In a repeated game principal offers a contract to an agent to produce output y in exchange for wage w. Per period utility for the principal is up =y−wandforagentisua =w−c(a). Ineveryperiod,iftheagent is hired and exerts effort a the output y is 1/2 with the probability a and 0 with the probability 1 − a. The effort costs the agent c(a) = a2/2. If he doesn’t work for the principal, he can get his per period outside wage w = 1/32 with zero effort. The principal makes zero profit without the agent. Both discount the future at an interest rate r. In every period the principal can offer the agent a contract consisting of salary s and bonus b that is payed if output is 1/2, that is, if y = 1/2 then w = s + b, otherwise w = s. Assume initially that the output is verifiable in court so that the principal cannot default on the bonus. (a) For a bonus level b, what effort a∗ does the agent choose? (b) What is the optimal b and s for the principal? What kind of contract is this?

 

Now assume that the output is not verifiable so the principal can default on the bonus.

(c) What is the critical interest rate r such that for r ≤ r the contract from (b) is sustainable in SPNE with trigger strategies whereby if the principal ever refuses to pay b the agent never exerts any effort thereafter? (Hint: find r such that rb equals the profit of the principal in (b)).

(d) Find the critical interest rate r ̄ such that for r > r ̄ no contract with a positive bonus can be signed. (Hint: at such r, the slope of the profit function of the principal with respect to b equals r, and rb equals her profit.)

(e) Show graphically for this example that for r ∈ (r,r ̄) the bonus b∗ that is sustainable is decreasing in r. Label the graph clearly.