Optimal first-period effort

If a principal anticipates that he will reemploy an agent after the first period with positive probability, this is clearly going to affect the contract that the principal offers in period one: The amount of effort that a principal implements in period one affects the agent’s wealth and his perceived ability in the next period. However, we can see that the structure of the first-period contract remains unchanged. Wage payments are pinned down by incentive compatibility and the participation constraint. As in the static setting, the participation constraint will always be binding: Otherwise it would be possible to reduce payments in period one and thus the agent’s future wealth in both states of the world, which is strictly beneficial. Hence, the introduction of a second period only affects the effort that a principal optimally implements in the first period.

Proposition 4 All policies under which a principal rehires an agent with positive prob- ability may lead to optimal first-period effort levels below or above the optimal one-period effort level. When policy N C is employed, effort in the first period equals the optimal one-period effort level.

Proof: Let us denote the principal’s expected periodtprofit by Πtand his overall surplus

by Π = Π1+ Π2. In order to gain an insight into the different determinants of period-

one effort we will start off by looking at a case where the principal reemploys the agent irrespective of his first-period success (AC). In this case, the return to setting slightly higher incentives in the first period is given by

dΠ de1 = period-one effect z }| { dΠ1 de1 +

direct period-two effect

z }| { Π∗₂ WH, θH ! −Π∗₂ WL, θL ! + (θ+e1) dΠ∗₂ dθH | {z } >0 dθH de1 | {z } <0 + dΠ ∗ 2 dWH | {z } <0 dWH de1 | {z } >0 + ! | {z }

indirect period-two effect after high output

+ (1−θ−e1) dΠ∗₂ dθL | {z } >0 dθL de1 |{z} <0 + dΠ ∗ 2 dWL | {z } <0 dWL de1 | {z } <0 ! | {z }

indirect period-two effect after low output

.

As in the static setting, an increase ine1has a direct effect on period-one profits by making

a success of the project more likely while increasing the agent’s expected compensation. Additionally, an increase ine1 affects the profits a principal can expect to make in the next

period via three distinct channels: First of all, it becomes more likely that the principal is faced with a successful agent in period two since a high period-one outcome is more likely. Since the principal does generally not make the same amount of profits with each type of agent, this is going to affect his expected profits. We will refer to this as the “direct” period-two effect. Secondly, there are indirect effects on period-two profits: A change in period-one incentives affects the profits a principal can expect to earn with either type of agent. If incentives are large, a positive outcome becomes less informative about the agent’s ability and a negative outcome becomes more informative. Successes will partly be attributed to higher effort, while failure despite increased effort is an even worse signal on ability. Hence, the expected ability of both types of agents decreases in e1.10 Moreover, an increase in period-one incentives increases the agent’s wealth in case of

success and it reduces his wealth after low outcomes. While the first effect reduces period- two profits, the second effect has a positive impact on the principal’s expected surplus. Whether optimal effort increases or decreases in comparison to one-period optimal effort is ambiguous, as the sign of the direct period-two effect may vary depending on (initial) ability and wealth. Additional ambiguity is introduced by the two indirect effects that may take either sign on aggregate.

Similar reasoning yields ambiguous effects under policiesHC orLC. Under those regimes

10_{Again, note that albeit both posteriors decrease with increased effort the agent’s expected ability}

the direct period-two effect must be positive for the former policy and negative for the latter as otherwise the principal could increase expected profits by not rehiring any agent. A change in e1 does not have any indirect effect in case the principal does not rehire his

old manager. For LC this is the case if the project was successful and for HC this is the case after a bad period-one outcome. Overall, the indirect effect will be negative for successful agents under HC and ambiguous for unsuccessful ones underLC, such that in each case the sum of the direct and indirect period-two effects may take either sign. Again, the optimal first-period effort might increase or decrease relative to the static problem in both cases.

Finally, in case the principal finds it optimal never to extend the employment contract of an agent, e1 is trivially the same as in the static problem: If the principal never rehires

the agent, changes in the agent’s period-two wealth or presumed ability do not affect the principal’s earnings. Also, second-period profits are independent of the outcome in period one, so there is no direct effect of first-period effort on second-period surplus. 2

Whether the optimal first-period effort level increases or decreases relative to the one- period problem will depend on the actual parameters and the shapes of the utility and the cost function. In the next section we will abstract from such issues by only allowing for binary effort levels and by assuming that the principal always wants to implement effort in the first period. This allows us to characterize the optimal employment policies more closely while preserving the key trade-offs of the more general model.