I can't solve problems 5.5-5.7

( Proportional payment), or Problem 5 Randal Graves works at RST Video, a video rental store. He's there 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 * = 1000 + 6de + 2, where f is a random variable with E() = 0 and Var(?) = 1/2. Its net profit is = = , - s, where's is the salary it pays Randal. Randal has expected utility given by Eu = E(s) - Var(a) - ode), where cle) = $ is his cost of effort. Problem 5.1 (First Best) 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(,) = at b, Problem 5.2 Given an arbitrary contract s(5,) = a + by, what effort e will Randal choose to marimize his expected utility? Problem 5.3 Write down Randal's participation constraint (meaning, for what a will he be willing fo secret the contract s(=) = + bug P Problem 5.4 Solve for the video store owner's second-best contract. (Hint: Remember to trans- form the problem to one of choosing which e to induce, given the participation and incentive con- struints you're derived. In other words, replace a and b with the appropriate functions of er You'll nerd a computer to solve for e.) Problem 5.5 Compare the first-best effort level to the effort level induced by the second-best con- fruct Problem 5.6 Compare Randal's expected utility in the first-best contract and second-best contract. Problem 5.7 Compare the company's expected profit in the first-best contract and second-best con- truck. Problem 5.8 Does the first-best outcome Pareto dominate the second-best outcome (mearaing, it's at least is good for everybody and strictly better for somebody)