MATH III
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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

Lecture 13

v


Uncertainty

Dixit: Optimization in Economic Theory

(Chapter 9)


  • 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


Risk Aversion

  • Y1, Y2 , p1, p2

  • Expectation of Y: p1 Y1 + p2 Y2

Y2

Y1


Insurance

  • Y1 < Y2

  • Premium $1 buys $b compensation in the bad state.

  • $x → $bx

  • Y1 – x + bx, Y2 – x


But: 1 = pb

(Competition in the insurance industry)

Full Insurance


Action to reduce the risk (Care)

  • Y1 < Y2

  • Cost z determines p1 = p(z).

  • p’(z) < 0.

+

Marginal benefit

Marginal cost


Care & Insurance

zero expected profit:


r a random variable


Safe and Risky asset


OR: interior solution:


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.


First Best (the owner can observe the manager’s quality)

His expected profit:

assume:

and:

Then Owner can get:


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


Owner’s expected payoff:

Make x, x-y small


Owner’s expected payoff:

same as First Best


and owner’s expected payoff:


owner’s expected payoff:

We assumed (high quality worker is better)

first best


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


+

+


+


End


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