• Two agents, i ∈ {1, 2}, form a partnership to produce an output y = f(a1, a2), ai is the private effort of agent I. 

• The cost of effort for agent i is given by gi(ai) = a 2 i 2. 

• The output of the team is given by f(a1, a2) = a1 + a2. • For each y, partnership specifies the payment wi(y) to agent i. such that w1(y)+w2(y) = y. • No measure of individual performance is available. That is, wi is contingent on the aggregate output y alone.

(a)  In the first-best level of effort, the team maximizes y − [g1(a1) + g2(a2)] by choosing a1, a2. What is the optimal effort choice, a ∗ 1, a∗ 2 in this problem? 

(b) Now, suppose that the agents agree to a payment rule wi(y), i ∈ {1, 2}. Given this payment rule, agent i chooses ai to maximize his utility, wi(y) − gi(ai). Can we have a payment rule, wi(y), so that agents choose ai = a ∗ i in this problem? Explain your answer. 

(c) Suppose that we relax the constraint that w1(y) + w2(y) = y to allow for w1(y) + w2(y) ≤ y. In particular, consider the following scheme: wi(y) = (y ∗ 2 if y ≥ y ∗ 0 otherwise where y ∗ = a ∗ 1 + a ∗ 2 . Is there an equilibrium where agents choose a ∗ i with this incentive scheme? Explain your answer.