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MATH III

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MATH III

Lecture 13

v

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

- Y1, Y2 , p1, p2
- Expectation of Y: p1 Y1 + p2 Y2

Y2

Y1

- 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

OR: interior solution:

- 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

- 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