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# MATH III - PowerPoint PPT Presentation

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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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

• \$x → \$bx

• Y1 – x + bx, Y2 – x

But: 1 = pb

(Competition in the insurance industry)

Full Insurance

• Y1 < Y2

• Cost z determines p1 = p(z).

• p’(z) < 0.

+

Marginal benefit

Marginal cost

zero expected profit:

r a random variable

• 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

Make x, x-y small

same as First Best

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

+