Anemployerisdesigningawagecontract. Therearetwo states, B andG. InstateB revenue is £400, and in state G revenue is £800. If the employee exerts a high level of effort, the

probability of state B is 0.4. If the employee exerts a low level of effort, the probability of

state B is 0.8. The employee maximises expected utility, with utility for wages u($) = ^$.

High effort decreases the employee’s expected utility by 6 units, and low effort decreases it by 1 unit. By not accepting the contract, the employee gets reservation utility of 12 units.

Question A.1: Assuming perfect information, explain and derive the contract that opti- mally implements low effort. (4 marks)

Question A.2: Assuming perfect information, explain and derive the contract that opti- mally implements high effort. (4 marks)

Question A.3: Find the optimal contract under perfect information. (4 marks) Now assume asymmetric information, in the sense that effort is unobservable.

Question A.4: Explain and derive the contracts that optimally implement low effort under

asymmetric information. (3 marks)

Question A.5: Explain and derive contract that optimally implements high effort under asymmetric information. (10 marks)

Question A.6: Find the optimal contract under asymmetric information. (5 marks)

Part B: Please answer ONE question (20 marks).

Question B.1: Explain the value of information in single-person decision problems and explain why this value is never negative. Give an example (from a market or game scenario) where the value of information is negative.

Question B.2: Explain what is meant by the “Nash solution” of a two-player bargaining

game and list the axioms that characterise the Nash solution. Explain the “dividing a dollar”

bargaining game assuming two players, A and B, with utilities uA  and uB  that are both strictly increasing, strictly concave, and uA (0) = uB (0) = 0. Using the Nash solution, show

that being “more risk averse” is detrimental in this bargaining game.

Question B.3: Consider the following competitive market for insurance.  There are two

states of the world: good G and bad B .  Consumers have endowment e = (eG , eB ) with eG   > eB .  Consumers are risk averse, expected utility maximisers, with common utility function u. There are two types of consumers: high risk types H and low risk types L. The probabilities that H and L types find themselves in the bad state are pH  and pL . There is probability p that a consumer picked at random is type L. Firms are risk-neutral expected profit maximisers.  The firm offers consumers a state-contingent contracts c  =  (cG , cB ) in exchange for their endowment e = (eG , eB ).  Explain what is meant by a competitive

equilibrium set of contracts and characterise these contracts under asymmetric information (assuming equilibrium exists).