Another look at equilibrium with asymmetric information
Econ 235, Spring 2013
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Prices as markets
• This setup can be traced back to Gale [1992, 1996]
• Guerrieri et al. [2010] re-stated it in terms of “competitive search”
• Guerrieri and Shimer [2012a,b], Chang [2011] are based on the same setup.
• Each possible price p defines a market
• Buyers can choose to demand in any market (i.e. price).
• Sellers can choose to suppy any quality of goods in any market (i.e. price)
• The amount of trade at any price is equal to min {S (p) , D (p)}. In Guerrieri et al. [2010],
instead, the amount of trade if given by the matching function m (S (p) , D (p)), of which
min {S (p) , D (p)} is a special case
• If demand is less than supply, all the goods supplied get rationed pro-rata.
1.1
Static case
• Sellers own a used car of quality q ∈ {q1 , q2 , . . . , qJ }
• Utility
(
u=
c + τq
c
if you keep car
if you sell car
• Buyers have an endowment of a goods
(
u=
c + tq
c
if you buy car
if you don’t buy car
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Econ 212, Spring 2013
Pablo Kurlat
• We’ll focus on the case where a is very large, so buyers consume part of their endowment /
the marginal value of wealth is 1
1.2
Equilibrium definition
1. Value vj for sellers who own car of quality qj
2. A function Θ : R+ → [0, ∞] that maps each market / price to a buyer/seller ratio
3. A function Γ : R+ → ∆J that maps each market to a distribution over car qualities
4. A measure F with support P
such that
1. Seller’s optimality:
vj = max [min {Θ (p) , 1} p + (1 − min {Θ (p) , 1}) τ qj ]
(1)
P
1
1
j γj (p) qj
,1 t
+ min
,1
=1
λ = max min
p∈R+
Θ (p)
p
Θ (p)
(2)
p∈R+
for all j
2. Buyer’s optimality
3. Equilibrium beliefs. For all markets p with
(a) Θ (p) < ∞, i.e. the buyer-seller ratio is less than infinity
(b) γj (p) > 0, i.e. there is positive weight on car-quality j
p must solve (1) for seller j
4. Active markets. p ∈ P only if it solves 2
5. “Consistency of supply with beliefs” / market clearing
ˆ
πj =
γj (p) dF (p)
P
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Econ 212, Spring 2013
1.3
Pablo Kurlat
Why we get a separating equilibrium
• Consider the case with two asset qualities
• Suppose there was pooling at price p, where the Buyer-Seller ratio is Θ (p)
• What happens at market p + ?
• Either:
– Θ (p + ) = ∞
or
– at least one of the types must find it weakly optimal to sell in market p + • Can we have Θ (p + ) = ∞ (infinite buyers per seller)?
– No: because then seller would sell for sure at that price
• Therefore Θ (p + ) must be such that one of the sellers must find it weakly optimal to sell
in market p + • By single crossing, if the low type finds it weakly optimal, the high type finds it strictly
optimal to sell in market p + . Therefore, either:
– it’s strictly optimal for the high type to sell in market p + , which contradicts the
assumption of pooling,
OR
– it’s weakly optimal for the high type to sell in market p + and strictly suboptimal
for the low type. In this case, buyers must believe that in market p + they will
only encounter high types, which means, for small enough, they would make profits.
Therefore in equilibrium they are not willing to buy at price p.
1.4
General pattern of equilibrium
• Full separation
• Quality qj sells in market p = tqj
• For all prices between tqj and tqj+1 , the buyer-seller ratio is such that type j is indifferent
and type j + 1 strictly prefers not to trade at those prices.
• Note that buyer-seller ratios are defined even in markets with zero buyers and zero sellers.
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Econ 212, Spring 2013
Pablo Kurlat
References
Briana Chang. Adverse selection and liquidity distortion in decentralized markets. Discussion
Papers 1513, Northwestern University, Center for Mathematical Studies in Economics and Management Science, August 2011.
Douglas Gale. A walrasian theory of markets with adverse selection. Review of Economic Studies,
59(2):229–55, April 1992.
Douglas Gale. Equilibria and pareto optima of markets with adverse selection. Economic Theory,
7(2):207–235, 1996.
Veronica Guerrieri and Robert Shimer. Dynamic adverse selection: A theory of illiquidity, fire
sales, and flight to quality. Working Paper 17876, National Bureau of Economic Research,
March 2012a.
Veronica Guerrieri and Robert Shimer. Markets with multidimensional private information. University of Chicago Working Paper, 2012b.
Veronica Guerrieri, Robert Shimer, and Randall Wright. Adverse selection in competitive search
equilibrium. Econometrica, 78(6):1823–1862, November 2010.
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