Motivating informed decisions

Abstract

This paper studies a principal-agent model where a risk-neutral principal delegates to a risk-neutral agent the decision of whether to pursue a risky project or a safe one. The return from the risky project is unknown and the agent can acquire costly unobservable information about it before taking the decision. The problem has features of moral hazard and hidden information since the acquisition of information and its content is unobservable to the principal. The optimal contract suggests that the principal should only reward the agent for outcomes that are significantly better than the safe return. It is also optimal to distort the project choice in favor of the risky one as a mechanism to induce the direct revelation of the uncertain state. In a managerial context, the findings explain why options induce better decision-making from CEOs, as well as why excessive risk taking might be optimal.

A Appendix

Proof (Lemma 1)

To show that $U\left( y_s\right) >\max \left\{ \mu _0,y_s\right\}$ integrate by parts the value of experimentation to obtain

$$\begin{aligned} U\left( y_s\right)= & {} y_s+\int _{y_s}^{1}\left( 1-F\left( x\right) \right) \mathrm{d}x \nonumber \\= & {} y_s+\int _0^{1}\left( 1-F\left( x\right) \right) \mathrm{d}x-\int _0^{y_s}\left( 1-F\left( x\right) \right) \mathrm{d}x \nonumber \\= & {} \mu _0+\int _0^{y_s}F\left( x\right) \mathrm{d}x \end{aligned}$$

(8)

Also note that

$$\begin{aligned} \frac{\partial U\left( y_s\right) }{\partial y_s}=F\left( y_s\right) >0 \end{aligned}$$

where the strict inequality comes from footnote 7. Convexity is also easily obtained since the second derivative is the probability of having a signal equal to $y_s$.

Proof (Lemma 2)

First note that $U\left( 0\right) =\mu _0$ and $U\left( 1\right) =y_s$. Therefore, $U\left( y_s\right) -c$ will cross at most once each of the outside options. It could cross once the constant $\mu _0$ from below since it is increasing in $y_s$. It could cross once $y_s$ from above since its first derivative with respect to $y_s$ is between 0 and 1. This in turn implies that $U\left( y_s\right)$ is farther from $\max \left\{ \mu _0;y_s\right\}$ precisely when $\mu _0=y_s$.

Since $U\left( y_s\right) -c$ is linear in c, there exists a $\hat{c}$ such that $U\left( y_s\right) -c=\mu _0=y_s$. Thus, for any $c<\hat{c}$, there exists $a_c,b_c\in \left( 0,1\right)$ such that $U\left( y_s\right) -c>\mu _0$ for any $y_s>a_c$, and $U\left( y_s\right) -c>y_s$ for any $y_s<b_c$. Obviously it must be the case that $\mu _0\in \left( a_c,b_c\right)$. Note that $a_c$ and $b_c$ are increasing and decreasing in c, respectively, precisely because the function crosses from below and above each of the corresponding outside options. Finally, for any $c>\hat{c}$, the interval is empty and the principal never experiments.

Proof (Lemma 3)

The principal is interested in obtaining a truthful report from the agent in order to take a decision between the safe project and the risky project. Let the set of signals that lead the principal to choose the risky project be denoted by R and the corresponding set that lead her to choose the safe project be denoted by S. Lets consider first the case when $x\in S$ and suppose the paid wage is $w\left( \hat{x},y_s\right)$. Since x is unrelated to $y_s$, the expected safe wage will not depend on x, only on $\hat{x}$. Thus, the agent will always prefer to report a $\hat{x}$ that maximizes $w\left( \hat{x},y_s\right)$, regardless of the observed x. Therefore, in order to have truthful revelation, the safe wage must not depend on $\hat{x}$ and can be expressed as $w_s$ given that $y_s$ is constant.

Now, consider the case where $x\in R$. Lets first consider any deviation from the agent such that $\hat{x}\in R$. Since $j\left( x\right) =r=j\left( \hat{x}\right)$, the principal is only interested in paying the agent the least possible as long as he reports a signal that leads her to choose the risky project. Formally, the principal solves the following problem:

$$\begin{aligned}&\min _{w\left( x,y_r\right) }\mathbb E\left[ w\left( x,y_r\right) |x\right] \\&\text {subject\;to } \mathbb E\left[ w\left( x,y_r\right) |x\right] \ge \mathbb E\left[ w\left( \hat{x},y_r\right) |x\right] \text { for all } \hat{x},x\in R\\&\text {and } \mathbb E\left[ w\left( x,y_r\right) |x\right] \ge w_s \text { for all }x\in R \end{aligned}$$

The solution to this program is to set $w\left( x,y_r\right) =w\left( y_r\right)$; thus, the first constraint will be always binding and the principal will not have to pay unnecessary informational rents.

Proof ( Lemma 4)

If the agent is hired, it must be the case that there exist signals $x_s$ and $x_r$ such that the safe project and the risky project are chosen, respectively. If this is not the case, and either project is always chosen, then the principal would not hire the agent since its information has no value. The existence of $x_e$ is therefore obtained using the intermediate value theorem given the continuity of the problem on x. Since payments for the principal are monotone nondecreasing, she wishes to implement a monotone decision. Such monotone decision can be implemented using expected monotone nondecreasing payments in x for the agent $\mathbb E\left[ w\left( y_r\right) |x\right]$, since an increase in x would benefit both individuals and the IR and IC constraints would continue to be satisfied. Those expected monotone nondecreasing payments exist and can be obtained through monotone nondecreasing payments, a consequence of the likelihood ratio order.

Proof (Proposition 1)

The problem and the constraints are linear in $w\left( y_r\right)$ and the Slater’s Condition is satisfied by the constraints. Therefore, Condition (3) is necessary and sufficient to obtain a maximum. We proceed in two steps. First we show that wages must be monotone nondecreasing. Since payments to the principal are also monotone nondecreasing, then optimal wages are option-like as suggested in the proposition. Finally, we prove that $z>y_s$.

For the first step it is sufficient to prove that Condition (3) is monotone increasing in $y_r$. The first two terms are independent of $y_r$. The third term is increasing in $y_r$ since the multipliers $\delta _r$ and $\delta _s$ are nonnegative, and the probability $1-F\left( x_e|y_r\right)$ is increasing in $y_r$ because the monotone likelihood ratio order induces first-order stochastic dominance. On the other hand, the hazard rate $\frac{f\left( x_e|y_r\right) }{1-F\left( x_e|y_r\right) }$ is decreasing in $y_r$ because the monotone likelihood ratio order induces log supermodularity (Athey 2002). Hence, it remains to prove that $\phi$ is nonnegative.

Suppose $\phi$ is negative and the wages are not monotone nondecreasing. Then there exists $y_L<y_H$ such that $w\left( y_L\right) >w\left( y_H\right)$. Consider a decrease of $w\left( y_L\right)$, denoted by $\Delta w_L<0$, compensated by an increase in $w\left( y_H\right)$, denoted by $\Delta w_H$, such that the total ex-ante utility of the agent does not change, and therefore, the IR constraint remains unaffected, as well as the objective function of the principal. Formally, it must be the case that:

$$\begin{aligned} \Delta w_L\left( 1-F\left( x_e|y_L\right) \right) g\left( y_L\right) +\Delta w_H\left( 1-F\left( x_e|y_H\right) \right) g\left( y_H\right) =0 \end{aligned}$$

This change continues to satisfy the IC1 constraint since $w_s$ does not change and $\mathbb E\left[ w\left( y_r\right) \right]$ decreases by

$$\begin{aligned} \Delta w_Lg\left( y_L\right) +\Delta w_Lg\left( y_H\right) =\Delta w_Hg\left( y_H\right) \left[ -\frac{1-F\left( x_e|y_H\right) }{1-F\left( x_e|y_L\right) }+1\right] <0 \end{aligned}$$

This reallocation also decreases $\mathbb E\left[ w\left( y_r\right) |x_e\right]$ by

$$\begin{aligned} \Delta w_Hf\left( y_H|x_e\right) \left( -\frac{\frac{f\left( y_L|x_e\right) }{1-F\left( y_L|x_e\right) }}{\frac{f\left( y_H|x_e\right) }{1-F\left( y_H|x_e\right) }}+1\right) <0 \end{aligned}$$

Moreover, for any x, the change on $\mathbb E\left[ w\left( y_r\right) |x\right]$ is given by

$$\begin{aligned} \Delta w_Hf\left( y_H|x\right) \left( -\frac{\frac{f\left( y_L|x_e\right) }{1-F\left( y_L|x_e\right) }}{\frac{f\left( y_H|x_e\right) }{1-F\left( y_H|x_e\right) }}\frac{\frac{f\left( y_L|x\right) }{f\left( y_h|x\right) }}{\frac{f\left( y_L|x_e\right) }{f\left( y_h|x_e\right) }}+1\right) \end{aligned}$$

where the term inside the brackets is increasing in x. Therefore, this reallocation leads to a minimization of the expected payments conditional on the signal, while maintaining to induce a monotone decision. Therefore, it must be the case that wages are monotone nondecreasing and that $\phi >0$. Given the monotonicity condition for the principal’s payoff, wages are option-like and it is also optimal for the principal to implement such monotone decision.

Finally, the cutoff z is given by the value of $y_r$ such that condition (3) is equal to zero. It must be greater than $y_s$, otherwise the principal prefers not to hire the agent and pursue the safe project since her net benefit from choosing the risky project would be $y_r- w\left( y_r\right) =\min \left\{ y_r,z\right\} <w_s$.

Proof (Lemma 5)

The first-order condition with respect to $x_e$ is given by:

$$\begin{aligned} f\left( x_e\right) \left( -x_e+\mathbb {E}\left[ w\left( y_r\right) |x_e\right] +y_s-w_s\right) -\phi \frac{\partial \mathbb {E}\left[ w\left( y_r\right) |x_e\right] }{\partial x_e}=0 \end{aligned}$$

Given the payment monotonicity for the principal and the agent, such condition becomes negative (positive) for any $x>(<)x_e$. Therefore, the problem is quasiconcave on $x_e$ and condition (4), which is obtained after using Condition (IC2*), characterizes a global maximum.