1
2 Uncertainty Dixit: Optimization in Economic Theory (Chapter 9)
3 1,2,3,….,m States of the world p 1, p 2,…..p m probabilities Y 1, Y 2,…..,Y m income in state i F(Y 1, Y 2,…..,Y m, p 1, p 2,…..p m ) - objective function Expected utility, U - von Neumann Morgenstern utility function
4 Risk Aversion Y 1, Y 2, p 1, p 2 Expectation of Y: p 1 Y 1 + p 2 Y 2 Y1Y1 Y2Y2
5 Insurance Y 1 < Y 2 Premium $1 buys $b compensation in the bad state. $x → $bx Y 1 – x + bx, Y 2 – x
6 But: 1 = pb (Competition in the insurance industry) Full Insurance
7 Action to reduce the risk (Care) Y 1 < Y 2 Cost z determines p 1 = p(z). p’(z) < 0. + Marginal benefit Marginal cost
8 Care & Insurance zero expected profit:
9
10 r a random variable
11 Safe and Risky asset
12 OR: interior solution:
13 Managerial Incentives Owner hires a Manager for a project Project (if it succeeds) yields V Probability of success is p or q ( p > q ). The manager determines the probability Cost of the higher probability p is e. Manager’s salary is w.
14 First Best (the owner can observe the manager’s quality) His expected profit: assume: and: Then Owner can get:
15 The owner cannot observe the manager’s quality If the owner pays the manager according to success or failure Pays x if success, and y if failure Incentive for manager Participation constraint indifference
16 Owner’s expected payoff: Make x, x-y small
17 Owner’s expected payoff: same as First Best
18 and owner’s expected payoff:
19 owner’s expected payoff: We assumed (high quality worker is better) first best
20 Cost-Plus Contracts Quantity produced q at cost c Government pays R >qc A firm with costs c 1 o r c 2 ( > c 1 ) Government knows prob. p 1 p 2 Government chooses R 1 R 2 c 1 c 2
21 + +
22 +
23 End