Post on 13-Jul-2020
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?
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 2 / 27
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), ...
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 3 / 27
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).
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 4 / 27
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).
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 5 / 27
(Static) Competitive Equilibrium
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 6 / 27
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:
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 7 / 27
Dynamic Competitive Equilibrum
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:
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 8 / 27
Dynamic Competitive Equilibrum
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� ):
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 9 / 27
Dynamic Competitive Equilibrum
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):
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 10 / 27
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.
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A Decentralized Market
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.
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 12 / 27
A Decentralized Market
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:
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 13 / 27
A Decentralized Market
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:
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 14 / 27
A Decentralized Market
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.)
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 15 / 27
A Decentralized Market
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:
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 16 / 27
A Decentralized Market
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 .
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 17 / 27
A Decentralized Market
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 .
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 18 / 27
A Decentralized Market
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.
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 19 / 27
A Decentralized Market
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.
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 20 / 27
A Decentralized Market
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
�:
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 21 / 27
A Decentralized Market
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 �.
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 22 / 27
A Decentralized Market
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.
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 23 / 27
A Decentralized Market
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
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 24 / 27
A Decentralized Market
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 .
Moreno-Wooders (UC3M-UTS) Market for Lemons UNSW, Dec. 2011 25 / 27
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.
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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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