Question

Hello , I cannot find the answer for this questions : Problem 3. (Signaling game: market for used cars) Consider the market for used cars and assume that the cars can be of one of two possible quality levels: low (the ’lemons’) and high (the ’plums’). The seller knows the true quality of the car (the seller’s type is given by the quality level) but the buyers only know the probability distribution of the cars in the market. Assume that the buyer’s initial (prior) beliefs are as follows: probability p for a car being of high quality, and probability (1 − p) for a car being of low quality. If the seller sells his car, then his payoff is equal to the price paid by the buyer. If the seller does not sell the car, his payoff will be equal to his valuation of the used car: $10000 if it is of high quality and $6000 if it is of low quality. The buyer’s valuation of cars with high quality is $12000, and his valuation of low-quality cars is $7000. However, he only learns the true quality after the purchase. If the buyer purchases a car, then his payoff is given by the difference between his true valuation and the price he pays. If he does not buy a car, his payoff is zero. (a) First, consider a market with complete information for both sides of the market. Explain why, given the buyers’ and sellers’ valuations, all cars will be sold. Is this outcome Pareto-optimal? (b) Now consider the market with incomplete information as described above. What would be a seller’s pooling strategy (to offer all the cars at the same price)? If sellers follow this pooling strategy, what would be their best response (strategy and beliefs) given their prior belief p? Given this best response of the buyers, is it optimal for the seller to follow the pooling strategy given p? (c) Find values for prior belief p such that there is a perfect Bayesian Nash equilibrium with pooling. Is this outcome Pareto-optimal, i.e. are all those cars sold that would be sold with complete information?

Thank you , Dania

Answer 1

a) If consider the market with incomplete information as described above

High quality used car: $10000

low quality used car $6000

buyer’s valuation of cars with high quality is $12000,

buyers valuation of low quality car  $7000

Expect worth of car to a risk neutral seller

1/3*10000+2/3*6000=7333.33

Expect worth of car to a risk neutral buyers

1/3*12000+2/3*7000=8666.66

market price would clear at the price of 866.66

the buyers’ and sellers’ valuations$ 7999.10

b) seller’s pooling strategy to offer all the cars at the same price because low and good quality car air mix at the time he sell all car air same price..

that the time buyer follow this pooling strategy is seller would be ready to sell low quality car for a price equal to or above $6000 and that is why such owner would his car at the expected worth of$8666.66 . thuse only bad car are able in the market but no good car. this point is considered by and thereforee, he knows that whatever car is going to buy is a bad car.

when the probability of existing of a high quality car=2/3 and the probability existing of a low quality car=1/3

expected worth to buyer

2/3*12000+1/3*7999.10=10666.36

c)all those cars sold that would be sold with complete information   

the owner of a high quality car would be ready to sell his car for a price at to or avobe $10000 and that is why such owner would be ready to sell car at the expected worth of $ 15000. The owner of a low quality car would be to sell his car price above $6000 and that time his expected worth of $ 6500.those bothh of car are ready in market.would be clear this price $6500.