Optimism, Overconfidence, and Moral Hazard

A.  Related Literature

The moral hazard model is the focus of a large literature, beginning with Mirrlees ([1975] 1999) and Holmström (1979).⁴ I contribute to the behavioral strand of this literature, which examines, among other things, how greater confidence or optimism on the part of the agent shapes optimal contracting.⁵ This literature takes the moral hazard model as given and studies its implications for optimal contracts; in this paper, I provide behavioral foundations for the model itself as well as for overconfidence and optimism.

My characterization of the testable implications of the moral hazard model is designed to respect the model’s two defining features: that effort is costly and that it is unobservable. This distinguishes my analysis from previous work on the behavioral content of the moral hazard model, which has either confined attention to the special case of costless effort (Drèze 1961, 1987) or else assumed that effort is observable (Karni 2006).

I also contribute to the theoretical literature on behavioral economics, which studies models of individual decision making that can accommodate various departures from standard economic behavior that have been documented in the lab (or sometimes in the field). One strand of this literature examines behavioral models from the axiomatic perspective of decision theory;⁶ another strand is concerned with overconfidence and optimism.⁷ This paper marries these two strands. Relative to other work in that general spirit,⁸ I give new and simple definitions that distinguish overconfidence from optimism and cash out their meaning in the moral hazard model.

Finally, at a high level, this paper provides axiomatic behavioral foundations for a standard economic model (namely, the moral hazard model). Other work in this spirit includes axiomatizations of Blackwell’s (1951, 1953) model of choice informed by a signal,⁹ of the canonical model of costly information acquisition,¹⁰ and of Kamenica and Gentzkow’s (2011) Bayesian persuasion model (Jakobsen 2021, 2025).

B.  Road Map

The rest of this paper is arranged as follows. In section II, I describe the environment and the moral hazard model. I introduce a canonical parsimonious moral hazard model in section III. In section IV, I characterize the moral hazard model in terms of six empirically testable axioms. I establish the model’s identification properties in section V. Finally, in section VI, I propose behavioral definitions of “more confident than” and “more optimistic than” and characterize their meaning in the moral hazard model.

A.  Definitions and Conventions

A contract $w\in W$ is constant if and only if $w(s)=w(s^{\prime })$ for all output levels $s,s^{\prime }\in S$. As is standard, we identify each constant contract (an element of W) with the random remuneration at which it is constant (an element of Δ(Π)). Similarly, we identify each degenerate random remuneration (an element of Δ(Π)) with the prize at which it is degenerate (an element of Π).

Any (utility) function $u:\mathrm{\Pi }\to \mathbf{R}$ defined on monetary prizes Π extends naturally to a linear (expected utility) function $u:\mathrm{\Delta }(\mathrm{\Pi })\to \mathbf{R}$ defined on random remunerations via

$$u(x)≔{\int }_{\mathrm{\Pi }}u(\pi )x(\mathrm{d}\pi )\text{ for every }x\in \mathrm{\Delta }(\mathrm{\Pi }).$$

Throughout the paper, we hold fixed an arbitrary pair ${\pi }_0<{\pi }_1$ of reference prizes in Π, and we call a (utility) function $u:\mathrm{\Pi }\to \mathbf{R}$ such that $u({\pi }_0)\ne u({\pi }_1)$ normalized if and only if $\{u({\pi }_0),u({\pi }_1)\}=\{0,1\}$. This is merely an (arbitrary) choice of units in which to measure utility.

Let Δ(S) denote the set of all output distributions, that is, probability distributions on S. For any nonempty set A, call a function $\varphi :A\to [-\infty ,\infty ]$ grounded if and only if ${\inf }_{a\in A}\varphi (a)=0$. In the sequel, subsets of Euclidean spaces (such as the simplex Δ(S)) are always equipped with the Borel σ-algebra, and “increasing” always means “weakly increasing.”

B.  The Standard Moral Hazard Model

In the moral hazard model (Mirrlees [1975] 1999; Holmström 1979), the agent chooses her effort from a nonempty, compact, and convex subset E of a Euclidean space, for example, $E=[0,\overline{e}]$ for some $\overline{e}\in (0,\infty )$. Each effort level $e\in E$ incurs a cost $C(e)\ge 0$ and produces a distribution $P_e\in \mathrm{\Delta }(S)$ of output, where the cost function $C:E\to {\mathbf{R}}_{+}$ is grounded and lower semicontinuous and the map $e\mapsto P_e$ is continuous. The distribution P_e need not be objective; all that matters is that the agent believes that output will realize according to the probability distribution P_e if she supplies effort e.

The agent’s payoff under a contract $w\in W$ given effort $e\in E$ is

$$-C(e)+\sum _{s\in S}u(w(s))P_e(s),$$

where $u:\mathrm{\Pi }\to \mathbf{R}$ is a strictly increasing and normalized utility function whose curvature captures the agent’s risk attitude. The agent chooses effort optimally, randomizing if desired. Her payoff under a contract $w\in W$ is therefore

$$\underset{\mu \in \mathrm{\Delta }(E)}{\sup }{\int }_E\left[-C(e)+\sum _{s\in S}u(w(s))P_e(s)\right]\mu (\mathrm{d}e),$$

where Δ(E) denotes the set of all probability measures on E.

C.  Data: Preferences over Contracts

The agent’s preference over contracts, ⪰, is a binary relation on W. As usual, we write $w\succ w^{\prime }$ if and only if $w⪰w^{\prime }⋡w$. We interpret the preference ⪰ as (potential) data on the agent’s choice behavior: in particular, her choices when presented with any given pair w, $w^{\prime }\in W$ of contracts. In other words, ⪰ is an empirical object.

A binary relation ⪰ on W is a moral hazard preference exactly if it may be represented by a moral hazard model: that is, if and only if there is a nonempty, compact, and convex (effort) set $E\subseteq {\mathbf{R}}^n$ (where $n\in \mathbf{N}$); a grounded and lower semicontinuous (cost) function $C:E\to {\mathbf{R}}_{+}$; a continuous (belief) map $e\mapsto P_e$ carrying E into Δ(S); and a strictly increasing and normalized (utility) function $u:\mathrm{\Pi }\to \mathbf{R}$, such that for any contracts $w,w^{\prime }\in W$, $w⪰w^{\prime }$ holds if and only if

$$\begin{array}{rl}& \underset{\mu \in \mathrm{\Delta }(E)}{\sup }{\int }_E\left[-C(e)+\sum _{s\in S}u(w(s))P_e(s)\right]\mu (\mathrm{d}e)\\ \ge & \underset{\mu \in \mathrm{\Delta }(E)}{\sup }{\int }_E\left[-C(e)+\sum _{s\in S}u(w^{\prime }(s))P_e(s)\right]\mu (\mathrm{d}e).\end{array}$$

We assume that ⪰ is the only data that are available. In particular, following the moral hazard literature, we assume that the agent’s effort choices $e\in E$ are unobservable.

A.  Basic Axioms

Our first four axioms are standard. The first three express basic economic properties, and the fourth is a technical regularity condition.

It is easily verified (using lemma 1) that any moral hazard preference satisfies these four axioms.

The first of our two substantial axioms is as follows.

Quasiconvexity says precisely that the agent is averse to mixing contracts: if two contracts are equally good, then the contract obtained by randomly selecting one of them (using an α-biased coin toss) is weakly worse. In a parsimonious moral hazard representation, this axiom is satisfied because tailoring a single effort level $p^{″}\in \mathrm{\Delta }(\mathrm{\Pi })$ to the mixture of two contracts w and w′ is more difficult than tailoring effort separately to each of the two contracts.¹³ A version of this observation was first made by Drèze (1961).¹⁴

Mathematically, quasiconvexity is identical to Schmeidler’s (1989) definition of uncertainty seeking (the opposite of uncertainty aversion).

B.  Independence

Every moral hazard preference has an expected utility attitude to risk: random remunerations $x\in \mathrm{\Delta }(\mathrm{\Pi })$ (i.e., constant contracts) are evaluated via the expectation $u(x)={\int }_{\mathrm{\Pi }}u(\pi )x(\mathrm{d}\pi )$. By the expected utility theorem (von Neumann and Morgenstern 1947), a relation ⪰ that satisfies axioms 1–4 has this objective expected utility property if it satisfies the following axiom.

Moral hazard preferences do not generally have an expected utility attitude to (Knightian) uncertainty. Equivalently (given axioms 1–4), moral hazard preferences do not generally satisfy the stronger Anscombe–Aumann independence axiom.¹⁶ They do, however, satisfy the following independence axiom from Maccheroni, Marinacci, and Rustichini (2006), which is stronger than vNM independence but weaker than Anscombe–Aumann independence.

MMR independence requires that mixing one random remuneration (y) into two contracts (w and w′) is much the same as mixing in another (y′): what matters is the probability ($1-\alpha$) with which the output-independent remuneration is awarded instead of whatever output-dependent remuneration the contracts (w and w′) specify. The reason why the probability α matters is that it affects the slope of contracts, that is, how they vary with output $s\in S$; by contrast, whether remuneration is (constant and equal to) y or y′ with probability $(1-\alpha )$ does not affect slope. In a parsimonious moral hazard representation, MMR independence is satisfied because replacing y with y′ does not affect the optimal choice of effort $p\in \mathrm{\Delta }(S)$.

C.  Axiomatic Behavioral Characterization

The following proposition characterizes the behavioral content of the moral hazard model.

To prove proposition 1, we begin with an observation. The inverse of a binary relation ⪰ on W is the binary relation ⊒ on W such that $w^{\prime }⊒w$ if and only if $w⪰w^{\prime }$ for all contracts $w,w^{\prime }\in W$. Given a binary relation ⊒ on W, a grounded, convex, and lower semicontinuous function $c:\mathrm{\Delta }(S)\to [0,\infty ]$, and a nonconstant function $v:\mathrm{\Pi }\to \mathbf{R}$, we say that (c, v) is a variational representation of ⊒ exactly if for any contracts $w,w^{\prime }\in W$, $w^{\prime }⊒w$ holds if and only if

$$\underset{p\in \mathrm{\Delta }(S)}{\min }\left[c(p)+\sum _{s\in S}v(w^{\prime }(s))p(s)\right]\ge \underset{p\in \mathrm{\Delta }(S)}{\min }\left[c(p)+\sum _{s\in S}v(w(s))p(s)\right].$$

Proof of proposition 1. It is easily verified (using lemma 1) that point 2 implies point 1. For the converse, suppose that ⪰ satisfies point 1 and let ⊒ be its inverse. By inspection, ⊒ satisfies axioms 1, 3, and 4, MMR independence, nondegeneracy (there exist contracts ${w,w}^{\prime }\in W$ such that $w\succ w^{\prime }$), and quasiconcavity (for any contracts $w,w^{\prime }\in W$ such that $w⊒w^{\prime }⊒w$, it holds that $\alpha w+(1-\alpha )w^{\prime }⊒w$ for all $\alpha \in (0,1)$). Thus, by theorem 3 in Maccheroni, Marinacci, and Rustichini (2006), ⊒ admits a variational representation (c, v). By axiom 2, −v is strictly increasing, so $u(\cdot )≔[v({\pi }_0)-v(\cdot )]/[v({\pi }_0)-v({\pi }_1)]$ is strictly increasing and normalized. Hence, by observation 1, (c, u) is a parsimonious moral hazard representation of ⪰, and thus ⪰ is a moral hazard preference by lemma 1. QED

A.  Confidence

A confident agent is one who believes she can significantly influence the distribution of output. In terms of contract choice behavior, greater confidence is thus expressed by a greater appetite for nonconstant contracts, under which pay depends on realized output. This motivates the following purely behavioral definition of relative confidence.

Note that the definition of relative confidence makes no reference to any objective facts about the agent’s actual ability to influence the distribution of output. Greater confidence may therefore reflect an increased subjective confidence on the agent’s part about her ability to influence output, an actual improvement in her ability, or a combination of the two.

Mathematically, definition 2 is identical to the standard definition of “less uncertainty averse than” (Epstein 1999; Ghirardato and Marinacci 2002).

In the moral hazard model, greater confidence is equivalent to uniformly lower costs:

Equivalently, (c, u) is more confident than (c′, u′) exactly if $u=u^{\prime }$ and the cost level sets are nested:

$$\left\{p\in \mathrm{\Delta }(S):c(p)\le k\right\}\supseteq \left\{p\in \mathrm{\Delta }(S):c^{\prime }(p)\le k\right\}\text{ for every }k\ge 0.$$

In other words, any output distribution that is (believed to be) attainable at cost ≤k under (c′, u′) is also (believed to be) attainable at cost ≤k under (c, u). The fact that $u=u^{\prime }$ is implied reflects a (conceptually desirable) separation of confidence from risk attitude.

Example 3. 

Let output be binary, $S=\{0,1\}$, so that each distribution $p\in \mathrm{\Delta }(S)$ may be identified with a single number, $p\equiv \mathrm{Pr}(\text{output}=1)$. Let $c(p)≔\alpha {(p-\beta )}^2$ for each $p\in [0,1]$, where $\alpha \in {\mathbf{R}}_{+}$ and $\beta \in [0,1]$ are parameters. As α falls (holding β constant), c shifts vertically downward (see fig. 1). By proposition 3, the agent becomes more confident.

Proof of proposition 3. The “if” part is trivial. For the “only if” part, let ⪰ and ⪰′ be moral hazard preferences with parsimonious representations (c, u) and (c′, u′), respectively, and let ⪰ be more confident than ⪰′. Write ⊒ and ⊒′ for the inverses of ⪰ and ⪰′, respectively. By observation 1, (c, −u) and (c′, −u′) are variational representations of ⊒ and of ⊒′, respectively. And by inspection, for any random remuneration $x\in \mathrm{\Delta }(\mathrm{\Pi })$ and any contract $w\in W$, $x{⊒}^{\prime }w$ implies $x⊒w$. Thus, $u=u^{\prime }$ and $c\le c^{\prime }$ by proposition 9 in Maccheroni, Marinacci, and Rustichini (2006). QED

B.  Optimism

An optimistic agent is one who expects output to be high. To capture this, assume that output levels are ordered: $S=\{s_1,s_2,\dots ,s_{\left|S\right|}\}$, where $s_1<s_2<\cdots <s_{\left|S\right|}$.

In terms of contract choice behavior, greater optimism is manifested by a greater propensity to choose steeper contracts, which pay relatively more following high output realizations than following low output realizations.

The appropriate formalization of “steeper” requires correcting for risk attitude by using units of utility rather than of money. Toward a definition, recall that by the expected utility theorem (von Neumann and Morgenstern 1947), if a relation ⪰ satisfies axioms 1–4 and vNM independence, then there is exactly one strictly increasing and normalized function $u:\mathrm{\Pi }\to \mathbf{R}$ such that for any random remunerations $x,x^{\prime }\in \mathrm{\Delta }(\mathrm{\Pi })$, $x⪰x^{\prime }$ holds if and only if ${\int }_{\mathrm{\Pi }}u(\pi )x(\mathrm{d}\pi )\ge {\int }_{\mathrm{\Pi }}u(\pi )x^{\prime }(\mathrm{d}\pi )$. Given any such relation ⪰ and any two contracts w, $w^{\prime }\in W$, we say that w is ⪰-steeper than w′ exactly if $s\mapsto u(w(s))-u(w^{\prime }(s))$ is increasing. Equivalently, $(1/2)w(s)+(1/2)w^{\prime }(s^{\prime })⪰(1/2)w(s^{\prime })+(1/2)w^{\prime }(s)$ holds for all output levels $s\ge s^{\prime }$ in S. This latter formulation furnishes a general definition that is applicable for any relation ⪰:

In other words, a more optimistic preference is one that is more disposed to prefer steeper contracts, where steepness is measured in utility units.

We shall characterize relative optimism in the moral hazard model in terms of upshiftedness of costs, defined as follows.

Despite its tricky definition, upshiftedness expresses a straightforward idea: that first-order stochastically higher output distributions are relatively cheaper under c than under c′. Indeed, c is upshifted from c′ exactly if under any contract $w\in W$ and for any strictly increasing and normalized $u:\mathrm{\Pi }\to \mathbf{R}$, the optimal choice of effort $p\in \mathrm{\Delta }(S)$ in the parsimonious moral hazard model (c, u) is first-order stochastically higher than optimal effort $p^{\prime }\in \mathrm{\Delta }(S)$ in the parsimonious moral hazard model (c′, u) (see Dziewulski and Quah 2024, sec. 5.2).²¹

Example 3 (Continued). 

Recall that $c(p)=\alpha {(p-\beta )}^2$ for each $p\in [0,1]$ and note that p FOSD p′ if and only if $p\ge p^{\prime }$. As β rises (holding α constant), c shifts horizontally rightward (see fig. 2); formally, c upshifts.²²

There is an intuitive necessary (though not sufficient) condition for upshiftedness in terms of the cost level sets

$$L_k≔\{p\in \mathrm{\Delta }(S):c(p)\le k\}\text{ and }{L}_k^{\prime }≔\{p\in \mathrm{\Delta }(S):c^{\prime }(p)\le k\}.$$

In other words, the set L_k of output distributions that are (believed to be) attainable at cost ≤ k under c is “FOSD higher” than the set $L_k^{\prime }$ of output distributions that are (believed to be) attainable at cost ≤k under c.²³

Proof. We prove the first half, omitting the analogous argument for the second half. Fix a $k\ge 0$ and a $p\in L_k$. Since c′ is grounded and lower semicontinuous, we may choose a $p^{\prime }\in \mathrm{\Delta }(S)$ such that $c^{\prime }(p^{\prime })=0$. By upshiftedness, there are $q,q^{\prime }\in \mathrm{\Delta }(S)$ such that p FOSD q′ and $c^{\prime }(q^{\prime })\le c(q)+c^{\prime }(q^{\prime })\le c(p)+c^{\prime }(p^{\prime })\le k$, so $q^{\prime }\in L_k^{\prime }$. QED

We now show that greater optimism is manifested in the moral hazard model by first-order stochastically higher distributions becoming relatively cheaper in the sense of upshiftedness.

The proof, given in the appendix, turns on a result due to Dziewulski and Quah (2024).

C.  Distinguishability of Confidence and Optimism

Relative confidence and relative optimism are logically independent: by example 3, a preference can be more confident than another without being more optimistic, and vice versa.²⁴ In other words, relative confidence is empirically distinguishable from relative optimism.

This conclusion rests in part on the richness of the data ⪰, which record the agent’s choices between all pairs $w,w^{\prime }\in W$ of contracts. On coarser data, distinguishability may fail. To explore this point, suppose that data are available only on choices between pairs of contracts belonging to $W^{\star }\subseteq W$, where W^⋆ contains the constant contracts (i.e., $W^{\star }\supseteq \mathrm{\Delta }(\mathrm{\Pi })$). For binary relations ⪰ and ⪰′ on W, say that ⪰ is more confident on W^⋆ than ⪰′ if and only if whenever $w{⪰}^{\prime }({\succ }^{\prime })x$ for a contract $w\in W^{\star }$ and a constant contract $x\in \mathrm{\Delta }(\mathrm{\Pi })$, we also have $w⪰(\succ )x$; and say that ⪰ is more optimistic on W^⋆ than ⪰′ if and only if whenever $w{⪰}^{\prime }({\succ }^{\prime })w^{\prime }$ for two contracts $w,w^{\prime }\in W^{\star }$ such that w is ⪰-steeper than w′, we also have $w⪰(\succ )w^{\prime }$.

Whether “more confident on W^⋆ than” implies “more optimistic on W^⋆ than” or vice versa depends on $W^{\star }\subseteq W$. Consider, for example, the set W^↑ of all increasing contracts, meaning those $w\in W$ such that w(s) first-order stochastically dominates w(s′) whenever $s\ge s^{\prime }$.

Thus, data on increasing contracts alone are insufficient to distinguish relative confidence from relative optimism.

Proof. Fix a contract $w\in W^{\star }$ and a constant contract $x\in \mathrm{\Delta }(\mathrm{\Pi })$ such that $w{⪰}^{\prime }({\succ }^{\prime })x$; we must show that if ⪰ is more optimistic on W^↑ than ⪰′, then $w⪰(\succ )x$. By the expected utility theorem (von Neumann and Morgenstern 1947), there is exactly one strictly increasing and normalized function $u:\mathrm{\Pi }\to \mathbf{R}$ such that for any random remunerations $x^{\prime },x^{″}\in \mathrm{\Delta }(\mathrm{\Pi })$, $x^{\prime }⪰x^{″}$ holds if and only if ${\int }_{\mathrm{\Pi }}u(\pi )x^{\prime }(\mathrm{d}\pi )\ge {\int }_{\mathrm{\Pi }}u(\pi )x^{″}(\mathrm{d}\pi )$. Since u is strictly increasing and w is increasing, w is ⪰-steeper than x. Thus, if ⪰ is more optimistic on W^↑ than ⪰′, then $w⪰(\succ )x$. QED

D.  Absolute Overconfidence and Optimism

A remaining question is what it means for an agent to be overconfident or optimistic in an absolute sense (rather than relative to some other agent). I conclude by offering one possible answer to this question, in the spirit of Epstein (1999) and Ghirardato and Marinacci (2002).

Whereas our definitions of “more confident than” and “more optimistic than” were purely behavioral, making no reference to objective facts about the agent’s ability to influence output, absolute overconfidence and optimism are concerned precisely with discrepancies between such objective facts and the agent’s subjective assessment of them. We must therefore fix an objective benchmark against which the agent’s behavior may be compared. Although it is natural to fix as our benchmark the first three parameters (E^⋆, C^⋆, $e\mapsto P_e^{\star }$) of a standard four-parameter moral hazard model, what will actually matter is only the objective output distribution cost $c^{\star }:\mathrm{\Delta }(S)\to [0,\infty ]$.²⁶

In other words, an overconfident (optimistic) preference is one that is more confident (more optimistic) than is objectively warranted.

Analogous definitions and characterizations may be given for (absolute) underconfidence and pessimism.