A manager and a worker interact as follows. The manager would like the worker to exert some effort on a project. Let e denote the worker's effort. Each unit of effort produces a unit of revenue for the firm; that is, revenue is e. The worker bears a cost of effort given by ae2, where a is a positive constant. The manager can pay the worker some money, which enters their payoffs in an additive way. Thus, if the worker picks effort level e and the manager pays the worker x, then the manager's payoff is e - x and the worker's payoff is x - ae2.
Assume that effort is verifiable and externally enforceable, meaning that the parties can commit to a payment and effort level. Imagine that the parties interact as follows: First, the manager makes a contract offer to the worker. The contract is a specification of effort en and a wage xn. Then the worker accepts or rejects the offer. If she rejects, then the game ends and both parties obtain payoffs of 0. If she accepts, then the contract is enforced (effort en is taken and xn is paid). Because the contract is externally enforced, you do not have to concern yourself with the worker's incentive to exert effort.
(a) Solve the game under the assumption that a is common knowledge between the worker and the manager. That is, find the manager's optimal contract offer. Is the outcome efficient? (Does the contract maximize the sum of the players' payoffs?) Note how the equilibrium contract depends on a.
Let e - and x - denote the equilibrium contract in the case in which a = 1/8 and let e and x denote the equilibrium contract in the case in which a = 3/8. Calculate these values. Let us call a = 3/8 the hightype worker and a = 1/8 the low-type worker.