Equilibrium Dynamics in Markets for Lemons€¦ · Equilibrium Dynamics in Markets for Lemons Diego...

Equilibrium Dynamics in Markets for Lemons

Diego Moreno and John Wooders

UNSW, Dec. 2011

Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 1 / 27

Introduction

Akerlof (1970): A (static) competitive market withadverseselection produces ine¢ cient outcomes.

. How do dynamic competitive markets with adverse selection perform?

. May a dynamic market recover from adverse selection?

. What determines the liquidity of an asset?

. Do some market structures perform better than others?

. What is the role of market frictions?

. Is there room for government intervention?


The Literature

Competitive Markets:

Akerlof (1970), ... Wooders (1998), Janssen and Roy (2002), ...

Matching and Bargaining markets:

Mini-micro foundations of CE: Binmore and Herrero (1988),Osborne and Rubinstein (1990), Wolinsky (1990), Gale (1996),Moreno and Wooders (2002), Blouin (2003), ...

Adverse Selection: Moreno and Wooders (2010), Kim (2011),Bilancini and Boncinelli (2011), Camargo and Lester (2011) ...

Adverse Selection in Finance:

Rocheteau (2009), Chang (2010), ..., Morris and Shin (2011), ...


The Market

An indivisible commodity whose quality is either high or low.

A measure qH 2 (0; 1) of sellers, each with a unit of high qualitygood whose opportunity cost is cH :

A measure qL = 1� qH of sellers, each with a unit of low qualitygood whose opportunity cost is cL:

A unit measure of buyers whose values for a unit of high and lowquality good are uH and uL; respectively.

Information:

� sellers know the quality of their unit� quality is unobservable to buyers (prior to purchase).


The Market

If a buyer and a seller trade at price p; then their payo¤s are u � p andp � c ; respectively, where:

u = uH and c = cH if the unit traded is of high quality,

u = uL and c = cL if the unit traded is of low quality.

Assumptions :

(i) uH > cH and uL > cL (there are gains to trade both qualities).

(ii) u(qH ) := qHuH +�1� qH

�uL < cH (Lemons problem).


(Static) Competitive Equilibrium


Dynamic Competitive Equilibrum

The market opens for T consecutive periods. Traders discount � 2 (0; 1]:

Supply

For p = (p1; :::; pT ) 2 RT+ and � 2 fH; Lg let

v � (p) = maxt2f1;:::;T g

f0; �t�1(pt � c� )g:

The supply of � -quality good, S� (p); is the set of sequences s� 2 RT+:

(S :1)XT

t=1s�t � q� ,

(S :2) s�t > 0 implies �t�1(pt � c� ) = v � (p), and

(S :3)�XT

t=1s�t � q�

�v � (p) = 0:



Demand

For p = (p1; :::; pT ) 2 RT+ and u = (u1; :::; uT ) 2 [uL; uH ]T let

vB (p; u) = maxt2f1;:::;T g

f0; �t�1(ut � pt)g:

The market demand, D(p; u), is the set of sequences d 2 RT+:

(D:1)XT

t=1dt � 1,

(D:2) dt > 0 implies �t�1(ut � pt) = vB (p; u), and

(D:3)�XT

t=1dt � qB

�vB (p; u) = 0:



A CE is a collection (p; u; sH ; sL; d); such that sH 2 SH (p); sL 2 SL(p),d 2 D(p; u); and for each t:

(CE :1) sHt + sLt = dt ; and

(CE :2) dt > 0 implies ut = (uH sHt + uLsLt )=

�sHt + s

Lt

�:

The surplus may be calculated as

SC =X

�2fH ;Lg

TXt=1

s�t �t�1(u� � c� ):



Prop. 1. Assume �T�1 > g. Then in every CE low quality tradesimmediately at the price uL and high quality does not trade; i.e.,

p1 = uL; sH1 = 0; sL1 = d1 = q

L; and

sHt = sLt = gt = 0 for t > 1.

Traders�payo¤s are

vL = uL � cL; and vH = vB = 0,

and the surplus is

SC = qL(uL � cL):


A Decentralized Market

The market opens for T consecutive periods. Traders discount� 2 (0; 1]:

Trade is bilateral.

Each date buyers and sellers in the market are matched withprobability � 2 (0; 1).

A matched buyer proposes a price at which to trade:

If the o¤er is accepted, then agents trade and leave the market.

If it is rejected, then agents split and wait for a new match.



A pure strategy for a buyer is a sequence of price o¤ers(p1; :::; pT ) 2 RT+.

Buyers�strategies are described by a sequence � = (�1; :::; �T ); where�t is c.d.f. with support on R+.

A pure strategy for a � -quality seller is a sequence of reservationprices r � = (r �1 ; :::; r

�T ) 2 RT+:

In equilibrium the reservation prices of sellers of the same type are thesame.



Let (�; rH ; rL) be a strategy distribution.

The probability that a seller of quality � 2 fH; Lg who is matched at datet trades is

��t =

Z 1

r�t

g�t :

The stock of � -quality sellers in the market at t + 1 is

m�t+1 = (1� ��t )m�t ;

and m�1 = q� :

The fraction of � -quality sellers in the market at each date is;

q�t =m�t

mHt +mLt:



The expected utility of a � -quality seller at date t is

V �t = �Z 1

r�t

(p � c� ) g�t(p) + (1� ��t )�V �t+1;

andV �T+1 = 0:



The expected utility at date t of a buyer who o¤ers (p1; :::; pT ) 2 RT+ is

Vt(p1; :::; pT ) = �X

�2fH ;Lgq�t I (pt ; r

�t )(u

� � pt)

+

241� � X�2fH ;Lg

q�t I (pt ; r�t )

35 �Vt+1(p1; :::; pT );and

VT+1(p1; :::; pT ) = 0:

(Here I (x ; y) is the indicator function.)



WriteV Bt = max

RT+Vt :

Since Vt is independent of (p1; :::; pt�1); we have

V Bt = maxR+

�X

�2fH ;Lgq�t I (pt ; r

�t )(u

� � pt)

+

241� � X�2fH ;Lg

q�t I (pt ; r�t )

35V Bt+1:



A strategy distribution (�; rH ; rL) is a DE if for each t:

(DE :�) r �t � c� = �V �t+1 for � 2 fH; Lg; and

(DE :B) Vt(pt) = V Bt for every pt in the support of �t :

The surplus in DE can be calculated as

SD = V B1 + qHV H1 + q

LV L1 .



Prop. 2. If (�; rH ; rL) is a DE, then

rHt = cH > rLt and q

Ht+1 � qHt for all t;

and the prices o¤ered with positive probability are cH , rLt , and below rLt .



Intuition:

Prices greater than cH are o¤ered with prob zero (Diamond paradox).Hence

V Ht = 0; and rHt = cH

cH is the maximum priced o¤ered, and

rLt = cL + �V Lt+1 � cL + �(cH � cL) < cH :

Prices pt 2�rLt ; c

H�are suboptimal.

Since rLt < cH = rHt ; H-sellers leave the market faster than L-sellers;

i.e., qHt is non-decreasing.



Prop. 2 implies that in a DE:

the high price (cH ) is o¤ered with probability �Ht = �Ht ;

the low price (rLt ) is o¤ered with probability �Lt = �

Lt � �Ht ;

negligible prices (p < rLt ) are o¤ered with probability 1��Ht + �

Lt

�:

A DE may be described by a pro�le (�H ; �L; rH ; rL):

Remark. If T = 1; then (�H ; �L; rH ; rL) = (0; 1; cH ; cL) is the unique DE.



Prop. 3. If T > 1 and frictions are small, then in the unique DE:

�H1 = 0; �L1 > 0; and 1� �L1 � �H1 > 0,

�Ht > 0; �Lt > 0; and 1� �Ht + �Lt > 0 for t 2 f2; :::;T � 1g

�HT > 0; �LT > 0; and 1� �HT + �LT = 0,

traders�payo¤s are V H1 = 0;

V L1 =�1� �T�1� (1� �q)

��uL � cL

�;

andV B1 = �

T�1� (1� �q)�uL � cL

�;

and the surplus is

SD =�qL + qH �T�1� (1� �q)

��uL � cL

�:



The assumption that frictions are small in Prop. 3 requires

�T�1� > g

and

��1� qH

�> 1�

�1� (1� �q) g

�

� qH�q:

Note that:

The payo¤ to buyers (low quality sellers) is above (below) theircompetitive payo¤, and decreases (increases) with T ; and increases(decreases) with � and �.

The surplus is above the competitive surplus and decreases with Tand increases with � and �.



Intuition. Assume that � and � are near one.

Since u(qH ) < cH ; then �H1 = 0:

�Ht < 1 for all t: (If �H�t = 1; then r

Lt > u

L for t < �t since � near one.)

Hence qHt � �q (otherwise �HT = 1):

�Lt < 1 prior to T (otherwise qHt+1 > �q since � near one)

�HT + �LT = 1.

�Ht + �Lt > 0 for all t:

Showing that �Ht+1; �Lt+1 > 0 and �

Ht + �

Lt < 1 for all t < T is more

involved.



An example:

qH = :2, uH = 1, cH = :6, uL = :4, cL = :2; � = � = :95t qHt rLt �Ht �Lt V Lt V Bt1 :2 :31426 0 :77179 :11426 :085742 :48375 :30975 :03818 :06386 :20276 :090253 :5 :2 :30402 :69598 :11552 :095

SD = :17715



Prop. 4. If T > 1; as � and � approach one the distribution of priceo¤ers approaches (~�H ; ~�L) given by

~�H1 = 0 < ~�L1 ;

~�HT > 0 = ~�LT , and

~�Ht = ~�Lt = 0 for all t 62 f1;Tg.

The payo¤ to buyers (low quality sellers) approaches

~V B = (1� �q)�uL � cL

�(resp. ~V L = �q

�uL � cL

�)

and the surplus approaches

~SD =�qL + qH (1� �q)

��uL � cL

�;

independently of T .


Long Lived Markets

Prop. 5. If �T�1 < g , then every pro�le (p; u; sH ; sL; g) such that

pt = uL + �T (�)�1(uH � pt) for t < T (�); pt 2 [cH ; uH ] for t � T (�),

ut = uL for t < T (�); ut = uH for t > T (�),

sL1 = g1 = qL; sH�T (�) = g �T (�) = q

H ; and sHt = sLt = gt = 0 otherwise,

is a CE. In these equilibria the surplus is

SC = qL(uL � cL) + qH �T (�)�1(uH � cH );

and approaches

~SC = qL(uL � cL) + qH g(uH � cH )

as � approaches one.


Long Lived Markets

Prop. 7. If T =1; �� > g ; and ��1� qH

�> 1� qH=q�, then the

"limiting" DE of a short lived market, given by

rHt = cH , rLt = u

L;

�H1 = 0; �L1 =

�q� � qH

�=��1� qH

�q��; and

�Lt = 0; �Ht = (1� �) g=�� for t > 1,

is a DE. In this DE traders�payo¤s are V B1 = VH1 = 0 and V L1 = u

L � cL;and the surplus is

SD1 = qL(uL � cL);

independently of the values of � and �: As � approaches one, �Htapproaches zero, and only low quality trades.


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