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Adverse Selection Econ 210C UCSB April 28, 2011

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Market Failure Due to Information Asymmetry Consider a market for used cars of two diﬀerent qualities, with one half worth \$5,000 and the other \$10,000. Each seller knows the quality but each potential buyer does not. What would be the market equilibrium? Buyer’s expected value of a car at price p is: E [ v | v p ] = ± 5, 000, 5, 000 p < 10, 000; 7, 500, p 10, 000. In market equilibrium: E [ v | v p ] = p . Econ 210C UCSB Paper
Suppose that the worths of the cars are independently and uniformly distributed over [5,000, 10,000]. What would be the market equilibrium? Buyer’s expected value of a car at price p is: E [ v | v p ] = 1 2 ( 5, 000 + p ) . In market equilibrium: E [ v | v p ] = p p = 5, 000 Econ 210C UCSB Paper

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Let v b and v s denote the worth of a car to a buyer and a seller, respectively. Assume v b = λ v s ( λ > 1 ) . What is buyer’s expected value of a car at price p ? E [ v b | v s p ] = λ E [ v s | v s p ] = λ 2 ( 5, 000 + p ) . What would be the market equilibrium? E [ v b | v s p ] = p p = 5, 000 λ 2 - λ . λ = 1.2 p = 7, 500. Econ 210C UCSB Paper
Signaling Game A signaling game is two-player game of incomplete information in which one player is informed. The informed player’s strategy is a type-contingent message and the uninformed player’s strategy is a message-continent action. Timing of moves: (i) nature draws a type t T for player 1 (the informed) according to a probability distribution p Δ ( T ) and reveals t to player 1 but not to player 2 (the uninformed); (ii) player 1 chooses an action (message) a 1 A 1 ; (iii) player 2 observes player 1’s choice and then chooses an action a 2 A 2 ; (iv) Player 1 receives payoﬀ u 1 ( a 1 , a 2 , t ) and player 2 receives payoﬀ u 2 ( a 1 , a 2 , t ) . Γ = { A 1 , A 2 , T , u 1 , u 2 , p } . Econ 210C UCSB Paper

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Player 1’s strategy: s 1 : T -→ A 1 . Player 2’s strategy: s 2 : A 1 A 2 . Perfect Bayesian equilibrium (PBE): (i) players’ strategies are best responses given beliefs; (ii) beliefs are derived from strategies via Bayes’ rule whenever applicable. Formally, a PBE in pure strategy is a pair ( s * , μ * ) with strategy proﬁle s * and belief system μ * = ( μ * ( ·| a 1 )) a 1 A 1 such that (i) for all t T , s * 1 ( t ) solves max a 1 A 1 u 1 ( a 1 , s * 2 ( a 1 ) , t ) and for all a 1 A 1 , s * 2 ( a 1 ) solves max a 2 A 2 t T u 2 ( a 1 , a 2 , t ) μ * ( t | a 1 ) ; (ii) μ * ( t | a 1 ) = p ( t ) / p ( t : s * 1 ( t ) = a 1 ) if p ( t : s * 1 ( t ) = a 1 ) > 0 and μ * ( t | a 1 ) is arbitrary otherwise.
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