A risk neutral principal hires a risk neutral agent to work

A risk neutral principal hires a risk neutral agent to work on a project. The project can either yield high output h or low output l. The probability of h depends on the agent’s unobservable effort e. With probability 1 e the output is low. The agent is protected by limited liability, so his compensation can never go below zero. Let t(h) and t(l) be the agent’s compensation as a function of the output. The agent’s outside option is zero and the cost of effort function is c(e) = e2/2. The principal offers to the agent a contract that maximizes the principal’s expected profit subject to the incentive compatibility, limited liability and participation constraints.

Find the optimal contract

Solution

We consider contracting at period zero with full commitment power from the principal. To deal with the agent’s hidden information at the time of contracting, the principal’s problem is, without loss of generality, to offer the agent a menu of dynamic contracts from which the agent chooses one. A dynamic contract specifies a sequence of transfers as a function of the publicly observable history, which is simply whether or not the project has been successful to date. To isolate the effects of adverse selection, we do not impose any limited liability constraints until Subsection 7.2. We assume that once the agent has accepted a contract, he is free to work or shirk in any period up until some termination date that is specified by the contract.14 Throughout, we follow the convention that transfers are from the principal to the agent; negative values represent payments in the other direction. Formally, a contract is given by C = (T, W0, b,l), where T N {0, 1, . . .} is the termination date of the contract, W0 R is an up-front transfer (or wage) at period zero, b = (b1, . . . , bT ) specifies a transfer bt R made at period t conditional on the project being successful in period t, and analogously l = (l1, . . . , lT ) specifies a transfer lt R made at period t conditional on the project not being successful in period t (nor in any prior period).15,16 We refer to any bt as a bonus and any lt as a penalty. Note that bt is not constrained to be positive nor must lt be negative; however, these cases will be focal and hence our choice of terminology. Without loss of generality, we assume that if T > 0 then T = max{t : either bt 6= 0 or lt 6= 0}. The agent’s actions are denoted by a = (a1, . . . , aT ), where at = 1 if the agent works in period t and at = 0 if the agent shirks.

No adverse selection or no moral hazard Our model has two sources of asymmetric information: adverse selection and moral hazard. To see that their interaction is essential, it is useful to understand what would happen in the absence of either one. Consider first the case without adverse selection, i.e. assume the agent’s ability is observable but there is moral hazard. The principal can then use a constant-bonus contract to effectively sell the project to the agent at a price that extracts all the (ex-ante) surplus. Specifically, suppose the principal offers the agent

k up to t Assumption 1. Experimentation is e!cient: for 2 {L, H}, !0\" > c. Note both tH > tL and tH < tL are robust possibilities ! Productivity vs. learning eects: • For given belief on good state, marginal benefit of eort higher for H • But at any point in time, given no success, belief lower for H Model – Environment (2) In each period t 2 {1, 2, . . .}, agent covertly chooses to work or shirk • Exerting eort in any period costs the agent c > 0 If agent works and state is good, project succeeds with probability ! • 1 > !H > !L > 0 If agent shirks or state is bad, success cannot obtain Project success yields principal payo normalized to 1 • No further eort once success is obtained Project success is publicly observable •

Results also hold if privately observed by agent but verifiable disclosure Figure 1 – The first-best stopping time. of type a constant-bonus contract C = (t , W 0 , 1), where W 0 is chosen so that conditional on the agent exerting effort in each period up to the first-best termination date (as long as success has not obtained), the agent’s participation constraint at time zero binds: U 0 C , 1 = t X t=1 t 0 1 t1 c (1 0)c + W 0 = 0,

Optimal Contracts when t H > tL We characterize optimal contracts by first studying the case in which the first-best stopping times are ordered t H > tL, i.e. when the speed-of-learning effect that pushes the first-best stopping time down for a higher-ability agent does not dominate the productivity effect that pushes in the other direction. Any of the following conditions on the primitives is sufficient for t H > tL, given a set of other parameters: (i) 0 is small enough, (ii) L and H are small enough, or (iii) c is large enough. We maintain the assumption that t H > tL implicitly throughout this section.

Assume t H > tL. There is an optimal menu in which the principal separates the two types using penalty contracts. In particular, the optimum can be implemented using a onetime-penalty contract for type H, CH = (t H, WH 0 , lH tH ) with l H tH < 0 < WH 0 , and a penalty contract for type L, CL = (t L , WL 0 ,l L), such that: 1. For all t {1, . . . ,t L }, l L t = (1 ) c L t L if t < t L , c L t L L if t = t L ; (6) 2. WL 0 > 0 is such that the participation constraint, (IRL), binds; 3. Type H gets an information rent: U H 0 (CH, H(CH)) > 0; 4. 1 H(CH); 1 L(CL); and 1 = H(CL).