ECO 340: Micro Theory
Optimal Contracts
Sami Dakhlia
U. of Southern Mississippi
microprof@gmail.com
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 salary
2. =1, <0 : tenant farming
3. 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
sharecropping
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 = pQH +(1-p)QL
•
* s.t. EYt=Eys, i.e., where a risk-neutral landlord
is indifferent between both contracts.
•
But since
YLs > YLt if Q=QL
and
YHs < YHt 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
UH
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
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
S=$0
S=$100
S=$400
a=0
0.6
0.3
0.1
a=5
0.1
0.3
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)
N
P
A
( S-w(S) , U(w(S),a) )
A
N
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  9
and 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 profit
270 - 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.