Game theory q2

Question 2. Consider the following principal agent problem: The agent chooses an effort level e c {1, 2). Effort level affects the distribution of output, R, according to the following table. Action R=2 R=10 P= PE p The agent has utility function depending on wage and effort: u(w, e) = vw - e. If the agent does not work for the principal the agent gets a utility of a = 0. A wage contract depends on performance: (w.(2), w(10)). (a) If the principal wants to induce the low effort level, show that w(2) = w(10) = 1 will achieve that. Specifically, show that this wage scheme satisfies the participation constraint with e = 1, so that with effort level e = 1, the agent has expected utility of at least 0. (b) Give the incentive compatibility constraint (e = 2 is preferred to e = 1), and the participation constraint requiring that e = 2 gives the agent an expected utility of at least 0. (c) Show that if the principal wants to induce e = 2, the wage contract (@(2), w(10)) = (2, ") achieves this. (d) Which contract yields the highest profit