Asymmetric Information in Bilateral Trade and in Markets: An Inversion Result Stephan Lauermann February 21, 2011

Abstract I consider a simple bilateral trading game between a seller and a buyer who have private valuations for an indivisible good. The seller makes a price o¤er which the buyer can either accept or reject. If the seller can observe the valuation of the buyer (if information is symmetric), then bilateral trade is trivially e¢ cient. If the seller cannot observe the valuation (if information is asymmetric), then bilateral trade is ine¢ cient. This bilateral trading game between a single buyer and a single seller is embedded into a matching market with a continuum of traders. I consider steadystate equilibria in stationary strategies. I show that, on the overall market level, the relation between the informational regime and e¢ ciency is inverted when frictions are small. In particular, if frictions are small and if information is asymmetric, the trading outcome in the market is close to being e¢ cient. If information is symmetric, however, the trading outcome can be very ine¢ cient— even if frictions vanish.

JEL Classi…cations:

C78, D82, D83

Keywords: Dynamic Matching and Bargaining Games, Decentralized Markets, Consumer Privacy, Search Frictions, Asymmetric Information.

University of Michigan, Department of Economics, [email protected]. This paper supersedes "Private Information in Bilateral Trade and in Markets" (2008b) and includes now the …rst part of "Price Setting in a Decentralized Market and the Competitive Outcome" (2008a). First versions of the paper date back to 2005, ssrn.com/abstract=664191. I bene…ted greatly from comments and suggestions by Georg Nöldeke, Tilman Börgers, Philipp Kircher, Yusufcan Masatlioglu, Art Shneyerov, Lones Smith, Gábor Virág, and seminar audiences at the Canadian Economic Theory Conference, the 2008 Meeting of the Society for Economic Dynamics, and the 2008 Meeting of the Society for Economic Design. I also bene…tted from very helpful suggestions of two referees and an associate editor.

1

Introduction

Asymmetric information prevents e¢ cient bilateral trade. This paper revisits this insight if bilateral trade is embedded in a large search market with pairwise matching and bargaining. There are search frictions and the level of these frictions can be interpreted as a measure for how well integrated the market is. If frictions are substantial, trading remains essentially bilateral. I show that, consequently, asymmetric information is detrimental to e¢ ciency if frictions are large. However, as frictions become smaller and the market becomes better integrated, I show that the conventional relation between e¢ ciency and information is reversed: The trading outcome is, in fact, more e¢ cient with asymmetric information than with symmetric information. Moreover, when frictions are vanishingly small, the trading outcome is guaranteed to be close to the e¢ cient outcome with asymmetric information but not with symmetric information. Thus, although asymmetric information interferes with e¢ cient bilateral trade between isolated agents, in markets, asymmetric information has a distinct, e¢ ciency enhancing role. To model a decentralized market with frictions, I use a dynamic matching and bargaining game with an exogenous in‡ow of traders, similar to the classical model in Gale (1987). I consider steady-state equilibria in stationary strategies. This restriction is discussed in Section 4. I consider bilateral trade between one buyer and one seller who each want to trade a single unit of an indivisible good. Bilateral trade is embedded into a dynamic market as follows. There is a continuum of buyers and sellers who are matched into pairs at the beginning of each period. Within each pair, they bargain over the terms of trade. The pairs are connected by allowing an unsuccessful trader to be matched with another partner in a new pair in the next period. However, there is a friction that makes waiting for the next period costly, so integration of the market is not perfect. Here, I follow McAfee (1993) and, in particular, Satterthwaite and Shneyerov (2008) and assume that this friction is an exogenous probability

2 (0; 1) that a trader cannot enter the next

period and exits (dies). On the individual level of the agents, the exit rate acts similar to a discount rate. On the aggregate level, the exit rate ensures that a steady state exists for all strategy pro…les. I …rst consider a particularly simple bilateral bargaining protocol within each match, which allows me to characterize equilibrium fully and makes the underlying economic forces transparent. The bargaining protocol is later generalized. Speci…cally, I assume that the seller makes a take-it-or-leave-it o¤er to the buyer. Following acceptance, the seller exchanges the good for money, and both traders leave the market. Following rejection, the buyer and the seller return to the stock and wait to be rematched with a new partner in the next period, unless forced to exit. I consider two variations of this bargaining protocol.

1

First, I consider symmetric information, assuming that a seller observes the true valuation of the buyer before making an o¤er. In the symmetric information regime and for given continuation payo¤s, bilateral bargaining within a pair is e¢ cient. Second, I consider asymmetric information, assuming that the seller does not observe the true valuation. The seller has a belief about the distribution of the buyer’s valuations and acts like a monopolist relative to this distribution. In the asymmetric information regime, bilateral

Surplus   

bargaining within a pair is ine¢ cient.

W W*

ˆ

Frictions δ Frictions δ

Figure 1: Realized surplus with symmetric and asymmetric information. How do the e¢ ciency properties within matches (the "local market conditions") relate to the e¢ ciency of the overall outcome (the "global market conditions")? As shown in the main result, the e¢ ciency properties of the local level are inverted on the global level when frictions are not too large. I denote by W sym ( ) and W asym ( ) the surplus in equilibrium in the symmetric and in the asymmetric information regime, respectively, given the exit rate . Let W denote the surplus in the …rst-best e¢ cient outcome that maximizes the expected payo¤s of the entering traders. Proposition 2 shows that, with symmetric information, the surplus is strictly larger when frictions are larger and the surplus is maximal when the exit rate is one, W sym (1) = W ; thus, frictions are welfare enhancing. With asymmetric information, the shape of the surplus function is inverted. Proposition 3 shows that the surplus is strictly smaller when frictions are large and the surplus is maximal when the exit rate vanishes to zero, W asym ( ) ! W when

! 0;

thus, frictions are welfare reducing. Figure 1 illustrates the shape of the realized surplus as

a function of frictions for the two informational regimes.1 As veri…ed in Corollary 1, there exists a cuto¤ ^ such that the realized surplus is larger with asymmetric information when < ^ (the market is well integrated) and the realized surplus is larger with symmetric 1

The equilibrium surplus for given is unique in both regimes as shown in the corresponding results. The …gure here is an illustrative sketch. For an exact graph for a particular example, see Figure 2.

2

information when

> ^ (trading is essentially bilateral).

The intuition for the results can be derived from considering the extreme cases when frictions are very large ( = 1) and when frictions are very small ( ! 0). When

= 1, sellers do not take the future into consideration when making an o¤er.

With symmetric information, this implies that sellers trade whenever the buyer’s valuation exceeds their private cost of selling; hence, the outcome is …rst-best e¢ cient. With asymmetric information, however, the outcome is necessarily ine¢ cient. When frictions are small, the essential intuition for the welfare comparison stems from the di¤erence in the sellers’ (continuation) pro…ts. With symmetric information, sellers engage in perfect (…rst-degree) price discrimination and extract the entire surplus from buyers.2 This is shown to imply that sellers’ continuation pro…ts stay positive even in the limit. Therefore, sellers are not willing to trade their good with buyers having low valuations, implying that the limit outcome is ine¢ cient. With asymmetric information, however, sellers’pro…ts are shown to vanish in the limit. Sellers are therefore willing to trade with all buyers, implying that the limit outcome is e¢ cient. I discuss the robustness of the welfare comparison with respect to the bargaining procedure in Section 3. In this section, I consider a slightly more general bargaining protocol in which both the seller and the buyer have some chance to make a take-it-or-leave-it price o¤er. The respective chance to make an o¤er parameterizes the distribution of bargaining power. I show that the inversion result is robust in the following sense: When buyers have some— but not much— bargaining power, asymmetric information is e¢ ciency enhancing when frictions are small. When frictions are large, asymmetric information is of course still e¢ ciency decreasing. For the case with small friction, the essential intuition is again that sellers enjoy larger pro…ts with symmetric information than with asymmetric information, implying that fewer buyers can trade. However, the technical argument for this result is di¤erent from before because the limit outcome is e¢ cient in both informational regimes. Instead, the welfare comparison for small frictions follows now from a comparison of the rates of convergence of the surplus to the e¢ cient level.3 The section provides not only a robustness check of the inversion result, but it also characterizes the outcome for all distributions of bargaining power. I show that if buyers have su¢ ciently large bargaining power, then it depends on the shape of the distribution of 2 Diamond (1971) was the …rst to point out that sellers can enjoy substantial market power even with small search friction. In his model, monopolistic pricing leads to ine¢ ciency of the equilibrium outcomes because of the combined assumption that (a) sellers must use linear prices while (b) individual buyers have elastic demand. Here, monopolistic pricing within a given match does not lead to ine¢ ciencies because I assume that buyers have unit demand (e.g., despite perfect price discrimination, the outcome is …rst-best e¢ cient when = 1); therefore, the source of the ine¢ ciencies is di¤erent. 3 In Section 3 I compare the rates of convergence to those found in Shneyerov and Wong (2010b).

3

valuations in the in‡ow whether or not asymmetric information is e¢ ciency enhancing for small frictions. Finally, I point out that convergence to the e¢ cient outcome is uniform in the distribution of bargaining power with asymmetric information but not with symmetric information. This failure of uniform convergence implies that the outcome can be far from the e¢ cient outcome when information is symmetric— even when frictions are small. The fact that asymmetric information can be welfare improving in a model with search and matching has been observed by Moreno and Wooders (2002) in a numerical example; they do not study this e¤ect systematically. Independently of the current paper, Shneyerov and Wong (2011) study the e¤ect of information on welfare in a search and matching environment explicitly. They consider a model with an entry stage and absolute search costs. They observe that asymmetric information implies lower pro…ts of marginal entering types and, thus, asymmetric information "deters" entry. Depending on the elasticity of the matching function such entry deterrence can be bene…cial implying that welfare can be higher with asymmetric information than with symmetric information. Their paper is similar in spirit by showing that the e¤ect of the informational regime on the distribution of rents can imply that less information can be welfare improving. However, they do not focus on the welfare e¤ect of information for di¤erent levels of frictions and for the convergence properties when frictions are small. Due to the presence of absolute search costs and an entry stage, the source of the ine¢ ciency and the structure of equilibrium are quite di¤erent. I provide a further review of related work in Section 4.

2

Model and Analysis

The model is set up to include both, the symmetric and the asymmetric informational regime as special cases. In the following section, I describe those features of the model that are common to both cases. The di¤erence lies in the determination of the sellers’ price o¤er, which is deferred to two separate subsections.

2.1

The Framework

I consider the steady state of a market with a continuum of buyers and sellers who have quasilinear preferences. Sellers are endowed with one unit of an indivisible good, and their cost of selling is zero. Buyers want to buy one unit of the good, and their valuations for the good are given by v 2 [0; 1]. Buyers and sellers interact in a repeated market over in…nitely

many periods. At the beginning of each period, there is a stock of buyers and sellers. All traders from the stock are matched into pairs consisting of one seller and one buyer. Within each pair, the seller makes a take-it-or-leave-it price o¤er p. The buyer announces whether

he accepts or rejects the o¤er. If he accepts, the seller receives a payo¤ p, and the buyer 4

receives v p. If the buyer rejects the o¤er, neither trader receives anything. Subsequently, those agents who have successfully traded leave the market together with a share of those who have not. After that, new traders enter the market. The in‡ow of buyers and the in‡ow of sellers has mass one each. The distribution of valuations among entering buyers is given by a cumulative distribution function G. With the in‡ow of new traders, the period ends, and the next period starts according to the same rules. The stock is described by M and

B.

M denotes the mass of buyers in the stock at the beginning of each period. This

mass is equal to the mass of sellers. The function types v in the stock. This distribution

B

B

describes the distribution of buyers’

is, of course, endogenous and depends on how

agents trade. The distribution of types in the in‡ow, on the other hand, is exogenous. There are an equal number of buyers and sellers in the stock. In particular, all the traders in the stock are matched, and the matching technology is (…rst-best) e¢ cient.4 Histories are private information and traders are almost surely never matched again. This motivates the restriction to stationary, history-independent strategies in the following analysis. This is a standard restriction in the literature with a continuum of agents. In Section 4 I review some of the work by Gale and Sabourian (2005, 2006) and others on models of search and bargaining without this restriction. The mass of entering buyers with valuations above v is given by (1 to this mass as "long-run demand." The mass (1

G (v)), and I refer

G (v)) is assumed to be strictly decreas-

ing and twice continuously di¤erentiable. The density g is assumed to be strictly positive everywhere. I assume that the Myerson virtual valuation v

(1

G (v)) =g (v) is strictly

increasing; this ensures regularity of the monopolistic pricing problem max p (1

G (p)).5

By the assumptions before, "long-run supply," that is, the mass of entering sellers is equal to one. Long-run demand and supply determine the feasibility constraints for the trading outcome. The main implication of these constraints is stated in Lemma 1: If sellers trade only with buyers with valuations above some cuto¤ v, the mass of sellers who can trade is determined by the long-run demand, (1

G (v)).

Sellers use a symmetric and stationary pricing strategy p ( ), where p (v) is the price o¤ered to a buyer having type v. (When considering asymmetric information, the pricing strategy must be a constant.) The payo¤ to a seller who uses a pricing strategy p ( ) can be derived as follows. Denote by D (p ( )) the probability of trading in any given period,6 and denote by q S (p ( )) the probability that a seller is able to trade at some time during his 4 Hence, a social planner who maximizes welfare subject to the matching constraints can implement the …rst-best. Indeed, the equilibrium outcome itself is shown to be …rst-best e¢ cient if the exit rate is one and if information is symmetric. 5 I am grateful for a referee’s observation that this condition is su¢ cient for the results. 6 Buyers accept a price if it is below their reservation price r (v) as de…ned below. Given r, the de…nition R is D (p ( )) p ( ) d B ( ). :p( ) r( )

5

lifetime; that is, q S denotes the probability that the seller ends up trading while 1

qS

denotes the complementary probability that the seller ends up being forced to exit the market without having sold the good. Given the per-period trading probability D, one can derive q S recursively from q S (p ( )) = D (p ( )) + (1

D (p ( ))) (1

) q S (p ( )). Since

there is no discounting, the seller does not care about when he conducts a trade but only about whether he is able to trade before he must exit and what price he gets. Denote by E [pjp ( )] the expected price conditional on being able to trade and let E [pjp ( )] = 0 if q S (p ( )) = 0. The expected pro…t is

, where

(p ( )) = q S (p ( )) E [pjp ( )].

To derive the optimal search strategy of a buyer, observe that he is essentially sampling without recall from a known and constant distribution of prices, given the offer strategy p ( ). It is well known that optimal search is characterized by a reservation price r. Given the reservation price and the price o¤er distribution, a buyer’s expected payo¤ depends on the trading probability and the price. The probability that a buyer with type v who is searching with a reservation price r trades some time during his lifetime is denoted by q B (r; v). The expected price conditional on trading is denoted E [pjp

r; v] = r if q B (r; v) = 0. Expected payo¤s are

r; v], where E [pjp

U B (r; v) = q B (r; v) (v

r; v]). Let V (v) = supr U B (r; v) be the maximized ex-

E [pjp

pected lifetime payo¤. At the reservation price r (v) buyers must be indi¤erent between acceptance and rejection, that is, r (v) = v

(1

) V (v) :

(1)

The expected payo¤ of the entering traders is the sum of the expected payo¤s of sellers and R1 buyers (p ( )) + 0 V (v) g (v) dv. The sum of the expected payo¤s of the newly arriving traders is the standard e¢ ciency benchmark in the literature; see, e.g., Satterthwaite and Shneyerov (2007). The market is in a steady state if the in‡ow equals the out‡ow. The in‡ow of buyers with valuations below v is G (v). The out‡ow consists of all buyers who trade plus those buyers who die. Equality of in- and out‡ows holds if G (v) = M

Z

v

1p(

) r( )

+ 1p(

)>r( )

d

B

( );

0

and similarly for sellers, 1 = M [D (p ( )) + (1

D (p ( )))].

Consider a market constellation (p ( ) ; r ( ) ; ; M ). A constellation is said to be a steady state if the stock

; M satis…es the steady-state conditions given the bargaining

pro…le p ( ) ; r ( ). The presence of the exit rate ensures that there exists a unique steadystate stock for every strategy combination p ( ) ; r ( ); see Nöldeke and Tröger (2009). 6

Suppose a steady-state market constellation is such that there is some cuto¤ v such that trade between a buyer and a seller happens if and only if the valuation of the buyer is above v. It turns out that the trading probability q S of sellers can be very nicely characterized by the long-run demand 1

G (v), that is, the mass of buyers with valuation

above v in the in‡ow. This relation provides a straightforward link between the in‡ow and the sellers’payo¤s, a link that is crucial for the results. In particular, if the cuto¤ exceeds zero so that low valuation buyers do not get to trade, the sellers’ trading probability is strictly smaller than one. A trading probability less than one is interpreted as rationing. The proof of this Lemma and the proofs of all subsequent results are in the appendix. Lemma 1 Suppose a bargaining pro…le is such that r (v) some cuto¤ v. Then, in steady state, sellers,

qS

=1

qB

(v) = 1 if v

p (v) if and only if v

v and

qB

v for

(v) = 0 otherwise. For

G (v).

A market constellation (p ( ) ; r ( ) ; ; M ) is said to be a steady-state equilibrium in stationary strategies if (a) the stock

; M is a steady state given the bargaining pro…le

p ( ) ; r ( ) and if (b) the bargaining pro…le p ( ) ; r ( ) corresponds to a perfect equilibrium in the bilateral bargaining game that is induced by the continuation payo¤s by the distribution

and V and

of buyer’s types. The preceding paragraphs have characterized the

steady-state conditions and the optimal acceptance strategy for buyers. Given a price o¤er strategy, these conditions are independent of the informational regime. In the following, I close the model separately for the two states, by introducing the appropriate equilibrium condition for the price o¤er strategy.

2.2

Symmetric Information

With symmetric information, a seller observes a buyer’s valuation before making an o¤er. Given continuation payo¤s (1

) , a price o¤er strategy is optimal if p (v)

r (v) 7 (1

)

Because a buyer accepts every price p

) p (v) = r (v) .

r (v) and (2)

r (v), o¤ering a price below r (v) would be

suboptimal. O¤ering p = r (v) is optimal if the seller’s pro…t at this price exceeds his continuation payo¤. A market constellation p ( ) ; r ( ) ; ; M is a steady-state equilibrium in stationary strategies with symmetric information if p ( ) satis…es the optimality condition (2), reservation prices satisfy (1), and the steady-state conditions hold. The next result shows that the cuto¤ v

(1

)

(p ( )) characterizes the equilibrium.

Equilibrium is essentially unique, except for the speci…cation of the unacceptable o¤ers and the o¤er to the cuto¤ type. 7

Proposition 1 For every exit rate, there exists a steady-state equilibrium with symmetric information. Equilibria are characterized by a cuto¤ v, where v is the unique solution to v = (1

)

Z

1

vg (v) dv.

(3)

v

Given the cuto¤ v, p (v) = v for all v > v, and p (v) > v for all v < v. Buyers’ payo¤ s V (v) = 0 and reservation price r (v) = v for all v. Given an exit rate , let v ( ) be the cuto¤ type. Since buyers’ payo¤s are zero, the total surplus with symmetric information, denoted W sym ( ), equals the sellers’ pro…ts, R1 W sym ( ) = v( ) vg (v) dv. The next proposition shows how the cuto¤ and the surplus change with the exit rate.

Proposition 2 In the symmetric information regime, the cuto¤ v ( ) is strictly decreasing in the level of frictions . At the extremes, lim trade when

!0 v (

) 2 (0; 1) and v (1) = 0. More buyers

is large. Therefore, realized equilibrium surplus W sym ( ) is strictly increasing

in the level of frictions (frictions are welfare enhancing). The market outcome for small frictions is ine¢ cient despite the fact that with symmetric information the bargaining protocol as well as the matching technology is fully e¢ cient. Matching is e¢ cient since every trader is matched immediately upon entry (see Footnote 2.1). Bargaining within pairs of traders is e¢ cient for given continuation payo¤s because the whole surplus that is available in any given match is actually realized in equilibrium.

2.3

Asymmetric Information

With asymmetric information, a seller does not observe the buyer’s valuation before making an o¤er. Therefore, the price cannot be conditioned on the type of the buyer. I formalize asymmetric information simply by requiring the price o¤er strategy to be a constant, i.e., the seller must choose the same price o¤er for every type. As before, attention is restricted to symmetric, stationary, and pure strategy equilibria in which all sellers offer the same price, denoted by p.7 Abusing notation, a constant price o¤er strategy is identi…ed with the constant itself, p ( ) = p. Consider the bilateral bargaining game between a buyer and seller that is induced by some constellation (p; r ( ) ; ; M ). The constellation determines the continuation payo¤s (p) and V (v) and the belief of the seller about the buyer’s type, which is identi…ed with the distribution

of types in the stock. The distribution of buyer’s types together with

the acceptance strategy of each type determines the probability D (p) with which a price 7

Lauermann (2008a) analyzes heterogeneous sellers and non-degenerate price distributions.

8

o¤er p is accepted. Given the seller’s continuation payo¤s, a price p is an optimal o¤er if and only if the price maximizes D (p) p

(1

D (p) (p

D (p)) (1

(1

)

)

(p), or, equivalently,

(p)) .

(4)

A market constellation (p; r ( ) ; ; M ) is a steady-state equilibrium in stationary strategies with asymmetric information if p maximizes (4), reservation prices satisfy (1), and the steady-state conditions hold. The objective function (4) is equivalent to the pro…t function of a monopolist who faces a demand function D and whose cost of selling is the foregone continuation payo¤ (1

)

(p). I show in the appendix that (4) is su¢ ciently regular so that the opti-

mal price solves the …rst-order condition.8 The …rst-order condition of the monopolistic price can be re-written as the familiar Lerner formula, relating the relative “mark-up” (p

(1

)

(p)) =p to the inverse elasticity of short-run demand D. Both— the mark-up

and the elasticity of demand— are endogenous equilibrium objects; however, these objects turn out to have a very simple and intuitive characterization. From Lemma 1, the pro…t is equal to p (1

G (p)). In the appendix it is show that the elasticity of endogenous short-

run demand D is equal to the elasticity of the exogenous long-run demand (1 scaled up by

1,

that is,

pD0 (p) =D (p)

1 (pg (p) = (1

=

G (p))

G (p))). An immediate im-

plication is that the function D is more elastic when frictions are smaller, discussed below. Substituting the equilibrium condition p = p, it follows that p

(1

) p (1 p

G (p))

=

(1

G (p)) . pg (p)

(5)

The equilibrium condition has a unique solution p for every . This price corresponds to the unique equilibrium with asymmetric information. Proposition 3 For every exit rate, there exists a unique equilibrium with asymmetric information. The equilibrium price o¤ er p is characterized by the unique solution to the Lerner formula (5). The monopoly price relative to the in‡ow (long-run demand) is denoted by pm = arg max p (1

G (p)). The monopoly price is unique by the regularity properties of G.

Let p ( ) be the equilibrium price when the exit rate is . The realized equilibrium surplus R1 R1 is W asym ( ) = q S p + p (v p) g (v) dv = p vg (v) dv. The next proposition shows how the price and the surplus change with the exit rate. 8

I show that the function is quasi-concave and continuously di¤erentiable. Quasi-concavity follows from monotonicity of virtual valuations. Note that continuous di¤erentiability is not trivial because it has to be shown that the per-period trading probability does not have a kink at p.

9

Proposition 4 In the asymmetric information regime, the equilibrium price o¤ er p ( ) is strictly increasing in the level of frictions . At the extremes, lim p (1) = W asym (

pm .

Less buyers trade when

!0 p (

) = 0 and

is large. Therefore, realized equilibrium surplus

) is strictly decreasing in the level of friction

(frictions are welfare decreasing.)

The comparative statics of the equilibrium follow from inspection of the Lerner formula (5). When the exit rate is one, the Lerner formula collapses to the pricing problem of a monopolist with zero marginal costs and demand given by (1 price must be

pm .

G). Hence, at

= 1, the

When the exit rate is smaller, the (absolute value of the) elasticity of the

trading probability D (p) is increasing and the price decreases. When

! 0, the elasticity

of the trading probability diverges to in…nity. Therefore, the inverse elasticity on the righthand side of the Lerner formula becomes zero. This implies that the "markup" on the lefthand side must become smaller and eventually vanish to zero. The markup is the di¤erence between the price and the continuation value. For small , the mark up is proportional to the "rationing" probability 1 p 1

q S , because p

q S . For the markup to vanish with

(1

) q S p is approximately equal to

! 0, the rationing probability 1

q S must

become zero— and that requires the price to be zero.

The intuition for why the elasticity of per-period demand D is increasing when

is

smaller is the following. Buyers having valuations above the equilibrium price become increasingly price-sensitive because they can easily wait for a lower price; buyers having valuations below the equilibrium price cannot trade and accumulate, hence, constituting a large share of the stock. Both imply that a small price change in either direction leads to a relatively large change in demand.

2.4

Comparing Symmetric and Asymmetric Information

Propositions 1 and 3 provide exact characterizations of the equilibrium outcomes with symmetric and with asymmetric information. An immediate consequence of Propositions 2 and 4 is that more surplus is realized with asymmetric information when

is small (the

market is well integrated) and more surplus is realized with symmetric information when is large (trading is essentially bilateral). Corollary 1 There exists a cuto¤ ^ such that surplus is higher with asymmetric information when < ^ and surplus is higher with symmetric information when > ^, W asym ( ) > W sym ( )

if

<^

and

10

W asym ( ) < W sym ( ) if

> ^.

3

General Bargaining Protocol

The purpose of this section is to investigate the robustness of Corollary 1. I consider a more general bargaining protocol, endowing the buyer with some bargaining power as well. The following discussion focuses on the case when is immediate when

3.1

is close to

is small; the welfare comparison

one.9

Symmetric Information

Suppose information is symmetric and suppose buyers themselves can make o¤ers as in Gale (1987). Within each buyer-seller pair, after observing the buyer’s type, with probability

the seller is the proposer of a price o¤er, and with probability (1

is the proposer. The main model corresponds to

) the buyer

= 1. The appendix contains the full

description of the model, including the de…nition of steady-state equilibrium in stationary strategies. As in the previous case, equilibrium is characterized by a cuto¤ type v ( ; ) that now depends on both the bargaining share

and the exit rate . The cuto¤ is the

unique solution to v = (1 When

(1 G (v)) ) + G (v) (1 ) (1

= 1, the …rst fraction reduces to (1

)

Z

v

1

v

g (v) dv. 1 G (v)

(6)

G (v)). Hence, (6) includes (3) as a special

case. Proposition 5 There exists a steady-state equilibrium in stationary strategies with symmetric information and intermediate bargaining power

2 [0; 1]. The equilibrium is char-

acterized by a cuto¤ v ( ; ) that is the unique solution to (6). A matched pair of agents trade if and only if v

v ( ; ).

The surplus that is realized in equilibrium depends only on the cuto¤ v ( ; ) and is R1 given by W ( ; ) = v( ; ) g (v) dv. Figure 2 provides an illustration of the comparative statics of the realized surplus; the surplus with symmetric information corresponds to the

straight lines. The graph suggests a number of properties: (i) the realized surplus for parameters

close to one can be well approximated by

= 1 whenever

is not too small;

(ii) the surplus is u-shaped in , and the surplus is increasing in the level of frictions when is not too close to zero; (iii) when the exit rate vanishes, the surplus becomes e¢ cient. The following result summarizes the comparative statics of W ( ; ) and is proven in the appendix. 9

The suggestions by the associate editor and two referees improved this section considerably. The suggestion to compare the rates of convergence was particularly helpful.

11

Welfare

0.50

Symmetric Information

0.45

Asymmetric Information 0.40

0.35

Frictions 0.2 N!

0.0 N&

0.4

0.6

0.8

1.0

Figure 2: Realized surplus as a function of the bargaining power and frictions when the distribution of valuations is uniform. Straight lines: Symmetric information when = 1 (thick), = 0:99 (thin), and = 0:95 (grey). Dashed lines (in increasing order): asymmetric information when = 1 (red), = 0:99 (purple), and = 0:95 (grey). Proposition 6 With symmetric information: (Continuity and Monotonicity in

.) For any given , the realized surplus W ( ; ) is

continuous and strictly decreasing in . (Monotonicity in .) For any given

, there exists a threshold

surplus is increasing in the level of frictions if When

is close to one, the threshold

(Convergence.) For all lim

!0 W

0(

( ; ) = W . When lim

W

)

0:5 such that

and decreasing if

) is close to zero.

2 [0; 1), the limit is e¢ cient, lim

proportional to ,

!0 v (

2 0;

0

.

; ) = 0 and, therefore,

2 (0; 1), the welfare loss converges to zero at a rate W sym ( ; )

!0

3.2

2

0; 1

0(

=

1 21

E [v] .

Asymmetric Information

Again, the buyer makes an o¤er with probability 1

, while the seller makes an o¤er

with probability , without observing the buyer’s type. Equilibrium is characterized by the price o¤er of the buyer and the seller, pB (v) and p, by the reservation price of the buyer, rb (v), and the reservation price of the seller, rS . The appendix contains the de…nition and complete characterization of equilibrium with asymmetric information and intermediate bargaining power. In equilibrium, a buyer’s acceptance strategy is characterized by a cuto¤ type ' so that rB (v)

pS if and only if v

' (I chose ' to distinguish it from the cuto¤ with 12

symmetric information, v.) Given equilibrium pro…ts rS

, the sellers’ reservation price is

) . Equilibrium is characterized by two cuto¤s, ' and rS .10

= (1

Proposition 7 There exists a steady-state equilibrium in stationary strategies with asymmetric information and intermediate bargaining power terized by two cuto¤ s, ', and (1

2 [0; 1]. Equilibrium is charac-

) : A matched pair of agents trade if and only if the

seller is chosen to make the o¤ er and v

' or if the buyer is chosen to make the o¤ er

and v

if

(1

) ; the cuto¤ ' > (1

)

< 1.

Buyers having su¢ ciently high valuations trade immediately while buyers having intermediate valuations wait until they are chosen as proposer. The probability that buyers with intermediate valuations end up trading is (1

) = (1

+

). Given

and , let

( ; ) and ' ( ; ) be the cuto¤s for some equilibrium. The realized surplus is W

asym

( ; )=

Z

1

vg (v) dv +

'( ; )

Z

1 1

Let w ( ) be the unique solution to w = (1

+ ) (1

'( ; )

vg (v) dv:

(1

) ( ; )

G (w)) =g (w).

Proposition 8 With asymmetric information: (i) For all

2 [0; 1], the limit is e¢ cient: lim

(ii) The cuto¤ s lim (iii) When lim

!0

!0 ' (

!0 W

; ) = w ( ) and lim

asym (

!0 (1

; )=W .

) ( ; ) = 0.

> 0, the welfare loss converges to zero at a rate proportional to , 1

W asym ( ; )) =

(W

1 ( w (1 2

1

G (w)) + G (w) E [vjv W asym is given by

With asymmetric information, the welfare loss W Z

0

(1

) ( ; )

vdG (v) +

(1

)+

w]).

Z

'( ; )

(1

vdG (v) ; ) ( ; )

This welfare loss is composed of two separate e¤ects: The …rst term measures the inef…ciency that stems from the fact that sellers receive pro…ts in the future, which makes sellers unwilling to trade with buyers who have low valuations. The second term measures the ine¢ ciency that stems speci…cally from asymmetric information: Whenever the seller is the proposer and the buyer has a valuation between (1

)

and ' ( ; ) there is no

transaction— despite the existence of gains from trade. The latter welfare loss is due to ine¢ cient bilateral bargaining with asymmetric information. 10 The cuto¤s are implicitly characterized by a solution to a …xed point problem; see (26) and (27). The fact that the trading pattern is not characterized by a single cuto¤ makes the characterization of this case somewhat more involved than the previous cases.

13

Shneyerov and Wong (2010b) derive the rate of convergence in a related model with asymmetric information, an entry stage, and absolute search costs. They have shown that in their model the welfare loss converges at a rate proportional to the search costs. Although the result is similar, the underlying mechanism is quite di¤erent. Speci…cally, in models with absolute search costs, the mechanism that is driving convergence and the structure of equilibrium are distinct from the current model; for a discussion of these di¤erences, see Satterthwaite and Shneyerov (2008, pp437-38).

3.3

Comparison of the Informational Regimes

The main result of this section is a general welfare comparison for the two informational regimes. The result is proven by comparing the rates of convergence derived before. Recall that pm denotes the monopolistic price which solves max p (1

G (p)). Given this price,

the following inequality is central for the result: Z

pm

(1

G (v)) dv

pm (1

G (pm ))

Z

1

(1

G (v)) dv. ( )

pm

0

The terms in condition ( ) have a simple economic interpretation, illustrated in Figure 3. Consider a monopolist with zero costs facing a demand function (1

G). Then the right-

hand side corresponds to the consumer surplus at the monopoly price while the left-hand side corresponds to the deadweight loss. Corollary 2 If ( ) holds, then there is a function ^ : (0; 1] ! (0; 1), s.t. W asym ( ; ) > W sym ( ; )

8 < ^( ) ;

If ( ) fails, then there is a cuto¤ ~ 2 (0; 1) and a function ^ : (0; 1] ! (0; 1), s.t. ? ~

)

W asym ( ; ) ? W sym ( ; )

8 < ^( ) .

When sellers have su¢ ciently much bargaining power, asymmetric information is always e¢ ciency increasing when

is small. When sellers have little bargaining power,

asymmetric information might or might not be welfare increasing, depending on the shape of the distribution G: If condition ( ) holds, asymmetric information is welfare increasing for all ; if ( ) fails there is a cuto¤ ~ such that asymmetric information is welfare increasing only if

> ~ and welfare decreasing otherwise.

To see that ( ) can either fail or hold depending on the distribution of valuations, note that if valuations are uniformly distributed, ( ) holds. In fact, the condition holds with equality, allowing a simple construction of an example where it fails, too: given a

14

v 1

A 

1–G(v)

A

B

1 pm pm

0

pm

1  G  v   dv 1  G  v   dv  w 1  G  w  

B 1–G(pm)

1

1–G

Figure 3: Suppose is su¢ ciently small. If area A is larger than area B, welfare is larger with asymmetric information, independent of the bargaining power . Otherwise, welfare is larger when is close to one and smaller when is close to zero. uniform distribution, one may bunch a set of types of the form [~ v ; 1] into an atom at v~. This decreases the right-hand side of ( ) but does not change the left-hand side if v~ is su¢ ciently large.11 In the remaining section, I compare the economic mechanism that underlie the results for the case when sellers have almost all the bargaining power ( close to one) and the case when sellers have all the bargaining power (

equal to one). In both cases, asymmetric

information is welfare improving. I show that the intuition for the result is essentially the same in either case: with asymmetric information, continuation pro…ts of sellers are smaller, implying that they trade with a larger set of buyers. Formally, I show that when sellers have almost all the bargaining power, the di¤erence in realized surplus between the informational regimes is almost entirely due to the di¤erence in pro…ts. Let

sym (

; ) and

asym (

; ) denote the sellers’ pro…ts in the two regimes. After

substituting the de…nitions, the inequality 1 W asym ( ; ) > 1 W sym ( ; ) becomes 1

Z

(1

(1

)

sym (

; )

vdG (v) > )

asym (

; )

(1

)+

Z

'( ; )

(1

vdG (v) .

(7)

) ( ; )

Each side of the inequality refers to a particular welfare e¤ect of the informational regimes: The left-hand side corresponds to the di¤erence in welfare loss that is due to the di¤erence in sellers’ pro…ts. With symmetric information, sellers have higher pro…ts. Therefore, sellers refuse to trade with a larger set of buyers. The right-hand side corresponds to the 11

For example, G (v) = v for v < 0:95, G (v) = 1 for v 2 [0:95; 1]. Although this example violates the assumption that the distribution has a continuous density, one can easily verify that this violation is not consequential for the characterization of equilibrium in either informational regime.

15

welfare loss that is speci…cally due to the ine¢ cient bargaining outcome with asymmetric information, described before. Using observations from the proofs of Proposition 6 and 8,12 the left-hand side of (7) can be shown to converge to

(1

1 )2

Z

1

vdG (v)

w (1

G (w)) ,

0

where w is the unique solution to w = (1

) (1

G (w)) =g (w). The term in brackets

has a simple economic interpretation. It corresponds to the di¤erence in the payo¤s of a monopolist who can perfectly price discriminate among buyers and a monopolist who sells at a price equal to w. The right-hand side of (7) converges to

(1

)

Z

w

vdG (v) .

0

The integral is the deadweight loss of selling at price w rather than at zero. Thus, the welfare comparison between symmetric and asymmetric information depends on the intuitive trade-o¤ between the distortion that is due to extra-pro…ts from price-discrimination relative to the deadweight-loss that is due to asymmetric information. The relative size of these distortions depends on the shape of the demand function and on . When

is close to one, the right-hand side of (7) (the welfare e¤ect of ine¢ cient R w( ) vdG (v) = 12 bilateral bargaining with asymmetric information) is bounded, (1 ) 0 = 1,13 while the left-hand side (the welfare e¤ect of the di¤erence in pro…ts) becomes arbitrarily large when gets close to one. Thus, when sellers have almost all the

when

bargaining power, the welfare-comparison is driven almost entirely by the higher pro…ts of sellers with symmetric information, as claimed. Remark: Failure of Uniform Convergence with Symmetric Information. In both informational regimes the trading outcome becomes e¢ cient if buyers have some bargaining power. However, in the symmetric information regime, convergence to the e¢ cient outcome is not uniform across . This follows from inspection of the proportionality factor for the rate of convergence in (6). Even when can be large when

is small, the e¢ ciency loss

is close to one. This observation re‡ects the fact that the e¢ ciency

loss does not converge to zero at all when

= 1, discussed before. With asymmetric

information, however, the e¢ ciency loss is guaranteed to be small when zero.14 12 13

lim 14

lim

is close to

Figure 2 illustrates the di¤erence between uniform convergence with asymmetric

From (21), observing that v ( ; ) = (1 ) sym( ; ) , and Step 4 of the proof of Proposition 8. R w( ) Using w ( ) = (1 ) (1 G (w)) =g (w), w ( ) ! 0 when ! 1, and lim !1 (1 ) 0 vdG (v) = !1 (1

The

!1 1

1 g )2

(0) (1 ) (1 G (0)) =g (0) = 12 . rate of convergence in Proposition 1 w ( ) (1 G (w ( ))) + G (w ( )) E [vjv 2

16

8 is bounded from 1 w ( )] = 2g(0) + 12 .

above,

noting

that

information and the failure of uniform convergence with symmetric information. Since convergence is not uniform with symmetric information, a researcher cannot use the limit result alone to predict the trading outcome when frictions are small— unless she also knows the distribution of bargaining power. In particular, knowing that the level of frictions

is close to zero does not allow a researcher to predict that the trading outcome

is close to being e¢ cient when information is symmetric.

4

Related Literature

The current model provides an example of a dynamic matching and bargaining game in which the outcome does not become competitive for vanishing frictions. Non-competitive limit outcomes have also been found in Serrano (2002), De Fraja and Sakovics (2001), Shneyerov and Wong (2010a), and Rubinstein and Wolinsky (1990). The sources of the persistence of a noncompetitive outcome with small frictions are di¤erent from the current model. Serrano (2002) models bargaining between a matched buyer and seller as a simultaneous double auction with a discrete set of prices. Due to the simultaneous bidding, sequential rationality does not restrict the set of equilibria.15 Thus, the failure of convergence is due to persistent ine¢ ciencies in the bargaining protocol. In De Fraja and Sakovics (2001) the stock is assumed exogenous and exiting agents are replaced by exact copies ("clones"). The cloning assumption implies that the feasibility constraints are endogenous; see Lauermann (2009).

Shneyerov and Wong (2010a) include a costly

entry stage; that is, new traders must choose whether they want to incur some costs to enter the stock and become active.16 They show that the market can "break down" even when frictions are small. Here, ine¢ ciencies result from a hold-up problem due to the fact that traders need to invest ex ante (before the match) into a costly search activity while receiving only some— potentially very small— fraction of the additional trading surplus that their entry generates. Finally, Rubinstein and Wolinsky (1990) show how to construct collusive equilibria that are subgame perfect in a model with a …nite, arbitrarily large number of agents with complete information. Non-competitive equilibria are supported by appropriately constructed punishments of deviating agents. In the current paper, history-independence of strategies is assumed, ruling out such punishments. This restriction is motivated by the assumption that agents do not observe each other’s histories and that agents are never matched again. However, the restriction to stationary strategies is not easily motivated outside a model with a continuum of anonymous agents. A growing literature investigates whether the restriction to stationary strategies can be 15 In my model, sequential rationality imposes restrictions on the acceptance decision that I use when deriving the reservation price (1). 16 See also Atakan (2007) for the analysis of a model with entry and similar e¤ects.

17

justi…ed, for example, by complexity consideration; see Chatterjee and Sabourian (2000), Gale (2000), Sabourian (2004), and Gale and Sabourian (2005, 2006). Lauermann (2009) contains a general characterization result that provides conditions under which the limit outcome of a dynamic matching and bargaining game is (or is not) competitive. In that paper, I also discuss the fact that the limit outcome in the current model is not competitive when information is symmetric and relate this to the failure of a particular condition called "no-rent extraction." These two papers di¤er in several aspects. The current paper considers a particular game and it provides a complete characterization of the trading outcome for all levels of frictions. In Lauermann (2009) I use an axiomatic approach for an abstract class of games and I ask whether or not the limit outcome is competitive for vanishing frictions. The characterization of the trading outcome with asymmetric information for the case when

= 1 is new. The main insight is that the elasticity of the relevant demand

function— the probability of trading— is determined in an intuitive manner by the level of the frictions and the elasticity of the long-run demand (the in‡ow). Asymmetric information implies that there is a natural trade-o¤ between the level of the price o¤er and the trading probability. To my knowledge, there is no comparable explicit characterization in the literature when agents are heterogeneous and types are private information.17 The fact that there has been no simple, explicit characterization of search with asymmetric information might explain why, for example, the applied literature on search in the labor market rarely uses asymmetric information. This paper is also related to the literature on embedding problems of "contract design" into (matching) markets; see, for example, Inderst (2001, 2004) or Felli and Roberts (2002). In my model, I show how a property of exchange between a small set of agents in isolation is altered when considered as part of the equilibrium of a market. Finally, the paper contributes to the economics of "consumer privacy" as discussed, for example, in Varian (1996); consumer privacy aims at withholding information from sellers. Conventional economic intuition derived solely from the detrimental e¤ects of asymmetric information for bilateral trade might give the misleading impression that there should be as little privacy as possible from an overall welfare perspective. The current paper demonstrates to the contrary that rents generated by price discrimination can lead to severe welfare-reducing distortions.

17

Characterization is either only implicit or the characterization is via so called "full-trade" equilibria in which all matches result in trade, as in Satterthwaite and Shneyerov (2007). With symmetric information, characterization is often done using a "constant surplus condition" as in Gale (1987, Theorem 11) and Mortensen and Wright (2002).

18

5

Appendix

Proof of Lemma 1: Substituting 1p(v) r(v) = 1 if and only if v v into the steady-state conditions yields G (v) = M (v) and (1 G (v)) = M (1 (v)). Adding up, M = G (v) = + (1

G (v)) .

(8)

The shares are B

(v) =

G (v) = M + (G (v) G (v) = M

Using the above observations, setting D = 1 qS =

D D+

D

=

1

G (v)) =M B

if v if v

v; v:

(9)

(v) and using the de…nition of q S ,

1 G (v) G (v) + G (v) + (1 G (v))

(1

G (v))

=1

G (v) . QED

Proof of Proposition 1: Step 1. (The reservation price r (v) = v.) This is immediate: From (2), p (v) r (v) implies E [pjp r; v] = r (v). Hence, V (v) = q B (r; v) (v r (v)), and, after substituting r (v) = v (1 ) V , I get V (v) = q B (r; v) ((1 ) V (v)), which implies V (v) = 0 because (1 ) < 1. From V (v) = 0, r (v) = v follows. Step 2. (Necessity) By de…nition of v and , v = (1 ) q S (p ( )) E [pjp ( )]. From (2) and V (v) = 0, r (v) p (v) if and only if v v. Hence, Lemma 1 applies, and the trading probability is q S (p ( )) = 1 G (v). The distribution of valuations conditional on v v is given by g (v) = (1 G (v)). This follows when inspecting the density of as determined by the cuto¤ v, see (9). (Intuitively, buyers having types v v trade immediately; hence, the distribution of their types is equal to the distribution Rof their types in the in‡ow.) There1 fore, observing that p (v) = v if v v, E [pjp ( )] = v vg (v) = (1 G (v)) dv. Together, R1 in equilibrium, it must be that pro…ts are (p ( )) = (1 G (v)) v vg (v) = (1 G (v)) dv. Canceling (1 G (v)) and multiplying by (1 ) yields (3). Step 3. (Existence and Su¢ ciency) There exists a solution to (3) by the intermediate value theorem: theR two sides of the equation are continuous in v on [0; 1]. At the bound1 aries, 0 (1 ) 0 vg (v) dv and 1 0. The solution is unique because the right-hand side of (3) is strictly decreasing in v whereas the left-hand side is strictly increasing in itself. Any solution to (3) induces an equilibrium where the bargaining strategies p and r are determined by v as in the proposition and the stock M; is determined by v as in equations (8) and (9). QED Proof of Proposition 2: The cuto¤ v ( ) is strictly decreasing. R (By contradiction.) 1 Suppose v ( ) would be weakly increasing. Then, v ( ) = (1 ) v( ) vg (v) dv cannot hold for two di¤erent . The left-hand side would be weakly increasing in , while the right-hand side would be strictly decreasing in . When = 1, the right-hand side of (3) is zero, and so v (1) = 0. When n ! 0, lim v ( n ) exists because v ( n ) is aR decreasing sequence bounded from below by 0. Let 1 v = lim v ( n ), and note that v = v vg (v) dv must hold, and this condition has a unique 19

solution R 1 v 2 (0; 1). In particular, f0; 1g are R 1 not solutions because (3) fails; if v = 0, then 0 < 0 vg (v) dv, and if v = 1, then 1 > 1 vg (v) dv = 0. QED

Proof of Proposition 3: The main step is to derive the trading probability D (p) and its derivative @D(p) given some p, using the steady-state conditions and the reser@p vation price policy (1). Buyers’ payo¤s are V (v) = max f0; v pg. Therefore, r (v) = min fv; v (1 ) (v p)g. The reservation price is increasing in v. Hence, the set of types accepting a price p is the interval [v (p) ; 1], where v (p) is the inverse of r (v) 8 if p r (1) ; < 1 1 1 p (1 ) p if p p r (1) ; (10) v (p) = : p if p p:

I have set v (p) = 1 for convenience when the price exceeds the highest reservation price. From (9), the steady-state conditions imply that the distribution of types for given p is B

(v) =

G (p) =M + (G (v) G (v) =M

G (p)) =M

if v if v

p; p:

(11)

The probability of trading at a price p is equal to the probability of being matched with a buyer having type v v (p), so, D (p) = 1 (v (p)). Taking derivatives, 8 0 if p > r (1) ; @D (p) < g (v (p)) =M if p p < r (1) ; = : @p g (v (p)) =M if p p: The derivative @D(p) is continuous on [0; r (1)). At p = p, (8) and (9) imply that @p (1 (v (p))) = (1 G (p)) =M . Thus, the inverse elasticity of D at p = p is as claimed, D (p) p @D(p) @p

=

1

G (p) pg (p)

:

From the observation before it follows that the objective function D (p) (p (1 ) (p)) is continuously di¤erentiable on the interval [0; r (1)). The maximizer must obviously be contained in the interval ((1 ) (p) ; r (1)). Using previous observations, the …rst derivative of the objective function is ( g(v(p)) (p (1 ) ) + (1 G(v(p))) if v p; M M g(v(p)) (1 G(p))+G(p) G(v(p)) (p (1 ) ) + if v p. M M I show that monotonicity of virtual valuations implies that the objective function is quasiconcave on the relevant interval ((1 ) (p) ; r (1)). Speci…cally, the function is locally concave at critical points where the …rst derivative is zero, proven now. Suppose the derivative is zero for some p p, that is, (p

(1

) )=

(1

G (p)) + G (p) g (v (p))

20

G (v (p))

.

The second derivative has the sign of g 0 (v (p)) (p (1 ) ) 2g (v (p)). Substituting for (p (1 ) ) from above and rewriting shows that the function is locally concave g 0 (v (p)) ( (1

G (p)) + G (p)

G (v (p)))

2g 2 (v (p))

0.

If g 0 > 0 this is immediate. If g 0 < 0, then it follows from monotonicity of the virtual valuations which implies that g 0 (v) (1 G (v)) 2g 2 (v) 0 for all v. Similar reasoning establishes local concavity at critical points p > p. Thus, the …rst-order condition is necessary and su¢ cient for a maximum. Setting the derivative of the objective function equal to zero, imposing the equilibrium ) (p), condition p = p, and substituting for @D(p) @p , D (p), and (1 p (1

(1

) (1

G (p))) =

1

G (p) . g (p)

(12)

By the preceding discussion, p is an equilibrium price if and only if p solves this condition. A solution to (12) exists and is unique. Existence follows from the intermediate value theorem: both sides are continuous in p. At p = 0, the left-hand side is zero, while the right-hand side is positive. At p = 1 the right-hand side is zero while the left-hand side is positive. The solution is unique: The left-hand side is strictly increasing in p. The right-hand side is decreasing in p: its derivative has the sign of g 2 (1 G) g 0 , which is negative by monotonicity of the virtual valuations. Dividing (12) by p implies (5). QED Proof of Proposition 4: Given some , denote the unique solution to (12) by p ( ). Rewriting (12) further p 1+

(1

)

G (p)

=

1

G (p) : g (p)

(13)

Monotonicity of virtual valuations implies that the solution p ( ) is increasing in . When = 1, the equilibrium condition (5) reduces to the optimality condition of a monopolist who maximizes (1 G (p)) p; hence, p (1) = pm . Given a sequence of exit rates k ! 0, the price p ( k ) = pk converges to some limit p because p ( k ) is a decreasing sequence. I show that p = 0 by contradiction. When k becomes smaller, the inverse elasticity on the right-hand side of the Lerner formula (5) becomes zero, (pk g (pk ) = (1 G (pk ))) 1 ! 0, given pk ! p > 0. Consider the lefthand side. Because p > 0 implies 1 G (p ) < 1, the limit of the left-hand side is p p (1 G (p )) > 0, which is not equal to the limit of the right-hand side. Contradiction. De…nition of Equilibrium with Symmetric Information. I characterize the equilibrium conditions for the stock ; M and for the bargaining pro…le. The bargaining pro…le consists of the price o¤er and reservation price strategies of sellers, denoted pS and rS , and the price o¤er and reservation price strategies of buyers, denoted pB and rB . Fix some steady state ; M and some bargaining pro…le. The steady-state conditions require

21

equality of in- and out‡ows, Z G (v) = M

v

0

+

1pS (

h

(1

) 1pB (

) rB ( )

) rS

+ (1

+ (1

) 1pS (

and similarly for sellers, Z

1=M

(1

0

+

1h

1pS (

) 1p B (

) rB ( )

+ (1

Payo¤s are de…ned recursively, Z 1h = (1 ) 1p B ( 0

+

1pS (

)

h

rB (

)

) rS

) 1pB ( i d B )>r ( )

pB ( ) + (1

pS ( ) + (1

V (v) = (1 ) 1pB (v) rS h + 1pS (v) rB (v) v

+ (1

) 1pS (

) rS

) 1pB ( i d B )>r ( )

) 1pS (

(14)

)
( );

(15)

)
) 1pB (

( ).

)
i

)

d

B

( ) i v pB (v) + (1 ) 1pB (v)rB (v) V (v) . )>rB (

The bargaining pro…le shall be a perfect equilibrium in the bargaining game induced by the continuation payo¤s. Therefore, a seller accepts a price o¤er if it exceeds his continuation payo¤ (1 ) ; the seller’s own o¤er is either equal to or above the reservation price of the buyer, depending on whether the reservation price exceeds the seller’s continuation payo¤s. Similarly, a buyer accepts a price if the payo¤ from acceptance exceeds his continuation payo¤; his own o¤er is either equal to or below the reservation price of the seller: 8 < = rB (v) S rB (v) p : > rB (v)

if rB (v) > (1 if rB (v) = (1 if rB (v) < (1 rS = (1

)

) ) )

8 < = rS B rS ; p (v) : < rS rB (v) = v

and

if v if v if v

rS > (1 rS = (1 rS < (1

(1

)V

)V )V )V

(16)

(17)

A constellation ; M; pS (v) ; rS ,pB (v) ; rB (v) is a steady-state equilibrium in stationary strategies with symmetric information and intermediate bargaining power if the steadystate conditions (14) and (15) and the optimality conditions (16) and (17) hold. Proof of Proposition 5. Let v = (1 ) . Using the optimality conditions (16) and (17) to rewrite the expected payo¤s yields the value functions =

Z

1

(v

(1

) V (v)) d (v) + (1

) (1

(1

v

V (v) = (1

) (v

(1

) ) + (1

22

) V (v)

if v

v,

(v)))

and V (v) = 0 if v

v. For v

v, buyers’payo¤s can be rewritten as a function of

V (v) =

(1 (1

1

)

(v

)

(1

) ).

, (18)

Rewriting sellers’pro…ts and using (18) to substitute for V (v), =

(1

(v))

Z

1

v

(1

)

v

(1 1 (1

) )

(v

(1

) )

(1

d (v) . 1 (v)

)

The optimality conditions imply that 1pB (v) rS = 1pS (v) rB (v) = 1 if v > v = (1 ) and 1pB (v) rS = 1pB (v) rS = 0 otherwise. Rewriting the steady-state conditions shows that the stock is determined by the cuto¤ v as in (8) and (9). In particular, (1 (v)) = (1 G(v)) G(v)+(1 G(v)) , which implies

1 Using

d (v) (1 (v))

=

(1 (1

g(v) 1 G(v)

=

) )

for v

=

(1 G (v)) + G (v) (1 ) (1

)

.

v from (9) and using (19), I solve for

(1 G (v)) + G (v) (1 ) (1

)

Z

v

1

v

(19) ,

g (v) dv. 1 G (v)

Therefore, equilibrium is characterized by (6), using the de…nition of v, v = (1

) .

Existence and uniqueness of the solution v ( ; ) to the equilibrium characterization (6) follows as in the proof of Proposition 1. Existence follows from the intermediate value theorem, and uniqueness follows by observing that the right-hand side is decreasing in v while the left-hand side is trivially increasing. As before, the cuto¤ v ( ; ) de…nes an equilibrium ; M; pS (v) ; rS ,pB (v) ; rB (v): the stock is determined by the cuto¤ v via conditions (8) and (9); the cuto¤ de…nes continuation payo¤s via v = (1 ) and (18). Therefore, the cuto¤ determines a bargaining pro…le via the optimality conditions (16) and (17). Proof of Proposition 6. Continuity and Monotonicity of v ( ; ) in . The righthand side of equation (6) is increasing in and decreasing in v. Hence, monotonicity follows: if v were weakly decreasing in , the right-hand side of (6) would be strictly increasing in , leading to a contradiction. Continuity of the solution is immediate by inspection. Convergence. I show convergence to e¢ ciency when is …xed at some value < 1. Suppose the claims is not true. Then there is a sequence of exit rates f k g that converges to zero and a sequence of cuto¤s v ( k ; ) that converges to some limit, denoted v , where v > 0. With vk = v ( k ; ), condition (6) requires (using continuity of the right-hand side

23

of (6) in

and v when v > 0) v = lim v ( k ; )

Z 1 g (v) (1 G (vk )) k v dv + G (v ) (1 ) (1 ) (1 G (vk )) k k k vk Z 1 (1 G (v )) 0 g (v) = lim v dv = 0. 0 + G (v ) (1 ) v (1 G (v )) = lim (1

k)

Hence, v > 0 implies a contradiction; therefore, lim v ( k ; ) = 0 for all

< 1.

Monotonicity. I show that v ( ; ) is "hump shaped" in for …xed and that it has a unique maximum (which implies that the surplus is u-shaped as claimed). Inspection of the identity (6) shows that the implicitly de…ned function v ( ; ) is continuously di¤erentiable. Given any 2 (0; 1), if 2 (0; 1), the cuto¤ must be strictly positive by (6). Therefore, v ( ; ) must be decreasing somewhere, since at = 1, the cuto¤ v (1; ) = 0. As shown before, v ( ; ) ! 0 when ! 0. Hence, v ( ; ) must also be increasing somewhere. Thus, there exists an interior maximum. Let 0 ( ) be some value for that maximizes v ( ; ) for given . At 0 , it must be that @@ v ( ; ) j = 0 ( ) = 0. I characterize 0 ( ) and show that it is the unique maximizer. From the implicit function theorem, ! Z 1 d v( ; ) = (1 ) vg (v) dv @ d 1 + G (v ( ; )) (1 ) (1 ) v( ; ) =

v (1

(1

)

) 1 + G (1

v 1 + G (1

)

(1

(1

)

v

+

) (1

1 + G (1

)

) (1

G (1

)

(1

)

)

)

vgv 0

gv 0

1 2

with v 0 = @@ v ( ; ), dropping the arguments on G and g, and using the de…nition of v. At 0 ( ), v 0 = 0 implies 0=

v (1

)

v

+ 1 + G (1

)

(1

)

G (1

)

1 2

,

and rewriting shows that this requires (1

0

)G v

The right-hand side is increasing in

0.

=

0;

From G v )

1 24

0

1

0

. 1, it must that at

2

0

(1

2

0

;

,

(20) 0,

0(

which implies an upper bound on 0

). Solving the inequality,

1 p 1

( )

(1

The bound is decreasing in and equal to zero at implies that lim sup 0 ( ) 0:5: lim sup

0

!0

( )

lim

!0

1 p 1

(1

) . = 1. When

) =

1 2

(1

! 0, L’Hôpital’s rule )

0:5

+1

1

= 0:5.

Uniqueness of the maximizer 0 follows from a monotonicity argument. At 0 , the derivative of v ( ; ) is zero, and v ( ; ) must be (weakly) decreasing in to the right (otherwise, 0 would not be a maximizer). Then, the right-hand side of (20) is strictly increasing while the left-hand is weakly decreasing. Therefore, there cannot be any other instance of to the right of 0 where the derivative is zero; therefore, 0 is the unique maximizer. By the same argument, v ( ; ) is strictly decreasing in to the right of 0 and increasing to the left of 0 (there can be saddle points). Together with the earlier observations, this proves that v ( ; ) is hump shaped and has a unique maximum at 0 . The cuto¤ v ( ; ) is weakly increasing in to the left of 0 and strictly decreasing in to the right of 0 . Speed of Convergence. The following immediate claim is used for the asymmetric information case, too. R v( ; ) Claim 1. If lim !0 v( ; )G(v( ; )) = C < 1, then lim !0 1 0 vg (v) dv = 21 C. "(v) Let " (v) g (v) g (0) and let ( ) supv2[0;v( ; )] g(0) . By assumption, g (0) > 0. Continuity requires lim !0 ( ) = 0. Rewriting de…nitions, lim

v ( ; ) G (v ( ; ))

!0

= lim

!0

= lim

v( ; )

!0

; )

(g (0) + " (v)) dv

0

(v ( ; ))2 g (0) k

!0

Therefore, lim Hence, lim !0

R v(

(v( ; ))2 g(0)

!0

( ),

( ) = 0 implies v( ; )

!0

R v( 0

; )

" (v) dv

1

Z

0

lim

R v(

; )

0

R v(

(v ( ; ))2

v( ; )

vg (v) dv = lim

!0

= lim

g (0)

!0

R v(

(v ( ;

!0

25

0

v( ; )G(v( ; ))

; )

vdv

))2 21 g (0)

+ lim

!0

lim

(v ( ; ))2 g (0)

!0

; )

0

" (v) dv

" (v)

( )

!0

The previous chain of equalities implies lim Therefore,

!0

+ lim

C. By de…nition of

k

lim

lim

v( ; )

1 = C. 2

0

; )

0

( ) g (0) dv.

= 0.

= lim R v(

R v(

; )

!0

(v( ; ))2 g(0)

v" (v) dv

= C.

.

R v( ; ) The last equality follows from 0 v" (v) dv observations. This establishes the claim. 1,

Multiplying (6) by v( ; )+

v( ; )

R v(

; )

0

" (v) d and the previous

substituting v = v ( ; ), and collecting terms,

v ( ; ) G (v ( ; )) (1

) (1

)

= (1

)

Z

1

vg (v) dv.

v( ; )

Since v ( ; ) ! 0, lim

v ( ; ) G (v ( ; ))

!0

=

1

E [v] .

(21)

Using the de…nitions and Claim 1, the desired result follows: lim

W

W sym ( ; )

= lim

!0

!0

R v( 0

; )

vg (v) dv

= lim

!0

1 v ( ; ) G (v ( ; )) 1 = 2 21

E [v] .

De…nition of Equilibrium with Asymmetric Information. As before, when = 1, the sole di¤erence between the symmetric and asymmetric information regime is the restriction that the price o¤er strategy of sellers be constant. A bargaining pro…le consists of an o¤er strategy and a reservation price strategy for buyers and for sellers, pB (v) ; rB (v) ; p; rS . Given a stock M , , a bargaining pro…le de…nes payo¤s V B (v) and recursively as before, replacing the function pS (v) by the constant p. Continuation payo¤s de…ne optimal reservation price strategies as in (17)). A price o¤er p is optimal for sellers if it satis…es p 2 arg max D (p) (p (1 ) ); (22) p

with D (p) = satis…es (16).

R

v:r(v) p d

(v). A price o¤er strategy pB (v) is optimal for buyers if it

A constellation ; M; pB (v) ; rB (v) ; p; rS is a steady-state equilibrium in stationary strategies with asymmetric information and intermediate bargaining power if the steadystate conditions (14) and (15) hold with pS (v) = p and the optimality conditions (16), (17), and (22) hold. Proof of Proposition 7. When = 0, existence and characterization of equilibrium is immediate. = 1 was considered before. Here, I consider 2 (0; 1). Step 1. Characterization of Equilibrium Payo¤s and Stocks. Let ' be the cuto¤ type that just accepts p, de…ned implicitly by rB (') = p. Such a type must exist in equilibrium and it must be interior. In equilibrium, rS < p since rS = (1 ) ; and the B S pro…t p. From the optimality conditions, buyers o¤er p (v) = r if v > rS and B S p (v) < r if v < rS . Thus, types with valuations v > ' trade immediately, either at pS (if chosen as responder) or at rS = (1 ) . Intermediate types v 2 ((1 ) ; ') wait S until they are chosen to propose and trade at r . Types v < (1 ) never trade.

26

Using these observations, 8 > ) (1 ) ) < v ( p + (1 (1 ) B (1 ) ) V (v) = 1 (1 ) (v > : 0

if if if

v > '; (1 ) v < (1

v

';

) .

Note that 1 (1(1 )) is the probability that a buyer who o¤ers an acceptable price but who rejects all price o¤ers ends up trading. The pro…t of a seller is =

1

(1 (1

(1

(')) ('))) (1

)

p.

The fraction on the right hand side is equal to the probability that a seller ends up trading if he o¤ers a price that is acceptable to types above ' but rejects all price o¤ers made to him. Rewriting the steady-state conditions yields that the total mass is M=

G ((1

) )

+

G (')

the distribution of types is given by 8 G((1 ) ) G(') G((1 ) ) + ( +(1 ))M + > M < G((1 ) ) G((1 ) ) (v) = + G(v) M ( +(1 ))M > : G(v)

G ((1 + (1

G(v) G(') M

M

) ) )

+1

G (') ;

if

v > ';

if

(1

if

v < (1

)

v

(23)

';

(24)

) .

Intuitively, the shares in the stock are inversely related to the exit rates. Speci…cally, the probability to exit in any given period is equal to for types below (1 ) , equal to ( + (1 )) for intermediate types, and equal to 1 for types above '. Step 2. Necessity and Su¢ ciency of the First Order Condition. I show that p0 2 arg maxp D (p) (p (1 ) ) if and only if the …rst order condition holds. The structure of the argument is analogous to the previous case with = 1. First, I characterize the function D (p) and show that it is continuously di¤erentiable. Using the characterization of buyers’payo¤s, reservation prices are 8 ) (1 ) ) if v > '; > < v + (1 2 ) ( p + (1 v + (1 ) (1 ) B r (v) = (25) if (1 ) v '; 1 (1 ) > : v if v < (1 ) . Let v (p) denote the inverse of r (p), which is de…ned on 0; rB (1) ; its derivative is

v 0 (p) =

8 < :

1 1

1

(1

)

if if if

p < p rB (1) ; (1 ) < p < p; 0 p < (1 ) .

Given the reservation prices and given the stock characterized before, the sellers’ per-

27

period trading probability as a function of the price o¤er 8 1 G(v(p)) > M < ') G(v(p)) 1 G(') + G( D (p) = M M (1 (1 ) ) > : 1 G(') G(') G((1 ) ) G((1 ) ) G(v) + M (1 (1 ) ) + M M

is if if

p p rB (1). (1 ) p p;

if

p

(1

) .

1 , which is continuous as claimed. Using the previous The derivative is D0 (p) = g(v(p)) M observation, the derivative of the objective function D (p) (p (1 ) ) is

8 > < > :

g(v(p)) M g(v(p)) M g(v(p)) M

(p (p

(1 (1

) )+ ) )+

(p

(1

) )+

1 G(v(p)) M 1 G(') ') G(v(p)) + G( M M (1 (1 ) ) 1 G(') ) ) + G(M')(1 G((1 M (1 ) )

p p rB (1). (1 ) p p; +

G((1

) ) G(v(p)) M

p

(1

) .

Necessity of the …rst order condition is immediate from the observation that the derivative is continuous and that the derivative is positive at the lower boundary and negative at the upper boundary of the relevant interval [(1 ) ; r (1)]. For su¢ ciency, I show that the objective function is quasi-concave. Quasi-concavity is shown to follow from the assumption that virtual valuations are monotone, together with the observations on r0 . Suppose that the derivative of the objective function is zero for some p in the lower range, (1 ) p p, that is, at p, 1

(p

(1

After substituting for (p negative at p if g 0 (v (p)) ((1

) )= (1

G (')) (1

G (') G (v (p)) (1 G (')) + . g (v (p)) g (v (p)) (1 (1 ) )

) ) and simple manipulation, the second derivative is

(1

) ) + (G (')

2 (g (v (p)))2

G (v (p))))

0.

Since p p implies v (p) ' this is clearly true if g 0 > 0. Suppose g 0 < 0. Since v (p) it is su¢ cient to show that g 0 (v (p)) (1

G (v (p)))

2 (g (v (p)))2

',

0.

This, however, is immediate from monotonicity of virtual valuations. A similar observation applies if the derivative is zero for some p in the upper range, p p r (1). Thus, the objective function is indeed quasi-concave, as claimed. Step 3. Fixed Point Argument and Existence. De…ne a function from the convex

28

n and compact set (v; ) 2 [0; 1]2 jv v 0 = min 1;

(1

G (v)) ((1 g (v)

o

(1

)

to itself as follows:

)+

) + (1

)

;

(26)

1 G(v) G((1

0

) )

= 1

+

G(v) G((1 ) ) +1 +(1 )

1 G(v)

(1

G((1

) )

+

v + (1 )2 (1 (1 )+ )

G(v)

G(v) G((1 ) ) +1 +(1 )

G(v)

) (1

)

.

(27)

The function that maps (v; ) into (v 0 ; 0 ) is continuous and has a …xed point (v ; ) by Kakutani’s theorem. Inspection shows that v (1 ) implies that v < 1 and < 1. The de…nitions are such that the …xed point v and corresponds to an equilibrium. De…ne a bargaining pro…le by setting pB (v) (1 ) if v > (1 ) and pB (v) < S (1 ) otherwise, r (1 ) , v +

p and, with ' B

r (v)

v , 8 > < v + (1 v +

> :

(1

1

)2 (1 )+

) (1

)

(1

)( p )2 (1 (1 )

(1

(1

)

v

)

)

,

if

v > ';

if if

(1 ) v < (1

< v < '; ) .

Let M and be the steady-state stock determined by the bargaining pro…le as in (23) and (24). I now show that the candidate ; M ; p; rS ,pB (v) ; rB (v) de…ned by v and satis…es the equilibrium conditions. This is immediate for the stock. The reservation price of the buyers satis…es the equilibrium condition, see (25). For sellers, note that the de…nition (')) of p implies that rB (') = p. Therefore, pro…ts are given by p 1 (1 (1 (1 (')))(1 ) . After substituting the steady-state conditions and the de…nition of p, the pro…t is exactly equal to the right hand side of (27), that is, the candidate constellation de…ned by (v ; ) is consistent with sellers earning pro…ts of . In particular, rS = (1 ) is indeed the optimal reservation price of sellers. It remains to show that o¤ering p is optimal for sellers. For this, note that (26) is equivalent to the original …rst order condition: The …rst order g(') G(') condition for an optimum at p is M (p (1 ) )+ 1 M = 0. Using the de…nition of p, this condition can be rewritten as ! 1 ' + (1 )2 (1 ) 1 G (') (1 ) =0 (1 )+ g (') ,

' (1

(1 )+

)

=

(1

G (')) g (')

(28)

Solving this condition for v yields (26), that is, if (26) holds, then the …rst order condition holds, too. Thus, the constructed constellation satis…es all conditions to be a steady-state equilibrium. QED 29

Proof of Proposition 8. Here, I consider the case where 2 (0; 1). When = 0, the claim is immediate. When = 1, the …rst part of the claim has been shown in Proposition 4; the second part follows both from the explicit characterization in Proposition 3 as well as from the reasoning below. Expanding the …rst order condition (28) by using the steady-state condition to substitute for yields the following helpful identity ' + (1 )2 (1 (1 )+ =

1

G((1

1

) 1

) ) G((1

+

1 1 G(') ) )

+

G(') G((1 ) ) 1 +(1 ) 1 G(')

1 1 G(')

+

+1

G(') G((1 ) ) 1 +(1 ) 1 G(')

+1

G (') . g (')

(29)

Step 1. lim !0 inf ' ( ; ) > 0 and lim !0 sup ' ( ; ) < 1. By contradiction. If, for some subsequence of ! 0, ' ( ; ) ! 0, then ' ( ; ) ( ; ) implies lim !0 ( ; ) = 0, too. Therefore, along this subsequence, lim !0 LHS(28)= 1 0. However, ' ( ; ) ! 0 implies that lim !0 RHS(28)= g(0) > 0, a contradiction. Suppose for some subsequence of ! 0, ' ( ; ) ! 1. Then, along this subsequence, lim !0 RHS(28)= 0. However, theR LHS cannot converge to zero because the pro…t is 1 bounded away from one, ( ; ) 0 vdG (v) < 1. Contradiction. Step 2. lim !0 ( ; ) = 0 for all . By contradiction. Let k ! 0 be some sequence such that limk!1 ; k > 0 for k k k k k some . Let 1 ; and ' ' ; . Substituting into (29) yields the condition that for all k, the following expression is bounded

lim sup

k!1

k 'k

+ (1

k

1 )+

G(

1

k

(1

k k

)

k

1

k

+

k

)

1 1 G('k )

G(

k k

)

+

G('k ) G( k ) 1 k +(1 ) 1 G('k )

1 1 G('k )

+

+1

G('k ) G( k ) 1 k +(1 ) 1 G('k )

. +1

When lim k > 0, the limit of the …rst term is strictly positive. The limit of the denominator of the second term is strictly positive and bounded, since 1 G1 'k is bounded by Step ( ) G( k ) k 1. The numerator diverges to in…nity when lim > 0, since lim 1 k 1 G1 'k = 1. ( ) Contradiction. G(w) Step 3. Let w 2 (0; 1) be the unique solution to w = (1 ) 1 g(w) . Then lim ' ( ; ) = w. This is immediate from (28) and lim !0 ( ; ) = 0. Uniqueness of w follows from monotonicity of virtual valuations.

Step 4. lim

(1 G(w))2 . g(w) 1 G(w) !0 RHS(29)= g(w) . Given any

( ; )G((1

) ( ; ))

=

By Step 3, lim sequence k ! 0, the previous steps imply that the limit of the denominators of the …rst and second term are (1 ) and 1,

30

1 G(w) g(w)

respectively. Therefore, k 'k

lim

+

k

k

1 (1

k!1

(1

2 (0; 1) must be equal to k

G

)1

k

)

G 'k G 1 + k k G (' ) + (1

1

! 1 +1 , G ('k )

k

) 1

k

and v k are de…ned as before. Since the …rst term converges to zero, the second G( k ) term must diverge to in…nity. Thus, ! 1. The in…nite term dominates the second k term. One can therefore simplify and rewrite further where

k 'k

lim

+

k

'k k!1 (1

k

(1

k!1

)

, lim

k

as claimed (recall

k

1

1 1 G (w) = 1 G (w) g (w) k (1 G (w))2 k G + k 1 = k g (w) k G (1 G (w))2 , = , lim k k k!1 g (w) k

k

G k

)

k

1

k

1G

k

;

(30)

).

Now, the proposition follows. Rewriting the de…nition of the surplus, W asym ( ; )

W

=

1

Z

Z

) ( ; )

(1

vdG (v) +

0

(1

)+

'( ; )

vdG (v) .

(1

) ( ; )

Claim 1 from the proof of Proposition 6 and Step 5 imply that lim

!0

1

Z

(1

) ( ; )

vdG (v) =

0

1 (1 G (w))2 = 2 g (w) 1

G(w) using that w = (1 ) 1 g(w) . From Step 3, lim Rw to (1 ) 0 vdG (v). Thus,

lim

W

W asym ( ; )

!0

=

1

1 w (1 2

!0 (1

)+

G (w)) +

(1

1 w (1 2 R '( (1

)

G (w)) ,

; ) ) ( ; ) vdG (v)

G (w) E [vjv

is equal

w] ,

as claimed. QED Proof of Corollary 2. When = 1, the welfare comparison asym follows from Corollary 1. ( ; ) W sym ( ; ) Let 2 (0; 1). It follows from Propositions 6 and 8 that lim W > 0 i¤ 1 1 E [v] > w (1 G (w)) + (1 G (w)) E [vjv w]; reordering terms and observing that 2 R2 w R1 G (w)) + w 1 G (v) dv, the inequality holds if 0 vg (v) = w (1 Z

w

(1

G (v)) dv

w (1

G (w)) <

Z

1

(1

G (v)) dv:

w

0

The rhs is decreasing in w and the lhs is increasing in w. By monotonicity of virtual valuations, w is a strictly decreasing function of and when = 0, w = pm . Hence, if ( ) 31

holds, then the above equation holds for all . If ( ) does not hold, then by continuity of both sides of the inequality there is a ^ 2 (0; 1) such that the equation holds with equality; the corollary now follows from the monotonicity observations before. QED

References [1] Atakan, Alp (2007): "E¢ cient Dynamic Matching with Costly Search," Mimeo. [2] Chatterjee, Kalan, and Hamid Sabourian (2000): "Complexity and Multi-Person Bargaining," Econometrica, 1491-1509. [3] De Fraja, Gianni; Sakovics, Jozsef (2001): "Walras Retrouve: Decentralized Trading Mechanisms and the Competitive Price," Journal of Political Economy, 842-863. [4] Diamond, Peter A. (1971): "A Model of Price Adjustment," Journal of Economic Theory, 158-68. [5] Felli, Leonardo; Roberts, Kevin (2002): "Does Competition Solve the Hold-up Problem?" CEPR Discussion Papers 3535. [6] Gale, Douglas (1987): "Limit Theorems for Markets with Sequential Bargaining," Journal of Economic Theory, 43, 20-54. [7] Gale, Douglas (2000): "Strategic Foundations of General Equilibrium: Dynamic Matching and Bargaining Games," Cambridge University Press. [8] Gale, Douglas; Sabourian, Hamid (2005): "Complexity and Competition," Econometrica, Vol. 73, 739-769. [9] Gale, Douglas; Sabourian, Hamid (2006): "Markov Equilibria in Dynamic Matching and Bargaining Games,” Games and Economic Behavior, 336-352. [10] Inderst, Roman (2001): "Screening in a Matching Market," Review of Economic Studies, Vol. 68, 849-68. [11] Inderst, Roman (2004): "Matching Markets with Adverse Selection," Journal of Economic Theory, 145-166. [12] Lauermann, Stephan (2008a): "Price Setting in a Decentralized Market and the Competitive Outcome," MPI Collective Goods Preprint, No. 6, Mimeo, Bonn. [13] Lauermann, Stephan (2008b): "Private Information in Bilateral Trade and in Markets," Mimeo, University of Michigan. [14] Lauermann, Stephan (2009): "Dynamic Matching and Bargaining Games: A General Approach," Mimeo, University of Michigan. [15] McAfee, Preston (1993): Econometrica,1281-1312.

"Mechanism

32

Design

by

Competing

Sellers,"

[16] Moreno, Diego and John Wooders (2002): "Prices, Delay, and the Dynamics of Trade," Journal of Economic Theory, 304-339. [17] Mortensen, Dale and Randall Wright (2002): "Competitive Pricing and E¢ ciency In Search Equilibrium," International Economic Review, 1–20. [18] Nöldeke, Tröger (2009): "Matching Heterogeneous Agents with a Linear Search Technology," Mimeo, University of Basel. [19] Rubinstein, Ariel; Wolinsky, Asher (1990): "Decentralized Trading, Strategic Behavior and the Walrasian Outcome," Review of Economic Studies, Vol. 57, 63-78. [20] Sabourian, Hamid (2004): "Bargaining and Markets: Complexity and the Competitive Outcome," Journal of Economic Theory, 189-228. [21] Satterthwaite, Mark; Shneyerov, Artyom (2007): "Dynamic Matching, Two-Sided Incomplete Information and Participation Costs: Existence and Convergence to Perfect Competition," Econometrica, Vol 75, 155 - 200. [22] Satterthwaite, Mark; Shneyerov, Artyom (2008): "Convergence of a Dynamic Matching and Bargaining Market with Two-sided Incomplete Information to Perfect Competition," Games and Economic Behavior, Vol 63, 435-467. [23] Serrano, Roberto (2002): "Decentralized Information and the Walrasian Outcome: A Pairwise Meetings Markets with Private Information," Journal of Mathematical Economics, Vol 38, 65-89. [24] Shneyerov, Artyom; Wong, Adam (2010a): "Bilateral matching and bargaining with private information," Games and Economic Behavior, 748-762. [25] Shneyerov, Artyom; Wong, Adam (2010b): "The Rate of Convergence to Perfect Competition of Matching and Bargaining Mechanisms," Journal of Economic Theory, 1164–1187. [26] Shneyerov, Artyom; Wong, Adam (2011):"The Role of Private Information in Dynamic Matching and Bargaining: Can It Be Good For E¢ ciency?" Mimeo. [27] Varian, Hal (1996): "Economic Aspects of Personal Privacy," U.S. Dept. of Commerce, Privacy and Self-Regulation in the Information Age, 1996.

33