Moral Hazard and Other-Regarding Preferences

Abstract

This chapter aims at obtaining new theoretical insights by combining the standard moral hazard models of principal-agent relationships with theories of other-regarding preferences, in particular inequity aversion theory. The principal is in general worse off, as the agent cares more about the wellbeing of the principal. When there are multiple symmetric agents who care about each other’s wellbeing, the principal can optimally exploit their other-regarding nature by designing an appropriate interdependent contract such as a “fair” team contract or a relative performance contract. The approach taken in this chapter can shed light on issues on endogenous preferences within organizations.

The original article first appeared in Japanese Economic Review 55, 18–45, 2004. A newly written addendum has been added to this book chapter.

Appendices

Appendix

In this appendix I show that in the model in Sect. 4, it is without loss of generality to restrict attention to contracts with $w_{\mathit{fs}}^{n} = w_{\mathit{ff }}^{n} = 0$. To this end, consider a symmetric contract (w, w) ∈ C in which at least one equality of w _fs  ≥ 0 and w _ff  ≥ 0 is strict. Note that to simplify the notation I drop the superscripts from the contracts. If w _sf  ≥ w _fs , the incentive compatibility constraint and the participation constraint are, respectively,

$$ \displaystyle\begin{array}{rcl} (w_{\mathit{sf }} - w_{\mathit{ff }}) + p_{1}(w_{\mathit{ss}} + w_{\mathit{ff }} - w_{\mathit{sf }} - w_{\mathit{fs}}) + [p_{1} - (1 - p_{1})\gamma ]\alpha (w_{\mathit{sf }} - w_{\mathit{fs}}) \geq \frac{d} {\Delta _{p}}& &{}\end{array}$$

(17.34)

$$\displaystyle\begin{array}{rcl} & & w_{\mathit{ff }} + p_{1}(w_{\mathit{sf }} - w_{\mathit{ff }}) + p_{1}(w_{\mathit{fs}} - w_{\mathit{ff }}) + p_{1}^{2}(w_{\mathit{ ss}} + w_{\mathit{ff }} - w_{\mathit{sf }} - w_{\mathit{fs}}) \\ & & \qquad - (1 - p_{1})p_{1}\alpha (1+\gamma )(w_{\mathit{sf }} - w_{\mathit{fs}}) \geq d. {}\end{array}$$

(17.35)

Similarly, if w _sf  < w _fs , they are

$$\displaystyle\begin{array}{rcl} (w_{\mathit{sf }} - w_{\mathit{ff }}) + p_{1}(w_{\mathit{ss}} + w_{\mathit{ff }} - w_{\mathit{sf }} - w_{\mathit{fs}}) + [p_{1}\gamma - (1 - p_{1})]\alpha (w_{\mathit{fs}} - w_{\mathit{sf }}) \geq \frac{d} {\Delta _{p}}& &{}\end{array}$$

(17.36)

$$\displaystyle\begin{array}{rcl} & & w_{\mathit{ff }} + p_{1}(w_{\mathit{sf }} - w_{\mathit{ff }}) + p_{1}(w_{\mathit{fs}} - w_{\mathit{ff }}) + p_{1}^{2}(w_{\mathit{ ss}} + w_{\mathit{ff }} - w_{\mathit{sf }} - w_{\mathit{fs}}) \\ & & \qquad - (1 - p_{1})p_{1}\alpha (1+\gamma )(w_{\mathit{fs}} - w_{\mathit{sf }}) \geq d. {}\end{array}$$

(17.37)

The expected payment to each agent is

$$\displaystyle{ W = w_{\mathit{ff }} + p_{1}(w_{\mathit{sf }} - w_{\mathit{ff }}) + p_{1}(w_{\mathit{fs}} - w_{\mathit{ff }}) + p_{1}^{2}(w_{\mathit{ ss}} + w_{\mathit{ff }} - w_{\mathit{sf }} - w_{\mathit{fs}}). }$$

(17.38)

Denote by C ^∗ ⊂ C the set of feasible contracts that satisfy the incentive compatibility constraint and the participation constraint.

Lemma 17.2

For a given contract (w,w) ∈ C ^∗ with w _sf < w _fs, there exists a contract (w′,w′) ∈ C ^∗ such that w′ _sf > w′ _fs holds and the principal’s expected payment under (w′,w′) is the same as that under (w,w).

Proof

Define the new contract by w′ _ss  = w _ss , w′ _ff  = w _ff , w′ _sf  = w _fs , and w′ _fs  = w _sf . Obviously, the new contract satisfies w′ _sf  > w′ _fs and the participation constraint. And the incentive compatibility constraint (17.34) is satisfied because (w, w) satisfies (17.36) and $p_{1} - (1 - p_{1})\gamma \geq p_{1}\gamma - (1 - p_{1})$ holds for γ ≤ 1. Finally, the expected payment under the new contract is equal to W. □ 

By Lemma 17.2, from now on I focus on contracts with w _sf  ≥ w _fs . Define a new pay scheme for each agent, $\hat{w} = (\hat{w}_{\mathit{jk}})$, that satisfies $\hat{w}_{\mathit{fs}} =\hat{ w}_{\mathit{ff }} = 0$, $\hat{w}_{\mathit{sf }} = w_{\mathit{sf }} - w_{\mathit{fs}}$, and

$$\displaystyle{ p_{1}\left (\hat{w}_{\mathit{sf }} + p_{1}(\hat{w}_{\mathit{ss}} -\hat{ w}_{\mathit{sf }})\right ) = W. }$$

(17.39)

Condition (17.39) implies that the principal’s expected payment under the new contract $(\hat{w},\hat{w})$ is equal to that under (w, w). It is easy to show $\hat{w}_{\mathit{ss}} > 0$.

I next show that the new contract $(\hat{w},\hat{w})$ satisfies the participation constraint

$$\displaystyle{ p_{1}\left (\hat{w}_{\mathit{ss}} + p_{1}(\hat{w}_{\mathit{ss}} -\hat{ w}_{\mathit{sf }})\right ) - p_{1}(1 - p_{1})\alpha (1+\gamma )\hat{w}_{\mathit{sf }} \geq d. }$$

(17.40)

The left-hand side is equal to

$$\displaystyle{ W - p_{1}(1 - p_{1})\alpha (1+\gamma )(w_{\mathit{sf }} - w_{\mathit{fs}}) }$$

which is the left-hand side of (17.35). Thus by (17.35), (17.40) holds and the new contract satisfies the participation constraint.

Finally, the incentive compatibility constraint is written as follows:

$$\displaystyle{ \hat{w}_{\mathit{sf }} + p_{1}(\hat{w}_{\mathit{ss}} -\hat{ w}_{\mathit{sf }}) + [p_{1} - (1 - p_{1})\gamma ]\alpha \hat{w}_{\mathit{sf }} \geq \frac{d} {\Delta _{p}}. }$$

(17.41)

By (17.39) and $\hat{w}_{\mathit{sf }} = w_{\mathit{sf }} - w_{\mathit{fs}}$, the left-hand side of (17.41) is equal to

$$\displaystyle{ \frac{W} {p_{1}} + (p_{1} - (1 - p_{1})\gamma )\alpha (w_{\mathit{sf }} - w_{\mathit{fs}}). }$$

(17.42)

Since w _fs  > 0 or w _ff  > 0 holds, it is easy to show that

$$\displaystyle{ \frac{W} {p_{1}} > (w_{\mathit{sf }} - w_{\mathit{ff }}) + p_{1}(w_{\mathit{ss}} + w_{\mathit{ff }} - w_{\mathit{sf }} - w_{\mathit{fs}}). }$$

(17.43)

Therefore by (17.43), (17.42), and (17.34), the new contract satisfies the incentive compatibility constraint (17.41). The result is summarized in the following proposition.

Proposition 17.6

For a contract (w,w) ∈ C ^∗ that satisfies w _fs > 0 or w _ff > 0, there exists a contract $(\hat{w},\hat{w}) \in C^{{\ast}}$ such that $\hat{w}_{\mathit{fs}} =\hat{ w}_{\mathit{ff }} = 0$ and the principal’s expected payment is equal to that under (w,w).

Addendum: Revisiting Moral Hazard and Other-Regarding Preferences

I would conclude that if behavioral economists want their revolution to occur, they might be well served to focus on producing applied theory papers that economists in various fields will want to teach their students. Glenn Ellison ²⁴

When I was writing a paper that was eventually published as Itoh (2004), I had never heard nor seen term “behavioral contract theory.”²⁵ There were already several psychology-based individual decision making models²⁶ that were developed following anomalies found in various experiments, and fields like “behavioral game theory” and “behavioral finance” were emerging. Partly inspired by Glenn Ellison’s discussion, I attempted to extend the standard model of the principal-agent relationship with hidden action by applying a particular theory of “caring-about-others” preferences, i.e. inequity aversion. My paper was one of the earliest in this direction, and fortunately has been well cited, provided that it is published in Japanese Economic Review.

Today research in behavioral contract theory is rapidly growing, as evidenced by the recent survey titled “behavioral contract theory” (Kőszegi 2013), that is forthcoming in Journal of Economic Literature. Recent theoretical work in the field incorporates not only inequity aversion but also other theories of individual decision making successfully into contract theory to address important issues on contract design (Bénabou and Tirole 2006; Sliwka 2007; Ellingsen and Johannesson 2008; Herold 2010; Herweg et al. 2010; Englmaier and Leider 2012), and I want to urge those interested in learning recent development in behavioral contract theory to read Kőszegi (2013). My modest purpose of this addendum is to revisit Itoh (2004) to supplement the part of Kőszegi (2013) that discusses inequity aversion and its applications to contract theory.

Kőszegi (2013) is partly motivated by results from laboratory experiments on gift exchange and discretionary bonus, such as Fehr et al. (2007), and shows nicely that inequity aversion itself works as an incentive device in the sense that the first-best action can be induced by either a fixed non-contingent wage or a voluntary bonus. On the other hand, Itoh (2004) is motivated to introduce inequity aversion into the standard principal-agent model with limited liability in which, in contrast to such experiments, the agent’s binary action only stochastically determines output.

The more important difference comes from the fact that the potential application I have in mind is to large organizations, that leads to the following two features of my analysis. First, in the single agent case, the agent does not take into account the cost of action when comparing his income with the principal’s. While it is reasonable to assume that the agent cares about his relative net income in laboratory experiments, employees in large organizations are not likely to compare their net incomes (wages minus the cost of action) to the income of the employer.

Second, the main focus of Itoh (2004) is on the analysis of the multi-agent case. Particularly in large organizations, employees tend to compare themselves with those with the same job titles, pay ranks, status, and so on. That is why I analyze a multi-agent setting in which each agent cares about his fellow agent rather than the principal.

Furthermore, comparing (the cost of) his action with (that of) the other agent is natural, and I show that the optimal “contracts” depend critically on whether or not the agents compare their actions, in the following sense. When the agents do not compare their actions, the main result is Proposition 17.4. Let me restate it in terms of α and β = α γ as Proposition 17.7 given below, since Proposition 17.4 of Itoh (2004) contains minor typos.

Proposition 17.7

The optimal contract $w^{{\ast}} = (w_{\mathit{ss}}^{{\ast}},w_{\mathit{sf }}^{{\ast}})$ is given as follows.

Case 1::

$w^{{\ast}} = (\hat{w}_{\mathit{ss}},0)$ if $\beta >\alpha p_{1}/(1 - p_{1})$ holds. It is an extreme team contract.

Case 2a::

$w^{{\ast}} = (0,\hat{w}_{\mathit{sf }})$ if both $\beta <\alpha p_{1}/(1 - p_{1})$ and $\beta \leq 1 -\alpha (\ell_{s}/\ell_{f})$ hold. It is an extreme relative performance contract.

Case 2b::

$w^{{\ast}} = (\overline{w}_{\mathit{ss}},\overline{w}_{\mathit{sf }})$ if both $\beta <\alpha p_{1}/(1 - p_{1})$ and $\beta > 1 -\alpha (\ell_{s}/\ell_{f})$ hold.

Figure 17.1 given below summarizes the optimal contract for the case of p ₁ < 0. 5. Case 1 applies to the grey region, Case 2a to the region with “extreme relative,” and Case 2b covers both regions with “relative” and “team” where a relative performance contract ($\overline{w}_{\mathit{ss}} < \overline{w}_{\mathit{sf }}$) or a team contract ($\overline{w}_{\mathit{ss}} > \overline{w}_{\mathit{sf }}$), respectively, is optimal.

Fig. 17.1

The optimal contract in Proposition 17.7 (case p ₁ < 0. 5)

Importantly, the principal’s expected payments to the agents are the highest and do not depend on (α, β) in the grey region where an extreme team contract is optimal. When the agents compare their actions as well, the optimal contract is still an extreme team contract in the grey region (Proposition 17.5). However, the principal’s expected payments are now the lowest in that region, and sometimes at the first-best level. Inequity-averse agents can thus benefit the principal only if they compare their actions.

I should note that my analysis is subject to the critical comments made by Kőszegi (2013). First, the choice of the reference group is still exogenous, and an important future research theme is to understand how each agent chooses his reference group. Second, although one of my interests is in the choice of the agents’ preferences by the principal, “comparative statics with respect to variables typically studied in economic analysis” are still missing. And finally, my focus on inequity aversion is obviously restrictive for the study of teams. Each agent working in a team is motivated by team-based monetary incentives, private benefits, reciprocity, self-image from public, self-image from his fellow agents, and so on. How these forces interact has to be carefully studied for our further understanding of teams.