Dynamic markets for lemons: Performance - Theoretical Economics

Dynamic markets for lemons: Performance - Theoretical Economics

Dynamic Markets for Lemons: Performance, Liquidity, and Policy Intervention Diego Morenoy John Woodersz March 2015 Abstract We study non-stationary...

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Dynamic Markets for Lemons: Performance, Liquidity, and Policy Intervention Diego Morenoy

John Woodersz

March 2015

Abstract We study non-stationary dynamic decentralized markets with adverse selection in which trade is bilateral and prices are determined by bargaining. Examples include labor markets, housing markets, and markets for …nancial assets. We characterize equilibrium, and identify the dynamics of transaction prices, trading patterns, and the average quality in the market. When the horizon is …nite, the surplus in the unique equilibrium exceeds the competitive surplus; as traders become perfectly patient the market becomes completely illiquid at all but the …rst and last dates, but the surplus remains above the competitive surplus. When the horizon is in…nite, the surplus realized equals the static competitive surplus. We study policies aimed at improving market performance, and show that subsidies to low quality or to trades at a low price, taxes on high quality, restrictions on trading opportunities, or government purchases may raise the surplus. In contrast, interventions like the Public-Private Investment Program for Legacy Assets reduce the surplus when traders are patient. We are grateful to George Mailath and three anonymous referees for useful comments. We gratefully acknowledge …nancial support from the Spanish Ministry of Science and Innovation, grant ECO2011-29762. y

Departamento de Economía, Universidad Carlos III de Madrid, [email protected]

z

Department of Economics, University of Technology Sydney, [email protected]

1

1

Introduction

We study the performance of decentralized markets for lemons in which trade is bilateral and time consuming, and buyers and sellers bargain over prices. These features are common in markets for real goods and …nancial assets. We characterize the unique decentralized market equilibrium, identify the dynamics of transaction prices, trading patterns, and the market composition (i.e., the fractions of units of the di¤erent qualities in the market), and study its asymptotic properties as traders become perfectly patient. Using our characterization of market equilibrium, we identify policy interventions that are welfare improving. We consider markets in which sellers are privately informed about the quality of the good they hold, which may be high or low, and buyers are homogeneous and value each quality more highly than sellers. Since we are interested in understanding dynamic trading when the lemons problem is severe, we assume that the expected value to buyers of a random unit is below the cost of a high quality unit.1 The market operates over a number of consecutive dates. All buyers and sellers are present at the market open, and there is no further entry. At each date a fraction of the buyers and sellers remaining in the market are randomly paired. In every pair, the buyer makes a take-it-or-leave-it price o¤er. If the seller accepts, then the agents trade at that price and exit the market. If the seller rejects the o¤er, then the agents split and both remain in the market at the next date. There are trading frictions since meeting a partner is time-consuming and traders discount future gains. In this market, equilibrium dynamics are non-stationary and involve a delicate balance: At each date, buyers’price o¤ers must be optimal given the sellers’reservation prices, the market composition, and the buyers’ payo¤ to remaining in the market. While the market composition is determined by past price o¤ers, the sellers’ reservation prices are determined by future price o¤ers. Thus, a market equilibrium cannot be computed recursively. We begin by studying the equilibria of decentralized markets that open over a 1

Under this assumption, in the unique static competitive equilibrium is ine¢ cient as only low

quality trades, and the entire surplus is captured by low quality sellers. We take the payo¤s and surplus at this static competitive equilibrium as the competitive benchmark. (We study dynamic competitive equilibrium in Appendix B.)

1

…nite horizon. Perishable goods such as fresh fruit or event tickets, as well as …nancial assets such as (put or call) options or thirty-year bonds are noteworthy examples. We show that if frictions are not large, then equilibrium is unique, and we calculate it explicitly. The key features of equilibrium dynamics are as follows: at the …rst date, both a low price (accepted only by low quality sellers) and negligible prices (rejected by both types of sellers) are o¤ered; at the last date, both a high price (accepted by both types of sellers) and a low price are o¤ered; and at all the intervening dates, all three types of prices –high, low and negligible –are o¤ered. Interestingly, as the traders’discount factor approaches one, there is trade only at the …rst and last two dates, and the market is completely illiquid at all intervening dates. In contrast to the competitive equilibrium, low quality trades with delay and high quality trades. The surplus realized in the decentralized market equilibrium exceeds the surplus realized in the competitive equilibrium: as we show, the gain realized from trading high quality units more than o¤sets the loss resulting from trading low quality units with delay. The surplus realized increases as frictions decrease, and thus a decentralized market open over an …nite horizon yields more than the competitive surplus (and traders’payo¤s are not competitive) even in the limit as frictions vanish. Holding market frictions …xed, the surplus decreases as the horizon becomes larger. As the horizon approaches in…nity, the trading dynamics become simple: at the …rst date buyers make low and negligible price o¤ers (hence only some low quality sellers trade), and at every date thereafter buyers make only high and negligible price o¤ers in proportions that do not change over time. In contrast to prior results in the literature, in this limiting equilibrium each trader obtains his competitive payo¤, and the competitive surplus is realized even when frictions are signi…cant. Moreover, all units trade eventually, and therefore the surplus lost due to trading low quality with delay exactly equals the surplus realized from trading high quality. Our characterization of decentralized market equilibrium yields insights into the e¤ectiveness of policies designed to increase market e¢ ciency and market liquidity. We take the liquidity of a good to be the ease with which it is sold, i.e., the equilibrium probability it trades. In markets that open over a …nite horizon, the liquidity of high quality decreases as traders become more patient and, somewhat counter-intuitively, as the probability of meeting a partner increases. Indeed, as the discount factor

2

approaches one, trade freezes at all but the …rst and the last two dates. In markets that open over an in…nite horizon, the liquidity of each quality decreases as traders become more patient, and is una¤ected by the probability of meeting a partner. We examine the impact on the market equilibrium of a variety of policies. Taxes and subsidies conditional on the quality of the good may alleviate or aggravate the adverse selection problem. When the horizon is …nite, providing a subsidy to buyers or sellers of low quality raises the (net) surplus, although a subsidy to buyers has a greater impact. In contrast, a subsidy to either buyers or sellers of high quality tends to reduce the net surplus – it does so unambiguously when traders are su¢ ciently patient. Regarding liquidity, a subsidy to buyers or sellers of low quality increases the liquidity of high quality, whereas a subsidy to buyers of high quality has the opposite e¤ect. Remarkably, when the horizon is in…nite, a tax on high quality raises revenue without a¤ecting either payo¤s or surplus, and hence increases the net surplus. We also study subsidies conditional on the price at which the good trades. We show that a subsidy conditional on trading at a low price increases the traders’payo¤s as well as the net surplus. When the horizon is in…nite the subsidy increases the liquidity of both qualities after the …rst date, as well as the net surplus. A subsidy conditional on trading at the high price increases (decreases) the payo¤s of buyers (low quality sellers). Interestingly, the liquidity of high quality decreases. When the horizon is in…nite, a subsidy is purely wasteful, whereas a tax raises revenue without a¤ecting payo¤s, thus raising the net surplus. In our setting, a Public-Private Investment Program (PPIP) such as the one implemented for legacy assets is e¤ectively a subsidy to buyers who purchase a low quality unit at the high price. We show that a PPIP has e¤ects analogous to subsidizing buyers of high quality: it increases the payo¤ of buyers and the surplus, decreases the payo¤ of low quality sellers and the liquidity of high quality, and as approaches one reduces the net surplus. We study the e¤ect of closing the market for some period of time. Such policies have been studied in the literature –e.g., Fuchs and Skrzypacz (2013) study it in a dynamic competitive setting. Our characterization of the market equilibrium shows that reducing the horizon over which the market opens (so long as the market remains open for at least two dates) increases payo¤s and surplus. We show that if the horizon

3

is not too long relative to the traders’discount factor, then closing the market at all dates except the …rst and the last has no e¤ect on payo¤s and surplus. If the horizon is long, however, by closing the market for some period of time separating market equilibria emerge in which the surplus is larger than when the market is open at all dates. Finally, we show that government purchases increase the payo¤ of low quality sellers and decrease the payo¤ of buyers; surplus increases provided the government values low quality nearly as highly as buyers, but decreases otherwise. Related Literature The recent …nancial crisis has stimulated interest in understanding the e¤ects of adverse selection in decentralized markets. Moreno and Wooders (2010) studies markets with stationary entry and shows that payo¤s are competitive as frictions vanish. In their setting, and in the present paper, traders observe only their own personal histories. Kim (2011) studies a continuous time version of the model of Moreno and Wooders (2010), and shows that if frictions are small and buyers observe the amount of time that sellers have been in the market, then market e¢ ciency improves, whereas if buyers observe the number of prior o¤ers sellers have rejected, then e¢ ciency is reduced. Thus, Kim (2011)’s results reveal that increased transparency is not necessarily e¢ ciency enhancing, and call for caution when regulating information disclosure. Bilancini and Boncinelli (2011) study a market for lemons with …nitely many buyers and sellers, and show that if the number of sellers in the market is public information, then in equilibrium all units trade in …nite time. For markets with one-time entry, the focus of the present paper, Blouin (2003) studies a market open over an in…nite horizon in which only one of three exogenously given prices may emerge from bargaining. Blouin (2003) shows that equilibrium payo¤s are not competitive even as frictions vanish. Although we address a broader set of questions, on this issue we …nd that payo¤s are competitive even when frictions are non-negligible. Camargo and Lester (2014) studies a model in which agents’discount factors are randomly drawn at each date from a distribution whose support is bounded away from one, and buyers may make only one of two exogenously given price o¤ers. It shows that in every equilibrium both qualities trade in …nite time. Moreover, liquidity, 4

i.e., the fraction of buyers o¤ering the high price, increases with the fraction of high quality sellers initially in the market. In contrast, in our model the unique equilibrium exhibits neither of these features: a positive measure of high quality remains in the market at all times, and marginal changes in the fraction of high quality only a¤ects the liquidity of low quality at date 1. Camargo and Lester (2014) also provides a numerical example demonstrating that a PPIP subsidy for has an ambiguous impact on liquidity as measured by the minimum time at which the market clears (taken over the set of all equilibria). We show that in our setting this policy decreases the liquidity of high quality, and we are able to evaluate its welfare e¤ects. In contrast to Blouin (2003) and Camargo and Lester (2014) our model imposes no restriction on admissible price o¤ers. Moreover, equilibrium is unique and is characterized in closed-form, which allows for a direct comparative static analysis of the e¤ect of changes in the parameters values on payo¤s, social surplus, and liquidity. The …rst paper to consider a matching model with adverse selection is Williamson and Wright (1994), who show that money can increase welfare. Inderst and Muller (2002) show that the lemons problem may be mitigated if sellers can sort themselves into di¤erent submarkets. Inderst (2005) studies a model where agents bargain over contracts, and shows that separating contracts always emerge in equilibrium. Cho and Matsui (2011) study long term relationships in markets with adverse selection and show that unemployment and vacancy do not vanish even as search frictions vanish. In their model, agents respond strategically to price proposals that are drawn from a uniform distribution. Lauermann and Wolinsky (2011) explore the role of trading rules in a search model with adverse selection, and show that information is aggregated more e¤ectively in auctions than under sequential search by an informed buyer. Our work also relates to a literature that examines the mini-micro foundations of competitive equilibrium. This literature has established that decentralized trade of homogeneous goods tends to yield competitive outcomes when trading frictions vanish. See, for example, Gale (1987, 1996) or Binmore and Herrero (1988) when bargaining is under complete information, and Moreno and Wooders (2002) and Serrano (2002) when bargaining is under incomplete information. There is also a growing literature studying dynamic competitive (centralized) mar-

5

kets with adverse selection. Janssen and Roy (2002) study a market that operates in discrete time and in which there is a continuum of qualities, and show that competitive equilibria may involve intermediate dates without trade before the market clears in …nite time. Fuchs and Skrzypacz (2013) study a market that operates in continuous time, and show that interrupting trade always raises surplus, while infrequent trade generates more surplus under some conditions. Philippon and Skreta (2012) and Tirole (2012) examine optimal government interventions in asset markets. In Appendix B we study the properties of dynamic competitive equilibria in our setting, compare the performance of centralized and decentralized markets, and discuss the di¤erential e¤ects of policy interventions.

2

A Decentralized Market for Lemons

Consider a market for an indivisible commodity whose quality can be either high (H) or low (L). There is a positive measure of buyers and sellers. The measure of sellers with a unit of quality

2 fH; Lg is m > 0. For simplicity, we assume that the

measure of buyers (mB ) is equal to the measure of sellers, i.e., mB = mH + mL .2

Each buyer wants to purchase a single unit of the good. Each seller owns a single unit of the good. A seller knows the quality of his good, but quality is unobservable to buyers prior to purchase. Preferences are characterized by values and costs: the value to a buyer of a unit of high (low) quality is uH (uL ); the cost to a seller of a unit of high (low) quality is cH (cL ). Thus, if a buyer and a seller trade at price p; the buyer obtains a utility of c, where u = uH and c = cH if the unit

p and the seller obtains a utility of p

u

traded is of high quality, and u = uL and c = cL if it is of low quality. A buyer or 2

This assumption, which is standard in the literature, e.g., it is made in all the related papers

discussed in the Introduction, simpli…es the analysis. With unequal measures the matching probability is endogenous and varies over time. We discuss this issue in Section 4, in connection to the impact of government asset purchases. Also, with this assumption …rst best e¢ ciency is achieved when all units trade, and the competitive equilibrium is ine¢ cient when adverse selection is severe, With unequal measures the characterization of …rst best e¢ ciency depends on the relative gains to trade of high and low quality, uH

cH and uL

cL , and the competitive equilibrium may be e¢ cient

even when adverse selection is severe.

6

seller who does not trade obtains a utility of zero. We assume that both buyers and sellers value high quality more than low quality (i.e., uH > uL and cH > cL ), and that buyers value each quality more highly than sellers (i.e., uH > cH and uL > cL ). Also we restrict attention to markets in which the lemons problem is severe; that is, we assume that the fraction of sellers of -quality in the market, denoted by q :=

m ; + mL

mH

is such that the expected value to a buyer of a randomly selected unit of the good, given by u(q H ) := q H uH + (1

q H )uL ,

is below the cost of high quality, cH . Equivalently, we may state this assumption as q H < q :=

cH uH

uL : uL

Note that q H < q implies cH > uL . Therefore, we assume throughout that uH > cH > uL > cL and q H < q. Under these parameter restrictions only low quality trades in the unique static competitive equilibrium, even though there are gains to trade for both qualities. For future reference, we describe this equilibrium in Remark 1 below. Remark 1. The market has a unique static competitive equilibrium. In equilibrium all low quality units trade at the price uL , and no high quality unit trades. Thus, the surplus, given by S = mL (uL

cL );

(1)

is captured by low quality sellers. In our model of decentralized trade, the market is open for T consecutive dates. All traders are present at the market open, and there is no further entry. Traders discount utility at a common rate

2 (0; 1], i.e., if at date t a unit of quality

at price p, then the buyer obtains a utility of a utility of

t 1

(p

t 1

(u

trades

p) and the seller obtains

c ). At each date every buyer (seller) in the market meets a

randomly selected seller (buyer) with probability

2 (0; 1]. In each pair, the buyer

o¤ers a price at which to trade. If the o¤er is accepted by the seller, then the agents 7

trade and both leave the market. If the o¤er is rejected by the seller, then the agents split and both remain in the market at the next date. A trader who is unmatched at the current date also remains in the market at the next date. An agent observes only the outcomes of his own matches. In this market, the behavior of buyers at each date t may be described by a c.d.f. t

with support in R+ specifying a probability distribution over price o¤ers. Likewise,

the behavior of sellers of each quality is described by a probability distribution with support on R+ specifying their reservation prices. Given a sequence

= ( 1; : : : ;

T)

describing buyers’ price o¤ers, the maximum expected utility of a seller of quality 2 fH; Lg at date t 2 f1; :::; T g is de…ned recursively as Z 1 Z 1 (p c ) d t (p) + 1 d t (p) Vt = max x2R+

x

Vt+1 ;

x

where VT +1 = 0. In this expression, the payo¤ to a seller of quality a price o¤er p is p

who receives

c if p is at least his reservation price x, and it is Vt+1 ; his

continuation utility, otherwise. Since all sellers of quality

have the same maximum

expected utility, then their equilibrium reservation prices are identical. Therefore we restrict attention to strategy distributions in which all sellers of quality use the same sequence of reservation prices r = (r1 ; :::; rT ) 2 RT+ .

2 fH; Lg

Let ( ; rH ; rL ) be a strategy distribution. For t 2 f1; : : : ; T g; the probability that

a matched seller of quality

2 fH; Lg trades, denoted by Z 1 d t (p): t =

t,

is (2)

rt

The stock of sellers of quality

in the market at date t + 1; denoted by mt+1 , is mt+1 = (1

t ) mt ;

where m1 = m : The fraction of sellers of high quality in the market at date t, denoted by qtH , is

mH t H mt + mLt L H 3 if mH t + mt > 0; and qt 2 [0; 1] is arbitrary otherwise. The fraction of sellers of low qtH =

quality in the market at date t, denoted by qtL , is qtL = 1 3

qtH :

Evaluating payo¤s requires specifying a value for qtH for all t. Lemma 2, part 1, implies that

H H L mH t > 0 for all t; and thus how qt is speci…ed when mt + mt = 0 does not a¤ect equilibrium.

8

The maximum expected utility of a buyer at date t 2 f1; :::; T g is de…ned recursively as VtB = max x2R+

8 < :

X

qt I(x; rt )(u

2fH;Lg

0

x) + @1

X

2fH;Lg

1

B qt I(x; rt )A Vt+1

where VTB+1 = 0. Here I(x; y) is the indicator function whose value is 1 if x

9 = ;

;

y; and

0 otherwise. In this expression, the payo¤ to a buyer who o¤ers the price x is u

x

when matched to a -quality seller who accepts the o¤er (i.e., when I(x; rt ) = 1), B , her continuation utility, otherwise. and it is Vt+1

De…nition. A strategy distribution ( ; rH ; rL ) is a decentralized market equilibrium (DME) if for each t 2 f1; : : : ; T g: rt for

(DM E: )

c = Vt+1

2 fH; Lg; and for almost all p in the support of t 0 1 X X B qt I(p; rt )(u p) + @1 qt I(p; rt )A Vt+1 = VtB : 2fH;Lg

(DM E:B)

2fH;Lg

Condition DM E: ensures that each type

seller is indi¤erent between accepting

or rejecting an o¤er of his reservation price. Condition DM E:B ensures that price o¤ers that are made with positive probability are optimal. The surplus realized in a decentralized market equilibrium can be calculated as S DM E = mB V1B + mH V1H + mL V1L .

(3)

Another feature of the market equilibrium worth identifying is the liquidity of each good, i.e., how easily each good can be bought or sold. In our setting, we de…ne the liquidities of high and low quality at each date t to be the equilibrium probabilities that these goods trade, which are given by

3

H t

and (

H t

+

L t ),

respectively.

Decentralized Market Equilibrium

Proposition 1 establishes basic properties of decentralized market equilibria. 9

Proposition 1. Assume that T < 1 and

< 1, and let t 2 f1; :::; T g: In a DME : qtH .

H (P1.1) rtH = cH > rtL , VtH = 0, and qt+1

(P1.2) Only the high price pt = rtH , or the low price pt = rtL ; or negligible prices pt < rtL may be o¤ered with positive probability. The intuition for these results is straightforward. Since the payo¤ of a seller who does not trade at date T is zero, sellers’ reservation prices at date T are equal to their costs, i.e., rT = c . Thus, price o¤ers above cH are suboptimal at date T , and are made with probability zero. Therefore the expected utility of high quality sellers at date T is zero, i.e., VTH = 0; and hence rTH

1

= cH : Also, since

is costly, low quality sellers accept price o¤ers below cH ; i.e., rTL

< 1, i.e., delay

1

< cH . A simple

induction argument shows that rtH = cH > rtL for all t. Obviously, prices above rtH , which are accepted by both types of sellers, or prices in the interval (rtL ; rtH ), which are accepted only by low quality sellers, are suboptimal, and are therefore made with probability zero. Moreover, since rtH > rtL then the proportion of high quality sellers in the market (weakly) increases over time (i.e., qtH ) as low quality sellers accept o¤ers of both rtH and rtL , and therefore exit

H qt+1

the market at least as fast as high quality sellers, who only accept o¤ers of rtH . In equilibrium, at each date a buyer may o¤er a high price p = rtH , which is accepted by both types of sellers, or a low price p = rtL , which is accepted by low quality sellers and rejected by high quality sellers, or a negligible price p < rtL ; which is rejected by both types of sellers. For

2 fH; Lg denote by

t

the probability of a

price o¤er equal to rt : Since prices greater than rtH are o¤ered with probability zero, the probability of a high price o¤er is

H t

=

H t .

(Recall that

t

is the probability

that a matched -quality seller trades at date t –see equation (2).) And since prices in the interval (rtL ; rtH ) are o¤ered with probability zero, then the probability of a low price o¤er is 1

(

H t

+

L t )

=1

L t

=

L t

H t :

Thus, the probability of a negligible price o¤er is

L t .

Proposition 1 thus allows a simpler description of a DME. Henceforth we describe a DME by a collection (

H

;

L

; rL ), where

= ( 1; : : : ;

T)

for

2 fH; Lg, and thus

ignore the distribution of negligible price o¤ers, which is inconsequential. Also we omit the reservation price of high quality sellers which is rtH = cH for all t by P 1:1. Proposition 2 establishes additional properties of DME. 10

Proposition 2. Assume that T < 1 and

< 1. Then in a DME:

(P2.1) At every date t 2 f1; : : : ; T g either high or low prices are o¤ered with positive probability, i.e.,

H t

+

L t

> 0.

(P2.2) At date 1 high prices are o¤ered with probability zero, i.e.,

H 1

= 0:

(P2.3) At date T negligible prices are o¤ered with probability zero, i.e., 1

H T

L T

= 0.

The intuition for P 2:2 is clear: Since at date 1 the expected utility of a random unit is less than cH by assumption, then high price o¤ers are suboptimal, i.e.,

H 1

= 0.

The intuition for P 2:3 is also simple: At date T the sellers’reservation prices are equal to their costs. Thus, buyers obtain a positive payo¤ by o¤ering either the low price rTL = cL (when qTH < 1), or the high price rTH = cH (when qTH = 1). Since a buyer who does not trade obtains zero, then negligible price o¤ers are suboptimal, i.e., H T

+

L T

= 1. The intuition for P 2:1 is as follows: Suppose to the contrary that

all buyers make negligible o¤ers at date t, i.e.,

H t

=

L t

= 0: Let t0 be the …rst

date following t where a buyer makes a non-negligible price o¤er. Since there is no trade between t and t0 , then the distribution of qualities is the same at t and t0 ; i.e., qtH = qtH0 . Thus, an impatient buyer is better o¤ by o¤ering at date t the price she o¤ers at t0 ; which implies that negligible prices are suboptimal at t; i.e., Hence

H t

> 0 and/or

L t

H t

+

L t

= 1:

> 0:

In a market that opens for a single date, i.e., T = 1, the sellers’reservation prices are their costs. The fraction of high quality sellers q^ :=

cH uH

cL ; cL

makes a buyer indi¤erent between an o¤er of cH and an o¤er of cL . It is easy to see that q < q^: Since q H < q by assumption, then q H < q^. Thus, if T = 1 only low price o¤ers are made (i.e.,

H 1

= 0 and

L 1

= 1) and only low quality trades, as implied by

P 2:1 and P 2:2. Remark 2 states these results. Remark 2. Assume that T = 1 and

< 1. Then the unique DME is (

H 1 ;

L L 1 ; r1 )

=

(0; 1; cL ). In equilibrium some low quality units trade at the price cL , and no high quality unit trades. Thus, the surplus realized, which is buyers. 11

mL (uL

cL ), is captured by

Proposition 3 below establishes that when frictions are not large a decentralized market that opens over a …nite horizon T > 1 has a unique DME. We say that frictions are not large when

and

are su¢ ciently near one that the following inequalities

hold: < min and (1

(1 +

uL ) (cH

(1 = )q H = )q H + (1 )(1

where

Inequality F:1 requires

cH ) (1

and

cL )

qH )

(F:1)

;1 ;

(F:2)

> q^;

uL cL := H : c cL be su¢ ciently close to one that a low quality seller

prefers to wait one period and trade with probability

at the price cH rather than

trading immediately at the price uL . The left hand side of F:1, =

, is an upper

bound of the probability that a high price is o¤ered at any date as we show in Lemma 2, part 6, in the Online Appendix. It is easy to see that F:1 holds for

and

near

one. Inequality F:2 requires that if all matched low quality sellers trade and at most a fraction =

of matched high quality sellers trade, then the fraction of high quality

sellers in the market at the next date is above q^: In Lemma 2, part 2, in the Online Appendix we show that this inequality implies that the low price is never o¤ered with probability one. Obviously, this inequality holds for

near one.

Write q^)(uL

:= (1 and for t 2 f1; : : : ; T g let t

Clearly

t

is increasing in

T t

=

cL );

:

and ; and approaches

as

approaches one; and

t

is decreasing in T; and approaches zero as T approaches in…nity. Proposition 3 establishes that when frictions are not large a market that opens over a …nite horizon has a unique DME, and provides a complete characterization of this equilibrium. Proposition 3. Assume that 1 < T < 1,

< 1; and inequalities F:1 and F:2 hold

(i.e., frictions are not large). Then the unique DME is given by: 12

(P3.1) High Price O¤ers:

H 1

= 0, H t

=

for all 1 < t < T , and H T

uL cH

cL uL +

cL (cH

cL )

1

uL

=

t

:

(P3.2) Low Price O¤ers: L 1

and

L T

=1

H T.

(1

2)

;

If T > 2, then L t

for all 1 < t < T

+ cH u(q H ) q H )(cH uL +

2

=

(1

H t )

= (1

)

uH

t+1

uH

cH

uL +

H T 1)

(1 )(u(^ q) H H q^(u c

t+1

uL cH

t

1, and L T 1

= (1

(P3.3) Reservation Prices: rtL = uL

t

for all t < T , and rTL = cL :

In equilibrium, the payo¤ to a buyer is V1B = V1H = 0 and V1L = uL

cL

1.

cH ) : T 1)

1,

and the payo¤s to sellers are

Thus, the payo¤ to a buyer (low quality seller)

is above (below) his competitive payo¤, decreases (increases) with T and increases (decreases) with

and . The surplus, which is given by S DM E = mL (uL

cL ) + m H

T 1

,

is above the competitive surplus S, decreases with T , and increases with Moreover, the liquidity of high quality decreases with

and .

and .

It is easy to describe the equilibrium trading patterns: at the …rst date only low and negligible prices are o¤ered, and thus some low quality sellers trade, but no high quality seller trades (i.e.,

H 1

=0<

negligible prices are o¤ered (i.e.,

H t ;

L 1 L t

< 1). At intermediate dates, high, low and > 0 and 1

H t

L t

> 0), and thus some

sellers of both types trade. At the last date only high and low prices are o¤ered (i.e., H T

+

L T

= 1 ), and thus all matched low quality sellers and some high quality sellers

trade. 13

Thus, both qualities trade with delay. Nevertheless, the surplus generated in the DME is greater than the competitive equilibrium surplus, S: the gain from trading high quality units more than o¤sets the loss from trading low quality units with delay. In contrast, in a market for a homogenous good the competitive equilibrium surplus is an upper bound to the surplus that can be realized in a DME –e.g., Moreno and Wooders (2002) show that this bound is achieved as frictions vanish. Price dispersion is a key feature of equilibrium: At every date but the …rst there is trade at more than one price since both high and low prices are o¤ered with positive probability. To see that price dispersion is essential, suppose instead that the high price cH is o¤ered with probability one at some date t. Since

and

are near one,

this implies that the reservation price of low quality sellers prior to t is near cH , and hence above the value of low quality uL (recall that uL < cH ). Thus, prior to t a low price o¤er (which if accepted buys a unit of low quality) is suboptimal, and therefore low price o¤ers are made with probability zero. Therefore sellers of both qualities leave the market at the same rate, and hence the fraction of high quality sellers remains constant, i.e., qtH = q H . Since q H < q; a high price o¤er is suboptimal at t, which is a contradiction. Hence high price o¤ers are made with probability less than one at every date. Likewise, suppose that the low price is o¤ered with probability one at some date t. Then at date t all matched low quality sellers trade, and no high quality seller trades. Since

is near one, this implies that the fraction of high quality sellers in the market

at date t + 1 is near one, and since this sequence in non-decreasing over time, the fraction of high quality sellers at the last date is above q^. (Recall that q^ is the fraction of high quality sellers that makes buyers indi¤erent between o¤ering the high and the low price at date T .) This implies that o¤ering cH is uniquely optimal and hence the high price is o¤ered with probability one at date T; which is a contradiction. Thus, low price o¤ers are made with probability less than one at every date. A more involved argument establishes that all three types of price o¤ers (high, low, and negligible) are made with positive probability at every date except the …rst and last –see the proof of Lemma 2, part 7, in the Online Appendix. Identifying the probabilities (

H t ;

L t )

is delicate: Their past values determine the

current market composition, qtH , and their future values determine the reservation 14

price of low quality sellers at date t. In equilibrium, at intermediate dates the market composition and the sellers’reservation prices must make buyers indi¤erent between o¤ering high, low or negligible prices, i.e., the equation u(qtH )

cH = (1

qtH )(uL

B B rtL ) + qtH Vt+1 = Vt+1

holds. We show in the proof of Proposition 3 in Appendix A that the system formed by these equations (together with the analogous equations for dates 1 and T , and the boundary conditions) admits a single solution. Establishing uniqueness of equilibrium requires showing that these properties are common to all market equilibria – see Lemma 2 in the Online Appendix. The comparative static properties of equilibrium relative to ;

and T are intu-

itive: Since negligible price o¤ers are optimal at every date except the last, the payo¤ to buyers is just their discounted payo¤ at the last date. Consequently, the payo¤ to a buyer increases with

and , and decreases with T . Low quality sellers capture

surplus whenever high price o¤ers are made, i.e., at every date except the …rst. The probability of a high price o¤er decreases with both

and , and increases with T ,

and thus the payo¤ to low quality sellers decreases with T . The surplus increases with

and , and increases with

and . Also, high quality is less liquid as the prob-

ability of meeting a partner increases or as traders become more patient. (Indeed, both qualities become completely illiquid at intermediate dates as

approaches one

–see Proposition 4 below.) Somewhat counter-intuitively, the surplus decreases with T , i.e., shortening the horizon over which the market opens is advantageous (so long as T > 2): Our assumption that frictions are small implies that in equilibrium buyers must be willing to o¤er negligible prices at every date but the last date. Hence their payo¤ is just their discounted expected utility at the last date.4 Thus, a longer horizon provides no advantage in screening sellers, and reduces the buyers’payo¤. The payo¤ to low quality sellers increases with T because the high price is o¤ered with higher probability at every date (except at the last date, at which it is o¤ered with a probability independent of T ). Further, since buyers’must remain willing to o¤er the low price, 4

In contrast, if traders are su¢ ciently impatient, then there is an equilibrium in which buyers

o¤er r1L at date 1, and then o¤er cH at every subsequent date. In this equilibrium, lengthening the horizon increases surplus when

< 1.

15

the increase in the payo¤ of low quality sellers exactly matches the decrease in the payo¤ of buyers. Therefore the surplus decreases with T since there are more buyers than low quality sellers, and is maximal when T = 2. A striking feature of equilibrium in decentralized markets is that the surplus realized exceeds the competitive equilibrium surplus: decentralized markets are more e¢ cient than centralized ones. While in a centralized market all units trade at a single market-clearing price, in a decentralized market several prices are o¤ered with positive probability, and di¤erent units trade at di¤erent prices. When

= 1, for example,

low quality units trade for sure – some at the high price and some at the low price – while high quality units trade with probability less than one. Thus decentralized trade generates an allocation closer to the surplus maximizing allocation, in which low quality sellers trade for sure, and high quality sellers trade with positive probability (less than one).5 Proposition 4 identi…es the limiting DME as traders become perfectly patient. A remarkable feature of the limiting equilibrium is that the market freezes at intermediate dates, and both qualities are completely illiquid: Low quality trades at the …rst and last two dates, and high quality trades only at the last date. The surplus is independent of the duration of the market. Proposition 4. Assume that 1 < T < 1, (i.e., frictions are not large). Then as

< 1, and inequalities F:1 and F:2 hold

approaches one the unique DME approaches

(~H ; ~L ; r~L ) given by: (P4.1) High Price O¤ers: ~H t = 0 for all t < T , and ~H T =

uL

cL (cH

cL )

.

(P4.2) Low Price O¤ers: ~L1 = 5

(1

(1

+ cH u(q H ) q H )(cH uL +

)

;

The (static) surplus maximizing menu contract is f(pH ; Z H ); (pL ; Z L )g, where pH = cH , Z H = q H )(uL

cL )=[cH

cL

q H (uH

cL )], pL = cL + Z H (cH

cL ) and Z L = 1. Here p is the

money transfer from seller to buyer and Z is the probability that the seller transfers the unit of good to the buyer, when the seller reports type . Even if H

trade with probability less than Z .

16

= 1, in the DME high quality sellers

and ~LT = 1

L ~H T : If T > 2, then ~t = 0 for all 1 < t < T

~LT

1

=

(1 )(u(^ q) H H q^(u c

(P4.3) Reservation Prices: r~tL = uL

1 and

cH ) : )

for all t < T , and r~TL = cL :

Moreover, (~H ; ~L ; r~H ; r~L ) is a DME of the market when = 1. In equilibrium, the payo¤ to a buyer is V~1B = , and the payo¤s to sellers are V~1H = 0 and V~1L = [1

(1

q^)](uL

cL ). Thus, the payo¤ to a buyer (low quality seller) remains

above (below) his competitive payo¤. The surplus, given by S~DM E = mL (uL

cL ) + mH

;

is independent of T and remains above the competitive surplus. Further, both qualities are completely illiquid at intermediate dates. When

= 1, time can no longer be used as a screening device (until the very last

period), and the market freezes at all dates but the last two. The DME identi…ed in Proposition 4 is not the unique market equilibrium. For example, there are DME in which buyers mix over low and negligible prices at dates prior to T in such a way that the total measure of low quality sellers that trades prior to T is the same as in the DME identi…ed in Proposition 4; then buyers o¤er high and low prices at date T L with probabilities ~H T and ~T , respectively.

We illustrate our …ndings in propositions 3 and 4 with an example. Example 1 Consider a market in which uH = 1, cH = :6, uL = :4, cL = :2, mH = :2, mL = :8, and T = 10. The graphs in the top row of Figure 1 show the evolution of the stocks of high quality sellers mH t in the market, and the fraction of high price o¤ers several di¤erent combinations of evolution of mLt and

L t .

H t

for

and . The graphs in the middle row show the

The bottom graph shows the evolution of the fraction of

high quality sellers in the market qtH .

17

UtH

mtH 0.20

0.60

0.50 0.15 0.40

0.10

0.30

0.20 0.05 0.10

0.00

0.00 1

2

3

4

5

6

7

8

9

10

11

1

2

3

t

HighQualityStocks

4

5

6

7

8

9

mtL

UtL

0.80

0.80

0.70

0.70

0.60

0.60

0.50

0.50

0.40

0.40

0.30

0.30

0.20

0.20

0.10

0.10

0.00

10

t

HighPriceOffers

0.00 1

2

3

4

5

6

7

8

9

10

LowQualityStocks

t

11

1

2

3

4

5

6

7

8

9

10

t

LowPriceOffers

qtH 1.00

0.80

0.60

D G  D G  D G 

0.40

0.20

0.00 1

2

3

4

5

6

7

8

9

ProportionofHighQualityintheMarket

Figure 1: Equilibrium Dynamics in a Decentralized Market

18

10

11

t

These graphs illustrate several features of equilibrium as frictions become small: high quality trades more slowly; low quality trades more quickly at the …rst date and at the last date, but trades more slowly at intermediate dates; the fraction qtH increases more quickly, but equals q^ = :5 at the market close regardless of the level of frictions. Decentralized Market Equilibria when the Horizon is Infinite We now consider decentralized markets that open over an in…nite horizon. In these markets, given a strategy distribution one calculates the maximum expected utility of each type of trader at each date by solving a dynamic optimization problem. The de…nition of DME remains otherwise the same. Proposition 5 identi…es the limiting DME as T approaches in…nity, and establishes that this limit is a DME of the market that opens over an in…nite horizon. In relating the formulae in propositions 3 and 5, it is useful to observe that

t

approaches zero

as T approaches in…nity. Proposition 5. Assume that

< 1; and inequalities F:1 and F:2 hold (i.e., frictions

are not large). Then as T approaches in…nity the unique DME approaches (^H ; ^L ; r^L ) given by: (P5.1) High Price O¤ers: ^H 1 = 0, and for all t > 1, ^H t =

uL cH

1

cL : uL

(P5.2) Low Price O¤ers: ^L1 =

q qH and ^Lt = 0 for all t > 1: q (1 q H )

(P5.3) Reservation Prices: r^tL = uL for all t. Moreover, if T = 1 then (^H ; ^L ; r^H ; r^L ) is a DME. In equilibrium, the traders’ payo¤s are the competitive payo¤s, i.e., V^1B = 0; V^1H = 0 and V^1L = uL cL , and the surplus is the competitive surplus S. Further, the liquidities of both qualities at dates t > 1 approach zero as the traders’become perfectly patient. As the horizon becomes in…nite, all units trade eventually. At the …rst date, some low quality units trade but no high quality units trade. At subsequent dates, 19

units of both qualities trade with the same constant probability. In the limit, the traders’payo¤s are competitive independently of

and , and hence so is the surplus,

even if frictions are non-negligible. Kim (2011) obtains an analogous result in a stationary setting. In contrast, the previous literature has established that payo¤s are competitive only as frictions vanish, e.g., Gale (1987), Binmore and Herrero (1988), and Moreno and Wooders (2002) for homogenous goods markets, and Moreno and Wooders (2010) for markets with adverse selection. The intuition for these results is simple: in the DME of a market that opens over a …nite horizon, the payo¤ to a buyer at the last date is VTB =

> 0, independently

of the horizon T . Since negligible prices are optimal at every date except the last, the payo¤ to a buyer is his discounted payo¤ at the last date,

T 1

, which approaches

zero as the horizon approaches in…nity. Thus, in a market that opens over an in…nite horizon the payo¤ to a buyer is zero. Hence low price o¤ers, if made with positive probability, must yield a payo¤ equal to zero, which implies that rtL = uL > cL . Then high prices must be o¤ered with positive probability at some dates. At these dates the proportion of high quality must be q in order for the expected payo¤ to a buyer o¤ering the high price to be zero. In a stationary equilibrium, the equation rtL = uL pins down the rate at which high price o¤ers are made, and q2H = q pins down the proportion of low price o¤ers at date 1. Since the payo¤s of buyers is zero, the proportion of high quality sellers in the market can not rise above q, and thus low price o¤ers are made with probability zero after date 1. When T = 1 there are multiple equilibria. Uniqueness of equilibrium when the

horizon is …nite justi…es focusing on the limiting DME identi…ed in Proposition 5.6 6

When T = 1 there is a continuum of DME that share the basic properties identi…ed in Propo-

sition 5:

H 1

= 0,

L 1

> 0 is such that q2H = q, and r1L = uL

rtL for all t > 1. In these DME, payo¤s

are competitive: V1B = 0 implies r1L = uL , and thus V1L = V2L = r1L

cL = uL

cL . In fact, we

conjecture that payo¤s are competitive in all DME. This conjecture is based on the idea that in all DME buyers make negligible price o¤ers with positive probability at every date, which implies that their payo¤ would diverge if it was positive. Proving this conjecture requires establishing versions of lemmas 1 and 2 when T = 1. (The proofs of these lemmas for T < 1 involve backward induction arguments that break down when T = 1.)

20

4

Policy Intervention

Our results allow an assessment of the impact of policies aimed at improving market e¢ ciency, such as subsidies, taxes, or other interventions such as the Public-Private Investment Program for Legacy Assets or closing the market for some period of time. Taxes and Subsidies Conditional on Quality L B

Suppose that the government provides a per unit subsidy of

> 0 to buyers

of low quality. Then the instantaneous payo¤ to a buyer who purchases a unit of low quality at price p is uL +

L B

p rather than uL

p. The impact of the subsidy

may therefore be evaluated as an increase in the value of low quality. Likewise, if the government provides a per unit subsidy of

L S

> 0 to sellers of low quality, then the

instantaneous payo¤ to a seller who sells a unit of low quality at price p is p (cL rather than p

L S)

cL , and therefore the impact of the subsidy may be evaluated as a

decrease in the cost of low quality. Such subsidies are feasible provided that quality is veri…able following purchase. Taxes are negative subsidies. When T < 1, the e¤ect of a subsidy may be determined using the formulae given in Proposition 3. For example, subsidizing buyers of low quality increases the net surplus: a marginal subsidy increases (gross) surplus by @S DM E = mL + mH @uL

T 1

d = mL + mH duL

T 1

(1

q^) ;

whereas the present value of the subsidy is at most mL since at most mL units receive the subsidy. Subsidizing sellers of low quality increases the net surplus as well since @S DM E = @cL

mL + mH

T 1

d = dcL

mL

mH

T 1

(1

q^)

uH uH

uL < cL

mL :

Comparing these two expressions reveals that subsidizing buyers has a larger e¤ect on surplus, i.e., @S DM E [email protected] > [email protected] DM E [email protected] j, since (uH

uL )=(uH

cL ) < 1. Corollary

1 below summarizes the e¤ect of subsidies to low quality on payo¤s and surplus. Its proof, which follows from di¤erentiating the formulae given in Proposition 3, is omitted. Corollary 1. Under the assumptions of Proposition 3, a subsidy to either buyers or sellers of low quality increases the payo¤s of buyers and low quality sellers, the net 21

surplus, and the liquidity of high quality. However, subsidizing buyers has a larger e¤ect on the payo¤ of buyers and on the surplus S DM E , and a smaller e¤ect on the payo¤ of low quality sellers, than subsidizing sellers. The intuition for the result that subsidies to low quality raise surplus is as follows: A subsidy, whether to buyers or sellers, raises the payo¤ to buyers at the last date, and therefore raises their payo¤ at every date. Consider a subsidy to buyers. Buyers must remain indi¤erent between low and negligible price o¤ers (i.e., uL

B rtL = Vt+1 )

B prior to date T . The value of low quality increases by the subsidy, whereas Vt+1

increases by less. Thus, the reservation price of low quality sellers, and hence their payo¤, must increase. This requires that high price o¤ers be made more frequently, which increases the liquidity of high quality and the surplus. A subsidy to buyers yields a greater increase in the payo¤ to buyers at the last date, and therefore leads to a greater increase in surplus, than does an equal-sized subsidy to sellers. Next we describe the impact of subsidies to buyers and sellers of high quality. When T < 1, the e¤ect of such subsidies may be assessed using the formulae of Proposition 3 as changes in the value or cost of high quality. Their impact on the net surplus is unclear in general as it is di¢ cult to calculate the present value of the subsidy, but as

approaches one the e¤ect is clear from Proposition 4: A subsidy to

buyers of high quality a¤ects surplus through its impact on q^: @ S~DM E = @uH

mH (uL

cL )

@ q^ = mH (uL H @u

cL )

cH (uH

cL : cL )2

Since high quality trades only at the last date, the marginal cost of the subsidy approaches mH ~H T . Thus the marginal e¤ect on the net surplus approaches @ S~DM E @uH

mH ~H = mH (uL T mH

uL uH

L cH cL cL Hu m (uH cL )2 (cH cL ) cH cL 1 u H cL

cL ) cL cL

< 0; where the weak inequality holds since

1. A subsidy to sellers of high quality also

reduces the net surplus since @ S~DM E = @cH

H

m

(u

L

@ q^ c ) H = @c L

22

m

H

uL uH

cL ; cL

and therefore @ S~DM E @cH

mH ~H T =

)mH

(1

uL uH

cL u H cL cH

cL cL

0:

We state these results in Corollary 2. Corollary 2. Under the assumptions of Proposition 3, subsidizing buyers or sellers of high quality increases the payo¤ of buyers and the surplus, and decreases the payo¤ of low quality sellers, although subsidizing sellers has larger e¤ects. Moreover, the liquidity of high quality increases (decreases) if sellers (buyers) are subsidized. As approaches one, either subsidy reduces the net surplus. A subsidy to buyers of high quality raises the payo¤s of buyers at every date. Since buyers make low price o¤ers at every date, the reservation price of low quality sellers, and therefore their payo¤, must decrease. Hence high price o¤ers are made less frequently, i.e., the liquidity of high quality decreases. A subsidy to sellers of high quality also raises the payo¤ of buyers at every date, but it has a direct negative e¤ect on the reservation price (and payo¤) of low quality sellers, since the high price o¤er decreases by an amount equal to the subsidy. High price o¤ers must be made more frequently in order to maintain the buyers’indi¤erence between high and low price o¤ers at every date. Table 1 below illustrates the e¤ect of several policies for the market described in Example 1 when

=

buyers of low quality,

= :95. The second row describes the e¤ect of a subsidy to L B

= :05. Relative to the equilibrium without any subsidy or

tax (…rst row), the volume of high quality sellers that trades increases 11:05 percentage points, and the net surplus increases 4% from :1720 to :1790. The third row shows the e¤ect of a subsidy to sellers of low quality. The di¤erential e¤ects of these two subsidies are consistent with Corollary 1. The fourth and …fth rows of Table 1 describe the e¤ects of subsidies to buyers and sellers of high quality, respectively. Both subsidies decrease the payo¤ of low quality sellers and increase the payo¤ of buyers and the (gross) surplus. Consistent with Corollary 2, these e¤ects are stronger for the subsidy to sellers than the subsidy to buyers. In the example, the negative e¤ect on net surplus of the subsidy to sellers is smaller. 23

The sixth row of Table 1 reports the e¤ects of an unconditional subsidy to buyers. (If quality is not veri…able after purchase, then a subsidy conditional on the quality of the good is not feasible.) The unconditional subsidy has a smaller positive e¤ect on the net surplus than a subsidy to buyers of low quality alone. The seventh row of Table 1 shows the e¤ects of taxing buyers of high quality, which are opposite to those of a subsidy. In particular, the measures of trade of both qualities and the net surplus increase.

Policy

Vol. Trade %

( = :05) None

Payo¤s

Surplus

Policy

H

L

mB V1B

mL V1L

S DM E

Net

47:90

99:09

.0599

.1121

.1720

.1720

.0000

Cost

Sub. Buyer

=L

58:95

99:20

.0748

.1401

.2150

.1790

.0360

Sub. Seller

=L

54:30

99:24

.0704

.1436

.2140

.1777

.0363

Sub. Buyer

=H

46:60

98:96

.0634

.1093

.1727

.1693

.0034

Sub. Seller

=H

51:20

98:88

.0673

.1061

.1735

.1697

.0038

Sub. Buyer

2 fH; Lg

57:40

99:09

.0792

.1366

.2159

.1761

.0398

49:40

99:20

.0559

.1153

.1712

.1748

:0036

PPIP

46:45

98:95

.0639

.1089

.1728

.1681

.0047

Sub. Low Price

60:50

99:30

.0704

.1796

.2141

.1821

.0320

Sub. High Price

45.17

98.81

.0673

.1061

.1735

.1702

.0033

Tax Buyer

=H

Table 1: Policy E¤ects Next we address the e¤ects of taxes and subsidies in a market that opens over an in…nite horizon. In such markets the e¤ects of subsidies on either quality are easily assessed by di¤erentiating the formulae provided in Proposition 5. Since low prices are o¤ered only at the …rst date, the liquidity of both qualities at each date H t > 1 is ^H t . Note that ^t is independent of , i.e., the liquidities of both goods

are una¤ected by changes in the probability of meeting a partner. Inspecting these formulae leads to an interesting …rst observation: in these markets subsidizing either buyers or sellers of

-quality has identical e¤ects on payo¤s and surplus. Corollary

3 describes the e¤ects of subsidizing low quality. The last statement in the corollary, 24

that as traders become perfectly patient the subsidy amounts to a transfer to low quality sellers, is established in the Appendix. Corollary 3. Assume that T = 1 and the assumptions of Proposition 5 hold. Subsidizing low quality has no e¤ect on the payo¤ of buyers, while it increases the payo¤ of low quality sellers, the net surplus, and the liquidities of both qualities at dates t > 1. As

approaches one, the subsidy has no e¤ect on the net surplus and

amounts to a transfer to low quality sellers. Interestingly, a tax on high quality raises revenue without a¤ecting either payo¤s or surplus, thereby increasing net surplus. A tax on buyers of high quality, for example, increases ^L1 while leaving ^Lt and ^H t unchanged for t > 1, thus accelerating trade at date 1 and leaving una¤ected the liquidities of both qualities at t > 1. We state this result in Corollary 4. Corollary 4. Assume that T = 1 and the assumptions of Proposition 5 hold. A tax on high quality raises revenue without a¤ecting payo¤s or surplus, thereby increasing the net surplus. The Public-Private Investment Program for Legacy Assets This program was designed to draw new private capital into the market for troubled real estate-related assets, comprised of legacy loans and securities, by providing equity co-investment and public …nancing. Its main objective was to reduce the perceived excessive liquidity discounts in legacy asset prices. The program provided private investors with non-recourse loans to purchase legacy assets. Investors had to provide only a small amount of equity (a fraction

= 1=14 of the purchase price).

An investor who purchased a low quality asset could choose to default on the loan and surrender the asset, losing only her equity (i.e., the fraction

of the price paid

for the asset).7 This policy may be framed in our setting as a subsidy to buyers who pay the high price cH for a low quality unit: under this program a buyer who purchases at the 7

The U.S. Treasury website (http://www.treasury.gov/initiatives/…nancial-stability/TARP-

Programs/credit-market-programs/ppip) provides abundant documentation about this program. See the White Paper released on March 23, 2009, which is reproduced in the Online Appendix.

25

high price, upon observing the quality of the unit acquired faces the choice to keep the unit and pay the loan, which is optimal if it is of high quality since uH > (1

) cH ;

or default and surrender the unit, which is optimal when it is of low quality provided uL < (1 Assume that

) cH :

is su¢ ciently small that this inequality holds. Then the payo¤ to a

buyer o¤ering the high price cH , denoted by P H ; which is a function of the fraction )cH

of high quality in the market is q and the e¤ective subsidy s := (1

uL ; is

given by P H (q; s) = q uH = u(q)

cH + (1 cH + (1

q)

cH

q)s:

Of course, the lemons problem can be solved altogether by setting a subsidy su¢ ciently large. Evaluating the impact of a small subsidy is somewhat more complex than a comparative statics exercise. However, the introduction of a small PPIP subsidy does not change the basic properties of equilibrium, and the formulae provided in propositions 3 to 5 describing the DME can be readily modi…ed to show the impact of this policy. Reviewing the proof of Proposition 3 reveals how the introduction of a subsidy s a¤ects the DME: The probabilities of high price o¤ers and reservation prices of low quality sellers, as well as the traders’ payo¤s and surplus, are not a¤ected directly by the subsidy, but only indirectly via its impact on the fraction q^(s) of high quality sellers in the market at the last date. Of course, q^(s) a¤ects in turn the entire sequence qtH , and the functions

t (s)

=

T t

(s); where (s) = (1

q^(s)) (uL

cL ). However,

the subsidy appears explicitly in the formulae describing the sequence of probabilities of low price o¤ers –we provide these formulae in the Online Appendix. Intuitively, the impact of this policy is as follows: In equilibrium, at date T buyers are indi¤erent between o¤ering the high or the low price, and therefore the fraction of high quality sellers qTH must be such that P H (qTH ; s) = (1 26

qTH )(uL

cL ):

The solution to this equation, qTH = q^(s), is decreasing in s. Hence introducing a PPIP subsidy s decreases the fraction of high quality in the market, and increases the buyers’payo¤, at the last date. It is easy to see the e¤ects of a PPIP subsidy in a market that opens only for two dates: the measure of low quality sellers that trades at date 1 decreases. Moreover, since buyers are indi¤erent between trading at date 1 or at date 2, and their expected utility is greater with the subsidy, the reservation price of low quality sellers at date 1 decreases with the subsidy, which in turn implies that the probability of a high price o¤er at date 2 decreases, and the measure of high quality sellers that trades decreases as well. Thus, this policy reduces the net surplus and makes both qualities less liquid. The analysis of the impact of a PPIP subsidy for a market that opens for more than two dates is more complex. However, its qualitative e¤ects, as well as the intuition for how it a¤ects the DME, are analogous to a subsidy to buyers of high quality (see corollaries 2 and 5). We summarize our conclusions in Corollary 5. Corollary 5. Under the assumptions of Proposition 3, a PPIP subsidy increases the payo¤ of buyers, and decreases the payo¤ of low quality sellers and the liquidity of high quality. Moreover, it reduces the net surplus as

approaches one.

For the market in Example 1, the row in Table 1 labeled PPIP shows the impact of this program: Its e¤ects are qualitatively the same as a subsidy to buyers of high quality – see the fourth row – but the PPIP program leads to a larger reduction of the net surplus due to its larger cost. In a market that opens over an in…nite horizon, the only impact of a PPIP program is to decrease the probability of low price o¤ers at the …rst date. Since surplus is una¤ected, it is purely wasteful: it causes an increase of the cost of delay in trading low quality that exactly o¤sets the subsidy. Camargo and Lester (2014) study the impact of the PPIP program on liquidity as measured by the minimum time (taken over the set of all market equilibria) required for all units of both qualities to trade. They show that if the initial fraction of high quality sellers is high, then the introduction of a su¢ ciently large PPIP subsidy gives rise to equilibria where all units trade earlier, thus increasing liquidity. (In 27

contrast, our analysis focuses on marginal subsidies, which do not change the basic structure of equilibrium. A su¢ ciently large PPIP subsidy eliminates the lemons problem entirely and hence increases liquidity in our setting as well.) Their results are obtained assuming that buyers may o¤er one of two exogenously given prices. Camargo and Lester (2014) also provide numerical examples showing that when price o¤ers are unrestricted a PPIP subsidy has ambiguous e¤ects on liquidity: it may decrease it when the lemons problem is severe, and increase it when it is not severe. In contrast, our Corollary 5 shows that a PPIP subsidy unambiguously reduces the liquidity of high quality. Our results, however, are obtained under the assumption that frictions are not large, whereas Camargo and Lester (2014) assumes that frictions are signi…cant. In our setting, when frictions are large one can construct equilibria in which all buyers o¤er the low price for dates t

t^, and then o¤er the high price for

t > t^. A PPIP subsidy may lead buyers to o¤er the high price earlier, i.e., increase the liquidity of high quality at date t^. Thus, the di¤erence in our results and those of Camargo and Lester (2014) arise as a consequence of our focus on marginal subsidies and small frictions. Taxes and Subsidies Conditional on Price Subsidies conditional on the quality of the good are feasible only if quality is veri…able following purchase. Hence it is useful to study the e¤ects of taxes and subsidies conditional on the price at which the good trades. The e¤ect of a small subsidy may also be assessed by modifying the formulae provided in propositions 3 to 5. It is interesting to observe that unlike subsidies conditional on the quality of the good, the e¤ects of a subsidy conditional on the price at which the good trades are the same whether it is given to buyers or sellers. Subsidies conditional on trading either at the high price cH or at a low price (i.e., a price below cH ), a¤ect the fraction of high quality sellers in the market at the last date, which becomes a function of the subsidy, as well as the functions and

t

involved in the formulae describing the DME. The formulae describing the

probabilities of high and low price o¤ers must be modi…ed appropriately – see the Online Appendix. These formulae reveal the e¤ects of these subsidies on traders’ payo¤s, surplus, and market liquidity. With a subsidy conditional on trading at a low price, for example, at the last 28

date the payo¤ to o¤ering the low price increases, and therefore the fraction of high quality in the market needed to preserve the indi¤erence between high and low price o¤ers increases. This has an impact on the probabilities of o¤ering high, low and negligible prices, as well as on the reservation prices of low quality sellers, at every date. Corollary 6 describes the impact of such a subsidy. The intuition for these results is analogous to that of a subsidy to low quality –see Corollary 1. Corollary 6. Under the assumptions of Proposition 3, a subsidy conditional on trading at a low price increases the payo¤s of buyers and low quality sellers, as well as the net surplus. The liquidity of low (high) quality at date 1 ( T ) increases. When T = 1 the subsidy increases the liquidity of both qualities after the …rst date, as well as the net surplus. The next to the last row of Table 1 describes the e¤ects of a subsidy conditional on trading at a low price for the market described in Example 1. This policy is the most e¤ective: relative to the DME without intervention (…rst row), the volume of trade of high quality increases 12.6 percentage points, and the net surplus increases by 5.9% from .1720 to .1821. Low quality sellers are the main bene…ciaries as their payo¤ increases by 60%, while the payo¤ of buyers increases by 17.5%. The e¤ects of a subsidy conditional on trading at the high price on payo¤s, surplus and liquidity are summarized in Corollary 7. This subsidy has e¤ects analogous to those of subsidies to buyers or sellers of high quality –compare rows 4 and 5 to row 10 in Table 1. Corollary 7. Under the assumptions of Proposition 3, a subsidy conditional on trading at the high price increases (decreases) the payo¤s of buyers (low quality sellers). The liquidity of high quality decreases. When T = 1 the subsidy is purely wasteful, whereas a tax raises revenue without a¤ecting payo¤s, thereby increasing the net surplus. Restricting trading opportunities Our results allow assessing other policies studied in the literature such as closing the market for some period of time: Since by Proposition 3 surplus is decreasing in T , closing the market altogether after date 2 increases the surplus. Intuitively, a longer horizon makes it more di¢ cult to screen low quality sellers. 29

Closing the market at intermediate dates has no e¤ect when traders are patient and the horizon is short. Suppose, for example, that the market is closed at all intermediate dates t 2 f2; :::; T replaced by

T 1

1g. If inequalities F:1 and F:2 hold when

is

, then the formulae of Proposition 3, particularized for T = 2 and

a discount factor equal to

T 1

, describe the market outcome. An inspection of

these formulae reveals that this intervention does not a¤ect payo¤s and surplus. The intuition for this result is as follows: Since buyers make negligible o¤ers at every date except the last, their payo¤ is their discounted utility at the last date, i.e., T 1

(1

q^)(uL

cL ), and is the same whether the market is open at intermediate

dates or not. Furthermore, since in both markets buyers obtain the same payo¤ and are indi¤erent between low and negligible price o¤ers at date 1, this implies that low quality sellers have the same reservation price at date 1, and thus the same payo¤ in both markets. When the time horizon is long, however, closing the market at intermediate dates = 1 and let T > t^; where t^ is

may increase surplus. For simplicity, assume that su¢ ciently large that uL

t^ 1

cL T 1

(uH

cL ):

(Hence F:1 fails if

is replaced by

dates t 2 f2; : : : ; t^

1g, and re-opens at dates t 2 ft^; : : : ; T g, there is an equilibrium

in which all buyers o¤er r1L =

t^ 1

(cH

.) If the market opens at date 1, closes at

cL ) + cL at date 1, and o¤er cH at every date

t 2 ft^; : : : ; T g. It easy to verify that the surplus realized in this equilibrium, mL (uL

cL ) + m H

t^ 1

(uH

cH );

is greater than the surplus in the DME when the market is always open, whether T < 1 or T = 1. Thus, for markets in which T is large or in…nite, closing the market after the …rst date for su¢ ciently long that the (separating) equilibrium described above can be sustained, raises welfare: closing the market prevents the wasteful delay that results when low quality sellers attempt to pool with high quality sellers. We summarize these results in Corollary 8. Corollary 8. Under the assumptions of Proposition 3, if F:1 and F:2 hold when is replaced by

T 1

, then closing the market for dates 2; : : : ; T 30

1 has no e¤ect on

payo¤s or surplus. If

is near one and uL

cL >

T 1

(uH

cL ), then closing the

market for some period of time may increase the surplus. Fuchs and Skrzypacz (2013) obtain related results for continuous-time dynamic competitive equilibrium with adverse selection and a continuum of qualities. In contrast to the …rst part of Corollary 8, they provide an example of a market that opens over a …nite horizon T; in which total surplus is higher when trade is “infrequent” (i.e., restricted to just two instants, date 0 and date T ) rather than taking place continuously over [0; T ]. Further, they show that it is never optimal for the market to be open continuously on [0; T ] and give conditions under which infrequent trade is optimal. In their model, sellers’ types are publicly revealed at date T , at which time there is no adverse selection, which drives the di¤erences in results. In a market that opens over an in…nite horizon, in both models closing the market for some time increases the surplus. Government Purchases We discuss the impact of government purchases. Assume that at the market open the government o¤ers to buy

units of the good, e.g., via a uniform price auction.

In equilibrium, the government acquires

units of low quality at a price equal to

the reservation price of low quality sellers in the market that follows, i.e., r1L . Our assumption that the matching probability is constant over time, and equal for buyers and sellers, is no longer appropriate since after the government intervention there are more buyers than sellers in the market. Let assume instead that the buyers’matching probability is a function of the market tightness, i.e., the ratio

t

L B = (mH t + mt )=mt .

Since equal measures of buyers and sellers trade and leave the market every date,

t

decreases over time. Hence a direct e¤ect of this program is to decrease the buyers’ matching probability at every date. Assuming that

is not so large as to alter the structure of the DME, the buyers’

payo¤ at the last date conditional on being matched is una¤ected since buyers must remain indi¤erent between o¤ering the high and the low price. However, since the buyers’matching probability is smaller, then their payo¤ at the last date decreases. Further, since buyers must be willing to o¤er high, low and negligible prices at every date but the …rst and last, and their expected utility at the last date is smaller, then 31

in order to reduce the payo¤ of low (high) price o¤ers in line to the decrease in the payo¤ of negligible price o¤ers, the reservation price of low quality sellers increases (the fraction of high quality sellers in the market decreases), i.e., the payo¤ of low quality sellers increases, which implies that high price o¤ers are made more frequently. Thus, a positive impact of the program is to increase the volume of trade of high quality. If government purchases crowd out private trade and the government’s value for low quality is less than the buyers’value, then the program also has a negative e¤ect on surplus, and the overall e¤ect is unclear. We examine the impact of this policy in a market that operates over two dates, and in which at every date t buyers are matched with probability

t.

In this market,

since the fraction of high quality in the market at date 2 is the same with and without the government purchases, the measure of low quality sellers who sell their good at date 1 (either to the government to private buyers) is also the same; and since all matched low quality sellers trade at date 2, the liquidity of low quality, and hence the volume of trade of low quality, are una¤ected. Since the buyers’payo¤ at date 2 is smaller, for buyers to be willing to o¤er negligible prices at date 1 the payo¤ to o¤ering the low price must decrease, which implies that the reservation price of low quality sellers increases, and therefore that the high price is o¤ered with a greater probability at date 2. Hence the volume of trade of high quality increases. The e¤ect on the net surplus depends on whether the surplus gained from the increase in the volume of trade of high quality is greater or less than the surplus loss due to the smaller value of low quality to the government. We summarize these conclusions in Corollary 9. The formal analysis is presented in the Online Appendix. Corollary 9. If T = 2 and the assumptions of Proposition 3 hold, then government purchases at the market open increase the payo¤ of low quality sellers and the liquidity of high quality, and decrease the payo¤ of buyers. If the value of low quality to the government is close to the buyers’value, then the net surplus increases. Tirole (2012) studies the design of government policies aimed at rejuvenating a market for a legacy asset frozen due to adverse selection. In a market with a continuum of qualities, he proposes a mechanism that, operating in conjunction with the private market, allows the government to cleanse the market of lower qualities 32

assets, thus liquidifying the private market. Our starting point in not a frozen market, but one in which the volume and timing of trade are suboptimal, and the intervention involves the government participating in the private market, buying low quality assets at the …rst date, and letting the market freely operate afterwards. As in Tirole (2012), this intervention increases liquidity and surplus, but unlike Tirole (2012) it is pro…table when the government values the asset the same as buyers. Philippon and Skreta (2012) study optimal government interventions in lending markets when every …rm has a positive NPV investment opportunity but requires outside funding to …nance it. Firms are privately informed about the quality of their legacy assets, and adverse selection may lead to ine¢ ciently low investment levels. They show that to implement any target investment level, the cost minimizing intervention involves the government o¤ering debt contracts. In their setting, an intervention that increases investment also increases surplus. In our setting, however, an intervention a¤ects surplus via its impact on the measures of high and low quality units that trade, and also through its impact on the timing of trade, and may reduce surplus.

5

Dynamic Competitive Equilibrium

When the horizon is …nite and frictions are not large, in the equilibrium of a decentralized market most low quality units as well as some high quality units trade, and the surplus is above the competitive surplus. When the horizon is in…nite all units of both qualities trade, although with delay, and payo¤s and surplus are competitive. We study on the Online Appendix the market described in Section 2 but where trade is centralized, i.e., trade is multilateral and agents are price takers. We show in Proposition 6 that if the horizon is …nite and traders are patient, then in a dynamic competitive equilibrium (DCE) all low quality units trade at the …rst date and no high quality units ever trade. Hence the surplus realized is the same as in the static competitive equilibrium. We show that subsidies, which are e¤ective in decentralized markets, are ine¤ective in centralized markets. Moreover, high (low) quality is more (less) liquid in decentralized markets than in centralized ones. These features hold even as frictions vanish. These results suggest that when the horizon is …nite,

33

decentralized markets perform better than centralized markets. We also show that if traders are su¢ ciently impatient or the horizon is in…nite, there are dynamic competitive equilibria in which all low quality units trade immediately at a low price and all high quality units trade with delay at a high price. These separating DCE, in which di¤erent qualities trade at di¤erent dates, yield a surplus greater than the static competitive surplus. Consequently, when the horizon is in…nite, centralized markets may perform better than decentralized markets. Interestingly, we show in Proposition 7 that as frictions vanish the surplus at a separating DCE of a market that opens over an in…nite horizon equals the surplus in the equilibrium of a decentralized market that opens over a …nite horizon. Intuitively this result holds since the same incentive constraints operate in both markets. In a separating DCE high quality trades with su¢ ciently long delay that low quality sellers are willing to trade immediately at a low price rather than waiting to trade at a high price. Likewise, in a DME high price o¤ers are made with a su¢ ciently small probability that low quality sellers are willing to immediately accept a low price, rather than waiting for a high price.

6

Appendix: Proofs

We begin by establishing a number of lemmas. In the proofs, we refer to previous results established in lemmas or propositions by using the letter L and P , respectively, followed by the number. The proof of lemma 1, which is straightforward, is provided in the Online Appendix. Lemma 1. Assume that 1 < T < 1 and

< 1, and let ( ; rH ; rL ) be a DME. Then

for each t 2 f1; : : : ; T g: (L1:1)

H L t (maxfrt ; rt g)

= 1:

(L1:2) rtH = cH > rtL ; VtH = 0 < VtB , and VtL H (L1:3) qt+1

qtH : H

(L1:4)

t (c

(L1:5)

t (p)

) = 1: =

L t (rt )

for all p 2 [rtL ; cH ): 34

cH

cL :

With these results in hand we prove propositions 1 and 2. Proof of Proposition 1. P 1:1 follows from L1:2 and L1:3, and P 1:2 follows from L1:4 and L1:5. Proof of Proposition 2. We prove P 2:3: Suppose by way of contradiction that H T

+

L T

< 1: Then negligible prices are optimal, and therefore VTB = VTB+1 = 0,

which contradicts L1:2. We prove P 2:1: Suppose contrary to P 2:1 that there is k such that P 2:3, k < T: Let k be the largest such date. Then 2 fH; Lg. If

H k+1

+

L k+1

cH ) + (1

> 0 and qk+1 = qk for

B ) Vk+2 :

B B Vk+2 (because the payo¤ to o¤ering a negligible price is Vk+2 ), then H L qk+1 uH + qk+1 uL

And since qk+1 = qk for qkH uH + qkL uL

cH

B Vk+1 :

B 2 fH; Lg; Vk+1 > 0 (by L1:2) and H L cH = qk+1 uH + qk+1 uL

cH

< 1, then

B B Vk+1 > Vk+1 :

Therefore a negligible price o¤er at k is not optimal, which contradicts that 0. Hence

H k+1

= 0, and thus L Vk+1 =

L k+1

L L k+1 (rk+1

> 0 and cL ) + (1

L k+1 )

L L Vk+2 = Vk+2 :

Therefore L rkL = cL + Vk+1

Since

L k+1

= 0: By

H is optimal, then > 0; i.e., o¤ering rk+1

H L B = (qk+1 uH + qk+1 uL Vk+1 B Since Vk+1

H k+1

H L k + k

L L L cL + Vk+1 = cL + Vk+2 = rk+1 :

L > 0; i.e., price o¤ers of rk+1 are optimal at date k + 1, we have L qk+1 (uL

L B qk+1 ) Vk+2

L rk+1 ) + (1

B Vk+2 :

Hence B Vk+2

uL

L rk+1

and B L Vk+1 = qk+1 (uL

L rk+1 ) + (1

35

L B qk+1 ) Vk+2

uL

L rk+1 :

H L k + k

=

Since

H k

L k

+

= 0; then the payo¤ to a negligible o¤er at date k is greater or equal

to the payo¤ to a low price o¤er at date k, i.e., qkL (uL

B Vk+1

Thus uL

rkL

rkL ) + (1

B qkL ) Vk+1 :

B B > 0 (by L1:2) and : Since Vk+1 Vk+1

uL

rkL

uL

B B < Vk+1 Vk+1

L < rkL , which is a contradiction. Hence i.e., rk+1

< 1, then L ; rk+1

L H k + k

> 0 for all k, which establishes

P 2:1. We prove P 2:2: Since q1H = q H < q by assumption and V2B > 0 by L1:2; then q1H uH + q1L uL

cH < 0 < V2B :

Hence o¤ering cH at date 1 is not optimal; i.e.,

H 1

= 0: Therefore

L 1

> 0 by P 2:1:

Lemmas 2 establishes properties that a DME has when frictions are not large. Recall that by assumption q H < q < q^ < 1: When

H t

+

L t

= 1 at some date t; then

the fraction of high quality sellers in the market at date t + 1 is H qt+1 =

mH t+1 = mH + (1 ) mLt+1 (1 t+1

H H (1 t )qt H H )(1 t )qt + (1

qtH )

= g(qtH ;

H t );

where the function g; given by g(x; y) :=

(1

(1 y)x y)x + (1 )(1

x)

is increasing in x and decreasing in y, and satis…es g(q H ; = Lemma 2. Assume that 1 < T < 1, all t 2 f1; : : : ; T g: H t

< 1.

(L2:2)

L t

< 1.

(L2:3)

H T

> 0;

L T

) > q^ by F:2.

< 1; and the inequalities F 1 and F 2 are

satis…ed (i.e., frictions are not large), and let (

(L2:1)

;

> 0; and qTH = q^.

(L2:4) VtL > 0. 36

H

;

L

; rH ; rL ) be a DME. Then for

(L2:5)

L t

> 0.

(L2:6)

H t

<

.

(L2:7) If t < T , then

L t

+

H t

< 1 and

H t+1

> 0.

The proof of lemma 2 is provided in the Online Appendix. Now we prove Proposition 3. Proof of Proposition 3. We show …rst that if (

H

;

L

; rH ; rL ) is a DME, then it is

given by P 3:1 to P 3:4, and the payo¤s and surplus are as given in Proposition 3. Since qTH = q^ by L2:3, then a buyer’s expected utility at T is VTB = (1

q^)(uL

cL ) =

T: H t

By L2:7 negligible o¤ers are optimal for all t < T , i.e., 1

L t

> 0. Then

B for t < T by DM E:B; and therefore for all t we have VtB = Vt+1

VtB =

(4)

t:

By L1:2 rtH = cH for all t: Since

H t

> 0; and 1

H t

L t

(5)

> 0 for 1 < t < T by L2:7, and

t+1

=

t

then qtH uH + (1

qtH )uL

B cH = Vt+1 =

t

by DM E:B. Hence for 1 < t < T we have qtH = Since

L t

H t

> 0 by L2:5 and 1 qtL (uL

cH u L + uH uL L t

t

:

(6)

> 0 for t < T by L2:7, then by DM E:B

rtL ) + (1

B B qtL ) Vt+1 = Vt+1 ;

i.e., uL

B = rtL = Vt+1

t:

Hence for t < T we have rtL = uL 37

t:

(7)

Moreover, since rTL

cL = VTL+1 by DM E:L; then rTL = cL :

(8)

We calculate the expected utility of low quality sellers. Since rtL

L for cL = Vt+1

all t by DM E:L; then equation (7) yields uL

L cL = Vt+1

t

for t < T: Reindexing we get VtL

1

=

u

L

H 1

for t 2 f2; : : : ; T g: And since

c

L

=

t 1

uL

cL

(9)

t;

= 0 by P 2:2, then

V1L = V2L = uL

cL

2

= uL

cL

(10)

1: H

Next we calculate the probabilities of high price o¤ers

. Since rtL

L cL = Vt+1

for all t by DM E:L; we can write the expected utility of a low quality seller as VtL =

H H t (c

cL ) + (1

VtL

L Vt+1 =

H H t (c

H t )

L Vt+1 ;

i.e.,

For 1 < t < T; since

t+1

=

cL

L Vt+1 ):

L then Vt+1 = uL

t;

cL

t

by equation (9), and

therefore VtL

L Vt+1 =

1

(uL

cL ):

Hence 1 and solving for

H t

(uL

cL ) =

H H t (c

cL

(uL

cL

t ));

yields H t

for 1 < t < T: Clearly (cH

H t

=

uL

1 cH

cL uL +

(11) t

> 0. Moreover, since uL +

t)

(cH

>

uL ) cH

(by F:1) > (1 +

) (1

)

= (1 +

) (1

) uL

> (1 38

) uL

cL ;

cL cL

then

H t

< 1.

Recall that

H 1

H T.

= 0 by P 2:2. We calculate VTL =

H H T (c

Since rT = cL by DM E:L, then

cL ):

Hence using (9) for t = T we have uL

cL

Solving for H T

=

H T

uL

and using

cL (cH

Substituting

T 1 L c )

=

T 1

and therefore

T

=

(1

=

T 1

= (1

(1

q^) (uL

H H T (c

=

T

1 uL cH

1 uL q^)) cH

= (1

H T

> 0: Moreover, since =

cL = (1 cL

cL = (1 cL

(1

< 1 by F:1, then L

We calculate the probabilities of low prices o¤ers H qt+1 =

Solving for

L t

(1

q^))

(12)

:

H H t )qt

+ (1

(

L t

q^)) H T

;

< 1.

. For each t we have

H H t )qt

(1 (1

cL yields

cL ) in this expression we get

H T

(1

q^) uL

(1 q^))

cL ):

H L t ))qt

+

:

we obtain L t

H for all t: Since qt+1

H qt+1 qtH H qt+1 (1 qtH )

H t )

= (1 H t

qtH by L1:3 and

L t

< 1, then

(13)

0: For t = 1 we have

H 1

=0

by P 2:2, and therefore L 1

=

2

(1

(u(q H ) cH ) q H )(cH uL +

where the inequality follows since u(q H ) Since

H T

+

L T

H T

H T

=1

T 1 : L c )

(15)

cH < 0.

If T > 2, then for t 2 f2; : : : ; T = (1

H t )

uL

=1

< 1 as shown above, we have

L t

(14)

= 1 by P 2:3; then L T

Since

> 0;

2)

L T

cL (cH

> 0.

2g; using equation (6) yields (1

cH

)

uL + 39

uH

t+1 t+1

uH

uL cH

> 0: t

(16)

Also qTH = q^ and equation (6) yields L T 1

u(^ q) q^(uH

H T 1)

= (1

cH cH

T 1 T 1)

:

Since q^)(uL

(1

cL ) = u(^ q)

cH ;

then uH

cH

T 1

= uH

cH

= uH

cH

u(^ q)

> uH

cH

uH

(1

) uH

= (1

q^)(uL

cL )

cH cH

cH > 0;

and u(^ q)

cH

cH

= u(^ q)

T 1

q^)(uL

(1

Hence L T 1

We show that

H t

H T 1 )(1

= (1 +

L t

)

+ (1

Hence q^(1 q1H ) > q^ q1H . Then in x, and q2H

H 1

T 1)

+

L 1

cH > 0:

(17)

> 0:

< 1. Since g(x; y) is

) > q^ (by F:2) then

q1H q1H

) u(^ q)

u(^ q ) cH q^(uH cH

< 1 for t < T: We …rst show

decreasing in y; q1H = q H , and g(q H ; = g(q1H ; 0) =

cL ) = (1

)(1

q1H )

> g(q H ; =

) > q^:

= 0 by P 2:2, (x q1H )=[ x(1 q1H )] is increasing

H 1

qTH = q^ by L2:3 and L1:3, imply H 1

For t 2 f2; : : : ; T

+

L 1

=

q2H q1H < q2H (1 q1H )

q^ q1H < 1: q^(1 q1H )

2g; from equation (11) we have H t

<

Also using equation (6), for 1 < t < T H qt+1 qtH = (1 H qt+1 (1 qtH )

)

uL cH

1

cL : uL

(18)

1 we have

(cH 40

t+1 uL +

uH t+1 ) u

H

uL cH

: t

Since in

t

t+1 ,

q^)(uL

< (1

cL ) for all t; and the ratio

t+1 =(c

H

uL +

is increasing

t+1 )

we have

H qt+1 qtH < (1 H (1 qtH ) qt+1

(1 q^)(uL cL ) uH uL cH uL + (1 q^)(uL cL ) uH cH (1 q^)(uL (1 q^)(uL cL ) uH uL ) cH u L uH cH (1 q^)(uL cL ) u L cL ) H ; c uL )

< (1 = (1

cL = u H

where the equality is obtained by substituting q^ = cH

cL )

cL : Using this

inequality and inequality (18) above we have H t

+

L t

H qtH qt+1 H (1 qtH ) qt+1 u L cL H H )(1 ) + (1 t t cH uL u L cL H 1 (1 ) H + (1 t c uL 1 uL cL uL 1 (1 ) cH uL cH 1 uL cL uL 1 (1 ) cH uL cH 1 uL cL (1 + ) cH uL (1 + ) (1 ) cH cL cH uL 1: H t

= < = < = < =

(by F:1) < As for t = T

H t )

+ (1

)

uL cH

cL uL

cL + (1 uL cL + uL

)

uL cH

cL uL

1; we have

H T 1

+

L T 1

=

H T 1

=

H T 1

H T 1)

+ (1

u(^ q) q^(uH

cH cH

T 1 T 1)

:

Rearranging yields H T 1

+

L T 1

H T 1

Substituting for u(^ q) H T 1

cH and +

L T 1

T 1

=

cH cH

u(^ q) q^(uH

1

T 1 T

1)

+

u(^ q) q^(uH

from equation (11) and using that

cH cH

= (1

T 1 T 1)

q^) uL

=

1 cH +

q^(uH

q^(uH

uL cL uL + cH

cH

) q^(uH

)

: 41

cH

u(^ q) cH

)

: cL = !

Since q^(uH

cH

)

cH

(u(^ q)

) = q^(uH = (1

cH

)

q^)(cH

uL +

(^ q uH + (1

q^)uL ) + cH +

);

then H T 1

+

L T 1

= =

Hence

H T 1

+

uL

cL (1 cH

q^)(cH uL +

uL +

)] <

q^(uH

cH

)

q^)] <

q^ uH

cH :

1 q^(uH (1

L T 1

cH

)

) [1 + (1 q^(uH cH

)

+

(1

)] : )

< 1 if and only if (1

) [1 + (1

i.e., 2

[1 Since q^

H

u

c

H

(1

q^ uH = 1 q^ uL

cH cH = H cL u

cL uH cH uL

cH 1 = ; L c

then this inequality becomes 2

1 which holds since

(1

q^) < 2

= > 1 by F:1 and 0 <

;

(1

q^) < 1.

The surplus can be calculated using (4), (10), and L1:2 as S DM E = mB V1B + mH V1H + mL V1L = (mL + mH ) = mH

1

L

+ mL (uL

+ mL (uL

Equations (11), (12) and P 2:2 identify and (15) identify

1

H

(19) cL

1)

cL ):

as given in P 3:1. Equations (14), (16)

as given in P 3:2. Equation (5) identi…es rH as given in P 3:3.

Equations (7) and (8) identify rL as given in P 3:4. The traders’payo¤s are identi…ed in equations (4) and (10), and in L1:2. The surplus is given in equation (19). Finally, as the construction above shows, the pro…le de…ned in P 3:1 to P 3:4 of Proposition 3 is indeed a DME. 42

)

!

Proof of Proposition 4. The unique DME as well as the traders’payo¤s and the surplus are given in Proposition 3. By P 3:1 H 1

lim

!1

= 0 = ~H 1 ;

and lim

!1

H t

= lim

uL

1

!1

cH

uL

uL

cL (cH

T

+

cL t (1 q^)(uL

cL )

= 0 = ~H t ;

for 1 < t < T; and also lim

!1

H T

= lim

!1

T 1 cL )

=

uL

cL (cH

cL )

= ~H T:

Since uH > uL > cL by assumption, then 0 < ~H T < 1: From equation (6) we have cH uL + !1 uH uL

lim qtH = lim !1

t

=

cH

uL + uH uL

:

for 1 < t < T . Also qTH = q^ implies lim qTH = q^: !1

P 3:2 implies lim

!1

L 1

= lim

cH

!1

uL + 2 q H (uH uL ) cH u L + q H (uH = (1 q H )(cH uL + 2 ) (1 q H )(cH uL +

and for 1 < t < T lim

!1

L t

uL ) = ~L1 ; )

1 (1 H t ) H c

= lim(1 !1

) t+1 uH uL uL + t+1 uH cH

= 0 = ~Lt ; t

and lim

!1

L T 1

= lim(1 !1

(1 H T 1)

q^(uH

q) ) u(^ H c

cH T 1)

=

(1 ) u(^ q) H H q^(u c

Also lim

!1

L T

H T

= lim 1 !1

=1

L ~H T = ~T :

L Thus, ~H T < 1 implies ~T > 0:

As for the traders’expected utilities, we have lim V1B = lim !1

!1

1

43

=

= V~1B ;

cH = ~LT 1 : )

and T 1

lim V1L = lim 1 !1

!1

q^) (uL

(1

cL ) = (1

cL ) = V~1L :

q^)) (uL

(1

Since VtH = 0; then lim VtH = 0 = V~tH : !1

H

L

It is easy to check that (~ ; ~ ; r~H ; r~L ) forms an equilibrium of the market when = 1: Finally, we have lim S DM E = lim mL (uL !1

!1 L

= m (uL

cL ) + mH

T 1

cL ) + mH (1

q^)(uL

(1

q^)(uL

cL )

cL )

= S~DM E :

Proof of Proposition 5. If frictions are not large, then the unique DM E is that given in Proposition 3. Thus, since limT !1 lim

T !1

t

= 0 for all t; we have

= 0 = ^H 1 ;

H 1

and for t > 1 we have H t

lim

T !1

Also lim

T !1

L 1

=

cH

= (1

)

uL cL = ^H t : (cH uL )

uL q H (uH uL ) = (1 q H )(cH uL )

q qH = ^L1 ; H q(1 q )

and for t > 1 we have lim

T !1

L t

= 0 = ^Lt :

Clearly limT !1 rtH = cH = r^tH , and limT !1 rtL = uL = r^tL : We show that the strategy distribution (^H ; ^L ; r^H ; r^L ) forms a DME when T = 1. Since (1 q H )q > q q H , then 0 < ^L1 < 1: Since

< 1; and

(cH cL ) > uL cL

by F:1, we have (cH

uL ) +

uL

cL >

(cH

uL ) +

Hence 0 < ^H t < 1 for all t > 1. 44

uL

cL =

(cH

cL ) > uL

cL :

Since r^tH = cH and r^tL = uL , then the (maximum) expected utility of high quality sellers is V^ H = 0 for all t. Hence r^H = cH for all t satis…es DM E:H. For t > 1 the t

t

expected utility of low quality sellers is uL V^tL =

cL

:

For t = 1 we have r^1L = cL + V^2L = uL . Hence r^tL = uL for all t satis…es DM E:L. Also V^1L = ^L1 (uL

^L1 ) V^2L = uL

cL ) + (1

cL :

L Using ^H 1 and ^1 we have

q2H =

qH ^L1 )(1

q H + (1

qH )

= q:

And since ^Lt = 0 for t > 1; then qtH = q2H = q: Hence qtH (uH

cH ) + (1

qtH )(uL

cH ) = 0

for t > 1; and therefore o¤ering the high price (cH ) leads to zero instantaneous payo¤ for all t > 1. Since q1H < q by assumption, then o¤ering the high price (cH ) at t = 1 leads to a negative instantaneous payo¤. Also since r^tL = uL for all t, then o¤ering the low price (uL ) yields a zero instantaneous payo¤. Thus, the buyers maximum expected utility is zero at all dates, i.e., V^ B = 0 for all t: Hence DM E:B is satis…ed. t

Proof of Corollary 3. We calculate the present value of a subsidy quality, which we denote for from below as

L

> 0 on low

< 1 by P V L ( ); and show that it approaches

L

mL

approaches 1. We have P V L( ) =

L

L L 1 m1

+

1 X

t 1 L

H L t mt :

t=2

Since

H t

is independent of t for t > 1 by P 5:1, denote

mL1 = mL ; and mLt = (1 P V L( ) =

L 1 )(1

L

mL

H t 2

)

L 1

+

H

H t

=

H

. Also, we have

mL for t > 1: Hence

(1

L 1)

1 X

t 1

(1

H t 2

)

t=2

=

L

mL

L 1

+

H

(1

L 1)

1 X t=1

45

t

(1

H t 1

)

!

! :

Since 1 X

t

H t 1

(1

)

=

t=1

= =

1 H)

(1 1 (1 1

1 X

H

(1

)

t

t=1

(1 H) 1 (1 H)

(1

H

) H)

;

then P V L( ) =

L

mL P ( );

where

H

P ( ) := Since 0 <

L 1

< 1 and

L 1

L 1)

+ (1

!1

P V L( ) =

+ (1

)

:

< 1; then P ( ) is a convex combination of 1 and a number

less than 1. Therefore P ( ) < 1 and P V L ( ) < then lim

H

L

L

mL : Further, since lim

!1

P ( ) = 1;

mL .

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48