPE σ is a specialization of φ such that(λ|ht ,α|ht ) and (λ|h′

t,α|h′

t) are a Nash equilibrium for every ht ∈ Ht , and h′

t ∈ H ′t . To ease the

notational burden, the superscripts λ, α and σ will be suppressed in the proofs whenever thereference is unambiguous. Finally, due to the possibility of randomization by the monopolist, allresults hold with probability 1.

Lemma 1. Fix any σ ∈ �. At every t, for any given ht ∈ Ht , either Dσt = ∅ or m(Dσt ) ≥ ε, forsome ε > 0.

Proof. The claim is trivially true for t = 0 (since D0 = Z). Suppose first that σ ∈�. Then, for any given θ0 and z ∈ Z, by definition (cf. (A.1)), V1(z, ξ1(θ0)) =supτ≥1

[δτ−1((γ + βz)qτ − pτ)|ξ1(θ0)

]. But z ≤ 1, q ≤ q, and γ > 0. Hence, if V1(z, ξ1(θ0)) =

+∞ for some z, then V1(z′, ξ1(θ0)) = +∞ for every z′ ∈ Z. In this case, V1(·, ξ1(θ0))is trivially continuous since it is a constant function. And if V1(z, ξ1(θ0)) is boundedabove for some z, then it is also bounded above for every z′ ∈ Z. Thus, V1(z, ξ1(θ0)) =maxτ≥1

[δτ−1((γ + βz)qτ − pτ)|ξ1(θ0)

], which is continuous on Z due to the continuity of the

max(·, ·) function. Now suppose that, contrary to the Lemma, there exists some θ0 such thatD1(ξ1(θ0)) �= ∅ butm(D1) = 0. Without loss of generality, letD1 = {zk}∞k=1 and pick any zk fromthis set. Next, let L1(z, ξ1(h′

0)) ≡ δV1(z, ξ1(θ0)) − [(γ + βz)q0 − p0]. L1(z, ·) is continuous in z,since V1(z, h1) is continuous in z for every h1 ∈ H1. Also, consumer rationality (cf. footnote 6)implies thatL1(z, ξ1(h′

0))|z=zk > 0. Thus, there exists some εk > 0 such thatL1(zk, ξ1(h′0)) ≥ εk.


But if zk is an interior point of Z, then from continuity there exists a neighborhood (zk1, zk2) suchthat L1(z′, ξ1(h′

0)) ≥ εk for every z′ ∈ (zk1, zk2). Thus, (zk1, zk2) ∈ D1, contradicting the hypoth-esis that m(D1) = 0. (The argument is easily adapted for the case where zk is either the upper orlower point of closure of Z.)

Now apply induction: fix any t ≥ 2, and any ht ∈ Ht , and suppose that Dt was is a countableunion of disjoint intervals in Z such that the complement of Dt is closed. But since σ ∈ � (byassumption), for any given h′

t = (ht, θt) and z ∈ Dt ,

Vt+1(z, ξt+1(h′t)) = sup

τ≥t+1

[δτ−(t+1)((γ + βz)qτ − pτ)|ξt+1(h′

t)].

Hence, an argument exactly analogous to one given above establishes that Vt+1(z, ht+1 =ξt+1(h′

t)) is continuous at z for any z ∈ (zk, zk+1) ⊆ Dt . (And if {0} ∈ Dt , then Vσt+1(z, ht+1)is right-continuous at z = 0, and Vt+1(z, ht+1) is left-continuous at z = 1 if {1} ∈ Dt .) Now, sup-pose that Dt+1 �= ∅ but m(Dt+1) = 0. Then, consumer rationality implies that for any zk ∈ Dt ,

Lt+1(zk, ξt+1(h′t))|z=zk ≡ {δVt+1(z, ξt+1(h′

t)) − [(γ + βz)qt − pt]}|z=zk > 0. (A.2)

If zk is an interior point ofZ, then from continuity there exist zk1 < z < zk2 such that (zk1, zk2) ⊆Dt+1, contradicting the hypothesis that m(Dt+1) = 0. The argument is easily adapted for thecase where zk is either the upper or lower point of closure of Z. Thus, the Lemma holds for anyσ ∈ �.

Suppose next that σ ∈ �. Again, the Lemma is trivially true for t = 0. For t = 1, if for agiven θ0, V

σ1 (z, ξ1(h′

0)) = +∞ for some z ∈ Z, then it is continuous in z on Z, for every h1 ∈ H1.So suppose that Vσ1 (z, ξ1(h′

0)) is bounded above, but for some h1 = ξ1(h′0), V σ1 (z, ξ1(h′

0)) isnot continuous at some interior point z′ of Z. Then, there exists some ε > 0 and some se-quence {zj}∞0 , limj→∞ zj = z′, such that |V1(z′, h1) − V1(zj, h1)| ≥ ε, for every j = 1, 2, . . ..Then, let, for any ht ∈ Ht, t ∈ I+, such that z, z′ ∈ Dt , the function fσt : D2

t ×Ht → R bedefined as:

fσt (z, z′;ht) ≡∫Cαt,Q

(z′,ht )

[β(z′ − z)qt

]dμλ

t,Q(ht) + δ

∫Cαt (z′,ht )

∫Cαt+1,Q(z′,ξt+1)

× [β(z′ − z)qt+1]

dμλt+1,Q(ξt+1(ht, θt)) dμλ

t (ht) + · · · (A.3)

Here, Cλt,Q and μλ

t,Q are the projections of Cλt and μλ

t on Q, respectively. Thus, fσt (z, z′;ht)computes the expected utility of buyer-type z if he mimics the strategy of buyer-type z′ in thecontinuation game σ|ht ∈ �ht . It is straightforward to show that

Vσt (z′, ht) − Vσt (z, ht) ≤ fσt (z, z′;ht) (A.4)

since buyer-type z can do at least as well in any continuation game by mimicking the strategy ofbuyer-type z′. (In the sequel, we again suppress the superscript σ for notational convenience.) Wenow claim that (A.3) and (A.4) imply that for every h1 ∈ H1, and every ε > 0, there exists somex(ε) > 0 such that |f1(z, z′;h1)| < ε, if |z− z′| < x(ε). Note that since q ≤ q,

βq|z− z′| ≥ suphτ∈Hτ

supq∈ϒτ,Q(hτ )

(βq|z− z′|) , τ ≥ 1. (A.5)


(Here,ϒτ,Q(hτ) is the projection ofϒτ(hτ) onQ.) Furthermore, since δ < 1, and buyers purchaseonly once, (A.3), (A.5) and the fact that Cα

τ,Q(z′, hτ) ⊆ ϒλτ,Q(hτ), imply that for any h1 ∈ H1,

|f1(z, z′;h1)| ≤ βq|z′ − z|. (A.6)

Then, choose x(ε) ≡ ε/(βq). Suppose that z < z′. Recalling that (A.4) applies for any ht ∈Ht, t ∈ I+, we have, V1(z, h1) ≥ V1(z′, h1) − f1(z, z′;h1). Hence, (A.6) and the definition ofx(ε) imply that V1(z′, h1) − V1(z, h1) ≤ ε/2 if (z′ − z) < x(ε/2). Then, for any given sequence{zj}, limj→∞ zj ↑ z, and for every ε > 0, there exists some natural number N+ such that(V1(z′, h1) − V1(zj, h1)) ≤ ε if j > N+. Now suppose that z′ < z. (A.3) and (A.4) then implythat (V1(z, h1) − V1(z′, h1)) ≥ f1(z, z′;h1). But since, in this case, f1(z, z′;h1) ≤ 0, (A.6) andthe definition of x(ε) also imply V1(z′, h1) − V1(z, h1) ≤ ε/2, if (z− z′) < x(ε/2). Hence, forany {zj}, limj→∞ zj ↓ z, and for every ε > 0, there exists some natural number N− such that(V1(zj, h1) − V1(z′, h1)) ≤ ε if j > N−. We have thus contradicted the hypothesis that V1(z′, h1)is not continuous at z′, when z′ is an interior point of Z. Similarly, it can be established thatfor every h1, V1(z′, h1) is left-continuous at z = 1 and is right-continuous at z = 0. Since theforegoing argument holds for any z′ ∈ Z and any h1, it follows that V1(z, h1) is continuous in z.Using an earlier argument, it also follows thatD1 is either empty or of positive measure for everyσ ∈ �. And an induction argument similar to that used above establishes that the Lemma is truefor any t ≥ 2 as well, for any σ ∈ �. �

Lemma 2. Fix any σ ∈ �. Suppose that there exists some period T ∈ I+ and hT ∈ HT such that,(i) DσT �= ∅, but (ii) DσT+1 = ∅. If θT ∈ ϒλ

T (hT ), then pT = (γ + βzT )qT .

Proof. If θT is the terminating offer, then profits are: T (hT , θT ) = (pT − b(qT )2/2)m(DT ).Now, for every τ ≥ T and hτ ∈ Hτ , put,Bα

τ (q, hτ |hT ) ≡ {p ∈ R|ατ(hτ, (p, q), z) = 1,∀z ∈ DT }.(If Dτ(hτ) �= DT , we set Bα

τ (q, hτ |hT ) = R). Let

Gα(q|hT ) ≡ supτ≥T

⎡⎣ ⋂hτ∈Hτ (hT )

{Bατ (q, hτ |hT )

}⎤⎦ . (A.7)

By construction, Gα(q|hT ) is the supremum of prices that would be accepted along σ|hT byevery z ∈ DT for any given quality q > 0, after any history hτ ∈ Hτ(hT ), τ ≥ T . Clearly, inany continuation game σ|hT , for any hτ ∈ Hτ(hT ), τ ≥ T , if θ′ = (p′, q′) ∈ ϒλ

τ (hτ), then p′ �<Gα(q′hT ), since the quality q′ is sure to be accepted by all (remaining) buyers at the priceGα(q′|hT ), and m(Dτ) > 0 (cf. Lemma 1).

We claim that Gα(qT |hT ) ≥ (γ + βzT )qT . Suppose not; fix any equilibrium σ|hT =(α|hT ,λ|hT ) in which there exists some θT ∈ ϒλ

T (hT ) that is accepted by every z ∈ DT , butpT = Gα(qT |hT ) < (γ + βzT )qT . Consider an alternative strategy for the monopolist, λ∗|hT . Foreach qT ∈ ϒλ

T,Q(hT ), λ∗|hT specifies the price offer, pT = (γ + βzT )qT + δGα(qT |hT ). And,

for τ ≥ T + 1, λ∗|hT is specified as follows. If hT+1 = ξT+1(hT , (pT , qT )) (≡ ξT+1), then letϒλ∗τ (hτ) = {(Gα(qT |hT ), qT )}, for every hτ ∈ Hτ(ξT+1). For every other hT+1 (i.e., for any

hT+1 �= ξT+1), let λ∗|hT+1 = λ|hT+1 . Note that if (α|hT ,λ|hT ) is a subgame perfect equilib-rium, then by construction λ∗

hT+1continues to be a best response strategy to α|hT+1 for every

hT+1 �= ξT+1. It remains to show that ϒλ∗τ (hτ) = {(Gα(qT |hT ), qT )} is a best response corre-

spondence for the monopolist for every hτ ∈ Hτ(ξT+1), τ ≥ T + 1. By construction, for each


z ∈ DT ,

(γ + βz)qT − pT − δVT+1(z, ξT+1) ≥ (γ + βz)qT − pT − δ[(γ + βz)qT −Gα(qT |hT )

]= (1 − δ)β(z− zT )qT ≥ 0. (A.8)

Since (A.8) applies for all z ∈ DT , every z ∈ DT accepts the price–quality offer (pT , qT ).Thus, DT+1(ξT+1) = ∅, and the offer θτ = (Gα(qT |hT ), qT ) is (trivially) subgame perfect forthe monopolist for every hτ ∈ Hτ(ξT+1), τ ≥ T + 1. But note that pT > Gα(qT |hT ), since(γ + βzT )qT > Gα(qT |hT ), by hypothesis. Sincem(DT ) > 0 (cf. Lemma 1), λ∗|hT strictly dom-inates λ|hT as a response to α|hT . Thus, the assumption that there exists a σ|hT = (α|hT ,λ|hT )in which Gα(qT |hT ) < (γ + βzT )qT , for some qT ∈ ϒλ

T,Q(hT ), is contradicted. Hence, in any

σ|hT , it must be the case that pT = Gα(qT |hT ) ≥ (γ + βzT )qT , for each qT ∈ ϒλT,Q(hT ). But

pT ≤ (γ + βzT )qT , since each buyer-type must receive non-negative utility from accepting anoffer. Thus, under the conditions annunciated in the statement of the Lemma, pT = (γ + βzT )qT(a.s.) for every θT ∈ ϒλ

T (hT ). �

Remarks. (i) Note that the Lemma applies whether zT ∈ DσT or otherwise. (ii) The proof extendsthe argument of Lemma 1 of Gul et al. (1986, p. 174) to the flexible quality case.

Lemma 3. Fix any σ ∈ �. For any t ∈ I+, and every ht ∈ Ht such that Dt �= ∅, αt(ht, (p, q)) =1, ∀z ∈ Dt , for any offer (p, q) such that p < (γ + βzt)q.

Proof. Suppose to the contrary. Then, there must exist a set D′ of positive measure of buyerswho reject the offer θt such that pt < (γ + βzt)qt . Thus, for every z ∈ D′ it must be the case that

δVσt+1(z, h′t) > (γ + βz)qt − pt. (A.9)

We now establish that for any such candidate rejection set D′, there exists some buyer type-zin D′ for whom (A.9) is violated for every continuation equilibrium path σ|h′

t. For expositional

ease, we first prove this claim for the pure strategy continuation equilibria, and then extend it tothe behavior strategy continuation equilibria. Then, fix any σ|h′

t∈ �|h′

t. There are two mutually

exclusive possibilities: (1) there exists some T ≥ t + 1 such that the market is dissipated on thatdate or (2)Dστ is non-empty for every τ ≥ t + 1 along σ|h′

t. Consider first the former case. Buyer

rationality requires that for every z ∈ DσT , δT−t[(γ + βz)qT − pT ] > (γ + βz)qt − pt . We knowfrom Lemma 2 that pT = (γ + βzT )qT . Thus, (A.9) requires that, for each z ∈ DσT ,

δT−t[β(z− zT )qT ] > (γ + βz)qt − pt > 0. (A.10)

Let at ≡ (γ + βzt)qt − pt . Since qT ≤ q, (A.10) implies δT−tβ(z− zT )q > at > 0; i.e., for anyz ∈ DσT , (z− zT ) > at/(δT−tβq). But, by definition of zT , for every ε > 0, there exists somez ∈ DσT such that (z− zT ) < ε. This fact contradicts the requirement that (z− zT ) > at/(δT−tβq),when we set ε < at/(δT−tβq). Thus, (A.10) cannot hold for every buyer-type in DσT . In fact, DσTmust be empty since for any zT ≥ zt , there exists a neighborhood (zT , zT + ε), ε > 0, such thatevery buyer z in this neighborhood would have preferred to accept θt at date t. Since this argumentholds for any σ|h′

t, and any candidate termination date T along this (continuation) equilibrium

path, we conclude that Dσt+1 = ∅ for all candidate continuation equilibrium paths with finitetermination dates.

Consider, next, continuation equilibrium paths that do not have finite termination dates; i.e.,σ|h′

tsuch that Dστ �= ∅ for every τ ≥ t + 1. In this case, there must exist some k ≥ 2 such that

m(Dσt+1) < m(Dσt+k). Suppose not; then, for every j ≥ t + 1, and for every hj ∈ Hj , the j-period


offer θj is accepted only by a buyer set of measure zero. However, by hypothesis and a foregoingargument, m(Dσj ) > 0 for every j ≥ t + 1. Thus, Dσj is an uncountable set. On the other hand,markets open only for a countably infinite set of dates. Therefore, there must exist some buyertype-z ∈ Dσt+1 who never purchases along σ|h′

t. But this buyer then receives zero utility along this

continuation equilibrium, violating buyer rationality since the buyer could accept θt and obtainpositive utility. Then, fix any D′ ⊆ Dσt+1, and let for every z ∈ D′, τ(z) ≥ t + 1 denote the dateat which z accepts the monopolist’s offer along σ|h′

t. Clearly, for any such z,

δVσt+1(z, h′t) − δτ(z)−t[(γ + βz)qτ(z) − pτ(z)] > (γ + βz)qt − pt ≥ at > 0. (A.11)

Since z ≤ q1, q ≤ q, (A.11) implies that δτ(z)−tpτ(z) < δτ(z)−t(γ + β)q− at . But the hypothesisthat Dστ �= ∅ for each τ ≥ t + 1, and the immediately preceding inequality, then imply that

δτ−tpτ < δτ−t(γ + β)q− at , ∀τ ≥ t + 1. (A.12)

But since δ < 1 and q is finite, for every ε > 0 we can find some natural number k such thatδk(γ + β)q < ε. However, at > 0 by hypothesis. Thus, (A.12) is valid for every τ ≥ t + 1 onlyif there exists some τ∗ ≥ t + 1 such that a negative price (pi < 0) is offered along σ|h′

tfor

every time-period i ≥ τ∗. However, no such τ∗ can exist along any equilibrium path since itimplies that the existence of some hτ∗ such that the monopolist’s present value of profits in thesubgame σ|hτ∗ , στ (h∗

τ ) < 0. This clearly violates subgame perfection for the monopolist, sincethe monopolist’s present value of profits in any subgame cannot be negative. For every σ|h′

t∈ �|h′

t,

for every D′ ⊆ Dσt+1 there exists some buyer-type in D′ for whom (A.11) is violated.To complete the proof, we need to show that the claim is true also for every σ|h′

t∈ �|h′

t.

Proceeding as before, we partition the set of behavior strategy continuation equilibria into twosubsets. The first subset is comprised of those σ|h′

t∈ �|h′

talong which the market is dissipated

by some T σ ≥ t + 1 with probability 1. The second set is the complement of the first subset. Nowchoose any element σ from the first subset, and with some abuse of notation let the market bedissipated by T (a.s.). By definition, there exists someBT ⊆ HT (h′t), m(BT ) > 0, such thatDσT �=∅ whenever hT ∈ BT . Furthermore, from Lemma 2, for each qT ∈ ϒλ

T,Q(hT ), pT = (γ + βzT )qT .Now pick any hT ∈ BT . Applying the foregoing arguments, there cannot exist any buyer-type inDt+1 that strictly prefers any offer in ϒλ

T (hT ) to the current offer. Thus, DσT (hT ) = ∅. Since thisargument applies for every hT ∈ BT , we have arrived at a contradiction from the hypothesis thatfor every hT ∈ BT , DσT (hT ) �= ∅.

Turn, next, to those σ|h′t∈ �|h′

tfor which T σ = +∞. Suppose that, contrary to the assertion

of the Lemma, there exists some σ|h′t∈ �|h′

tfor whichDσt+1 �= ∅. From consumer rationality, for

every z ∈ Dσt+1,

δVσt+1(z, ξt+1(h′t)) > (γ + βz)qt − pt ≥ at > 0. (A.13)

We now prove that (A.13) is invalid for any z ∈ Dσt+1. We will now show that (A.13) is valid onlyif, with positive probability, the monopolist makes price offers less than −(γ + β)q. However, itis readily established that such offers will be made with zero probability along any equilibriumpath. Suppose to the contrary, and fix any σ ∈ � such that for some hτ ∈ Hτ, τ ∈ I+, there existssome θτ ∈ ξτ+1(h′

τ), such that pτ < −(γ + β)q. But then pτ < γqτ , since −(γ + β)q < 0 ≤ γqτ ,for every qτ ∈ Q. Hence, Lemma 2 implies that θτ is not a terminal offer. It also follows fromconsumer rationality that for each z ∈ Dστ+1,

δVστ+1(z, ξτ+1(h′τ)) > (γ + βz)qτ − pτ > (γ + β)q. (A.14)


The last inequality follows from the fact that pτ < −(γ + β)q. Now, let Eστ (p, z|ξτ+1(h′τ)) be

the expected price that buyer-type z will accept along the continuation game σ|h′τ. Similarly, we

define Eστ (q, z|ξτ+1(h′τ)). Then, (A.14) implies that

Eστ (p, z|ξτ+1(h′τ)) < −(γ + β)q+ (γ + βz)Eστ (q, z|ξτ+1(h′

τ)) < 0. (A.15)

The last inequality in (A.15) follows because z ≤ 1 and no quality offer can exceed q. Thus,(A.14) is valid only if the expected price paid by buyer-type z is negative along the continuationequilibrium path. Since this argument applies for every z ∈ Dτ+1, it follows that (A.14) is validonly if the monopolist’s expected profits along σ|h′

τare negative, since each remaining buyer pays

a negative expected price for a positive quality product. Thus, we have arrived at a contradiction.Now fix any σ|h′

t∈ �|h′

t, and let ht+1 ≡ ξt+1(h′

t). Also, for every z ∈ Dt+1, and τ ≥ t + 1, let,

�τ(z, hτ ; at) ≡ {s′ ∈ Cτ(z, hτ)|δτ−t[(γ + βz)q′ − p′] ≥ at} be the set of offers that z will ac-cept in period τ, given hτ , that are superior in the sense that they give at least at greater util-ity than the current offer. (The complement of this set will be denoted by �′

τ(z, hτ ; at)). Alsolet �τ(z, F ; at) ≡ ∪hτ∈F �τ(z, hτ ; at), for any F ⊆ Hτ . Finally, for any given F ⊆ Hτ, GF ⊆�τ(z, F ; at), let ντ(z, F,GF |ht+1) denote the probability that (along σ|ht+1

) buyer-type z willaccept a superior offer from the setGF , F. We claim that if (A.13) is valid, then there exist naturalnumbers w(z) and w(δ), with t + 1 ≤ w(z) ≤ t + 1 + w(δ), ⊆ Hw(z), and GF ⊆ �w(z)(z, F ; at)such that νw(z)(z, F,GF |ht+1) ≥ ε(z) > 0, for some ε(z) > 0. Suppose not; but we know thatprices along the equilibrium path are uniformly bounded below by −(γ + β)q. Hence, for everyθτ ∈ ϒλ

τ (hτ), τ ∈ I+, (γ + βz)qτ − pτ ≤ (2γ + β(1 + z))q, for each z ∈ Z. Therefore,

Vt+1(z, ht+1) < (2γ + β(1 + z))q

[ ∞∑τ=t+1

δτ−(t+1)ντ(z,Hτ, �τ(z,Hτ ; at)|ht+1)

]+ x(z)at ,

(A.16)

where x(z) ≡∑∞τ=t+1 δ

τ−(t+1)ντ(z,Hτ, �′τ(z,Hτ ; at)|ht+1) < 1, since only one offer can be ac-

cepted by the buyer. Now, the hypothesis is that for any given ε > 0, for each τ ≥ t + 1, any F ⊆Hτ , and GF ⊆ �τ(z, F ; at), ντ(z, F,GF |ht+1) ≤ ε. Then, choose ε(z) ≤ at(1 − x(z))[(2γ +β(1 + z))q]−1. In this case, (A.16) implies that δVt + 1(z, ht+1) < at , which contradicts the as-sumption that (A.13) is valid. Thus, the probabilities of accepting superior offers (i.e., ντ) mustbe strictly bounded below for all τ ≥ t + 1.

Next, fix any τ ≥ t + 1, and suppose that for some hτ ∈ Hτ(h′t), there exists a θτ ∈

�τ(zτ, hτ ; at). From Lemma 2 we know that θτ is not a terminating offer, since the infemumbuyer-type in the market cannot get positive utility from such an offer. Thus, there must exist aresidual buyer setDτ+1 ⊆ Dτ , such thatm(Dτ+1) > 0 (cf. Lemma 1). And every z ∈ Dτ+1 musthave positive expected utility along the continuation game that exceeds the utility from acceptingθτ . Put, aτ ≡ (γ + βzτ)qτ − pτ . By hypothesis, for every z ∈ Dτ+1,

δVτ+1(z, ξτ+1(hτ, θτ)) > (γ + βz)qτ − pτ ≥ aτ > at > 0. (A.17)

Applying the immediately preceding argumentation we are led to conclude that (A.13) is valid onlyif there exist an infinite sequence of dates and corresponding (extensive) marginal buyer-types,


{ij}∞j=0, and {zij+1}∞j=0, respectively (with i0 = t + 1, zi0+1 = zt+1), and

i1 ≡ inf{τ ≥ t + 1|∃F ⊆ Hτ,G ⊆ �τ(zt+1, F ; at) : ντ(zt+1, F,G|ht+1) ≥ εi1 > 0}i2 ≡ inf{τ ≥ t + 1|∃F ⊆ Hτ,G ⊆ �τ(zi1+1, F ; ai1) : ντ(zi1+1, F,G|hi1+1) ≥ εi2 > 0}

...

(A.18)

We now contradict (A.13) by showing that the sequence ({ij}∞j=0) cannot exist along any σ|h′t∈

�|h′t. By definition, any θi1 ∈ �i1(zt+1, hi1; at) must be such that δi1−t[(γ + βzt+1)qi1 − pi1] ≥

at . That is, pi1 ≤ (γ + βzt+1)qi1 − (at/δi1−t). Hence, for each z ∈ Di1, (γ + βz)qi1 − pi1 ≥β(z− zt+1)qi1 + (at/δi1−t). Now let ai1 ≡ (at/δi1−t). Then, by definition, at date i2, any of-fer θi2 ∈ �i2(zi1+1, hi2; ai1) must be such that δi2−i1[(γ + βzi1+1)qi2 − pi2] ≥ ai1. Using thedefinition of ai1, this implies that pi2 ≤ (γ + βzi1+1)qi2 − (at/δi2−t). Inductively, at date ik,

pik ≤ (γ + βzi(k−1)+1)qik − (at/δik−t). (A.19)

Note that the quantity at/δik−t is strictly increasing in the index k = 1, 2, . . .. Then, let,

y∗ ≡ [ln(at) − ln(3(γ + β)q)]/ ln δ. Since pt ≥ −(γ + β)q, at < 3(γ + β)q. And, since δ < 1,and γ , β and q are well defined constants, y∗ is well defined. Also, by construction, [at/δy∗] ≥3(γ + β)q. Now let ij∗ be the first date such that ij∗ ≥ (y∗ + 1). Then, any offer θij∗ ∈�ij∗ (zi(j∗−1)+1, hij∗ ; ai(j∗−1)) is such that

pij∗ ≤ (γ + βzi(j∗−1)+1)qij∗ − ai(j∗−1) ≤ −2(γ + β)q < −(γ + β)q. (A.20)

However, (A.20) contradicts the fact that prices are strictly bounded below by −(γ + β)q alongthe equilibrium path. Thus, (A.13) is invalid, and the proof of the Lemma is completed. �

Lemma 4. For any σ ∈ �, at any ht ∈ Ht, t ∈ I+, eitherDσt = ∅ orDσt = [0, zt(1)) ∪ �t , where�t is a countable union of intervals whose complement is closed.

Proof. This claim is trivially true for t = 1. So suppose that it was false for some t ≥ 2. Then,by hypothesis, zt > 0. Suppose first that zt ∈ Dσt . But Lemma 3 establishes that Vσt (zt, ht) = 0for any ht ∈ Ht . Now let θ(0) = (p(0), q(0)) be the price–quality offer that was accepted by themarginal buyer (z = 0) at some date τ < t. Clearly, p(0) ≤ γq(0). But then if the buyer type-zthad accepted θ(0) it would have received the utility δτ[(γ + βzt)q(0) − p(0)] > 0. This violatesbuyer rationality for type-zt , and thus the hypothesis leads to a contradiction. Meanwhile, supposethat zt �∈ Dσt . But since Vσ(zt, ht) = 0 for every ht ∈ Ht , it follows from the continuity of buyerexpected utility in z (cf. Lemma 1) that for every ε > 0 there exists some z ∈ Dσt such thatVσt (zt, ht) < ε. But we also know that for every z ∈ Dσt ,

δτ[(γ + βz)q(0) − p(0)] ≥ δτ[(γ + βzt)q(0) − p(0)] > 0,

implying a contradiction. Thus, for every σ ∈ �, for every t, and ht ∈ Ht , there exists some z > 0such that [0, z) ∈ Dσt , andDσt = [0, z) ∪ �t , where�t is a countable disjoint collection of intervalswhose complement is closed. �

Lemma 5. Fix any σ ∈ �(δ) and any ht ∈ Ht, t ∈ I+, such that m(Dσt ) > 0. Then, there existsan integer κ(δ) such that m(Dσt+κ(δ)) ≤ (1 − ν)m(Dσt ), ν ≡ [γ2/2(γ + β)2] > 0.


Proof. We know that for any σ ∈ �, any ht ∈ Ht , with mt = m(Dt), the present value of profitsalong σ|ht , σt (ht) ≤ mt[(γ + β)2/2b]. This follows since (γ + β)2/2b are the monopolist’s per-fectly discriminating profits if the entire market were composed of the inframarginal buyer-type’s(z = 1) valuation. Hence, if along σ|ht , the monopolist sells some quantity ε > 0 between the datest and t + j, then,

σt (ht) ≤ (γ + β)2

2b[ε+ δj(mt − ε)]. (A.21)

Now put, ν ≡ γ2/2(γ + β)2. Then, note that,

(γ + β)2

2b[νmt + δjmt(1 − ν)] = mt

(γ2 + δj[(γ2 + 2β(β + 2γ))]

4b

).

Now put,

κ(δ) ≡ SGI

(ln(γ2) − ln(γ2 + 2β(β + 2γ))

ln δ

),

where SGI(x) denotes the smallest integer greater than or equal to x. Since the right hand sideof (A.21) is strictly increasing in ε (cf. δ < 1), it follows that if ε < νmt and j ≥ κ(δ), then σt (ht) < (γ2/2b)mt . This contradicts the fact that σt (ht) ≥ (γ2/2b)mt in every subgame alongevery equilibrium path (cf. Lemmas 3 and 4), since γ2/2b is the profit-margin in the pooling offer(θ). �Lemma 6. Fix any 0 < m < 1. Then, there exists some finite date T (m, δ) such thatm(DσT (m,δ)) ≤m for every σ ∈ �(δ).

Proof. From Lemma 5, along any σ ∈ �(δ), m(Dσκ(δ)) ≤ 1 − ν, m(Dσ2κ(δ)) ≤ (1 − ν)2, etc.Clearly, (1 − ν)x ≤ m if x ≥ ln m/ ln(1 − ν). Then, put T (m, δ) = SGI (ln m/ ln(1 − ν))κ(δ). �Lemma 7. Fix any σ ∈ �(δ). There exists some z(δ) > 0 such that for any ht ∈ Ht, t ∈ I+, ifDσt = [0, z′), with z′ ≤ z(δ), then Dσt+1 = ∅.

Proof. Fix any σ ∈ �, a date t, and any ht ∈ Ht , such thatDσt = [0, z) ∪ �t andm(Dσt ) = m; i.e.,m(�t) = m− z. Recalling that (γ + βz)2/2b are the perfect price-discrimination profit marginfor buyer type-z, we have

σt (ht) ≤[(γ + βz1)2(z2 − z1) + δ

∫ z10 (γ + βr)2 dr + δ

∫ zz2

(γ + βr)2 dr]

2b

+ (m− z)(γ + β)2

2b(A.22)

Put the first term in the right-hand-side of (A.22) as M(z1, z2, z). Then, consider the difference,

M(z1, z2, z) = (γ2/2b)z−M(z1, z2, z) = z1M1(z1, z2) + (z− z2)M2(z2, z), (A.23)

where

M1(z1, z2) = (2b)−1[γ2(1 − δ) − z1β[β((δ/3)z1 + (z2 − z1)) + γ(δz1 + 2(z2 − z1))]

];

M2(z2, z) = (2b)−1[γ2(1 − δ) − δβ[(β/3)(z2 + z2

2 + zz2) + γ(z+ z2)]].


It can be verified that M2 > 0 whenever z ≤ [γ/(β√δ)](1 − √

δ). Put z∗(δ) ≡min(1, [γ/(β

√δ)](1 − √

δ)), and note that z∗(δ) > 0 for each 0 ≤ δ < 1. Now M1(z1, z2)is strictly decreasing in z2. Hence, for any given z1, M1(z1, z2) is no smaller than M1(z1, 1).But M1(z1 = 0, 1) > 0. Since M1 is a continuous function of z1, it follows that there existssome z∗∗(δ) such that M1(z1, z2) > 0 whenever 0 < z1 ≤ z∗∗(δ). Thus, M(z1, z2, z) > 0 (cf.(A.23)) whenever z ≤ z(δ) ≡ min(z∗(δ), z∗∗(δ)). However, M(z1, z2, z) overstates the profitsfrom serving [0, z) along any continuation equilibrium path. Therefore, if Dt = [0, z), z ≤ z(δ),then Dt+1 = ∅, along any σ|ht . �Lemma 8. For a given δ, there exists a finite date T (δ) such that Dστ = [0, zτ(1)) ∪ �τ, zτ(1) ≤z(δ), ∀τ ≥ T (δ). Furthermore, if Dστ = [0, zτ(1)) ∪ �τ , then a buyer set B ⊂ [0, zτ(1)), withm(B) > 0, must accept an offer from the seller in a finite number of periods from τ.

Proof. The first part of the Lemma follows from Lemmas 4 and 7. Fix any σ ∈ �(δ). FromLemma 4, we know that Dστ = [0, zτ(1)) ∪ �τ , generically, along any equilibrium path. And,from Lemma 7, we know that m(DσT (z(δ),δ)) ≤ z(δ). Thus, it is certainly the case that zτ(1) ≤ z(δ)if τ ≥ T (z(δ), δ). Suppose, next, that the latter part of the Lemma is false. Then, it must bethat m(Dστ+k) ≥ zτ(1) for every k = 1, 2, . . .. However, Lemma 7 again assures us that thereexists a finite integer T ′ = T ((zτ(1))/2, δ) such that m(DσT ′ ) ≤ (zτ(1))/2, which contradicts thehypothesis. �Lemma 9. For every t ∈ I+, for any given ht ∈ Ht and θt ∈ ϒλ

t (ht), if 0 < z−t (h′t) < z+t (h′

t) < zt ,then, upto closure,

Dσt+1 = Dσt − Aσt (h′t) = {z ∈ Dσt |z < z−t (h′

t)}⋃{

z ∈ Dσt |z > z+t (h′t)}.

Proof. Fix any hτ ∈ Hτ, τ ∈ I+. To prove the Lemma, we show that for any given θτ ∈ ϒ(hτ),there cannot exist any triple z1 < z2 < z3 such that,

(γ + βz1)qτ − pτ ≥ δVτ+1(z1, ξτ+1(h′τ)); (γ + βz2)qτ − pτ < δVτ+1(z2, ξτ+1(h′

τ));

(γ + βz3)qτ − pτ ≥ δVτ+1(z3, ξτ+1(h′τ)). (A.24)

That is, buyer-types z1 and z3 accept θτ while buyer-type z2 rejects the offer. Subtracting thesecond inequality from the first in (A.24) yields,

β(z1 − z2)qτ − δ[Vτ+1(z1, ξτ+1(h′τ)) − Vτ+1(z2, ξτ+1(h′

τ))] > 0. (A.25)

And subtracting the second inequality in (A.24) from the third one gives,

β(z3 − z2)qτ − δ[Vτ+1(z3, ξτ+1(h′τ)) − Vτ+1(z2, ξτ+1(h′

τ))] > 0. (A.26)

Let Eqστ+1(z|hτ+1) denote the expected product quality accepted by z ∈ Dστ (hτ+1) along anyσ|hτ+1 ∈ �|hτ+1 . From Lemma 1 (cf. Eq. (A.4)) we know that,

δ[Vτ+1(z1, ξτ+1(h′τ)) − Vτ+1(z2, ξτ+1(h′

τ))] ≥ β(z1 − z2)Eqτ+1(z2|ξτ+1(h′τ));

δ[Vτ+1(z3, ξτ+1(h′τ)) − Vτ+1(z2, ξτ+1(h′

τ))] ≥ β(z1 − z2)Eqτ+1(z2|ξτ+1(h′τ)). (A.27)

But (A.27) implies that (A.25) is valid only if β(z1 − z2)[qτ − Eqτ+1(z2|ξτ+1(h′

τ))]> 0. That is,

ifqτ < Eqτ+1(z2|ξτ+1(h′

τ)), since z1 < z2. However, (A.27) also implies that (A.26) is valid only ifβ(z3 − z2)

[qτ − Eqτ+1(z2|ξτ+1(h′

τ))]> 0. That is, if qτ > Eqτ+1(z2|ξτ+1(h′

τ)), since z3 > z2.Hence, starting from (A.24), we have arrived at a contradiction. �


Lemma 10. Fix any σ ∈ �(δ). Take any t ∈ I+, and any ht ∈ Ht such that Dσt = [0, zt(1)) ∪ �t(cf. Lemma 4). If zt(1) ≤ z(δ) (cf. Lemma 6), and there exists some θt ∈ ϒλ

t (ht) and some B ⊂[0, zt(1)), with m(B) > 0, such that αt(ht, θt, z) = 1, ∀z ∈ B, then Dσt+2 = ∅.

Proof. Fix any σ ∈ �, and ht ∈ Ht . Partition the support of the monopolist’s offers,ϒλt (ht), into (1) the set of offers accepted by the supremum buyer-type zt : �λ

t (ht) ≡ {θ ∈ϒλt (ht)|αt(ht, θ, zt) = 1}, (2) the set of offers accepted by the marginal buyer-type z = 0 :

�λt (ht) ≡ {θ ∈ ϒλ

t (ht)|αt(ht, θ, z = 0) = 1}, and (3) the remainder: �λt (ht) ≡ {θ ∈ ϒλ

t (ht)|θ �∈�t ∪�t(ht)}. In fact,�λ

t (ht) = {(p = γ2/b, q = γ/b) ≡ θ}, since θ is the profit-maximizing ter-minal offer that must be accepted by z = 0. To establish the Lemma, it is sufficient to show thatany θt ∈ ϒλ

t (ht) accepted by a set of positive measure in the interior of [0, zt(1)) must either be θor must belong to the set �λ

t (ht). In the former case,Dσt+1 = ∅, and the Lemma holds. In the lattercase, it Dσt+1(ξσt+1(ht, θt)) = [0, z+t ), where z+t ≤ zt(1) ≤ z(δ). (Recall that z−t = inf Aσt (ht, θt)and z+t = supAσt (ht, θt).) But Lemma 6 then implies that the market is dissipated in period t + 1with probability 1, so that Dσt+2 = ∅.

We note first that, Eqσt (z|ht), the expected quality accepted by buyer-type z in σ|ht , is a non-decreasing function of z. From the argument in Lemma 1, for any z′, z′′ ∈ Dσt , z′ > z′′,

β(z′ − z′′)Eqσt (z′′|ht) ≤ δVσt+1(z′, ·) − δVσt+1(z′′, ·) ≤ β(z′ − z′′)Eqσt (z′|ht). (A.28)

Thus, Eqσt (z′′|ht) ≤ Eqσt (z′|ht), since z′ > z′′. Hence, Eqσt (z|ht) is a.e. continuous with welldefined left- and right-hand limits, Eqσt,L(z|ht) and Eqσt,R(z|ht), respectively. Lemma 7 im-plies that for any θt ∈ �λ

t (ht), Dσt+1(ht, θt) = Dσt − Aσt (ht, θt) = {z ∈ Dσt |0 ≤ z < z−t } ∪ {z ∈Dσt |z > z+t }. Let, for any θt ∈ �λ(ht), s

+t ≡ inf{z ∈ Dσt − Aσt (ht, θt)|z > z+t (ht, θt)}, and s−t ≡

sup{z ∈ Dσt − Aσt (ht, θt)|z < z−t (ht, θt)}.(We now fix a σ ∈ �, and eliminate the σ superscript.) Put, ηt(z, θt) ≡ (γ + βz)qt − pt . Take

any θt ∈ �λt (ht), and pick any z′ ∈ At(h′

t) and z ∈ Dt+1(ξt+1(h′t)) such that z < z−t . Such a buyer-

type exists given the assumption that θt ∈ �λt (ht). Hence,

ηt(z′, θt) − δVt+1(z′, ξt+1(h′

t)) − [ηt(z, θt) − δVt+1(z, ξt+1(h′t))]> 0. (A.29)

And since z′ > z, we also know from Lemma 1 that,

δVt+1(z′, ξt+1(h′t)) ≥ [β(z′ − z)Eqt+1(z|ξt+1(h′

t)) + δVt+1(z, ξt+1(h′t))]. (A.30)

Hence, (A.29) is valid only if qt > Eqt+1(z|ξt+1(h′t)) for every z ∈ Dt+1(ξt+1(h′

t)) such thatz < z−t . Let,

Eqt+1(s−t |ξt+1(h′t)) =

{Eqt+1(s−t |ξt+1(h′

t)) if s−t < z−tEqt+1,L(z−t |ξt+1(h′

t)) if s−t = z−t .

Note that qt > Eqt+1(s−t |ξt+1(h′t)). We claim for any θt ∈ �λ

t (ht) such that z−t < z+t , along anyσ|ξt+1(h′

t ),

δVt+1(s+t , ξt+1(h′t)) ≥ [β(s+t − s−t )Eqt+1(s−t |ξt+1(h′

t)) + δVt+1(s−t , ξt+1(h′t))]. (A.31)

From the definition of s+t and consumer rationality it follows that,

δVt+1(s+t , ξt+1(h′t)) ≥ ηt(s

+t , θt) = ηt(z

+t , θt) + β(s+t − z+t )qt. (A.32)

Consumer rationality also implies that

ηt(z+t , θt) ≥ δVt+1(z+t , ξt+1(h′

t)), (A.33)


and we also know from Lemma 1 that,

δVt+1(z+t , ξt+1(h′t)) ≥ [β(z+t − s−t )Eqt+1(s−t |ξt+1(h′

t)) + δVt+1(s−t , ξt+1(h′t))]. (A.34)

Suppose that s+t > z+t . Then, substituting (A.33), (A.34) into (A.32) yields,

δVt+1(s+t , ξt+1(h′t)) ≥ [β(z+t − s−t )Eqt+1(s−t |ξt+1(h′

t)) + δVt+1(s−t , ξt+1(h′t))]

+β(s+t − s−t )qt >[β(s+t − s−t )Eqt+1(s−t |ξt+1(h′

t)) + δVt+1(s−t , ξt+1(h′t))],

which establishes (A.31) for the case s+t > z+t . Next, suppose that s+t = z+t . We argue that (A.34)cannot hold as an equality in this case as well. Suppose not; that is, assume that

δVt+1(z+t , ξt+1(h′t)) = [β(z+t − s−t )Eqt+1(s−t |ξt+1(h′

t)) + δVt+1(s−t , ξt+1(h′t))]. (A.35)

Note that (A.33) and (A.34) are mutually consistent only if pt = (γ + βz+t )qt −δVt+1(z+t , ξt+1(h′

t)). Hence, for each z ∈ At(h′t), z < z+t ,

β(z− z+t )qt −[δVt+1(z, ξt+1(h′

t)) − δVt+1(z+t , ξt+1(h′t))] ≥ 0. (A.36)

Substituting (A.35) in (A.36) and recalling that

δVt+1(z, ξt+1(h′t)) ≥ [β(z− s−t )Eqt+1(s−t |ξt+1(h′

t)) + δVt+1(s−t , ξt+1(h′t))],

we conclude that (A.34) is valid only if

β(z− z+t )qt + β(z+t − s−t )Eqt+1(s−t |ξt+1(h′t)) ≥ β(z− s−t )Eqt+1(s−t |ξt+1(h′

t)). (A.37)

But since z < z+t , (A.37) is valid only if qt ≤ Eqt+1(s−t |ξt+1(h′t)), contradicting the earlier

established fact that qt > Eqt+1(s−t |ξt+1(h′t)). Thus, (A.35) is not valid when s+t = z+t , and

(A.31) is established. Next, under the hypotheses of this Lemma, for any τ ≥ t such thatDτ �= ∅, Dτ = [0, zτ(1)) ∪ �τ , zτ(1) ≤ zt(1), and �τ ⊆ �t . Define,

�τ(hτ) ≡ {θτ ∈ �(hτ)|m(Aτ(hτ, sτ)) > 0, z−τ < zτ(1)}. (A.38)

Thus, if θ ∈ �τ is such that there exists some B ⊂ Aτ(hτ, θ) ∩ [0, zτ(1)), and m(B) > 0, thenθ ∈ �τ(hτ). By definition also, z−τ < z+τ , for any θ ∈ �τ(hτ |zτ(1)). Hence, (A.31) implies thatfor any θt ∈ �t(ht),

δVt+1(s+t , ξt+1(h′t)) − [β(s+t − s−t )Eqt+1(s−t |ξt+1(h′

t)) + δVt+1(s−t , ξt+1(h′t))] ≡ ζt > 0.

(A.39)

We now show that if zt(1) ≤ z(δ) (cf. Lemma 6), then along any σ|ξt+1(h′t )

, the L.H.S.of (A.39) is strictly less than ζ for every ζ > 0. This will contradict (A.39) and the as-sumption that θt ∈ �t(ht). Thus, �t(ht) is empty (if zt(1) ≤ z(δ)), and it will followfrom (A.38) that �t(ht) is also empty, establishing the Lemma. Fix any σ|ξt+1(h′

t )for

which (A.39) is valid. Put, for any z ∈ Dτ(hτ), �+τ (z, hτ) ≡ {θ ∈ �τ(hτ)|z = z+τ (hτ, θ)},

�−τ (z, hτ) ≡ {θ ∈ �τ(hτ)|z = z−τ (hτ, θ)}, �−

τ (z, hτ) ≡ {θ ∈ �τ(hτ)|z = z−τ (hτ, θ)}. Thus,�+τ (z, hτ) is the subset of offers in �τ(hτ) for which the buyer-type z is the supremum

of the accepting set of buyers. It follows from Lemma 7 that for each τ ≥ t + 1, andany hτ ∈ Hτ(h′

t), if θ ∈ �τ(hτ) ∩ Cτ(s−t , hτ), then θ ∈ Cτ(s+t , hτ), as well. Similarly, ifθ ∈ Cτ(z, hτ) ∩ Cτ(s−t , hτ), for any z ∈ Dτ such that 0 ≤ z < s−t , then θ ∈ Ct(s−t , hτ).Also, if θ ∈ �+

τ (s−t , hτ), then θ ∈ Cτ(s−t , hτ). And, if θ ∈ �−τ (s+t , hτ) or θ ∈ �−

τ (s+t , hτ),then θ ∈ Cτ(s−t , hτ). In fact, Cτ(s

−t , hτ) = Cτ(s

+t , hτ) ∩ Cτ(s−t , hτ) ∪ �+

τ (s−t , hτ) and


Cτ(s−t , hτ) = Cτ(s

+t , hτ) ∩ Cτ(s−t , hτ) ∪ �−

τ (s+t , hτ) ∪ �τ(s+t , hτ). Using these observations,straightforward computations show that (A.39) is valid only if the following condition holds,

C1. There exists some date t + k, k = 1, 2, . . ., and some sets of histories and offers F ⊆Ht+k(ξt+1(h′

t)) and G ⊆ �t+k(s+t , F ) ∪ �−t+k(s

+t , F ), respectively, such that for every ht+k ∈ F

and each θt+K ∈ G,

δk{ηt+k(s+t , θt+k) − [β(s+t − s−t+k)Eqt+k+1(s−t+k|ξt+k+1) + δVt+k+1(s−t+k, ξt+k+1)

]} ≥ ζt.

(A.40)

We now argue that the condition (C1) cannot be satisfied along any σ|h′t, given the hypothesis

that zt(1) ≤ z(δ). The argument proceeds by partitioning the set of continuation equilibrium paths(�|h′

t) into those where the remaining market is dissipated with probability 1 by some dateT ≥ t +

1, and those where no such (maximal) termination date exists. The former case is handled in detailin Lemma 12 below. It is shown there that for any τ ∈ I+ (such that Dτ �= ∅), if there exists anyσ|hτ where the market is dissipated with probability 1 by some date T ≥ τ, then, for every hτ+k ∈Hτ+k(hτ), 0 ≤ k ≤ T − τ, (i) ϒλ

τ+k(hτ+k) = �τ+k(hτ+k) and (ii) ∀θτ+k ∈ ϒλτ+k(hτ+k), pτ+k =

η(s−τ+k, θτ+k) − δVστ+k+1(s−τ+k, ξτ+k+1(hτ+k, θτ+k)). These properties clearly invalidate (C1).It remains then to consider whether (C1) can hold for those continuation equilibria where the

residual market is not completely served with probability 1 by some finite date. Then, considerany such candidate σ|h′

t. We first establish that this condition does not apply for the class of

equilibrium offers �−t+j(s

+t , F ), for any j ≥ 1. It is sufficient to show that following any θt+j ∈

�−t+j(s

+t , F ), the market is dissipated with probability 1 by some finite date t + j + T . Fix any

θt+j ∈ �−t+j(s

+t , F ). By definition, Dt+j+1(ξt+j+1(h′

t+j)) = {z ∈ Dt+j|z ≤ s−t }, which can bewritten as,

{z ∈ Dt+j|z ≤ s−t } = [0, zt+j+1(1))⋃[∪Kk=2

(zt+j+1(k), zt+j+1(k + 1)

)], (A.41)

with zt+j+1 = zt+j+1(K + 1) ≤ s−t < z(δ). Here, K is some integer not exceeding j + 1, sinceonly one offer is accepted in each period and [0, s−t ) ⊂ Dt+1. Now, fix any σ|ξt+j+1 . We show that

for every τ ≥ t + j + 1, �τ(hτ) = ∅, for every hτ ∈ Hτ(ξt+j+1). Put z− ≡ z−τ (hτ, θτ) and z+ ≡z+τ (hτ, θτ). Two cases need to be considered. First, suppose z+ < zτ(1). In this case,Aτ(hτ, θτ) =[z−, z+). Using the generic representation in (A.41), and following the argument in Lemma 6,let,

M ≡[

(γ + βz)2(z+ − z−) + δ

∫ z−

0(γ + βy)2 dy + δ

∫ zτ (1)

z+(γ + βy)2 dy

+K∑k=2

δ

∫ zτ (k+1)

zτ (k)(γ + βy)2 dy

]. (A.42)

And compute the difference,

M(z−, z+,Dτ) ≡ (2b)−1[(γ2)m(Dτ) −M] ∝K∑k=2

(zτ(k + 1)

−zτ(k))

[γ2(1 − δ) − δβ

[β

3(z2τ (1) + (z+)2 + zτ(1)z+) + γ(zτ(1) + z+)

]]


+ (zτ(1) − z+)

[γ2(1 − δ) − δβ[

β

3(z2τ (k + 1) + z2

τ (k) + zτ(k)zτ(k + 1))

+ γ(zτ(k + 1) + zτ(k)]

]+ z−

[γ2(1 − δ) − z−β[β(

δ

3z− + (z+ − z−))

+ γ(δz− + 2(z+ − z−))]

]≡ E1 + E2 + E3. (A.43)

By assumption, zτ ≤ s−t < z(δ). And, E3 > 0, since z− ≤ z+ < zt+j+1(1) and zt+j+1(1) <zt(1) ≤ z(δ), while E2 > 0, since zτ(1) ≤ zt+j+1(1) < z(δ). Finally, E1 ≥∑K

k=2(zτ(k + 1) −zτ(k))

[γ2(1 − δ) − δβ(z2

τ + 2γzτ)]> 0. Thus, M > 0. Since τ(hτ) ≤ M, it follows that M >

τ(hτ), and hence immediate dissipation is optimal whenever z+ < zτ(1). The other case iswhen zτ(1) ≤ z+. In this case, Aτ(hτ, θτ) = [z−, zτ(1)) ∪∑K

k=2(zτ(k), zτ(k + 1)), zτ(K + 1) =z+ < zτ . An argument similar to that used in (A.42) shows that the terminating offer strictlydominates any candidate offer θτ ∈ �τ(hτ) in this case as well. Thus, we have establishedthat along any given σ|ξt+j+1 , if τ ≥ t + j + 1 is the first date at which there exists somehτ ∈ Hτ(ξt+j+1) and θτ ∈ ϒτ(hτ) such that every buyer-type z ∈ B ⊂ [0, zτ(1)), m(B) > 0,accepts θr, then Dτ+2 = ∅ (a.s.). But Lemma 6 implies that along any given σ|ξt+j+1 , someB ⊂ [0, zt+j+1(1)), m(B) > 0, must accept an offer by the date t + j + T , where T is boundedabove by SGI

([κ(zt+j+1(1), δ)(1 − zt+j+1(1))]/ψ(zt+j+1(1), δ)]

).

Meanwhile, suppose that (C1) is valid for the class of offers �−t+j(s

+t , F ). Put ζt+j = ζt/δ

j

(where ζt is quantified in (A.39)). Then, (A.40) implies that for each θt+j ∈ G,

(γ + βs+t )qt+j − ζt+j − [β(s+t − s−t )Eqt+j+1(s−t |ξt+j+1) + δVt+j+1(s−t , ξt+j+1)] ≥ pt+j.

(A.44)

But s+t > s−t ≥ s−t+j , and every buyer-type receives non-negative utility along any equilibrium

path. Thus, (A.44) implies that pt+j ≤ (γ + βs+t )qt+j − ζt+j . SinceG ⊆ �−t+j(s

+t , F ) by defini-

tion, it follows that for every ht+j ∈ F , and any t+j ∈ G, there exist some s+t ≤ s+t+j(ht+j, θt+j) <zt+j and some 0 < s−t+j < s+t such thatDt+j+1(ξt+j+1(h′

t+j)) = {z ∈ Dt+j|s+t+j < z < zt+j, 0 ≤z < s−t+j}. Hence, using (A.44),

δVt+j+1(s+t , ξt+j+1) > ηt+j(s+t+j, θt+j) ≥ β(s+t+j − s+t )qt+j + ζt+j

+ [β(s+t − s−t )Eqt+j+1(s−t |ξt+j+1) + δVt+j+1(s−t , ξt+j+1)] ≥ ζt+j

+ [β(s+t − s−t )Eqt+j+1(s−t |ξt+j+1) + δVt+j+1(s−t , ξt+j+1)]. (A.45)

The reason for the last inequality in (A.45) is that if θt+j ∈ �t+j(s+t+j, ht+j), then qt+j ≥Eqt+j+1(s−t |ξt+j+1). To prove this, notice that if θt+j ∈ �t+j , then δVt+j+1(s−t , ξt+j+1) ≤ηt+j(s−t , θt+j). And, this inequality is of the opposite sign for buyer-type s+t . Hence,

β(s+t+j − s+t )qt+j − [δVt+j+1(s−t , ξt+j+1) − δVt+j+1(s+t , ξt+j+1)] ≤ 0. (A.46)

But we know that, δVt+j+1(s+t , ξt+j+1) − δVt+j+1(s−t , ξt+j+1) ≥ β(s+t − s−t )Eqt+j+1(s−t+j|ξt+j+1). Thus, since s−t > s−t+j , (A.46) is valid only if qt+j ≥ Eqt+j+1(s−t |ξt+j+1).Next, applying a previous argument, (A.45) is valid only if there exists some date t + j + r andF ⊆ Ht+j+r and G ⊆ �t+j+r(s+t+j, F ) ∪ �−

t+j+r(s+t+j, F ), such that for every ht+j+r ∈ F and


θt+j+r ∈ G,

δr{ηt+j+r(s+t+j, θt+j+r) −

[β(s+t+j − s−t+j+r)Eqt+j+r+1(s−t+j+r|ξt+j+r+1)

+ δVt+j+r+1(s−t+j+r, ξt+j+r+1)]}

≥ ζt+jδr

≡ ζt+j+r. (A.47)

But ifG ⊆ �−t+j+r(s

+t+j, F ), then (A.47) implies that 0 < s−t+j+r < s+t+j ≤ s+t+j+r < zt+j+r, and{

δVt+j+r+1(s+t+j+r, ξt+j+r+1) −[β(s+t+j+r − s−t+j+r)Eqt+j+r+1(s−t+j+r|ξt+j+r+1)

+ δVt+j+r+1(s−t+j+r, ξt+j+r+1)]}

≥ ζt+j+r+1. (A.48)

Extrapolating this line of argument forward, (A.48) is valid only if there exists someσ|ξt+j+r+1(h′

t+j+r) for which there is some date k ≥ t + j + r + 1 and some F ⊆ Hk and

G ⊆ �k(s+t+j+r, F ) ∪ �−

k (s+t+j+r, F ), such that for every hk ∈ F and θk ∈ G, a condition

analogous to (A.40) applies. But if θk ∈ �k(s+t+j+r, F ), then Dk+1(h′k) = [0, zk+1(1)) ∪[∪Ww=2 (zk+1(w), zk+1(w+ 1))

], and zk+1 = zk+1(W + 1) ≤ s−t < z(δ). (Note that if, as hypoth-

esized, θt+j ∈ �t+j(s+t , F ), then from Lemma 7 each buyer-type s+t ≤ z ≤ s+t+j accepted theoffer θt+j in period t + j.) Thus, from a previous argument, the condition (C1) applied to thebuyer-type s+t+j cannot be satisfied in the class of offers �k(s

+t+j+r, F ).

We are now in a position to provide a necessary condition for (C1) to be valid. Fix any σ ∈ �,any date hτ, τ ∈ I+, and offer θτ = (pτ, qτ) ∈ ϒτ(hτ). Then, for each j = 1, 2, . . ., and any givenF ⊆ Hτ+j and G ⊆ �τ+j(z, F ) ∪ �−

τ+j(z, F ), let νστ+j(z, F,G|h′τ) denote the probability that,

conditional on h′τ , and along σ|h′

τ, buyer-type z ∈ Dτ will exit by accepting an offer in G that is

superior (in terms of utility) to the offer θτ . The argument given above implies that (C1) is validonly if there exists an infinite sequence of dates {jk}∞k=0 and sets of buyer-types {zjk}∞k=0, j0 ≡ t,with s+j(w+1) ≥ s+jw, w ≥ 1, such that

j1 ≡ inf{i ≥ t + 1|∃F ⊆ Hi(ξt+1),G ⊆ �−i (s+j0, F ) : νi(s

+j0, F,G|ξt+1) > 0}

j2 ≡ inf{i ≥ j1 + 1|∃F ⊆ Hi(ξj1+1),G ⊆ �−i (s+j1, F ) : νi(s

+j1, F,G|ξj1+1) > 0}

...

(A.49)

We now show that (A.49) cannot exist along any σ|h′t. Let, i∗ ≡ SGI([ln(ζt) −

ln(3(γ + β)q)]/ ln δ), and let jk∗ ≡ inf{k|jk∗ ≥ i∗}. By construction, ζjk∗ ≥ 3(γ + β)q. Hence,for any i ≥ jk∗, and any given F ⊆ Hτ(ξt+1(h′

t)), every θi ∈ �−i (s+j(k∗−1), F ) must be such that

pi ≤ −2(γ + β)q. (This follows from the definition of ζi in (A.40).) Thus, (A.39) is valid only ifthe condition (C1) applies, which in turn is valid only if, along every σ|ξt+1(h′

t ), for some i ≥ t + 1,

there exists a class of offers θi such that pi < −(γ + β)q. But it is established in Lemma 3 thatsuch offers occur with zero probability along any equilibrium path. Thus, we have arrived at acontradiction from the assumption that (C1) is valid, and hence it must be that �τ(hτ |zτ(1)) = ∅(where �τ is defined in (A.38)), for any hτ ∈ Hτ, τ ∈ I+, whenever zτ(1) ≤ z(δ). �

Lemma 11. Fix any σ ∈ �(δ). There exists some T (σ) ∈ I+ such that DσT (σ)+1 = ∅.

Proof. The conclusion of the Lemma follows immediately from Lemmas 8 and 10. Lemma 10specifies sufficient conditions on the nature of the residual market set Dσt that ensures complete


market depletion in two more offers. However, Lemma 8 ensures that this sufficient condition issatisfied with probability 1 in a finite number of time periods, along any σ ∈ �. �

Lemma 12. Fix any σ ∈ �(δ). For any given ht ∈ Ht, 0 ≤ t ≤ T σ , if θt ∈ ϒλt (ht), then

(i) m(At(h′t)) > 0; (ii) At(h′

t) = {z ∈ Dt(ht)|z−t (h′t) ≤ z < zt(h′

t)}; (iii) pt = (γ + βz−t (h′t))qt −

δVσt+1(h′t , z

−t (h′

t)); (iv) qt = (γ + βz−t (h′t))/b > qt+1.

Proof. The proof proceeds with backward induction from the terminal offer date T, whose exis-tence is assured by Lemma 11. Fix anyhT ∈ HT such thatDT �= ∅, butDT+1 = ∅. From Lemma 1,we know thatm(DT ) > 0. And, from Lemma 3, we know thatϒT (hT ) = {(p = γ2/b, q = γ/b)}.Thus, properties (i)–(iv) are satisfied (since qT+1 = 0, and hence VT+1(h′

T , ·) = 0). The monopo-list’s dissipating profits are T = (γ2/2b)mT (where mT ≡ m(DT )), and every z ∈ DT receivesthe current-value utility of (γβ/b)z.

Take any hT−1 ∈ HT−1 and offer θT−1 ∈ ϒT−1(hT−1) such that DT (hT−1,

ξT−1(hT−1, θT−1)) �= ∅ (i.e., DT−1(hT−1) − AT−1(h′T−1) �= ∅). Such an offer will be called a

continuation offer. We first argue that m(AT−1(h′T−1)) > 0, for any continuation offer. Suppose

not; then, by hypothesis, m(DT ) = m(DT−1), and hence T−1(hT−1) = δ(γ2/2b)mT−1.However, from Lemmas 2 and 3 we know that the monopolist can always dissipate the marketin period T − 1 with the profits (γ2/2b)mT−1, which contradicts subgame perfection forthe monopolist, since δ < 1. This establishes property (i). We argue next that qT−1 > δqT ,for any continuation offer. Suppose not; given the period-T equilibrium price–quality offer,VT (·, z) = (γβ/b)z, z ∈ DT−1. Then, z−T−1 ≡ inf AT−1 is given by,

(γ + βz−T−1)qT−1 − pT−1 = δγβz−T−1

b. (A.50)

Since, qT−1 ≤ δqT , by hypothesis, it is easy to check that: AT−1 = {z ∈ DT1 |z ≤ z−T−1}. Inparticular, the marginal buyer-type z = 0 ∈ AT−1 and hence z = 0 �∈ DT . But from Lemma4 we know that z = 0 ∈ Dt , for all Dt �= ∅. Thus, we have arrived at a contradiction startingfrom the hypothesis that δqT ≥ qT−1. We conclude that, for every continuation offer in periodT − 1, DT = {z ∈ DT−1|z < z−T−1}, establishing property (ii).

We now argue that property (iii) must hold for any continuation offer in T − 1. Suppose not,and let

pT−1 < (γ + βz−T−1) − δ

(γβz−T−1

b

). (A.51)

There are two cases to consider. First, let z−T−1 ∈ AT−1. Then, the monopolist could raise pT−1

slightly, and yet not affect the set AT−1 due to the fact that DT = {z ∈ DT−1|z ≤ z−T−1} aslong as (A.51) holds weakly. Thus, there exists a strictly improving perturbation on pT−1 forthe monopolist, since m(AT−1) > 0. Meanwhile, suppose that z−T−1 �∈ DT−1. But we knowDT−1 is a countable union of intervals, so that for every ε > 0, there exists some z′ ∈ AT−1such that (z′ − z−T−1) ≤ ε. Then, let the monopolist increase pT−1 by the quantity ε > 0. Us-ing the arguments above, every z ∈ AT−1 will still accept the offer (pT−1 + ε, qT−1), sinceηT−1(z, θT−1) + ε− (δγβz)/b ≥ 0, for ε small. (Recall that ηt(z, θt) ≡ (γ + βz)qt − pt .) Thus,this perturbation on pT−1 is also strictly improving. Hence, (A.51) cannot hold for any (T − 1)-period offer. Meanwhile, the opposite strict inequality also violates subgame perfection for themarginal buyer-type in the acceptance set. We can now also derive the optimal quality offer inperiod (T − 1). Let, for any z′ in the closure ofDT−1 (≡ DT−1), DT−1(z′) ≡ {z ∈ DT−1|z < z′}.


Then, the optimal (T − 1)-period offer is derived from the solution to,

maxqT−1∈Q,z−T−1∈DT−1

{[(γ + βz−T−1)qT−1 − δγβz−T−1

b− bq2

T−1

2]m(DT−1 −DT−1(z−T−1))

+ δγ2

2bm(DT−1(z−T−1))

}. (A.52)

Since m(AT−1) > 0, (A.52) yields the optimal quality offer, qT−1 = (γ + βz−T−1)/b > qT . TheLemma is then established for period (T − 1).

We now use the induction principle. Notice that present-value of the monopolist’s profitsalong the equilibrium path in periods (T − 1) and T (i.e., T−1(·) and T (·)) depend on historyonly up-to DT−1 (or DT ); i.e., the present value of profits in any subgame are independent ofthe price–quality history θj = (θ0, . . . , θ

j−1), j = T − 1, T . Then, fix any date τ ≤ r ≤ T − 2,and suppose that for every hr+j ∈ Hr+j, j = 1, . . . , T − r, the properties established abovefor period T − 1 apply: (1) r+j(hr+j) depends only on Dr+j , (2) qr+j = (γ + βz−r+j)/b,

where z−r+j (≡ inf Ar+j) = supDr+j+1, (3) pr+j = (γ + βz−r+j)qr+j − δVr+j+1(z−r+j), and (4)

Ar+j = {z ∈ Dr+j|z−r+j < z < z+r+j} with z−r+j = zr+j+1. Under these hypotheses, the monopo-list’s optimization problem is,

supθr∈P×Q

[(pr − b(qr)

2/2)m(Ar(hr, θr)) + δ r+1(Dr+1(hr, θr))]

subject to,

(γ + βz)qr − pr − δVr+1(z,Dr+1) ≥ 0, ∀z ∈ Ar(hr, θr). (A.53)

We claim thatm(Ar(hr, θr)) > 0, for any candidate solution θr to (A.53). Suppose not; by hypoth-esis,m(Ar(·, θr)) = 0. Hence,Dr andDr+1(hr, θr) differ by sets of buyers of measure zero. Also,by the induction hypotheses, r+1 depends only on Dr+1; is independent of θr. Thus, under themaintained hypothesis, r(hr) = δ r+1(Dr+1(hr, θr)). Now, we also know from the inductionhypothesis that every θr+1 ∈ ϒr+1(ξr+1(hr, θr)) is such thatm(Ar+1(·, θr+1)) > 0. Then, considerthe defection where the monopolist offers any θr+1 ∈ ϒr+1(ξr+1(hr, θr)) in period r itself. Usingthe foregoing argument and the induction hypotheses on for r + 2, . . . , T − 2, this defection im-proves the monopolist’s payoffs by the amount (1 − δ)(pr+1 − b(qr+1)2/2)m(Ar+1) > 0, since(pτ − b(qτ)2/2) ≥ γ2/2b > 0 (cf. Lemma 2). Hence, the hypothesis that m(Ar(hr, θr)) = 0, forany θr ∈ ϒr(hr), contradicts subgame perfection for the monopolist, and property (i) is estab-lished.

Suppose, next that for any candidate solution to (A.53), ηr(z−r (h′r), θr) > δVr+1(z−r (h′

r), ξr+1);i.e., property (iii) does not apply. First, let z−r ∈ Ar(hr, θr). Then, the monopolist could raise prslightly, and yet not affect the set Ar due to consumer rationality. Furthermore, this perturbationdoes not affect the present value of profits from period r + 1 onwards since (by the inductionhypothesis) r+1 depends only onDr+1 and we have just shown that the said perturbation leavesDr+1 unaffected. Meanwhile, if z−r �∈ Dr, then by the definition of the infemum, there exists someε > 0 such that ηr(z′, θr) > δVr+1(z′, ξr+1), for every z′ ∈ (z−r , z−r + ε) and z′ ∈ Ar(h′

r, θr). Theimmediately preceding argument can then be applied to show that there exists a strictly improvingperturbation pr + ε, where for ε > 0 and sufficiently small, Ar (and hence Dr+1) is unaffected.Property (iii) is thus proved. Then, we can re-write the monopolist’s optimization problem along


any σ|hr in the following form:

maxqr,z

−r

[(γ + βz−r )qr − δVr+1(z−r , ξr+1(hr, θr) − b(qr)

2/2)m(Ar(hr, θr))

+ δ r+1(Dr+1(hr, θr))]. (A.54)

But r+1(Dr+1) is independent of price–quality histories by hypothesis. The first-ordercondition for the optimal quality offer in period r is, qr = (γ + βz−r )/b. Suppose that qr <E[qr+1|ξr+1(hr, θr)

]; i.e., property (iv) is false. Since all behavioral strategy equilibrium paths are

mixtures over pure strategy equilibrium paths in games of perfect recall (cf. Kuhn (1953)), theremust exist a pure strategy continuation equilibrium σ|h′

rsuch that qr < δqr+1. But by the induction

hypothesis, for every (qr+1, pr+1) ∈ ϒr+1, qr+1 = (γ + βz−r+1(hr+1, θr+1))/b. Thus, under thehypothesis that qr < δqr+1, it must be the case that z−r+1 > z−r . But we now establish that underthe induction hypotheses on σ|hr+1 , for every θr ∈ ϒ(hr), Ar(hr, θr) = {z ∈ Dr|z−r ≤ z ≤ zr},which rules out z−r+1 > z−r , since zr ≡ supDr. We recall from Lemma 10 that, sr+1 ≡ inf{z ∈Dr − Ar(hr, θr)|z > z+r (hr, θr)}, and s−r ≡ sup{z ∈ Dr − Ar(hr, θr)|z < z−r (hr, θr)}. So supposethat z+r < zr. This implies that the buyer sets, {s+r+1 < z ≤ zr} and {0 ≤ z < s−r+1} reject θr.And since σ|hr+1 is a pure strategy continuation equilibrium, by assumption, there must ex-ist some date r + 1 ≤ j ≤ T when buyer-type s+r+1 accepts the offer θj = (pj, qj). Now, weshow that it must be the case that: s+r+1 = z−j (≡ inf Aj). Suppose not; i.e., z−j < s+r+1. Then,

by hypothesis, we have three inequalities: (1) ηr(z−r , θr) ≥ δj−rηj(z−r , θj); (2) ηr(s+r+1, θr) <

δj−rηj(s+r+1, θj); (3) ηr(z−j , θr) < δj−rηj(z−j , θj), where z−j < z−r < s+r+1. Subtracting (2) from

(1) yields β(z−r − s+r+1)(qr − δj−rqj) > 0 so that qr < δj−rqj . But subtracting (3) from (1) yields,β(z−r − z−j )(qr − δj−rqj) > 0, which contradicts the fact that qr < δj−rqj . Since s+r+1 = z−j (cf.

property (iv)), δVr+1(s+r+1, ξr+1(hr, θr)) = δj+1−rVj+1(s+r+1, ξj+1(h′j)). But the induction hypoth-

esis is that qj > δqj+1 > δqj+2 . . . > δqT . Then, by Lemma 13 below the market depletion ismonotonic along σ|hj+1 ; i.e., for k = j + 1, . . . , T , zk ∈ Ak and z−k > z−k+1. Hence, if s+r+1 re-jects θj it will accept θj+1−r along any continuation equilibrium path. Thus, we conclude that

δVr+1(s+r+1, ξr+1(hr, θr)) = δj+1−r[β(s+r+1 − z−j+1)qj+1 + δVj+2(z−j+1, ξj+2(h′j+1))].

(A.55)

Since buyer-type z−r also has the option of rejecting θr and accepting θj+1, and since there mustexist some z−j+1 < z < s+r+1 that accepts θj+1 (as m(Aj+1) > 0), it then must be the case that,

(γ + βz−r )qr − pr ≥ δj+1−r[β(z−r − z−j+1)qj+1 + δVj+2(z−j+1, ξj+2(h′j+1))];

(γ + βs+r+1)qr − pr < δj+1−r[β(s+r+1 − z−j+1)qj+1 + δVj+2(z−j+1, ξj+2(h′j+1))];

(γ + βz)qr − pr ≥ δj+1−r[β(z− z−j+1)qj+1 + δVj+2(z−j+1, ξj+2(h′j+1))]. (A.56)

Using a previous argument, the three inequalities in (A.56) are inconsistent. Thus, we haveshown that every candidate buyer-type s+r+1 will accept the current offer θr. Thus, Ar = {z ∈Dr|z ≥ z−r }. Hence, (γ + βz−r )/b > δ(γ + βz)/b for every z ∈ Dr+1. We conclude that thereexists no pure strategy continuation equilibrium σ|hr+1 in which qr < δqr+1, and hence qr >δE[qr+1|ξr+1(hr, θr)]. Properties (ii) and (iv) are also then established. �


Lemma 13. Fix any σ ∈ �, and suppose that for every t, qσt ≥ qσt+1. Then, for every period τ,and z, z′ such that z′ > z, if z �∈ Dστ , then z′ �∈ Dστ as well.

Proof. Suppose not; in particular, suppose that there exists some period y and hy ∈ Hy such thatz ∈ Dy accepts θy but z′ > z rejects θy. By assumption, it then must be the case that,

(γ + βz)qy − py ≥ δVy+1(z, hy+1(hy, θy));

(γ + βz′)qy − py < δVy+1(z′, hy+1(hy, θy).) (A.57)

However, under the maintained hypothesis that qt is non-increasing, it must also be the case that,Vy+1(z′, hy+1(hy, θy)) − Vy+1(z, hy+1(hy, θy)) ≤ β(z′ − z)qy. The reason is that in the continu-ation equilibrium path σ|hy+1 , buyer-type z can always follow the strategy of accepting the sameprice–quality offer that is accepted by the buyer-type z′. Let this offer be (py+j, qy+j) in someperiod y + j. (Note that buyer-type z′ must accept some offer in the continuation equilibrium: ifhe does not accept any offer, then he gets zero utility, whereas he can guarantee positive utilityby accepting the current offer (py, qy).) Then, clearly

Vy+1(z, hy+1(hy, θy)) − Vy+1(z′, hy+1(hy, θy)) ≥ δjβ(z− z′)qy+j ≥ β(z− z′)qy,

since δ < 1, and by assumption qy ≥ qy+j and z′ > z. But then it is also the case that

[(γ + βz′)qy − py − δVy+1(z′, hy+1(hy, θy))]

− [(γ + βz)qy − py − δVy+1(z, hy+1(hy, θy))] = β(z′ − z)qy

− δ[Vy+1(z′, hy+1(hy, θy)) − Vy+1(z, hy+1(hy, θy))] > 0,

since δ < 1. Thus, the assumption that z accepts θy while z′ > z rejects this offer is contradicted(cf. (A.57)). �

Proof of Theorem 2. Fix any σ ∈ �, and take any date r ∈ I+, and history hr ∈ Hr. FromTheorems 1 and 2, the present expected value of the monopolist’s profits satisfy the dynamicprogramming (DP) equation,

r(hr) = supz−r ∈Dr

{[(γ + βz−r )2

2b− δVr+1(z−r ,Dr(z

−r ))

]nr(z

−r ) + δ r+1(Dr(z

−r ))

}.

(A.58)

Here, nr(z−r ) ≡ m(Dr −Dr(z

−r )). Let zr(hr) denote the solution correspondence of (A.58). zr(hr)

clearly depends on hr only through Dr. Now let,

z∗r (Dr) ≡ argminz∈zr(Dr)

[β(zr − z)

(γ + βz)

b+ δVr+1(z,Dr(z))

]. (A.59)

Here, the minimand is the utility of the supremum buyer-type zr in period r. Then, z∗r (Dr) is thesolution correspondence that determines the optimal market depletion, and price–quality offerin period r, for any hr ∈ Hr. Any other z ∈ (zr − z∗r ) is suboptimal, since offers in z∗r are moreprofitable in period r, and have the same present expected value of profits from period r + 1onwards. And since m(Ar(hr, θr)) > 0, the optimal quality is qr = (γ + βz−r )/b, z−r ∈ z∗r (Dr).Finally, (iii) of Lemma 12 must apply for the optimal pr. Hence, the monopolist’s optimal price–quality offer in period r satisfies Eqs. (7) and (8) in the text. �


Proposition 1. limδ↑1 zk(δ) = 0 for each k = 1, 2, . . . .

Proof. The proof proceeds by backwards induction on z1, . . .. From Lemma 7, z1 > 0 for each0 < δ < 1. We first prove that in fact,

Lemma 14.

z1(δ) =

⎧⎪⎨⎪⎩

min(

1, γ2β

)if 0 < δ ≤ 3

4

min(

1, 2γ[√

1−δ−(1−δ)]β

)if 3

4 < δ < 1.(A.60)

Proof. From Theorems 1 and 2, we know that in any σ ∈ �(δ), and for any given statez′ ∈ ψσ1 (δ), p(z′) = γ2/b, and q(z′) = γ/b. Hence, (z′) = (γ2/2b)z′ and V ∗(z′) = (γβ/b)z′.Consider now any state z′ ∈ ψσ2 (δ). Put,

Y (z, z′) ≡[−β

2z3

2b+ β

b(βz′

2− γ(1 − δ))z2 + (1 − δ)

γ

b(βz′ − γ

2)z

]. (A.61)

Then, in period (T σ − 1), given the state z′, the monopolist’s optimization problem is to choose0 < z < z′ to maximize:

F (z, z′) = Y (z, z′) + δ(γ2/2b)z′. (A.62)

However, we know that if the market is completely served for any state 0 < z′ ≤ 1, then themonopolist obtains profits of (γ2/2b)z′. Thus, it follows from (A.62) that z = 0 is the globalmaximum of F (z, z′) if and only if Y (z, z′) ≤ 0 for every 0 ≤ z < z′.

Then, let o(z′) denote the set of zeroes of the function Y (z, z′) (where we suppress the implicitdependence of o(z′) on δ for notational convenience). Clearly, 0 ∈ o(z′) for each 0 < z′ ≤ 1. Butit is also true that Y (z′, z′) ∝ −(γ2/2)z′(1 − δ) < 0. From the continuity of Y , it follows that(for any z′ and any δ) if there exists some x ∈ o(z′), x > 0, then there must also exist someo < z′′ < z′ such that Y (z′′, z′) > 0. Then, suppose that for some z′, there exists 0 < z′′ < z′ suchthat z′′ ∈ o(z′). And because z′′ is a critical point of Y (z, z′), it must be implicitly defined by,

−β2(z′′)2 + βz′′(βz′ − 2γ(1 − δ)) + γ(1 − δ)γ(2βz′ − γ) = 0. (A.63)

However, (A.63) has only one non-negative root:

z+1 (z′) ≡ βz′ − 2γ(1 − δ) +√K(z′)

2β, K(y) = β2y2 + 4γβy(1 − δ) − 4γ2δ(1 − δ).

(A.64)

But if 0 < δ ≤ 3/4, then z+1 (z′ = γ/2β) = 0. In fact, for all 0 < δ ≤ 3/4, the unique non-negativereal component of o(γ/2β) is zero. But it is seen from (A.64) that z+1 (·) is a strictly increasingfunction, and hence z+1 (z) > 0 for all z > γ/2β if 0 < δ ≤ 3/4. From a previous argument, thismust imply that for discount factors in the said range and z > γ/2β, Y (·, z′) > 0 on some openinterval in (0, z′). Conversely, for any z < γ/2β, Y (·, z′) < 0 everywhere on (0, z′). Hence, for0 < δ ≤ 3/4, z1(δ) = min(1, γ/2β).

We turn now to the case where 3/4 < δ < 1. For discount factors in this range, there exists apositive real component of o(z′); viz., (2δ− 3/2). Hence, 0 �∈ o(z′ = γ/2β), and therefore z1(δ) <γ/2β whenever δ > 3/4. To compute z1(δ) for δ > 3/4, we return to the function F (z, z′) (cf.


(A.62)), and note that the critical points of this function are given by the (real) solutions to thefirst-order condition:

Fz(z, z′) = −3γ2z2

2+ βz(βz′ − 2γ(1 − δ)) + (1 − δ)(γβz′ − γ2

2) = 0. (A.65)

Now, put,

w(δ) ≡ (γ/β)[√

3(1 − δ)δ− (1 − δ)]. (A.66)

If z′ ≥ max[0, w(δ)], then (A.65) has two real roots given by

βz′ − 2γ(1 − δ) ± √K(z′)

3β, K(z′) ≡ β2(z′)2 + 2γβz′(1 − δ) + γ2(1 − δ)(1 − 4δ).

(A.67)

Then, let,

z+1 (z′) ≡[βz′ − 2γ(1 − δ) + √

K(z′)]

3β. (A.68)

It is readily checked that Fzz(z+1 (z′), z′) < 0, and hence z+1 (z′) is the unique relative maximum of

F (z, z′).14 We claim that for 3/4 < δ < 1, z1(δ) is (implicitly) defined by,

Y (z′) ≡ Y (z+1 (z′), z′) = 0 (A.69)

(if the solution to (A.69) is less than 1, and is set equal to 1 else.) First, we argue that z1(δ) ≥ w(δ)(cf. (A.66)). By definition (cf. (A.61)), Y (z′, z′) < 0 and Y (0, z′) = 0. Hence, if z′ < w(δ), thenY (z, z′) < 0 for every 0 < z < z′ because there is no non-negative critical point for any z′ < w(δ).Thus, z1(δ) ≥ w(δ). Now suppose that there exists some 0 < z∗ < 1 that satisfies (A.69). Then,clearly Y (z′) > 0 for each z′ > z∗, because Y (·) is an increasing function. Hence, by definitionof z1(δ), z∗ ≥ z1(δ). On the other hand, for every w(δ) ≤ z′ < z∗, Y (z′) < 0. But then for suchz′, Y (z, z′) < 0 for each 0 < z < z′. This is because Y (0, z′) = 0, for every z′, there are only twostationary points of F (z, z′), and z+1 (·) is the relative maximum.

Now define, ρ(δ) ≡ (2γ/β)(√

1 − δ− (1 − δ)). Notice that if z′ = ρ(δ), then from (A.64),K(z′) = 0. In fact, z+1 (z′ = ρ(δ)) = (γ/β)(

√(1 − δ) − 2(1 − δ)) > 0 whenever 3/4 < δ < 1.

Thus, if z′ < ρ(δ), then there exists no zero of the function Y (z, z′) that is a real and positivenumber. Meanwhile,

ρ(δ) − w(δ) = (2γ

β)[√

1 − δ−√

(1 − δ)3δ− (1 − δ)]> 0,

whenever 3/4 < δ < 1. Hence, z+1 (z′) is well-defined for z′ = ρ(δ). Thus, if it is the case thatY (ρ(δ)) = 0, then we can conclude that z1(δ) = ρ(δ). This is because, as noted above, for everyz′ < ρ(δ), there exists no real and positive component of o(z′), i.e., the set of zeroes of Y (z, z′).And this implies that for all such z′, z = 0 is the unique global maximum of Y (z, z′) because Y (·)is strictly increasing. Meanwhile, for any z′ > ρ(δ), Y (z′) > 0. Thus, z1(δ) �> ρ(δ).

14 One can verify the prior computed value of z1(δ) for 0 < δ ≤ 3/4. If δ ≤ 3/4, then z+1 (z′) ≤ 0 whenever z′ ≤ γ/2β.Thus, if 0 < δ ≤ 3/4, then D = [0, z′) is immediately dissipated whenever z′ ≤ γ/2β. Hence, z1(δ) = min(1, γ/2β) fordiscount factors in this range.


Then, to establish that Y (ρ(δ)) = 0, it is sufficient to show that z+1 (ρ(δ)) = z+1 (ρ(δ)). Substi-tuting the approriate expressions, we compute,

z+1 (ρ(δ)) − z+1 (ρ(δ)) =[βρ(δ)

6− γ(1 − δ)

3+√K(ρ(δ))

2−

√K(ρ(δ))

3

]∝ −√

1 − δ

+ 2(1 − δ) +√

5 − 9δ− 4√

1 − δ(1 − δ) + 4δ2 = 0 (A.70)

(because (√

1 − δ− 2(1 − δ))2 = 5 − 9δ− 4√

1 − δ(1 − δ) + 4δ2). This verifies that Y (ρ(δ)) =0, and hence z1(δ) = ρ(δ). It is straightforward to show that for any 3/4 < δ < 1, t given the statez1(δ), the optimal correspondence for the succeeding state (cf. Eq. (6)) is, z∗(z1(δ)) = {0}. Hence,z1(δ) = min(1, ρ(δ)) whenever 3/4 < δ < 1, and this establishes the Lemma. �

We now complete the Proof of Proposition 1. Clearly, limδ↑1 z1(δ) = 0 (cf. (A.60)).Moreover, z1(δ) is piecewise continuously differentiable, and we compute, z′1(δ) ∝ [(1 −(1/2)(

√(1 − δ))−1/2], for 3/4 < δ < 1. Hence, limδ↑1 z

′1(δ) = −∞. Since z

′1(δ) is continuous

on (3/4, 1), it follows that z′1(δ) < 0 for δ in some neighborhood of 1. We also compute,

limδ↑1

[z1(δ)z′1(δ)] = lim

δ↑1[2γ2

β2 (−1 + 3√

1 − δ− 2(1 − δ))] = −2γ2

β2 . (A.71)

Because z+1 (z′) is continuous and strictly increasing in z′ (cf. (A.68)), it is invertible. We claim

that z2(δ) = z+−1

1 (z1(δ)). Suppose not; then, there exists some σ ∈ �(δ), and some z′ ∈ ψσ2 (δ)such that z′ > z2(δ). But along the equilibrium path if zj = z′ ∈ ψσ2 (δ) at some stage j, then themonopolist’s optimal policy is derived from maximizing F (z, z′) (cf. (A.62)), for which we knowthere is a unique solution such that the next-period state is, zj+1 = z+1 (z′). But by hypothesis,z+1 (z′) > z1(δ), and subgame perfection then requires that zj+2 > 0, contradicting the hypothesisthat z′ ∈ ψσ2 (δ). Hence, for every σ ∈ �(δ), supψσ2 (δ) ≡ zσ2 (δ) ≤ z2(δ). In fact, using (A.65) and(A.68), z2(δ) is implicitly defined by,

βz2(δ) − 2γ(1 − δ) +√K(z2(δ)) − 3βz1(δ) = 0. (A.72)

Let, K(x) ≡ √K(x). Application of the implicit function theorem to (A.72) yields,

z′2(δ) =

{−[

(2γ − 3βz′1(δ)

β + K(z)

]+ [γβz2(δ) − γ2/2(8δ− 5)]

βK(z) + β(βz+ γ(1 − δ))

}∣∣∣∣∣z=z2(δ)

. (A.73)

The first term of (A.73) is negative as δ → 1. The denominator of the second term is positive for 0 <δ < 1. (A.72) and K(z2(δ)) > 0, imply that for any 3/4 < δ < 1, z2(δ) < [3βz1(δ) + 2γ(1 − δ)].Thus,

βz2(δ) − (γ/2)(8δ− 5)) < β[3βz1(δ) + 2γ(1 − δ)) − (γ2/2)(8δ− 5)].

But the latter term is negative as δ → 1, since z1(δ) is monotonically declining to 0 in aneighborhood of 1, (8δ− 5) > 0 for 5/8 < δ < 1, and γ > 0. (A.73) then implies that we canfind some δ∗ such that z2(δ) is monotonically declining on δ∗ < δ < 1. Furthermore, z2(δ) isclearly uniformly bounded on 0 < δ < 1. Hence, limδ↑1 z2(δ) exists. In fact, using the chainrule of limits and (A.72) and (A.73), it follows that limδ↑1 z2(δ) = 0. Thus, ψ2(δ) is vanishinglysmall as δ → 1. And it is readily computed that, limδ↑1[z1(δ)z

′2(δ)] and limδ↑1[z1(δ)K(z2(δ))]

both exist and are finite. Next, z2(δ) inherits piecewise continuous differentiability from z1(δ).


Since z+−1

1 (z) is a strictly increasing function, we compute limδ↑1 z′2(δ) = −∞. Also, note that

limδ↑1[z2z′2] = limδ↑1[z

′1z2] limδ↑1[z

′2/z

′1]. The latter product is finite since the first limit is known

to be finite and we can compute limδ↑1[z′2/z

′1] = 117/72.

Now, let, 2 and V ∗2 be mappings from (z1(δ), z2(δ)] to the non-negative reals such that for

every z′ ∈ (z1(δ), z2(δ)],

V ∗2 (z′) = (γ + βz′)q(z+1 (z′)) − p(z+1 (z′)), (A.74)

2(z′) =[p(z+1 (z′)) − b(q(z+1 (z′)))2

2

](z′ − z+1 (z′)) + δ

(γ2

2b

)z+1 (z′). (A.75)

V ∗2 and 2 are smooth since z+1 (·) is smooth. Now for any z2(δ) < z′ ≤ 1, consider the optimization

problem,

maxz1(δ)<z<z2(δ)

[((γ + βz)2/2b− δV ∗

2 (z))(z′ − z) + δ 2(z)], (A.76)

and let z2(z′; δ) denote the set of relative maxima of (A.76). A typical element of z2(z′; δ) isdenoted by z+2 (z′). Hence, by definition, z+2 (z′) is implicitly defined by,

−�a2(z+2 (z′); δ) +�b2(z+2 (z′); δ)(z′ − z+2 (z′)) = 0, (A.77)

where for any z ∈ Z,

�a2(z; δ) ≡[γ2

2b(1 − δ) + β2

2b(z2 − δ(z+1 (z))2) + γβz(1 − δ) − δβ2(z− z+1 (z))z+1 (z)

b

](A.78)

�b2(z; δ) ≡ β[(γ + βz+1 (z)) − δV ∗′(z+1 (z))]

b. (A.79)

Note that by the implicit function theorem z+2 (z′) is a locally smooth function since �j2 aresmooth functions, for j = a, b (see, e.g., Blume et al., 1982). And it is readily shown that z+2 (z′)is also locally a strictly increasing function. Hence, in terms of the notation at hand, ψσ3 (δ) isthe connected interval (zσ2 (δ), zσ3 (δ)], and zδ(δ) is the supremum of ψσ3 (δ) over all σ ∈ �3(δ).Now, let, z+2 (z′; δ) ≡ sup z2(z′; δ) (whenever it is non-empty). Since, z3(δ) ≡ sup ψ3(δ), it followsthat z3(δ) ≡ supz2(δ)<z′≤1 z

+2 (z′; δ). But every element of �3 is a specification of the functions

{z+j (z′)}2j=1, where z+j (z′) : (zj(δ), zj+1(δ)] → (zj−1(δ), zj(δ)], for j = 1, 2. Hence, by construc-

tion and using previously established facts, z3(δ) is implicitly defined by,

−�a2(z2(δ); δ) +�b2(z2(δ); δ)(z3(δ) − z2(δ)) = 0. (A.80)

We now use (A.77)–(A.80) to evaluate limδ↑1 z3(δ). From (A.80), we have,

limδ↑1

z2(δ) = limδ↑1

{�a2(z2(δ); δ)

�b2(z2(δ); δ)+ z2(δ)

}. (A.81)

From (A.78) and the fact that limδ↑1 zJ (δ) = 0, j = 2, it immediately follows thatlimδ↑1�

a2(z2(δ); δ) = 0. Next, upon totally differentiating V ∗

2 (z′) (cf. (A.74)), we get,

limδ↑1�b2(z2(δ); δ) ∝ lim

δ↑1[γ(1 − δ) + β(z− z+′

1 (z)) + δz+′1 (z)(γ(1 − δ)

−β(z− 2z+1 (z)))]|z=z2(δ), (A.82)


where by direct computation from Eq. (A.68), z+′1 (z) ∝ [β + (β2z+ γβ(1 − δ))/K(z)]. Clearly,

the limit value in (A.82) depends on the quantity, limδ↑1z+′1 (z2(δ)). We now show that this limit

exists and is finite. From Eq. (A.67) and the definition of K(·) (≡ √K(·)), limδ↑1K(z2(δ)) = 0.

Hence, the second term of limδ↑1z+′1 (z) has a 0/0 term. To analyze this limit further, we compute

some useful limits. We find that limδ↑1[z1(δ)z′1(δ)] = −2(γ2/β2). Then, it must be the case thatlimδ↑1z

′1(δ) < 0 (since z1(δ) > 0 for every 0 < δ < 1). Hence, (A.72) implies that for every ε > 0

there exists some δ(ε) < 1 such that if δ(ε) < δ < 1, then

z′1(δ)

[z2(δ) − 2γ(1 − δ) + K(z2(δ))

β

]+ 6γ2

β2 < ε. (A.83)

But straightforward computation shows that limδ↑1[z′1(δ)(1 − δ)] ∝ limδ↑1[√

1 − δ(√

1 − δ−1/2)] = 0. Hence, (A.83) requires that for any ε > 0, there exists some neighborhood (δ(ε), 1)such that for every δ ∈ (δ(ε), 1):

z′1(δ)

[z2(δ) + K(z2(δ))

β

]+ 6γ2

β2 < ε.

Thus, the limiting values limδ↑1(z′1(δ)z2(δ)) and limδ↑1(z′1(δ)K(z2(δ))) exist and are fi-nite. Indeed, judicious application of the algebra of limits shows that limδ↑1[z′2z2] =−(5.3)(γ2/β2); limδ↑1(z′1K(z2)/β) = −(39/12)(γ2/β2); limδ↑1(z′2/z

′1) = 117/72. Then, return-

ing to (A.72), it follows from the fact that the just annunciated limits are finite that limδ↑1z′2(δ) =

−∞.Next, expanding out the terms in the product:

[δz+′

1 (z)[γ(1 − δ) − β(z− 2z+1 (z))]] |z=z2(δ), and

using previously established facts, we find that,

limδ↑1

[δz+′

1 (z)[γ(1 − δ) − β(z− 2z+1 (z))]] |z=z2(δ) = lim

δ↑1

[(βz2 + γ(1 − δ))2

K(z2)

]. (A.84)

From L’Hopital’s rule,

limδ↑1

[(βz2 + γ(1 − δ))2

K(z2)

]= lim

δ↑1

[1 − 3δ(1 − δ)

[βz2 + γ(1 − δ)]4

]−1/2

. (A.85)

But in view of the algebra of limits and L’Hopital’s rule,

limδ↑1

[3δ(1 − δ)

[βz2 + γ(1 − δ)]4

]

= limδ↑1(3 − 6δ)

limδ↑1{

4(βz2 + γ(1 − δ))2(β2z2z′2 − 2γβz2 − γ2(1 − δ))

} = −∞. (A.86)

Hence, the second term in (A.82) is zero, implying that limδ↑1�b2(z2(δ); δ) = 0. Thus, return-

ing to (A.81), we use L’Hopital’s rule to evaluate the limit limδ↑1[�a2/�b2]. Judicious substitu-

tion of the various limits computed above shows that limδ↑1[�a′

2 (z2(δ); δ)] = −(1.33)(γ2/b), andlimδ↑1[�b

′2 (z2(δ); δ)] = −∞. It then follows from (A.80) that, limδ↑1z3(δ) = 0.

We now apply the induction principle and show that the show that the previous set of argumentsapplies for every n in the limit as δ ↑ 1. Fix any n ≥ 4. Since we are dealing with a pure strategycontinuation equilibrium, the equilibrium path specifies the market depletion strategy for theremaining stages; viz., the profile {zj(z′)}n−1

j=1 , so that along the continuation equilibrium path,


Dj−1 = [0, zj(z′)) whenever Dj = [0, z′). (Note that the implicit coordination role of price–quality histories is already represented in the specification of the market depletion functions,whenever z∗j (z′) is multi-valued.) We take as an induction hypothesis that the functions zj(z′) aresmooth, since this has shown to be true for j ≤ n− 1. This hypothesis implies that the functions j(z′) and V ∗

j (z′) are smooth, and we define, for any state z3(δ) < z′ ≤ 1,

�aj (z′; δ) ≡ −

[(γ + βzj−1(z′))2

2b− δV ∗

j−1(z′)]

+ δd j−1(zj−1(z′))

dz;

�bj (z′; δ) ≡

[(β/b)(γ + βzj−1(z′)) − d V ∗

j−1(zj(z′))dz

]. (A.87)

We inductively assume: I(1) limδ↑1 zj(δ) = 0, limδ↑1 z′j(δ) = −∞; I(2) limits of the functions

z′j(δ)zj(δ), z′j−1(δ)zj(δ), z′j(δ)zj−1(δ), as δ ↑ 1, exist and are finite; I(3) limδ↑1�kj(zj+1(δ); δ) =

0, k = a, b; I(4) 0 < limδ↑1 z′(zj(δ)) ≤ 1; I(5) limδ↑1�

a′j (zj(δ); δ) �= 0, and finite;

I(6)−∞ < limδ↑1[�b′j (zj(δ); δ)�bj (zj(δ); δ)] < 0; I(7) −∞ < limδ↑1[�b′j (zj(δ); δ)�aj (zj(δ); δ) −(β2/b)�a′j (zj(δ); δ)] < 0. We note that each of these hypotheses has been proven above for n ≥ 4.

Now, I(1) immediately implies that limδ↑1 zn−1(δ) = 0. Replicating foregoing arguments, thesmoothness hypothesis implies that zn(δ) is implicitly defined recursively by the relation,

−�an(zn−1(δ)) +�bn(zn−1(δ))(zn(δ) − zn−1(δ)) = 0. (A.88)

It follows from (A.88) and the induction hypotheses that limδ↑1 zn(δ) = 0, so that the firstpart of I(1) holds for j = n as well. But this also implies that limδ↑1�

an(zn(δ); δ) = 0, and

hence the first part of I(3) applies for n as well. We now show that under the induction hy-potheses, limδ↑1 d(�bn(zn(δ); δ)/dδ = −∞. Note first that the maintained hypotheses imply that�kr (δ); δ), 2 ≤ r ≤ n− 1, k = a, b, are smooth functions of δ on an open neighborhood of 1, say(δr, 1), and are also left-continuous at 1. It is then straightforwardly shown that these proper-ties apply for �kr (zr(δ); δ), k = a, b, as well. Restricting attention to δr < δ < 1, and completelydifferentiating (A.87) w.r.t. δ, yields,

�b′r zr(δ); δ) =[

(β/b)[−γ + βz′r−1(1 − δ)] + (β2/b)(z′r − z′r−1 − δzr−1(δ))

+ d[δErz′(zr(δ))]dδ

], (A.89)

where Er ≡ [�br (zr−1(δ); δ) − (β2/b)(zr(δ) − zr−1(δ))]. But for any r ≤ n− 1,

zr(δ) = zr−1(δ) +�r(zr−1(δ); δ); �r(zr−1(δ); δ) ≡ �ar (zr−1(δ); δ)

�br (zr−1(δ); δ). (A.90)

Hence, total differentiation of (A.90) with respect to δ implies that, z′r = z′r−1 +�δ(zr−1(δ); δ).But then the generalized mean value theorem (see, e.g., Buck, 1978, p. 118) implies that

limδ↑1

�δ(zr−1(δ); δ) = − limδ↑1

[�(zr−1(δ); δ)

1 − δ

]. (A.91)

But a straightforward application of L’Hopital’s rule shows that the limit in the R.H.S. of (A.91)diverges to +∞. Hence, limδ↑1(z′r − z′r−1) = −∞, r ≤ n− 1. Also, by the generalized mean


value theorem,

limδ↑1

{d[z′(zr(δ))Er]/dδ} = − limδ↑1

[z′(zr(δ))] limδ↑1

(Er/(1 − δ)) = −∞.

This establishes that limδ↑1�b′r (zr(δ); δ) = −∞, for any 2 ≤ r ≤ n− 1. But then it also fol-

lows that limδ↑1 z′n = −∞, so that I(1)(ii) also holds for r = n. We claim next that I(2)

also holds for r = n. In light of the just established results, it is sufficient to show that

limδ↑1

[�b

′n−1(zn−1(δ); δ)�bn−1(zn−1(δ); δ)

]�= 0. However, this result is an immediate conse-

quence of I(5)–I(7). Next, complete differentiation of �an(zn(δ); δ) w.r.t δ yields

�a′n (zn(δ); δ) ∝

{[−γ2 + β2(z′nzn(δ) − (zn−1(δ))2−δz′n−1zn−1(δ))+γβz′n(1−δ)−γβzn(δ)]

−β2[(zn(δ) − zn−1(δ))zn−1(δ) + δ(z′nzn−1(δ) + z′n−1(zn(δ) − 2zn−1(δ)))]}.

(A.92)

But if I(2) holds for r ≤ n, then it follows from (A.92) that limδ↑1�a′n (zn(δ); δ) is fi-

nite. In fact, under the induction hypotheses, this limit must also be non-zero. Hence, I(5)also holds for r = n. Next, we know from (A.88) that for δn−1 < δ < 1, zn(δ) = zn−1(δ) +�an(zn−1(δ); δ)/�bn(zn−2(δ); δ). Application of the implicit function theorem on this relation gives,z′(zn(δ)) = −�bn(zn−1(δ); δ)/Fzz(zn−1(δ), zn(δ)). Straightforward expansion and the applicationof limits shows that the limit of the latter term as δ ↑ 1 exists, is positive, and cannot exceed 1.Hence, I(4) also holds for j = n. Next, the fact that I(1)–I(5) hold for j = n, imply that I(6) alsoholds for j = n, and this in turn implies that I(7) holds for j = n as well.15 Thus, by the inductionhypotheses, limδ↑1zj(δ) = 0 for every j = 1, 2, . . .. �

Proof of Theorem 3. From Proposition 1, it follows that for every ε > 0, and every z′ ∈ Z,there exists δ(ε, z′) < 1 such that for δ > δ(ε, z′), (z′ − zσ(z′, θ; δ)) < ε for every history θ ∈ �∞and every σ ∈ �. Suppose not; then, by hypothesis, there exists some history (z′, θ), somesubsequence {δk}∞k=1 where limδ↑1δk = 1, and some ε > 0 such that (z′ − zσ(z′, θ; δ)) ≥ ε.But from Theorem 1 for every natural number k there exists another natural number j(k)such that zσ(z′, θ; δ) ∈ ψσj(k)−1(δk). But then it is certainly the case that (z′ − zσ(z′, θ; δk)) ≤ρ(ψσj(k)−1(δk)) + ρ(ψσj(k)(δk)), where ρ(X) denotes the diameter of the set X. However, we alsoknow from the foregoing that for any {δk} ↑ 1, for each ε′ > 0, there exists some natural numberK(ε′) such that for every k > K(ε′), ρ(ψj(δ)) < ε′, for every j ≥ 1. But then the hypothesis thatlimδ↑1(z′ − zσ(z′, θ; δ)) ≥ ε yields a contradiction when we choose, for instance, ε′ = ε/3.

Now we also know from Lemma 12 and Corollary 1 that in any pure strategy equilibriumσ ∈ �(δ), qσ(z′, θ; δ) is linear in zσ(z′, θ; δ) for each (z′, θ). It then follows that for every ε > 0,there also exists some δ(ε, z′) < 1 such that for all δ(ε, z′) < δ < 1,

|q∗(z′) − qσ(z′, θ; δ)| ≤ ε. (A.93)

Next, for any 0 < δ < 1, fix some σ ∈ �(δ), and any history (z′, θ). We denote the utility of theinframarginal buyer-type z′ along the equilibrium path as V σ(z′, θ; δ). From Theorems 1 and 2,

15 Details are available on request.


we know that V σ(z′, θ; δ) is monotone (strictly) increasing in z′, and hence a.e. differentiable inz′. Since, zσ(z′, θ; δ) is also a strictly increasing function, we can compute (a.e.)

∂V σ(z′, θ; δ)∂z′

= βq(zσ(z′, θ; δ)) − ∂zσ(z′, θ; δ)∂z′

×[βq(zσ(z′, θ; δ)) − β2

b(z− zσ(z′, θ; δ)) − δ

∂V σ(z = zσ(z′, θ; δ), θ)∂z

]

(A.94)

Here, the function q(·) has been defined in Eq. (5) of the text. However, we now establish frombackwards recursion and the foregoing results that for any history (z′, θ) with the state z′ andevery ε > 0 there exists some δ(ε, z′) < 1 such that∣∣∣∣∣∂V

σ(z′, θ; δ)∂z′

− βq(zσ(z′, θ; δ))

∣∣∣∣∣ < ε. (A.95)

Consider the final offer. We know that for every z′ ∈ (0, z1(δ)), V σ(z′, θ; δ) = (γβ/b)z′, and hence(A.95) holds for every ε and every δ. Next, by direct computation, for every history such thatz′ ∈ ψσ2 ,

∂V σ(z′, θ; δ)∂z′

= βq(z+1 (z′)) − z+′1 (z′)

[�b2(z+1 (z′); δ) − (β2/b)(z′ − z+1 (z′))

]. (A.96)

Thus, based on the previously established facts that, limδ↑1[z+′1 (z′)�b2(z+1 (z′); δ) = 0 and

limδ↑1[z+′

1 (z′)(z′ − z+1 (z′))] = 0, we conclude that, limδ↑1

∣∣∣ ∂V σ (z′,θ;δ))∂z′ − βq(zσ(z′, θ; δ))

∣∣∣ = 0.

Hence, (A.95) holds for all pneultimate offer periods as well. Now, fix any state 0 < z′ ≤ 1and suppose that limδ↑1|∂V σ(z∗, θ; δ)/∂z∗ − βq(zσ(z∗, θ; δ))| = 0 for every 0 < z∗ < z′. Then,taking the limit of the right hand side of (A.94) as δ ↑ 1, and using the induction hypothesis

and other results established above, we find that, limδ↑1[ ∂Vσ (z∗,θ;δ)(∂z∗) ] = βq(zσ(z′, θ; δ)). Therefore,

(A.95) is established.Now we use the theorem that if J(x, ·) is a function that is bounded and monotone non-

decreasing in x over a compact domain X = [x, x], then ∂J(x, ·)/∂x is equal (in X) to a

summable function. Moreover, for any x < x′ < x,∫ x′x

[∂J(x, ·)/∂x] dx ≤ J(x′, ·) − J(x, ·) (see,

e.g., Royden, 1968, p. 96). But we know from Theorem 1 that 0 ≤ V σ(z∗, θ; δ) ≤ (γ + β)q forevery σ ∈ �, and every history. Hence, it follows from (A.95) that for any given history (z′, θ),for every ε > 0, there exists some δ′(ε, z′) < 1 such that for δ > δ′(ε, z′),

V σ(z′, θ; δ) ≥∫ z′

0βqσ(zσ(r, θ; δ)) dr − ε = (β/b)

[γzσ(z′, θ; δ) + β(zσ(z′, θ; δ))2

]− ε.

(A.97)

In (A.97), we have used the fact that V σ(0, θ; δ) = 0 for every δ and any price–quality history,along any equilibrium path (cf. Theorem 1). However, we also know that for every ε′ > 0, ifδ > δ(ε′, z′) (where δ(ε′, z′) has been quantified above), then (z′ − zσ(z′, θ; δ)) ≤ ε′ (for any price–quality history). Then, for every (ε, z′) ∈ R++ × Z, let δ(ε, z′) = max{δ(ε, z′), δ(ε, z′), δ(ε, z′)}.


It then follows that for δ > δ(ε, z′),

V σ(z′, θ; δ) ≥ (β/b)(γz′ + β(z′)2/2) − ε[(β/b)(γ + β) + 1

]. (A.98)

From Theorems 1 and 2, along any σ ∈ �(δ), pσ(zσ(z′, θ; δ)) = [(γ + βzσ(z′, θ; δ))/b−δV σ(zσ(z′, θ; δ); δ)]. Now, put η ≡ [(β/b)(γ + β) + 1]. Then, it follows from (A.98) that forδ > δ(ε, z′),

γ + βzσ(z′, θ; δ)b

− δV ∗σ(zσ(z′, θ; δ); δ) ≤ γ2 + β(γz′ + β(z′)2/2)

b+ ηε

+ (1 − δ)V σ(zσ(z′, θ; δ); δ) = p∗(z′) + ηε+ (1 − δ)V σ(zσ(z′, θ; δ); δ) ≤ p∗(z′)

+ ηε+ (1 − δ)(γ + β)q (A.99)

In (A.99), we have used the function p∗(·) as defined in Eq. (9) and the fact that V σ(z, ·; δ) ≤(γ + β)q, for every (z, θ) (cf. Theorem 1). Eq. (14) of the theorem then follows for � from (A.99)when we let g(ε) ≡ ε/2η, and set δ∗(ε, z′) ≡ max{δ(g(ε), z′), 1 − [ε/(2(γ + β)q)]}. Notice thatδ∗(ε, z′) < 1 for each (ε, z′) since g(ε) > 0 for every ε > 0 and δ(ε′, z′) < 1 for every ε′ > 0. Thetheorem is thus established for all σ ∈ �(δ). But any σ ∈ �(δ) is a probability mixture over �(δ).Hence, it must apply for behavior strategy equilibrium paths as well, with a judicious re-calibrationof the function δ∗(ε, z′). �

References

Aghion, P., Howitt, P., 1992. A model of growth through creative destruction. Econometrica 60, 323–351.Aumann, R., 1964. Mixed versus behavior strategies in infinite extensive games. Annals of Mathematics Studies 52,

627–630.Ausubel, L., Deneckere, R., 1989. Reputation in bargaining and durable goods monopoly. Econometrica 57, 511–531.Bagnolli, M., Salant, S., Swierzbinski, J., 1989. Durable-goods monopoly with discrete demand. Journal of Political

Economy 97, 1459–1478.Blume, L., Easley, D., O’Hara, M., 1982. Characterization of optimal plans for stochastic dynamic programs. Journal of

Economic Theory 28, 221–234.Buck, R., 1978. Advanced Calculus. McGill, New York.Bulow, J., 1982. Durable-goods monopolists. Journal of Political Economy 90, 314–332.Coase, R., 1972. Durability and monopoly. Journal of Law and Economics 15, 143–149.Deneckere, R., McAfee, P., 1996. Damaged goods. Journal of Economics and Management Strategy 5, 149–174.Ellison, G., Fudenberg, D., 2000. The Neo–Luddite’s lament: excessive upgrades in the software industry. Rand Journal

of Economics 31, 253–272.Fishman, A., Rob, R., 2000. Product innovation in durable-good monopoly. Rand Journal of Economics 31, 237–252.Fudenberg, D., Levine, D., Tirole, J., 1985. Infinite horizons of bargaining with one-sided compelete information. In:

Roth, A. (Ed.), Game Theoretic Models of Bargaining. Cambridge University Press, Cambridge.Fudenberg, D., Tirole, J., 1998. Upgrades, tradeins, and buybacks. Rand Journal of Economics 29, 235–254.Gul, F., Sonnenschein, H., Wilson, R., 1986. Foundations of dynamic monopoly and the Coase conjecture. Journal of

Economic Theory 40, 119–154.Hendel, I., Lizzeri, A., 1999. Interfering with secondary markets. Rand Journal of Economics 30, 1–21.Kahn, C., 1986. The durable goods monopolist and consistency with increasing costs. Econometrica 54, 275–294.Kuhn, H., 1953. Extensive games and problems of information. Annals of Mathematics, vol. 28. Princeton University

Press, Princeton.Kuhn, K.U., Padilla, J., 1996. Product line decisions and the Coase conjecture. Rand Journal of Economics 27, 391–414.Kumar, P., 2002. Price and quality discrimination in durable goods monopoly with resale trading. International Journal

of Industrial Organization 30, 1313–1339.Liang, M., 2000. The Coase Conjecture and Durable Goods Monopoly with Depreciation. University of Western Ontario,

Mimeo.


Mussa, M., Rosen, S., 1978. Monopoly and product quality. Journal of Economic Theory 28, 310–317.Royden, H., 1968. Real analysis. MacMillan, New York.Rust, J., 1985. Stationary equilibrium in a market for durable assets. Econometrica 53, 783–805.Stokey, N., 1981. Rational expectations and durable goods pricing. Bell Journal of Economics 12, 112–128.Swan, P., 1970. Durability of consumption goods. American Economic Review 60, 884–894.Waldman, M., 1996. Durable goods pricing when quality matters. Journal of Business 69, 489–510.Wang, G., 1998. Bargaining over a menu of wage contracts. Review of Economic Studies 65, 295–305.

Intertemporal Price-Quality Discrimination and the Coase Conjecture

Documents

Transcript of Intertemporal Price-Quality Discrimination and the Coase Conjecture