# 1 . 2 Uncertainty Dixit: Optimization in Economic Theory (Chapter 9)

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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

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