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Optimal Contracts
• Principal/Agent Game: Principal (boss) has objective (maximize profit) and must provide agent (worker) with proper incentives to work hard.
• We study two examples:– sharecropping vs. land lease– optimal commission for salespeople
Sharecropping vs. Lease
• Confine attention to linear contracts, where the farmer’s income is Y=Q+, where Q is output, is the share of crop, and is fixed income.
3 typical contracts:1. =0, >0 : fixed salary2. =1, <0 : tenant farming3. 0<<1, =0: sharecropping
Sharecropping vs. Lease
• How do these contracts spread risk between principal (owner) and agent (farmer)?
• How does this compare with incentives to work hard?
RISK fixed salary
tenant farming
share-cropping
Landlord high low medium
Farmer low high medium
Sharecropping vs. Lease
• 2 states of nature: good weather w/ prob p and bad weather w/ prob (1-p).
tenant system: EYt = p(QH-)+(1-p)(QL- )
sharecrop syst: EYs = pQH +(1-p)QL
* s.t. EYt=Eys, i.e., where a risk-neutral landlord is indifferent between both contracts.
• But since YLs > YL
t if Q=QL
and YHs < YH
t if Q=QH
a risk-averse farmer will prefer sharecropping.
Sharecropping vs. Lease
• Now let’s focus on the moral hazard problem: Farmer can put in two levels of effort, EL and EH.
• As before, if effort is high, prob(Q=QH if E= EH)=p.
high effort low effort
good weather p q
bad weather 1-p 1-q
(Of course, p>q.)Principal must now come up with a contract that provides incentive to work hard!
Sharecropping vs. Lease
This contract must satisfy
1. the participation constraint:
p U(wH) +(1-p) U(wL) - E ≥ U(wR)i.e., UH ≥ (UR+E)/p - (1-p)/p UL
2. the incentive constraint:
p U(wH) +(1-p) U(wL) - E ≥
q U(wH) +(1-q) U(wL)
i.e., UH ≥ E/(p-q) + UL
UH
UL
Principal/Agent Problem
• Suppose a salesperson’s (agent’s) utility is U(w,a) = (w) - a
• a A={0,5}; reservation utility u=9
• Finite set of outcomes (sales):
prob a=0 a=5
S=$0 0.6 0.1
S=$100 0.3 0.3
S=$400 0.1 0.6
Principal/Agent Problem
• Principal is risk neutral: B(a)=pa(S).S• Hence B(0)=$70 and B(5)=$270• Must design a contract, i.e., a function that
maps effort into wage (w:S)
P A A
N
N
reject
accept
(0,u)
a=5
a=0
0
0
100
100400
400( S-w(S) , U(w(S),a) )
expected utilities: ( B(a)-pa(a,S)w(S) , pa(S)U(w(S),a) )
Principal/Agent Problem
Quick computations:• To get A to work at low effort, P must offer wage
s.t. (w) - 0 9, i.e., w $81.But since low effort only generates expected revenue B(0) = $70, there will be no deal.
• To get A to work hard, P must offer wage s.t.(w) - 5 9, i.e., w $196. Harder work would generate expected revenue B(5) = $270, so deal is potentially possible.
Principal/Agent Problem
• We assume that trust will not work (so offering $196 without further stipulations will not garantee high effort.)
• Neither can contract be made contingent on effort, since it is not observable/enforceable.
• Therefore contract must be made contingent on sales result. This means that agent must share some risk: Pay:
• w0 if S=$0• w1 if S=$100• w2 if S=$400
Principal/Agent Problem• So A’s utility is
– U = 9 if he refuses contract;– U = 0.6 (w0) + 0.3 (w1) + 0.1 (w2) - 0 if E=0;– U = 0.1 (w0) + 0.3 (w1) + 0.6 (w2) - 5 if E=5.
• P wants to minimizes wages paid subject to– participation constraint (A agrees to be hired)– incentive constraint (A puts in high effort)
• Formally: MIN 0.1 w0 + 0.3 w1 + 0.6 w2 s.t. 0.1 (w0) + 0.3 (w1) + 0.6 (w2) - 5 9and 0.1 (w0) + 0.3 (w1) + 0.6 (w2) - 5 0.6 (w0) + 0.3 (w1) + 0.1 (w2) - 0
Principal/Agent Problem• Solution
– w0 = $29.46– w1 = $196.00– w2 = $238.04
• Expected wage bill 0.1 w0 + 0.3 w1 + 0.6 w2 = $204.56
• Expected profit270 - 204.56 = $65.44
Principal/Agent Problem Question: what would happen if agent was risk
neutral? For instance, what if his utility function was U(w,a) = w - a and his reservation utility equal to 81?
Answer: A will work hard if w - 5 ≥ 81, i.e., w ≥ $86.Profit to P is then 270 - 86 = $184.Contract: A is free to choose effort, but must pay P a
fixed rent of $184; all risk is borne by A.
