LEMON OR PEACH?:
A DYNAMIC TRADING PROCESS UNDER ASYMMETRIC
INFORMATION
IN-KOO CHO AND AKIHIKO MATSUI
Abstract. This paper investigates the dynamic market for lemons (Akerlof (1970)),
which is modeled as a random matching model between buyers and sellers with the option
of initiating a long term relationship (e.g., Burdett and Wright (1998) and Mortensen
and Wright (2002)) with informational friction. In contrast to the prediction from most
existing dynamics models of the market for lemons, we demonstrate that the equilibrium
payoff of the low quality seller is significantly affected by the presence of the high quality
seller in the market. The presence of high quality sellers is necessary to generate the
informational rent for a low quality seller by providing an opportunity for imitation, while
also increases the competitive pressure among sellers. If players are impatient, then the
buyer can take advantage of the competitive pressure among sellers to extract positive
surplus from trading, which may happen even if the buyer’s mass is larger than the mass
of low quality sellers. If players are patient, then the informational rent dominates the
competitive pressure among sellers so that the low quality seller can extract the entire
gain from trading, which may happen even if the mass of low quality seller is larger than
the mass of buyers. In such a case, a positive mass of buyers chooses to stay in the
matching pool for a positive amount of time to look for a better deal with a seller, even
though the frequency of meeting a seller increases indefinitely.
Keywords: Matching, Search, Undominated equilibrium, market for lemons, peaches,
common value
1. Introduction
Akerlof (1970) demonstrated that if low quality product (“lemon”) exists together with
high quality product (often called “peach”), and if consumers cannot differentiate high
quality product from low quality product at the time of purchase, the market for high
quality good may collapse. This problem is known as lemons problem. This paper sets
forth a dynamic decentralized trading model for lemons and peaches and discovers economic implications of peaches, which have been overlooked in the existing literature.
A common feature among equilibria in the literature on dynamic markets for lemons
(e.g., Vincent (1989), Wolinsky (1990), Blouin and Serrano (2001), Guerrieri, Shimer,
and Wright (2010)) is that the low quality seller’s equilibrium payoff is set according
to the corresponding equilibrium payoff for the games with complete information, and
then, construct an equilibrium payoff for the high quality seller, subject to the incentive
Date: October 12, 2012.
We are grateful for helpful conversation with Asher Wolinsky who first suggested this problem. The
financial support from National Science Foundation is gratefully acknowledged.
1
2
IN-KOO CHO AND AKIHIKO MATSUI
constraint. Since the high quality seller needs to prevent the low quality seller from
imitating the high quality seller, the trading opportunity and consequently, the equilibrium
payoff of the high quality seller get reduced, compared to the model under complete
information. The presence of high quality seller has little impact on the low quality
seller’s equilibrium payoff, because it is the low quality seller who wants to imitate the
high quality seller, but not vice versa. One might conclude that the presence of high
quality good should not affect the equilibrium payoff of the low quality seller, computed
under the assumption of complete information about the quality of the good.
The main findings of the present paper show otherwise in a dynamic decentralized
trading model. The presence of peach substantially affects the equilibrium outcome of the
market. In particular, the equilibrium payoffs of the low quality sellers and the buyers can
significantly differ from what the existing models on lemons market predict.
Suppose that the trading occurs only between a buyer and a seller with low quality
good, as Akerlof (1970) claims. By the end of the day, all low quality good will be sold
out, while no high quality good is sold. A rational buyer can infer that the average quality
of the good left in the market is almost as high as the high quality good. Thus, if a buyer
deviates to wait one day, and offers a price slightly higher than the production cost of the
high quality good next day, all sellers with high quality good are willing to sell the good,
and the deviating buyer can receive higher payoff than in the alleged equilibrium.
The possibility of trading at multiple prices over time was shut off in Akerlof (1970)
to focus on the (non-)existence of a single market equilibrium clearing price. Even if one
allows that trading can occur at more than a single price as in Guerrieri, Shimer, and
Wright (2010), however, a buyer cannot fully exploit the additional trading opportunity
with the high quality seller, unless he can wait and make an offer in the next day. The
main innovation in our model is to impose no restriction on the set of prices at which
agent can trade (cf. Wolinsky (1990) and Blouin and Serrano (2001)), and no restriction
on the number of opportunities the agents can trade over time (cf. Akerlof (1970) and
Guerrieri, Shimer, and Wright (2010)), while maintaining the stationary structure of the
market in contrast to Vincent (1989) and Blouin and Serrano (2001).
While the presence of the high quality good may provide the buyer with an additional
opportunity, it also offers a seller with low quality good an opportunity to imitate the high
quality seller, accruing the informational rent and possibly generating expected payoff
higher than what the static model predicts. Also, these opportunities may change the
bargaining powers of respective agents, and the delivery price of the low quality price
may differ from the corresponding equilibrium price in the existing models. The focus
of our analysis is on how the two effects of additional trading opportunity for the buyer
and informational rent for the low quality seller interact with each other to determine
the equilibrium outcome in a dynamic decentralized trading model with lemons problem
without entry.
Because the two key impacts stemming from the presence of high quality sellers arise
only in the dynamic context, we examine a dynamic model in contrast to a static model of
Guerrieri, Shimer, and Wright (2010). As we admit possibly more than a single delivery
price, the properties of equilibrium delivery prices are our main interest. In contrast to
Wolinsky (1990) and Blouin and Serrano (2001), where the delivery prices are exogenously
fixed, we assume that any positive number can be a delivery price, and the delivery
DYNAMIC LEMONS PROBLEM
3
prices are endogenously determined as a part of an equilibrium. We essentially admit any
incentive compatible trading rule to determine the delivery prices.
More precisely, our model is built on the decentralized trading model of Burdett and
Wright (1998), but assumes the asymmetric information about the quality of the good
traded. In each period, a seller who knows the quality of the good is randomly matched to
a buyer who does not observe the quality, and then randomly draw a price, at which the
good is delivered. If either party disagrees, then the two players return to the pool, waiting
for another chance to be matched to another player. If both parties agree, then the trade
occurs and the two players leave the pool of unmatched players (but not the economy),
generating surplus from trading in each period while the agreement is in place. The long
term agreement is dissolved by the decision of either party or by an exogenous shock.
Upon dissolution of the long term relationship, both players return to the respective pool
of players. The objective function of each agent is the expected discounted average payoff.
We examine stationary equilibria in which trading occurs with a positive probability. In
order to crystallize the impact of the asymmetric information, we examine a sequence of
stationary equilibria as the friction, quantified as the time span of each period, vanishes.
We obtain a complete characterization of the outcomes as the friction vanishes.
In any sequence of stationary equilibria in which trading occurs with positive probability,
the market can sort out the quality of the good only partially. The proportion of the low
quality is strictly between zero and 1, even in the limit as the friction vanishes. The failure
of complete sorting is the main reason for inefficient equilibrium allocation.
The market outcome, especially the equilibrium surplus of the low quality seller, is
determined by the informational rent of the low quality seller, and the pressure among
sellers competing for the buyers. The ratio between the discount rate and the rate at
which the long term relationship dissolves exogenously is found to be the most convenient
parameter to summarize the influence of these two forces on the equilibrium outcome.
If the ratio is large (so that the agents have a large discount rate compared to the
dissolution rate), then the players are less patient. In this case, the buyer extracts positive
surplus, exploiting the additional trading opportunity with a high quality seller at a high
price. It may happen even if the mass of buyers is larger than the mass of the low quality
sellers so that under complete information, the low quality seller should have extracted
the entire gain from surplus.
On the other hand, if this ratio is small, then the informational rent of the low quality
seller dominates so that the low quality sellers extract the entire gain from trading. It
may happen even if the mass of low quality seller is larger than the buyer so that under
the complete information, the buyer should have extracted the entire gain from trading.
More surprisingly, some buyers may not be matched to form a long term relationship with
a seller, even if the total mass of the sellers is larger than the mass of buyers. As the
buyer’s payoff is depressed, the buyer has incentive to remain in the market for a better
deal in the future. The market outcome exhibits a positive amount of excess demand and
a positive amount of excess supply at the same time.1
1
Guerrieri, Shimer, and Wright (2010) construct a directed search theoretic model where buyers choose,
post, and commit to a pair of price and the probability of trade. They show that each buyer either posts
a low price and the trade probability of one, or a high price and a trade probability less than one so that
the low quality sellers are indifferent between two offers. Even though there appear two prices, a smart
4
IN-KOO CHO AND AKIHIKO MATSUI
The rest of the paper is organized as follows. Section 2 formally describes the static
and the dynamic models of lemons. Section 3 states the main result. Section 4 proves
the main result, while establishing a series of preliminary results, which also offer useful
insight into the structure of an equilibrium. While we focus on the case in which the
lemons problem is severe, Section 5 examines the remaining case to provide the analysis
for all set of parameter values. Section 6 demonstrates that our result is robust against
small changes of parameter values. Section 7 concludes the paper.
2. Model
In a baseline model, we consider an economy which is populated by 2 unit mass of
infinitesimal (infinitely-lived) sellers and 1 unit mass of infinitesimal (infinitely-lived) buyers. One unit mass of sellers have production cost sl , called low quality sellers, while the
remaining one unit of sellers have production cost sh , called high quality sellers. Assume
sh > sl . The quality of output is positively correlated with the production cost. For this
reason, we refer to a high (low) cost seller as a high (low) quality seller. If the production
cost is sh , then the marginal utility for a buyer is φh , and if the cost is sl , then the marginal
utility is φl , where φh > φl . Each seller produces at most one unit of the good, and each
buyer consumes at most one unit of the good.
After analyzing the baseline model, we vary the mass of low quality sellers to demonstrate that the conclusion of this paper is robust against the proportion of low quality
sellers in the economy.
Note that the high cost seller produces a high quality good, and the low cost seller
produces a low quality good. We make the following three assumptions on the parameter
values, which are critical for capturing the lemons problem.
A1. φh > sh > φl > sl , which implies that the existence of the gains from trading under
each state is common knowledge. Because sh > φl , if the buyer believes that the
quality is high, then he may end up paying more than φl .
A2. φh − sh > φl − sl so that it is socially efficient for the high quality sellers to deliver
the good to the buyers.
We first examine the case in which the lemons problem is severe so that no single
competitive market clearing price can exist.
φh + φl
< sh so that the lemons problem is severe in the sense that random transA3.
2
actions lead to a negative payoff either to a buyer or to a high quality seller. This
assumption also ensures that a single market clearing price cannot exist.
After presenting a complete analysis for the case satisfying A1 − A3, we examine in
section 7 the remaining case, in which A1 and A2 hold, but A3 may not hold so that the
lemons problem is not severe.
If p is the delivery price of the good, and φy is the quality of the good, sx seller’s profit
is p − sx and buyer’s surplus is φy − p where x, y ∈ {h, l}. Under the assumptions we
made, one can show that no single price can clear the market. It is often asserted that
buyer still can wait for the market to clear and look for a seller after many low quality sellers are matched
away. In this sense, Guerrieri, Shimer, and Wright (2010) suffers from the same issue as the static model.
DYNAMIC LEMONS PROBLEM
5
s, p, φ 6
φh
sh
φl +φh
2
φl
sl
-
0
1
2
quantity
Figure 1. Lemons market
the presence of asymmetric information about the quality can drive out high quality good
out of the market.
Let us embed the classic lemons market into a decentralized dynamic trading model.
Time is discrete, and the horizon is infinite. When a buyer and a seller is initially matched
at period t. Conditioned her type x ∈ {h, l}, the seller reports her type, possibly in a
randomized fashion, to a third party (or mechanism) which draw p according to probability
density function fνx where νx is a probability measure over R. Without loss of generality
we assume that the support of νx is [sl , φh ].
We assume
(2.1)
∀x ∈ {h, l}, ∀p ∈ [sl , φh ], fνx (p) > 0 and is continuous.
Conditioned on p drawn by the mechanism, Each party has to decide whether or not to
form a partnership. After forming the partnership, the buyer can purchase the good at
the agreed price, and the seller can sell the good at the same price to the buyer. If the
good is delivered at p, the seller’s surplus is p − sx and the buyer’s surplus is φy − p where
x, y ∈ {h, l}.
Then, at the end of the period, either one of two events will occur. The partnership
breaks down with probability 1 − δ, and then, both players are dumped back to the
respective pools. The partnership continues with probability δ without the true quality
being revealed.
We assume that the true quality of the good is not revealed to the buyer during the long
term relationship, like a life insurance. This assumption is mainly to simplify exposition.
One can extend the analysis by assuming that the true quality is revealed to the buyer
during the long term relationship according to a Poisson distribution with intensity λs .
Upon the revelation of the true quality, the buyer can decide to continue or terminate
the existing long term relationship. As we focus on the case where λs = 0, we essentially
6
IN-KOO CHO AND AKIHIKO MATSUI
examine the worst possible case of lemons problem. As long as λs < ∞, lemons problem
exists and the main conclusion from the analysis of the baseline model applies. As λs →
∞, low quality good is dumped back to the pool more quickly, which alleviates lemons
problem, as the market completely sorts out lemons from peaches in the limit.
In each period, the buyer and the seller in a partnership can choose to maintain or to
terminate it. If one of the players decides to terminate the partnership, both players return
to the pool, waiting for the next round of matching. If both players decide to continue
the partnership, the partnership continues with probability δ = e−dΔ where d > 0, and
with probability 1 − δ, the partnership dissolves, and the two players are forced to return
to the pool. We call 1 − δ layoff probability (Burdett and Wright (1998)).
The objective function of each player is the long run discounted average expected payoff:
(1 − β)E
∞
β t−1 ui,t
t=1
where ui,t is the payoff of player i in period t and β = e−bΔ is the discount factor.
We focus on undominated stationary equilibrium which is stationary equilibrium where
no dominated strategy is used, to exclude “no trading equilibrium” in which every player
refuses to reach agreement. We simply refer to an undominated stationary equilibrium as
an equilibrium, whenever the meaning is clear from the context.
Let ϕ(·|x) be an equilibrium reporting strategy, which is a probability distribution over
{h, l} conditioned on true type x ∈ {h, l}. Given the equilibrium reporting strategy, we
can compute the equilibrium probability distribution νxe over [sl , φh ], conditioned on true
type x ∈ {h, l}. Invoking the revelation principle, we assume that each type of the seller
truthfully report her type x. Conditioned on the truthful report, the mechanism selects a
price according to νxe ∀x ∈ {h, l}. Since νx satisfies (2.1), so does νxe .
To simplify exposition, however, we assume for the rest of the paper that p is drawn
from [sl , φh ] according to the uniform distribution regardless of the report of the seller.
The extension to the case where the price is drawn from a general distribution satisfying
(2.1) is cumbersome but straightforward (Cho and Matsui (2011)).
3. Main Results
Wsh (p),
Wsl (p),
and Wb (p) be the continuation game payoffs of a high quality seller,
Let
a low quality seller, and a buyer, respectively, after the two players agree on p. Also, let
Wsh , Wsl , and Wb be the continuation values of respective agents after they fail to form
a long term relationship. Given the equilibrium value functions, let us characterize the
optimal decision rule of each player. In the sequel, we write x ≤ O(Δ) if
lim
Δ→0
x
< ∞.
Δ
Let zsl and zsh be the mass of sl and sh sellers in the pool. Similarly, let zb be the mass
of buyers in the pool. Since one buyer is matched to one seller,
2 − (zsh + zsl ) = 1 − zb .
DYNAMIC LEMONS PROBLEM
7
Let
zsh
zsh + zsl
be the proportion of high quality sellers in the pool of sellers, and let μl = 1 − μh be the
proportion of low quality sellers in the pool. Also, let
zb
zb
=
ζ= h
zs + zsl
1 + zb
be the fraction of buyers relative to that of sellers in the pool.
Let Πhs be the set of prices that a high quality seller and a buyer agree to accept, and
let πsh = P(Πhs ). For p ∈ Πhs , we can write
Wsh (p) = (1 − β)(p − sh ) + β δWsh (p) + (1 − δ)Wsh .
μh =
The first term is the payoff in the present period. At the end of the present period,
with probability 1 − δ, the partnership dissolves, and the high quality seller’s continuation
payoff is Wsh . With probability δ, the high quality seller continues the relationship, of
which continuation value is given by Wsh (p).
A simple calculation shows
(1 − β)(p − sh ) + β(1 − δ)Wsh
.
1 − βδ
The high quality seller agrees to form a partnership with delivery price p if
(3.2)
Wsh (p) =
Wsh (p) > Wsh
which is equivalent to
p > sh + Wsh .
(3.3)
On the other hand, Wsh is given by
(3.4)
Wsh = βζπsh E[Wsh (p)|Πhs ] + β(1 − ζπsh )Wsh .
Substituting (3.2) into (3.4), we obtain, after some calculation,
(3.5)
Wsh =
βζπsh
E[p − sh − Wsh |Πhs ].
1 − βδ
Wsl =
βζπsl
E[p − sl − Wsl |Πls ],
1 − βδ
Similarly, we obtain
(3.6)
where Πls is the set of prices that a low quality seller and a buyer agree to accept, and let
πsl = P(Πls ). In any undominated equilibrium, sl seller accept p if
p > sl + Wsl .
Imitating the behavior of high quality sellers, a low quality seller can always obtain
a higher (or equal) continuation value than a high quality seller.2 Therefore, we have
2
If the true quality is revealed with a positive probability after the good is delivered, then we cannot
invoke the same argument to prove the inequality. Yet, the main result is carried over.
8
IN-KOO CHO AND AKIHIKO MATSUI
Wsl ≥ Wsh . Now, we would like to claim that the threshold price for a low quality seller is
lower than that for a high quality seller.
Lemma 3.1.
sh − sl > Wsl − Wsh .
Proof. Suppose that a high quality seller imitate a low quality seller, then the long run
expected payoff from the deviation is
βπsl
.
1 − βδ + βπsl
Since the deviation payoff is less the equilibrium payoff,
Wsl − (sh − sl )
Wsl − Wsh ≤ (sh − sl )
βπsl
< sh − sl
1 − βδ + βπsl
as desired.
Let Πls and Πhs be the set of prices where sl and sh sellers trade with a positive probability. Lemma 3.1 says
sl + Wsl = inf Πls < sh + Wsh = inf Πhs .
In an equilibrium, we can partition the set of prices into three regions, Πs , Πp , and the
rest:
Πs = Πls \ Πhs ,
Πp = Πls ∩ Πhs ,
Πs is the set of the prices at which trade occurs only with low quality sellers (the subscript
stands for separating), while Πp is the set of the prices at which trade occurs with both low
and high quality sellers (the subscript stands for pooling). Finally, the remaining region
is the one in which no trade occurs. We have yet to show that Πs = ∅ and Πs = ∅. Note
that we have
Πs ⊂ [sl + Wsl , sh + Wsh ],
Πp ⊂ [sh + Wsh , ∞).
Let πs = P(Πs ) and πp = P(Πp ). Since we focus on an equilibrium in which trading
occurs with a positive probability,
πs + πp > 0
in an equilibrium.
Definition 3.2. If πp = 0 in an equilibrium, we call such an equilibrium a separating
equilibrium. If πs = 0, then the equilibrium is called a pooling equilibrium. If πs > 0 and
πp > 0, then it is called a semi-pooling equilibrium.
Let us calculate the value function of a buyer. In the private value model in which a
buyer knows exactly how valuable the objective is (Cho and Matsui (2012)), the informational content of p is irrelevant for a buyer to decide whether or not accept p. In contrast,
in the common value model like the lemons problem, the expected quality conditioned
DYNAMIC LEMONS PROBLEM
9
on p is a critical factor for a buyer to make a decision on p.3 Let φe (p) be the expected
quality if p is the price to be agreed upon. If p ∈ (sl + Wsl , sh + Wsh ), then only low quality
sellers agree to accept the price, and therefore, we have φe (p) = φl . On the other hand, if
p > sh + Wsh holds, then both low and high quality sellers agree to do so, and therefore,
we have
φe (p) = φ(μl ) ≡ μl φl + μh φh .
If a buyer and a seller agree to form a partnership at price p, then the expected continuation
value of the buyer is given by
Wb (p) = (1 − β)(φe (p) − p) + β [δWb (p) + (1 − δ)Wb ] .
Therefore, we have
(1 − β)(φe (p) − p) + β(1 − δ)Wb
.
1 − βδ
Also, the continuation value after no match is given by
Wb (p) =
Wb = βμl πs E [Wb (p)|Πs ] + βπp E [Wb (p)|Πp ] + β(1 − μl πs − πp )Wb .
After substitutions and tedious calculation, we obtain
βπp
βμl πs
E [φl − p − Wb |Πs ] +
E [φ(μl ) − p − Wb |Πp ] .
(3.7)
Wb =
1 − βδ
1 − βδ
A buyer is willing to accept p if
Wb (p) > Wb ,
or equivalently,
φe (p) − p > Wb .
Since φe (p) is a function of p, the buyer’s equilibrium decision rule may not be characterized
by a single threshold. Combining these results and including the endpoints as they are
measure zero events, we have
[sl + Wsl , φl − Wb ] if sl + Wsl ≤ φl − Wb ,
Πs =
∅
otherwise,
[sh + Wsh , φ(μl ) − Wb ] if sh + Wsh ≤ φ(μl ) − Wb ,
Πp =
∅
otherwise.
The size of population of each type of the players is determined by the balance equations:
πp ζ
πs ζ
+
zsl
(3.8)
1 − zsl =
1−δ 1−δ
πp ζ h
z
(3.9)
1 − zsh =
1−δ s
πp
πs μl
+
zb .
(3.10)
1 − zb =
1−δ 1−δ
3
Even if each individual is infinitesimally small, the informational content of p affects the decision of all
buyers. In this sense, each individual is not “informationally small” in the sense of Gul and Postlewaite
(1992).
10
IN-KOO CHO AND AKIHIKO MATSUI
From (3.8) and (3.9),
(3.11)
μh
zh
μh
=
= sl =
1 − μh
μl
zs
πs ζ
1−δ
+
πp ζ
1−δ
πp ζ
1−δ
+1
+1
which determines μh . Similarly, since
ζ=
(3.12)
1=
zb
,
1 + zb
μl πp ζ
μh πp ζ
μl πs ζ
+
+
+ 2ζ
1−δ
1−δ
1−δ
which determines ζ. We can stack up the value functions of each type of players as
⎡ h⎤
⎤
⎤
⎡
⎡
0
Ws
ζE(p − sh − Wsh |Πp )
βπ
p ⎣
⎣ Wsl ⎦ = βπs ⎣ ζE(p − sl − Wsl |Πs ) ⎦ +
(3.13)
ζE(p − sl − Wsl |Πp ) ⎦
1 − βδ
1 − βδ
μl E(φl − p − Wb |Πs )
Wb
E(φ(μh ) − p − Wb |Πp ).
An equilibrium is characterized by (μh , ζ, Wb , Wsl , Wsh ) that solves (3.11), (3.12) and (3.13).
Before stating the main result for the decentralized dynamic model with lemons problem,
let us state, as a benchmark, the prediction from the static counter part, and the prediction
from the model populated only with the low quality sellers. In a static model with lemons
problems, satisfying A1 − A3, no high quality seller can trade with a positive probability,
if one insists that the trading occurs at a single price. Trading will occur between a
low quality seller and a buyer and the market clearing price should be p ∈ [sl , φl ]. The
multiplicity of the market clearing price is a direct consequence of a feature of the baseline
model in which the mass of the low quality seller and the mass of the buyer are equal to
1. Naturally, if the mass of the low quality seller is larger (smaller) than the buyer’s mass,
then the trading should occur at sl (φl ). The equilibrium surplus is
φl − sl
which is smaller than the maximum surplus
φh − sh .
Hence, the market outcome is inefficient, but all buyers trade with a seller with probability
1. By the end of the day, only the high quality sellers are left in the market.
If the dynamic decentralized market is populated only with the low quality sellers and
buyers, then Cho and Matsui (2011) provides complete characterization of stationary
equilibria in which trading occurs. If the mass of each part is equal, then the prices at
which trading occurs converge to (φl + sl )/2. Otherwise, the prices must converge to the
static counter part, depending upon the relative size of the low quality sellers and buyers.
Our interest is whether and how the equilibrium payoff of the low quality seller changes,
if we “add” one unit of high quality sellers to the model.
The main result of the paper is to characterize completely the limit of the equilibrium
value functions as Δ → 0. Let us state the asymptotic properties of the equilibrium payoffs
for the case where A1 − A3 hold.
DYNAMIC LEMONS PROBLEM
11
Theorem 3.3. For any sequence of undominated stationary equilibria,
lim Wsh = 0
Δ→0
d
lim Wsl = min φl − sl , [φh − sh − (φl − sl )]
Δ→0
b
d
lim Wb = max 0, (φl − sl ) − [φh − sh − (φl − sl )] .
Δ→0
b
Theorem 3.3 implies
(3.14)
lim Wb + Wsl = φl − sl
Δ→0
which is exactly what the static model predicts. But, the equilibrium payoff the low quality
seller is significantly different from what the static model predicts.
The presence of high quality sellers has two different effects. First, the low quality
seller can imitate the high quality seller, thus accruing informational rent. Second, as
more sellers are in the market, the sellers are subject to more competitive pressure, which
enhance the bargaining position of the buyer. The equilibrium payoff is determined as a
result of interactions between these two forces.
A high quality seller does not want to imitate low quality seller, and consequently,
cannot earn any informational rent. The competitive pressure among sellers wipe out the
high quality seller’s profit. The low quality seller’s ability to exploit her ability to imitate
a high quality seller requires a high price, at which a high quality seller is willing to trade.
Thus, if a low price, at which only a low quality seller is willing to trade, is drawn, then
the low quality seller can assert a strong bargaining position over a buyer.
To do so, the low quality seller must be sufficiently patient to wait for a high price to be
drawn, in case the buyer refuses to form a long term relationship. Indeed, if the players are
patient, then the low quality seller can extract the entire gain from surplus. Surprisingly,
it can happen, even if the low quality seller’s mass is larger than the buyer’s mass. On
the other hand, if the players are less patient, then the competitive pressure dominates
the informational rent of the low quality seller. As a result, a buyer can extract a positive
surplus from trading, and a low quality seller’s equilibrium payoff may fall below the level
predicted by the static model.
To better understand the equilibrium outcome, we examine the asymptotic property of
ζ, which represents the ratio of the mass of buyers over the mass of sellers. Since the mass
of sellers in the economy is twice as large as the mass of buyers, ζ → 0 if and only if a
buyer is matched away almost immediately when he is dumped into the pool, as Δ → 0.
Positive ζ is yet another evidence that the equilibrium allocation of a dynamic model can
substantively differ from the static equilibrium outcome.
Note that the mass of sellers is strictly larger than the mass of buyers in the economy
as a whole, and in fact, in the matching pool, because one buyer is matched away with
one seller. If the market is efficient, then all buyers should be matched away, leaving
only sellers in the matching pool as the friction vanishes. Indeed, Cho and Matsui (2012)
demonstrated that even if each agent has private information about his own valuation,
this sort of asymmetric information does not hinder every agent in the short side from
12
IN-KOO CHO AND AKIHIKO MATSUI
being matched away, as the decentralized market allocation is asymptotically efficient as
friction vanishes.
We are asking whether the nature of asymmetric information can affect the efficiency
of matching mechanism. Note that in the lemons market, a buyer does not observe how
valuable the object is, which is a feature commonly found in common value models of
auctions. Our goal is to delineate completely the set of parameter values under which
b
some buyers remain in the matching pool so that ζ = z l z+z
h is bounded away from 0, even
s
s
if the frequency of meeting a seller increases to infinity.
To find whether ζ can be bounded away from 0 in the limit of a sequence of equilibria
as Δ → 0, first note that
lim Wsl = φl − sl and
Δ→0
lim Wb = 0.
Δ→0
if and only if
φh − sh
(φh − sh ) − (φl − sl )
b
=
−1
<
C4.
d
φl − sl
φl − sl
holds.4 That is, the equilibrium payoff in the limit is affected by two ratios: one between
the impatience of an agent and the rate at which an existing partnership dissolves, and
the other between the social surplus under two different states. For example, C4 holds,
if an agent is very patient, compared to the probability that the existing partnership is
dissolved, or if the potential gain from trading high quality goods is much larger than
the potential gain from trading low quality goods. In either case, sl seller has a stronger
incentive to imitate sh seller than otherwise.
Theorem 3.3 says that lim Wb = 0 if C4 holds and lim Wb > 0 if C4 is violated with
l −sl )
. Indeed, C4 delineates the set of parameter values
strict inequality: db > (φh −sφhl)−(φ
−sl
under which the low quality seller can extract entire gain from trading between a buyer
and a low quality seller.
Theorem 3.4.
φh − sh − b+d
d (φl − sl )
lim ζ = max 0,
Δ→0
φh − φl
In particular, lim ζ > 0 if and only if C4 holds. If lim ζ > 0, then lim Wb = 0 and
lim Wsl = φl − sl .
As Theorem 3.4 indicates, a positive mass of buyers may remain unmatched over an
extended time, even as the frequency of matching increases indefinitely. It is a quite
striking inefficiency in matching, because the mass of buyers is one half of the mass of
sellers, and still, some buyers are not matched. Since sl seller can imitate sh seller and
some transaction occurs at a price agreeable to sh seller, sl seller can extract profit by
trading with a buyer, while revealing the low quality, but also can generate profit by
pooling with sh seller, which generates at least sh − sl > 0 amount of profit. Since sl
seller has an opportunity to pool with sh seller to generate large profit, sl seller bargains
aggressively against a buyer, which results in lim Wb = 0. As a buyer is forced to accept
a low payoff, he is willing to remain in the pool, waiting for more agreeable outcome.
4
Note that the right hand side of the above inequality is positive due to A1 − A3.
DYNAMIC LEMONS PROBLEM
13
4. Analysis
4.1. Existence. We first show the existence of an undominated stationary equilibrium.
An important corollary of the existence result is that any equilibrium must be a semipooling equilibrium. That is, in any undominated equilibrium, 0 < πs , πp < 1 ∀Δ > 0.
Balance equations can be rewritten as
πp ζ
μl πs ζ
(4.15)
+
+ 2ζ
1 =
1−δ
1−δ
πp ζ
+1
μl
(4.16)
.
= π ζ 1−δ πp ζ
s
1 − μl
+
+1
1−δ
We also rewrite
Wsh ,
Wsl ,
1−δ
and Wb as
βAs (πp )2 ζ
1 − βδ
2
βA
βπp ζ
s (πs ) ζ
(4.18)
+
E[p − sl − Wsl |Πp ]
Wsl =
1 − βδ
1 − βδ
βAb (πs )2 μl βA(πp )2
Wb =
(4.19)
+
1 − βδ
1 − βδ
for some constants As , Ab > 0,which depend upon the density function of ν .
Finally, rewrite πs and πp as
(4.17)
Wsh =
(4.20)
πs = C[φl − sl − Wb − Wsl ]
(4.21)
πp = C[φ(μl ) − sh − Wb − Wsh ]
for some constant C > 0, which depends upon the density function of ν.
Let Φ : [0, 1]2 → [0, 1]2 be given by
(4.22)
Φ(πs , πp ) = (max{min{πs , 1}, 0}, max{min{πp , 1}, 0}) .
Note that μl , ζ, Wb , Wsl , and Wsh are determined as continuous functions of (πs , πp ).
Therefore, Φ is a continuous mapping from [0, 1]2 into itself. Thus, by Brouwer’s fixed
point theorem, Φ has a fixed point (πs∗ , πp∗ )
We demonstrate that (πs∗ , πp∗ ) is not on the boundary of [0, 1]2 , which implies that any
undominated stationary equilibrium must be a semi-pooling equilibrium.
Lemma 4.1. There exists Δ > 0 such that for all Δ ∈ (0, Δ), (πs∗ , πp∗ ) ∈ (0, 1)2 .
Proof. Case 1: (πs∗ , πp∗ ) = (0, 0)
Suppose (πs∗ , πp∗ ) = (0, 0). Then Wb = Wsl = Wsh = 0, since no agreement is reached.
Then, (4.20) implies πs∗ = C[φl − sl ] > 0, which is a contradiction.
Case 2: πs∗ = 0
Suppose πs∗ = 0. Then, (3.11) implies μl = 1/2 from which φ(1/2) − sh < 0 follows.
Therefore, (4.21) implies πp∗ = 0, which is reverted to the case of (πs∗ , πp∗ ) = (0, 0).
Case 3: πp∗ = 1
Suppose πp∗ = 1. Then, for a sufficiently small Δ > 0, we have Wb > φh , which implies
πs∗ = 0. This case is reverted to the second case.
14
IN-KOO CHO AND AKIHIKO MATSUI
Case 4: πs∗ = 1
Suppose πs∗ = 1. Then, from (4.18) and (4.19), we have ζ ≤ O(Δ) and μl ≤ O(Δ); for if
not, either limΔ→0 Wsl = ∞ or limΔ→0 Wb = ∞ holds, and therefore, πs∗ = 0 = 1. Then,
the left hand side of (4.16) converges to zero, while its right hand side is bounded away
from zero. A contradiction.
Case 5: πp∗ = 0
Suppose πp∗ = 0. Then, From (3.11) and (3.12), one can show
ζ = μl .
Thus,
Wsh = 0
Aζ(πs∗ )2
Wsl = Wb =
1 − βδ
This together with πs∗ = 0 (otherwise, we can apply step 2) implies Wb <
we have
φl − sl
∗
.
πp > C φ(μl ) − sh −
2
Under the hypothesis of πp∗ = 0,
φl −sl
2 .
Therefore,
φl − sl
< 0.
2
φ(μl ) − sh −
Since
φl − sl
φl − sl
= φh − sh −
> φh − sh − (φl − sl ) > 0,
2
2
μl must be uniformly bounded away from zero. From (3.11), one can show
φ(0) − sh −
1 − 2ζ
πs∗
=
.
1−δ
ζμl
Since ζ = μl and μl is uniformly bounded away from 0, the right hand side is uniformly
bounded from above as Δ → 0. It then implies
πs∗ ≤ O(Δ).
(4.23)
But, (4.23) implies that
Wsl = Wb =
A(πs∗ )2 ζ
→ 0.
1 − βδ
Thus,
πs = C[φl − sl − Wb ] → C[φl − sl ] > 0,
which contradicts with (4.23).
This lemma directly implies the following theorem and corollary.
Theorem 4.2. There exists an equilibrium.
Corollary 4.3. The only type of equilibrium is semi-pooling.
DYNAMIC LEMONS PROBLEM
15
4.2. Preliminary Results. We examine the asymptotic properties of an undominated
stationary equilibrium, which are useful in calculating the equilibrium value function√in
the limit as Δ → 0. First, we observe that πp vanishes at a rate no slower than Δ
vanishes.
Lemma 4.4. (πp )2 ≤ O(Δ)
Proof. The second term of (4.19) and Wb < ∞ imply the statement.
We can then identify the rate at which ζπp vanishes.
Lemma 4.5. limΔ→0
ζπp
1−βδ
Proof. Suppose limΔ→0
< ∞.
ζπp
1−βδ
= ∞. We can rewrite the balance equation as follows:
(4.24)
zsl =
(4.25)
zsh =
Then, (4.24) and (4.25) imply that both
one-to-one matching implies
1
πs ζ
1−δ
+
1
πp ζ
1−δ +
zsl and
πp ζ
1−δ
1
+1
.
zsh converge to zero as Δ → 0. Note that
(1 − zsl ) + (1 − zsh ) = 1 − zb ≤ 1.
If zsl , zsh → 0, then the left hand side converges to 2, and the equality cannot hold. This
contradiction proves the lemma.
Based upon these two observations, we conclude that the high quality seller’s equilibrium
payoff vanishes as Δ → 0.
Lemma 4.6. limΔ→0 Wsh = 0.
Proof. Apply Lemmata 4.4 and 4.5 to (4.17).
Since πs > 0, an sl seller and a buyer trades with a positive probability, which imposes
an upper bound on Wsl + Wb .
Lemma 4.7. Wsl + Wb < φl − sl .
Proof. This statement is directly implied by (4.20) and πs > 0.
The next lemma shows that the low quality seller cannot be completely sorted out in
a semi-pooling equilibrium, even in the limit as Δ → 0. As the pool contains a nonnegligible portion of low quality sellers, the buyer needs to sort out the sellers, which is
costly for the buyer and for the society as a whole, even if the friction vanishes. On the
other hand, the low quality seller has an option to imitate the high quality seller, which
provides significant bargaining power to a low quality seller when she is matched to a
buyer.5
5
The next lemma relies on the assumption that the Poisson rate λs at which the true quality is revealed
to the buyer is finite. As λs → 0, all high quality goods are kept but all low quality goods are dumped to
the market. As a result, μl → 1.
16
IN-KOO CHO AND AKIHIKO MATSUI
Lemma 4.8. limΔ→0 μl > 0.
Proof. Suppose limΔ→0 μl = 0. Then limΔ→0 φ(μl ) = φh holds. Thus, from (4.21), Lemmata 4.6 and 4.7 together with Wsl ≥ 0, we have
lim πp = lim C[φh − sh − Wb − Wsh ] ≥ C[(φh − sh ) − (φl − sl )] > 0,
Δ→0
Δ→0
which contradicts with Lemma 4.4.
As in Lemma 4.5, we can compute the rate at which ζπs vanishes.
Lemma 4.9. limΔ→0
ζπs
1−βδ
< ∞.
ζπs
= ∞. Then from Lemma 4.5 and (4.16), limΔ→0 μl = 0
Proof. Suppose limΔ→0 1−βδ
holds, which contradicts to Lemma 4.8.
The next lemma is seller’s counter part of Lemma 4.4.
Lemma 4.10. πs ≤ O(Δ).
Proof. This statement is directly implied by Lemma 4.8 and (4.19).
A corollary of Lemma 4.10 is that the sum of the long run average payoffs of a buyer
and sl seller converges to φl − sl .
Lemma 4.11. limΔ→0 Wsl + Wb = φl − sl .
Proof. From Lemma 4.10 together with (4.20), we have
lim πs = lim C[(φl − sl ) − (Wb + Wsl )] = 0.
Δ→0
Δ→0
From Lemma 4.5, we know that
it is bounded away from zero.
Lemma 4.12. limΔ→0
ζπp
1−βδ
Proof. Suppose that limΔ→0
ζπp
limΔ→0 1−βδ
is bounded. We would like to show that
>0
ζπp
1−βδ
= 0. By (3.9),
zsh → 1.
By (3.12),
μl πs ζ
→ 1 − 2ζ.
1−δ
Since lim μl > 0 by Lemma 4.8,
0≤
1 − 2ζ
πs ζ
→
< ∞.
1−δ
μl
From (3.8),
1 ≥ zsl →
1
1+
1−2ζ
μl
> 0.
DYNAMIC LEMONS PROBLEM
17
Then,
1
zl
μl = h s l →
zs + zs
1+
1+
By dividing both numerator and denominator by
1−2ζ
μl
1
1+
.
1−2ζ
μl
1
1+ 1−2ζ
μ
, the right hand side of the above
l
equation becomes
1
1+
1−2ζ
μl
+1
=
μl
.
2μl − 2ζ + 1
Thus, in the limit
μl =
μl
2μl + 1 − 2ζ
which implies
μl − ζ → 0.
Recall that μl > 0. Thus, under the hypothesis of the proof, we have
πp
= 0.
lim
Δ→0 1 − βδ
Hence,
Wsl − Wb → 0.
We can now invoke the essence of the argument used for Case 5 of the proof for Lemma
4.1 to derive the contradiction.
Lemma 4.13. limΔ→0 Wsl > 0
Proof. Recall the equilibrium value function of Wsl , and observe that the second term of
the value function is strictly positive, even in the limit as Δ → 0.
Lemma 4.14. 0 < limΔ→0
πs
πp
<∞
Proof. We can write (3.11) as
μh
=
1 − μh
πs
πp
1−δ
πp ζ
1−δ
πp ζ
+1+
1+
.
If πs /πp explodes, then μh → 1 since lim 1−δ
πp ζ < ∞ holds due to Lemma 4.12. If πs /πp → 0,
then μh → 1/2. In the case of semi-pooling equilibrium under A3, we know
1
< lim μh < 1.
2 Δ→0
Thus, πs /πp must be bounded from above and away from zero in the limit of Δ going to
zero.
Lemma 4.15. limΔ→0
ζπs
1−βδ
>0
18
IN-KOO CHO AND AKIHIKO MATSUI
Proof. We can invoke the proof of Lemma 4.12, since πs vanishes at the same rate as πp
vanishes.
Lemma 4.16. limΔ→0 E[p|Πp ] = sh .
Proof. Apply Lemmata 4.6, 4.14, and 4.15 to (3.5).
Lemma 4.17. limΔ→0 ζWb = 0
Proof. Note
Aζπs2 + ζπp E(p − sl − Wsl |Πp )
Wsl
=
.
Wb
Aμl πs2 + πp2
Thus,
(4.26)
Aμl πs ππps + μl E(p − sl − Wsl |Πp )
Aμl ζπs2 + μl ζπp E(p − sl − Wsl |Πp )
μl Wsl
=
=
.
ζWb
Aμl ζπs2 + ζπp2
Aμl πs ππps + πp
From Lemma 4.16 and lim Wsl ≤ φl − sl , the denominator converges to zero, while the
numerator converges to a value greater than or equal to μl (sh − φl ) > 0. Therefore, since
limΔ→0 μl Wsl > 0, ζWb → 0.
4.3. Proof of Theorem 3.3. We are ready to prove Theorem 3.3. To illuminate the
relationship among ζ, Wb and C4, we examine two cases: (i) lim ζ > 0 and (ii) lim ζ = 0.
While examining each case, we prove Theorem 3.4.
4.3.1. Case (i). Let us state a couple of useful results.
Lemma 4.18. If lim ζ > 0, then limΔ→0 Wb = 0.
Proof. Apply Lemma 4.17.
Lemma 4.19. If lim ζ > 0, then limΔ→0 Wsl = φl − sl .
Proof. This statement follows directly from Lemmata 4.11 and 4.18.
Lemma 4.20. If lim ζ > 0, then C4 holds.
Proof. Suppose lim ζ > 0. A sequence of equilibria is determined by a sequence {(μl , ζ, πs , πp )}
that satisfies (4.15) and (4.16), and the following conditions:
(4.27)
μl φl + (1 − μl )φh → sh
Wsl → φl − sl .
(4.28)
ζπs
< ∞ holds, we have πs ≤ O(Δ). Therefore, from the value function of Wsl
Since lim 1−δ
together with (4.28), we have
lim Wsl = lim
πp ζ
(sh − φl ) = φl − sl .
1 − βδ
Thus,
lim
b + d φl − sl
πp ζ
=
.
1−δ
d sh − φl
DYNAMIC LEMONS PROBLEM
From (4.15),
19
1
d + b φl − sl
πs ζ
→
1 − 2ζ −
.
1−δ
μl
d sh − φl
After substituting μl , πs ζ/(1 − δ) and πp ζ/(1 − δ) as functions of ζ, (4.16) becomes
d+b
φl − sl
1+
φh − sh
d
s − φl
h
=
.
φl − sl
d+b
sh − φl
sh − φl
(1 − 2ζ) −
+ 2(1 − ζ)
φh − sh
d
φh − sh
The denominator is meaningful only if it is positive. However, if ζ = 1, then the denominator becomes negative, which implies that if a solution exists at all, it must be strictly
less than 1. Notice that the right hand side is an increasing function of ζ over a domain
where the denominator is positive. If ζ = 0, the right hand side can be written as
sh − φl + d+b
φh − sh
d (φl − sl )
.
sh − φl
sh − φl + 2(φh − sh ) − d+b
d (φl − sl )
A strictly positive solution exists if and only if the denominator is positive and the above
−sh
. A simple calculation reveals that
equation is strictly less than the left hand side, φshh −φ
l
a strictly positive solution exists if and only if C4 holds.
4.3.2. Case (ii). Suppose that ζ → 0 in a sequence of equilibria. By individual rationality,
lim Wb ≥ 0. In this case, we may admit multiple equilibria, all of which have the same
asymptotic properties.
An equilibrium must satisfy (4.15) and (4.16). From the value function of Wsl and
Lemma 4.14,
d πp ζ
(sh − sl )
→ 0.
Wsl − d + b 1 − δ
d πp ζ
1+
d+b1−δ
From Lemma 4.16, we have
μl φl + (1 − μl )φh − Wb → sh
which implies
(4.29)
Wb → μl φl + (1 − μl )φh − sh ≥ 0.
Since πs → 0 holds, we have the third, and the final, equation that characterizes the
equilibrium in the limit.
(4.30) 0 = lim Wb + Wsl − (φl − sl )
⎡
⎤
d πp ζ
(sh − sl )
⎥
⎢
= lim ⎣(μl φl + (1 − μl )φh − sh ) + d + b 1 − δ
⎦ − (φl − sl ).
d πp ζ
1+
d+b1−δ
20
IN-KOO CHO AND AKIHIKO MATSUI
In an equilibrium, (x, s, p, μl ) must satisfy (4.15), (4.16) and (4.30), while satisfying the
inequality of (4.29).
Since we have 4 unknowns, but 3 equations, the system may admit multiple solutions
along the sequence {Δ} that converges to zero. However, (πs , πp , ζ) always appear as
πs ζ πp ζ
πs ζ πp ζ
, 1−δ ). Thus, we can solve ( 1−δ
, 1−δ , μl ) uniquely from (4.15), (4.16) and (4.30) for
( 1−δ
the limit of any sequence of equilibria:
sh − sl
d+b
(4.31)
μl =
1−
b
φh − φl
while
πp ζ
1−δ
πs ζ
lim
1−δ
lim
=
=
μl
1 − μl
1 − 2μl
.
μl (1 − μl )
Lemma 4.21. If lim ζ = 0, then C4 is violated.
Proof. Substituting (4.31) into (4.29), we obtain
sh − sl
φh − sh
d+b
1−
≤
b
φh − φl
φh − φl
which is simplified as
φh − φl − (sh − sl )
b
≥
d
φl − sl
so that C4 must be violated.
5. Complete Characterization
5.1. When lemons problem is not severe. Let us examine the case where the lemons
problem is not severe. Instead of A3, let us assume
φh + φl
≥ sh .
2
The analysis indicates that under A3, a pooling equilibrium cannot exist and the only
equilibrium must be a semi-pooling equilibrium. It would be an interesting question if A3
no longer holds.
(5.32)
Proposition 5.1. An equilibrium exists and must be a pooling equilibrium, if both (5.32)
and
sh − sl
b + 2d
>
(5.33)
2(b + d)
φh − φl
hold.
We prove the proposition for the rest of this subsection. To prove the first statement,
observe that the construction of the semi-pooling equilibrium does not rely on whether or
not (5.32) holds.
DYNAMIC LEMONS PROBLEM
21
To prove the second statement, we first write the equilibrium value functions.
⎤
⎤
⎡
⎡
⎡ h⎤
ζπp
ζE(p − sh − Wsh |Πp )
Ws
βπ
Aβπ
p ⎣
p ⎣
⎣ Wsl ⎦ =
(5.34)
ζE(p − sl − Wsl |Πp )⎦
ζE(p − sl − Wsl |Πp ) ⎦ =
1 − βδ
1 − βδ
πp
Wb
E(φ(μh ) − p − Wb |Πp )
for some A > 0, which depends upon fν .
We prove a series of intermediate results.
√
Lemma 5.2. πp ≤ O( Δ)
Proof. Since Wb < ∞ uniformly,
lim sup
Δ→0
πp2
<0
1 − βδ
which implies that πp has to vanish at a rate no slower than
√
Δ.
√
This result is weaker than we need, since it admits the possibility that πp < O( √Δ) so
that lim Wb = 0. The next step is to eliminate this possibility to establish πp = O( Δ).
√
Lemma 5.3. In any pooling equilibrium, πp = O( Δ).
√
Proof. Suppose that πp < O( Δ). Then, Wb → 0 and also, Wsh → 0 as Δ → 0. Recall that
in any pooling equilibrium, μl = 1/2, and the expected quality of the good is (φh + φl )/2,
which is strictly larger than sh under (5.32). For a sufficiently small Δ > 0, the buyer can
generate a positive surplus by accepting p ∈ [sh + Wsh , (φh + φl )/2] and therefore,
lim inf Wb > 0
Δ→0
which is a contradiction.
Recall that
ζ=
zb
and zb πp = (1 − zb )(1 − δ).
1 + zb
Thus,
πp
ζπp
=
1−δ
2(1 − δ) + πp
which implies the follow lemma.
√
Lemma 5.4. ζ = O( Δ) and
ζπp
= 1.
Δ→0 1 − δ
lim
Proposition 5.5. In any pooling equilibrium,
lim Wsh = 0,
Δ→0
lim Wb =
Δ→0
φh + φl
− sh and
2
lim Wsl =
Δ→0
d
(sh − sl ).
b + 2d
22
IN-KOO CHO AND AKIHIKO MATSUI
√
√
Proof. Since πp2 = O(Δ) and ζ = O( Δ), Wsh = O( Δ). Note that πp = ν([sh +
l
Wsh , φh +φ
− Wb ]). Since πp → 0, and Wsh → 0,
2
φh + φl
− sh → 0
Wb −
2
as Δ → 0.
We can write
Wsl = Wsh +
βζπp
(sh − sl ).
1 − βδ + βζπp
Note that
βζπp
d
→
.
1 − βδ
b+d
Thus,
lim Wsl =
Δ→0
d
(sh − sl ).
b + 2d
In order to be a pooling equilibrium, it is necessary that πs = 0 or equivalently,
Wb + Wsl > φl − sl .
This inequality holds in the limit as Δ → 0 if and only if
sh − sl
b + 2d
>
.
2(b + d)
φh − φl
5.2. Characterization. We have shown that if
φh + φl
≥ sh
2
and
sh − sl
b + 2d
≥
(5.35)
,
2(b + d)
φh − φl
then an equilibrium exists which must be a pooling equilibrium. It would be instructive
to identify an equilibrium for the entire domain of the parameters of (φh , φl , sh , sl , b, d).
We need to consider three boundaries.
B1.
b φh − sh
−
+1=0
d
φl − sl
B2.
φh + φl
− sh = 0
2
B3.
sh − sl
b + 2d
−
=0
2(b + d) φh − φl
DYNAMIC LEMONS PROBLEM
23
B1 is the boundary for condition C4, while B2 and B3 are the boundaries for conditions
A3 and (5.35), respectively.
Let us add superscript + or − to represent the area of the parameters where ≥ or <
holds instead of =. For example,
b φh − sh
+
B1 =
−
+1≥0
d
φl − sl
and
−
B1 =
b φh − sh
−
+1<0 .
d
φl − sl
Note that B1− is the area of parameters in which C4 holds, while C4 fails over B1+ .
It is instructive to see that the three equations of B1, B2 and B3 are not independent.
Note that if B2 and B3 hold, then B1 holds. Also, one can show that over B1+ , if B2
holds, then
sh − sl
b + 2d
−
< 0.
2(b + d) φh − φl
Similarly, over B1− , if B3 holds, then
φh + φl
− sh < 0.
2
Thus,
B2+ ∩ B3+ ∪ B2− ∩ B1− ∪ B3− ∩ B1+
covers the entire domain of feasible values of (φh , φl , sh , sl , b, d).
We have shown that over B2+ ∩ B3+ , the equilibrium must be a pooling equilibrium.
The construction of the semi-pooling equilibrium is carried over for the rest of the parameter values. Over B2− ∩ B1− , the equilibrium must be a semi-pooling equilibrium in
which lim inf Δ→0 ζ > 0. Over B2− ∩ B1+ , the equilibrium is a semi-pooling equilibrium
in which lim inf Δ→0 ζ = 0.
6. Robustness
It is easy to see that a slight change in xhs does not affect the result, because few high
quality sellers have opportunities to trade with buyers. On the other hand, it is less obvious
whether or not a slight change of xls affects the equilibrium outcome in a discontinuous
manner. For some parameter values, only a small fraction of low quality sellers remain in
the market, and therefore, a small difference in population size between buyers and low
type sellers may be magnified in the relative size of population in the pool. We show,
however, that this is not the case. Our results are robust against this type of perturbation
of the model.
Let xls = 1 + ε where ε is a parameter that can be either positive or negative. We
assume the following condition that corresponds to A3:
(1 + ε)φl + φh
< sh .
A3 .
2+ε
24
IN-KOO CHO AND AKIHIKO MATSUI
Note that if A3 holds, then there exists ε̂ > 0 such that for all ε ∈ [−ε̂, ε̂], A3 holds.
To begin with, the balance equation (3.8) is modified to
πp ζ
πs ζ
l
+
zsl
(6.36)
1 + ε − zs =
1−δ 1−δ
Then balance equations (4.15) and (4.16) are now modified to
(6.37)
πp ζ
μl πs ζ
(1 + ε) +
(1 + ε) + (2 + ε)ζ
1−δ
1−δ
πp ζ
1−δ + 1
(1 + ε).
πp ζ
πs ζ
+
+
1
1−δ
1−δ
1 =
μl
1 − μl
(6.38)
=
Lemmata up to Lemma 4.11 are carried over to the current case, though some proofs
are to be slightly modified. To prove Lemma 4.12, we need the following lemma, which is
a stronger version of Lemma 4.8.
Lemma 6.1. lim μl ≥
Δ→0
(φh − sh ) − (φl − sl )
.
φh − φl
Proof. Lemmata 4.6 and 4.7 imply
lim (Wb + Wsh ) ≤ φl − sl .
Δ→0
From (4.21) and limΔ→0 πp = 0, we have
lim (φ(μh ) − sh ) = lim (Wb + Wsh ).
Δ→0
Δ→0
Combining the above two, we have
lim (φ(μh ) − sh ) ≤ φl − sl .
Δ→0
Since φ(μh ) is defined as μl φl + (1 − μl )φh , we have the statement of the lemma as desired.
Using this lemma, we can prove the statement that corresponds to Lemma 4.12.
ζπp
> 0.
Δ→0 1 − βδ
Lemma 6.2. ∃ε > 0 ∀ε ∈ (−ε, ε) lim
Proof. Suppose that limΔ→0
ζπp
1−βδ
= 0. By (3.9),
zsh → 1.
By a properly modified version of (3.12),
1
2+ε
μl πs ζ
→
−
ζ.
1−δ
1+ε 1+ε
Since lim μl > 0 by Lemma 4.8,
0≤
1 − (2 + ε)ζ
πs ζ
→
< ∞.
1−δ
μl (1 + ε)
DYNAMIC LEMONS PROBLEM
25
From (6.36),
1 ≥ zsl →
1+ε
1+
1−(2+ε)ζ
μl (1+ε)
> 0.
Then,
1+ε
zl
lim μl = lim h s l = lim
Δ→0
Δ→0 zs + zs
Δ→0
1+
1−(2+ε)ζ
μl (1+ε)
1+ε
1+
1+
.
1−(2+ε)ζ
μl (1+ε)
By multiplying both numerator and denominator of the right hand side of the above
equation by (1 + ε)μl + 1 − (2 + ε)ζ, we obtain
(1 + ε)2 μl
.
Δ→0 (2 + ε)(1 + ε)μl + 1 − (2 + ε)ζ
lim μl = lim
Δ→0
This implies
lim ζ = lim −ε + (1 + ε)μl ,
(6.39)
Δ→0
Δ→0
which is strictly positive for a sufficiently small ε in absolute value due to Lemma 6.1.
Thus, under the hypothesis of the proof, we have
πp
= 0.
Δ→0 1 − βδ
lim
Hence, the second terms of Wsl in (4.18) and Wb in (4.19) converge to zero as Δ → 0.
Also, (6.36) becomes
lim
Δ→0
πs
(1 − μl )(1 + ε) 1
= lim
− < ∞.
1 − δ Δ→0
μl ζ
ζ
Therefore, we have
(πs )2
= 0,
Δ→0 1 − δ
lim
which implies that the first terms of Wsl and Wb converge to zero as Δ → 0. Thus, both
Wsl and Wb converge to zero as Δ → 0. This is a contradiction to the statement that
corresponds to Lemma 4.11.
The rest of the lemmata can be proven in the same manner as in the case of ε = 0 with
Condition C4 being modified to
(φh − sh ) − (φl − sl )
b
ε φh − φl
<
−
,
C4 .
d
φl − sl
1 + ε φl − sl
and we have the following modified theorem.
26
IN-KOO CHO AND AKIHIKO MATSUI
Theorem 6.3. There exists ε > 0 such that for all ε ∈ (−ε, ε), and for any sequence of
undominated stationary equilibria, we have
lim Wsh = 0
Δ→0
lim
Δ→0
Wsl
lim Wb
Δ→0
d
ε
= min φl − sl ,
(φh − sh ) − (φl − sl ) −
(sh − φl )
b
1+ε
d
ε
(φh − sh ) − (φl − sl ) −
(sh − φl ) .
= max 0, (φl − sl ) −
b
1+ε
7. Concluding Remarks
The excess demand and supply are important objects of investigation in the labor
market search models. A typical labor market search model (e.g., Mortensen and Pissarides
(1994)) assumes a matching function m(u, v) which specifies the rate at which unemployed
workers (u) are matched to the vacant positions (v). Indeed, Blanchard and Diamond
(1989) pointed out that the matching function itself presumes the simultaneous existence
of a positive excess supply (u) and a positive excess demand (v).
We have demonstrated that if the labor market is subject to lemons problem, then the
equilibrium outcome can entail positive excess supply and demand at the same time. We
believe it is an important first step to provide a micro foundation for Beveridge curve,
that is a stable relationship between the unemployment and the job vacancy in the steady
state labor market (Blanchard and Diamond (1989)).
DYNAMIC LEMONS PROBLEM
27
References
Akerlof, G. A. (1970): “The Market for ”Lemons”: Quality Uncertainty and the Market Mechanism,”
Quarterly Journal of Economics, 84(3), 488–500.
Blanchard, O. J., and P. Diamond (1989): “The Beveridge Curve,” Brookings Papers on Economic
Activity, 1989(1), 1–76.
Blouin, M. R., and R. Serrano (2001): “A Decentralized Market with Common Values Uncertainty:
Non-Steady States,” Review of Economic Studies, 68, 323–346.
Burdett, K., and R. Wright (1998): “Two-Sided Search with Nontrasferable Utility,” Review of
Economic Dyanmics, 1(1), 220–245.
Cho, I.-K., and A. Matsui (2011): “Search Theory, Competitive Equilibrium, and Nash Bargaining
Solution,” University of Illinois and University of Tokyo.
(2012): “Competitive Equilibrium and Matching under Two Sided Incomplete Information,”
University of Illinois and University of Tokyo.
Guerrieri, V., R. Shimer, and R. Wright (2010): “Adverse Selection in Competitive Search Equilibrium,” Econometrica, 78(6), 1823–1862.
Gul, F., and A. Postlewaite (1992): “Asymptotic Efficiency in Large Exchange Economies with
Asymmetric Information,” Econometrica, 60(6), 1273–1292.
Mortensen, D. T., and C. A. Pissarides (1994): “Job Creation and Job Destruction in the Theory of
Unemployment,” Review of Economic Studies, 208, 397–415.
Mortensen, D. T., and R. Wright (2002): “Competitive Pricing and Efficiency in Search Equilibrium,”
International Economic Review, 43(1), 1–20.
Vincent, D. R. (1989): “Bargaining with Common Values,” Journal of Economic Theory, 48(1), 47–62.
Wolinsky, A. (1990): “Information Revelation in a Market with Pairwise Meetings,” Econometrica, 58(1),
1–23.
Department of Economics, University of Illinois, 1206 S. 6th Street, Champaign, IL 61820
USA
E-mail address: [email protected]
URL: https://netfiles.uiuc.edu/inkoocho/www
Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan
E-mail address: [email protected]
URL: http://www.e.u-tokyo.ac.jp/~amatsui