Análisis de correspondencias de una tabla Disyunta Completa

Consider the following principal-agent relationship: A principal needs to hire an agent to realize a project. The agent is one of two types, trustworthy (T) or untrustwor- thy (U); the proportion of trustworthy types in the economy,π, is strictly between zero and one. The trustworthy type of agent is intrinsically interested in the success of the project,9 _{whereas the untrustworthy type does not care about the project per se.}

The principal cannot observe the agent’s type. She receives, however, a binary, private signals ∈ {+,−}about the type of the agent. In case the agent is trustworthy, the principal receives a positive signal with the exogenously given probability σT. In case the agent is untrustworthy, the principal receives a positive signal with the exogenously given probability σU, with σU < σT. These two moves by “nature” are illustrated in figure 3.1

By Bayes’ rule a principal with a positive signal believes that she faces the trustwor- thy type with probability π+ = _π+σ_Uπ

σ_T (1−π)

> π and a principal with a negative signal believes that she interacts with a trustworthy type with probabilityπ−= π

π+1−σU

1−σ_T (1−π)

<

9_{Instead of being intrinsically interested in the project the agent may also have preferences for}

Figure 3.1: Nature’s moves u u ©©©© ©©©© © u H H H H H H H H H u ¡¡ ¡¡ ¡ u @ @ @ @ @ u ¡¡ ¡¡ ¡ u @ @ @ @ @ Nature π (1−π) Nature σT (1−σT) σU (1−σU)

State of the world: (T,+) (T,−) (U,+) (U,−) with 0< σU < σT <1.

π. Notice that π+ > π−, i.e. a principal with a positive signal has a stronger belief in the agent’s trustworthiness. In other words, the principal with a positive signal trusts the agent more strongly than the principal with a negative signal.

The project consists of two parts, the contractible part 1 and the non-contractible part 2. The project’s success in the contractible part 1, B1 ∈ {0, B1} depends only

on an unobservable efforte1 ∈ {0, e1} by the agent. High efforte1 benefits the project

deterministically by B1 ≡ B1(e1) > e1. Low effort e1 = 0 leads to B1 = 0. A

contract can condition on the outcome B1(e1) which is realized at the very end of the

relationship, i.e. after both players have chosen their actions in the second part of the relationship. Although efforte1 is not directly observable, it can be inferred from the

realized value of B1.10 A sufficiently harsh punishment in case of B1 = 0 therefore

implements a high effort level e1 =e1 in this contractible part 1.

The success of the project in part 2 depends firstly, on an unobservable effort decisione2 ∈R+0 of the agent, secondly, on a decision of the principal whether to rely

on the agent or whether to play safe, and finally, on an unobservable choice by the agent, whether to workhonestly ordishonestly.

Consider again our introductory example of the scientist and the research assistant. Part 2 of the relationship then corresponds to a jointly beneficial research project.

10_{By assuming thate}

1is only ex post (indirectly) observable we avoid complication ofe1signalling

Firstly, the assistant has to decide how much (unobservable) effort he wants to exert in studying the literature of the research topic. Then, the scientist has to decide whether to share an interesting idea with the assistant and to start a mutually beneficial joint research project. Joint work can be mutually beneficial. Yet, if the the assistant does not work reliably this could ruin the scientist’s reputation. If the scientist prefers to play it safe she can instead choose a projet in which the assistant’s trustworthiness is of minor importance.

More generally, in case the principal chooses to rely on the agent, the project’s success is very sensitive to the agent’s honesty. If the agent workshonestly the project benefits by Bh(e2), where Bh(·) is a twice differentiable function of the agents ef- fort e2 ∈ R+0, with Bh0 > 0, Bh00 <0, Bh(0) ≥ Bh > 0, lime2→∞Bh(e2) ≤ Bh ∈ R and

lime2→0B

0

h(e2) = ∞, where Bh > Bh > 0 are two, exogenously given boundaries. If the agent worksdishonestly he gains a private benefit ofMd>0, but causes a damage to the project of Bd > Md. In case the principal chooses to play safe, the agent’s decision is only of minor importance: all payoffs are multiplied by a small constant ², with 0< ²¿1. The interaction is illustrated in figure 3.2.

Figure 3.2: Reliance Game

u u ©©©© ©©©© © u H H H H H H H H H u ¡¡ ¡¡ ¡ u @ @ @ @ @ u ¡¡ ¡¡ ¡ u @ @ @ @ @ Principal rely safe Agent

honestly dishonestly honestly dishonestly

Payoffs: Project: Agent privately: B0h(e2) −MBdd ²Bh(e2) 0 −²M²Bdd with Bd> Md >0, Bh > Bh(·)> Bh >0 and 1> ² >0.

Let B2(e2,·) denote the project’s success resulting from this interaction in part 2.

Due to this second non-contractible part the principal has an interest to signal her trust in the agent.

The total successB of the project is the sum of the successes, B1(e1) and B2(e2,·),

in both parts of the relationship, i.e.

B(e1, e2,·) ≡ B1(e1) +B2(e2,·). (3.1)

Timing of events

The timing of events is illustrated in Figure 3.3.

Figure 3.3: Timeline 0 1 2 3 H H_©t © Principal receives private signal about the agents type.

Contracting Stage

Agent’s unobserv- able effort choices e1 and e2.

Reliance Game

After nature has randomly chosen the agent’s type and the principal’s signal, princi- pal and agent sign a contract. Then, the agent chooses his effort levelse1 ande2. Both

effort choices are unobservable. Finally, principal and agent interact in the reliance game.

Contracting Stage

At the contracting stage, the principal proposes a contract. This contract can, by assumption, only be conditional on B1(e1), the project’s success in the first part. The

most relevant feature of this contract is, whether the contract enforces high effort e1

by a sufficiently harsh punishment in case of B2(e2) = 0. In general, however, the

principal can design a sophisticated wage scheme and the agent’s decision whether to accept or reject the proposal may potentially reveal information about his type.

Here, we want to demonstrate our main point as concise as possible. For the moment, we thus restrict the set of contractual choices of the principal to a binary choiceC ∈ {contract (c),no contract (n)}. In appendix 3.5.2, we demonstrate that our main argument remains valid if we allow for more general contracts and if we take care

of the agent’s participation constraint.

The contract prescribes high effort e1 = e1 in the contractible part 1. A court

enforces the contract by the threat of a sufficiently harsh punishment in case ofB1(e1) =

0. In case of writing no-contract the principal refrains from such an enforcement. A possible interpretation for this simple setting is that principal and agent are working together already, and that there exists a binding agreement fixing the wage. In particular, let this existing agreement give the principal the discretion to enforce high effort of the agent in the contractible part 1 of the relationship through a contract

or to abstain from doing so. Notice that in this simple setting, the agent takes no observable action. After the first exogenous signal s, the principal’s belief (i.e. her trust) about the agent’s type remains fixed at π±.

Preferences

For simplicity, let the principal and the agent be risk neutral. The untrustworthy type of agent maximizes his private, monetary and non-monetary, payoffsM minus his total effort costs e≡e1+e2. He does not care about the project.

The utility-function of theuntrustworthy-agent is given by

UU(M, e, B(·)) = M−e. (3.2)

Thetrustworthytype of agent is intrinsically interested in the success of the projectB(·). We allow for the possibility that the trustworthy agent puts a lower weight, κ≤1, on the project’s success than the principal (in monetary units). The utility function of a

trustworthy agent is therefore

UT(M, e, B(·)) = M −e+κB(·). (3.3)

We need, however, that the trustworthy agent sufficiently cares about the project to deliberately exert high effort e1 in part 1 and to work honestly in part 2. This is

Assumption 8 1≥κ≥max ½ e1 B1 ,Md Bd ¾ . (3.4)

The principal’s utility is given by

V(WP, B(·)) =B(·)−WP, (3.5)

whereWP is the wage payed by the principal. Wage payments are relevant only for the extended version in appendix 3.5.2. Here, we normalizeWP = 0. Hence, the principal maximizes simply the expected total success of the project B(·) .