Multiperiod Contracting and Salesperson Effort Profiles: The Optimality of “Hockey Stick,” “Giving Up,” and “Resting on Laurels”

Abstract

The authors study multiperiod sales force incentive contracting in which salespeople can engage in effort gaming, a phenomenon that has extensive empirical support. Focusing on a repeated moral hazard scenario with two independent periods and a risk-neutral agent with limited liability, the authors conduct a theoretical investigation to understand which effort profiles the firm can expect under the optimal contract. The authors show that various effort profiles that may give the appearance of being suboptimal, such as postponing effort exertion (“hockey stick”) and not exerting effort after a bad or a good initial demand outcome (“giving up” and “resting on laurels,” respectively) may indeed be induced optimally by the firm. This is because, under certain conditions that depend on how severe the contracting frictions are and how effective effort exertion is in increasing demand, the firm wants to concentrate rewards on extreme demand outcomes. Doing this induces gaming and reduces expected demand but also makes motivating effort cheaper, thus saving on incentive payments. On introducing dependence between time periods, such as when the agent can transfer demands between periods, this insight continues to hold and, furthermore, “hockey stick,” “giving up,” and “resting on laurels” can be optimal for the firm even under repeated short time horizon contracting. The results imply that one must carefully consider the setting and environmental factors when making inferences about contract effectiveness from dynamic effort profiles of agents.

Keywords

sales force compensation dynamic incentives effort gaming “hockey stick,” sales push out and pull in

Introduction

Sales force expenditures account for 10%–40% of the revenues of U.S. firms (Albers and Mantrala 2008), which is in the order of hundreds of billions of dollars annually (Zoltners et al. 2008). Compensation contracts used by firms to reward salespeople are usually comprised of a fixed part (e.g., base salary) and a variable part (e.g., commissions on sales, discrete bonuses awarded for achieving a quota of sales in a specified time period). According to Joseph and Kalwani (1998), who conducted a survey of Fortune 500 firms, over 90% use quota-based rewards in their compensation plans. The advantage of quota-based reward plans is that they provide stronger incentives to salespeople to reach a high level of sales (Hedges 2015), as variable compensation is rewarded only in that case.

Firms employ salespeople for extended periods of time, and when quota-based incentives are used in such a multiperiod, long time horizon setting, the issue of dynamic gaming of effort arises. This is because in a multiperiod scenario the agents may strategically adjust their effort exertion over time on the basis of how uncertain outcomes are realized and how the contract will determine reward in current and future periods. One might intuit that in a multiperiod scenario, the principal would always want the agent to consistently exert high effort (the leftmost plot in Figure 1). However, empirical research has carefully documented the effort exertion profiles of agents induced by different types of contracts in multiperiod scenarios, and consistent effort exertion is often not the case.

Figure 1.

Canonical effort profiles in a multiperiod sales scenario.

As a canonical example, consider a scenario in which outcomes are measured every quarter and the salesperson is paid a bonus if a particular sales quota is reached in six months (i.e., two quarters). To reach their six-month quota with minimum effort, the agent may strategically shirk work in the first quarter hoping for a high demand outcome without much effort and exert greater effort only in the second quarter on the basis of the outcome of the first quarter. Such effort postponement is often observed in reality and is sometimes known as the “hockey stick” pattern because effort exertion is flat in early periods and increases sharply in later periods, thus taking the shape of a hockey stick, as the middle plot in Figure 1 illustrates (Chen 2000). Oyer (1998) analyzes aggregate data (from the Survey of Income and Program Participation) spanning multiple industries in which quota-based plans are used, and he detects that firms’ sales increase at the end of a fiscal year, suggesting that salespeople postpone effort exertion until the end of a compensation window to meet their quotas and get bonuses. Steenburgh (2008) analyzes individual salesperson-level data from a Fortune 500 company that sells durable office products and uses quota-based plans, and he finds similar patterns to those reported in Oyer (1998). Misra and Nair (2011) analyze data from a Fortune 500 contact lens manufacturer and find evidence of agents shirking effort in the early part of the compensation cycle. As in Oyer (1998) and Steenburgh (2008), Misra and Nair (2011) find higher sales at the ends of quarters compared with early in the quarters, which again suggests that agents tend to increase effort as they get closer to the end of a compensation window. Chung and Narayandas (2017) conduct a field experiment and also report delaying effort as an issue of concern in long time horizon contracts.

For similar strategic reasons related to effort gaming, we can observe a “giving up” effort profile under a quota-based contract (Chung and Narayandas 2017; Chung, Steenburgh, and Sudhir 2014; Jain 2012; Steenburgh 2008). Suppose that the salesperson exerts effort in the first quarter, but demand realization is low. Because they are far away from achieving the six-month quota, the salesperson does not exert effort in the second quarter (i.e., the salesperson “gives up”). Chung, Steenburgh, and Sudhir (2014) analyze data from a Fortune 500 office durable goods manufacturer and find that weak performers may give up if they realize that sales quotas under the long time horizon contract become unachievable. Chung and Narayandas (2017) also find empirical evidence that under a monthly quota plan, salespeople who had a series of bad draws early in the month may decide to give up late in the month because there is no chance that they can meet or exceed the quota set by the firm.

Alternatively, sales agents who have already achieved sales close to the quota in early sales cycles may not have the incentive to put in much subsequent effort and may “rest on laurels” instead (Chung and Narayandas 2017; Chung, Steenburgh, and Sudhir 2014; Misra and Nair 2011). For example, suppose that in the first quarter the salesperson exerts effort and demand turns out to be high, bringing the salesperson reasonably close to the six-month quota. The salesperson then does not exert much effort in the second quarter, being almost assured of the bonus at the end of the two quarters. Chung, Steenburgh, and Sudhir (2014) present evidence that the best performers will reduce productivity after getting close to or attaining quotas. On a similar note, Misra and Nair (2011) show that agents may shirk after they bring in enough sales to bring them close to their quotas. In both the “giving up” and the “resting on laurels” scenarios, an agent’s effort level is expected to decline over time, as the rightmost plot in Figure 1 illustrates.

An agent’s effort gaming—postponement of effort exertion and shirking after either bad or good early outcomes—is usually considered an undesirable outcome from the firm’s point of view because it involves the agent not exerting effort before they reach the sales quota. In this article, we conduct a theoretical investigation on whether these effort profiles are necessarily suboptimal for the firm. A different way of asking this question is: Is it always optimal for the firm to consistently induce high effort from the agent? If not, what effort profile(s) will be induced under the optimal incentive contract? An associated question here is: What is the optimal contract structure? Specifically, should the firm reward only extreme sales outcomes to motivate effort, or should they also reward intermediate sales outcomes? If the firm only rewards extreme sales outcomes, it would be more susceptible to gaming. But would it be too costly to prevent gaming by rewarding intermediate sales outcomes as well?

To answer these questions, we built a stylized principal–agent model with moral hazard in which a firm interacts with a salesperson for two time periods. The firm uses a contract that is determined at the start of the first period and pays once at the end of the second period on the basis of the outcomes of the two periods. We assume the demand outcome in each period to be stochastically dependent on the effort exerted in that period and assume the demand outcomes in the two periods to be independent of each other. The reward to the agent is based on the history of demand outcomes in the two periods. Under the two-period contract, the agent can dynamically adjust their effort level in the later period on the basis of the early period’s demand outcome, which, in turn, also influences their first-period effort exertion decision. We assume that the firm and the salesperson are risk neutral and that the agent has limited liability. Limited liability can be thought of as protection from downside risk for the salesperson. In other words, the salesperson will be guaranteed a minimum payment even in the case of an unfavorable market outcome (which is a robust feature of real-world compensation plans). This assumption aligns well with industry practice, even though limited liability introduces contracting frictions.

We find that, under different conditions, the optimal two-period contract can be a “gradual contract” that rewards the salesperson for all demand realizations (with higher reward for higher demand realizations) or an “extreme contract” that rewards the salesperson only for the maximum possible demand realization across the two periods. Interestingly, we find that different effort exertion profiles are possible under the different optimal contracts. When the optimal contract is a gradual contract, the agent exerts effort in both periods. However, when the optimal contract is an extreme contract, then we observe two types of effort patterns that, as discussed earlier, are characterized by effort gaming: (1) no effort exertion in the first period followed by effort exertion in the second period only in the case of high demand realization in the first period (the “hockey stick” effort profile) and (2) effort exertion in the first period followed by effort exertion in the second period only in the case of high demand realization in the first period (the “giving up” effort profile). Note that both of these effort profiles are characterized by probabilistic effort exertion in the second period based on the outcome of the first period. Our results show that observing less than high effort exertion either early or late may not necessarily imply that the contract is “not effective” in inducing demand. In fact, such contracts may be optimal for the firm in the sense that they provide exactly the effort profile the firm expects when choosing the optimal contract. Our results imply that one must carefully understand and consider the setting and environmental factors when making inferences about contract effectiveness from dynamic effort profiles of agents. In particular, in the presence of contracting frictions, and upon observing less than high effort exertion by salespeople in some time periods, one should not rush to the conclusion that this must mean that the system is in a suboptimal state and that the firm should want to change something to induce consistently high effort exertion.

Which effort profile we would expect depends on the optimal contract, which, in turn, is based on the tension between two countervailing forces: the “demand effect” (which refers to how much demand is expected from that contract; the firm wants this to be high) and the “incentive effect” (which refers to how much the firm will have to pay per unit of expected demand under that contract; the firm wants this to be low). Extreme contracts that induce effort gaming reduce the expected demand but also reduce the expected incentive cost per unit of expected demand. This is because, when firms expect agents to exert less effort (under moral hazard), detecting whether agents have exerted effort becomes easier, meaning that the incentive firms must provide to stimulate effort is smaller. In other words, an extreme contract that rewards only for high demand outcomes achieves effort exertion more efficiently than a gradual contract that rewards for all demand outcomes. The insight that bonuses for high demand outcomes motivate more effort is recognized in the industry (Hedges 2015), and it also aligns with the findings in Chung, Steenburgh, and Sudhir (2014) that bonuses, especially end-of-year bonuses, perform well through higher effort motivation despite the presence of gaming effects.

The hockey stick pattern is especially interesting as it implies that under the optimal contract, the firm induces delaying of effort. The hockey stick pattern is generated when the optimal contract is an extreme contract that makes implementing effort cheaper in the second period when the first period outcome is already revealed. Furthermore, inducing low effort in the first period also reduces the expected incentive payment because there is a lower probability of the extreme sales quota being met. Although this effort postponement is typically interpreted negatively (Chen 2000) and as something to avoid, our analysis shows that it indeed can be generated under an optimal contract and even with independent periods, because even though the firm reduces expected demand generated, it also reduces the expected incentive payment to generate this demand.

A number of articles, including Oyer (1998), Steenburgh (2008), Misra and Nair (2011), Jain (2012), and Chung, Steenburgh, and Sudhir (2014), document another kind of gaming (in addition to effort gaming) in a dynamic incentives setting. They show that in a multiperiod setting with nonlinear contracts, sales agents transfer demand between periods. Specifically, they may pull in orders from future periods if they would otherwise fall short of a sales quota in one period, and they may push out orders to the future if quotas are either unattainable or have already been achieved. We extend our basic model to study strategic sales pull in and push out behavior, which makes the two periods interdependent. We find that if sales pull in and push out is possible, the “hockey stick” and “giving up” effort profiles can also be generated under short term horizon contracting (in our model, when the firm uses a sequence of period-by-period contracts). In other words, the agent’s incentive to transfer sales under a sequence of short-term contracts plays a role similar to the agent’s dynamic gaming under a nonlinear two-period contract in terms of being used by the principal to save on incentive cost.

Finally, we extend our basic model in another way to make the two time periods interdependent. Specifically, we introduce the idea of an exogenous and limited amount of product inventory that can be sold across the two periods. This makes the contract design decisions for the principal in the two time periods dependent. We find that with limited inventory, the principal has lower incentive to induce effort in the first period. In other words, to save on incentive cost, the principal may optimally desire the agent to postpone their effort in the hopes of achieving a high demand outcome in first period, and if the first-period demand outcome is low, then the agent can exert effort in the second period. Naturally, this again leads to a hockey stick effort profile.

Our research is related to the literature seeking to explain an agent’s dynamic effort gaming under a long time horizon contract. Consider the hockey stick pattern characterized by an increasing effort profile over time, which has attracted much attention. Existing explanations for this phenomenon are, broadly speaking, of three types: (1) those that invoke behavioral explanations or argue that this is due to suboptimal behavior from either the principal or the agent, (2) those that take a behavioral goal attainment perspective, and (3) those that rely on demand shifting across time periods to explain this effort profile. Chen (2000) shows that if quotas are not in line with an agent’s level of productivity, the salesperson may find it optimal to wait before exerting effort to resolve uncertainty over the realization of early demand shocks (i.e., the agent will postpone effort). Chung, Steenburgh, and Sudhir (2014) focus on suboptimal gaming behaviors committed by the agent and discover from a counterfactual analysis that effort concentration in later periods can arise from agents’ myopic behaviors, whereas a forward-looking agent would smooth out efforts over time to take into account the uncertainty in future demand shocks. Also taking a behavioral perspective, Jain (2012) studies a scenario in which agents lack self-control. In these situations, the firm can take advantage of an agent’s lack of self-control to maximize its profits by paying a single bonus at the end, which essentially encourages effort postponement. The goal attainment literature also adopts a behavioral perspective to explain an agent’s procrastination. Kivetz et al. (2006) show the goal gradient phenomenon wherein agents work harder toward a goal as they get closer to achieving it. Heath, Larrick, and Wu (1999) use goal serving as a reference point to explain effort postponement in agents. The idea of goal serving is that goals have diminishing returns, and thus combining multiple short-term goals into a long-term goal will result in less effort exertion earlier on. Distinct from these explanations, our work provides the novel insight that agents’ gaming behaviors can be optimally induced by the firm to improve its profits.¹ The hockey stick effort pattern can also be observed if salespeople pull in orders from future periods when they are short of sales to meet the current quota. This has been documented by a number of articles on salesforce compensation (Kishore et al. 2013; Oyer 1998; Steenburgh 2008), accounting (Healy 1985) and navy recruitment (Asch 1990). However, we show that demand borrowing is not necessarily needed to obtain the hockey stick profile (recall that our basic model assumes that there is no demand borrowing but still produces the hockey stick profile).

Our research is related to the body of work on dynamic incentives with repeated moral hazard. One stream of this work assumes the firm to be risk neutral but agents to be risk averse, which leads to contracting frictions. Under this paradigm, a seminal article, Hölmstrom and Milgrom (1987), shows that a linear contract is optimal for the principal when a number of other assumptions hold. We note that the gradual two-period contract that we derive as optimal for the firm under certain conditions can be interpreted as a linear contract as well, but it is only optimal for a certain region of the parameter space (recall that we assume the agent to be risk neutral). A number of articles in the risk aversion paradigm revisit the assumptions of Hölmstrom and Milgrom (1987) and show the optimality of nonlinear contracts (Hellwig and Schmidt 2002; Rubel and Prasad 2015; Rogerson 1985; Sannikov 2008; Schättler and Sung 1993; Spear and Srivastava 1987; Sung 1995).

A second stream of the work on dynamic incentives assumes agents to be risk neutral with limited liability, which is a different source of contracting friction (our article falls under this paradigm). Bierbaum (2002) studies how to induce high effort from the agent in each of two periods (which may not be profit maximizing for the principal), whereas we allow different effort profiles to be induced by the optimal contracts under different conditions. Kräkel and Schöttner (2016) study the firm’s choice between commissions and bonuses and determines conditions under which one or the other (or a combination) is optimal in a dynamic setting involving a special case when an agent’s outside option is equal to his limited liability. Schöttner (2016) studies optimal contracting when the agent’s effort costs change over time. These articles, however, assume equal values of the reservation utility and the limited liability, which is not without loss of generality. In fact, our most interesting results on effort patterns are obtained for the case when the value of the outside option is greater than the value of the limited liability, which is a reasonable assumption. We also note that Carroll (2015) assumes limited liability with risk neutrality and shows that a linear contract is optimal, but that this result is driven by the assumption that the objective of the principal is to write a robust contract to optimize the worst-case outcome rather than to optimize the expected profit (which we assume and is the more standard assumption).

More broadly, our research adds to the extensive literature on salesforce incentives in marketing, which, in addition to the articles already cited, includes Raju and Srinivasan (1996), Godes (2004), Simester and Zhang (2010), and Zhang (2016), among many others. Our extension involving limited inventory is related to the work on salesforce compensation when operational considerations are important (Chen 2000; Dai and Jerath 2013, 2016, 2019; Plambeck and Zenios 2003; Saghafian and Chao 2014).

The rest of this article is organized as follows: In the next section, we describe the model. Then, we conduct the analysis, obtain our key insights regarding the different forces at play, and show under what conditions the principal can optimally induce an agent’s dynamic gaming. Following this, we allow the agent to push out and pull in sales between periods, and we allow for periods to be dependent by assuming that the principal has limited inventory to be sold in the two periods. In the final section, we conclude with a discussion. The proofs are provided in the Appendix and Web Appendix (indicated appropriately).

Model

We develop a stylized agency theory model in which a firm (the principal) hires a salesperson (the agent) to exert demand-enhancing effort. There are two time periods denoted by $t \in {1,2}$ .² Demand in both periods is uncertain, and the outcomes in each period are assumed to be independent of the other period, conditional on the effort exerted in that period. Let $D_{t}$ be the demand realization in period $t$ , which can be either $H$ or $L$ with $H > L > 0$ . The agent’s effort increases the probability of realizing high demand levels. The effort level in period $t$ , denoted by $e_{t}$ , can be either $1$ or $0$ (i.e., the agent either “works” or “shirks” in each period); however, the principal does not observe the effort level. For instance, we can think of effort level $0$ as a salesperson making a client visit (which is observable and verifiable) and effort level $1$ as the salesperson’s additional effort spent in talking to and convincing the client to make the purchase (which the firm cannot observe or verify). Without effort exertion ( $e_{t} = 0$ ), demand is realized as $H$ with a probability of $q$ , and with effort exertion ( $e_{t} = 1$ ), this probability increases to p ( $0 < q < p < 1$ ). A larger p implies greater effectiveness of the salesperson’s effort, and $q$ can be interpreted as the natural market outcome. We assume that all the demand created can be met and each unit sold gives a revenue of $1$ and has a marginal cost of 0. The cost of effort is given by $ϕ > 0$ for $e_{t} = 1$ and is normalized to zero for $e_{t} = 0$ .

We assume that both the firm and the salesperson are risk neutral. Unlike the firm, however, the salesperson has limited liability, implying that they must be protected from downside risk. Specifically, we assume that the salesperson has a limited liability of $K$ in each period. In other words, to employ the agent for one period, the principal must guarantee a compensation of at least $K$ under any demand outcome. Limited liability is a widely observed feature of salesforce contracts in the industry, and this assumption is a standard one in the literature to prevent the firm from being sold to a risk-neutral agent (Carroll 2015; Dai and Jerath 2013; Kim 1997; Laffont and Martimort 2009; Oyer 2000; Park 1995; Sappington 1983; Simester and Zhang 2010). The limited liability assumption also implies the existence of a wage floor for the salesperson, which is aligned with industry practice. We assume that the salesperson’s reservation utility is $U$ for each period, which can be thought of as coming from the agent’s outside option. The agent’s limited liability can be different from his reservation utility (Kim 1997, Oyer 2000). For instance, if the salesperson’s alternative employment opportunities are attractive, then the agent’s reservation utility will be higher than their limited liability. In the analysis, we assume this to be the case and, in accordance with this, we assume $U \geq K$ .³

The agent is reimbursed for effort using an incentive contract. Effort is unobservable to the firm and demand is random but can be influenced by effort, so the firm and the agent sign an outcome-based contract. The firm can propose a long time horizon contract, which in this case is a two-period contract that is determined at the beginning of the first period and pays once at the end of the second period on the basis of the outcomes of the two periods.^4,5

The timeline of the game is as follows. At the beginning of Period 1 (i.e., $T = 1$ ), the principal proposes the contract and the agent decides whether to accept the offer. If accepted, the agent then decides on their effort in the first period, $e_{1}$ . At the end of $T = 1$ , the agent and the principal observe the demand outcome for the first period, $D_{1}$ . The agent then chooses their second period effort, $e_{2}$ . At the end of $T = 2$ , the agent and the principal observe the second period demand outcome $D_{2}$ . The firm realizes the profit,⁶ and the agent gets paid according to the contract. We make the assumption that when the agent is indifferent between exerting effort or not, they will choose to exert effort. We now proceed to the analysis.

Analysis

Benchmark 1: First-Best Scenario

We start by presenting the first-best solution (e.g., if the agent’s effort is observable). In this case, the two periods are independent and equivalent, and it is sufficient to study just one period. We obtain the following first-best solution (details of the formulation and the analysis are provided in the “First-Best Solution” section of the Appendix).

Result 1 (optimal first-best solution): The first-best contract instructs the agent to exert effort $e_{FB}^{*} = 1$ and pays a fixed salary equal to the agent’s cost of effort $s_{FB}^{*} = ϕ$ , if and only if $H - L ⩾ \frac{ϕ}{p - q}$ . Otherwise, the principal chooses not to direct effort exertion (i.e. $e_{FB}^{*} = 0$ ), and $s_{FB}^{*} = 0$ .

From Result 1, we can infer that the principal would like the agent to exert effort when the upside market potential is large, or when the effectiveness of the agent’s effort is high. To rule out the trivial case in which the firm is not interested in motivating effort even in the first-best scenario, we only consider the parameter space with $H - L ⩾ \frac{ϕ}{p - q}$ in the rest of the article.

Benchmark 2: One-Period Scenario

Suppose that the firm could only contract with the agent one period at a time, but there is moral hazard and limited liability. This is a standard textbook problem, and we obtain the following result (details of the formulation and the analysis are provided in the “Optimal One-Period Contract” section of the Appendix).

Result 2 (optimal one-period solution): The optimal one-period contract gives the salesperson a fixed salary of $max{K,U - \frac{q}{p - q} ϕ}$ , no bonus if the realized demand is $L$ , and a bonus of $\frac{ϕ}{p - q}$ if the realized demand is $H$ . Under this contract, the salesperson exerts effort.

In this case, the firm gives the salesperson a fixed salary to cover the salesperson’s limited liability while ensuring their participation. The notable feature of this contract is that a bonus, equal to $\frac{ϕ}{p - q}$ , is given only if the realized demand is $H$ . This bonus decreases as the effectiveness of the agent’s effort, $p$ , increases, because a larger $p$ implies that effort exertion is easier to detect.

Two-Period (Long Time Horizon) Contract

In this scenario, the firm proposes a long time horizon two-period contract at the beginning of the first period and pays once at the end of the second period on the basis of the outcomes of the two periods. A key feature of this scenario introduced due to unobservability of effort and the contract paying at the end of two periods is that the agent can “game” the system: the agent can choose their effort in Period 2 on the basis of the outcome of Period 1 (and, realizing this, can also choose the effort in Period 1 strategically). We denote the two-period effort profile by $〈 e_{1}, e_{2}^{H}, e_{2}^{L} 〉$ , where the second period’s effort $e_{2}^{D_{1}}$ is contingent on the first period’s demand realization, $D_{1} \in {H, L}$ .

In full generality, this contract involves a guaranteed salary for employing the agent for two periods, plus a bonus issued at the end of the two periods that is contingent on the whole history of outputs. We denote the fixed salary by $S$ and denote the bonus paid at the end of $T = 2$ by $b_{2} (D_{1}, D_{2})$ . Such a contract thus stipulates four possible bonuses: $b_{2} {(L,L), b}_{2} {(L,H), b}_{2} (H,L),$ and $b_{2} (H, H)$ .⁷ To prevent the agent from restricting sales to $L$ when demand is $H$ , we impose a constraint on the two-period contract given by $b (H, H) \geq max {b (H, L), b (L, H)}$ (i.e., the bonus paid when demand in both periods is realized as $H$ should be no lower than the bonus paid when demand in only one of the periods is realized as $H$ ). (Note that our key insights hold even without this assumption, as shown in the Optimal Two-Period Contract Without Nondecreasing Constraint section of the Appendix, but this is a natural assumption and simplifies the analysis.) Under this constraint, we obtain the following lemma (the detailed proof is in the General Two-Period Contract section in the Appendix):

Lemma 1: In the weakly dominant two-period contract, $b_{2} (H, L) = b_{2} (L, H)$ .

Lemma 1 directly implies the following lemma:

Lemma 2: At the end of two periods, it is optimal for the principal to pay the agent a bonus according to cumulative sales (which can be $2 L, H + L$ or $2 H$ ).

Lemma 2 significantly simplifies the analysis.⁸ We denote the fixed salary by $S$ , normalize the bonus payment when the total sales across two periods are $2 L$ as $0$ , denote the bonus payments when the total sales are $H + L$ and $2 H$ by $B_{1}$ and $B_{2}$ , respectively, and define $D = D_{1} + D_{2}$ . We formulate the principal’s problem as follows:

\begin{array}{l} max_{B_{1}, B_{2}} & E [D | e_{1}, e_{2}^{H}, e_{2}^{L}] - E [S + B_{1} + B_{2} | e_{1}, e_{2}^{H}, e_{2}^{L}] \\ s.t. & U_{A} (e_{2}^{H}) > U_{A} ({\tilde{e}}_{2}^{H}) & {(IC}_{2}^{H}) \\ ​ & U_{A} (e_{2}^{L}) > U_{A} ({\tilde{e}}_{2}^{L}) & {(IC}_{2}^{L}) \\ ​ & U_{A} (e_{1} | e_{2}^{H}, e_{2}^{L}) > U_{A} ({\tilde{e}}_{1} | e_{2}^{H}, e_{2}^{L}) & {(IC}_{1}) \\ ​ & U_{A} (e_{1}, e_{2}^{H}, e_{2}^{L}) \geq 2 U & (PC) \\ ​ & S, S + B_{1}, S + B_{2} \geq 2 K & (LL) \end{array}

In this formulation, $U_{A} (\cdot)$ denotes the agent’s expected net utility. ( $I C_{2}^{H}$ ) stands for the agent’s incentive compatibility constraint in the second period following $D_{1} = H$ , where $U_{A} (e_{2}^{H})$ represents the agent’s net payoff in Period 2 upon exerting effort $e_{2}^{H}$ . (Then, if the agent exerts effort, they will get net utility $S + B_{2} - ϕ$ with probability $p$ and $S + B_{1} - ϕ$ with probability 1−p. Without exerting effort, they will get net utility $S + B_{2}$ with probability $q$ and $S + B_{1}$ with probability 1−q.) To induce $e_{2}^{H}$ , the principal needs to ensure that the agent gets a higher payoff upon exerting effort $e_{2}^{H}$ compared with a different effort level ${\tilde{e}}_{2}^{H}$ . Similarly, ( $I C_{2}^{L}$ ) stands for the incentive compatibility constraint for inducing effort level $e_{2}^{L}$ in the second period following $D_{1} = L$ . ( $I C_{1}$ ) represents the incentive compatibility constraint in the first period. $U_{A} (e_{1} | e_{2}^{H}, e_{2}^{L})$ denotes the agent’s expected net utility across two periods upon exerting $e_{1}$ in the first period, given that the agent is induced to exert effort $e_{2}^{H}$ or $e_{2}^{L}$ in the second period on the basis of the outcome of the first period. If $e_{1} = 1$ , his total net utility will be $U_{A} (e_{2}^{H}) - ϕ$ with probability $p$ and $U_{A} (e_{2}^{L}) - ϕ$ with probability $1 - p$ . If $e_{1} = 0$ , his expected net utility will be $U_{A} (e_{2}^{H})$ with probability $q$ and $U_{A} (e_{2}^{L})$ with probability $1 - q$ . To induce $e_{1}$ , the principal needs to ensure that the agent gets a higher total net utility on exerting $e_{1}$ compared with a different effort level ${\tilde{e}}_{1}$ . ( $P C$ ) is the agent’s participation constraint, which ensures that the agent’s expected net utility across the two periods is no less than twice their per-period reservation utility, $2 U$ . ( $L L$ ) is the agent’s limited liability constraint, which ensures that the agent’s two-period compensation is no less than their limited liability over two periods, (i.e. $2 K$ ).

To arrive at an optimal contract for the principal, it is crucial to understand how the agent’s effort profile in the two periods changes with the bonuses $B_{1}$ (which is provided if demand is $H + L$ ) and $B_{2}$ (which is provided if demand is $2 H$ ) in the two-period contract. The following proposition describes the expected effort profile $(e_{1}, E [e_{2}])$ , where $e_{1}$ is an agent’s effort level in the first period and $E [e_{2}]$ is an agent’s expected effort level in the second period (since $e_{2}$ depends on $D_{1}$ , which is random).⁹ For instance, if the agent exerts effort in Period 1 and will exert effort in Period 2 only if the outcome in Period 1 is $H$ , then $e_{2} = 1$ with probability $p$ and $e_{2} = 0$ with probability $1 - p$ , so we write this expected effort profile as $(1, p)$ .

Proposition 1: (Agent’s Response to Two-Period Contract) Given $B_{1}$ and $B_{2}$ , the agent’s expected effort profile $(e_{1}, E [e_{2}])$ is as follows (the regions referred to are as per Figure 2, and they are analytically defined in Table A2 in the “Agent’s Effort Responses” section of the Appendix):

A constant work expected effort profile, $e = (1, 1)$ , is induced when both $B_{1}$ and $B_{2} - B_{1}$ are large (Region IV), and a constant shirk expected effort profile, $e = (0, 0)$ , is induced when both $B_{1}$ and $B_{2} - B_{1}$ are small (Region I).

A “hockey stick” expected effort profile, $e = (0, q)$ or $e = (0, 1 - q)$ , where the agent exerts more effort in expectation in the second period than in the first period is induced when $B_{1}$ is small and $B_{2} - B_{1}$ is moderate (Region II) or when $B_{1}$ is moderate and $B_{2} - B_{1}$ is small (Region VI), respectively.

A “giving up” expected effort profile, $e = (1, p)$ , where the agent exerts effort in the first period and in the second period upon a high first-period demand outcome, is induced when $B_{1}$ is small and $B_{2} - B_{1}$ is large (Region III).

A “resting on laurels” expected effort profile, $e = (1, 1 - p)$ , where the agent exerts effort in the first period and in the second period upon a low first-period demand outcome, is induced when $B_{1}$ is large and $B_{2} - B_{1}$ is small (Region V).

Figure 2 illustrates Proposition 1 graphically. The $x$ -axis, $B_{1}$ , is the incremental reward when total sales increase from $2 L$ to $H + L$ , and the $y$ -axis, $B_{2} - B_{1}$ , is the incremental reward when total sales increase from $H + L$ to $2 H$ . In line with Proposition 1, if both rewards are small, there is no effort exertion in either period, denoted by $e = (0, 0)$ , which is Region I. If both rewards are large, the agent will put in effort in both periods (i.e., $e = (1, 1)$ ), which is Region IV. For other regions, the effort exertion decisions are more nuanced. If the demand outcome in Period 1 is $L$ and the agent does not secure the bonus $B_{1}$ , they will not exert additional effort in Period 2 if $B_{1} \leq \frac{ϕ}{p - q}$ . If the demand outcome in Period 1 is $H$ and the agent secures the bonus $B_{1}$ , they will not exert additional effort in Period 2 if $B_{2} - B_{1} \leq \frac{ϕ}{p - q}$ . In other words, $B_{1}$ and $B_{2} - B_{1}$ motivate the agent to exert effort in the second period if demand in the first period turns out to be $L$ and $H$ , respectively.

Figure 2.

Agent’s optimal effort responses based on bonuses $B_{1}$ (given when final sales are $L + H$ ) and $B_{2}$ (given when final sales are $2 H$ ). The regions are defined in Table A2 in the “Agent’s Effort Responses” section of the Appendix.

Furthermore, the agent’s effort exertion at $T = 1$ depends on the values of both $B_{1}$ and $B_{2} - B_{1}$ . In Regions II and VI, the agent does not work in Period 1 and chooses to “ride their luck” in Period 1. This results in a “hockey stick” effort profile in which the expected effort level is higher in the second period than in the first period. The difference between Regions II and VI is that in Region II, the agent works in Period 2 if the demand outcome in Period 1 is favorable (i.e., $H$ ), and in Region VI, the agent works in Period 2 if the demand outcome in Period 1 is unfavorable (i.e., $L$ ). In Regions III and V, the agent works in Period 1. However, in Region III, they work in Period 2 only if the demand outcome in Period 1 is favorable (i.e., $H$ ). This results in a “giving up” effort profile in which the agent does not exert effort in Period 2 if the early demand outcome is unfavorable and the bonus becomes unattainable. In Region V, the agent works in Period 2 if the demand outcome in Period 1 is unfavorable (i.e., $L$ ). This results in a “resting on laurels” effort profile in which the agent does not exert effort in Period 2 if the early demand outcome is favorable (i.e., $H$ ).

We now determine the optimal compensation plan for the salesperson, which balances the expected revenue $E [D]$ and the expected compensation cost $E [S + B_{1} + B_{2}]$ . The following proposition characterizes the optimal two-period contract for the principal.

Proposition 2: (Optimal Two-Period Contract) The optimal two-period contract is as follows (the regions referred to are as per Figure 3, and they are analytically defined in Table A4 in the Optimal Two-Period Contract section of the Appendix):

In Region I, the optimal contract is a “zero-bonus contract” that only offers a fixed salary $S_{I}^{*} = 2 U$ .

In Region II, the optimal contract is an “extreme, low-powered contract” that rewards bonuses only at $2 H$ , with $B_{1} = 0, B_{2}^{*} = \frac{ϕ}{p - q}$ , and a fixed salary of $S_{I I}^{*} = max {2 U - \frac{q^{2}}{p - q} ϕ,2 K}$ .

In Region III, the optimal contract is an “extreme, high-powered contract” with $B_{1}^{*} = 0, B_{2}^{*} = \frac{1 + p - q}{p (p - q)} ϕ$ , and a fixed salary of $S_{I I I}^{*} = max {2 U - \frac{q}{p - q} ϕ,2 K}$ .

In Region IV, the optimal contract is a “gradual contract” that rewards bonuses at both $H + L$ and $2 H$ , with $B_{1}^{*} = \frac{ϕ}{p - q}, B_{2}^{*} = 2 \frac{ϕ}{p - q}$ , and a fixed salary of $S_{I V}^{*} = max {2 U - \frac{2 q}{p - q} ϕ,2 K}$ .

Figure 3.

Optimal two-period contracts and effort outcomes. The optimal contract in Region II is an extreme, low-powered contract. In Region III, the optimal contract is an extreme, high-powered contract, and in Region IV, the optimal contract is a gradual contract (in Region I, the agent is not offered any incentive payment). The regions are defined in Table A4 in the “Optimal Two-Period Contract” section of the Appendix.

The regions of the parameter space under which different contracts are optimal are illustrated in Figure 3, and the contract forms themselves and the outcomes are illustrated graphically in detail in Figure 4. Corollary 1 presents an interesting implication of Proposition 2.

Corollary 1: Under a two-period contract, the firm may optimally induce both a “hockey stick” expected effort profile given by $(0, q)$ and a “giving up” expected effort profile given by $e = (1, p)$ .

Proposition 2 and Corollary 1 provide interesting and important implications for agents’ observed effort profiles. First, under certain conditions, a pattern of delaying effort (i.e., the “hockey stick” effort profile) is optimally induced from the agent under the optimal contract chosen by the firm. Here, the effort profile is $〈 0, 1, 0 〉$ and the expected effort profile is $(0, q)$ (i.e., effort is not exerted in the first period but is then exerted in the second period with probability greater than zero, only if the first period demand is realized as high). This happens when the firm finds it optimal to use an extreme yet low-powered contract (see the leftmost column of Figure 4). Specifically, this is a contract in which $B_{1} = 0$ and $B_{2} = \frac{ϕ}{p - q}$ . (In other words, it gives a bonus only if realized demand is $2 H$ , but this bonus is small. Note from the benchmark one-period analysis that $\frac{ϕ}{p - q}$ is the bonus given under the optimal contract on obtaining demand $H$ in one period, whereas here this bonus is given on obtaining demand $2 H$ in two periods.)

Figure 4.

Each column of this figure corresponds to one of the optimal contracts: the left column corresponds to the extreme low-powered contract (Region II of Figure 3), the middle column corresponds to the extreme high-powered contract (Region III of Figure 3), and the right column corresponds to the gradual contract (Region IV of Figure 3). The figures in the top row illustrate the final bonuses, $B_{0}$ , $B_{1}$ , and $B_{2}$ (at the leaves of the trees) and the induced effort levels, $e_{1}, e_{2}^{L},$ and $e_{2}^{H}$ (in the circular nodes), for each contract. The figures in the second row depict $e_{1}$ and $E [e_{2}]$ for each contract. The figures in the third row illustrate the form of the contract. The fixed salaries from the left column to the right are $S_{I I} = max {2 U - \frac{q^{2}}{p - q} ϕ,2 K}, S_{I I I} = max {2 U - \frac{q}{p - q} ϕ,2 K}, and S_{I V} = max {2 U - \frac{2 q}{p - q} ϕ,2 K},$ respectively.

Under certain conditions, the firm also optimally induces a pattern of “giving up” in the second period. Here, the effort profile is $〈 1, 1, 0 〉$ and the expected effort profile is $(1, p)$ (i.e., effort is exerted in the first period but then exerted in the second period with probability less than 1, only if the first period demand is realized as high). The optimal contract to induce such an effort profile is the extreme and high-powered contract (see the middle column of Figure 4) that provides a bonus $B_{1} = 0$ and a large bonus $B_{2} = \frac{1 + p - q}{p (p - q)} ϕ$ (which is larger than $\frac{ϕ}{p - q}$ ).

Finally, in certain parameter spaces, the principal finds it optimal to motivate the effort profile $〈 1, 1, 1 〉$ and the expected effort profile $(1, 1)$ (i.e., effort is induced under all conditions in both periods). The contract adopted for this is a “gradual contract” (see the rightmost column of Figure 4) that rewards bonuses at both $H + L$ and $2 H$ (specifically, $B_{1} = \frac{ϕ}{p - q}$ and $B_{2} = 2 \frac{ϕ}{p - q}$ ).

To understand the intuition behind these results, we discuss two effects that are operative: namely, the “demand effect” and the “incentive effect.” Note that we can rewrite the principal’s profit, given by $E [D] - E [S + B_{1} + B_{2}]$ , as a product of $E [D]$ , which is the expected demand generated, and $1 - \frac{E [S + B_{1} + B_{2}]}{E [D]}$ , which is the net expected profit for inducing each unit of expected demand. The former term is relevant to the demand effect and the latter term is relevant to the incentive effect.

First, we discuss the demand effect. In terms of the expected demand generated under each contract, implementing the “hockey stick” expected effort profile $e = (0, q)$ generates the least expected demand $E [D]$ , followed by the “giving up” expected effort profile $e = (1, p)$ , followed by constant expected effort profile, $e = (1, 1)$ .

Next, we discuss the incentive effect. The term $\frac{E [S + B_{1} + B_{2}]}{E [D]}$ can be interpreted as the expected incentive payment per unit of expected demand, and the firm prefers this to be small. It turns out that inducing the “hockey stick” expected effort profile $e = (0, q)$ has the smallest $\frac{E [S + B_{1} + B_{2}]}{E [D]}$ , followed by inducing the “giving up” expected effort profile $e = (1, p)$ , followed by inducing the constant expected effort profile $e = (1, 1)$ . To see this, note that essentially the principal needs to implement three effort choices: $e_{1}$ , $e_{2}^{H}$ , and $e_{2}^{L}$ . Among these, it is most efficient to implement $e_{2}^{H} = 1$ , followed by $e_{1} = 1$ , followed by $e_{2}^{L} = 1$ . Intuitively, it is “cheaper” to implement $e_{2}^{H} = 1$ and $e_{1} = 1$ (rather than $e_{2}^{L} = 1$ ) because in both cases the principal can concentrate reward at the highest sales level of $2 H$ by adopting an extreme contract. Furthermore, comparing between $e_{2}^{H} = 1$ and $e_{1} = 1$ , it is cheaper to implement $e_{2}^{H} = 1$ because the first period outcome is already revealed to be $H$ . For $e_{1} = 1$ , the first period outcome will be $H$ only with probability $p$ if the salesperson works. It is “costliest” to implement $e_{2}^{L} = 1$ because the first period outcome is already revealed to be $L$ and the principal therefore must offer a bonus $B_{1} > 0$ at the less extreme sales level $H + L$ . This gives the agent an incentive to ride their luck in the first period, hoping for a high demand outcome to secure bonus payments, and it makes it harder for the principal to motivate effort. To obtain the expected effort profiles $(0, q)$ , $(1, p)$ , and $(1, 1)$ , the principal must implement $〈 0, 1, 0 〉$ , $〈 1, 1, 0 〉$ , and $〈 1, 1, 1 〉$ , respectively. Given the preceding discussion, one can see that $(0, q)$ ranks as most efficient according to the incentive effect, followed by $(1, p)$ , followed by $(1, 1)$ .

Therefore, the hockey stick, giving up, and constant expected effort profiles (induced by the extreme low-powered, extreme high-powered, and gradual contracts, respectively) are ordered from worst to best by the demand effect and in exactly the opposite direction by the incentive effect¹⁰ (i.e., the demand effect and the incentive effect are countervailing forces acting in opposite directions, thus creating a tension in the model). Combining the demand effect and the incentive effect, we obtain the overall outcomes of which contract is optimal under different parametric conditions, as shown in Figures 3(a) and 3(b). Consider, for both figures, how the contracts and outcomes change as we vary $U - K$ (following the horizontal dotted lines). When the limited liability $K$ is sufficiently high (i.e., in Region I in which $U - K$ is small), the friction from moral hazard is large enough such that the principal does not want to induce effort exertion. As limited liability decreases into Region II, the friction due to moral hazard becomes smaller, and the principal starts to motivate effort using the extreme two-period contract, which provides effective incentives. However, since the limited liability is still relatively high in this scenario, the principal only induces $e_{2}^{H} = 1$ (which, as discussed earlier, is the cheapest way to have effort exerted) through a small ultimate bonus, and the expected effort profile is the hockey stick profile $e = (0, q)$ .

As the limited liability continues to decrease further into Region III, the principal implements $e_{1} = 1$ through a high ultimate bonus to leverage the demand effect, and the expected effort profile is the giving up effort profile $e = (1, p)$ . When limited liability becomes small enough (i.e., in Region IV), the demand effect dominates the incentive effect. The principal implements the constant expected effort profile $e = (1, 1)$ using the gradual two-period contract.

Next, using Figure 3(a), we discuss the optimal contract against $p - q$ , which increases with the effectiveness of the agent’s effort. We vary $p - q$ in Figure 3(a) along the vertical dotted line, keeping $H - L$ and $U - K$ fixed. Generally speaking, the firm needs to give a more effective agent a lower incentive to work because the outcome is a better signal of the effort the agent exerts and is therefore more easily detectable. In other words, as $p$ increases, the frictions due to moral hazard decrease. Consequently, in line with the intuition discussed earlier, the principal starts to implement $e = (0, q)$ , followed by $e = (1, p)$ , followed by $e = (1, 1)$ . However, if $p$ becomes very large, the principal will induce $e = (1, p)$ again. This is because the demand losses due to no effort exertion in the second period upon low demand realization are insignificant when $p$ is very large. When $p$ increases, the demand loss, given by $(p - q) (H - L)$ if it happens, gets larger, but the probability of effort not being exerted in the second period, given by $(1 - p)$ , gets smaller.

Finally, using Figure 3(b), we discuss the optimal contract against $H - L$ , which is another metric of the impact of the agent’s effort. We vary $H - L$ in Figure 3(b) along the vertical dotted line, keeping $p - q$ and $U - K$ fixed. As $H - L$ increases, the demand effect becomes stronger. Therefore, the principal starts to implement $e = (0, q)$ , followed by $e = (1, p)$ , followed by $e = (1, 1)$ .

To summarize, in contrast to the lay view that the “hockey stick” phenomenon is somehow suboptimal for the firm, our analysis suggests that the firm can optimally induce it even though the firm could choose a contract in which consistently high effort could be motivated. The intuition is that delaying effort can be optimal for the agent when a low ultimate bonus payment is associated with a high quota level, and inducing delayed effort can simultaneously be optimal for the firm to save on incentive cost when the moral hazard friction is high (as in the case of high limited liability). Similarly, we can also explain the optimality of the “giving up” effort profile. We highlight that we obtain these results under the assumption that periods have independent demand outcomes (conditional on effort).

Discussion on Contract Structures

It is worth noting that the gradual long time horizon contract is essentially a commission-based contract. This is because every additional sales outcome $H$ brings in the same additional reward $\frac{ϕ}{p - q}$ . With linear sales commissions, there is a constant incentive to work and no incentive to delay effort or give up. However, if we assume ex ante that only a commission contract is allowed, then the optimal long-term contract will be either a commission-based contract (i.e., the gradual contract) that induces $e = (1, 1)$ or a contract that pays only a base salary and induces $e = (0, 0)$ . Referring to Figure 3, in Regions I and IV the outcomes would be the same, whereas in Regions II and III the principal would make less profit with only linear contracts allowed than it would with nonlinear contracts allowed, but there would be no effort gaming.

Furthermore, the gradual contract is a replicate of a sequence of optimal short time horizon contracts. In other words, it is a sequence of period-by-period contracts in which the principal specifies a one-period contract at the beginning of the first period and then specifies another one-period contract at the beginning of the second period (see Result 2). Therefore, whenever the principal prefers to induce constant effort exertion in the two-period contract, the principal is indifferent between a sequence of period-by-period contracts and a two-period contract. Note that there is no effort gaming in this situation. We state this as a corollary.

Corollary 2: With independent periods, whenever the principal finds it optimal to induce constant effort exertion in the two-period contract, the principal is indifferent between a sequence of period-by-period contracts and a two-period contract. In this situation, there is no effort gaming.

A point of discussion in the salesforce compensation literature is the time horizon of contracts (i.e., whether a firm should use long- or short-term contracts). In a recent review article, Coughlan and Joseph (2012) list this as a very important yet underresearched issue in salesforce management. Although it is straightforward that a fully flexible long time horizon contract will be weakly dominant over a sequence of short time horizon contracts, our result above shows that under certain conditions long time horizon contracting will be strictly better than short time horizon contracting, whereas under other conditions both will achieve the same result.

Extensions

Until now, we have assumed that the two time periods are independent of each other. In this section, we study scenarios in which the two periods are interdependent. There may be many ways in which the periods can be interdependent. We consider two such ways: (1) demand transfer in the presence of sales push out and pull in that introduces dependence due to the agent’s incentive to transfer sales and (2) having a limited amount of product to sell across the two periods, which introduces dependence due to an external constraint. We show that with interdependent periods, new forces emerge that make it optimal for the principal to choose a contract that induces a “hockey stick” effort profile under long time horizon contracting and even in short time horizon contracting.

Sales Push Out and Pull In Between Periods

Salespeople working under quota-based plans may resort to modifying demand in particular periods to meet quotas in those periods. Oyer (1998) empirically demonstrates the existence of demand pull in and push out between fiscal cycles when salespeople face nonlinear contracts. In particular, Oyer (1998) reveals that sales agents will pull in orders from future periods if they would otherwise fall short of a sales quota in one cycle, whereas they push out orders to the future if quotas are either unattainable or have already been achieved. In previous sections, we dismissed such sales push out and pull in phenomena by assuming that the agent cannot shift sales between two periods (yet they still obtain the hockey stick and giving up effort profiles).

In this section, we relax this assumption and allow the agent to push extra sales to, or borrow sales from, the later period. When the agent has this ability, the two-period optimal contract (which pays at the end of the two periods) is not affected, but the period-by-period contract (which pays in the interim) has to be reanalyzed. We provide a sketch of the analysis herein, with details provided in the “Period-by-Period Contract with Sales Push Out and Pull In” section of the Appendix. A main takeaway from this analysis is that in the presence of sales push out and pull in effects, we can observe effort dynamics (e.g., delayed effort under optimal short time horizon contracting) as well.

We assume that in the period-by-period contract, at the end of the first period, the agent observes the actual sales $D_{1}$ ahead of the principal. The agent can then strategically push out sales to, or pull in sales from, the second period. The principal only observes the sales level after the agent’s manipulation, which we denote by $D_{1}^{'}$ , and the principal pays the agent according to $D_{1}^{'}$ . For instance, the principal will observe $D_{1}^{'} = H$ if $D_{1} = H$ , or if $D_{1} = L$ and the agent pulls in $H - L$ from the second period.¹¹ Likewise, observing $D_{1}^{'} = L$ may imply that $D_{1} = L$ , or that $D_{1} = H$ and the agent pushed out the demand $H - L$ to the second period.

To show the incentives and dynamics at play with sales push out and pull in, we provide the following illustration. Consider first the principal’s problem at $T = 2$ . At the beginning of the second period, a new contract is initiated. To induce a specific effort level in the second period, the principal will set the second period’s quota level and bonus value on the basis of the observed earlier outcome $D_{1}^{'}$ , which we denote by $χ_{2} (D_{1}^{'})$ and $b_{2} (D_{1}^{'})$ , respectively. Although we relegate the details to the “Period-by-Period Contract with Sales Push Out and Pull In” section of the Appendix, the result is that to induce $〈 e_{2}^{H}, e_{2}^{L} 〉 = 〈 1, 0 〉$ , the principal sets $χ_{2} (D_{1}^{'}) = 2 H - D_{1}^{'}$ , and to induce $〈 e_{2}^{H}, e_{2}^{L} 〉 = 〈 0, 1 〉$ , the principal sets $χ_{2} (D_{1}^{'}) = H + L - D_{1}^{'}$ . In both cases, $b_{2} (D_{1}^{'}) = \frac{ϕ}{p - q}$ . This implies that the principal readjusts the quota level but not the bonus amount to achieve a desired effort profile in the second period. Anticipating this, if the first period’s quota level is not high enough, the agent, rather than exerting effort, prefers to pull in sales from the second period to achieve the first-period quota and obtain the first-period bonus. In turn, to induce $e_{1} = 1$ , the principal must set $χ_{1}$ high enough ( $H + L$ , for instance) such that it would become impossible for the agent to simply secure an early bonus by pulling in sales if $D_{1} = L$ , but it would still achievable when $D_{1} = H$ . Once we recognize the above incentives, the intuition for the rest of the analysis is quite similar to the main model. Table A5 in the “Period-by-Period Contract with Sales Push Out and Pull In” section of the Appendix presents the optimal short-term contract to induce different effort profiles. We summarize the results in the following proposition and illustrate them in Figure 5.

Proposition 3: In the presence of sales push out and pull in, under short time horizon contracting, the “hockey stick” expected effort profile, $e = (0, q)$ , the “giving up” expected effort profile, $e = (1, p)$ , and the “resting on laurels” expected effort profile, $e = (1, 1 - p)$ , are optimal for the principal to induce in the respective regions in Figure 5.

Figure 5.

Effort outcomes under the optimal contracts with sales push out and pull in. The optimal contracts and regions are defined in Table A5 in the “Period-by-Period Contract with Sales Push Out and Pull In” section of the Appendix.

Proposition 3 suggests that with sales push out and pull in, agents’ dynamic gaming can be expected in certain parameter spaces, even if the principal designs their contract optimally by accounting for agents’ gaming behaviors. If we compare the effort outcomes in Figure 5 and Figure 3, we find that short-term contracting in the presence of sales push out and pull in achieves the same outcome as long-term contracting in the parameter space where the “hockey stick” expected effort profile, $e = (0, q)$ , or the “giving up” expected effort profile, $e = (1, p)$ , is optimally induced. This suggests that even if each contract signed only spans one period, the principal can take advantage of the agent’s push out and pull in behaviors to induce dependence between two periods and provide incentives that are as efficient as a two-period contract provides.

Essentially, the principal wants to make the agent’s later effort choice dependent on earlier outcomes to save on incentive cost. In the long time horizon contract with independent periods, this is achieved through an agent’s dynamic gaming in response to a nonlinear two-period contract. In this section, it is achieved by an agent’s incentive to push out and pull in sales in response to a sequence of short time horizon contracts. Therefore, in this part of the parameter space, the agent’s ability to push out and pull in sales benefits the principal, even when the principal can only sign period-by-period contracts, by enabling more efficient effort exertion than would be possible if this ability were absent.

Furthermore, a “resting on laurels” expected effort profile $e = (1, 1 - p)$ is optimal for the principal under short time horizon contracting when agents can transfer sales in the parameter space in which, roughly speaking, constant effort exertion $e = (1, 1)$ is optimal under long time horizon contracting. To see this, note that in the presence of sales push out and pull in, the short time horizon contract fails to induce $e = (1, 1)$ . This is because when the principal anticipates that agents may push out and pull in sales, and agents also realize that the principal will respond optimally to it, the principal is unable to induce consistently high efforts. It is also worth noting that the period-by-period contract in the presence of sales push out and pull in still performs weakly worse for the principal than the two-period contract. Namely, it performs the same as the two-period contract for inducing $e = (1, p)$ , $e = (0, q)$ , and $e = (0, 0)$ , but it fails to induce $e = (1, 1)$ . This is because, given any period-by-period contract, the agent can always rely on pushing out or pulling in sales (rather than exerting effort in both periods) to obtain the same expected bonus as they would get by exerting effort in both periods.

Interdependent Periods with Limited Inventory

In this section, we assume that the principal has a limited amount of product to sell across the two periods, such that the demand outcome in the first period can change incentive provision for inducing demand in the second period. We extend the model by assuming that the principal has limited inventory, denoted by $Ω > 0$ , to be sold across two periods. The inventory cannot be replenished before Period 2 starts, and any demand more than $Ω$ is lost (i.e., actual sales $\bar{D} = min {D_{1} + D_{2}, Ω}$ ). Therefore, the two periods become dependent through $Ω$ . We assume zero inventory costs for simplicity. We keep everything else in the model the same as in the independent Two-Period Contract section without sales push out and pull in. To focus on the interesting cases, we only consider the case in which $H - L \geq \frac{p}{{(p - q)}^{2}} ϕ$ so that the market upside potential is large enough to justify effort induction in both periods given unlimited inventory. In this case, we solve the firm’s contracting problem by replacing the total demand $D$ by its truncated value $\bar{D} = min {D_{1} + D_{2}, Ω}$ . Recall that we assume compensation cannot be decreasing in total sales (i.e., the bonus paid when demand in both periods is realized as high should be no lower than the bonus paid when demand in only one of the periods is realized as high). We obtain the following proposition:

Proposition 4: With limited inventory, the “hockey stick” expected effort profile, $e = (0, 1 - q)$ , and the “resting on laurels” expected effort profile, $e = (1, 1 - p)$ , are the expected effort profiles that the firm optimally induces under the long time horizon contract in the respective regions in Figure 6(a).

Figure 6.

Effort outcomes and optimal contracts under limited inventory. (a) Effort outcomes under the optimal two-period contract. The regions are defined in Table A6 in the “Interdependent Periods with Limited Inventory” section of the Appendix. (b) Contract comparison. The regions are defined in Table A7 in the “Interdependent Periods with Limited Inventory” section of the Appendix.

The proof of the preceding proposition is in Section OA2 in the Web Appendix. We find that under a two-period contract with limited inventory, the firm can optimally induce both the “hockey stick” effort profile $e = (0, 1 - q)$ and the “resting on laurels” effort profile $e = (1, 1 - p)$ . The expected effort profile $e = (0, 1 - q)$ indicates that the agent does not exert effort in the first period and exerts effort in the second period only if the demand outcome of the first period is low (note the difference from the hockey stick expected effort profile $e = (0, q)$ obtained in the main analysis, in which the agent does not exert effort in the first period and exerts effort in the second period only if the demand outcome of the first period is high). The expected effort profile $e = (1, 1 - p)$ indicates that the agent exerts effort in the first period and exerts effort in the second period only if the demand outcome of the first period is low. This can be interpreted as “resting on laurels” in the second period, as no effort is exerted if the first period is a success (note that this is also different from the “giving up” effort profile).

A key insight is that when the working environment is easy enough for the agent (i.e., the total amount of product to be sold, $Ω$ , is small or $H$ or $p$ is large), the principal may have the incentive to induce agents to postpone effort (i.e., work hard only in the late period but only if the first period demand outcome is unfavorable). Furthermore, as a result of limited inventory, the conclusion from the basic model that firms weakly prefer two-period contracting over period-by-period contracting does not always hold true. In the current scenario, the period-by-period contract outperforms the two-period contract when limited liability is very small or very large, since it gives the principal more flexibility in adjusting the contracts (note that we maintain the assumption that compensation is nondecreasing in sales). Specifically, the principal prefers the period-by-period contract to the two-period contract in Regions III and IV in Figure 6(b) (the detailed solution is in Section OA3 in the Web Appendix). In Region III, where the agent’s limited liability is large, the period-by-period contract implements $e = (0, 1 - q)$ and performs the best for the principal. This is because, under the constraint that compensation cannot be decreasing in sales, the two-period contract rewards the agent more than the period-by-period contract when demand outcomes at both periods are $H$ . In Region IV, where the agent’s limited liability is small, the period-by-period contract implements $e = (1, 1 - p)$ and performs the best for the principal. The two-period contract cannot replicate the period-by-period contract for inducing $e = (1, 1 - p)$ because it suffers from the agent’s dynamic gaming. To ensure early effort exertion, the principal pays a higher bonus under the two-period contract when demand outcomes in both periods are $H$ than under the period-by-period contract. In those scenarios, a period-by-period contract that gives the principal flexibility to adjust quota levels performs better than a two-period contract.

Conclusions and Discussion

Firms employ and reward salespeople over multiple time periods. We address a question that arises in this context: Is an agent’s dynamic gaming under a long time horizon contract (“delaying effort,” “giving up,” and “resting on laurels”) necessarily suboptimal for the firm? We employ a two-period repeated moral hazard framework with stochastic demand and unobservable effort, and we assume the agent to be risk neutral with limited liability.

We show that the two-period expected effort profile under the optimal contract may not always be high effort exertion in both periods. Under different conditions, the optimal effort induced by the principal may be such that, in one of the periods, the level of effort exerted is expected to be lower than the highest level of effort exertion. Figure 1 shows three canonical expected effort profiles that the firm may optimally induce: the leftmost plot indicates high effort in both periods, the middle plot indicates no effort in the first period and low expected effort exertion in the second period (the “hockey stick” profile), and the rightmost plot indicates high effort in the first period but low expected effort in the second period (the “giving up” or “resting on laurels” profile).¹² The latter two expected effort profiles appear to be suboptimal agent behaviors, even though the firm may want exactly these effort profiles under optimality. In essence, we show that a number of different effort profiles are possible under the optimal contract, and high effort exertion in every period is actually not always desired by the principal. Therefore, one has to be careful in making inferences about contract effectiveness from agents’ realized multiperiod effort exertion profiles.

The firm induces effort profiles characterized by gaming when they use a contract over a long time horizon that enables them to reward the salesperson only when the salesperson reaches a relatively high quota level. However, the firm always has the option to implement a contract that prevents gaming by rewarding the agent for intermediate demand outcomes. The reason that the firm may choose an extreme contract and not prevent gaming is because the extreme contract enables the firm to efficiently implement effort (the “incentive effect”) even though it reduces the expected demand (the “demand effect”). We show that our results are strengthened if we make the different time periods interdependent by allowing the salesperson to shift demand between periods or by assuming that a specific amount of inventory must be sold across the two periods.

We conclude with a brief discussion of some of our assumptions and limitations. We have assumed binomial demand and binary effort levels. However, since our main insights are driven by the tradeoff between providing incentives at the cost of gaming losses, we expect them to hold in a continuous setting as well. By the same token, if we allow periods to be dependent in other ways (e.g., a high demand outcome in the early period makes a high demand outcome in the second period more or less likely), our key insights will hold. Finally, we have not considered phenomena such as “racheting” in which future quota targets for a salesperson are determined on the basis of past performance. We leave these considerations, which fall under an asymmetric information paradigm, for future research.

Supplemental Material

Supplemental Material, jmr.18.0141-File003 - Multiperiod Contracting and Salesperson Effort Profiles: The Optimality of “Hockey Stick,” “Giving Up,” and “Resting on Laurels”

Supplemental Material, jmr.18.0141-File003 for Multiperiod Contracting and Salesperson Effort Profiles: The Optimality of “Hockey Stick,” “Giving Up,” and “Resting on Laurels” by Kinshuk Jerath and Fei Long in Journal of Marketing Research

Footnotes

Appendix

Associate Editor

Duncan Simester

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Online supplement:

Notes

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