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October 26, 2004
Capacity Choice Counters the Coase Conjecture*
by
R. Preston McAfee
100 Baxter Hall
MC 228-
Humanities & Social Sciences
California Institute of Technology
Pasadena, CA 91125
and
Thomas Wiseman
Department of Economics
University of Texas at Austin
1 University Station C
Austin, TX 78712
Abstract: The Coase conjecture (1972) is the proposition that a durable goods monopolist, who sells over time and can quickly reduce prices as sales are made, will price at marginal cost. Subsequent work has shown that in some plausible cases thatconjecture does not hold. We show that the Coase conjecture does not hold in any plausible case. In particular, we examine that conjecture in a model where there is avanishingly small cost for production capacity, and the seller may augment capacity in every period. In the “gap case,” a ny positive capacity cost ensures that in the limit, asthe size of the gap and the time between sales periods shrink, the monopolist obtains profits identical to those that would prevail when she could commit ex ante to a fixedcapacity , given a standard condition on demand. Those profits are at least 29.8% of the full static monopoly optimum.
- We thank Ken Hendricks and Philip Reny for their stubborn unwillingness to beconvinced by incorrect arguments. We also thank seminar audiences for helpful comments.
1. Introduction In 1972, Nobel laureate Ronald Coase startled the economics profession with a counterintuitive proposition, which came to be known as the Coase conjecture, concerning the monopoly seller of a durable good. Coase’s original example was the hypothetical owner of all land in the United States. The monopolist maximizes profits by identifying the monopoly price and selling the quantity associated with that price. Having sold that quantity, however, the monopolist now faces a residual demand and she is induced to try to sell some additional units to the remaining buyers at a price that is lower than the initial price. Such logic entails a sequence o f sales at prices falling toward marginal cost. Rationally anticipating falling prices causes most potential buyers to wait for future lower prices. Provided that the monopolist can make sales and cut price sufficiently rapidly, Coase conjectured that the monopolist’s initial offer would be approximately marginal cost, and that the monopoly would replicate the competitive outcome. Intuitively, the monopolist competes with future incarnations of herself. Even when facing a monopolist, buyers have an alternative supplier: the monopolist in the future. That the power of such a substitution possibility might render the monopoly perfectly competitive remains a captivating idea even for an audience accustomed to the fact that subgame perfection (or time consistency) restricts equilibria in dramatic ways. Arguably, Coase’s conjecture remains the most extreme example of the power of subgame perfection. Not surprisingly, perfect durability (which ensures that high value buyers exit the market) and rapid transactions are not the only assumptions required to prove the Coase conjecture. The first formal proofs of the Coase conjecture are given by Jeremy
seller. Such a threat guarantees that the seller won’t deviate from anything at least as profitable as the low-profit Coasian equilibrium. Ausubel and Deneckere demonstrate that there are many non-stationary equilibria that can be constructed by the threat of reversion to the Coasian equilibrium. In addition to more profitable equilibria which exist in the no gap case, there are a variety of other means for a durable goods monopolist to escape the grim logic of the Coase conjecture. Leading the list is renting, which is a means identified in Coase’s original article. A seller who rents, rather than sells, has no incentive to expand output beyond the monopoly quantity, for such an expansion entails a price cut not just to the new customers, but also to existing customers. By allowing existing customers to renegotiate, rental serves a means of committing to a “most favored customer” c lause, in which early buyers are offered terms no worse than later buyers. Renting as a means of commitment has been offered as an explanation for IBM’s rental of business machines (Wilson, 1993), although evidence is scant. Other solutions offered in the literature include return policies or money -back guarantees, destroying the production facilities, making the flow costs of staying in the market expensive (e.g. by renting the factory), concealing the marginal cost from buyers to interfere with their expectations about future prices, and planned obsolescence to eliminate the requisite perfect durability. (See Tirole (1988).) We offer a very different, more general limitation on the Coase conjecture. Focusing on the gap case, we show that the Coase conjecture is not robust to a modification of the game that would be relevant in almost any practical application. In particular, we consider a small cost of capacity, so that selling a given amount over a smaller span of time costs more. We envision a perfectly durable capacity, so that, once
bought, the capacity is never purchased again. For example, if the good is produced in a factory, faster sales entail a larger factory, which creates a one -time larger cost. Even Coase’s hypothetical seller of all U.S. land, who faces no production cost, still bears a capacity cost: To sell the land rapidly requires a large number of sales agents, so the seller must incur hiring and training costs. Whenever there is an increased cost of increased speed, there is an effective capacity cost.^1 If a monopolist chose production capacity at the beginning of time and could not augment it later, the monopolist would use capacity as a commitment device, setting a low capacity and dribbling output into the market. That approach has the advantage of ensuring that prices are high early and fall slowly, as high value buyers pay more for early acquisition of the good. Indeed, we will demonstrate that in such a “commitment game” the monopolist obtains at least 29.8% of the static monopoly profits, no matter how fast the stages of the game occur; increasing the speed of the game induces the monopolist to cut capacity in a way that keeps the flow of goods to buyers constant. Note that the monopolist could not achieve anything higher than static monopoly profits even if she could commit ex ante to a sequence of prices. Stokey (1979) shows that she would optimally set the static monopoly price in each period, and thus earn static monopoly profits. That is, the ability to discriminate dynamically does not help the monopolist. While the specificity of the lower bound of profits is remarkable, the fact that profits don’t converge to zero is not; such a model endows the seller with an extraordinary commitment ability – the ability to commit at the beginning of the game
(^1) Potential congestion in sales, so that selling twice as fast incurs more than twice the costs, creates an (^) A constant marginal cost of selling does not create a capacity cost, but just an ordinary marginal cost. effective capacity cost.
The intuition behind the theorem suggests a defect in Coase’s reasoning. The Coasian price path requires a seller to sell all of the demand very rapidly. Because buyers will wait for prices close to marginal cost (since these are coming rapidly), the opening price is close to marginal cost, and most sales take place in the first few minutes. In the limit as the time periods get arbitrarily close, all sales take place immediately. In environments requiring production or some transaction medium, that outcome requires the seller to produce a very large production facility or high- bandwidth transaction facility, so that the flow of sales can be extremely large for a very short period of time. If the cost of capacity is high relative to the size of the market, then the seller will not purchase so much capacity. In fact, for any positive cost, the fact that the seller will not increase capacity near the end of the game, when few buyers remain, allows her to credibly commit to a low level of capacity at the beginning. The logic of backwards induction compels buyers to believe that she will not increase capacity in the future, and thus that prices will fall slowly. Another way to see that intuition is as follows: Suppose that in the “commitment game,” where the seller chooses capacity once and for all at the beginning, the optimal capacity increases with the size of the market. As sales are made, then, the desired commitment capacity falls. Thus, a seller who chooses an initial capacity slightly below the “desired” capacity won’t be later tempted to increase it, because the slight reduction will still exceed the subsequent desired capacity. This means that the seller has local commitment ability – she can effectively commit to a slight reduction or increase in capacity around the equilibrium opening level. But the ability to commit to a small change is sufficient to ensure that profits are maximized as a function of capacity because the first order conditions hold; that is, that the level of profits when capacity is
augmentable is identical to that when capacity is chosen once and for all. Local is as good as global in ensuring the first-order conditions hold. Our result that capacity choice in each period delivers the same profits as the commitment version holds in the limit of the gap case, as the gap shrinks to zero. That case, where the gap is positive but small, had been the only setting left where the Coase conjecture had bite, and the monopolist made no profit. In the no gap case, Ausubel and Deneckere (1989) show the existence of equilibria where the seller makes high profits. In the gap case, the monopolist sells at a price near the lowest consumer’s valuation, but if the gap between marginal cost and the lowest valuation is large, then that price entails high profits. In this paper, we show that the seller can earn substantial profits even when the gap is vanishingly small. Thus, in any relevant economic situation Coase’s conclusion does not hold: A durable goods monopolist can make profits. Alternatively, our result can be interpreted as a way to select from Ausubel and Deneckere’s (1989) continuum of equilibria in the no gap case: When the monopolist chooses costly capacity, equilibria where the monopolist makes very low profits are not robust to the introduction of a small discontinuity of buyers’ valuations just above marginal cost. (Such a discontinuity could result if, for example, prices can be set only at discrete levels.) Note that without capacity choice, the selected equilibrium is very different. With a very small gap, the unique equilibrium entails very low profits. The rest of the paper proceeds as follows. In the next section, we set up the model and, as a preliminary, analyze the commitment version of the game. We present the main theorem in the third section, showing that the outcome with capacity choice in each period mirrors the outcome with initial capacity commitment. In the fourth section we consider the robustness of our result, and we conclude in the fifth section.
Consumers’ valuations are bounded above by vH^ ≡ p (0) + g and below by g (= p ( q 0 ) + g ).
This is the “gap case,” where the lowest valuation among the buyers is strictly greater than the monopolist’s marginal cost. At the beginning of each sales period, the monopolist publicly chooses how much additional capacity to purchase. She then announces a price P for that period, and must sell to any buyer who wants to buy at that price, up to a maximum of K / N units. We will assume that the rationing rule is to serve higher v aluation buyers first. Equivalently, we could allow costless resale among the buyers. (We discuss alternative rationing rules in Section 4.)
Assumption 1: If the quantity of consumers who wish to buy in any sales period z is greater than Kz / N , then sales will be made to the subset of size Kz / N of potential buyers with the highest valuations.
The goal of the monopolist is to maximize the discounted value of revenue, minus the discounted value of expenditures on capacity. The consumers seek to maximize their discounted surplus. The surplus to consumer q who buys in sales period z at price Pz is δ z / N^ [ v ( q ) − Pz ]. As is standard in the literature on the Coase conjecture, we will consider only equilibria where deviations by a zero mass set of consumers have no effect on continuation play. In the absence of capacity constraints, Fudenberg, Levine, and Tirole (1985) and Gul, Sonnenschein, and Wilson (1986) show that there is generically a unique subgame perfect equilibrium in the gap case. That equilibrium satisfies the Coase conjecture, in the sense that as the number of periods N per unit time goes to infinity, the monopolist
earns profits close to gq 0 by setting an initial price close to g and selling to the entire market nearly instantaneously. In that equilibrium, which we will call the Coase equilibrium, all consumers are served in a finite number of sales periods, which implies that prices in each period can be determined by backwards induction. When there are no capacity constraints, the equilibrium path has the “skimming” property. That is, in any period there is a cutoff valuation v such that all consumers with valuations greater than v have already bought, and all consumers with valuations less than v have yet to buy. (See Fudenberg, Levine, and Tirole’s (1985) Lemma 1 and Ausubel and Deneckere’s (1989) Lemma 2.1.) The intuition is that if a consumer with
valuation v is willing to buy at price P , then so is any consumer with valuation v′ > v. Both buyers get the same benefit (in the form of lower future prices) from waiting, but the cost of delaying consumption is greater for the high-valuation consumer. Thus, in any period the remaining market can be characterized completely by q , the volume of consumers who have been served so far. Let SC ( q , g , N ) and PC ( q , g , N ) denote the quantity sold and price offered, respectively, in any period on the Coase path when the quantity served so far is q , the size of the gap is g , and there are N offers per unit time, and let πC ( q, g , N ) be the remaining Coase profits to the monopolist. Define
S (^) max C^ ( q , g , N )as the maximum quantity sold in any period along the Coase path when
the market served so far is q , the gap is g , and there are N offers per unit time. Note that
if K / N is greater than S (^) max C^ ( q , g , N ), the capacity constraint will never bind along the
Coase path. In that case, capacity is no longer relevant, and the only subgame perfect continuation is the Coase equilibrium.
gap is g , and the monopolist commits to capacity K , which costs c. Let RcomN ( K , q , g , c )
be the associated revenue. The quantity sold in a period on the commitment path when
the served market is q is given by S comN ( K , q , g , c ). Let K comN ( q , g , c )be the capacity
level that maximizes the commitment profits π (^) Ncom ( K , q , g , c ), and denote by
Π comN ( q , g , c )the value of the maximized profit. How large is the maximized commitment profit relative to the static monopoly profits? (Remember that the static monopoly profit is the highest that the monopolist could attain even with the ability to commit to future prices.) We examine the limit of the profits as both the time between offers and the capacity cost c shrink to zero. (For simplicity, we assume in this analysis that g = 0 – the no-gap case. The profits for the gap case, where g > 0, can be no lower.) Consider the monopolist who sells at a rate K per period and in a continuous time fashion, which is the limit of the discrete case when the intervals get short. Here we simplify the profit expression and prove a global minimum for profits (at least 29.8% of the static monopoly profits). Let MR denote marginal revenue:
MR ( q )= qp ′( q )+ p ( q ).
MR ( qm ) = 0 defines the static monopoly quantity qm. Market saturation occurs at quantity q 0. It is permitted in the analysis for q 0 = ∞. Given capacity K , the quantity
sold through time t is Kt. The market is saturated at T = q 0 / K ; at this point, the price is
zero. The value of the buyer who buys at time t is p ( Kt ). Let P ( Kt ) be the price charged by the seller. A buyer of type Kt who buys at time s has utility
u ( Kt )= max s e −^ rs ( p ( Kt )− P ( Ks )).
From the envelope theorem, since this buyer chooses to buy at t ,
u ′(^ Kt )= e − rt^^ p ′( Kt ).
Thus, since u ( KT ) = 0,
T t
T rs t
e rt^ ( p ( Kt ) P ( Kt )) u ( Kt ) u ( KT ) u ( Ks ) Kds e p ( Ks ) Kds
e p ( y ) dy e p ( Kt ) Kre p ( y ) dy.
KT Kt
rt ryK KT Kt
−∫ − ryK ′ = − − ∫ −
Thus, setting z = Kt ,
0 P z e Kre ryKpy dy
q z
= rt^ ∫ −
The firm’s profits are
=max 1 − (^1 m + m )≈ 0. 298425
aq a aq
e m aq , which is approximately 191
Let that lower bound on the ratio π* / R ( qm ) be denoted γ :
Definition. Define the constant γ as the value m m
aq a aq max 1 −^ e^ m^ (^1 + aq ).
That bound is tight. Suppose that a mass 1 − ε of consumers have valuation 1, and
a mass ε has valuation ε. (A smoothed version of that demand system can satisfy our
differentiability assumptions and belong to the set Ψ. The following reasoning will still
hold.) For low ε , the monopoly price is 1 and the monopoly quantity qm is 1 − ε. As ε
shrinks, the revenue from selling to consumers beyond qm becomes arbitrarily small. Also, for q ≤ qm , R ( q ) = ( q/qm ) R ( qm ). Thus, both inequalities in the derivation of the
bound γ are arbitrarily close to equalities for small enough values of ε.
The profits with a capacity constraint may be arbitrarily close to the static monopoly profits. Appendix 2 demonstrates that, in the constant elasticity of demand case, in the limit as the elasticity converges to 1, the ratio of the capacity constrained profits to the static monopoly profits converges to 1. This profit ratio is graphed in Figure 1. As the elasticity of demand diverges, the ratio converges to roughly 0.673478, which is also the ratio that prevails when p ( q ) = log( q ), which arises with a* ≈ 3.47845.
When demand is linear p ( q ) = 1- q , MR = 1-2 q , and numerical computation shows that a * ≈ 2.688 and profits are 55.74% of the static monopoly profits of 0.25. At an annual
interest rate of five percent, that value implies that the monopolist sells to approximately 1.86 percent (5 / 2.688) of the market per year.
0 5 10 15 20
expensive. The formal proof relies on two more lemmas. Lemma 2 shows that the
monopolist can commit not to purchase capacity past K max N^ ( g , c ), defined as
K max N ( g , c ) = (^) q ∈max[ 0 , q 0 ] KcomN ( q , g , c ).
Capacity K max N^ ( g , c )is the maximum optimal commitment capacity over all market
sizes no greater than q 0. Intuitively, if the seller chooses that capacity in the first period, Lemma 1 guarantees that she will never be tempted to increase it. Thus, she can earn the profit associated with committing to that capacity forever.
Lemma 2: If Assumption 1 holds, then for any capacity cost c > 0 and any gap size g > 0, there exists an integer N ( c ) such that for all N > N ( c ), any SPE of the game GN (0, 0,
g , c ) gives the monopolist a p rofit of at least π (^) Ncom ( K max N^ ( g , c ), g , c ).
The third lemma demonstrates that as the gap g shrinks and the number of sales periods per unit time N increases, the value of the maximum optimal commitment
capacity K max N^ ( g , c )converges to Kcom (0, 0, c ), the optimal commitment capacity at the
beginning of the (continuous-time) game, given Assumptions 1 and 2.
Lemma 3: If Assumption 1 holds, then for any ε > 0, there exist a real number c ( ε ) > 0
and real-valued functions g ( c , ε ) > 0 and N ( c , ε ) > 0 such that whenever c < c ( ε ), g < g ( c ,
ε ), and N > N ( c , ε ), then | K max N^ ( g , c )− Kcom^ (0, 0, c ) | < ε.
Now we can state our main result: In the limit, the monopolist earns commitment profits, even when she can augment capacity in every period. Theorem 1 follows immediately from the three lemmas.
Theorem 1: If Assumption 1 holds, then for any ε > 0, there exist a real number c ( ε ) >
0 and integer-valued functions g ( c , ε ) > 0 and N ( c , ε ) > 0 such that whenever c < c ( ε ), g
< g ( c , ε ), and N > N ( c , ε ), then any SPE of the game GN (0, 0, g , c ) gives the monopolist a
profit of at least π com^ (0, 0, c ) − ε.
We showed in Section 2 that revenues in the commitment version of the game are at least a fraction γ (≈ 0.298) of static monopoly revenue R ( qm ). That result implies the
following Corollary of Theorem 1.
Corollary 1: If Assumptions 1 and 2 hold, then for any ε > 0, there exist a real number
c ( ε ) > 0 and integer-valued functions g ( c , ε ) > 0 and N ( c , ε ) > 0 such that if c < c ( ε ), g <
g ( c , ε ), and N > N ( c , ε ), then any SPE of the game GN (0, 0, g , c ) gives the monopolist a
profit of at least γ R ( qm ) − ε.
