Andrzej Skrzypacz

March 30, 2010

Abstract We study dynamic bargaining with asymmetric information and interdependent values. We show that as the gap between the cost and value of the weakest type shrinks to zero, the continuous time limit of equilibria changes dramatically from rare bursts of trade with long periods of inactivity to a smooth screening down the demand function, independent of the distribution. If we interpret the model as a durable goods problem with experience curve eﬀects, then the monopoly problem is consistent with perfect competition. In other words, even though the monopoly outcome is ineﬃcient, it converges to the competitive equilibrium outcome as the commitment power disappears. The competitive equilibrium is ineﬃcient because of the positive externality caused by the experience eﬀects.

1

Introduction

Coase (1972) asked as an example what price would a monopolist who owned all the land in the US be able to charge when selling it. He asserted: “. . . [T]he competitive outcome may be achieved even if there is but a single supplier.” Furthermore, he conjectured “the whole process would take place in the twinkling of an eye”. That observation led to a large literature on what is now known as the Coase conjecture. The literature that has formalized Coase’s original ideas (starting with Bulow (1982), Stokey (1981), Fudenberg, Levine and Tirole (1985) and Gul, Sonnenschein and Wilson (1986), Ausubel, and Deneckere (1989)) has shown that as the monopolist with a constant marginal cost of production loses commitment power (i.e. makes frequent oﬀers): ∗

Fuchs: University of California Berkeley, Haas School of Business. e-mail: [email protected]. Skrzypacz: Stanford University, Graduate School of Business. e-mail: [email protected]. We thank Jeremy Bulow, Peter DeMarzo and Songzi Du for comments and feedback on this project.

1

a) in the “no gap case” (i.e. if the demand crosses the cost curve) all stationary equilibria have eﬃcient trade at competitive prices — both parts of the Coase conjecture hold.1 b) in the “gap case” (i.e. if the lowest possible value is above the marginal cost) the equilibrium is generically unique, trade is eﬃcient trade but prices are above the competitive level (they are equal to the lowest valuation). Hence, in this case the durable goods monopolist makes strictly positive profits and only the eﬃciency part of the conjecture holds. A related literature on bargaining with one-sided private information and interdependent valuations, Evans (1989) and Vincent (1989), and Deneckere and Liang (2006, henceforth DL), has shown that if the cost of serving a consumer is increasing in his valuation (as could be the case if the good in question is insurance) and average cost is higher than the lowest valuation, then trade cannot take place immediately. The intuition is that if all types traded quickly, all buyers would have to pay a price corresponding to the lowest valuation and hence the seller would lose money on average. In addition, as shown in DL, in the gap case (i.e. if the value of the lowest type is strictly above the cost of serving him), the seller generically makes strictly positive profit. Hence, in the gap case neither trade is eﬃcient nor the prices are competitive. Since the bargaining model and the durable goods model are equivalent, these results apply also to a durable good monopolist with a cost function that decreases in past sales due to experience eﬀects. To resolve the contrast between the independent and interdependent models, in this paper we study the model with interdependent valuations and no gap. We show existence of a stationary equilibrium that has ineﬃcient delay but nevertheless coincides with the competitive outcome. In this model the competitive equilibrium is ineﬃcient (has too much delay) because of an externality: the experience eﬀects imply that selling additional units today reduces the cost of serving future customers, generating additional value that current sellers cannot capture. On the technical side, our approach is to take a limit of the DL model shrinking the gap in valuations and costs for the lowest type and characterizing the limit of the sequence of the DL equilibria. Along the sequence, DL showed that equilibrium dynamics exhibit a very unusual pattern: trade in equilibrium takes place in short, isolated bursts of activity, interrupted by extensive quiet periods when no trade takes place. We show that as the gap at the lowest value disappears, the DL equilibria change dramatically. The atoms disappear and the trade takes place gradually over time. That in turn implies that the prices converge to the competitive prices. An additional advantage is that the no-gap dynamics are much easier to describe than the DL equilibria. Each type pays a price equal to the cost of serving that type and prices drop 1

Figure II in Coase (1972) shows the no gap case.

2

slowly over time in a way that is independent of the distribution of types (depends only on the shape of the cost function and the range of values) and trade happens smoothly over time.2

2

The Model

Consider the following dynamic bargaining game with interdependent values. There is a seller and a buyer. The seller has a unit of an indivisible good. The buyer type is c ∈ [c, c] and type c values the good at v (c) . The function v (.) is common knowledge, continuously diﬀerentiable and satisfies

v (c) > c for all c > c v (c) = c (the no-gap case) v0 (c) ∈ [A, B] for all c for some A > 1, B < ∞. Note that v0 (c) being bounded implies that v(c)−c ∈ [A, B] for all c > c. c−c Type c is buyer’s private information and it is distributed according to a distribution with c.d.f. F (c) which is continuous and strictly increasing with density f (c) which is continuous and strictly positive for all c. F is common knowledge. The seller’s cost of selling the good is c (which the seller does not know). Note that we use a diﬀerent normalization than in most of the literature - we call the type of the buyer c, rather than v, but since v (c) is strictly increasing, the costs and values are mapped one-to-one. This diﬀerent normalization yields a simpler notation. This model, with the exception of the no-gap assumption, is the same as the one on DL.3 There are two ways to interpret v (c) being increasing (rather than constant as in the classic Coase conjecture papers). First, the actual cost of providing the good/service may be increasing in the value of the good to the buyer. For example, in case of insurance, buyers with a higher probability of having a claim value insurance more and at the same time cost more to insure. Second, increasing v (c) can represent the opportunity cost of serving the buyer, instead of waiting for some news to arrive, as in Fuchs and Skrzypacz (2009): in one of the applications of that model there is a Poisson arrival of information that reveals the value of the buyer (or arrival of a second buyer with identical value) which allows the seller to extract full surplus upon arrival. 2

DeMarzo and Urosevic (2006) obtained a similar simplification of equilibrium dynamics of the continuous-time limit in a game with moral-hazard and dynamic trade. 3 DL actually present a model with the private information on the seller side, but as they point out in Section 7, these models are equivalent. We focus on the private information on the buyer side to relate this model to the durable goods problem.

3

λ In that example, c = λ+r v since by serving the customer the seller forgoes the option of waiting 4 for the arrival. Time is discrete with periods of length ∆ and horizon is infinite. In every period the seller oﬀers a price p, the buyer decides whether to accept current price or reject it. Both players discount payoﬀs at a rate r, so that if there is trade at time t ∈ {0, ∆, ...} and at price p, the payoﬀs are:

π s = (p − c) e−rt

π b = (v (c) − p) e−rt

As usual, in any (Perfect Bayesian Nash) equilibrium the skimming property holds, that is, if type c accepts with positive probability a price p, then all types c0 > c accept that price with probability 1. As a result, for any history of the game the seller’s beliefs are a truncated version of the prior F (c) : there is a cutoﬀ k such that the seller believes that the remaining types are (c) . distributed over the range [c, k] according to FF (k) This cutoﬀ is a natural state variable and is used to define stationary equilibria: Definition 1 A stationary equilibrium is described by a pair of functions P (k; ∆), κ (p; ∆) which represent the seller’s price given the belief cutoﬀ k and buyer acceptance rule (i.e. the cutoﬀ type that accepts price p, which does not depend on the history of the game), such that (1) Given κ (p; ∆) the seller maximizes his time-zero expected payoﬀs by following P (k; ∆) 2) Given the path of prices implied by P (k; ∆) and κ (p; ∆), for any buyer type c it is optimal to accept prices p if and only if c ≥ κ (p; ∆) In our analysis we also use an implied function keeping track of the equilibrium belief cutoﬀs, K (t; ∆) , which formally is defined as follows K (0; ∆) = c and K (t + ∆; ∆) = κ (P (K (t; ∆)) ; ∆) . Take a sequence of games indexed by ∆i → 0 and take a selection of stationary equilibrium functions P (k; ∆i ), K (t; ∆i ) . For any converging subsequence we call P (k) = lim∆i →0 P (k; ∆i ) and K (t) = lim∆i →0 K (t; ∆i ) the limit of these equilibria.

3

Atomless Limit of (DL) Equilibria

A crucial element of the equilibrium dynamics is the probability of trade in any given period. Since F (c) is atomless, F (K (t − ∆; ∆)) − F (K (t; ∆)) → 0 if and only if K (t − ∆; ∆) − λ Where λ+r is the present value of a dollar that arrives with Poisson intensity λ and r is the continuously compounded interest rate used for discounting. 4

4

K (t; ∆) → 0, that is, the unconditional probability of trade in any given period converges to zero if and only if the cutoﬀs change continuously. Therefore we define: Definition 2 A sequence of equilibria has an atomless limit if: lim K (t − ∆; ∆) − K (t; ∆) = 0

∆→0

for all t ∈ {∆, ...} (in other words, if the K (t) function is continuous). One can be also interested in the probability of trade conditional on reaching given period: F (K (t; ∆)) − F (K (t − ∆; ∆)) . ∆→0 F (K (t − ∆; ∆)) lim

Clearly, if the conditional probability converges uniformly to zero, then the unconditional probability does as well, but the opposite does not have to be true. We now prove that our game has a stationary equilibrium with an atomless limit. We do so by taking a sequence of games with types c being distributed between [c0 , c] with c0 > c (so that c0 < v (c0 ) - there is a gap for the lowest type). Such games are studied in DL. We take c0 →c (i.e. the gap to zero) and characterize the limit of the DL equilibria as we take ∆ and gap to zero. Along the sequence DL provide us with an existence of a stationary equilibrium, the limit of which is an equilibrium of our game. Even though the DL equilibria have atoms on the equilibrium path (even as ∆ → 0!), we show that as the gap disappears, the (unconditional) atoms disappear. That yields our main result. The proof is complicated by the fact that the conditional probability of trade at the bottom of the distribution does not become atomless, as the proof and the example in the next section illustrate. Consider a sequence of games with a gap: a sequence of games indexed by ε (with ε → 0). For a given ε, c is distributed over [c0 , c] where c0 = c + ε. The p.d.f. of c is g (c; ε) =

f (c) 1 − F (c0 )

that is, for any ε, g (c; ε) is computed as a truncated distribution. DL provide us with the following result, translated here to our notation: Proposition 0 (DL Proposition 3) For every ε and ∆ the game has a stationary equilibrium. As ∆ → 0, there exists a stationary equilibrium with a limit described by two sequences {cn } and {pn } with c0 = c+ε and p0 = v (c0 )

5

and the remaining elements defined recursively by: pn+1 pn

(v (cn+1 ) − cn+1 )2 = v (cn+1 ) − v (cn+1 ) − pn Z cn+1 cg (c) dc = G (cn+1 ) − G (cn ) cn

(1) (2)

(Given the starting values p0 and c0 , equation (2), which implicitly defines cn+1 , is used to compute c1 . Then equation (1) is used to compute p1 and so on.) The DL equilibrium path limit is derived from these two sequences as follows. Given {cn } , let N be the largest n such that cn < c. Then in the DL equilibrium limit the seller oﬀers price pN at t = 0. It is accepted by types (cN , c]. Then there is no trade for some amount of time and the next price is pN−1 (the delay is such that type cn is indiﬀerent between buying immediately at pN and waiting for the lower price pN−1 ). Price pN −1 is accepted by types (cN −1 , cN ]. This atom of trade is followed by another period of no trade and then price pN−2 , accepted by types (cN −2 , cN−1 ], another quiet period and so on. We call these DL equilibria. An interesting feature of these equilibria is that Equation (2) implies that the types that accept price pn for n < N cost the seller on average pn , so that the seller makes on average no profit beyond the first oﬀer pN at t = 0 (which generically yields a strictly positive expected profit since generically cN+1 > c). As emphasized by DL, for any ε > 0, in the limit ∆ → 0, equilibrium trade is very discontinuous: bursts (atoms) of trade are interrupted by sizable quiet intervals of no trade. The atom can be computed as follows: type cn is indiﬀerent between trading at a price pn and waiting τ n to get price pn−1 : v (cn ) − pn = e−rτ n (v (cn ) − pn−1 ) ¶2 µ v (cn ) − cn −rτ n = <1 ⇒ e v (cn ) − pn−1 where we used (1) to get the equality and the observation in the sequence pn−1 < cn to get the inequality. Our main technical result is that as ε → 0, these atoms and quiet intervals disappear and trade becomes continuous. At the same time, since the seller’s profit stems only from the first atom, we obtain that in the limit (since that atom disappears as well) the seller makes zero profit and every type pays a price equal to the cost of serving that type. For a given ε we write the corresponding DL sequences as {cn (ε)} , {pn (ε)}. Proposition 1 (Atomless Limit) As the gap between valuations and costs disappears, the 6

limit of DL equilibria becomes atomless. Formally, for every ψ > 0 and c ∈ (c, c], there exists ε00 > 0 small enough so that for any ε ≤ ε00 and any n (which can change with ε) such that cn (ε) ≥ c, we have cn (ε) − cn−1 (ε) ≤ ψ. We prove the proposition via a sequence of lemmas. We define three new sequences: cn+1 (ε) − c cn (ε) − c v (cn (ε)) − c γ n (ε) = ∈ [A, B] cn (ε) − c pn (ε) − c yn (ε) = cn (ε) − c xn (ε) =

The sequences {xn (ε)}, {γ n (ε)} and {yn (ε)} allow us to track the solutions to (1) and (2) n (ε) so to prove that atoms disappear in the limit, in an easy way. xn (ε) is equal to 1 + cn+1cn(ε)−c (ε)−c we need to show that xn(ε) (ε) → 1 as long as cn(ε) (ε) → c > c (where we write n (ε) to allow n to grow with ε). Of course, proving it for all n would be suﬃcient (since it would imply that the conditional probability of trade converges uniformly to zero in every period), but as we will see below, it is not true in the limit: for any fixed n, cn (ε) → c and xn (ε) → β n > 1 (where β n are defined below). Yet, as we increase n, β n → 1, yielding the claim. Note also that equation (2) can be re-stated as: pn =

Z

cn+1

cn

cf (c) dc F (cn+1 ) − F (cn )

(3)

Our first step is to show that the sequences {xn (ε)} , {γ n (ε)} converge as ε → 0 to a uniformly bounded limit: ∞ Lemma 1 There exist sequences {β n }∞ n=0 and {αn }n=0 such that for any n, xn (ε) → αn and yn (ε) → β n (as ε → 0 then). The limiting sequences are uniformly bounded: for all n : 1 ≤ β n ≤ αn ≤ α for some α < ∞.

The next lemma shows that sequences αn and β n decrease asymptotically to 1 : Lemma 2 The sequences αn and β n defined in the previous lemma are decreasing and converge to 1 as n → ∞. The remaining lemma states that for any fixed n, cn (ε) →c:

7

Lemma 3 For any fixed n lim cn (ε) = c

ε→0

That allows us to finish the proof of the proposition. Proof of Proposition 1. Suppose the limit is not atomless. Then, there exists ψ > 0 such that even as εk → 0, there exists c > c and nk (changing with ε) such that cnk (εk ) − cnk −1 (εk ) > ψ and yet cnk (εk ) ≥ c. Now, since for all finite n, cn (ε) → c (from Lemma 3), we must have that nk → ∞ (i.e. we can find εk small enough that nk is arbitrarily large to satisfy cnk (εk ) > c). Lemma 2 then implies that for any ε∗ > 0 we can find n∗ large enough such that for all n > n∗ , cn (ε)−c < 1 + ε∗ . That means that for all nk > n∗ , such that cnk (εk ) ≥ c limε→0 cn−1 (ε)−c lim cnk (εk ) − cnk −1 (εk ) < ε∗ (cnk −1 (εk ) − c) < ε∗ (c − c)

εk →0

Choosing ε∗ =

3.1

1 ψ 2 c−c

yields a contradiction. Hence, indeed the limit is atomless.

Example: Uniform Distributions

To provide intuition for our result we now present a direct computation for the double-uniform case. Suppose c˜U [1, c] and v = ηc for some η > 0. In the uniform case, instead of taking c0 → 0, we can also take c → ∞ since a problem with domain [1, c] is equivalent to a problem ¤ £ with domain cc0 , 1 by changing the units. As a result, the probability of trade conditional on reaching a period is invariant to that transformation. That means that along the sequence, the conditional atoms of trade do not converge uniformly to zero. The conditional k − th last atom (for any k) stays constant, but is decreasing to zero in k. However, as c0 → 1, the number of atoms in equilibrium grows without bound and the conditional atoms further and further away from the end of the game become smaller and smaller, yielding our result. The benefit of the (double) uniform case is that equations (1) and (2) can be simplified to:

pn pn

(γ − 1)2 c2n = ηcn − γcn − pn−1 cn+1 + cn = 2

which can be combined, to: cn+1

4 (η − 1)2 c2n = (2η − 1) cn − (2η − 1) cn − cn−1

8

(4) (5)

3

zn 2

1 0

5

Figure 1:

10

cn+1 cn

15

25

30

in the uniform case for v = 2c.

Divide both sides by cn and let zn =

cn+1 cn

to get a simple formula:

zn = (2η − 1) − z0 =

20

4 (η − 1)2 1 (2η − 1) − zn−1

(6)

c1 = 2η − 1 > 1 c0

It can be shown (see the Appendix) that the sequence zn is decreasing and converges to 1. For example, for η = 2, zn is shown in Figure 1: ³ ´ cn −cn+1 cn −cn+1 cn = 0. After re-scaling the = limn→∞ Since zn → 1, we have that limn→∞ c cn c £ c0 ¤ problem to c , 1 , it implies that as the gap disappears, the atoms disappear as well.

3.2

Dynamics of the Atomless Limit

The limit of DL equilibria (as the gap disappears) being atomless (as we have just shown), implies that in the limit the seller makes zero profit (recall that in the DL limit the seller makes money only on the first atom). Also, since we have shown that lim

ε→0,n→∞

yn (ε) =

pn (ε) − c =1 ε→0,n→∞ cn (ε) − c lim

we get that in the limit P (k) = k In words, given the seller believes that the remaining types are distributed on a range [c, k] , he asks price k. Since there are no atoms in the limit, that price is accepted by type k only, hence 9

each type pays a price equal to the cost of serving that type. In the gap case the prices are quite complicated and depend on the details of the distribution. As the gap disappears, prices become very simple and independent of the distribution! To finish describing the limit we only need to pin down K (t) . It is described by a diﬀerential equation that guarantees that no type of the buyer wants to postpone or speed up trade: ∂pt ∂t for c = K (t)

r (v (c) − c) = −

where pt = P (K (t)) is the equilibrium path of prices over time. This indiﬀerence condition can be written as: r (v (K (t)) − K (t)) = −K˙ (t) and with a boundary condition K (0) = c (since there is no atom at time 0), it uniquely defines K (t) . For example, when v (c) = ηc for some η > 1, c = 0 (so that v (c) = c)) and we normalize c = 1 then we get: K (t) = e−r(η−1)t = pt

1 0.8 0.6 0.4 0.2

0

1

2

t

3

4

5

pt for r = 1 and η = 2

4

Durable Goods and the Coase Conjecture.

We can interpret the model also as a durable good problem in the following way. Let v be distributed according to G (v) (which can be obtained from v (c) and the distribution F (c)) and let e c ≡ v −1 . Then 1 − G (v) is the demand function where the total mass of customers has been normalized to 1. Let V (Q) be the corresponding inverse demand function. Let qt be the quantity 10

sold at time t, and Qt be the cumulative sales including time t : Qt = Qt−1 + qt Q0 = q0 Assume the following industry-wide experience eﬀects: if the industry sold Qt−1 units by time t and sells qt units at time t then the cost per unit of production at time t is Z

Qt

e c (V (q)) dq if qt > 0 qt Qt−1 C (Qt−1 , qt ) = e c (V (Qt−1 )) if qt = 0

C (Qt−1 , qt ) =

For example, if demand is linear, V (Q) = 1−Q, and the underlying function from the bargaining problem is e c (v) = ηv for some η > 1, then C (Qt−1 , qt ) =

1³ qt ´ 1 − Qt−1 − η 2

Our assumption that v 0 (c) > 0 implies that the costs of serving customers fall with the cumulative sales. C (Qt−1 , qt ) here corresponds in the bargaining model to the average cost of serving the buyer conditional on a sale, completing the usual analogy between the durable goods and bargaining problems. A competitive equilibrium in this model is described by a sequence of prices {pt }∞ t=0 and ∞ quantities {qt }t=0 such that: pt = C (Qt−1 , qt ) if qt > 0 pt ≤ C (Qt−1 , qt ) if qt = 0 V (Qt ) − pt = δ (V (Qt ) − pt+1 ) The first condition is that if there are any sales in period t then firms make zero profit. The second condition means that firms are always willing to supply the product at prices above the current marginal cost, so for sales to be zero it must be that prices are below current marginal costs. The last condition captures buyer optimality: given the sequence of prices buyers choose optimally when to buy. Note that even if the DL equilibrium has zero profit in the limit (which is a non-generic case), it does not satisfy the conditions of competitive equilibrium: during the quiet periods prices are necessarily higher than C (Qt−1 , qt ) . Finally, we stress the positive externality in this market: a firm producing more today reduces 11

the marginal cost to other firms both in the current period and in the future. Since a firm does not capture that increase of total surplus, the competitive equilibrium outcome is ineﬃcient: Qt is growing too slowly. It is eﬃcient to produce immediately at the level where e c (V (Q)) = V (Q) , but the competitive equilibrium is only slowly converging to that total output. c (V (Qt )) and In the continuous-time limit prices change continuously over time with pt = e cumulative sales follow the following process: Q0 = 0 r (V (Qt ) − e c (V (Qt ))) = − e c0 (V (Qt )) Q˙ t | {z } =−p˙ t

That implies:5

Proposition 2 (Weak Coase Conjecture) Consider a durable good problem. In the "no gap" case with experience eﬀects, there exists a stationary equilibrium of the monopolist problem that converges to the competitive equilibrium outcome as ∆ → 0. To reiterate, the Coase Conjecture is not about eﬃciency of the outcome (as it is often interpreted) but about the monopolist acting in the limit as a competitive industry would. With no experience eﬀects, the competitive equilibrium is eﬃcient, but here it is not. This is consistent with the original Coase (1972) claim that the monopolist without commitment would act no diﬀerent than competitive sellers.6 Finally, we call it the "Weak" Coase Conjecture, since we are not able to prove that all stationary equilibria of the monopolist problem converge to the competitive equilibrium outcome. It is an open question whether a "Strong" version of the conjecture is true, i.e. if all stationary equilibria of the monopolist problem converge to the competitive outcome as ∆ → 0.7 The methods in Fuchs and Skrzypacz (2009) can be used to establish that all stationary equilibria with an atomless limit do converge, but we do not know how to prove if there exist any stationary equilibria of the monopoly problem with atoms of trade in the limit. 5

This claim was informally discussed in Fuchs and Skrzypacz (2009), but the proof is original to this paper. Since in the standard model competitive equilibrium is eﬃcient, the Coase conjecture is sometimes interpreted as a claim that the monopolist with commitment will also achieve eﬃciency. Yet, Coase (1972) stresses the comparison between the competitive and monopolistic markets. 7 Even if all stationary equilibria satisfy the Coase conjecture in the no-gap case there can also also be nonstationary equilibira that violate the Coase Conjecture. See Ausubel and Deneckere (1989) for the construction of such equilibria for the constant marginal cost case. 6

12

5

Appendix

Claim: zn defined in equation (6) is decreasing and converging to 1 Proof: Let φ (z) be: φ (z) = (2η − 1) −

4 (η − 1)2 (2η − 1) − 1z

µ

¶2

so that zn+1 = φ (zn ) . We have 2 (η − 1) φ (z) = (2η − 1) z − 1 φ (1) = 1, φ0 (1) = 1, 0

>0

φ0 (z) < 1, φ (z) > z for all z > 1. That implies that if z0 > 1 then zn > 1 for all n. Also, since φ (z) < z for all z > 1, we get that the sequence zn is decreasing. Finally, fix any any z ∗ > 1. For all z ≥ z ∗ , z − f (z) is uniformly bounded away from 0 (to see this, note that φ0 (z) is uniformly bounded away from 1 for all z ≥ z ∗ ). That implies that from any zk ≥ z ∗ there exists a finite l such that zk+l < z ∗ . Therefore, the decreasing (and bounded, hence converging) sequence zn converges to 1. Proof of Lemma 1. The proof is by induction. STEP 1. Take n = 0 we have p0 (ε) = v (c0 (ε)) so y0 (ε) =

p0 (ε) − c v (c + ε) − v (c) → v0 (c) ≡ β 0 ∈ [A, B] . = c0 (ε) − c ε

Using (3) let cˆ1 (c0 ) for any c0 ≥ c be a solution to: (v (c0 ) − c) (F (c1 ) − F (c0 )) − Note that cˆ1 (c) = c. Hence x0 (ε) =

c1 (ε)−c c0 (ε)−c

=

c1

c0

(c − c) f (c) dc = 0

cˆ1 (c+ε)−ˆ c1 (c) ε

and

∂ˆ c1 (c0 ) ≡ α0 c0 →c ∂c0

lim x0 (ε) = lim

ε→0

Z

13

Using the implicit function theorem: ∂ˆ c1 (c0 ) c1 (c0 )) − F (c0 )) − f (c0 ) (v (c0 ) − c − (c0 − c)) v 0 (c0 ) (F (ˆ = lim − c0 →c c0 →c ∂c0 f (ˆ c1 (c0 )) (v (c0 ) − c − (ˆ c (c ) − c)) ³ ´ 1 0 (c0 )) cˆ1 (c0 )−c0 )−c v0 (c) (F (ˆccˆ11(c(c00))−F − f (c0 ) v(cc00−c −1 )−c0 c0 −c ´ ³ = lim − )−c )−c c0 →c f (ˆ c1 (c0 )) v(cc00−c − cˆ1c(c00−c

α0 = lim

=

β 0 α0 − 2β 0 + 1 α0 − β 0

Solving it for α0 yields α0 = 2β 0 − 1 ≥ β 0 .Clearly, both α0 and β 0 are bounded. STEP 2. Now, assume that for all indices k ≤ n − 1 we have xk (ε) → ak and yk (ε) → β k and 1 ≤ β k ≤ αk ≤ α. We prove that it implies the same for index n. Note that our inductive hypothesis implies that cn (ε) − c ≤ αk (ε) → 0 (using xn−1 → an−1 ) and pn−1 (ε) − c ≤ α (cn−1 (ε) − c) → 0, so that for the fixed k, the cutoﬀs and prices converge to c. Using (1) again we get

yn (ε) =

pn (ε) − c v (cn (ε)) − c = − cn (ε) − c cn (ε) − c

³

(v(cn (ε))−c) cn (ε)−c

(v(cn (ε))−c) cn (ε)−c

−

´2 −1

(pn−1 (ε)−c) cn−1 (ε)−c cn−1 (ε)−c cn (ε)−c

Taking the limit ε → 0 we get (γ n − 1)2 yn (ε) → γ n − ≡ βn γ n − β n−1 /αn−1 Note that since cn (ε) → c, γ n → v0 (c) = β 0 and we can simplify: βn = β0 −

(β 0 − 1)2 β 0 − β n−1 /αn−1

and bound the new term β n from above and below: (β 0 − 1)2 (β 0 − 1)2 (β 0 − 1)2 2β − 1 ≤ βn = ≤ β0 − = 0 1 = β0 − β0 − 1 β 0 − β n−1 /αn−1 β0 β0 Finally, to show that αn is bounded, use again (3) (pn (cn ) − c) (F (cn+1 ) − F (cn )) −

14

Z

cn+1

cn

(c − c) f (c) dc = 0

(7)

to define a function cˆn+1 (cn ) . We have again cˆn+1 (c) = c. Since by the inductive hypothesis cn → c, we can apply the same reasoning as for x0 : β ((αn − 1) − (β n − 1) ∂ˆ cn+1 (cn ) = n cn →c ∂cn αn − β n

αn = lim xn (ε) = lim ε→0

where to take into account that pn changes with cn we used ∂(pn (cn )−c) → βn. ∂cn (ε) Solving for αn yields αn = 2β n − 1 ≥ β n Since we computed a uniform bound β n ≤

2β 0 −1 , β0

1 ≤ β n ≤ αn ≤

pn (ε)−c cn (ε)−c

→ β n which implies

(8)

that gives us also a uniform bound for αn : 3β 0 − 2 ≡α β0

and finished the proof. Proof of Lemma 2. Combining equations (7) and (8) we get βn = β0 − Let

(β 0 − 1)2 ¡ ¢ β 0 − β n−1 / 2β n−1 − 1

Γ (y) = s −

(9)

(s − 1)2 s − y/ (2y − 1)

for any s ≥ 1. This function has the following properties: 1) The unique fixed point of this function is y = 1 (Γ (1) = 1). 2) For all y > 1, we have 0 < Γ0 (y) < 1 and hence y > Γ (y) > 1 As a result, the sequence β n starting with β 0 = v0 (c) > 1 is strictly decreasing and converges to 1. Since αn = 2β n − 1, the same is true of αn . Proof of Lemma 3. From the definition of xn (ε) : cn (ε) − c = xn−1 (ε) (cn−1 (ε) − c) =

n−1 Y k=0

xk (ε) (c0 (ε) − c) ≤ αn ε | {z } ε

where we used the uniform upper bound xn (ε) ≤ α from Lemma 1. Taking ε → 0 completes the proof.

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References [1] Ausubel, Lawrence M., and Raymond J. Deneckere. 1989. “Reputation in Bargaining and Durable Goods Monopoly.” Econometrica, 57(3): 511-531. [2] Bulow Jeremy. 1982 “Durable Goods Monopolists." Journal of Political Economy 90, no. 2, (April 1982):314-32. [3] Coase, Ronald H. 1972. “Durability and Monopoly.” Journal of Law and Economics, 15 (1): 143-149. [4] DeMarzo, Peter M. and Branko Urosevic. 2006. “Ownership Dynamics and Asset Pricing with a Large Shareholder.” Journal of Political Economy, 114(4): 774-815. [5] Deneckere, Raymond J., and Meng-Yu Liang. 2006. “Bargaining with Interdependent Values.” Econometrica, 74(5): 1309-1364. [6] Evans, Robert. 1989. “Sequential Bargaining with Correlated Values.” Review of Economic Studies, 56(4): 499-510. [7] Fuchs, William and Andrzej Skrzypacz. 2009 “Bargaining with Arrival of New Traders.” [8] Fudenberg, Drew, David Levine, and Jean Tirole. 1985. “Infinite-horizon models of bargaining with one-sided incomplete information.” In Game Theoretic Models of Bargaining, ed. Alvin E. Roth, 73-98. Cambridge, MA: Cambridge University Press. [9] Gul, Faruk, Hugo Sonnenschein, and Robert Wilson. 1986. “Foundations of dynamic monopoly and the Coase Conjecture.” Journal of Economic Theory, 39(1): 155-190. [10] Stokey, Nancy L. 1981. “Rational Expectations and Durable Goods Pricing.” Bell Journal of Economics, 12(1): 112-128. [11] Vincent, Daniel R. 1989. “Bargaining with Common Values.” Journal of Economic Theory, 48(1): 47-62.

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