6.2.1 Preliminaries
This section describes the general setup of the game. Some of the basic ideas are closely related to the work by Bénabou and Tirole [2002a] and Souvorov [2003].
Where assumptions differ, this is made clear in the text.
In the game, there are two players: a principal and an agent. There is a finite number of periods, for simplicity set to two. The case with an infinite horizon has been studied briefly by Bénabou and Tirole [2002a] in a slightly different model.
The agent has to decide on his effort level. He chooses to make efforts or not:
e ∈ {0, 1}. If the agent undertakes the task, i.e. e = 1, he incurs a cost of c in terms of disutility. Depending both on effort, e, and ability, θ, the outcome of the effort can be either a success, S, or a failure, F . The probability of success is given by:
prob(S | e) = eθ. (6.1)
In other words, ability and effort are complements. No effort induces a failure with certainty. In case of success, it yields a payoff equal to V to the agent, and W to the principal. A failure yields a payoff equal to zero for both. Both parties are risk-neutral and the agent is protected by limited liability.
The principal has to select a reward policy. In each period, he can offer a bonus b∈ R+ to the agent.
6.2.2 The main assumptions
Most of the above description is relatively standard for a principal-agent game.
The model, however, departs from most conventional models in several respects.
Each of these are discussed in more detail.
Imperfect self-knowledge. Although it is not a usual assumption, the idea that people have only imperfect knowledge about their personal characteristics is
plau-sible (see for instance Bénabou and Tirole [2002b]). First because retrospective evaluations of past utilities are known not always to be reliable (Kahneman [1994]). Thus, based on retrospection, people make incorrect estimates about how they will feel about certain matters. Moreover, some situations are new to people. In this case, they do not have enough information about themselves to infer their ability. Someone who tries to quit smoking for the first time is unlikely to be able to guess how persistent he will be. This requires some learning, but learning opportunities are usually limited.
Imperfect self-knowledge in the current context means that people are not perfectly informed about their ability. They cannot foresee their ability to make a success out of it. The task they have to undertake is for example relatively new to them, or they have forgotten how well they did on this or a comparable task in the past. They do form an estimate about their ability. Based on this estimate, they form an estimate of their chances to succeed, which represents their self-confidence.
To make things concrete: suppose that the agent can be either one of two possible types, a high type with ability θH or a low type with ability θL < θH.
His prior on being a high type is given by ρ. His self-confidence is then given by:
ρθH + (1− ρ)θL. (6.2)
Clearly, self-confidence is increasing in ρ. In the remainder of the chapter the parameters θH and θL are kept fixed, and with slight abuse of terminology, self-confidence is identified with the parameter ρ.
Non-contractibility of the bonus. The principal has the possibility to give a reward b to the agent. However, a crucial assumption in the model will be that the outcome is not observable to the agent or an outside party. The outcome is therefore private information to the principal. It follows that a reward contingent on the outcome cannot be specified in a contract, because the agent or third party would not be able to verify the truthfulness of the principal’s claim. That is, the principal can always report a failure and no party can contest this claim.
The non-contractibility is one of the main departures from the model of Bén-abou and Tirole [2002a]. They have analyzed the case where a contract can be written that specifies the bonus in advance. Of course, they also have to assume that the output is verifiable to the agent. The case of noncontractibility is inter-esting because first, in reality there are many situations where the bonus is indeed
noncontractible, and second, evidence from experiments show that the relation between bonuses and motivation differs depending on whether or not a bonus is specified in advance.
Intrinsic motivation. Even though no contract that specifies a bonus can be written, it is still assumed that agents have a motivation to make efforts. Econo-mists usually takes rewards as the motivation to work. According to Frey [1997], many psychologists emphasize that the motivation to undertake a task can come from within the person. If they are motivated without apparent reward or en-vironmental control, they are said to be intrinsically motivated. In the words of Deci and Ryan [1985, 43]:
”Intrinsic motivation is the innate, natural propensity to engage one’s interests and exercise one’s capacities, and in so doing, to seek and conquer optimal challenges. Such motivation emerges spontaneously from internal tendencies and can motivate behavior even without the aid of extrinsic rewards or environmental controls”.
It is undisputed that people are intrinsically motivated to do certain things:
playing football, solving a puzzle, the list is endless. An assumption in this chapter is that people are indeed motivated for the task they have to undertake, even if they get no current rewards. This is not a completely innocent assumption. Even if people are intrinsically motivated to perform certain tasks, it does not follow that they are intrinsically motivated to do all possible thinkable tasks. However, the assumption is not crucial in the sense that the agent may also be motivated for expected rewards in the future, despite the absence of current rewards. The model allows for both interpretations.
The motivation of the agent is modeled as the value V in the model. To make things interesting, one additional assumption has to be made on V, namely that it cannot be directly observed. In other words, it is assumed that the agent is motivated to do a task for which the benefits come later in life. Thus, one can interpret V as the discounted value of payoffs later in life, be it extrinsic or intrinsic. The agent may be a pupil learning to play the piano. First, he needs to practice all kind of chords, a rather dull activity. The reward only comes when he is able to play a decent piece. The agent may also be a student studying for an exam, or writing an essay, not pleasant tasks for many people. His benefits
may be to get a job afterwards that he really likes. Or he may be a worker, who undertakes the task with the prospect of getting a promotion afterwards.
Asymmetric information. As explained, the agent is not sure about his ability to bring the task to a successful end. Moreover, the focus in on situations where even afterwards he does not get to know directly for sure whether it was a success or a failure. He only gets an imperfect signal about the outcome. On the other hand, the principal is able to observe the outcome. For instance, the pupil learning to play piano cannot really tell whether he is talented after a few sessions, but the principal can tell, having seen many pupils trying before this pupil. The same is true for the student, whose grade will only be imperfectly informative about his ability. This is certainly the case where the grade is dependent on the subjectivity of the teacher, as with an essay. For a worker, it may be the case that this is the first time he undertakes the task, or that his task is only a small part of a bigger whole he is part of, so that he is not able to judge the outcome based on his own information only.
Note that this assumption is contrary to most conventional principal-agent models, where the agent has more information rather than less. For example, in the classic job-market signalling model of Spence [1973] it is the agent who knows his ability, whereas the principal only knows the distribution of abilities among the population.
The private signal that the agent gets is given by σ ∈ [0, 1]. This signal has a conditional distribution G(σ | y) = Gy(σ) and density g(σ | y) = gy(σ), where y is the outcome of the task: y ∈ {S, F }. A higher σ is interpreted as good news in the sense that it is more likely that a success has occurred. To capture this idea, it is assumed that the likelihood function l(σ) with
l(σ)≡ gS(σ)
gF(σ), (6.3)
is an increasing function in σ. This is the monotone likelihood ratio property (MLRP).
The next section examines equilibrium behavior of the principal and agent. To focus on interesting cases, the following additional assumptions are made.
Assumption 3 Were the agent to know his type, then he would only undertake the task without a bonus if he is a high ability type: θLV < c < θHV.
As will be demonstrated shortly, no bonus is offered by the principal in the second period. If assumption 3 did not hold, then either the agent would never work in period 2, or he would always work, independent of his self-confidence. In both cases, there is no role for the principal to increase self-confidence. Thus, no bonus would be given in the first period either.
Furthermore, the following restriction is put on the likelihood ratio:
Assumption 4 The likelihood ratio l(σ) is continuous in σ and has full sup-port on [0, +∞). Furthermore, the monotone likelihood ratio property (MLRP) is satisfied: l(σ) is everywhere increasing in σ.
The full support assumption simplifies matters. It is also used by Bénabou and Tirole [2002a] and Souvorov [2003] in related settings. The MLRP is an essential assumption in many models with asymmetric information.
Assumption 5 In period 1, the agent undertakes the task: e = 1 in period 1.
This last assumption is made to focus on the interesting aspect of the model, which is the behavior of the agent in period 2. Although conditions can be derived under which e = 1 is an equilibrium strategy in period 1, not much insight is gained from doing that.
6.2.3 Timing and summary of the game
Each period is divided in two subperiods. In each first subperiod, the agent decides to make effort or not: e ∈ {0, 1}. Effort costs c in terms of disutility. At the end of the first subperiod, the principal observes the outcome (y ∈ {S, F }) and the agent receives a private signal σ about the outcome. A success occurs with probability eθ and gives a payoff V to the agent and W to the principal. In the second subperiod, the principal determines his reward policy b ∈ R+.
Note also the following: at the beginning of the game, both the principal and the agent have the same prior ρ that the agent is of the high type, θH.To simplify, both of them observe effort. The signal σ is private information to the agent, but the conditional distribution functions are common knowledge.