ECN611 Handout9

ECN611 Handout9 - E c o n o m ic s 6 1 1 H a n d o u t 9 T...

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Unformatted text preview: E c o n o m ic s 6 1 1 H a n d o u t # 9 T H E E C O N O M IC S O F A S Y M M E T R IC IN F O R M A T IO N Chap. 14: Asymmetric info after the time of contracting: Moral Hazard Standard example: Principal-agent model “A valid contract cannot commit either party to transfer money under conditions that cannot be independently verified by both.” Employment model Employees HOMOGENEOUS Effort: E, 0￿E￿1 Von Neumann-Morgenstern utility function: U = W ￿ (1 – E) Reservation utility: W > 0 (Exogenous) Probability caught shirking: ð(E) = 1 – E Firm Revenue function: V(0) = 0; Profit function: V(LE) = V(L ￿ E) V￿ > 0, V￿ < 0 R(W, L) = V(L ￿ E) – W ￿ L 611.09 - 1 We seek a subgame perfect Nash equilibrium via backwards induction: Employee: MaxE ª[U(W, E)] = ð(E) ￿ W + (1 – ð(E)) ￿ W ￿ (1 – E) = (1 – E) ￿ W + E ￿ W ￿ (1 – E) Firm: Choose W and L (or LE) to maximize R. Example: V(LE) = (LE)1/2. W* = 2W; L* = [64W2]–1; E* = 1/4. At a solution, W*, we must have W* > W. But W* is a market clearing wage, i.e., at full employment, W = W*. Therefore, the market clearing wage is NOT part of a subgame perfect Nash equilibrium. Exercise: Do it all again with improved detection: ð(E) = 1 – E2. 611.09 - 2 Unemployment Insurance - - without Moral Hazard Assume a risk-averse individual with income Y faces unemployment with fixed probability ð. Insurance option: Indemnity, I ; Premium, P. Individual maximizes expected utility: U(Net Income), U￿ > 0; U￿ < 0 ª(U(Net Income)) = ð U(I – P) + (1 – ð) U(Y – P) Insurance Co.: Expected profit R(P,I) = ð (P – I) + (1 – ð) P Actuarially fair: Expected profit = 0: P*(I) = ð I Individual: ª(U(Net Income) = ð U(I – ð I) + (1 – ð) U(Y – ð I) Derivative with respect to I: ð U￿(I – ð I) (1 – ð) + (1 – ð) U￿(Y – ð I) ( – ð) Setting equal to 0 at solution I*: ð (1 – ð) U￿(I* – ð I*) = (ð) (1 – ð) U￿(Y – ð I*) so: U￿(I* – ð I*) = U￿(Y – ð I*) ￿ by monotonicity of U￿, I* = Y, full insurance. Example, ð = .25, Y = 400 and U(￿) = (￿)½: P = .25 I for actuarially fair insurance, so ª(U(I)) = .25(I – .25 I)½ + .75(400 – .25 I)½ 611.09 - 3 Insurance - - with Moral Hazard Assume a risk-averse individual with income Y faces unemployment with fixed probability (.25) but then can search for a new job, failing in that search with probability ￿(E) where E is the effort (measured in foregone utility) to prevent that failure. Insurance option: Indemnity, I ; Premium, P. Individual: If laid off, the expected utility of search is ª(U(P,I,E)) = ￿(E)[U(I – P) – E] + (1 – ￿(E))[U(Y – P) – E] = U(Y – P) – ￿(E)[U(Y – P) – U(I – P)] – E = U(Y – P) – ￿(E)N(P,I) – E where N(P,I) is the net benefit of finding a new job when holding a (P,I) policy Solve the maximization problem for E*(P,I) Insurance Co.: Expected profit: R(P,I) = .25￿(E*(P,I))(P – I) + (1 – .25￿(E*(P,I))) ￿ P Set equal to 0 and solve for actuarially fair premium: P*(I) = .25￿(E*(P*(I), I)) ￿ I Individual: Maximizes ª(U(P,I)) = .75U(Y – P) + .25{[(1 – ￿(E*(P,I)))[U(Y – P) – E*(P,I)] + ￿(E*(P,I))[U(I – P) – E*(P,I)]} = U(Y – P) – .25[￿(E*(P,I))N(P,I) + E*(P,I)] Example: Y = 400, U(￿) = (￿)½, and ￿(E) = e –E/4 611.09 - 4 (exponential) [270 is the solution to N(.25I, I) = 4] 611.09 - 5 ...
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