Econ 452, RA agent
Principal - Agent (PA) II: risk averse manager
Spse manager is RA in $$:
U ( x, y) = B( y) + x = B( y) + Ω − e
Simplify: effort, e ∈{eL , eH }, eH > eL
Profit: random: high or low
- probability depends on e
- Prob { R H | eH } = 3/4
- Prob { R H | eL} = 1/4
Compensation: y = θ R + F
Owner receives…?
ASSUME: in first best world, high effort is optimal
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Econ 452, RA agent
Constraints ensuring high effort?
a) participation / individual rationality constraints
0.75B(θ R H + F ) + 0.25B(θ R L + F ) + Ω − eH ≥ u0
b) incentive compatibility constraint
- better off (expected) if provide high effort than low
0.75B(θ R H + F ) + 0.25B(θ R L + F ) + Ω − eH
≥ B(θ R H + F )/3 + 2B(θ R L + F )/3 + Ω − eL
Owner chooses parameters to max E{(1−θ ) R − F}
Properties of solution?
1. participation constraint binds (why?)
- focus on {θ , F} combos yielding EU manager = u 0
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Econ 452, RA agent
Consider iso-EU curves, for manager (agent), for
EU manager = u 0 , in state space ( here, ( y L , y H )
space)
- slope of curves?
- depends on level of effort
- if high effort:
EU manager = 0.75B( y H ) + 0.25B( y L ) + Ω − eH = u 0
then dEU = 0.75B '( y H )dy H + 0.25B '( y L )dy L = 0
and
dy H = − 0.25B '( y L ) = − B '( y L )
dy L
0.75B '( y H )
3B '( y H )
Thus, along certainty ray, slope = -1/3
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Econ 452, RA agent
If low effort?
EU manager = B( y H )/3 + 2B( y L )/3 + Ω − eL = u0
dEU = ( B '( y H )/3)dy + (2B '( y L )/3)dy L = 0
H
H
L
and dy L = − 2B '( yH )
dy
B '( y )
Hence, along certainty line, slope = -2 if low effort
So iso-EU for low effort everywhere steeper than that
for same value (of utils), for high effort. Intuition?
Characterize contracts?
- on certainty line, θ = 0, so y = F
- above certainty line, θ > 0
- below certainty line, θ < 0
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Econ 452, RA agent
Contracts satisfying constraints?
1. Consider point A: on EU = u0 for high effort;
income not state dependent.
This contract is not incentive compatible.
Why? Yields higher utility than D ( on low effort
iso-EU for reservation utility)
2. Which contracts satisfy both constraints?
3.
Principal's preferred contract in feasible
set?
-
P is rn - cares only about expected profits.
-
Eprofits = ER(e,ξ ) − Ey
-
for fixed e and, hence, fixed ER( ),
principal indifferent between all contracts
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Econ 452, RA agent
yielding same expected income (Ey) to A
So: P's iso-EU curves in ( y L , y H ) space?
if high effort, Ey = 0.74 y H + 0.25 y L
-
slope of P's IC?
-
dy H = − 1
3
dy L
for e = eH
- similarly, if low effort, Ey = ( y H + 2 y L )/3
-
H
dy
so
= −2
dy L
for e = eL
Both cases: P's iso-EU curves are
a) linear
b) tangent to A's iso-EU along certainty
line (for relevant effort level)
c) higher EU (for P) as move closer to
origin
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Econ 452, RA agent
So: optimal contract when P is rn, A is ra?
- A gets EU = reservation utility (1st best)
- A just indifferent between high and low effort
- A must bear some risk, to induce high effort
- P bears some risk, and has EU lower than
would obtain in first best
Cost of private information?
- loss of social surplus (in utility terms):
- at optimal contract, P values additional unit of
Ey more than A does
- b/c A is risk averse, must receive more than P
requires to accept additional risk
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