Randal Graves works at RST Video, a video rental store. He's t

here for 8 hours, but may not work hard for all 8 hours. Let e be the number of hours he works hard. The video store's gross profit is _g = 1000 + 64e + , where is a random variable with E( ) = 0 and V ar( ) = 1/2. Its net profit is = _gs, where s is the salary it pays Randal. Randal has expected utility given by Eu=E(s) 1 /2 V ar(s) c(e), where c (e) = e³/3 is his cost of effort.

a. suppose the video store owner can videotape Randal's performance. The owner offers a contract saying he is paid s* if Randal works hard for at least e* hours. What e* and s*

will the owner's "first-best" contract?

Consider the "second best" case where the firm cannot monitor Randal's choice of e. Suppose the firm offers a linear contract of the form s(_g) =a +b_g.

_b. Given an arbitrary contract s(_g) =a + b_g, what effort will Randal choose to maximize his expected utility?

c. Write down Randal's participation constraint (meaning, for what a will he be willing to accept the contract s(_g) =a+b_g?.

d. Solve for the video store owner's second-best contract.

(Keep in mind to transform the problem to one of choosing which e to induce, given the participation and incentive constraints you've derived. In other words, replace a and b with the appropriate functions of e. You'll need a computer to solve for e.)

e. Compare the first-best effort level to the effort level induced by the second-best contract.

f. Compare Randal's expected utility in the first-best contract and second-best contract.

g. Compare the company's expected profit in the first-best contract and second-best contract.

h. Does the first-best outcome Pareto dominate the second-best outcome (meaning, it's at least as good for everybody and strictly better for somebody)?