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 Consider principal (P) who employ an agent (A) for the production of a good the quality of which is measure by q. P pays A p each period, and renews A’s contract for the next period with probability q.
(a) If A is providing quality q in every period, what is the expected lifetime of the contract
(b) Assume that A’s utility depends on p and q such that
u (p, q) = p – 2/1q
Find what is the value of the contract to A (u(p,q) ).
(c) Assume that if A loses their job then they will receive an unemployment benefit with value 1. What is A’s enforcement rent?
(d) Find A’s incentive compatibility constraint.
(e) Assuming that P wants to maximize q/p find the contract’s Nash Equilibrium (price, quantity and time period of the contract.)
 Consider the previous set up but assume, that u(p,q) = p 2/1q and the unemployment benefit is 0 (no benefit). Find the Nash Equilibrium of the contract.
 An employer hires a worker to perform a task which requires the worker to exert effort, measured in working hours (h). The workers utility is
u = w – h,
where w is the salary (in income units) that the workers gets if not fired. If the workers exerts effort at least equal to 5 hours a day, then their contract will not be terminated. If the worker exerts loss effort then there is a 50% chance that the employer will find out and terminate the workers contract.
(a) Explain why, the worker will only choose to work either for 0 or for 8 hours a day.
(b) if the worker loses their job, then with probability 1/3 they find another job, where they get paid half from what they would be paid in the previous job, otherwise they get nothing. Express this situation in a tree form
(c) How much should the employment offer (w) be, such that the worker has an incentive to put effort?
 An employer hires a worker to perform a task which requires the worker to exert effort, measured in working hours (h). The workers utility is
U = w – h,
where w is the salary (in income units) that the worker gets if not fired. If the worker exerts loss effort at least equal to 4 hours a day, then their contract will not be terminated. If the worker exerts less effort then there is a 50% chance that the employer will find out and terminate the workers contract. Assume that there is very high unemployment, such that if the worker gets fired, then they do not find a job but get unemployment benefit B = income units.
(a) Express this situation in a tree form.
(b) How much should the employment offer (w) be, such that the worker has an incentive to put effort?
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