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2

Uncertainty

Dixit: Optimization in Economic Theory(Chapter 9)

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• 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

m

1 2 m 1 m i ii=1

F(Y ,Y , ...Y , p , ...p ) = p U Y

Expected utility,

U - von Neumann Morgenstern utility function

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Risk Aversion

• Y1, Y2 , p1, p2

• Expectation of Y: p1 Y1 + p2 Y2

1 2 1 2U pY + 1 - p Y > pU Y + 1 - p U Y

Y1 Y2

U Y

U'' Y < 0

m

m

i i i ii=1 i=1

U p Y > p U Y

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Insurance

• Y1 < Y2

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

• $x → $bx

• Y1 – x + bx, Y2 – x

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max 1 2 pU Y - x + bx + 1 - p U Y - xx

1 2p b - 1 U' Y - x + bx 1 - p U' Y - x = 0

But: 1 = pb (Competition in the insurance industry)

pb - p = 1 - p

1 2U' Y - x + bx = U' Y - x

1 2Y - x + bx = Y - xU'' < 0 →→ 2 1bx = Y -YFull Insurance

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Action to reduce the risk (Care)

• Y1 < Y2

• Cost z determines p1 = p(z).

• p’(z) < 0.

1 2(z) = p z U Y - z + 1 - p z U Y - z

'

2 1

1 2

z (z) = -p' z U Y - z -U Y - z

- p z U' Y - z 1 - p z U' Y - z

+Marginal benefit

Marginal cost

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Care & Insurance

1 2(x,z) = p z U Y - z - x + bx + 1 - p z U Y - z - x

x (x, z) = 0

1 2p z U' Y - z - x + bx b - 1 1 - p z U' Y - z - x = 0

zero expected profit: bp z = 1

1 2U' Y - z - x + bx = U' Y - z - x

1 2Y - z - x + bx = Y - z - x 0= Y

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'

2 1

1 2

z (x,z) = -p' z U Y - z - x -U Y - z - x + bx

- p z U' Y - z - x + bx 1 - p z U' Y - z - x

'

0 0

0 0

z (x, z) = -p' z U - U

- p z U' 1 - p z

Y

Y U'

Y

Y 0= -U' Y < 0

2 1bx = Y -Y z = 0

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r a random variable

r r,r , density function f r

r

r E U Y = U Y r f r dr

r

r E Y = Y r f r dr

U'' < 0 U E Y > E U Y (Jensen's Inequal ity)

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Safe and Risky asset

0 0 W = W - x + x 1+ r = W + r 0 x 0,W

r

0rE U W = U W + xr f r dr x =

r

0r' x = rU' W + xr f r dr = 0

r

0r' = rU' W f r dr0

r

0 0r= U' W rf r dr = U' W E r

E r > 0 x > 0

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0if r > 0 then ' x > 0 x = W

r

0 0r' x = rU' W + xr f r dr = 0, x 0,W

OR: interior solution:

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Managerial Incentives

• Owner hires a Manager for a project

• Project (if it succeeds) yields V• Probability of success is p or q (p > q).

• The manager determines the probability

• Cost of the higher probability p is e.

• Manager’s salary is w.

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

His expected profit:

pV - (w + e) or qV - w

assume: pV - (w + e) > qV - w p - q V > e

and: pV > (w + e)

Then Owner can get:

pV - (w + e)

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The owner cannot observe the manager’s quality

If owner pays the manager ,

he cheats and the owner gets :

e

qV - (w + e)

If the owner pays the manager according to success or failure

Pays x if success, and y if failure

px + 1 - p y - e > qx + 1 - q y

p - q x - y > e

Incentive for manager

p - q x - y e

px + 1 - p y w + e Participation constraint

indifference

y + p x - y w + e

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p - q x - y e y + p x - y w + e

Owner’s expected payoff:

= pV - px + 1 - p y

= pV - y - p x - y

Make x, x-y small

ex - y =

p - q

ey + p w + e

p - q

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ey + p w + e

p - q

ex - y =

p - q

e= w -

qy

p - q

e 1 -+

qx = w

p - q

Owner’s expected payoff:

= pV - px + 1 - p y

= pV - w + e

same as First Best

ybut 0 is ??

0

w + e e<

p p - q

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p - q x - y e y + p x - y w + e

x

px

choo

w +

se y = 0

e

p - q e

x

w + ex

p

e

p - q

w + e e<

p p - q

ex =

p - q

and owner’s expected payoff:

ep= pV -

p - q

epw + e <

p - q

< pV - w + e????? 0 <

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pV - (w + e) > qV - w p - q V > e

owner’s expected payoff:

ep= pV -

p - q < pV - w + e????? 0 <

We assumed (high quality worker is better)

ep= pV -

p - q < pV - w + e 0 <

first best

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Cost-Plus Contracts

• Quantity produced q at cost c• Government pays R >qc

• A firm with costs c1 or c2 ( > c1 )

• Government knows prob. p1 p2

• Government chooses R1 R2 c1 c2

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1 1 1 2 1 2R - c q R - c q

2 2 2 1 2 1R - c q R - c q

B q Gov. benefits from quantity q.

1 1 1 2 2 2

expected net benefit :

p B q - R + p B q - R

2 1 1 2c - c q - q 0

+ +

1 1 1 2 2 2R c q R c q

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1 1 1 2 1 2R - c q R - c q2 2 2R - c q 0

1 1 1 2 1 2R - c q R - c q

2 2 2 1 2 1R - c q R - c q

1 1 1 2 2 2R c q R c q

2 2 2R - c q 0 ? ??, 1 2R + R +

2 2 2R = c q

1 1 1 2 2 2R c q R c q

1 1 1 2 1 2R c q + c - c q 1 1 1 2 1 2R c q + c - c q

max 1 1 1 2 2 2p B q - R + p B q - R

+

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1 1 1 2 2 2p B q - R + p B q - R

2 2 2R c q

1 1 1 2 1 2R c q + c - c q

1 1 1 1 2 1 2 2 2 2 2p B q - c q - c - c q + p B q - c q

1 1B' q = c

12 2 2 1

2

pB' q = c + c - c

p

End