1. 1 

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

3. 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. 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. 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. 6 But: 1 = pb (Competition in the insurance industry) Full Insurance

7. 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. 8 Care & Insurance zero expected profit:

9. 9

10. 10 r a random variable

11. 11 Safe and Risky asset

12. 12 OR: interior solution:

13. 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. 14 First Best (the owner can observe the manager’s quality) His expected profit: assume: and: Then Owner can get:

15. 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. 16 Owner’s expected payoff: Make x, x-y small

17. 17 Owner’s expected payoff: same as First Best

18. 18 and owner’s expected payoff:

19. 19 owner’s expected payoff: We assumed (high quality worker is better) first best

20. 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. 21 + +

22. 22 +

23. 23 End