El interés informativo

The following is the main proposition of the chapter, which relates the reward to initial self-confidence and motivation in period 2⁶:

Proposition 14 Under assumptions (3)-(6), there always exists is a unique con-tinuation equilibrium depending on the initial self-confidence. In particular, there exists a threshold level ρ^∗ such that for values ρ ∈ [˜ρS,˜ρ_F):

(i) for ρ < ρ^∗, the unique equilibrium is a semi-separating equilibrium in which the principal always offers a bonus bS = ˆθ_F(1− G^F(˜σ_S))W after success ( xS = 1), and randomizes between bS and bF = 0 after a failure with probabilities xF = _A(ρ)^l(˜σ) and 1 − x^F respectively. The threshold ˜σ_S is positive and determined by l(˜σ_S) = ^ˆθ_ˆθ^F

S, and ˜σ_F = 1.

(ii) for ρ ≥ ρ^∗, the unique equilibrium is a pooling equilibrium where no bonus is ever offered. The threshold ˜σ is positive and determined by l(˜σ) = A(ρ).

The threshold level ρ^∗ is determined by the ρ for which⁷

ˆθ_F(ρ^∗)/ˆθ_S(ρ^∗) = A(ρ^∗). (6.22)

6The proposition states the equilibrium conditions for values of ρ such that ρ ∈ [˜ρ_S,˜ρ_F).Recall that it was already established that for values of ρ /∈ [˜ρ_S,˜ρ_F)no bonus is ever offered. For ρ < ˜ρ_S,σ˜_S= ˜σ_F = 1(the agent never works) and for ρ ≥ ˜ρ_F,˜σ_S= ˜σ_F = 0(the agent always works).

7The existence of a point ρ^∗ is obvious: A(ρ) decreases from infinity to 0 on the interval [˜ρ_S,˜ρ_F]and r(ρ) =^ˆθ_ˆ^F(ρ)

θ_S(ρ) is positive and bounded away from 0 on this interval. Implicitly we assume uniqueness as well. For this it is sufficient (but not necessary) to require that r(ρ) be non-decreasing on the relevant interval. Multiple ρ^∗s would slightly modify the proposition in a straightforward manner.

That the unique equilibrium for sufficiently high initial self-confidence is a pooling equilibrium with no bonus is intuitive: if self-confidence is high, the agent is likely to work in the second period, and it becomes too costly for the principal to signal a success. The main point of the proposition is however that there is a region where the principal does have an incentive to give a bonus, and that this bonus increases self-confidence. In this region, the agent is relatively unlikely to make efforts in the second period. In this case, the principal has an incentive to make a costly signal to the agent to make clear to him that a success has occurred.

It is also possible to show that for ρ < ρ^∗ the probability of a reward increases in initial self-confidence. Since ˆθ_F/ˆθ_Sis increasing in ρ, so must l(˜σ_S).It is then easily seen that x_F must be increasing, since A(ρ) is decreasing in ρ. The probability that an agent works, on the other hand, is decreasing, as ˜σ_S increases in ρ. The change in the size of the bonus for a higher initial self-confidence is ambiguous.

There are two opposing effects: first ˆθ_F increases, since a higher probability of a high type increases the success of the agent in the second period. Secondly, 1− GF(˜σ_S)decreases since ˜σ_S increases. The total effect depends on the relative sizes of these two opposing effects.

The size of the bonus in the region where self-confidence is relatively low, that is ρ < ρ^∗, is proportional to the payoff for the principal in case of success, W.

This means that the scope of applications is not limited to situations where the stakes are high for the principal. For example, it would be enough if the principal derives a small benefit from observing a successful performance of the agent, say out of altruistic feelings. For smaller stakes, the corresponding equilibrium bonus will be lower.

6.5 Discussion

When contracts are absent in a relationship, one easily ends up with the argument that no bonus will ever be given, and neither that efforts will be made. The cause is the strong backward induction argument: the agent knows that the principal has no incentive to give a reward in the last period and so he makes no effort, after which the principal realizes that rewarding in the before-last period makes no sense, and so the agent will not work in that period either, and so on until the very first period. This chapter sheds some light on why rewards and efforts may be observed after all.

There is no shortage of empirical and experimental evidence that shows the existence of rewards which are not conditioned on performance, and also that there is a positive relationship between rewards and efforts. Akerlof [1982], for in-stance, has noted that labour markets can often be characterized as gift exchange relationships. The employers give wages above the minimum wage, and work-ers make more efforts than is required. Laboratory experiments show the same positive relationship between wages and efforts, even in the absence of explicit performance incentives (Fehr and Gächter [2000]). Deci and Ryan [1999] survey the psychological literature on rewards and intrinsic motivation. They present some studies which find a positive effect, although not all studies which used unexpected rewards find a positive effect, and on average they find no significant relation⁸.

The positive relationship between rewards and efforts is also called positive reciprocity. Chapters 2 and 3 provide a more detailed exposition and extend the phenomenon to other environments. These chapters give other explanations of this relationship. For example, it is advanced that people are reciprocal by virtue of their fair nature: they are driven by the moral obligation to reward generous behavior by generous behavior (see e.g. Falk et al. [1999] and section 2.3.3). Another possibility is that people care about social approval and that generous behavior elicits generous behavior (see chapter 3). The current chapter adds another explanation to the existing literature, by focusing on the role of self-confidence.

Obviously, the proposed mechanism can only be valid as long as the main assumptions are satisfied. An important assumption is that the principal has more information about the expected payoffs than the agent. This makes the theory more applicable to situations where agents are in their learning phase: at school or at new jobs. A second important assumption is the sorting condition that is implicit in the model. The principal must obtain a higher marginal benefit from rewarding an agent after a success than after a failure. Otherwise, the principal would be tempted to reward the agent after a failure as well, disturbing the proposed equilibrium.

8However, in these studies it is not clear to the participants what the benefits of the experimentator are.

The setup of the experiments do therefore not completely fit the current model.

The model is also extendable to other situations with asymmetric information.

For example, it extends to situations where the agent is unsure about his own payoff rather than his ability. Another possibility is that the agent cares about the principal’s payoff (e.g. through altruism), but is unaware of how much utility the principal derives from his effort.