ECO 340: Micro Theory Optimal Contracts Sami Dakhlia U. of Southern Mississippi

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 share- cropping Landlordhighlowmedium Farmerlowhighmedium

Sharecropping vs. Lease 2 states of nature: good weather w/ prob p and bad weather w/ prob (1-p). tenant system:EY t = p(Q H -  )+(1-p)(Q L -  ) sharecrop syst:EY s = p  Q H +(1-p)  Q L  * s.t. EY t =Ey s, i.e., where a risk-neutral landlord is indifferent between both contracts. But since Y L s > Y L t if Q=Q L and Y H s < Y H t if Q=Q H 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, E L and E H. As before, if effort is high, prob(Q=Q H if E= E H )=p. high effortlow effort good weatherpq bad weather1-p1-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(w H ) +(1-p) U(w L ) - E ≥ U(w R ) i.e., U H ≥ (U R +E)/p - (1-p)/p U L 2. the incentive constraint: p U(w H ) +(1-p) U(w L ) - E ≥ q U(w H ) +(1-q) U(w L ) i.e., U H ≥ E/(p-q) + U L UHUH ULUL

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): proba=0a=5 S=$ S=$ S=$

Principal/Agent Problem Principal is risk neutral: B(a)=  p a (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= ( S-w(S), U(w(S),a) ) expected utilities: ( B(a)-  p a (a,S)w(S),  p a (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: w 0 if S=$0 w 1 if S=$100 w 2 if S=$400

Principal/Agent Problem So A’s utility is –U = 9 if he refuses contract; –U = 0.6  (w 0 )  (w 1 )  (w 2 ) - 0 if E=0; –U = 0.1  (w 0 )  (w 1 )  (w 2 ) - 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 w w w 2 s.t. 0.1  (w 0 )  (w 1 )  (w 2 ) - 5  9 and 0.1  (w 0 )  (w 1 )  (w 2 ) - 5  0.6  (w 0 )  (w 1 )  (w 2 ) - 0

Principal/Agent Problem Solution –w 0 = $29.46 –w 1 = $ –w 2 = $ Expected wage bill 0.1 w w w 2 = $ Expected profit = $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 = $184. Contract: A is free to choose effort, but must pay P a fixed rent of $184; all risk is borne by A.