Pareto Gains from Limiting Compensation Options

It is well known that when a worker is privately informed about his ability to perform a key task, an employer often can increase her expected profit by offering the worker a choice among compensation schedules. Through careful design of the set of possible compensation schedules, the employer can reduce the rent the worker secures when he has high ability. The purpose of this research is to demonstrate that a very different conclusion can arise under arguably plausible conditions. Specifically, an employer can sometimes enhance her expected profit by implementing a single payment structure policy (SPP) that precludes the employer from offering the worker any choice among compensation schedules. This restriction also benefits the worker, so it can generate Pareto gains.

Pareto gains can arise from implementing an SPP when the worker's ability is endogenous. Implementing an SPP reduces the employer's ability to limit the rent the worker secures when he has high ability. The SPP thereby enhances the worker's incentive to undertake ability-enhancing investment in human capital. The resulting increase in the worker's expected ability increases the maximum potential expected total surplus, which can increase the employer's expected profit despite her more limited ability to control the agent's rent.

To induce the worker to labor diligently in our model, the employer links the worker's payment to his realized performance, as under a profit-sharing arrangement, for example. In the absence of an SPP, the employer can offer the worker a choice among profit-sharing plans. To illustrate, the worker might be permitted to choose between a plan with limited profit sharing and a plan with substantial profit sharing. The former plan might set a relatively high base wage, but offer limited potential for additional financial reward, even when the worker's performance is exceptional. The latter plan might set a relatively low base wage, but offer substantial potential for additional compensation in the event of strong performance. The additional compensation might reflect the impact of stock options, sales commissions, or performance bonuses, for example.

In practice, many employers offer their employees little or no choice among compensation schedules. However, there are exceptions. To illustrate, some companies offer their workers a choice regarding the fraction of their compensation that is fixed and the fraction that varies with their individual performance (e.g., sales commissions) or with the company's performance (e.g., stock options).

See Hallock and Olson (2009) and Bommaraju and Hohenberg (2018), for example.

Offering workers a choice among compensation schedules does not always serve primarily to limit the rent that a worker with high ability commands from his private information, as it does in our model.

See Tropman (2001) and Zoltners et al. (2006), for example.

Many other studies also examine the optimal design of reward schedules in the presence of both moral hazard and adverse selection. However, these studies generally take the distribution of the worker's ability to be exogenous. For example, Escobar and Pulgar (2017) (EP) demonstrate that although an SPP always reduces the employer's expected profit when the distribution of the worker's ability is exogenous, the reduction can be relatively small under plausible conditions.

Reichelstein (1992), Bower (1993), McAfee (2002), and Rogerson (2003) provide related conclusions. Whereas these studies report that the principal's loss from employing a single reward structure can be relatively small, we demonstrate that the principal can gain from committing herself to offer only a single reward structure.

Lundberg (1991)'s analysis is more similar to ours in the sense that both admit endogenous worker ability. An imperfect signal of a worker's ability is available in Lundberg's analysis. If this signal is more accurate for workers of one type than for workers of another type, then the latter workers will undertake less ability-enhancing investment in human capital. This underinvestment distortion can be eliminated by requiring employers to implement the same relationship between wages and measured ability for all workers. However, the requirement can reduce welfare by limiting the ability of employers to match job assignments to workers abilities. Our analysis does not include differentiated tasks or signals about worker ability with accuracies that vary across worker types. In our model, employers can observe perfectly the expected ability that each worker secures via human capital investment. However, each worker ultimately becomes privately informed about his actual ability (and only the worker observes the effort he delivers to enhance his performance).

Our analysis proceeds as follows. Section 2 describes the key elements of our model. Section 3 identifies the payment structure the employer would implement in the absence of adverse selection. Section 4 examines the effects of an SPP when worker ability is exogenous in the setting with adverse selection and moral hazard. Section 5 extends the analysis of Section 4 to the setting where worker ability is endogenous. Section 6 concludes. The Appendix provides the proofs of all formal conclusions.

An employer (or “principal”) hires a worker (an “agent”) to operate a project. The project either succeeds and generates payoff x¯ \bar x or fails and generates payoff x_(<x¯) {\underline x} ( < \bar x) . The project succeeds with probability p ∈ [0,1]. The agent can increase p by exerting personally costly effort. C (p, θ_i) ≥ 0 is the agent's cost of ensuring success probability p when his ability is θ_i ∈ {θ_L < θ_H}, where θ_L < θ_H, We assume Cp,θi=p22θi for θi∈θL,θH C\left( {p,{\theta _i}} \right) = \frac{{{p^2}}}{{2{\theta _i}}}\;{\rm{for}}\;{\theta _i} \in \left\{ {{\theta _L},{\theta _H}} \right\} ,

This assumption is stronger than necessary to prove Lemmas 1–3 and Proposition 1 below. As is apparent from their proofs, these findings all hold if C(p, θ_H) < C(p, θ_L), C_p(p, θ_H) < C_p(p, θ_L), C_pp(p, θ_H) < C_pp(p, θ_L), and C_ppp(p, θ_i) is sufficiently close to 0 for all p ∈ (0,1), for i = L, H.

The agent's ability is the realization of a binary random variable. The principal knows that the agent's ability is θ_L with probability ϕ_L ∈ (0,1), and θ_H with probability ϕ_H, where ϕ_L + ϕ_H = 1. These probabilities are exogenous parameters in the setting with exogenous ability. The agent learns his ability before he interacts with the principal. In contrast, the principal never observes the agent's ability.

Thus, in the present setting, factors beyond the agent's control determine whether his ability when working for the principal is high or low. The principal understands the stochastic process that determines the agent's ability (i.e., the principal knows ϕ_L and ϕ_H). However, only the agent knows the outcome of the stochastic process. Section 4 considers a different setting in which the agent can in influence θ_H, the likelihood that he ultimately has high ability when working for the principal.

When she is not precluded from doing so, the principal optimally offers the agent a choice between two payment structures, w_L,w¯L \left( {{{\underline w}_L},{{\bar w}_L}} \right) and w_H,w¯H \left( {{{\underline w}_H},{{\bar w}_H}} \right) . It is convenient to assume the agent chooses payment structure w_j,w¯j \left( {{{\underline w}_j},{{\bar w}_j}} \right) by reporting his ability to be θ_j ∈ {θ_L, θ_H}. w¯j {\bar w_j} is the payment the principal delivers to the agent when the project succeeds after the agent reports his ability to be θ_j. w_j {\bar w_j} is the corresponding payment when the project fails.

We associate the agent's choice of a payment structure with a report of his ability for expositional ease. Our findings are unchanged if the principal simply offers the agent a choice between two payment structures, w_L,w¯L, w_H,w¯H \left\{ {\left( {{{\underline w}_L},{{\bar w}_L}} \right),\;\left( {{{\underline w}_H},{{\bar w}_H}} \right)} \right\} , and commits to implement the payment structure the agent chooses.

The principal seeks to maximize her expected profit, which is the difference between the expected project payoff and her expected payment to the agent. Formally, the principal's problem, [P], is: (3) Maximizew_i, w¯i∑i∈L,Hϕipix¯−w¯i+1−pix_−w_i \mathop {{\rm{Maximize}}}\limits_{{{\underline w}_i},\;\;{{\bar w}_i}} \sum\nolimits_{i \in \left\{ {L,H} \right\}} {{\phi _i}\left\{ {{p_i}\left[ {\bar x - {{\bar w}_i}} \right] + \left[ {1 - {p_i}} \right]\left[ {{\underline x} - {\underline w_i}} \right]} \right\}} subject to, for i, j ∈{L, H}(j ≠ i): (4) Uθi,θi≥0 and U\left( {{\theta _i},{\theta _i}} \right) \ge 0\;{\rm{and}} (5) Uθi,θi≥Uθj,θi U\left( {{\theta _i},{\theta _i}} \right) \ge U\left( {{\theta _j},{\theta _i}} \right) where pi≡ argmaxp∈0,1 uip,θi {p_i} \equiv {\rm{\;}}\mathop {{\rm{arg\;max}}}\limits_{p \in \left[ {0,1} \right]} {\rm{\;}}{u_i}\left( {p,{\theta _i}} \right) .

The participation constraint, (4), ensures the agent secures nonnegative expected utility when he reports his ability truthfully. The incentive compatibility constraint, (5), ensures the agent prefers to report his ability truthfully than to misrepresent it. The revelation principle (e.g., Myerson, 1979) ensures this formulation of the principal's problem is without loss of generality.

For expositional ease and to avoid uninteresting “corner solutions”, we restrict attention to model parameters that ensure the principal optimally contracts with the agent and induces him to deliver an “interior” success probability (i.e., one that is strictly positive and strictly less than 1) both when θ = θ_L and when θ = θ_H. After agreeing to work for the principal, the agent with ability θ ∈ {θ_L, θ_H} will implement a strictly positive success probability if C_p (0, θ_i) = 0. This equality holds when Cp,θi=p22θi C\left( {p,{\theta _i}} \right) = \frac{{{p^2}}}{{2{\theta _i}}} . The agent's preferred success probability will be strictly less than 1 if C_p (1, θ_i) exceeds the incremental reward the agent receives for success rather than failure. This will be the case if CP1, θi>x¯−x_ {C_P}\left( {1,\;{\theta _i}} \right) > \bar x - {\underline x} (see Lemmas 2 and 3 below). The principal will optimally attract the agent and induce him to deliver an interior success probability when θ= θ_L (and when θ = θ_H) if ϕ_L is sufficiently close to 1 and θ_H – θ_L is sufficiently close to 0.

The interaction between the principal and the agent in the setting with exogenous ability proceeds as follows. First, the agent (privately) learns his ability. Then the principal specifies the payment structures from which the agent can choose by reporting his ability. Next, the agent reports his ability and then chooses the (unobserved) success probability. Finally, the project performance is observed publicly and the principal delivers the promised payment to the agent. The interaction between the principal and the agent is not repeated. This timing is summarized in Figure 1.

Figure 1

The same timing prevails in the presence of a single payment structure policy (SPP), i.e., when the principal can only offer a single payment structure to the agent. However, the second and third steps in the timing are trivial when an SPP prevails. The principal specifies a single payment structure and the agent simply decides whether to accept the single payment structure or terminate his relationship with the principal. Consequently, in the presence of an SPP, the principal's problem, [PS], is problem [P] with the additional restrictions: (6) w¯H=w¯L and ​w_H=w_L. {\bar w_H} = {\bar w_L}\;{\rm{and}}\;{{\underline w}_H} = {{\underline w}_L}.

As a benchmark, consider the payment structure the principal would implement in the full information setting, where she shares the agent's knowledge of his ability (θ_i) from the outset of their interaction.

In the full information setting, the principal awards the agent the full incremental value of success, i.e., she sets w¯i−w_i=x¯−x_ {\bar w_i} - {{\underline w}_i} = \bar x - {\underline x} . Doing so induces the agent to implement the success probability that maximizes expected total surplus, i.e., pis≡ argmaxp∈0,1 px¯+1−px_−Cp,θi p_i^s \equiv {\rm{\;}}\mathop {\arg \max }\limits_{p \in \left[ {0,1} \right]} {\rm{\;}}\left\{ {p\bar x + \left[ {1 - p} \right]{\underline x} - C\left( {p,{\theta _i}} \right)} \right\} . The principal also maximizes her expected profit by setting the payments to eliminate the agent's rent.

Now consider the setting of primary interest where the agent is privately informed about his ability from the outset of his interaction with the principal. The payment structures the principal implements in this setting in the absence of an SPP are characterized in Lemma 2. The lemma refers to: (i) Δx≡x¯−x_ {\Delta _x} \equiv \bar x - {\underline x} , the incremental payoff from project success; and (ii) Δj≡w¯j−w_j for i∈{1,2} {\Delta _j} \equiv {\bar w_j} - {{\underline w}_j}\;{\rm{for}}\;i \in \left\{ {1,2} \right\} , the agent's incremental reward for success when he reports his ability to be θ_j. Throughout the ensuing analysis, we will refer to the agent as “the high-ability agent” when his ability is θ_H, and as the “low-ability agent” when his ability is θ_L.

Lemma 2 reflects standard considerations.

These considerations arise in the aforementioned models of Escobar and Pulgar (2017) and Balmaceda (2020). Also see Ollier and Thomas (2013), Gary-Bobo and Trannoy (2015), and Rietzke and Chen (2020), for example.

In the presence of an SPP, the principal can only implement a single payment structure, w_,w¯ \left( {{\underline w},\bar w} \right) , where w_ {\underline w} (respectively, w¯ \bar w ) is the payment the agent receives when the project fails (respectively, succeeds). An SPP thereby makes it more costly for the principal to reduce the incremental reward for success, Δ≡w¯−w_ \Delta \equiv \bar w - {\underline w} , below Δ_x in order to reduce the rent of the high-ability agent. The reduction in Δ reduces expected total surplus below its maximum possible level both when the agent has low ability and when he has high ability. Consequently, in the presence of an SPP, the principal increases Δ above Δ_L (the incremental reward for success the low-ability agent selects in the absence of an SPP). The principal sets w_ {\underline w} to ensure U (θ_L), the expected utility of the low-ability agent, is zero.

Formally, Uθi≡ maximump∈0,1p w¯+1−pw_−Cp, θi for θi∈θL,θH U\left( {{\theta _i}} \right) \equiv {\rm{\;}}\mathop {{\rm{maximum}}}\limits_{p \in \left[ {0,1} \right]} \left\{ {p\;\bar w + \left[ {1 - p} \right]{\underline w} - C\left( {p,\;{\theta _i}} \right)} \right\}\;{\rm{for}}\;{\theta _i} \in \left\{ {{\theta _L},{\theta _H}} \right\} .

Lemmas 2 and 3 underlie the conclusions in Proposition 1. The proposition reports that in the setting with exogenous ability, the presence of an SPP: (i) does not alter the expected utility of the low-ability agent; (ii) increases the expected utility of the high-ability agent; and (iii) reduces the principal's expected profit. The proposition refers to E{Π}, the maximum expected profit the principal can secure in the absence of an SPP, and to E{Π_s}, the principal's corresponding maximum expected profit in the presence of an SPP.

Formally, E{Π} is the value of expression (3) at the solution to [P]. E{Π_S} is the corresponding value of the principal's objective function at the solution to [PS].

An SPP increases the expected utility of the high-ability agent in the setting with exogenous ability because Δ > Δ_L. The larger incremental reward for success that the principal implements in the presence of an SPP generates greater rent for the high-ability agent. An SPP reduces the principal's expected profit by restricting her ability to limit the rent of the high-ability agent. This restricted ability compels the principal to deliver more rent to the high-ability agent and induces the principal to reduce expected total surplus below its maximum feasible level when the agent has high ability.

An SPP can either increase or reduce expected welfare. Proposition 2 identifies conditions under which an SPP increases expected welfare by increasing the agent s expected utility more than it reduces the principal's expected profit.

To understand the qualitative conclusions in Proposition 2, recall that an SPP increases expected welfare when the agent has low ability and reduces expected welfare when the agent has high ability (because Δ_H = Δ_x from Lemma 2 and Δ ∈(Δ_L, Δ_x) from Lemma 3). Therefore, an SPP increases expected welfare when θ_L is relatively likely, i.e., when ϕ_H is sufficiently small.

Also, when θHθL \frac{{{\theta _H}}}{{{\theta _L}}} is sufficiently large, the principal often sets Δ_L well below Δ_x in the absence of an SPP to limit the agent's rent. An SPP can induce the principal to increase Δ considerably above Δ_L in this case to avoid an excessive reduction in total expected surplus when the agent has high ability. The corresponding increase in expected total surplus when the agent has low ability increases expected welfare.

The analysis to this point has taken the agent's ability (θ) to be exogenous. We now consider the possibility that the agent might exert personally costly effort to increase the likelihood that his (unobserved) ability is high (θ_H). This effort might entail working diligently to develop or hone skills that enhance the agent's expected performance in working for the principal, for example.

Formally, in the present setting with endogenous ability, the agent can undertake a costly investment in human capital before he interacts with the principal. The more the agent invests in his human capital, the greater is ϕ_H, the probability that he ultimately will be a high-ability agent. We will denote by K (ϕ_H) ≥ 0 the personal cost the agent must incur to ensure he ultimately becomes a high-ability agent with probability ϕ_H. K (·) is a strictly increasing, strictly convex function of ϕ_H. We assume K (·) is such that the agent always implements ϕ_H > 0.

Sufficient conditions are K(0) = 0 and limϕH→0K′ϕH=0 \mathop {\lim }\limits_{{\phi _H} \to 0} K'\left( {{\phi _H}} \right) = 0 We continue to assume Cp,θI=p22θI for i∈L,H C\left( {p,{\theta _I}} \right) = \frac{{{p^2}}}{{2{\theta _I}}}\;{\rm{for}}\;i \in \left\{ {L,H} \right\} in the setting with endogenous ability.

The principal observes the agent's expected ability (by observing ϕ_H) before she designs the payment structure(s) for the agent. However, the principal does not observe the agent's actual ability, which the agent learns (privately) before the principal specifies the reward structure(s) under which the agent can operate. Consequently, the information structure in the present setting with endogenous ability parallels the structure in the setting with exogenous ability. The key difference in the present setting is that ϕ_H, and thus the agent's expected ability, is endogenous.

The timing of the interaction between the principal and the agent in the setting with endogenous ability is summarized in Figure 2.

Figure 2

Although the principal observes the realized value of ϕ_H in the setting with endogenous ability, this value is not contractible.

Formally, when ϕ_H is not contractible, an entity that might enforce the terms of a contract between the principal and the agent (e.g., a court) cannot observe the realized value of ϕ_H.

Although the principal cannot link payments to ϕ_H in the present setting, she can link payments to the realized project performance and to the agent's reported ability, just as in the setting with exogenous ability. Let Ũ(θ_i ;ϕ_H) (respectively, Ũ_S (θ_i ;ϕ_H)) denote the agent's expected utility at the solution to [P] (respectively, [PS]) when his realized ability is θ_i (i ∈{L, H}) and when ϕ_H is the ex ante probability that he will ultimately be a high-ability agent. Then in the setting with endogenous ability, in the absence of an SPP, the agent implements:

In essence, the agent becomes the Stackelberg leader in the setting with endogenous ability, choosing his preferred point on the principal's reaction function through his choice of ϕ_H.

In the presence of an SPP in this setting, the agent implements: ϕHS*=argmaxϕHϕH U˜SθH,ϕH+1−ϕHU˜SθL;ϕH−KϕH. \phi _{HS}^* = \mathop {\arg \max }\limits_{{\phi _H}} \left\{ {{\phi _H}\;{{\tilde U}_S}\left( {{\theta _H},{\phi _H}} \right) + \left[ {1 - {\phi _H}} \right]{{\tilde U}_S}\left( {{\theta _L};{\phi _H}} \right) - K\left( {{\phi _H}} \right)} \right\}.

When the agent's ability is endogenous, an SPP can motivate the agent to increase his expected ability (by choosing a higher value for ϕ_H, the probability that he will ultimately be a high-ability agent). This is the case because an SPP induces the principal to implement a payment structure that increases the difference between the equilibrium expected utility of the high-ability agent and the low-ability agent. (Recall Proposition 1.) This increased incremental reward for securing high ability induces the agent to increase his expected ability, as Lemma 4 reports.

An SPP also increases the agent's expected utility in the setting with endogenous ability. It does so by limiting the principal's ability to extract the rent of the high-ability agent. This conclusion is recorded formally in Lemma 5. The lemma refers to EU (ϕ_H), the agent's ex ante expected utility in the setting with endogenous ability in the absence of an SPP when the agent implements probability ϕ_H. The lemma also refers to EU_S (ϕ_H), the agent's corresponding expected utility in the presence of an SPP. Formally: EUϕH≡ϕH U˜θH;ϕH+1−ϕHU˜θL;ϕH−KϕH andEUSϕH≡ϕH U˜SθH;ϕH+1−ϕHU˜SθL;ϕH−KϕH. \begin{array}{*{20}{l}}{EU\left( {{\phi _H}} \right) \equiv {\phi _H}\;\tilde U\left( {{\theta _H};{\phi _H}} \right) + \left[ {1 - {\phi _H}} \right]\tilde U\left( {{\theta _L};{\phi _H}} \right) - K\left( {{\phi _H}} \right)\;{\rm{and}}}\\{E{U_S}\left( {{\phi _H}} \right) \equiv {\phi _H}\;{{\tilde U}_S}\left( {{\theta _H};{\phi _H}} \right) + \left[ {1 - {\phi _H}} \right]{{\tilde U}_S}\left( {{\theta _L};{\phi _H}} \right) - K\left( {{\phi _H}} \right).}\end{array}

The impact of an SPP on the principal's expected profit is less apparent because an SPP introduces two countervailing effects. First, an SPP constrains the principal's ability to extract the agent's rent, which serves to reduce the principal's expected profit. Second, an SPP effectively enables the principal to commit to deliver additional rent to the high-ability agent. The prospect of greater rent induces the agent to increase his expected ability, thereby increasing the expected efficient surplus.

The expected efficient surplus is ϕ_L S (θ_L)+ ϕ_HS (θ_H), where Sθi≡ maxp p x¯+1−px_−Cp,θi for i∈L,H S\left( {{\theta _i}} \right) \equiv {\rm{\;}}\mathop {\max }\limits_p {\rm{\;}}\left\{ {p\;\bar x + \left[ {1 - p} \right]{\underline x} - C\left( {p,{\theta _i}} \right)} \right\}\;{\rm{for}}\;i \in \left\{ {L,H} \right\} .

For emphasis, the Pareto gains identified in Lemmas 5 and 6 are summarized in Proposition 3.

Proposition 3 can be viewed an illustration of the principle of the second-best (Lipsey and Lancaster, 1956): in the presence of one friction (the noncontractibility of ϕ_H), an additional friction (an SPP) can sometimes enhance aggregate expected welfare. In addition to increasing expected welfare in the present setting, an SPP can sometimes generate Pareto gains by increasing both the agent's expected utility and the principal's expected profit.

If the principal could explicitly link the agent's payment to his expected ability (ϕ_H), she would not gain by restricting herself to an SPP. This restriction would limit her ability to control the rent of the high-ability agent without enhancing her ability to secure her preferred level of ϕ_H.

If ϕ_H were contractible, the principal would (in addition to implementing the optimal payment structures, given ϕ_H) simply promise to pay the agent KϕHP K\left( {\phi _H^P} \right) if and only if he sets ϕH=ϕHP {\phi _H} = \phi _H^P , where ϕHP \phi _H^P is the level of ϕ_H preferred by the principal.

Lemma 6 and Proposition 3 indicate that an SPP increases the principal's expected profit and therefore generates Pareto gains when the potential variation in the agent's ability is sufficiently pronounced. As θHθL \frac{{{\theta _H}}}{{{\theta _L}}} increases, the rent the high-ability agent can secure under an SPP increases. The prospect of securing greater rent when he has high ability induces the agent to increase ϕ_H. The corresponding increase in the expected efficient surplus enhances the potential for the principal to secure increased expected profit under an SPP, despite her more restricted ability to limit the agent's rent.

It remains to better understand the meaning of the term “sufficiently large” in Proposition 3. It can be shown that Pareto gains arise if θHθL≥max2.8,10.212K′0.5θLx¯−x_2 \frac{{{\theta _H}}}{{{\theta _L}}} \ge {\rm{\;max\;}}\left\{ {2.8,\frac{{10.212K'\left( {0.5} \right)}}{{{\theta _L}{{\left[ {\bar x - {\underline x} } \right]}^2}}}} \right\} .

See Pal et al. (2022) for a formal proof of this conclusion. This condition is discussed further below.

Table 1 reports that for the specified parameter values, an SPP increases the agent's expected utility in Example 1 by between 4% (when θ_H = 0.30) and 140% (when θ_H = 0.80). An SPP reduces the principal's expected profit when θ_H = 0.30, but increases her expected profit when θ_H ≥ 0.35.

In Example 1, 10.212K′0.5θLx¯−x_2=10.2120.10.50.2512=2.0424 \frac{{10.212K'\left( {0.5} \right)}}{{{\theta _L}{{\left[ {\bar x - {\underline x} } \right]}^2}}} = \frac{{10.212\left[ {0.1} \right]\left[ {0.5} \right]}}{{\left[ {0.25} \right]{{\left[ 1 \right]}^2}}} = 2.0424 . Therefore, max2.8,10.212K′0.5θLx¯−x_2=2.8 {\rm{\;max\;}}\left\{ {2.8,\frac{{10.212K'\left( {0.5} \right)}}{{{\theta _L}{{\left[ {\bar x - {\underline x} } \right]}^2}}}} \right\} = 2.8 . In contrast, Table 1 reports that an SPP secures Pareto gains in Example 1 if θHθL>1.4 {\rm if}\,\frac{{{\theta _H}}}{{{\theta _L}}} > 1.4 .

The finding that an SPP generates Pareto gains if θHθL≥max2.8,10.212K′0.5θL[x¯−x_]2 {\rm if}\,\frac{{{\theta _H}}}{{{\theta _L}}} \ge {\rm{\;max\;}}\left\{ {2.8,\frac{{10.212K'\left( {0.5} \right)}}{{{\theta _L}{{[\bar x - {\underline x} ]}^2}}}} \right\} suggests that θHθL^ \widehat {\frac{{{\theta _H}}}{{{\theta _L}}}} , the smallest value of θHθL \frac{{{\theta _H}}}{{{\theta _L}}} above which an SPP ensures Pareto gains, may increase as it becomes more costly for the agent to increase ϕ_H.

This finding also suggests that θHθL^ \widehat {\frac{{{\theta _H}}}{{{\theta _L}}}} may decline as x¯−x_ \bar x - {\underline x} increases, so the incremental value of project success increases. Numerical solutions confirm that this relationship holds under a broad set of conditions.

We have examined the effects of a single payment structure policy (SPP) in the presence of moral hazard and adverse selection. We have shown that the effects of an SPP can vary according to whether the agent's ability is exogenous or endogenous. When the agent's ability is exogenous, an SPP reduces the principal's expected profit and increases the agent's expected utility. It does so by restricting the principal's ability to limit the rent of the high-ability agent. When the agent's ability is endogenous, an SPP can increase both the principal's expected profit and the agent's expected utility. This is the case because by restricting the principal's ability to limit the rent of the high-ability agent, an SPP induces the agent to increase ϕ_H, the probability that the agent ultimately has high ability. The corresponding increase in the expected efficient surplus admits the possibility of Pareto gains. Such gains can be particularly likely to arise when the potential variation in the agent's ability is relatively pronounced, when it is not very costly for the agent to increase ϕ_H, and when the principal lacks alternative means to credibly promise to reward the agent for enhancing his (noncontractible) human capital (i.e., for increasing his expected ability in performing the principal's idiosyncratic task).

Our analysis suggests that a common view of optional compensation structures may merit more nuanced consideration in some settings. The common view is that employers often can gain by encouraging employees to choose among simple, carefully-designed compensation structures (e.g., Bommaraju and Hohenberg, 2018, p. 122). This conclusion is certainly correct when employee ability is largely exogenous. The conclusion also can be correct when employee ability is endogenous. However, as we have shown, the conclusion is not always correct in this case. Consequently, when designing compensation structures, employers should consider not only the optimal design of optional compensation structures, but also whether it might be preferable to preclude any choice among compensation schedules in order to motivate employees to undertake ability-enhancing investment in human capital.

We have shown that an SPP can generate Pareto gains when it induces the agent to increase ϕ_H, the probability that the agent ultimately has high ability. An SPP would not secure Pareto gains if the principal, rather than the agent, were the party that could devote personally costly effort to increase ϕ_H. In such a setting, an SPP compels the principal to deliver more rent to the high-ability agent, thereby reducing the principal's incentive to increase ϕ_H.

We have focused on the interaction between an employer and a worker, for concreteness. However, our analysis is relevant more broadly. For example, our conclusions also hold in a setting where a regulator attempts to motivate a regulated firm to improve its performance (e.g., reduce its operating costs or improve it service quality). When the regulator cannot reward the firm directly for enhancing its expected ability to improve its performance (because the expected ability is not contractible), the regulator can motivate the firm to undertake ability-enhancing effort by adopting an SPP.

In practice, regulators often allow the firms they regulate little or no choice among compensation structures. However, such choice sometimes prevails in the telecommunications and electricity sectors, for example (Sappington, 2002; Joskow, 2014; Hellwig et al., 2020).

We have considered a binary adverse selection problem for analytic tractability. However, the same forces we have identified will prevail more generally. An SPP will continue to reduce the principal's expected profit when the agent's expected ability is exogenous. In contrast, by enhancing the agent's rent for the higher ability realizations, an SPP can encourage the agent to increase the likelihood of the highest possible ability realizations. The corresponding enhanced expected efficient surplus introduces the potential for Pareto gains from an SPP.

In addition to considering non-binary adverse selection settings, future research might explore competition among principals.

Future research might also examine the optimal design of reward structures when an imperfect measure of the agent's expected utility (ϕ_H) is contractible. The presence of such an imperfect measure likely will reduce the principal's expected gain from committing to implement an SPP.