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: Principalagent 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, 0E1 Von NeumannMorgenstern 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 riskaverse 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 riskaverse 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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 Fall '08
 Kelly,J
 Game Theory, Utility, subgame perfect Nash, perfect Nash equilibrium, T H E E C O N O M IC S O F A S Y M M E T R IC

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