Asymmetric Information and Optimal Contracts in Labor Markets - Prof. Charles H. Nelson, Study notes of Agricultural engineering

Download Asymmetric Information and Optimal Contracts in Labor Markets - Prof. Charles H. Nelson and more Study notes Agricultural engineering in PDF only on Docsity! ACE 501 Fall, 2008 ACE 501 Lecture Notes — 11/07/08 Carl H. Nelson 1 Asymmetric information — hidden information with productive action The screening model in the previous lecture has the unrealistic attribute that the task assigned to the worker does not have productivity for the firm. The only function served by the task is to differentiate worker types. A more realistic model is one where the task or resource allocation decision that is assigned, has value to the principal making the assignment. We will consider the MWG model of a firm hiring a manager, then we will consider a monopoly facing a market of buyers with unknown valuation. This order of material skips sections 14.A an 14.B of MWG, which presents the model of moral hazard. We will come back to this material. The model that we will consider now produces some general results. The moral hazard model, in contrast, does not produce general results. In the labor market model, a firm wants to hire a manager. The profit of the firm depends on the effort, e, exerted by the manager. So we assume the profit of the firm, π(·), is characterized by: π(0) = 0 π′(e) > 0 π′′(e) < 0 There is asymmetric information because the manager has an unknown cost of effort, which he does not learn until he takes the position. In the simplest model presented in section 14.C of MWG there are two possible values of a parameter, θ, that indexes the cost of effort (θH > θL). The cost of effort is a function of effort and θ with the following properties: ∂g(e, θ) ∂e > 0 ∂2g(e, θ) ∂e∂e > 0 ∂g(e, θ) ∂θ < 0 ∂2g(e, θ) ∂e∂θ < 0 As in the screening and signalling models, the last of these properties, geθ(e, θ) < 0, is important for the properties of the solution. Notice that the properties of 1 g(e, θ) mean that type θH has a lower cost and lower marginal cost of effort than type θL. Assume that it is common knowledge that Prob(θH) = λ. 1.1 Complete information Pareto optimality The payoff function of the firm is π(e) − w. The manager is assumed to be risk averse, so the payoff function of the manager is v(w − g(e, θ)), where v′ > 0 and v′′ < 0. As with the screening model, we will first consider resource allocation under complete information in order to find first-best Pareto optimal resource al- location. This means that the firm can offer the contracts (wH , eH), (wL, eL) and verify that only θH managers take the (wH , eH) contract and only θL managers take the (wL, eL). The firm sets these contracts ex ante before they know which type of manager they are hiring so they are specified to solve: max wH,eH wL,eL λ [π(eH)− wH ] + (1− λ) [π(eL)− wL] s.t. λ v (wH − g(eH , θH)) + (1− λ)v (wL − g(eL, θL)) ≥ ū where ū is the manager’s opportunity cost (utility in next best alternative). The contract needs to satisfy this constraint in order for the manager to voluntarily participate in the contract. Let γ be the LaGrange multiplier on the constraint. Then, assuming an interior solution, the four first order conditions are: −λ+ γλv′ (w∗H − g(e∗H , θH)) = 0 −(1− λ) + γ(1− λ)v′ (w∗L − g(e∗L, θL)) = 0 λπ′(e∗H)− γλv′ (w∗H − g(e∗H , θH)) ge(e∗H , θH) = 0 (1− λ)π′(e∗L)− γ(1− λ)v′ (w∗L − g(e∗L, θL)) ge(e∗L, θL) = 0 MWG provide Kuhn-Tucker conditions which allow for solutions where effort levels may be binding at 0. The first two first order conditions imply: v′ (w∗H − g(e∗H , θH)) = v′ (w∗L − g(e∗L, θL)) = 1 The first equality is the condition for Pareto optimal risk sharing. The marginal utility of the risk averse manager is the same in each state. The second equal- ity implies that the third and fourth first order conditions can be re-written as: π′(e∗H) = ge(e ∗ H , θH) and π ′(e∗L) = ge(e ∗ L, θL) These are the conditions for Pareto 2 this more formally. If we substitute our two binding constraints into the objective function we obtain: λ [ π(eH)− [( g(eH , θH) + v −1(ū) ) + (g(eL, θL)− g(eL, θH)) ]] +(1− λ) [ π(eL)− (g(eL, θL) + v−1(ū)) ] We can find the optimal contract by solving this unconstrained optimization prob- lem, ignoring the incentive compatibility constraint of the low type, and check that the solution satisfies the constraint. Taking first-order conditions with respect to eH and eL we obtain: π′(eH) =ge(eH , θH) π′(eL) =ge(eL, θL) + λ (1− λ) (ge(eL, θL)− ge(eL, θH)) ge(eL, θL) − ge(eL, θH) > 0 and π′′ < 0. Therefore the eL that solves these first order conditions is smaller than e∗L from first best Pareto optimality, π ′(eL) = ge(eL, θL). This solution is frequently referred to as no distortion at the top mean- ing that the resource allocation of the best type is first best Pareto optimal and the resource allocation of all worse types is distorted down from first best Pareto optimality. Now the reason for distorting eL to a level below first best Pareto optimality is explained in figure 14.C.8 in MWG. In frame (a) they show that the lost profit from sub-optimal eL is relatively small. Notice that eL is adjusted by moving down the indifference curve of type L that passes through (w∗L, e ∗ L). In frame(b) it is shown that the movement enables the firm to move the type H manager to a lower indifference curve than the one that passes through (w∗L, e ∗ L). As depicted this movement creates a relatively large increase in profit. The intuitive idea is that the firm must pay type H managers an informational rent to prevent them from taking the contract for type L managers. The larger the resource allocation required of the type L manager the larger the rent that must be paid to the type H manager. Thus the firm distorts the resource allocation of the type L manager to the point where the marginal cost of the distortion equals the the marginal benefit of the distortion. We will see this more formally in the next model we consider. The informational rent paid to the typeH manager means thatwH−g(eH , θH) > ū. Therefore the utility of the infinitely risk averse manager is not the same in both 5 states of nature, and the optimal contract under asymmetric information does not provide Pareto optimal risk sharing. In summary, the optimal contract under asymmetric information has several properties that turn out to be common to models with unknown types. The IR constraint for the worst type is binding. The IR constraints for types of above the worst type can be ignored because they are implied by IC constraints. The IC constraint of the best type is binding. The resource allocation of the best type is first-best Pareto optimal. Every type below the best type has their resource allocation distorted below first-best Pareto optimality. The resource allocation distortion is done to lower the informational rents paid to better types. The optimal distortion is characterized by equating the marginal cost of the distortion to the marginal value of the distortion. 6