;y

Adam Chi Leung Wongb;z

a

CIREQ, CIRANO and Department of Economics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada b School of International Business Administration, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, China 200433 January, 2010x

Abstract In a dynamic matching and bargaining market with costly search, we nd that private information typically deters entry. But, the welfare can actually be higher under private information. Keywords: Matching and Bargaining, Search Frictions, Two-sided Incomplete Information JEL Classi cation Numbers: C73, C78, D83.

1

Introduction

Can private information improve the e ciency of bargaining? In this paper, we show that, yes, it can if bargaining is imbedded in a dynamic matching market. We show this by comparing equilibrium outcomes of two models. The rst is the full information bargaining model of Mortensen and Wright (2002).1 The second is its private information replica, a model recently proposed by Shneyerov and Wong (forthcoming 2009). Speci cally, we study the steady state of a market with continuously in owing cohorts of buyers and sellers who are randomly matched pairwise and bargain under either full or private information. The in owing traders are heterogeneous in their types (valuations for buyers and costs for sellers). The types are drawn independently once and remain unchanged throughout their Corresponding author. Tel.: +1 514 848 2424 ext 5288. Fax: +1 514 848 4536 E-mail address: [email protected] z E-mail address: [email protected] x The results appearing in this paper were originally contained in the 2006 version of Shneyerov and Wong (forthcoming 2009). 1 Their model is an extension of Rubinstein and Wolinsky (1985) and Gale (1986) to general bilateral matching technologies. y

1

lifetime. There are frictions due to costly search, and time discounting at the rate r 0. The process of matching is bilateral and random. Once a buyer is matched to a seller, each trader proposes a price with some positive probability.2 If the o er turns out to be unacceptable, the match is dissolved and the traders go back to the pool of unmatched traders and search again. In both models, equilibria exist in which all matches result in trade. Even more: If r is small enough, there is a unique equilibrium and it involves full trade. This can be easiest understood by noting that with r = 0, the costs of delay are independent of a trader's types. Therefore, reservation prices do not depend on valuations and if a price is accepted, it is accepted by all traders. Similarly, optimal price o ers do not depend on valuations. This is a special feature of search models without discounting. In particular, since the types do not a ect preferences over prices and acceptance decisions, in models with r = 0 the information structure does not matter.3 However, when r is positive, the information structure a ects the distribution of the surplus between the traders and thereby it a ects the entry decisions. As we show, introducing private information reduces entry - an "entry deterrence e ect" - because "marginal entrants" receive a smaller share of the surplus under incomplete information. The marginal entrants are the lowest-value active buyers and the highest-cost active sellers. With full information, traders obtain positive rents when they propose, and zero rents when they accept o ers that are only marginally good to them. With private information, traders obtain smaller rents when they propose, but larger rents when they respond. But the marginal entrants in both models have zero rents when they respond. This means that they get smaller rents under private information, which softens the incentives to enter the market. We show that entry deterrence may increase or decrease welfare. When r = 0, entry in a model with full information is constrained e cient (i.e. the trading surplus is maximized subject to the constraint of the matching process) if the ratio of the probabilities of making an o er satis es the "Hosios (1990) condition". When the Hosios condition does not hold and when r is positive but close to zero, entry decisions under full information are ine cient, and reducing entry might actually improve welfare. We show yes, it can. We derive a precise mathematical condition for this welfare enhancing e ect of entry. There is an extensive related literature on dynamic matching and bargaining games. Until very recently, most of it has considered full information bargaining: e.g. Mortensen (1982), Rubinstein and Wolinsky (1985, 1990), Gale (1986), Gale (1987).4 Satterthwaite and Shneyerov (2007), Satterthwaite and Shneyerov (2008), and, in a bilateral setting, Shneyerov and Wong (forthcoming 2009) and Shneyerov and Wong (forthcoming 2010) have recently introduced two-sided private information in a dynamic matching market in a steady state, and have shown that the presence of private information does not a ect convergence to perfect competition.5 Several papers have compared complete and incomplete information 2

This is a natural generalization of the Nash bargaining solution to a private information setting. In addition, Atakan (2009) shows that even if traders are allowed to o er general mechanisms, they can do no better than making take-it-or-leave-it o ers. 3 This observation is also made in Atakan (2008). 4 A notable exception is the unpublished manuscript Butters (1979). 5 Other papers that have incorporated private information in some form are Wolinsky (1988), De Fraja and Sakovics (2001), Serrano (2002) and Moreno and Wooders (2002). Several recent papers have explored convergence under private information in more detail: Satterthwaite and Shneyerov (2008) show convergence

2

in search markets, e.g. Inderst (2001), Inderst and M• uller (2002), Moreno and Wooders (2002), and Lauermann (2008). The last two papers show that more information might be worse for some parameter constellations. Moreno and Wooders (2002) consider a bilateral matching and bargaining market with two-sided private information and one-time entry. They construct a numerical example in which the surplus is (slightly) higher with private information. In Lauermann (2009b), there is no cost of search, and all potential traders enter. There is no discounting either. The only friction is exogenous exit rate . The sellers have all the bargaining power, and all have the same cost. As we do here, Lauermann (2009b) considers both private and full information. He shows that, as the friction is removed ( ! 0), under private information all equilibria converge to perfect competition. In particular, the price o ers converge to the sellers' cost. But under full information, there are equilibria that do not converge to perfect competition. The intuition is that sellers are able to price discriminate even when is small. As a result, the lowest price o er made by the sellers is above the cost and some buyers who should be included on e ciency grounds are now excluded. Since none of the papers that have compared full and private information have done so in a setting with costly search, the channels for welfare e ects they have identi ed are very di erent from the one considered here. This paper is the rst to identify and study the e ect of private information on entry, and through it, on the e ciency of matching.

2

The Models

Here, we review the models of Mortensen and Wright (2002) (hereafter MW) and Shneyerov and Wong (forthcoming 2009), but focus only on the essential details of these models and their equilibria. The agents are potential buyers and sellers of a homogeneous, indivisible good. Each buyer has a unit demand for the good, while each seller has unit supply. All traders are risk neutral. Potential buyers are heterogeneous in their valuations (or types) v of the good. Potential sellers are also heterogeneous in their costs (or types) c of providing the good. We assume v; c 2 [0; 1]. Time is continuous and in nite horizon. The agents discount future utility at the instantaneous rate r. Potential buyers and sellers are continuously born at rates b > 0 and s > 0. The types of new-born traders are drawn at the time of their birth and remain the same throughout their life in the market. Speci cally, buyers draw their valuations i.i.d. from distribution F and the sellers draw their costs i.i.d. from distribution G. Once born, each trader decides whether to enter the market. Those who do not enter get zero payo . Those who enter incur the search cost continuously at the rates B > 0 and S > 0, until they leave the market. We assume that F and G have densities bounded away from 0 and 1. in the model that is a replica of Satterthwaite and Shneyerov (2007) except that it has exogenous exit rate. Lauermann (2008) shows convergence even if one side of the market has all the bargaining power, and Lauermann (2009b) shows that in that case, the welfare under private information may be higher than under full information. Atakan (2009) provides a generalization to multiple units. Shneyerov and Wong (forthcoming 2010) establish the rate of convergence for the model of this paper, and also demonstrate the lack of convergence for the double auction mechanism. Lauermann (2009a) derives a set of general conditions for convergence.

3

We model the process of search by the means of a matching function M (B; S) that gives the rate of matching of the mass of active buyers B and the mass of active sellers S. The matching function M is (i) continuous on R2+ , (ii) nondecreasing in each argument, (iii) satis es M (0; S) = M (B; 0) = 0, and (iv) exhibits constant returns to scale (i.e. is homogeneous of degree one). We denote by a (steady-state) ratio of buyers to sellers, i.e. B=S. Since the matching function exhibits constant returns to scale, the arrival rates for buyers and sellers are `B ( )

M ( ; 1)= ;

`S ( )

M ( ; 1):

(1)

Once a pair of buyer and seller is matched, they bargain either observing (under full information) or not observing (under private information) the type of their partner. With probability S 2 (0; 1), the seller makes a nal o er to the buyer, and the buyer chooses whether to accept or reject. And with probability B = 1 S , the buyer proposes and the seller responds. If a type v buyer and a type c seller trade at a price p, then they leave the market with payo s v p, and p c respectively. If the bargaining breaks down, both traders return to the pool of unmatched traders and search again.

3

Full Trade Equilibria

Turning to the properties of market equilibria, let WB (v) and WS (c) be traders' equilibrium values of search. A buyer of type v will accept any price o er p such that v p > WB (v), and similarly any seller of type c will accept any price o er p such that p c > WS (c). The responding strategies are therefore characterized by the reservation prices v WB (v) for the buyers and c + WS (c) for the sellers. The proposing strategies pB and pS di er according to whether the information is full or private. Under full information, the traders can do no better than propose the reservation price of their partner, pB (v; c) = c + WS (c) and pS (v; c) = v WB (v) if v c > WS (c) + WB (v); otherwise the matched pair does not trade. Under private information, the proposing strategies pB (v) and pS (c) are chosen by buyers and sellers optimally given the market distributions of their partners' reservation prices. In this paper, we focus on a simple class of nontrivial steady-state equilibria, full trade equilibria, in which every meeting results in trade. We therefore do not require a full equilibrium characterization and refer the reader to MW and Shneyerov and Wong (forthcoming 2009) for further details. The only equilibrium components that will matter for us here are (i) v and c, the marginal participating types of buyers and sellers v

inf fv : WB (v) > 0g ;

c

sup fc : WS (c) > 0g ;

and (ii) , the steady state ratio of the market stocks of buyers (B) and sellers (S). MW and Shneyerov and Wong (forthcoming 2009) show that the Walrasian price pW , de ned as the unique solution to sG (p) = b [1 F (p)], falls in between the marginal participating types: pW 2 (c; v)

3.1

Private Information Model

Shneyerov and Wong (forthcoming 2009) prove several properties of a full trade equilibrium 4

1

v − WB (v )

pS (c )

v

pB (v )

c

c + WS (c )

,c v 0

1

v

c

Figure 1: A full trade equilibrium under private information

in the private information model. Refer to Figure 1. First, the supports for active buyers' types and active sellers' types are separate, i.e. v > c. This is because otherwise a buyer with v will not trade if he meets a seller with c: the seller will not propose or accept anything less than c, while the buyer will only propose or accept something below v. Second, under private information, the lowest buyer's (and hence all active buyers') o er is exactly at the level just acceptable to all active sellers, i.e. is equal to c; and similarly, the highest seller's (and hence all active sellers') o er is exactly at the level just acceptable to all active buyers, i.e. equal to v. We index the private (resp. full) information equilibrium objects by p (resp. by f ). Under private information, the full trade equilibrium admits a very simple characterization. The marginal traders must be indi erent between participating or not, which leads to the following two indi erence equations: `B ( p )

B

vp

cp

=

B;

(2)

`S ( p )

S

vp

cp

=

S:

(3)

The intuition of (2) is that, rst, the marginal buyers make positive pro t only when they propose. Second, they propose cp . Because marginal buyers make 0 expected net pro t, their expected pro t rate from bargaining, `B ( p ) B v p cp , must be equal to B , the rate of their search cost. The intuition for (3) is parallel.

5

From (1), `S ( p )=`B ( p ) =

p,

so (2) and (3) can be easily solved for =

p

vp

S

S

B

cp = K

where K( )

B

B

`B ( )

+

p

and v p

cp :

;

(4)

;

(5)

p S

`S ( )

:

A necessary condition for existence of a full trade equilibrium is that K p < 1, which we assume throughout this paper. Since successful traders leave the market in matched pairs, in steady state, the in ow of active buyers must equal the in ow of active sellers: b[1

F v p ] = sG (cp ) :

(6)

Since v p cp is determined from (5), v p and cp are uniquely pinned down by (6). Note that neither the marginal participating types nor p depend on r. The triple p ; v p ; cp may not always characterize an equilibrium, because the marginal buyers may have an incentive to o er prices below cp , and marginal sellers { to o er prices above v p . However, Shneyerov and Wong (forthcoming 2009) show that these solutions do in fact characterize an equilibrium if r is smaller than a positive threshold, and the equilibrium is unique.6 Further, the search values of buyers and sellers are given by WBp (v) =

`B r + `B

p

v

vp ;

WSp (c) =

p

`S r + `S

p

(cp

The total utility ow of the arriving buyers and sellers is Z cp Z 1 WSp (c)dG (c) : WBp (v)dF (v) + s Wp (r) = b vp

3.2

c) :

(7)

p

(8)

0

Full Information Model

The characterization of a full trade equilibrium in the full information model is slightly more complicated because the payo s of the marginal types depend on the market distribution of partner types, not only on the marginal partner type. Because each meeting results in trade, these distributions are truncations of the original distributions of types in the arriving ows, i.e. are equal to 1 FF(v)v for buyers and GG(c) for sellers. Now f , v f and cf ( f) (cf ) are functions of r implicitly determined by the equations parallel to the private information model: Z cf dG (c) v f c WSf (c) `B f B = B; (9) G (cf ) 0 Z 1 dF (v) `S f S [v WBf (v) cf ] = S; (10) 1 F vf vf b 1 6

F (v f )

In Proposition 3 of Shneyerov and Wong (forthcoming 2009).

6

= sG(cf );

(11)

where WBf (v) =

B `B

r+

f

B `B

v

vf ;

WSf (c) =

f

S `S

r+

S `S

f

(cf

c) :

f

MW (in Proposition 1) show that, for r > 0 small, there is a unique solution f ; v f ; cf to (9) - (11) and it has v f > cf . This solution characterizes a unique equilibrium. The total utility ow of the arriving buyers and sellers now is Z 1 Z cf WSf (c) dG (c) : (12) Wf (r) = b WBf (v)dF (v) + s vf

4

0

How Private Information Can Be Good For E ciency

When r = 0, the buyer-seller ratio and the marginal participating types in full trade equilibria of both models are equal: f (0) = p , v f (0) = v p and cf (0) = cp . In a privateinformation full-trade equilibrium, v p and cp do not depend on r. Under full information, the following lemma shows that for r > 0 the marginal types are closer to each other (and to the Walrasian price pW ) than they are under private information. Lemma 1 For r > 0, we have v p > v f and cp < cf .7 Proof. From the in ow balance equations (6) and (11), the two inequalities v p > v f and cp < cf are equivalent. Therefore it su ces to prove v p cp > v f cf . We will consider `B ( f ) since `B two cases: p f and p < f . Suppose p f rst. Then `B ( p ) is a nonincreasing function. Now the indi erence conditions (2) and (9) for the marginal buyers imply Z cf dG(c) `B ( p ) B (v p cp ) = `B ( f ) B v f [c + WSf (c)] ; G(cf ) 0 vp

cp

vf > vf

Z

cf

[c + WSf (c)]

0

dG(c) G(cf )

cf ;

where the last inequality follows from c + WSf (c) < cf + WSf (cf ) = cf for c < cf .8 Now suppose p < f . Applying symmetric logic to the sellers' marginal equations (3) and (10), we again obtain v p cp > v f cf . For intuition, rst notice that, in both models, the marginal entrants get zero rent from bargaining when they respond. Under full information, the marginal entrants extract full rents from the partners when they propose. In contrast, under private information, the 7

In this form, this result rst appeared in Chapter 3 of the second author's Ph.D. dissertation, Wong (2009). The 2006 version of Shneyerov and Wong (forthcoming 2009) contained a weaker version of this result, for su ciently small r > 0. 8 This is because r > 0 implies c + WSf (c) is strictly increasing in c.

7

marginal entrants are only able to extract the rents of the most ine cient partner type when they propose. If r = 0, this makes no di erence because there is no heterogeneity in the partners' reservation prices.9 As r increases away from 0, the distributions of reservation prices becomes more heterogeneous, and there are more rents to be extracted under full information. Of course, the marginal entrants have to be indi erent between entering or not. Hence under full information less e cient types of traders enter. Let WB0

b

Z

1

v

v p dF (v) = b

vp

WS0

s

Z

cp

(cp

c) dG (c) = s

0

Z

1

v

v f (0)

Z

v f (0) dF (v) ;

cf (0)

(cf (0)

c) dG (c)

0

be the total search values of the arriving ows of buyers and sellers when r = 0. When there is no discounting, they are the same in both models. Therefore, in order to compare welfare under full and private information for small r, it is su cient to compare the slopes Wp0 (r) and Wf0 (r) evaluated at r = 0. Direct calculations using (8) and (12) show that

and Wf0 (0) =

1

WB0

B `B

WS0

WB0

Wp0 (0) =

`B

WS0

1

S `S

p

`S

p

(13)

p

v 0f (0)]:

+ sG (cp ) [c0f (0)

(14)

p

We will show that whether Wp0 (0) or Wf0 (0) is greater depends on the signs of the slopes 0 p ) and f (0).

K 0(

Lemma 2 The slope K 0 ( p ) is of the same sign as S

S;

p

(15)

where S ( ) SM2 (B; S) =M (B; S) is the elasticity of the matching function with respect to the mass of sellers. The slope 0f (0) is of the same sign as WS0

WB0

S

:

(16)

B

Proof. To prove (15), note that K( ) = K0 ( ) = 9

When r = 0, v

+ S ; M ( ; 1) 1 [ M ( ; 1)2 B

BM

( ; 1)

M1 ( ; 1) (

WB (v) and c + WS (c) are constants in both models.

8

B

+

S )] :

For

=

p,

B

0 S = S = S , so K ( M1 ( ;1) M ( ;1) is the elasticity

+

) has the same sign as

B

M1 M p

=

B

B

p

,

of the matching function with respect to the mass of where B ( ) buyers. Since the matching technology has constant returns to scale, the matching function 0 is homogeneous of degree 1, therefore B = 1 S , hence the slope K p has the same sign as S p . S To prove (16), divide the buyers' marginal type equation (9) through by `B f , apply integration by parts to the integral in the left-hand side, di erentiate through at r = 0, and rearrange: " Z " # " # # cf rc + S `S f cf dG (c) d d B vf = B dr G (cf ) dr `B f r + S `S f 0 r=0 r=0 " # Z cf G (c) d r v f cf + dc = B B ( p ) 0f (0) B dr r + S `S f 0 G (cf ) r=0 " # WS0 0 0 cf (0) + (17) = B B ( p ) 0f (0) B v f (0) s S `S p G (cp ) where B ( p)

1 `B ( )

d d

= =

p

`0B

p

`B

p

2

> 0:

Working with the sellers' marginal type equation (10) in the same fashion, we have " # WB0 0 0 0 cf (0) + = S v f (0) S S ( p ) f (0) b B `B p 1 F v p where S( )

1 `S ( )

d d

= =

p

`0S

p

`S

p

2

Equations (17) and (18) can be solved for v 0f (0) c0f (0) and from the characterizing equations for p ; v p ; cp , we get 0 f

(0) =

K p sG (cp )

S S ( p)

B B ( p)

+

S

Wp0 (0) =

> 0: 0 f

(0). After some rewriting,

WS0

B

Sum (17) and (18), insert the resulting c0f (0) obtain after some algebra: Wf0 (0)

1

(18)

S

WB0

:

(19)

B

v 0f (0) into (14), and cancel terms, we

sG (cp ) K 0 ( p )

0 f

(0) :

(20)

Under private information, increasing the discount rate from 0 to a small dr > 0 leads to welfare loss given by the r.h.s. of (13) times dr because buyers and sellers in the arriving ow have expected search times until the next pro table trade given by 1=`B p and 1=`S p , and have their utilities discounted proportionately over those time periods. But 9

under full information, there is also an indirect entry e ect of discounting: Lemma 1 implies that the full information model has more entry. The new entrants congest the market for their own side, and provide a thicker market for the other side. The entry of traders could increase or decrease the buyer-seller ratio f , which in turn a ects the expected searching times. The r.h.s. of (20) is the indirect e ect on the total accumulated search costs incurred by the ow of buyers and sellers. In the presence of matching externalities, more entry may or may not be socially desirable. Our main result is the following corollary. Corollary 1 For all su ciently small r > 0, the private information welfare Wp (r) is higher (resp. lower) than the full information welfare Wf (r), if WS0 S

p

WB0

S S

(21)

B

is positive (resp. negative). The di erence Wp0 (0) Wf0 (0) can be either positive or negative, depending on the elasticity of the matching function. For example, if b = s, the new-born distributions F and G are both p uniform [0; 1] (implying WS0 = WB0 ), and the matching function is Cobb-Douglas BS (implying S = 1=2), then the sign of Wp0 (0) Wf0 (0) is the same as 1 S ( B S ). In other words, when discount rate is positive but small, the private 2 information welfare is higher (resp. lower) than the full information welfare if the side with greater bargaining power incurs higher (resp. lower) search costs. To get more insight, recall that traders' entry imposes a positive externality on the opposite side of the market and a negative externality on the same side. When r = 0, MW show that the positive and negative externalities completely cancel out only when the Hosios (1990) condition holds, i.e. S = S . If, for example, the elasticity of matching function with respect to the mass of sellers, S , is larger than sellers' bargaining weight S , then the equilibrium buyer-seller ratio is higher than the constrained optimal level, hence decreasing would be welfare enhancing. On the other hand, Lemma 2 implies that for small positive r, if WS0 > WB0 , then the private information model has a smaller . S B Therefore, private information can enhance welfare if S > S and WS0 > WB0 . S B

10

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