In this problem, we will consider a multitasking problem where the principal can only incentivize the agent on one task, but where there is a "crowding-in" effect: the agent's effort in one task reduces the agent's marginal cost of effort in the other task.

There are two tasks (task 1 and task 2). The principal benefits from the agent's effort in both tasks: Up = y1 +y2 w where y1 = e1 and y2 = e2. The agent has payoff function Ua = w 12 (e21 + e2 e1e2).

(Note that the agent may choose negative effort levels, potentially resulting in negative output.)

The principal cannot reward the agent for total output; instead, he can only reward the agent for his performance in the first task. That is, the principal can offer the agent an incentive scheme of the form w = + x, where x = e1.

The timing is as usual:

step 1: Principal offers agent an incentive scheme w = + x. step 2: Agent may accept or reject the offer. If he rejects, he receives an outside option of

zero. step 3: If agent accepts, then he chooses e1 and e2.

step 4: Principal pays agent w = + x.

We'll go through the problem step-by-step.

1) For step 3, given the principal's offer (w = + x), write down the agent's maximization problem, and calculate his payoff-maximizing effort choices e1 and e2 as a function of and .

2) For step 1, write down the principal's maximization problem, and calculate his payoff- maximizing choice of incentive scheme ( and ).

3) What effort levels does this incentive scheme induce in the agent?

4) Calculate the efficient effort levels (i.e. the effort levels e1,e2 that maximize total payoffs

Up + Ua).

5)Explain, in words, why your answers to (3) and (4) differ (if they indeed differ).