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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
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