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Optimization in Economic Theory: Uncertainty and Risk Management Strategies

This lecture delves into the principles of optimization under uncertainty within economic theory. It covers various states of the world, with associated probabilities and potential incomes in each scenario. The focus is on expected utility, the von Neumann-Morgenstern utility function, and the implications of risk aversion. Topics such as insurance premiums, competition in the insurance industry, risk-reduction actions, and managerial incentives are explored. By examining these concepts, we gain insight into decision-making processes in uncertain environments and the role of contracts in economic governance.

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Optimization in Economic Theory: Uncertainty and Risk Management Strategies

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Presentation Transcript

  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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