6.4 The Basic Scheme when the Agent is Risk Averse

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1 100 OPTIMAL COMPENSATION SCHEMES such that Further, effort is chosen so that ˆβ = ˆα = 0 e = δ The level of utility in equilibrium is strictly larger than the outside otion (which was normalized to zero in this case). 6.4 The Basic Scheme when the Agent is Risk Averse The resence of risk aversion changes the comensation schemes. In the baseline contract, the worker (or the agent) bears all the uncertainty associated with the luck comonent of outut. The firm gets a fixed ayment while the worker is obliged to observe fluctuations in her income. When the realization of the luck comonent is high, the comensation increases, while it falls when the realization of the luck comonent is low. As long as the worker is risk neutral, she does not care about income variability. This is no longer true in the case of risk aversion. When the worker is risk averse (and the firm risk neutral), the otimal distribution of risk calls for the firm to take u some of the risk. Indeed, as we will see, the otimal comensation imlies a reduction in the bonus comonent and an increase in the fixed ayment. In other words, the rincial rovides some insurance to the agent. Providing insurance, however, has an imact on effort and efficiency, which we know to be maximum under a ure bonus scheme. In other words, the otimal contract under risk aversion is the one that solves this efficiency/insurance trade-off THE AGENT PREFERENCES General Agent Preferences. We now assume that the agent likes higher exected wage w e but he dislikes income variability Var(w). In this section we use a mean-variance utility function, whereby utility increases with the exected value w e and decreases with the variance of the comensation and with the effort U (w, e) = w e λvar(w) δ e

2 OPTIMAL COMPENSATION SCHEMES 101 The difference between risk aversion and risk neutrality on the agent s references is analysed next. A risk-neutral agent: A risk-neutral agent is somebody who does not care about income variability, as was analysed in the first art of the chater, so that for a risk-neutral guy λ = 0or U (w, e) = w e δ e = α + βe δ e Risk-averse guy with mean variance references: A risk-averse guy with mean variance references is somebody who does care about income variability, so that his utility function is U (w, e) = w e λvar(w) δ e The larger λ is, the more the worker is averse to income variability. The linear comensation that we are analysing (w = α +βx) has a variance that is simly given by Var(w) = β Var(x). Since the variance of the random variable x is constant and equal to v the variance of the comensation is Var(w) = β v so that the utility can be written as U (w, e) = α + βe λβ v δ e 6.4. THE PROBLEMS OF THE AGENT We now solve the two roblems of the agents under the case of risk aversion with mean variance references. We begin with the choice of effort under the general bonus contract: In the bonus contract w = α+βx so that w e = α+βe and Var(w) = β v and the utility level is Max e : U (w, e) = α + βe λβ v δ e which imlies that the otimal choice of effort is the condition (IC) above, or that e = β δ (IC: Incentive Comatibility Constraint)

3 10 OPTIMAL COMPENSATION SCHEMES The choice of the otimal effort does not deend on the resence of risk aversion. The second roblem of the worker is whether articiating in the contract is convenient, since the worker can enjoy an outside otion which yields exected utility equal to u. The utility of the agent is U (w, e) = α + βe λβ v δ e since e = β δ it follows that articiation is convenient if and only if U (w(e ), e )) u which imlies that α + β δ β δ λβ v u α + β δ λβ v u so that α + β [ 1 λvδ ] u δ ((P) Particiation Constraint-Risk Aversion) The resence of risk aversion changes the articiation constraint. The resence of risk aversion makes articiation in a bonus scheme a less attractive otion to the worker. A bonus scheme is associated with variable wages, since the luck comonent of the outut does affect the worker s utility. As one can see from the articiation constraint required, the larger the risk aversion comonent v, for given β and α, the less likely is articiation in the deal. We now move to the roblem of the rincial THE PROBLEM OF THE PRINCIPAL WITH A RISK-AVERSE AGENT: THE OPTIMAL BONUS SCHEME The rincial must now chooses β and α where we indicate variables with a symbol to indicate that we are analysing the case of risk aversion.

4 OPTIMAL COMPENSATION SCHEMES 103 Choice of α. Let s first consider the otimal choice of α given β. Forgiven choice of β the rincial will want to ay a salary so that the agent will choose to work. The agent chooses to work for this rincial as long as he is just as well off as at his next-base oortunity. Thus, the smallest base ay α that the rincial can offer, if she offers β, and still get the agent to work for her is α + β δ λ β v = u α = u β [ 1 λvδ δ Choice of β. Let s study the rofits of the rincial when she offers β and the agent is going to work by offering a level of α as indicated above. The exected rofits of the agents are E[ ] =E[x] w e s.t. e = β δ = e α βe and α = u β δ + λ β v Substituting the two constraints in the objective functions one has E[ ] = β δ u β δ λ β v ] The rincial will maximize rofit with resect to β so that the first-order condition solves δ β δ λ βv = 0 β = 1 + λδv which imlies that the otimal β is less than the rice (as long as λ>0, υ>0 and δ>0). For a risk averse guy, the otimal commission rate is never the franchising comensation scheme, i.e. 0 < β <

5 104 OPTIMAL COMPENSATION SCHEMES Summing u With a risk-averse agent the scheme works as follows β = 1 + λδv [ ] 1 λvδ α = u β δ [ ] 1 λvδ α = u (1 + λδv) δ b APPENDIX 6.1. WHAT IF THE PRINCIPAL AND THE RISK-AVERSE AGENT COULD CONTRACT ON EFFORT Let s start by defining the surlus from the deal as the sum of the utility of each arty, net of the resective outside otion. This is identical to the difference between the revenues from the deal and the workers cost of eliciting effort. The surlus is then S = ( 0) + (U u) = e w + w δe u = e δe u and let us assume that the two arties could contract directly on effort, rather than relying on the outut x. Let s see what the effort choice is that maximizes the surlus from the job. In other words, the otimal effort is the one that makes the marginal surlus zero (i.e. S e = 0). We call the otimal level of effort e0 and it is easy to see that δe o = 0 from which it follows that e o = δ The level of effort that maximizes the surlus is e 0 = /δ. So that the maximum value of the surlus is (evaluating S at e o ) S(e o ) = δ δ δ = δ and the job is efficient as long as S(e o )>0 δ > u