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The Journal of Law, Economics, and Organization
, Volume 34 (1) – Mar 1, 2018

29 pages

/lp/ou_press/incentive-contracts-and-downside-risk-sharing-iMoTXSW7AN

- Publisher
- Oxford University Press
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- © The Author 2017. Published by Oxford University Press on behalf of Yale University. All rights reserved. For Permissions, please email: journals.permissions@oup.com
- ISSN
- 8756-6222
- eISSN
- 1465-7341
- D.O.I.
- 10.1093/jleo/ewx014
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- See Article on Publisher Site

Abstract This paper seeks to characterize incentive compensation in a static principal–agent moral hazard setting in which both the principal and the agent are prudent (or downside risk averse). We show that optimal incentive pay should then be “approximately concave” in performance, the approximation being closer the more downside risk averse the principal is compared with the agent. Limiting the agent’s liability would improve the approximation, but taxing the principal would make it coarser. The notion of an approximately concave function we introduce here to describe optimal contracts is relatively recent in mathematics; it is intuitive and translates into concrete empirical implications, notably for the composition of incentive pay packages. We also clarify which measure of prudence—among the various ones proposed in the literature—is relevant to investigate the tradeoff between downside risk sharing and incentives. 1. Introduction Under moral hazard, it is well-known that optimal compensation trades off risk sharing and incentives. The involved parties’ risk preferences, particularly their respective risk aversion, therefore matter for explaining the shape of incentive pay. In certain circumstances, one also has to look beyond risk aversion. It is by now well understood that prudence (or aversion to downside risk) affects an agent’s effort choice directly (Eeckhoudt and Gollier 2005; Peter 2017), and when the principal subjects the agent to contingent monitoring (Fagart and Sinclair-Desgagné 2007) or a principal–agent relationship is exposed to background risk (Ligon and Thistle 2013), the agent’s prudence must be considered. In many other situations, it seems reasonable to expect the principal’s prudence to be relevant. As she interacts with her doctor/agent, for example, prudence will likely play a role in explaining a patient’s willingness to prevent the occurrence of sickness (Courbage and Rey 2006). Prudence might also characterize an insurance customer seeking advice from her broker/agent (Hau 2011). A regulator who adopts a precautionary stand—as it is the case in Europe and the United States in public health, food safety, and environmental policy (Barrieu and Sinclair-Desgagné 2006; Wiener et al. 2011)—shows evidence of prudent risk preferences (Gollier et al. 2000). A firm’s executive who contemplates entry in a foreign market might do so with prudence, since running a foreign subsidiary/agent can have major downside risk implications (Reuer and Leiblein 2000). In corporate governance, finally, jurisprudence and the law endow board members with fiduciary duties of loyalty and care toward their corporation (Clark 1985; Gutiérrez and Gutiérrez 2003; Lan and Heracleous 2010; Corporate Law Committee 2011). The duty of care confers board of directors a key role in preventing and managing crisis situations (Mace 1971; Williamson 2007; Adams et al. 2010).1 When establishing the CEO/agent’s compensation, they should accordingly weigh the risks on the downside differently from those on the upside, thereby exerting “(…) that degree of care, skill, and diligence which an ordinary, prudent man would exercise in the management of his own affairs” (Clark 1985: 73; emphasis added). This paper now examines the shape of incentive compensation in a principal–agent setting in which the principal is prudent, or downside risk averse, as this attribute is currently understood in economics. Formally, whereas someone exhibits risk-aversion when her utility function is concave, someone is prudent when her marginal utility function is convex (Menezes et al. 1980; Kimball 1990).2 A prudent decision maker prefers additional volatility to be associated with good rather than bad outcomes (Eeckhoudt and Schlesinger 2006; Denuit et al. 2010). She dislikes mean and variance-preserving transformations that skew the distribution of outcomes to the left (Crainich and Eeckhoudt 2008; Ebert and Wiesen 2011).3 Our main result is that incentive compensation should then be “approximately concave” in performance (in the formal sense due to Páles 2003), the approximation being closer the more prudent the principal is relative to the agent. This proposition sheds light on the tradeoff between downside risk sharing and incentives. Roughly speaking, as they approach concavity, incentives become more sensitive to outcomes in the range where they are mediocre, thereby transfering the agent greater downside risk; a more prudent principal would likely afford such incentives. Making this intuition precise, however, required us to first invoke, in a principal–agent moral hazard setting, Modica and Scarsini (2005)’s “coefficient of downside risk aversion”—instead of Kimball (1990)’s better known “coefficient of prudence”—and to make use of the relatively recent notion (unused so far in economics and finance) of an approximately concave function. There is an extensive literature dealing with the shape of optimal remuneration in various principal–agent settings.4Holmström and Milgrom (1987), for instance, provide a rationale for linear incentive schemes, Hemmer (1993) validates piecewise linear arrangements, Schmitt and Spaeter (2005) and Gutiérriez Arnaiz and Salas-Fumás (2008) support convex–concave contracts, and Osband and Reichelstein (1985) justify concave schemes. Among the articles looking specifically at the impact of downside risk preferences,5Hemmer et al. (2000) and Chaigneau (2015) show that a risk neutral principal would propose a convex contract to a prudent agent, Hau (2011) that a prudent principal would offer concave remuneration to a risk neutral agent. Looking beyond the polar cases where either the principal or the agent is prudent, Hemmer et al. (2000) also add that the benefit of convexity declines as the distribution of outcomes exhibits lower skewness, and Chaigneau (2015) argues that no convex contract can be optimal if the principal is sufficiently prudent relative to the agent. So far, however, optimal compensation when both the principal and the agent can be prudent has eluded characterization. This paper is now addressing this gap. Moreover, while the above results predict rather stylized payment schemes that might not show up in practice (CEO compensation, for example, involves capped bonuses and similar concave devices, as well as call options that tend to “convexify” remuneration), our description of the pay–performance relationship as approximately concave has concrete empirical implications: approaching concavity more closely actually means that the relative weight of concave items in the overall pay package goes up. The rest of the paper unfolds as follows. Section 2 presents the benchmark model—a static principal–agent model where the agent is risk averse and exerts costly effort while the principal is both risk averse and prudent. Adopting the first-order approach is justified using Jung and Kim (2015)’s recent and very general assumptions. Our central result—that the optimal contract should in that case be approximately concave, thereby seeking a balance in the downside risk respectively borne by the agent and the principal—is established in Section 3. The section ends with a pair of numerical examples corroborating this statement. Section 4 then extends our main result, showing how two frequent contextual elements—limiting the agent’s liability or taxing the principal—might affect the downside-risk/incentives tradeoff: limited liability brings compensation closer to concavity in the range where it varies with performance, taxes have the opposite effect. It also discusses what happens when it is the agent who is more downside risk averse or when neither the principal nor the agent are prudent. Section 5 looks next briefly into empirical matters. Section 6 offers some concluding remarks. All mathematical proofs are in the Appendix. 2. The Model Consider an agent—standing for a physician, a financial advisor, a regulated firm, a foreign subsidiary, a CEO, etc.—whose preferences can be represented by a Von Neumann–Morgenstern utility function u(⋅) defined over monetary payments. We assume this function is three-times differentiable, increasing, and strictly concave, formally u′(·)>0 and u″(·)<0, so the agent is risk averse. This agent can work for a principal—namely a patient, an individual investor, a regulator, a multinational’s executive, a corporate board, etc.—whose preferences are represented by the Von Neumann–Morgenstern utility function v(⋅) defined over net final wealth. We suppose this function is increasing and strictly concave, i.e., v′(·)>0 and v″(·)<0, so the principal is risk averse. Moreover, the marginal utility v′(·) is convex, i.e., v‴(·)>0, which means that the principal is downside risk averse or (equivalently) prudent (Menezes et al. 1980; Kimball 1990). The principal’s profit depends stochastically on the agent’s effort level a. The latter cannot be observed, however, while the agent incurs a cost of effort c(a) that is increasing and convex ( c′(a)>0 and c″(a)≥0). The principal only gets a verifiable signal s, drawn from a compact subset S=[s̲,s¯] of ℝ, which is correlated with the agent’s effort through a conditional probability distribution F(s; a). This distribution has a density f(s; a) which is differentiable in a and strictly positive on S. The function LF(s;a)=fa(s;a)f(s;a) then denotes the likelihood ratio of signal s. Based on the realized value of s, the principal receives a benefit π(s) expressed in monetary terms, which we suppose increasing and concave or linear in s ( π′(s)>0 and π″(s)≤0), and she pays the agent a compensation w(s). The principal’s problem is to find a reward schedule w(s) that maximizes her expected utility, under the constraints that the agent will then maximize his own expected utility (the incentive compatibility constraint) and must ex ante receive an expected utility that is not inferior to some external one U0 (the participation constraint). This can be written formally as maxw(s),a∫s∈Sv(π(s)−w(s))dF(s;a)subject to a∈ argmaxe∫s∈Su(w(s))dF(s;e)−c(e).∫s∈Su(w(s))dF(s;a)−c(a)≥U0 (1) Without losing generality, we shall concentrate on smooth (i.e., twice continuously differentiable) contracts w(⋅) such that, for D1, D2, … , Dn a finite partition of S, the derivatives w′(s)<Mi on each Di, with Mi a positive real number. At any performance signal s, the agent’s marginal revenue will therefore be bounded. This amounts to saying that the allowed incentive schemes are locally Lipschitz continuous: for any w(s) on each set Di, |w(x)−w(s)|≤Mi|x−s| for all x,s∈Di. This restriction, which will be useful later in the proofs, seems rather innocuous, since the exogenous number n and ceilings Mi's are arbitrary, and since any continuous function can be approximated as closely as wanted by a sequence of locally Lipschitz maps (Miculescu 2000). 2.1 The First-Order Approach For tractability reasons, one usually replaces the incentive compatibility constraint by a relaxed constraint based on the first-order necessary condition for the agent’s utility-maximizing effort a. This transforms the principal’s initial problem into the following one: maxw(s),a ∫s∈Sv(π(s)−w(s))dF(s;a)subject to ∫s∈Su(w(s))dFa(s;a)−c′(a)≥0, (γ)∫s∈Su(w(s))dF(s;a)−c(a)≥U0, (μ) (2) where γ and μ denote the constraints’ respective Lagrange multipliers. This so-called “first-order approach” delivers for sure a solution to problem (1) under the following hypotheses. Assumption 1. Under the distribution F(s;a), π(s) is such that, for a2 > a1, Pr[π(s)≤π¯|a1]≥ Pr[π(s)≤π¯|a2] for all π¯, where the inequality holds strictly on a set of positive measure. Definition 1. A subset Z⊆ℝn is an increasing set if, for any zo∈Z, any z ≥ zo (i.e., zi≥zio for every i = 1,…, n) also belongs to Z. Assumption 2. For any increasing set ZπLF of pairs (π(s),fa(s;a)f(s;a)), Pr[s∈T(ZπLF)|a] is concave in a, where T(⋅) is the mapping defined as T(Z)={s∈S| (π(s),fa(s;a)f(s;a))∈Z}, Z⊆ℝ2. Assumption 1 is a first-order stochastic dominance (FOSD) condition with respect to π. It basically says that higher effort by the agent makes higher profits more likely. Assumption 2 is a concave increasing set probability (CISP) condition with respect to (π,LF), which then adds that the agent’s effort is subject (probabilistically) to decreasing returns. The next statement corresponds to Jung and Kim (2015)’s Proposition 11. Lemma 1. Problem (2) is equivalent to problem (1) if F(s; a) satisfies Assumptions 1 and 2. Jung and Kim (2015) demonstrate this lemma. They also show, in their Proposition 12, that the assumptions made allow more probability distributions (hence are weaker) than any of those used so far in the literature. 2.2 Downside Risk Aversion Let us write Ru=−u″u′ and Rv=−v″v′ for the standard Arrow–Pratt measure of absolute risk aversion corresponding to the agent’s and the principal’s utility functions u and v, respectively. In a well-known article, Kimball (1990) introduced the analogous “coefficients of absolute prudence” Pu=−u‴u″ and Pv=−v‴v″. Using the agent’s utility function, for illustration, Pu is proportional to the amount ς determined by the first-order necessary condition for optimal “precautionary” savings: u′(x−ς)=E[u′(x+ς+ε∼)], where E(ε∼)=0. These coefficients thus indicate the agent’s and the principal’s respective propensity to prepare in the face of an inevitable risk. In what follows, our analysis makes use of the products PuRu=u‴u′=du and PvRv=v‴v′=dv.6 The alternative coefficients du and dv have been proposed by Modica and Scarsini (2005) to capture the intensity of downside risk aversion. In contrast with Pu (using again the agent’s utility function, for illustration), du relates to the usual equality for establishing a risk premium ξ: i.e., u(x−ξ)=E[u(x+ε∼)], under mean and variance-preserving lottery transformations. The higher du, the larger the compensation the agent requires to be inflicted additional risk at unfavorable outcomes (Crainich and Eeckhoudt 2008). For that reason, trading off downside risk sharing and incentives, while managing agency costs, rests on the relative magnitudes of du and dv.7 Now, let Dom(u) and Dom(v) refer to the respective domains of the utility functions u and v. We shall introduce a key definition. Definition 2. For some constant real number k ≥ 1, the principal is said to be more downside risk averse than the agent by a factor k if, for any real numbers x∈Dom(u) and y−x∈Dom(v), we have that k·du(x)≤dv(y−x). In other words, the principal is more downside risk averse than the agent by a factor k≥1 if, for any amount y to be split between the two and any agent’s share x of this amount, the principal’s coefficient of downside risk aversion dv taken at her wealth level (y– x) is at least k times bigger than the agent’s own coefficient du(x). Intuitively (see Crainich and Eeckhoudt 2008), this means that the principal with wealth level (y– x) would demand k times the amount that the agent receiving x would require to accept bearing more downside risk. Definition 2 is trivially met if the agent’s utility function is quadratic, since u‴(x)=0 and the agent in this case displays no downside risk aversion ( du(x)=0). The definition also holds if the agent and the principal respectively have constant absolute risk aversion (CARA) utility functions u(x)=− exp (−a·x) and v(y−x)=− exp (−b·(y−x)), where a > 0 and b>k·a with k≥1. One more example is when the parties respectively have constant relative risk aversion (CRRA) utility functions u(x)=x1−α1−α and v(y−x)=(y−x)1−β1−β, with 0<α<1, β>α, x∈[b1,b2], b1>0 and y−x∈(0,b2]. The downside risk aversion coefficients are then du(x)=(1+α).αx2 and dv(y−x)=(1+β)·β(y−x)2, so for all x, y – x we have that k·du(x)≤dv(y−x) when β>k·(b2b1)2(1+α)−1, k≥1. A concrete situation where the principal might be more downside risk averse than the agent is when the latter is an expert (a scientist, a physician, an insurance broker, say) advising an interested party (a politician, a patient, a customer, say). Corporate governance and executive compensation might also constitute an analogous case, as we argue in Section 5. Let us now proceed to characterize the optimal incentive scheme. 3. The Optimal Contract This section will establish that a principal who is more downside risk averse than the agent should set an incentive compensation package that is approximately concave in outcome. We shall first define approximate concavity, then state and prove our central result, then illustrate it with two examples. 3.1 Approximate Concavity The following definition is adapted from Páles (2003).8 Definition 3. Let I be a subinterval of the real line ℝ and δ,ρ some nonnegative real numbers. A function g:I→ℝ is called (δ,ρ)-concave on I if tg(x)+(1−t)g(y)≤g(tx+(1−t)y)+δt(1−t)|x−y|+ρ for all x,y∈I and t∈[0,1]. The function g is of course concave when δ=ρ=0. For other values of δ and ρ, note that not every monotone increasing function on I can be (δ,ρ)-concave, so approximate concavity is not the same as quasi-concavity.9 Examples of approximately concave functions are pictured in Section 3.3. Basically, the graph of an approximately concave function must “wrap around” that of a concave function. Combining Páles (2003)’s theorems 4 and 5 provides a natural description of a (δ, 0)-concave function, which is the situation we will encounter here. Lemma 2. Let I be a subinterval of the real line ℝ and δ a nonnegative number. A function g:I→ℝ is (δ,0)-concave on I if there exists a nonincreasing function q:I→ℝ such that g(y)≤g(x)+q(x)(y−x)+δ2|y−x| for all x,y∈I. For completeness, a proof of this lemma is given in the Appendix.10 One may notice that the above function q bears a close resemblance to a subgradient. The literature indeed says that g is nonincreasingly (δ, 0)-subdifferentiable when such a function exists. 3.2 Approximately Concave Incentive Schemes In order to characterize the agent’s incentive contract, we need to make the following assumption, which implies Assumption 1 when the signal s is one-dimensional (Whitt 1980; Sinclair-Desgagné 1994). Assumption 3. The likelihood ratio LF(s;a)=fa(s;a)f(s;a) is nondecreasing and concave in s for any a. This is the so-called concave monotone likelihood ratio property (CMLRP), an assumption which is rather common in principal–agent analysis. It is satisfied by many familiar distributions, such as the Poisson with mean a, the gamma with mean κa, and the chi-squared with degree of freedom parameter a.11 It “(…) suggests that variations in output at higher levels are relatively less useful in providing “information” on the agent’s effort than they are at lower levels of output” (Jewitt 1988: 1181). Let us stress that assuming a concave likelihood ratio does not make the optimal contract trivially concave in outcome. Hemmer et al. (2000), for instance, need Assumption 3 to justify the first-order approach; yet, their analysis supports the use of convex incentive devices. Now, the Kuhn–Tucker necessary and sufficient conditions require that a solution to program (2) meet the equation v′(π(s)−w(s))u′(w(s))=μ+γfa(s;a)f(s;a), ∀s (3) The multiplier γ being positive, Assumption 3 entails that the right-hand side of Equation (3) is increasing in the signal s. This allows to say the following. Lemma 3. The optimal reward schedule w∗(s) is increasing in the performance signal s. To prove this lemma (see the Appendix), we take the first derivative with respect to s of the left-hand side of Expression (3), then seek conditions which are necessary to make it positive, since the right-hand side’s derivative with respect to s is positive by Assumption 3. Similarly taking the second derivative of Expression (3)’s left-hand side and using the same line of argument yields the central result of this paper. Theorem 1. Suppose that the principal is more downside risk averse than the agent by a factor k. Then the optimal wage schedule w∗(s) is (δ(k),0)-concave at any s∈S. The number δ(k) decreases with k and tends to 0 as k grows. The proof in the Appendix suggests that convergence to concavity may not be asymptotic: when (π′(s)−w′(s)w′(s))2≥1k, i.e., the agent’s earnings do not grow too fast with respect to the principal’s net benefit,12 then we have δ(k)=0 so w∗(s) is concave at s. In the region where it is not perfectly concave, moreover, the optimal wage schedule has a nonincreasing (δ(k),0)-subgradient which is precisely the derivative π′(s) of the principal’s benefit function; the contracted payment is thus aligning—albeit imperfectly—the agent’s incentives on the principal’s interests. The theorem’s conclusion holds vacuously—hence the optimal incentive scheme is concave—when the agent is not prudent (since u‴≤0 implies du≤0, k can then be as big as wanted). If the agent is prudent (i.e., u‴>0), the theorem says that the pay–performance relationship will only get closer to concavity as the principal’s downside risk aversion grows.13 For concreteness, the upcoming subsection provides some numerical examples. 3.3 Two Examples Let the principal’s benefit function takes the specific form π(s)=s, s∈[0,+∞), and let the distribution F(s;a), parameterized by the agent’s discrete effort levels a ∈ {1,2,3}, be a gamma distribution with mean κa, 0<κ. 3.3.1 The Agent Is Not Prudent First, consider a risk averse and prudent principal and a risk averse but non-prudent agent. Let their respective utility functions be given by u(x)=x−bx2 and v(z)=ln(z+10), taking b > 0 so that u′(x)>0 on the relevant domain. The first-order condition (3) comes up to (s−w∗(s)+10)−11−2bw∗(s)=A(s)1=A(s)⋅(1−2bw∗(s))⋅(s−w∗(s)+10)0=2bw∗2(s)−(1+2b⋅(s+10))⋅w∗(s)+(s+10−A−1(s)), (4) where A(s)=μ+γs−κaa2. Two roots exist; we choose the one so that w′(s)>0. Hence, the optimal reward function is w∗(s)=B(s)−B(s)2−8b(s+10−A(s)−1)4b, (5) with B(s)=(1+2b⋅(s+100)). It can be checked that this increasing reward function satisfies w∗″(s)<0, so the optimal incentive scheme is concave, as expected. This scheme is shown in Figure 1, using parameters μ=0.2,γ=0.1,κ=1,b=0.035. The optimal level of effort is a∗=2; it is obtained by introducing Equation (5) in the objective function of the agent and by solving program (2). The scheme is indeed quite steep over the region of lower outcomes, so the non-prudent agent bears significant downside risk. Figure 1. View largeDownload slide w∗(s) When the Agent Is Risk Averse But Non-Prudent ( u‴(·)=0). Figure 1. View largeDownload slide w∗(s) When the Agent Is Risk Averse But Non-Prudent ( u‴(·)=0). 3.3.2 The Principal and the Agent Have CRRA Utility Functions Now, suppose the agent and the principal have respective CRRA utility functions u(x)=x1−α1−x and v(z)=z1−kα1−kα, with 0<α<1 and k≥1. For k = 1, the first-order condition (3) becomes (s−w(s)w(s))−α=μ+γs−κaa2, ∀s. Write again A(s)=μ+γs−κaa2. We have that w1∗(s)=s1+A(s)−1/α. (6) When k = 2, Condition (3) amounts instead to (s−w(s))2=A(s)−1/α⋅w(s), ∀s. The only root of the latter equation which is increasing in s is w2∗(s)=(2s+A(s)−1/α)−(2s+A(s)−1/α)2−4s2. (7)Figure 2 portrays the two incentive schemes (6) and (7), using the values μ=0.2, γ=0.1, κ=1, and α=0.5. The bold (respectively, soft) curve is the optimal reward function obtained in the case k = 1 (respectively, k = 2) with optimal level of effort a∗=2 (respectively, a∗=3). Figure 2. View largeDownload slide The Incentive Scheme for CRRA Utility Functions, When k=1 (Bold Line) and k = 2 (Soft Line). Figure 2. View largeDownload slide The Incentive Scheme for CRRA Utility Functions, When k=1 (Bold Line) and k = 2 (Soft Line). The reward function approaches a concave function more closely when k = 2 than when k = 1, which corroborates statement (ii) of the above Theorem. The optimal wage schedule when k = 2 is also steeper (flatter) at low (high) outcomes than the optimal wage schedule when k = 1; indeed, the agent bears more downside risk under a more downside risk averse principal. In this example, furthermore, the incentive scheme remains convex at both values of k when the signal s is small. This feature is intuitive. Transfering more downside risk upon a downside risk averse agent is costly to the principal, who must then compensate the agent in order to satisfy the participation constraint. Here, the principal always finds it too expensive to implement a contract which has a downside risk averse agent bear all (or almost all) downside risk. This would not be the case in general, though, as the remark following the above Theorem and the discussion of the upcoming Section 4.3 (which respectively deal with k large and k < 1) explains. 4. Theoretical Extensions An important issue for the theory of incentives is whether, and to what extent, an incentive contract is influenced by its contextual background. Two customary elements of this background are limited liability and taxation. To check the robustness of our characterization, and thereby sharpen intuition on the tradeoff between downside risk sharing and incentives, this section will now successively consider these elements. It will be shown that they effectively alter the tradeoff, but in opposite ways. The section will end on discussing what happens to incentives when the converse of Definition 2 holds, so the principal is less downside risk averse than the agent. 4.1 Limiting the Agent’s Liability Suppose the agent’s revenue is bounded from below, so he cannot bear very high penalties when performance is bad. Remuneration is frequently subject to this type of constraint.14 An agent with limited wealth, for instance, can file for bankruptcy if he cannot afford paying some penalty. In other contexts, institutions that prevent an agent from breaching his contract under bad circumstances might simply not exist. Without loss of generality, let us then normalize the agent’s minimum revenue to zero. The principal’s optimization problem becomes maxw(s),a ∫s∈Sv(π(s)−w(s))dF(s;a)subject to ∫s∈Su(w(s))dFa(s;a)−c′(a)≥0, (γ1)∫s∈Su(w(s))dF(s;a)−c(a)≥U0, (μ1)w(s)≥0,∀s (λ(s)), (8) where λ(s) is the Lagrange multiplier associated with the nonnegative wage constraint at signal s. The Kuhn–Tucker conditions for a solution to this problem are this time given by the equation v′(π(s)−w(s))u′(w(s))=μ1+γ1fa(s;a)f(s;a)+λ(s)f(s;a)u′(w(s)), ∀s (9) with λ(s)w(s)=0 at all s. Arguments and computations similar to those of Section 3 lead to the following statement. Proposition 1. Suppose that the principal is more downside risk averse than the agent by a factor k. If the agent is protected by limited liability, then: The optimal wage schedule is such that w1(s)=0 for any signal s below a threshold s1 and w1(s) is (δ1(k),0)-concave when s>s1. The number δ1(k) decreases with k and tends to 0 as k grows. For s>s1 and any given k, δ1(k)≤δ(k) so w1(s) is closer to being concave than the incentive scheme w∗(s) obtained in Theorem 1. On the range where pay varies with performance, the proposition describes an optimal incentive scheme analogous to the one outlined in the above Theorem. But part (iii) adds an intuitive feature. Since limited liability shelters the agent from the worst outcomes, the prudent principal can afford having him bear more downside risk across the range where compensation is linked to outcomes. On this range, she then selects a wage schedule that is closer to a concave one than the schedule she would choose under no limited liability constraint.15 The same logic actually stands in a straightforward extension. Suppose that the interval of performance signals s at which w(s) = 0, noted [s ,s1], is now exogenous (it is set by law, say). The bound δ(k,s1) then depends on the upper threshold s1. As the latter goes up, so the range of “poor” circumstances over which the floor wage must prevail expands, one can show (see the proof of Proposition 1) that δ(k,s1) diminishes, so incentive compensation is then closer to being concave in the range where pay varies with performance. 4.2 Taxing the Principal’s Benefits Suppose now that the principal’s net benefits are subject to a constant tax rate θ which applies when they are positive. The incentive scheme chosen by the principal must be a solution to maxw(s),a ∫s∈S2¯v((1−θ)(π(s)−w(s)))dF(s;a)+∫s∈S2̲v(π(s)−w(s))dF(s;a)subject to∫s∈Su(w(s))dFa(s;a)−c′(a)≥0, (γ2)∫s∈Su(w(s))dF(s;a)−c(a)≥U0, (μ2) (10) where S2¯={s∈S;π(s)−w(s)≥0} and S2̲={s∈S;π(s)−w(s)<0} are endogenous sets. The Kuhn–Tucker conditions in this case are given by two equations: (1−θ)v′((1−θ)(π(s)−w(s)))u′(w(s))=μ2+γ2fa(s;a)f(s;a), ∀s∈S2¯ , (11) v′(π(s)−w(s))u′(w(s))=μ2+γ2fa(s;a)f(s;a), ∀s∈S2̲ . (12) The next proposition, which is derived using the same proof arguments as before, expresses how the principal then trades off downside risk sharing and incentives. Proposition 2. Suppose that the principal is more downside risk averse than the agent by a factor k. If a constant tax rate θ applies to the principal’s net benefits when they are positive, then: the optimal wage schedule is (δ2(k),0)-concave when net benefits are negative, where the number δ2(k) gets smaller with k and tends to 0 as k grows; the optimal wage schedule is (δ2(k,θ),0)-concave when net benefits are positive, where the number δ2(k,θ) gets smaller with k and tends to 0 as k grows; δ2(k,θ) increases with θ, so the higher the tax rate the cruder the pay–performance concavity. Statements (i) and (ii) characterize the chosen incentive scheme over the complementary domains where net benefits are respectively negative and positive; in both cases, the optimal wage schedule mirrors the one described in Theorem 1. Part (iii), however, raises the possibility of incentive pay leaning away from concavity. Recall that downside risk aversion makes the principal worry more about the variability of net benefits in bad states than in good states. When positive net benefits are taxed, the principal associates positive net benefits with states which are relatively less good; she then becomes more sensitive to risk at those states. As the tax rate increases, willingness to have the agent bear more of this risk grows, so the pay–performance relationship turns further aside from concavity. 4.3 When the Principal Is Less Downside Risk Averse Than the Agent What happens to incentives if Definition 2 does not hold? Assume, for instance, that the situation described in the next definition, which is the converse of Definition 2, is the one that prevails. Definition 4. For some constant real number k≥1, the principal is said to be less downside risk averse than the agent by a factor k if, for any real numbers x∈Dom(u) and y−x∈Dom(v), we have that k·dv(y−x)<du(x). Would a statement symmetric to Theorem 1 ensue? The formal conclusion would then be that the optimal wage schedule is approximately convex in performance, with the approximation improving as the gap between the principal’s and the agent’s respective downside risk aversion increases.16 The answer is negative. The principal–agent literature usually assumes that the principal is risk neutral, so with a (even slightly) downside risk averse agent Definition 4 is satisfied for any k. In this context, Chaigneau et al. (2017) recently showed that the curvature of the optimal wage schedule with respect to outcome depends on the sum of the likelihood ratio’s curvature ∂2∂s2LF(s;a)∂∂sLF(s;a) and the difference Pu(w(s))−2Ru(w(s)), weighted by the positive wage gradient w′(s), between the agent’s coefficient of absolute prudence and twice his index of absolute risk aversion. With a linear likelihood ratio (so ∂2∂s2LF(s;a)=0), the optimal wage schedule is convex if and only if du(w(s))>2Ru(w(s))2 . (13) Many utility functions, such as the CARA utility function u(x)=− exp (−a·x) and decreasing absolute risk aversion (DARA) utility functions where the coefficient of prudence Pu is moderate, violate this inequality.17 With an agent whose risk preferences are captured by a CARA utility function, and a linear likelihood ratio, the optimal contract is then even concave in performance (Hemmer et al. 2000: 315). This is contrary to what the first example in Section 3.3 might have hinted at. Another limit case which might counter expectations is the one where both the principal and the agent have quadratic preferences, so du(x)=dv(y−x)=0 for all x∈Dom(u) and y−x∈Dom(v). As shown in the Appendix, the optimal wage schedule is then concave. Downside risk sharing does not matter here, of course, but only risk sharing. The risk averse (but downside-risk neutral) principal is now only interested in smoothing her net revenue without mitigating the agent’s incentives. Since the benefit function π(·) is concave in the signal s, an increasing and concave incentive contract allows to do just that. 5. Empirical Considerations An overall concern with principal–agent theory (see, e.g., Allen and Lueck 1999; Prendergast 2002; Chiappori and Salanié 2003; Dittman and Maug 2007; Edmans and Gabaix 2016; among many others) is whether it yields empirically testable and valid predictions. The previous analysis delivered the following propositions: P1a. When the principal is more downside risk averse than the agent, the optimal contract is approximately concave in performance. P1b. As the gap between the principal’s and the agent’s respective downside risk aversion increases, the pay–performance relationship gets closer to concavity. P2. Taxing the principal will worsen the approximation. P3. Limiting the agent’s liability will improve the approximation. We submit that these statements are indeed both testable and potentially useful to explain certain facts. 5.1 Measurability Several standard hurdles to empirically testing principal–agent theory (endogeneity problems, notably) are beyond the scope of this paper. One potential obstacle can be addressed here, however: measurability. First, approximate concavity can formally be related to the composition of incentive contracts. According to Páles (2003, corollary 4), a function g:I→ℝ is (δ,ρ)-concave on I if it can be represented as the sum g=h+ℓ+b, where h:I→ℝ is a concave function, ℓ:I→ℝ is δ/2-Lipschitz continuous, i.e., |ℓ(x)−ℓ(y)|≤δ/2|x−y| for all x,y∈I, and b:I→ℝ is bounded so ∥b∥ ≤ρ/2 on I. This result suggests and supports a practical way to capture the (δ(k),0)-convavity of an incentive contract: simply decompose the contract into its concave component (made of, say, downside penalties, clawbacks, put options, or capped linear bonuses) and its Lipschitz-continuous non-concave one (call options, e.g.). The Lipschitz constant attached to the latter component, which is proportional to the relative weight of this item in the overall pay package, will correspond to half the coefficient δ(k). Regarding downside risk aversion, several measurements for populations of interest already exist in the literature. In a recent article, for instance, Brenner (2015) makes use of option exercising data from 1996 to 2008 to elicit the risk preferences of U.S. executives; as these preferences are presumed to be captured by some CRRA utility function u(x)=x1−α1−α, the estimated parameter α can be used to assess the downside risk aversion coefficient du(x)=(1+α).αx2. Surveys (as in Eisenhauer and Ventura 2003) and experimental methods (as in Ebert and Wiesen 2011) aimed at estimating prudence have also been proposed. Finally, note that a principal’s downside risk aversion is frequently inferred (albeit qualitatively) without explicitly naming it. In the vast empirical literature on CEO compensation, for instance, Murphy (1999) and Frye et al. (2006), respectively, report that regulation and “social responsibility” add to fiduciary duties to ultimately make corporate boards/principals more prudent. In corporate governance, conservatism—the practice of producing finer information at lower earnings levels—can also be associated with downside risk aversion (Kirschenheiter and Ramakrishnan 2010). 5.2 Explaining Incentives Concrete incentive contracts are usually made of various items. In their study of compensation policies in the hotel industry, Freedman and Kosová (2014) find that the composition of contracts has to do with ownership (i.e., whether an individual hotel is company-managed or franchised). Considering hospitals, Roomkin and Weisbrod (1999) report that the relative importance of bonuses versus base salary varies according to whether the organization is a for-profit or a nonprofit one. In his more recent survey of executive compensation, Murphy (2013) attributes the changing structure of CEOs’ pay packages to political factors and government intervention in the form of, notably, tax policies, accounting rules, securities laws, and listing requirements. Most empirical findings on the composition of incentive contracts, however, often lack theoretical support. In a recent review of the executive compensation literature, Edmans et al. (2017: 12–13) acknowledge that18: (…) most executive pay packages contain five basic components: salary, annual bonus, payouts from LTIPs, restricted option grants, and restricted stock grants. In addition, top executives often receive perks, defined-benefit pension plans, and severance payments upon departure. The relative importance of these compensation elements has changed considerably over time. (…) Explaining these drastic changes in the structure of pay since the 1980 s, especially the surge in option pay and their subsequent replacement by (performance-based) restricted stock, remains a challenge.19 [emphasis added] Our claim is that principal–agent models which (i) allow for the predicted pay–performance relationship to approximate rather than fit exactly certain somewhat “pure” patterns (like convexity or concavity), and (ii) draw attention to the agent’s and/or the principal’s higher-order risk preferences (such as prudence), might actually deliver relevant insights.20 A stylized account of the observed changes in CEO compensation would then run as follows. According to Brenner (2015), notwithstanding sharp differences across individuals, industrial sectors and firm characteristics, the risk preferences of U.S. executives have remained relatively stable over time.21 Meanwhile, in the aftermath of the 2001 fraudulent bankruptcies and the 2008 financial debacle, financial distress (Chang et al. 2015), renewed long-term emphasis (Boschen and Smith 2014), and some regulatory measures—such as providing shareholders with “Say-on-Pay” clauses, fostering the presence of females and independent directors on corporate boards, and making directors personally liable—have likely enhanced the prudence of corporate boards.22 As the gap between CEOs’ and boards’ respective downside risk aversion widened, CEOs’ incentive contracts have come closer to concavity (by reducing the relative importance of call options), as statement P1b above predicts. 6. Concluding Remarks When the principal or the agent are prudent (i.e., downside risk averse), an optimal contract trades off downside risk sharing and incentives. This paper first pointed out that the appropriate measure of downside risk aversion to investigate this tradeoff is the coefficient introduced by Modica and Scarsini (2005). This coefficient—the ratio of the third over the first derivative of the utility function—precisely indicates how much someone would require to accept facing greater volatility over bad outcomes. Next, we showed that an optimal incentive package will be approximately concave in the relevant outcomes (in the precise mathematical sense due to Páles 2003), the approximation being closer the more the principal is downside risk averse compared with the agent. Intuitively, approaching a concave function shifts more downside risk upon the agent, since remuneration is then more variable on the whole in adverse circumstances. This result is qualitatively robust to limiting the agent’s liability or taxing the principal, although these common contextual features might affect the approximation in opposite directions (making it finer or coarser, respectively). Our analysis adds to the theoretical literature in providing a first characterization of incentive pay in circumstances when both the principal and the agent can be prudent. The fact that our characterization is qualitatively robust to changes in certain modeling assumptions also answers a major concern with principal–agent theory, that “seemingly innocuous features of the modelling setup, often made for tractability or convenience, can lead to significant differences in the model’s implications” (Edmans and Gabaix 2016: 1232). Finally, as we argued in Section 5, the notion of approximate concavity we introduced might help explain many compensation schemes observed in practice. Such schemes are often compound packages made of various items, some concave (like capped bonuses) and other convex (like stock call options) in outcome; and coming closer to concavity actually means that the non-concave items have become relatively less significant in the overall pay package. All in all, this paper can be seen as fitting a research program that studies how compensation is shaped by plausible attributes of the principal’s preferences. Pursuing such a program might deliver other empirically testable insights on remuneration. It would also allow, in some cases, a normative analysis of incentive pay. In some situations, indeed, certain risk attitudes by the principal may be called for. In health care, for instance, one may take the Hyppocratic oath as a traditional call for medical prudence (see Linden 1999). In corporate governance, one may interpret the fiduciary duty of care that corporate boards/principals must obey as imposing “prudence” on their decision making (see Clark 1985). In these contexts, statements like the ones contained in our theorem and propositions should be seen as requirements (not predictions) on the pay–performance relationship, and those requirements as policy outcomes potentially achievable through influencing the principal instead of regulating compensation directly. Conflict of interest statement. None declared. We thank Claude d’Aspremont, Pierre Chaigneau, Bruno Deffains, Dominique Demougin, Frédéric Deroïan, Dominique Henriet, René Kirkegaard, Bertrand Koebel, Henri Loubergé, David Martimort, Frédéric Robert-Nicoud, Harris Schlesinger, and Marie-Claire Villeval for their valuable remarks and suggestions. Perceptive observations and questions from audiences at the 2012 North American Summer Meeting and the 2015 World Congress of the Econometric Society, CORE-Université Catholique de Louvain, the University of Liverpool, LAMETA-University of Montpellier, the 2014 Canadian Economic Theory conference, École polytechnique-Paris, Aix-Marseille School of Economics (AMSE), the University of Geneva, Washington State University, Paris School of Economics (PSE), the University of Lille, GATE-University of Lyon, the University of Guelph, and “Risk and Choice: A Conference in Honor of Louis Eeckhoudt” at the Toulouse School of Economics (TSE) are gratefully acknowledged. Constructive comments from the Editor, Wouter Dessein, and two competent and careful referees also helped to improve the content and presentation of the paper. This work benefitted from financial support through project CSES of the Initiative of Excellence (IDEX) at the University of Strasbourg. Appendix Proof of Lemma 2. Since we work here with approximately concave rather than approximately convex functions, the proof combines and adapts our notation and context to the arguments underlying theorems 4 and 5 in Páles (2003). Suppose there is a nonincreasing function q:I→ℝ such that g(y)≤g(x)+q(x)(y−x)+δ2|y−x| for all x,y∈I. The latter inequality can be rewritten as q(x)(y−x)≥g(y)−g(x)−δ2|y−x| for all x,y∈I. Take the primitive function of q, which is Q(x)=∫x0xq(t)dt where x0 is an arbitrary fixed element of I. This function Q:I→ℝ is concave since q is non-increasing. Let x=t0<t1<⋯<tn=y be an arbitrary grid on I. Applying the above inequality successively to ti−1 and ti for i=1,…,n and summing up gives ∑i=1nq(ti)(ti−1−ti)≥∑i=1n(g(ti−1)−g(ti)−δ2(ti−ti−1))=g(x)−g(y)−δ2(y−x). If maxi (ti−ti−1) is small enough, the latter implies that Q(x)−Q(y)≥g(x)−g(y)−δ2(y−x). Similarly, apply the above inequality to ti and ti−1 for i=1,…,n and take the sum. This yields ∑i=1nq(ti−1)(ti−ti−1)≥∑i=1n(g(ti)−g(ti−1)−δ2(ti−ti−1))=g(y)−g(x)−δ2(y−x). Hence, if the grid is again fine enough, we have that Q(y)−Q(x)≥g(y)−g(x)−δ2(y−x). Define ℓ:I→ℝ as ℓ(x)=g(x)−Q(x). The upshot is that |ℓ(x)−ℓ(y)| ≤δ2(y−x) for all x<y in I. For t∈[0,1], this conclusion and the triangle inequality entail that tℓ(x)+(1−t)ℓ(y)=ℓ(tx+(1−t)y)+t(ℓ(x)−ℓ(tx+(1−t)y))+(1−t)(ℓ(y)−ℓ(tx+(1−t)y))≤ℓ(tx+(1−t)y)+|t(ℓ(x)−ℓ(tx+(1−t)y))+(1−t)(ℓ(y)−ℓ(tx+(1−t)y))|≤ℓ(tx+(1−t)y)+t|ℓ(x)−ℓ(tx+(1−t)y)|+ (1−t))|ℓ(y)−ℓ(tx+(1−t)y)|≤ℓ(tx+(1−t)y)+tδ2(1−t)|x−y|+(1−t)δ2t|x−y|=ℓ(tx+(1−t)y)+t(1−t)δ|x−y| for all x,y∈I. Now, since Q is concave, we have tQ(x)+(1−t)Q(y)≤Q(tx+(1−t)y). Adding this inequality and the latter one leads to tg(x)+(1−t)g(y)≤g(tx+(1−t)y)+δt(1−t)|x−y| for all x,y∈I and t∈[0,1]. Hence, g is (δ,0)-concave. ♦ Proof of Lemma 3. Risk aversion of at least one player is sufficient to obtain that w∗′(s)≥0. Indeed we have, with v(π(s)−w(s)) denoted as v(·) and u(w(s)) denoted as u(·): ∂∂s(v′(π(s)−w(s))u′(w(s)))=(π′(s)−w′(s))⋅v″(⋅)u′(⋅)−v′(⋅)⋅u″(⋅)⋅w′(s)(u′(·))2=−w′(s)⋅(v″(⋅)⋅u′(⋅)+v′(⋅)⋅u″(⋅))+π′(s)⋅v″(⋅)⋅u′(⋅)u′(⋅)2 The latter must be positive, by Assumption 3, in order to satisfy Equation (3). A necessary condition for this is w′(s)≥0. ♦ Proof of Theorem 1. Let us now compute the second derivative of the left-hand side term in Equation (3). Assumption 3 entails it must be negative. ∂2∂s2(v′(π(s)−w(s))u′(w(s)))=1(u′(·))4·({−w″·(v″u′+v′u″)−w′·[(π′−w′)v‴u′+w′v″u″+(π′−w′)v″u″+w′v′u‴]+π″v″u′+π′[(π′−w′)v‴u′+v″w′u″]}·u′2+2u″u′·w′·[w′(v″u′+v′u″)−π′v″u′])=1u′3·({−w″·(v″u′+v′u″)+(π′−w′)2v‴u′+(π′−w′)w′v″u″−w′2v′u‴+π″v″u′}·u′−(π′−w′)v″u″w′u′(u′+2)+2u″2w′2v′)=1u′3·({−w″·(v″u′+v′u″)+(π′−w′)2v‴u′−w′2v′u‴+π″v″u′}·u′−2w′u″·((π′−w′)v″u′−v′u″w′)=1u′2·[−w″·(v″u′+v′u″)+(π′−w′)2v‴u′−w′2v′u‴+π″v″u′]+2w′Ru·((π′−w′)v″u′−v′u″w′)u′2 The last term here is in fact ∂∂s(v′(π(s)−w(s))u′(w(s))), which must be positive by Equation (3) and Assumption 3. Then: ∂2∂s2(v′(π(s)−w(s))u′(w(s)))=2w′Ru·∂∂s(v′(π(s)−w(s))u′(w(s)))+1u′2·[−w″·(v″u′+v′u″)+(π′−w′)2v‴u′−w′2v′u‴+π″v″u′]=2w′Ru·∂∂s(v′(π(s)−w(s))u′(w(s)))+v′u′·[w″·(Rv+Ru)+(π′−w′)2PvRv−w′2PuRu−π″Rv]=2w′Ru·∂∂s(v′(π(s)−w(s))u′(w(s)))+v′u′·[w″·(Rv+Ru)−π″Rv+(π′−w′)2PvRv−w′2PuRu] The sign of this last expression depends on the sign of (π′−w′)2PvRv−w′2PuRu, which writes explicitly as (π′(s)−w′(s))2Pv(π(s)−w(s))Rv(π(s)−w(s))−w′(s)2Pu(w(s))Ru(w(s))=(π′(s)−w′(s))2·dv(π(s)−w(s))−w′(s)2·du(w(s)) . Two cases are possible. Either (π′(s)−w′(s)w′(s))2≥1k , and 1k≥du(w(s))dv(π(s)−w(s)) by assumption, and Definition 1 makes it necessary that w″(s)<0 (so w is concave at s); or (π′(s)−w′(s)w′(s))2<1k . (A1) Recall that the derivatives w′(s) are locally bounded by assumption. Thus, there exist positive real numbers Mi associated with each Di, i=1,…,n, of a finite partition D1,D2,…,Dn of S such that w′(s)<Mi on Di. Let M= maxiMi and take δ(k)>0 so that (δ4M)2=1k. Inequality (A1) is equivalent to (π′(s)−w′(s))2<(δ(k)4M)2(w′(s))2 . For all x∈S, x≠s, then: |(π′(s)−w′(s))(x−s)| <(δ(k)4M)w′(s)|x−s| < δ(k)4|x−s|. Hence, w′(s)(x−s) <π′(s)(x−s)+δ(k)4|x−s|. Now, since w is differentiable at s, we have that w(x)=w(s)+w′(s)(x−s)+r(x) with the residual r(x) satisfying limx→s r(x)x−s=0. The last inequality entails that w(x)<w(s)+π′(s)(x−s)+(r(x)|x−s|+δ(k)4)|x−s|≤w(s)+π′(s)(x−s)+δ(k)2|x−s| if x is sufficiently close to s. Since π′(s) is decreasing in s by assumption, applying Lemma 2 yields that w(s) is (δ(k),0)-concave on a subinterval of S that contains s. Since this is to be true at any point s, keeping the same number δ(k), then w∗(s) is (δ(k),0)-concave on S. This proves assertion (i). Assertion (ii) is immediate when considering the above construction of δ(k), as (δ4M)2=1k. ♦ Proof of Proposition 1. The conclusion of Lemma 3 still holds here, so the optimal wage schedule w1(s) must be increasing in its argument s. There is therefore a threshold s1 such that w1(s)=0 for s≤s1 and w1(s)>0 for s>s1. For any signal s≤s1, the limited liability constraint is binding so w(s) = 0. For s>s1, we have λ(s)=0; Equation (9) is then similar to Equation (3), and the proof of the Theorem can be replicated to conclude that w(s) is (δ1(k),0)-concave. This proves Assertion (i) of the proposition. Assertion (ii) is immediate since δ1 satisfies (δ14M′)2=1k. To show (iii), notice that the proof of (i) uses the bound M′= maxDi ∩ S1 ≠ ØMi, considering only the neighborhoods Di having a nonempty intersection with S1={s∈S ; s>s1}. Clearly, M′≤M= maxall DiMi. Hence, (δ4M)2=1k=(δ14M′)2 implies that δ1(k)≤δ(k) for any given k. ♦ Proof of Proposition 2. When s∈ S2̲, net benefits are negative so the principal is not taxed. Theorem 1 then applies and the optimal wage schedule w2(s) is (δ2(k),0)-concave at s, where the number δ2(k) decreases with k and tends to 0 as k grows. This proves assertion (i). Now, for any s∈S2¯, ∂∂s(v′((1−θ)(π(s)−w(s)))u′(w(s)))=(1−θ)(π′−w′)v″u′−v′u″w′u′2=−w′(1−θ)(v″u′+v′u″)+(1−θ)π′v″u′u′2=v′(Ruw′−CRv)u′ (A2) where C(s)=∂∂s[(1−θ)(π(s)−w(s))]=(1−θ)(π′(s)−w′(s)). From Equation (A2), w′(s)>0 is a necessary condition for ∂∂s(v′((1−θ)(π(s)−w(s)))u′(w(s)))>0. Computation of the second derivative gives ∂2∂s2(v′((1−θ)(π(s)−w(s)))u′(w(s)))=C′v″+C2v‴+(Ru′v′+RuCv″)w′+Ruv′w″u′−(Cv″+Ruv′w′)u″w′(u′)2=C′v″+C2v‴+(Ru′v′+RuCv″)w′+Ruv′w″+(Cv″+Ruv′w′)Ruw′u′=v′(C2RvPv−C′Rv)+(Ru′−RuCRv)v′w′+Ruv′w″+(Ruw′−CRv)Ruw′v′u′=v′u′.(C2RvPv−C′Rv+(Ru′−RuCRv)w′+Ruw″+(Ruw′−CRv)Ruw′)=v′u′.[Rv(C2Pv−C′)+(Ru′−RuCRv)w′+Ru(w″+(Ruw′−CRv)w′)], (A3) where dRuds=Ru′=−w′(du−Ru2). The component within brackets of Equation (A3) reduces to Rv(C2Pv−C′)+(R′u−RuCRv)w′+Ru(w″+(Ruw′−CRv)w′)=C2dv−C′Rv−w′2du+w′2Ru2−RuCRvw′+Ruw′·(Ruw′−CRv)+Ru·w″=((1−θ)2(π′−w′)2·dv−w′2du)+2w′·Ru·(Ruw′−CRv)−(1−θ)π″Rv+((1−θ)Rv+Ru)·w″. (A4) The second term in Equation (A4) is equal to 2Ruw′(s)u′(s)v′(s).∂∂s(v′((1−θ)(π(s)−w(s)))u′(w(s))) which is positive. The third term is positive by assumption. Hence, the sign of w″(s) for any s∈S2¯ depends on the sign of the first term, (1−θ)2(π′−w′)2⋅dv−w′2du. As in the proof of the theorem, two cases are possible. Either ((1−θ)⋅(π′(s)−w′(s))w′(s))2≥1k , and, since 1k>dudv, the expression (1−θ)2(π′−w′)2⋅dv−w′2du is positive. It is then necessary that w″(s)<0, so w2 is concave at s∈S2¯. In the opposite situation, ((1−θ)⋅(π′(s)−w′(s))w′(s))2<1k . This inequality can be written (π′(s)−w′(s)w′(s))2<1k·(1−θ)2 . Let M″= maxDi ∩ S2 ¯≠ ØMi, and take δ2(k,θ)>0 so that (δ2(k,θ)4M″)2=1k·(1−θ)2 . (A5) The latter inequality reduces to (π′(s)−w′(s))2<(δ2(k,θ)4M″)2(w′(s))2 . For all x∈S2¯, x≠s, it follows that |(π′(s)−w′(s))(x−s)| <(δ2(k,θ)4M″)w′(s)|x−s| <δ2(k,θ)4|x−s|, so w′(s))(x−s) <π′(s)(x−s)+δ2(k,θ)4|x−s|. Since w is differentiable at s, w(x)=w(s)+w′(s)(x−s)+r(x) with the residual r(x) such that limx→s r(x)x−s=0. The last inequality entails that w(x)≤w(s)+π′(s)(x−s)+(r(x)|x−s|+δ2(k,θ)4)|x−s|≤w(s)+π′(s)(x−s)+δ2(k,θ)2|x−s| if x is sufficiently close to s. Since π′(s) is decreasing in s by assumption, applying Lemma 2 yields that w2(s) is (δ2(k,θ),0)-concave in s∈S2¯. This shows assertion (ii). To demonstrate assertion (iii), note that, as it is defined in Equation (A5), δ2(k,θ) must increase with θ. ♦ Proof that u‴(x)≡0and v‴(y−x)≡0entails w∗(·)concave: Suppose the principal and the agent have quadratic utility functions, so they are downside-risk neutral and u‴(x)=0 for x∈Dom(u), v‴(y−x)=0 for y−x∈Dom(v). The second-order derivative shown in the proof of Theorem 1 now simplifies into ∂2∂s2(v′(π(s)−w(s))u′(w(s)))=2w′Ru∂∂s(v′(π(s)−w(s))u′(w(s)))+v′u′[w″(Rv+Ru)−π″Rv] Since π″≤0 by assumption and ∂∂s(v′(π(s)−w(s))u′(w(s)))≥0 by Lemma 3, this derivative can only be negative if w″≤0, so the optimal wage schedule must be concave. ♦ Footnotes 1. Quoting Adams et al. (2010: 58): “People often question whether corporate boards matter because their day-to-day impact is difficult to observe. But when things go wrong, they can become the center of attention. This was certainly true for the Enron, Worldcom, and Parmalat scandals. The directors of Enron and Worldcom, in particular, were held liable for the fraud that occurred: Enron directors had to pay $168 million to investor plaintiffs, of which $13 million was out of pocket (not covered by insurance); and Worldcom directors had to pay $36 million, of which $18 million was out of pocket.” 2. It is well-known that decreasing absolute risk aversion implies prudence. However, prudence is orthogonal to risk aversion: Crainich et al. (2013) point out that “even risk lovers can be prudent,” and Deck and Schlesinger (2014) report experimental evidence of this. 3. It can be shown that prudence implies skewness seeking, but not the other way around. Experimental evidence of this can be found in Ebert and Wiesen (2011). 4. Notwithstanding the numerous related pieces considering the use of specific devices, like shares, options, bonuses, etc. 5. Aside from risk preferences, features which researchers have examined that have an influence on the shape of incentives include monitoring systems (formally represented by likelihood ratio distributions of outcomes) and regulatory constraints (such as limited liability). 6. This was also seen (yet not explained) by Belhaj et al. (2014), in a related but different context. 7. Incidentally, note that neither the prudence coefficient indices Pu, Pv nor the downside risk aversion coefficients du, dv retain similar global properties as the corresponding Arrow–Pratt measures of risk aversion. An alternative index which does increase under monotonic downside risk averse transformations of the utility function was proposed by Keenan and Snow (2002, 2016). Nevertheless, one can show that du, dv increase as their respective utility functions u, v become more concave while the marginal utilities u′, v′ are more convex (Crainich and Eeckhoudt 2008). 8. Páles (2003) defines and works with approximately convex functions. One has to be careful, because some of his results do not straightforwardly carry over after reversing the inequality sign. 9. Let X be a convex subset of ℝn. Recall that a function g:X→ℝn is quasi-concave on X if g(tx+(1−t)y)≥min(g(x),g(y)) for all x,y∈X and t∈[0,1]. Clearly, when n = 1, any monotone increasing function (even a convex one) is quasi-concave. 10. Note that the direct converse of Lemma 2 is not true, so this lemma does not convey an if-and-only-if characterization of (δ, 0)-concave functions. See Remark 3 in Páles (2003: 250) for details. 11. These distributions can be transformed into ones that have compact support and satisfy the previous assumptions by suitably reallocating the probability mass of the tails. 12. The optimal wage gradient w′(s) is of course endogenous. As Equation (3) shows, however, it is proportional to the sensitivity of the likelihood ratio LF(s;a)=fa(s;a)f(s;a) with respect to s, which indicates how accurately the principal can infer the agent’s effort (see, e.g., Kim 1995). 13. To be clear, the term “closer” we have used so far, and will be using later on, does not refer to greater curvature, but to being more like some concave function (which turns out here to be an affine transformation of the principal’s benefit function). 14. Hence, since Holmström (1979) and especially Sappington (1983)’s seminal works, analyzing the impact of an agent’s limited liability remains a rather well-covered topic in the principal–agent literature. For a recent account of this literature, see Poblete and Spulber (2012). In most articles, both the principal and the agent are assumed to be risk neutral. 15. One possible way to implement the incentive contract which is proposed here might be to use “collar options.” A collar option grants a put option to an agent/employee owning company stocks, thereby sheltering him from downside risk, while subjecting these stocks to a call option from the principal, imposing thereby a cap on the agent’s earnings. In a recent paper, Carey and Sun (2015) analyze the feasibility of such a scheme. 16. From Páles (2003): let I be a subinterval of the real line ℝ and δ,ρ some nonnegative real numbers, a function g:I→ℝ is called (δ,ρ)-convex on I if g(tx+(1−t)y)≤tg(x)+(1−t)g(y)+δt(1−t)|x−y|+ρ for all x,y∈I and t∈[0,1]. 17. If u(·) is a CARA utility function, we have du(x)=Ru(x)2. If u(·) is a DARA utility function, then Pu(x)=−u‴(x)u″(x)>−u″(x)u′(x)=Ru(x), which is necessary but not sufficient for Condition (13) to hold. 18. Frydman and Jenter (2010: 82) come to the same conclusion in their own earlier survey: “Explaining the stark changes in the structure of compensation since the 1980s remains an open challenge. (…) This intriguing shift away from options has not yet attracted much attention in the academic literature, and further research is needed to determine its causes and consequences.” 19. LTIP stands for “long-term incentive plans.” 20. 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