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MATH III

MATH III. Lecture 13. v. Uncertainty. Dixit: Optimization in Economic Theory (Chapter 9). 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

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MATH III

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  1. MATH III Lecture 13 v

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

  3. 1,2,3,….,m States of the world • p1, p2,…..pm probabilities • Y1, Y2,…..,Ym income in state i • F(Y1, Y2,…..,Ym , p1, p2,…..pm) - objective function Expected utility, U - von Neumann Morgenstern utility function

  4. Risk Aversion • Y1, Y2 , p1, p2 • Expectation of Y: p1 Y1 + p2 Y2 Y2 Y1

  5. Insurance • Y1 < Y2 • Premium $1 buys $b compensation in the bad state. • $x → $bx • Y1 – x + bx, Y2 – x

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

  7. Action to reduce the risk (Care) • Y1 < Y2 • Cost z determines p1 = p(z). • p’(z) < 0. + Marginal benefit Marginal cost

  8. Care & Insurance zero expected profit:

  9. r a random variable

  10. Safe and Risky asset

  11. OR: interior solution:

  12. Managerial Incentives • Owner hires a Manager for a project • Project (if it succeeds) yieldsV • Probability of success is p or q(p > q). • Themanager determines the probability • Cost of the higher probability p is e. • Manager’s salary isw.

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

  14. The owner cannot observe the manager’s quality If the owner pays the manager according to success or failure Pays x if success, and yif failure Incentive for manager indifference Participation constraint

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

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

  17. and owner’s expected payoff:

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

  19. Cost-Plus Contracts • Quantity produced q at cost c • Government pays R >qc • A firm with costs c1or c2 ( > c1 ) • Government knows prob. p1p2 • Government chooses R1 R2 c1 c2

  20. + +

  21. +

  22. End

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