K. SudhirYale School of Management, New Haven, Connecticut 06520, [email protected]
We estimate a dynamic structural model of sales force response to a bonus-based compensation plan. Thispaper provides substantive insight into how different elements of the compensation plan enhance pro-
ductivity. We find evidence that (1) bonuses enhance productivity across all segments; (2) overachievementcommissions help sustain the high productivity of the best performers, even after attaining quotas; and (3) quar-terly bonuses help improve performance of the weak performers by serving as pacers to keep the sales forceon track in achieving its annual sales quotas. The paper also introduces two main methodological innovationsto the marketing literature: First, we implement empirically the method proposed by Arcidiacono and Miller[Arcidiacono P, Miller RA (2011) Conditional choice probability estimation of dynamic discrete choice modelswith unobserved heterogeneity. Econometrica 79(6):1823–1867] to accommodate unobserved latent-class hetero-geneity using a computationally light two-step estimator. Second, we illustrate how discount factors can beestimated in a dynamic structural model using field data through a combination of (1) an exclusion restric-tion separating current and future payoff and (2) a finite-horizon model in which there is no forward-lookingbehavior in the last period.
History : Received: July 24, 2011; accepted: August 4, 2013; Preyas Desai served as the editor-in-chief andMiguel Villas-Boas served as associate editor for this article. Published online in Articles in AdvanceNovember 7, 2013.
1. IntroductionPersonal selling is one of the most important ele-ments of the marketing mix, especially in the con-text of business-to-business (B2B) firms. An estimated20 million people work as salespeople in the UnitedStates (Zoltners et al. 2008). Sales force costs averageabout 10% of sales revenues and as much as 40% ofsales revenues for certain industries (Heide 1999). Inthe aggregate, U.S. firms spent more than $800 billionon sales forces in 2006, a sum three times larger thanadvertising spending (Zoltners et al. 2008).
Marketing researchers routinely create responsemodels for marketing mix instruments such as price,sales promotion, and advertising. Meta-analyses ofvarious research studies estimate that the sales forceexpenditure elasticity is about 0.34 (Albers et al. 2010),relative to about 0.22 for advertising (Assmus et al.1984) and about −2.62 for price (Bijmolt et al. 2005).
Relative sales force expenditure elasticity is usefulin determining the relative effectiveness of differentinstruments in the marketing mix, but little insight isgiven on how to design a sales force compensationplan, which is widely understood to be the primarytool by which firms can induce their sales forces toexert the optimal levels of effort and thus to optimizethe use of sales force expenditures.
A compensation plan can consist of many compo-nents: salary, commissions, and bonuses on achiev-ing a certain threshold of performance called quotas.Figure 1 shows a variety of compensation plans thatinclude combinations of these components. Which ofthese different types of contracts should a particu-lar firm offer to its sales force to maximize prof-its? What combination of salary, commission, and/orquota-based bonuses should one use? Having chosenthe compensation components, what should be the
Plan F: Commission + Bonus+ Overachievement commission
Ear
ning
s
Sales
Plan D: Commission withfloor and ceiling
Ear
ning
s
Sales
Plan A: Pure commission
specific parameters for commission rate, quotas, andbonus levels? Furthermore, what is the right frequencyfor quota targets? For example, should there be quar-terly or annual quotas? A firm needs to understandhow the sales force will respond to different elementsto develop an appropriate compensation plan.
Thus, this paper has two substantive goals: First, itseeks to gain insight into how a firm should designits compensation plan. For example, should a firmoffer quotas and bonuses in addition to commis-sions? Second, we aim to determine how often quotasshould be set and bonuses paid. For example, shoulda firm implement a monthly, quarterly, or annualbonus? Should it offer a quarterly bonus in addi-tion to an annual bonus? In the education literature,researchers have argued that frequent testing leads tobetter performance outcomes (Bangert-Drowns et al.1991). Can quarterly quotas serve a similar roleto improve outcomes? As in education, where fre-quent exams help students to be better prepared forthe comprehensive final exam, frequent quota-bonusplans may serve as a mechanism to keep the salesforce motivated to perform well in the short run soas to be within striking distance of the overall annualperformance quota.
Quotas and bonuses are widely used by firms.According to Joseph and Kalwani (1998), only about24% of firms use a pure commission-based plan; therest used some form of quotas. As per the 2008 Incen-tive Practices Research Study by ZS Associates, 73%,85%, and 89% of firms in the pharma/biotech, med-ical devices, and high-tech industries, respectively,use quota-based compensation (Training 2008). Yet,despite the ubiquity of quota-based compensation,there is considerable controversy in both the theoret-ical and empirical literatures about the effectiveness
of quotas and bonuses relative to straight linear com-mission plans.
We begin with a discussion of the relevant theoreti-cal literature. Using the principal-agent framework ofHolmstrom (1979), Basu et al. (1985) and Rao (1990)find that the combination of salary and commission(usually nonlinear with respect to sales) is optimal.In this light, quota-bonus plans can be seen as anapproximation to a continuous nonlinear plan thatalso takes into account heterogeneity in territory salespotential. However, under the assumption of linearexponential utility and normal errors, Holmstrom andMilgrom (1987) and Lal and Srinivasan (1993) showthat a linear commission scheme can achieve the first-best outcome. Yet why do we see quota-bonus plans?Raju and Srinivasan (1996) suggest that even though acommission-over-quota plan may not be theoreticallyoptimal, it provides the best compromise betweenefficiency and ease of implementation. Others arguethat quota-based plans offer high-powered incentivesthat can motivate salespeople to work harder (e.g.,Darmon 1997). Park (1995) and Kim (1997) demon-strate that a quota-bonus plan may lead to the first-best outcome, but in their framework, a quota-bonusplan is just one of many possible plans that can lead tofirst-best outcomes. Oyer (2000) shows that when par-ticipation constraints are not binding, a quota-bonusplan with linear commissions beyond quotas can beuniquely optimal because it can concentrate the com-pensation in the region of effort where the marginalrevenue from effort minus the cost of compensationis maximized.
In terms of empirical work, Ferrall and Shearer(1999) and Paarsch and Shearer (2000) estimate staticstructural models of worker behavior given linear
contracts. Oyer (1998) is the first to empirically inves-tigate quota-based plans. Using aggregate sales acrossdifferent industries in different quarters, he con-cludes that quota-based plans encourage salespeopleto maneuver the timing of orders; this negative effectoverwhelms the positive benefit of quotas. Usingreduced-form analysis of individual-level data, Steen-burgh (2008) finds that the net improvement in rev-enues from effort dominates the inefficiencies inducedby intertemporal dynamic considerations and showsthat an aggregate analysis might have led to the oppo-site conclusion similar to that of Oyer (1998).
Copeland and Monnet (2009) estimate the firstdynamic structural model of worker productivity in acheck-sorting environment with nonlinear incentives.Unlike sales force productivity, where we observeonly aggregate sales, the outcomes associated withevery processed check are observed in their envi-ronment; hence, they have a qualitatively differentmodel. Our paper shares many similarities with arecent paper by Misra and Nair (2011), who alsoestimate a dynamic structural model of sales forcecompensation, although they use a very differentquota-compensation scheme. In contrast to our focuson quotas with bonuses (plan F in Figure 1), Misraand Nair analyze quotas with floors and ceilings oncommissions (plan D in Figure 1). They conclude thatquotas reduce performance. This is because of twocharacteristics of their quotas: First, the quota ceil-ing (beyond which salespeople receive zero additionalcompensation) limits the effort of the most produc-tive salespeople, who would normally have exceededthat ceiling. Second, the company followed an explicitpolicy of ratcheting quotas based on past productiv-ity. This reduces salespeoples’ incentives to work hardin any given period, because hard work is penalizedthrough higher future quotas. In contrast, we find thatquotas coupled with bonuses enhance performance.In the plan we consider, the company offers extraoverachievement commissions for exceeding quotasand uses a group quota updating procedure that min-imizes ratcheting effects. Thus, these two papers offercomplementary perspectives that enhance our under-standing of how quotas impact performance.
From a methodological perspective, our paper intro-duces two key ideas to the marketing literature. First,we accommodate latent-class heterogeneity within thetwo-step conditional choice probability (CCP) estima-tion framework—an issue that has been an economet-ric challenge for the literature for nearly two decades.Misra and Nair (2011) avoid the unobserved hetero-geneity issue by estimating each salesperson’s util-ity function separately.1 Although the use of two-step
1 This is similar to estimating individual-level utility functions inconjoint analysis or scanner panel data, when there are a large
estimation approaches has recently gained popular-ity (Hotz and Miller 1993, Bajari et al. 2007) becauseof their ease of computation relative to traditionalnested fixed-point estimation approaches (e.g., Rust1987), their use in empirical applications has been lim-ited by their inability to accommodate unobservedheterogeneity. Arcidiacono and Miller (2011) proposean algorithm that accommodates latent-class hetero-geneity within the two-step framework. Our paperintroduces this idea to the marketing literature andillustrates its implementation with an empirical appli-cation that accommodates unobserved heterogeneityusing two-step dynamic structural estimation.2
Second, unlike Misra and Nair (2011)—and ofbroader importance to the dynamic structural mod-eling literature—we estimate rather than assume dis-count factors. It is well known in the literatureon dynamic structural models that discount fac-tors cannot be identified in standard applicationsbecause there are no instruments that provide exclu-sion restrictions across current and future period pay-offs (Rust 1994). Hence the standard approach is toassume discount factors. We illustrate how the bonussetting provides us exclusion restrictions allowing usto estimate discount factors—thus demonstrating howfield data can indeed be used to estimate discountfactors.3 Since bonus payoffs occur only at the endof each quarter or year, in the nonbonus periods, theprobability of achieving the quota and receiving abonus will not affect the current payoff, only futurepayoffs. Only a forward-looking person (i.e., one witha nonzero discount factor) would be concerned withwhether she is close to the quota in nonbonus peri-ods. We demonstrate through reduced-form evidencethat such behavior exists in the data, and then weexploit this exclusion restriction to identify the dis-count factor. In addition, the finite-horizon nature ofthe sales force problem implies that, in the last period,the model reduces to a static model, for which it iswell known that utility can be identified. This fur-ther facilitates identification of the discount factor. Yaoet al. (2012) apply our idea to a cellphone usage con-text to identify discount factors. In their setting, how
number of observations per individual. More importantly, theapproach requires that salespeople exert effort equally across allcustomers—an assumption they show is valid in their data butunlikely to hold in general. Our latent-class approach works in themore common situation where there are limited observations perindividual.2 Finger (2012) and Beauchamp (2012) are two concurrent work-ing papers implementing Arcidiacono and Miller’s approach ineconomics.3 Use of exclusion restrictions to empirically infer discount factorsis important because, prior to this paper, the conventional wisdomwas that field data cannot be used for discount factor identificationand that one needs to use surveys (e.g., conjoint data as in Dubéet al. 2012).
close the user is to exceeding his or her monthly quotaserves as an exclusion restriction, much like in oursetting. Similar to the last period of our data, wherethe model reduces to a static model, they have datafor a period in which pricing is linear and thereforereduces to a static model.
There are three specific modeling and estimationchallenges in the structural estimation of response tocompensation plans, especially those with quotas andbonuses. First, we do not observe the effort of thesales force, only the outcome of the agent’s effort (i.e.,sales), which is correlated with effort. This requiresa modeling assumption on the link between salesand effort. The identification of effort poses particularchallenges in our application as a result of seasonal-ity and potential sales substitution across quarters.4
We discuss these issues in §2.3. Second, compensa-tion plans do not change over time. Here, we drawon an empirical insight from Steenburgh (2008) foridentification. In any given period, a sales agent’soptimal effort depends on her state—how close theperson is to achieving her quota. A sales agent mayfind it optimal to reduce her effort when she is closeor very far from achieving the quota, but she maystretch herself to reach the quota when she has a mod-erate chance of achieving it. Thus changes in salesin response to changes in the agent’s state (distanceto quota) within and across agents facilitate identifi-cation. Finally, quotas and bonuses induce dynamicforward-looking behavior; an agent choosing to exerteffort has to be concerned not just with the currentpayoff but with the effect of that effort on the likeli-hood of obtaining a bonus in the future. This requiresa dynamic structural model.
We estimate a dynamic structural model of salesforce response to various features of the compensationplan using sales force output and compensation datafrom a Fortune 500 firm that sells office durable goods.This firm uses plan F in Figure 1. In addition, bonusesare provided at two different frequencies: quarterlyand annual. As the compensation structure of thefocal firm features almost all dimensions in typicallyused compensation plans, we observe how the salesforce responds to the plan’s different dimensions. This“rich” plan provides us with two key benefits: First,the presence of bonuses helps us identify and esti-mate discount factors. Second, even though in theoryone can perform counterfactuals of any type of com-pensation plan if able to estimate structural param-eters (other than discount factors) for a salespersonwith a less rich compensation plan, an analyst or
4 This issue has parallels in empirical channel response models.For example, Sudhir (2001) makes an inference about manufactureractions (wholesale prices) from the observed retail price and salesto infer competition between manufacturers.
manager should have greater faith in the counterfac-tuals based on parameters that were estimated fromobserved responses to different elements of the com-pensation plan.
The rest of the paper is organized as follows. Sec-tion 2 discusses the institutional details of the com-pensation plan at the firm, provides a numericalexample that offers insights into how bonuses induceeffort, and provides some model-free evidence thatfacilitates model building. We present the model andthe estimation methodology in §§3 and 4, respec-tively. Section 5 discusses the estimation results andthe counterfactual analysis. Section 6 concludes.
2. Institutional Details andDescriptive Analysis
We first describe the details of the bonus-based com-pensation plan, followed by a numerical example toclarify how bonuses can help motivate the salesper-son and induce intertemporal effort that assists uswith identification. We then provide model-free evi-dence of forward-looking behavior, seasonality, andthe absence of sales substitution across quarters.
2.1. The Compensation PlanThe focal firm under study is a highly regarded multi-national Fortune 500 company that sells durable officeproducts primarily using its own direct sales force.Each sales agent is given an “exclusive” territory; thefirm traditionally does not encourage group work orteam cooperation among the sales force. The firm alsohas an indirect sales force through “rep” firms who donot compete with the direct sales force. They are paidpurely on commission, unlike the regular sales force.5
Our analysis focuses on sales performance datafrom 348 salespeople from the regular sales force dur-ing the three-year period 1999–2001. The firm’s com-pensation structure follows the pattern in plan F ofFigure 1, and the details of the compensation schedulefor the period of analysis are described in Table 1. Weprovide descriptive statistics of the data in Table 2.
Every month, salespeople receive a fixed monthlysalary (on average, $3,585) and a commission of 1.5%of the revenues generated in that month. In the firstthree quarters, a quarterly lump-sum bonus of $1,500is paid if the quarterly quotas are met. At the end ofthe year (i.e., end of the fourth quarter), an annuallump-sum bonus of $4,000 is paid if the annual quotais met. Furthermore, an overachievement commis-sion of 3% is paid for any excess revenues beyondthe annual quota. There are no caps on revenues forwhich an agent could obtain commissions or over-achievement commissions. Overall, for a salespersonthat meets all quotas, the salary component will beroughly 50% of total compensation.
5 Such rep firms are the focus of Jiang and Palmatier (2009).
Quarterly bonus $1,500 awarded if the quarterly revenue exceedsquarterly quota
March, June, September
Annual bonus $4,000 awarded if the annual revenue exceedsannual quota
December
Base commission About 1.5%a paid in proportion to the revenuegenerated each month
Every month
Overachievementcommission
About 3%a paid in proportion to the total cumulativerevenue surpassing the annual quota
December
aThese numbers are approximate for confidentiality reasons.
In building annual and quarterly quotas for itssales force, the company uses internal metrics called“monthly allocated quotas” (based on expectedmonthly revenues, given seasonality, and territorialcharacteristics), though these are not used for com-pensation. We do not use these quotas for our mod-eling and estimation, but we use them to benchmarkperformance in the reduced-form analysis.
The most important element in performance eval-uation within the firm is the annual quota; i.e., thefirm views a salesperson as having a successful yearif the annual quota is met. From Table 2, we see thatsalespeople meet their annual quota roughly 50% ofthe time. All quotas, including the quarterly quotas,are updated annually and reset in January of the fol-lowing year. However, managers at the firm informedus that they were sensitive to the fact that ratchetingquotas based on individual performance could leadthe sales force to purposefully reduce performance.To avoid such an adverse impact on sales, the quotaadjustment year after year was done based only ongroup performance, where each individual’s currentperformance would have minimal direct impact onher future quota. Nevertheless, we test for statisticalevidence of ratcheting in the data. Table 3 reports theresults of the tests. In Model 1, we regress the percent-age increase in the annual quota on whether the sales-person met her quota in the previous year. We findno significant effect of meeting the previous year’squota on the current year’s quota. In Model 2, we
Table 2 Descriptive Statistics
No. of salespeople 348Average salary (US$) 3,585Average tenure (years) 11.8
Average quota % achieving Average sales(US$ ’000) quota (US$ ’000)
Dependent variable 4AQ y −AQ y−15/AQ y−1 log4AQ y 5 log4AQ y 5
I4Salesy−1 ≥ AQ y−15 0.0364 0.0033(0.0229) (0.0299)
log4%AQ y−15 0.0347(0.0196)
Year fixed effects Yes Yes YesSalesperson fixed effects NA Yes Yes
Notes. Standard errors are shown in parentheses. y represents each year inthis analysis.
regress the logarithm of the annual quota on whethera salesperson met her quota in the previous year, thistime controlling for salesperson fixed effects. Again,we find no significant effect. Finally, in Model 3, weregress the logarithm of quota on the logarithm ofthe previous year’s performance relative to the annualquota. Again, we find no significant effects. Thus,consistent with the intuition of the firm’s managers,who sought to avoid the negative impact of ratch-eting on employee performance, we find no directstatistical evidence of ratcheting. For this application,we therefore abstract away from quota ratcheting andtreat each salesperson’s quarterly and annual quotasas exogenous. To the extent that a salesperson’s quotais partly updated based on her own performance,there are still potential ratcheting effects that may bedetectible with longer panel data.6
2.2. Numerical ExampleBefore building an empirical model of salesperson’sresponse to a bonus-based compensation scheme, weprovide a numerical example that illustrates how aquota-bonus scheme can be more effective than purecommissions in generating more effort. The exam-ple will also show that a person’s distance to quota
6 The assumption of no ratcheting allows us to make the finite-horizon argument for identification of the discount factor discussedin §4.3. The exclusion restriction argument for identification will bevalid even if we allow for ratcheting.
can induce variations in optimal effort (and gener-ated sales)—providing intuition for our identificationstrategy.
Let the utility function of the salesperson thattrades off effort (e) and income from sales (s) and whohas sold S units at the beginning of the new period be
U4s1 e1 S5= −de2+ rs +BI8s+S≥Q91
where −d is the disutility parameter and r is the com-mission rate (d > 01 r > 0), and B is the bonus forreaching quota (Q). For illustration, let us assume adirect match between sales and effort; i.e., s = e.
First, we illustrate the effectiveness of bonusesusing a static model. For simplicity, assume S = 0.In the pure commission case with no bonus, whered = 1, r = 10, and B = 0, the optimal effort is e∗ = 5.In the bonus case, where Q = 10 and B = 30, the opti-mal effort is higher, at e∗ = 10, and the compensationcost to the firm is $130. To achieve the same levelof sales and effort (e∗ = 10) from a pure commissionplan, the commission rate r has to increase to 20 andcosts more for the firm, at $200. Figure 2(a) illustratesthese results graphically. Thus, the quota-bonus planin this context induces more effort and sales for thesame level of compensation. We are not claiming thata bonus-based plan is the optimal compensation plan.Through the example above, we are simply illustrat-ing that a nonlinear plan can be more efficient than alinear plan, suggesting that the optimal compensationplan is possibly nonlinear and that the bonus plan
Figure 2(a) How Quotas and Bonus Serve as Stretch Goals
–20
0
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40
0 2 4 6 8 10
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d = 1, r = 10, B = 0, e* = 5
–20
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d = 1, r = 10, B = 30, e* = 10
020406080
100120
0 2 4 6 8 10
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d = 1, r = 20, B = 0, e* = 10
Figure 2(b) Effort as a Function of Distance to Quotas
d = 2, r = 10, B = 30,S = 0, e* = 2.5
d = 2, r = 10, B = 30S = 5, e* = 5
d = 2, r = 10, B = 30S = 7, e* = 3
–150
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can be a way to approximate the optimal nonlinearincentive plan.
Second, we show how distance to quota inducesvariation in effort. Let d = 2, r = 10, Q = 10, andB = 30. Consider three scenarios of distance to quotas:S = 0 (far away from quota), S = 5 (moderately closeto quota), and S = 7 (close to quota). Figure 2(b)shows that the optimal effort levels are e∗ = 205, 5,and 3, respectively, for the three cases; the salesper-son exerts maximum effort at S = 5, when moderatelyfar away from quota, all else being equal. We use thisvariation of optimal levels of effort (thus, sales) withinagents across time to identify their preferences in ourmain model.
2.3. Model-Free AnalysisWe consider three features of the data that informmodel development. First, we provide evidence offorward-looking behavior induced by bonuses andhence the need to develop a dynamic model. Sec-ond, we demonstrate the presence of seasonality inthe data, which therefore requires us to accommodatethis in the model. Third, we show that sales substi-tution across quarters by sales agents appears to belimited in our data; we therefore abstract away frommodeling the timing of sales bookings by sales agents.
2.3.1. Forward-Looking Behavior. As discussedpreviously, bonuses provide an exclusion restriction inthat they do not impact current payoffs, only futurepayoffs. To the extent that a sales agent’s perfor-mance is affected by variables relating to proximity
to bonuses, this is evidence of forward-lookingbehavior.7 But proximity to bonus quota will impactperformance only if agents have a reasonable chanceof making the quota. Figure 3 shows the graph of theprobability of reaching the annual quota, conditionalon the cumulative fraction of annual quota (%AQ)achieved by November. It is clear that there is verylittle chance of achieving quota if %AQ < 005. Wetherefore divide agents by their state %AQ< 005 and%AQ> 005 to test whether the state affects sales andestimate regressions on sales performance in Novem-ber as a function of their states. Table 4 reports theresults of the regressions. Consistent with forward-looking behavior, the state %AQ is significant only foragents with %AQ> 005.8
For additional evidence of forward-looking behav-ior, we show scatterplots and the best-fitting non-parametric smoothed polynomial (and its 95% con-fidence interval) of sales revenues normalized bymonthly allocated quotas in the quarterly bonusmonths (March, June, September, and December)against the percentage of quota attained by the pre-vious month in Figure 4(a). For March, June, andSeptember, the x axis is the percentage of quarterlyquota completed (%QQ), whereas for December, thex axis is the percentage of annual quota completed(%AQ). The vertical line shows the %QQ and %AQ atwhich salespeople, on average, achieve their monthlyallocated quotas.
7 A similar argument is made in providing evidence of forward-looking behavior by students in the textbook market by Chevalierand Goolsbee (2009).8 The qualitative conclusions are robust in the range of thresholdsof %AQ from 0.4 to 0.6. Note that if there were no overachievementcommission, agents very close to the quota may reduce their effort.
Table 4 Sales Performance in November
Estimate
%AQ< 005 %AQ> 005
Intercept 0005∗∗∗ 0006∗∗∗
4000185 40001235
%AQ 0006 0004∗∗∗
4000495 4−0001265
Note. Standard deviations are shown inparentheses.
∗∗∗p < 0001.
Two key elements stand out from Figure 4(a). First,across the board there is little reduction in effort whensalespeople are close to achieving quota, most likelyas a result of the overachievement commission rate.Second, there is a steady increase over time in the%QQ and %AQ threshold beyond which salespeoplereach their monthly targets. The threshold is about25% in March, 35% in June, 45% in September, andclose to 70% in December. Early in the year, even ifbelow targets, salespeople still have hopes of receiv-ing a large annual bonus by working hard. As onegets closer to year-end, the chances of reaching thequota become less likely, and salespeople respond byreducing effort even at higher levels of %AQ and%QQ. Since annual bonuses should have no impacton current payoffs in March, June, or September,only on future payoffs, this is suggestive of forward-looking behavior.
The next set of graphs presented in Figure 4(b)shows the same relationship in the pre-bonus months(February, May, August, and November) and providesadditional evidence for forward-looking behavior. InFebruary, May, and even August, at all levels of %QQ,the salesperson on average sells above the monthlyallocated quota. This is because hard work (and somegood luck in the form of positive sales shocks) maygive the salesperson a reasonable chance of attainingthe smaller quarterly targets. However, in November,only at a very high level of %AQ does the salespersonsell above the monthly allocated quota, because onehas a very limited chance of closing the large gap injust two months. In the pre-quarterly bonus months,the immediate future quarterly bonus impacts behav-ior, even though it has no impact on the current pay-off; this indicates more conclusive forward-lookingbehavior.
This evidence leads to a natural question: Shouldthe large annual bonus be split into a quarterly bonus(as in other months) and an annual bonus? The quar-terly bonus can prevent salespeople from giving up inNovember, even if they do not have a chance of reach-ing the annual quota. But with such a quarterly quota,early in the year, agents may have limited incentive towork hard after reaching quarterly quotas. How these
Figure 4(a) Sales and Percentage Quota Achieved—Bonus Months
0
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ar s
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% quarterly quota sold by Feb
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0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
% annual quota sold by Nov
two issues trade off is an empirical question that wesubsequently address in the counterfactual analysis.
2.3.2. Seasonality. Figures 5(a) graphs the aver-age revenues over the months for the regular sales
Figure 4(b) Sales and Percentage Quota Achieved—Pre-Bonus Months
0
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eb q
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0 0.2 0.4 0.6 0.8 1.0 1.2% quarterly quota sold by Jan
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% quarterly quota sold by Apr
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% annual quota sold by Oct
force. There are clear peaks at the end of each quar-ter. These peaks could be due to either seasonalityor bonuses at the end of each quarter. Figure 5(b)shows the index of indirect sales revenue (ISR) for
Notes. The ISR index is on a base of 1, reflecting the indirect sales in January1999. Here, we have averaged the index across three years (1999–2001).
Figure 5(c) Revenues from the Regular Sales Force and ISR IndexMultiples
0
50
100
150
200
250
300
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Revenues ISR*30
ISR*40 ISR*50
the pure commission indirect sales force.9 Becausethe ISR index is not contaminated by bonuses, weuse it to control for seasonality and isolate the effectof bonuses on salesperson effort and revenues. Tobuild intuition for how the ISR index can help con-trol for seasonality and isolate effort, see Figure 5(c),which graphs the average revenues of the direct salesforce and multiples of the ISR index. At a multiple ofaround 50, the ISR virtually mimics the average rev-enues, making the revenues from the commissionedand bonus sales force close to identical. This sug-gests that bonuses are not effective in inducing addi-tional effort. When the ISR index has a multiple of 30or 40, even after the overall seasonality is accounted
9 For the indirect sales force, the firm only provided us with an indexof revenues. The ISR index is set to a base of 1 for January 1999.
Table 5 Testing for Sales Substitution Across Quarters
Model 1 Model 2
Other months of quarter 168087∗∗∗ 101076∗∗∗
4340575 4340355
First month of quarter 147079∗∗∗ 101086∗∗∗
4340875 4340525
Other months of quarter× 91009∗∗∗ 56058∗∗∗
Previous month % distance to quota 450625 450905
First month of quarter× 2059 0015Previous month % distance to quota 470025 460935
ISR (indirect revenue) index 38072∗∗∗
420265
Salesperson fixed effects Yes Yes
Note. Standard deviations are shown in parentheses.∗∗∗p < 0001.
for, there is a gap in revenue that we interpret asinduced by effort. It is interesting that these gapsare larger at the end of the quarter, suggesting thevalue of bonuses in inducing effort. We empiricallyestimate the multiple for the ISR index to control forseasonality. We later show in §5.1 that the ISR mul-tiple for our model is about 25, given that the aver-age of lagged annual quota is 1,639. We acknowledgethat accounts handled by the direct and indirect salesforce are likely systematically different, but our main-tained assumption is that the seasonality multiplierson sales are identical across the two types of accounts.We acknowledge that this assumption is a limitationto fully control for seasonality.
2.3.3. Sales Substitution Across Quarters. Onepossibility is that salespeople giving up at the endof the quarter may be doing so to increase the oddsof meeting quotas in subsequent quarters by simplynot booking the sales in the current quarter. Alter-natively, a salesperson who is meeting the quarterlyquota may simply shift sales to the next quarter toincrease the odds of meeting next quarter’s quota.In either case, we should see a positive relationshipbetween the first month of the quarter and the previ-ous quarter’s percentage cumulative performance toquota (%QQ or %AQ). However, in Table 5, we findthat the coefficient of First month of quarter× Previousmonth (%) distance to quota is statistically insignificantand very small in magnitude.10 We therefore abstractaway from the challenges of modeling sales substi-tution in our application. We acknowledge that ourtest may have limited power in identifying timingbehavior; future research should explore more seri-ously the issue of sales timing and the attendant mod-eling challenges.
10 Managers of the focal firm stated that they monitor sales agentsextensively to prevent such time shifting of sales bookings. Also,customers play a major role in determining when sales occur inthis environment.
3. ModelBased on the model-free evidence, we build adynamic model of sales force response to the quota-based compensation scheme. The timing of the modelis as follows:
1. At the beginning of each year, the firm choosesan annual compensation plan.
2. Each month, agents observe their current stateand exert effort in a dynamically optimal manner.
3. An idiosyncratic sales shock is realized; theshock plus the agent’s effort determines the agent’srealized sales for the period. The agent receives com-pensation.
4. The realized sales of the current period affect theagent’s state of the next period. Steps 2 and 3 arerepeated each month until the end of the year, andSteps 1–3 are repeated each year.
We describe the model in five parts: (i) the com-pensation plan, (ii) the sales agent’s utility function,(iii) state transitions, (iv) the effort as a function ofstate variables, and (v) the optimal effort choice bythe sales agent.
3.1. Sales Response ModelWe model the sales revenue function (Sit) for sales-person i at time t in two parts: (1) a base level ofsales independent of effort, parameterized by demandshifters (zDit ), and (2) sales induced as a result of effort(eit), parameterized by effort shifters that include ter-ritory and salesperson characteristics (zEit):
Sit = f 4zDit 5+ eit4zEit5+ �it1 (1)
where �it is an additive sales revenue shock not antic-ipated by the salesperson when choosing effort.
As discussed previously, the market potential variesacross territories and across time. To account for thecross-sectional variation in market potential, we usethe annual quota from the previous year (AQi1 y−1). Toaccount for seasonality of demand across months, weuse the ISR index (ISRt). We also include an interac-tion between the two variables to allow for season-ality to have a different impact on differently sizedterritories.
For effort shifters in eit , we use the followingvariables: given that effort is a function of demandshifters, we include both AQi1 y−1 and ISRt in zEit . Asdiscussed in the motivation, the salesperson’s statewith respect to achieving her quota will have animpact on the effort she expends. We therefore usethe cumulative percentage of quarterly and annualquotas completed till time t (%QQit1%AQit) as vari-ables that affect effort. In addition, we use the time(month type) within quarterly and annual quota peri-ods to allow for different effort levels at different tem-poral distances to bonus payments; thus, the effort
policy function is different for each month. Further-more, we allow a time-invariant salesperson-specificvariable, tenure with the firm (�), to moderate thelevel of effort.
Note that unlike the demand-shifter function f 4 · 5,which is common across all salespeople, the effortfunction will vary across salespeople. Specifically, weallow for salespeople to belong to one of multiple dis-crete segments; hence these effort functions will beestimated at the segment level. We estimate the effortfunction nonparametrically using Chebyshev polyno-mials of the variables described above.
3.2. Compensation PlanThe compensation plan has three components: (i) themonthly salary wit ; (ii) the end-of quarter bonus Biqt
received for achieving the corresponding quarterlyquota Qiqt , and the end-of-year bonus Biyt received forachieving the corresponding annual quota Qiyt ; and(iii) a commission rate, rit , per dollar’s worth of salesand an overachievement commission rate, r ′
it , givenat the end of the year for sales over and above theannual quota for each individual i at time t. We rep-resent the compensation plan for a salesperson i bythe vector �it = 8wit1Qiqt1Qiyt1Biqt1Biyt1 rit1 r
′it9.
3.3. Salesperson’s Per-Period UtilityIn each period t, salesperson i receives a positive util-ity of wealth Wit earned based on realized sales anda disutility C4eit3�i5 from exerting effort eit . Thus theutility function is defined as
U4eit1Sit3�i1�i1�i5
=E6W4Sit3�i57−�ivar6W4Sit3�i57−C4eit3�i51
where �i and �i are the risk aversion and disutilityparameters, respectively, for salesperson i.11
Given the sales levels and the compensation plan,the wealth for individual i, Wit , can be computed;Wit arises from four components: the per-periodsalary component wit , the lump-sum bonus compo-nent Bit , the commission component Cit , and theoverachievement commission component OCit . Thedetailed expressions of wealth is as follows:
Wit = wit +Bit +Cit +OCit1
Bit = IqtI
(
zi1t +sit4eit4z
Eit51 z
Dit3�i5+ �it
Qiqt
> 1)
Bqt
+ IytI
(
zi2t +sit4eit4z
Eit51 z
Dit3�i5+ �it
Qiyt
> 1)
Byt1
11 In the case of the constant absolute risk-aversion utility function(exponential utility function) with normal errors and a linear com-pensation plan, this functional form represents the certainty equiv-alent utility of the agent. Here, we consider the utility function tobe a second-order approximation to a general concave utility func-tion with a constant level of risk aversion.
where zi1t and zi2t are the percentage of quarterly andannual quotas completed, respectively, by salesper-son i until time t; Iqt and Iyt are indicators for whethertime t is a quarterly or annual bonus period.
In our empirical analysis, we use a quadratic func-tional form for the disutility function; specifically,C4e3�i5 = �ie
2. Thus the set of structural parametersof the salesperson’s utility function that need to beestimated is �i = 4�i1�i5.
3.4. State VariablesAs discussed previously, the nonlinearity of the com-pensation scheme with quotas and bonuses intro-duces dynamics into the sales agent’s behaviorbecause there is an additional trade-off between thedisutility of effort today and a higher probability of alump-sum bonus and overachievement commissionstomorrow. To incorporate the dynamics of the model,we consider the following key state variables: the per-centage of annual quota completed, the percentageof quarterly quota completed, month type within thequarterly quota period, and month type within theannual quota period (time of the year). These statevariables evolve as follows:
1. Percentage of quarterly quota completed (%QQ)
zi1t =
0 if t is the start of the quarterlyquota period1
zi14t−15 +Si4t−15
Qiqt
otherwise3
2. Percentage of annual quota completed (%AQ)
zi2t =
0 if t is the start of the annualquota period1
zi24t−15 +Si4t−15
Qiyt
otherwise3
3. Month type within quarterly quota period
zi3t =
1 if t is the start of the quarterlyquota period1
zi34t−15 + 1 otherwise3
4. Month type within annual quota period
zi4t =
1 if t is the start of the annualquota period1
zi44t−15 + 1 otherwise0
Whereas the first two state variables evolve stochas-tically, conditional on the effort levels and revenueshocks of the previous periods, the latter two evolvein a purely deterministic manner. Other state vari-ables would include time-varying demand shifters,the ISR index, and territory characteristics for whichwe use the previous year’s annual quota. We usetenure with the focal firm (�) as an individual statevariable that impacts effort. These state variables arecollected in a state vector,
zEit ={
zi1t1 zi2t1 zi3t1 zi4t1 ISt1AQi4y−151 �i}
0
3.5. Optimal Choice of EffortGiven the parameters of the compensation scheme �,as well as the state variables and their transitions,each sales agent would choose an effort level con-ditional on her states to maximize the discountedstream of expected future utility flow. Alternatively, ifthis value function is below the reservation wage, thesalesperson may choose to leave the firm.
The stream of utility flow, under the optimal effortpolicy function, conditional on staying at the firm, canbe represented by a value function:
V 4z3�1ì5
= maxe
{
U4e1z3�1�5+ �E[
maxe′
U4e′1 z′3�1�5
+ �E[
maxe′′
U4e′′1 z′′3�1�5+ · · ·
]]}
1
where ì= 4�1�5 is the set of primitives or structuralparameters of the underlying utility function � andthe discount factor �. The expectation of the valuefunction is taken with respect to both the present andfuture sales shocks.
4. EstimationTraditionally, the nested fixed-point algorithm (NFXP)developed by Rust (1987) is used to estimate dynamicmodels. However, NFXP estimators are computa-tionally burdensome because one has to solve thedynamic program numerically over each guess of theparameter space for every iteration. The two-step esti-mation first introduced by Hotz and Miller (1993) andextended by Bajari et al. (2007) can serve to reduce thecomputational burden. In this approach, the modelestimation proceeds in two steps. In the first step, weestimate the conditional choice probabilities of choos-ing a certain action as a flexible nonparametric func-tion of state variables. Then, in the second step, theseconditional choice probabilities are used to estimatethe structural parameters of the sales agent’s utilityfunction.
Until recently, it was believed that the accurateestimation of conditional choice probabilities for an
agent is impractical when there is unobserved het-erogeneity. Arcidiacono and Miller (2011) propose anexpectation-maximization algorithm-based approachto accommodate unobserved heterogeneity in the firststep of the two-step estimation procedure. We provideone of the first applications of this approach illustrat-ing the empirical validity of the approach in practicalapplications. We now discuss the details of the two-step estimation procedure.
4.1. Step 1: Estimating CCPsIn this step, we need to estimate a flexible nonpara-metric mapping between observable states and thesalesperson’s actions; this requires a nonparametricmodel of the monthly effort function, ei4zEit5, that linkseffort and states in Equation (1). We model the effortfunction nonparametrically as a combination of basisfunctions of the state variables. Thus the nonparamet-ric effort function is
eit =L∑
l=1
�l4zEit5�il1 (2)
where the lth basis function is �l4zEit5. In this applica-
tion, the lth basis function is the lth-order Chebyshevpolynomial.12
From Equations (1) and (2), we have the followingsales response function to estimate:
Sit = f 4zDit 5+L∑
l=1
�l4zEit5�il + �it0
For zDit , which is a subset of zEit (hereafter referred toas zit5, we use two variables: (i) the lagged annualquota for salesperson i and (ii) the ISR index. Weuse the direct linear effect of these variables to con-trol for cross-sectional variations of territory charac-teristics and temporal variations in monthly seasonal-ity. The interaction effects of these variables with theother state variables go into the polynomial functionin (2).13
12 Chebyshev polynomials are a sequence of orthogonal polyno-mials, defined by the recurrence relation T04x5 = 1, T14x5 = x, andTn+14x5 = 2xTn4x5 − Tn−14x5. By using the orthogonal Chebyshevpolynomials to approximate the policy function, rather thanapproximate the policy function with the standard 1, x, x2 poly-nomials, we avoid multicollinearity issues. In our approximationof the effort policy function (reported in Table 6(b)), we use up tothird-order Chebyshev polynomials; i.e., T04x5= 1, T14x5= x, T24x5=
2x2 − 1, and T34x5= 4x3 − 3x. For more details on Chebyshev poly-nomials, we refer the reader to Judd (1998).13 Separating sales effects stemming from seasonality or territorialcharacteristics and effort is a difficult challenge given the unobserv-ability of effort. Our maintained assumption that the main effect ofzDit isolates the territory and seasonal effects from the effort policyfunction is valid if the firm sets quotas in proportion to territorycharacteristics and seasonality (zDit ). This remains an important issueto explore in future research.
The lagged annual quota takes into account generalterritory characteristics that are likely to be generatedwith limited effort (i.e., market size). The revenuesfrom the indirect sales force capture market seasonal-ity, independent of the nonlinear nature of the com-pensation plan. We assume that the revenue shocks(�it) come from an independent and identically dis-tributed normal distribution.14
If it were possible to estimate the sales response andeffort response functions at the level of each individ-ual, then one could simply obtain the individual-levelparameters of the effort and sales policy functions bymaximizing the log-likelihood of the sample as
äi =argmaxäi
T∑
t=1
log{
Li
(
Sit−f 4zDit3�i5−L∑
l=1
�l4zEit5�il
)}
1
(3)where the vector äi = 8�i1�i1�i9 contains the set ofparameters of the sales response and effort policyfunctions and the distribution of sales shocks, where
Li4�5=1
�i42�51/2e−41/254�/�i5
20 (4)
We accommodate unobserved heterogeneity byallowing for discrete segments. Assume that salesper-son i belongs to one of K segments, k ∈ 811 0 0 0 1K9with segment probabilities qi = 8qi11 0 0 0 1 qiK9. Let thepopulation probability of being in segment k be �k.Let L4Sit � zit1 k3äk5 be the likelihood of individual i’ssales being Sit at time t, conditional on the observ-ables zit and the unobservable segment k, given seg-ment parameters äk. Then the likelihood of observ-ing sales history Si over the time period t = 11 0 0 0 1 T ,given the observable history zi and the unobservablesegment k, is given by
Lk4Si � zi3äk1�k5=�k
( T∏
t=1
Likt
)
1 (5)
where Likt = L4Sit � zit1 k3äk5. As noted earlier, weassume the distribution of the revenue shocks to benormally distributed and hence use the normal likeli-hood for Equation (5) as in Equation (4). The param-eter äk = 8�k1�k1�k9 is the vector of segment-levelparameters of the sales response and effort policyfunctions where each �k is the parameter that indexesthe effort policy for segment k, and �k is the param-eter for the distribution of the revenue shocks forsegment k.
By summing over all of the unobserved statesk ∈ 811 0 0 0 1K9, we obtain the overall likelihood ofindividual i:
L4Si � zi3ä1�5=
K∑
k=1
Lk4Si � zi3äk1�k51
14 It is noteworthy that Mirrlees (1999) shows that the first-best canbe approximated with normal errors in the sales response functionin a one-shot model setting if unbounded punishments are feasible.
and hence the log-likelihood over the N sample ofindividuals becomes
N∑
i=1
log(
L4Si � zi3ä1�5)
=
N∑
i=1
log( K∑
k=1
�k
T∏
t=1
Likt
)
0 (6)
Directly maximizing the log-likelihood in (6) iscomputationally infeasible because the function isnot additively separable. So we use the approach ofArcidiacono and Jones (2003) and Arcidiacono andMiller (2011) to iteratively maximize the expected log-likelihood in Equation (7):
N∑
i=1
K∑
k=1
T∑
t=1
qik logL4Sit � zit1 k3äk51 (7)
where qik is formally defined below as the probabilitythat individual i is of segment type k given parametervalues ä = 8ä11 0 0 0 1äK91where äk = 8�k1�k1�k9, andsegment probabilities � = 8�11 0 0 0 1�k9, conditional onall of the observed data of individual i:
Pr4k � Si1 zi3ä1�5 = qik4Si1 zi3ä1�5
=Lk4Si � zi3äk1�k5
L4Si � zi3ä1�50 (8)
The iterative process is as follows: We start with aninitial guess of the parameters ä0 and �0. A naturalcandidate for such starting values would be to obtainthe parameters from a model without unobservedheterogeneity and slightly perturbing those values.15
Given the parameters 8äm1�m9 from the mth iteration,the update of the (m+ 1)th iteration is as follows:
(a) Compute q4m+15ik using Equation (8) with äm
and �m.(b) Obtain ä4m+15 by maximizing (7) evaluated
at q4m+15ik .
(c) Update � 4m+15 by taking the average over thesample such that
�4m+15k =
1N
N∑
i=1
q4m+15ik 0
We iterate (a)–(c) till convergence.For the basis functions in the effort policy, we use
Chebyshev polynomials of state variables to approxi-mate effort, as described in footnote 12. From the esti-mation, we obtain the vector of parameters for thebasis functions (�), the vector of parameters forthe sales policy (�), and the parameters of the revenueshocks (�) for each segment k. We also obtain the pop-ulation segment probabilities (�) for each segment.
15 We started the initial values from one-tenth of the standard errorfrom the parameter values obtained from a single-segment model.The initial values of the segment probabilities were set equallyacross segments.
Thus, this procedure gives us the sales revenue func-tion S4 · 5 and effort policy function e4 · 5 for each seg-ment. We use these segment-level policies to obtain thestructural parameters of each segment. A caveat withthe two-step estimation procedure is that the first-stage policy function estimates can be biased (and thusgenerate bias in the structural parameters) if the statevariables in the policy function are correlated with thefirst-stage errors.
4.2. Step 2: Estimating Structural ParametersThe key idea of the two-step estimation is that inthe first stage we observe the agent’s optimal actions.Using these observed optimal actions, we are ableto construct estimates of the value function, whichenables us to estimate the primitives of the model thatrationalize these optimal actions.
Let the value function of a representative agent atstate z that follows an action profile e—conditional onthe compensation plan �, the sales profile S, and theprimitives of the utility function and discount param-eters ì= 4�1�5—be represented as
Vt4z3 e3�1S1ì5
= E
{ T∑
t=0
D4t5U4e4zt51 zt1�t3�5∣
∣
∣
z0 = z3�1S1ì
}
1 (9)
where D4t5= �t is the discount factor, and the expec-tation operator would be over the present and futuresales shock �t .
Using the estimated sales and effort policy functionand the distribution of the sales shocks in the firststage, we are able to forward-simulate the actions ofsales agents to obtain the estimate of the value func-tion. The detailed simulation procedure is as follows:
(a) From initial state of zt , calculate the optimalactions as e4zt5.
(b) Draw sales shock �t from f 4�5.(c) Update state zt+1 using the realized sales
s4e4zt55+ �t .(d) Repeat (a)–(c) until t = T .By averaging the sum of the discounted stream
of utility flow over multiple simulated paths, wecan get the estimate of the value function, V 4z3 e4z53�1S1ì5.16
Let es4z5 be any deviation policy from a set offeasible policies that is not identical to the optimalpolicy, and, by using the same simulation methodproposed above, let the corresponding estimate ofthe value function be called the suboptimal valuefunction, V 4z3 es4z53 �1S1ì5. Since e4z5 by definitionis the effort policy and thus at an optimum, thenany deviations from this policy rule would generate
16 For each segment, we drew 400 simulation draws over eachperiod and computed the value functions.
value functions of lesser or equal value to that of theoptimal level.
Let us define the difference in the two value func-tions as
Q4v3�1S1ì5 = V 4z3 e4z53�1S1ì5
−V 4z3 es4z53�1S1ì51
where v ∈ V denotes a particular 8z1 es4z59 combina-tion.17 Then if e4z5 is the optimal policy, the functionQ4v3�1S1ì5 would always have a value greater thanor equal to zero. Thus our estimate of the underlyingstructural parameters ì would satisfy
ì= arg min∫
(
min8Q4v3�1S1ì5109)2dH4v51
where H4v5 is the distribution over the set V of in-equalities. Our empirical counterpart to Q4v3�1S1ì5would be
Q4v3�1 S1ì5 = V 4z3 e4z53�1 S1ì5
− V 4z3 es4z53�1 S1ì50
As a result, our estimates of the structural parametersare obtained from minimizing the objective functionin Equation (10):18
1NI
NI∑
j=1
(
min8Q4vj3�1 S1ì5109)20 (10)
The above procedure is performed for eachsegment with the segment-specific effort policiesobtained in Step 1. This allows us to estimate thestructural parameters for each segment.19 In practice,since the objective function in Equation (10) is rela-tively flat with respect to changes in the discount fac-tor, it is difficult to pin down the discount parameterwith traditional gradient-based optimization. In ourspecification, we first estimate the structural param-eters by minimizing the objective function in Equa-tion (10) over a grid of discount factors. Second, we
17 As indicated in Bajari et al. (2007), there are multiple ways todraw these suboptimal policy rules. Although the method of select-ing a particular perturbation will have implications for efficiency,the only requirement necessary for consistency is that the distri-bution of these perturbations has sufficient support to yield iden-tification. We chose to draw a deviation policy from a normaldistribution with mean zero and a quarter of the variance from therevenue shock distribution; i.e., es4z5= e4z5+�.18 We drew 200 deviation strategies to construct the objective func-tion, and hence NI =200.19 In addition, we used a second set of moment inequalities to reflectthe participation constraint that employees continued to work at afirm because they at least obtained a reservation value (normalizedto zero); i.e., min8V 4z3 e4z53�1S1ì5109. It turns out these inequali-ties are nonbinding and do not affect our estimates.
simulate data with the estimated structural param-eters associated with each discount factor to com-pute the mean absolute percentage error (MAPE) perperiod with respect to the observed data. We choosethe final estimates as the discount factor and the asso-ciated structural parameters with the lowest MAPE.20
4.3. IdentificationThere are a couple of major identification challenges.First, we do not observe effort. Hence the linkbetween effort and sales cannot be identified nonpara-metrically. Second, in dynamic structural models, itis usually impossible to identify discount factors sep-arately from the utility function. Below, we discusshow we address these issues.
Realized sales are a function of demand shifters,effort, and additive sales shocks. Conditional onobserved demand shifters and given multiple obser-vations of sales at different states, we can separatelyidentify nonparametrically the density of sales shocksand a deterministic function of effort. We assume adeterministic (but flexible) relationship between effortand observable states (%QQ and %AQ and demandshifters) for each segment. Finally, because we do notobserve effort, we need a strictly monotonic paramet-ric relationship between sales and effort. As we esti-mate a flexible relationship between observable statesand effort, we model the relationship between salesand effort to be linear.
We rely on the variation in states across agents andwithin agents for identification. The average coeffi-cient of variation of states %AQ within agents (i.e.,CV = 1/N
∑Ni=1 �
%AQi /�%AQ
i , where �AQi and �AQ
i arethe mean and standard deviation, respectively, of thedistance to annual quotas over three years for agent i)over the three years is 0.55. This allows us to separateeffort from the individual fixed effects. The averagecoefficients of variation across agents and months on%AQ and %QQ are 0.63 and 0.91, respectively.21 Thevariation in states across agents helps separate agenteffort from the monthly fixed effects.
The discount factor is not identified separately fromthe utility function in standard dynamic structuralmodels because, typically, there are no variables thatdo not affect contemporaneous utility, only future
20 The statistical properties of the grid-search estimator based onMAPE for discount factors are not known. Hence, we computedthe standard errors using subsample bootstrap methods, wherewe subsampled the data and estimated the model multiple timesto construct a bootstrap distribution (Efron 1979). In her analysisof entry and exit in the washing machine industry in the UnitedStates, Shen (2013) also uses grid search to estimate the discountfactor of firms; she uses parametric bootstrapping to estimate thestandard error.21 There is substantial variation in the across-agent CV of %AQ and%QQ across months.
utility (Rust 1994, Magnac and Thesmar 2002). In theabsence of such an exclusion restriction, this impliesthat if an agent exerts low effort in a period, it is notpossible to distinguish whether this is due to highdisutility for effort or because she discounts futureutilities very heavily.22
Two aspects of our setting allow us to identify util-ity functions separately from the discount factor. First,we have a finite-horizon setting, where at the endof the year, the quotas are reset and all agents startwith a fresh quota for the following year. This meansthat every December, the agent faces a static optimiza-tion problem, conditional on the sales agent’s state(%AQ). Utility parameters are well identified for astatic model, and hence the agent’s choice in the lastperiod should allow us to nonparametrically identifythe agent’s utility function. Given this, a variation insales (that is monotonically linked to effort) in thelast period and variations in wealth should help iden-tify the effort disutility and risk-aversion coefficientwithin the utility function.
Second, the bonus setting generates exclusionrestrictions between the current and future utility; i.e.,we have instruments in nonbonus periods that do notaffect current utility, only future utility. As we demon-strated with reduced-form evidence earlier, that anagent’s performance in November is related to hisproximity to the annual bonus given in Decemberindicates forward-looking behavior. We also collectedother evidence of how agents respond to quarterly orannual quotas even though they do not affect currentpayoffs. This allows us to estimate the discount factor.
Beyond these conceptual arguments, we report inthe appendix the results of a simulation study thatdemonstrate that the structural parameters of the util-ity function and discount factor can be identifiedusing our empirical strategy.
5. ResultsWe first report the first-stage estimates of the demandshifters and effort policy function for the salesresponse model; then we report estimates of struc-tural parameters of sales agents’ utility functions fromthe second-stage estimation. We then perform severalcounterfactual simulations to address the substantivequestions we seek to answer.
22 Another reason why an agent might exert low effort is that shemay have incorrect expectations about the transition density offuture states; i.e., she might be very pessimistic about future goodstates. Similar to other dynamic structural modeling papers, weassume rational expectations for the transition densities of states.In this case, this translates into a rational expectations assumptionon sales shocks.
5.1. First-Stage EstimatesThe parameter estimates for the demand shifters inthe sales response function is reported in Table 6(a).We find that lagged annual quota and the interactionterm between it and indirect sales revenue are statis-tically significant. Thus larger markets tend to have abigger sales multiplier independent of effort in high-demand periods.
We estimate segment-level effort policy functionsby estimating the nonparametric relationship betweensales and state variables through Chebyshev poly-nomials of the state variables. We estimated up tofourth-order Chebyshev polynomials with an alterna-tive number of segments and chose the best-fittingmodel based on the Bayesian information criterion.The best fit was for the model with up to third-orderChebyshev polynomials, allowing for three segments.The estimates of the best-fitting polynomial functionand the standard deviations of the revenue shocksfor each segment are reported in Tables 6(b) and 6(c).As the coefficients associated with the Chebyshevpolynomials have no intuitive meaning, for intuition,we show graphs of the effort policy function for thethree segments as a function of the percentage annualquota (%AQ) and percentage quarterly quota (%QQ)for select months in Figure 6(a). Both %AQ and %QQare normalized across sales agents, such that 1 impliesmeeting quota and 0.9 indicates 10% below quota and1.1 indicates 10% above quota.
Table 7 shows the share of the three segmentsand their descriptive characteristics. Segment 2 is thelargest with a share of 47%; Segments 1 and 3 haveshares of 32% and 21%, respectively. The averagetenure with the firm is not very different across seg-ments, at approximately 12 years. Segment 3 has thehighest annual quotas, followed by Segments 2 and 1.It is worth noting that Segments 2 and 3, with largerquotas, achieve their quota targets more often thanSegment 1, which has trouble meeting its quota.
Figure 6(a) shows that Segment 3 exerts the mosteffort and is the most productive segment, and Seg-ment 1 exerts the least effort and is the least pro-ductive segment. This is consistent with the allocatedquotas and percentage of time quotas are achieved inTable 7. We also see a positive relationship betweenexerted effort and %AQ for all months shown. As for
Table 6(a) Parameter Estimates—Sales Response
Lagged annual quota −00022∗∗∗
4000075
Indirect sales −009554603075
Indirect sales× Lagged annual quota 00023∗∗∗
4000035
Note. Standard deviations are shown in parentheses.∗∗∗p < 0001.
Note. Standard deviations are shown in parentheses.∗p < 001; ∗∗p < 0005; ∗∗∗p < 0001.
%QQ, we see an increasing but concave relationshipin March, implying that once a salesperson is wayabove the quarterly quota, she starts to graduallyslow down. Given that the average states in Marchfor each segment were 0.55, 0.58, and 0.62, respec-tively, not a lot of salespeople are in a position wherethey can slow down. Effort in December does notfall off even if the salesperson has already reached orexceeded her quota (%AQ> 1), likely because of theoverachievement commissions in preventing sales-people from lowering effort after achieving quotas.Our results are consistent with Steenburgh (2008),who finds that salespeople “give up” when far awayfrom achieving their quota, such as for all segmentsin our case, but they do not slow down much oncetheir quota is reached.
Figure 6(b) shows the effect of tenure on effort forall segments. Salespeople in Segments 2 and 3 ini-
Table 6(c) Revenue Shock Distribution
Segment 1 Segment 2 Segment 3
Sigma 80.59 141.84 271.73
tially increase effort with experience, but this tapersoff with time. This is probably because in the earlyyears of their careers, they want to work hard not onlyfor monetary payments from increased wages but alsofor other intangible incentives, such as promotions ortransfers to better job titles. However, after a certainnumber of years, these intangibles do not matter asmuch, and the effort levels tend to taper off. It is note-worthy that Segment 1, the lowest-productivity seg-ment, does not gain in productivity from experience.
Figure 6(a) Effort Policy by Segment as a Function of Percentage of Quota
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2000.
25
0.33
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5.2. Second-Stage Structural Parameter EstimatesThe first column of Table 8 shows the estimates ofthe dynamic model with exponential discounting. Wefind the discount factor � to be 0.9 and highly sig-nificant. Frederick et al. (2002) have a comprehensivesummary of the estimated discount factors from pre-vious studies. The summary shows that the estimateddiscount factors vary extensively, ranging from as lowas a mere 0.02 to no discounting at all, with a discountfactor of 1. For purely monetary values, the estimateddiscount factor seems rather low. But as Fredericket al. point out, for behavioral aspects such as pain—and in our case, effort—the discount factors tend tobe low. Hence, our estimate appears to be reasonable.
The parameters for disutility of effort are negativeand significant for all three segments. Their relativemagnitudes are consistent with the effort policy func-tions estimated in the first stage. Segment 3, whichproduces the greatest sales on average, has the lowestdisutility for effort. Segment 1, which has the low-est sales, has the greatest disutility. The risk-aversioncoefficients for all segments are insignificant, show-ing no direct evidence of risk aversion by the salesagents. This may be because in the range of incomes
Figure 6(b) The Effect of Tenure on Effort
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1 2 3 4 5 6 7 8 9 10 11 12 13
Effo
rt
Tenure (years)
Seg 1
Seg 2
Seg 3
earned by the sales force, risk aversion is not a seriousconcern. The estimated model fits the observed salesrevenue data reasonably well with a MAPE of 9.4%.
5.3. Assessing the Value of aDynamic Structural Model
How important is it to model the dynamics of sales-person behavior? In a static model, any effort wouldbe attributed to the current payoff, not accountingfor the large future bonuses. This will downwardbias the salesperson’s disutility parameters and over-state the effects of compensation on productivity. Thesecond column of Table 8 reports the estimates ofthe myopic model, where the discount factor is setto zero. As expected, the disutility parameters aresmaller in magnitude relative to the forward-lookingmodel for all segments. For Segment 3, the downwardbias is as much as 28%. The myopic model also has a
Table 8 Structural Parameters
Dynamic Myopic
Discount factor 0090∗∗∗
400045Segment 1
Disutility −00375∗∗ −00284∗∗
4001895 4001145Risk aversion −000001 −000001
40000205 40000035Segment 2
Disutility −00145∗∗∗ −00111∗∗∗
4000385 4000325Risk aversion −000001 −000001
40000055 40000055Segment 3
Disutility −00077∗∗∗ −00056∗∗∗
4000165 4000105Risk aversion 000000 000000
40000025 40000045MAPE 00094 00167
Note. Standard errors are shown in parentheses.∗∗p < 0005; ∗∗∗p < 0001.
Figure 7 Simulated Revenue and Effort—Static vs. Dynamic
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poorer fit: a MAPE of 16.7% relative to MAPE of 9.4%for the dynamic model.
We next compare the revenue and effort predictionsbetween the dynamic and myopic models. To isolatethe effects of forward-looking behavior, we simulatebased on the disutility estimates from the dynamicmodel but set the discount parameter to zero for themyopic model. Figure 7 compares the predicted rev-enues and effort of the myopic and dynamic models.The myopic agent has systematically lower revenuesbecause she does not take into account the effect offuture bonuses and overachievement commission incurrent effort. In contrast, the forward-looking agentanticipates that in an uncertain environment, there isa chance of bad shocks later that may prevent herfrom meeting the quota, so she prepares for such arainy day by working harder early on so that she iswithin striking distance of her quota even if a badsales shock occurs.
The effort graph in Figure 7 enables us to iso-late out the sales revenue cyclicality and focus onthe differences in effort across dynamic and myopicagents. The myopic salesperson concentrates muchmore effort in the bonus period, but the forward-looking salesperson smooths effort over time, giventhe uncertainty in future demand shocks. The effortpeaks in the bonus periods are not as pronouncedfor the dynamic agent. The observed effort smooth-ing is similar to consumption smoothing by forward-looking consumers facing uncertain incomes in thedevelopment economics literature.
5.4. Counterfactual SimulationsWe now perform a series of counterfactual simula-tions that address the two sets of substantive ques-tions we wish to answer. First, we address the issueof how valuable different components of the com-pensation plan are. The overall change in revenuesunder the alternative conditions is reported in Table 9and the effect by segment in Table 10. Second, wecompare the role of bonus frequency—how quarterlyand annual bonuses affect performance. In runningthe counterfactual simulation studies, given a regimechange, we randomly chose 25 representative sales
agents from each segment (a total of 75 salespeople)and simulated 1,000 simulation paths for each of themto obtain the productivity of each segment. Then, weweighted each segment based on the estimated seg-ment sizes reported in Table 7.
5.4.1. Value of Nonlinear Incentive Components.We compare changes in revenues and profits whenthe firm moves from the current compensation plan toa pure commission-only plan. We consider two cases:(1) where the commission rate is the same as the cur-rent commission rate and (2) where a higher commis-sion rate is such that total compensation is exactlyequal to the current compensation. We find that therevenues are about 17.9% greater with the currentcompensation plan compared with a pure commissionplan. Not surprisingly, given the lower incentives inthe absence of quotas, bonuses, and overachievementcommissions, all segments suffer from substantiallypoorer performance, as shown in Table 10. Even afteradjusting commission rates to be higher to make totalcompensation identical to the current plan, revenuesare 2.4% higher with the current compensation plan,suggesting that the nonlinear plan shifts salespeopleinto high-power areas of incentives. To understandthe role of different components, we next investigatecounterfactuals based on individual components oneat a time.
Table 9 Impact of Alternative Bonus Plans on Sales Revenues
Counterfactual Change in revenues (%)
Only pure commissions −1709Only pure commissions −204
(adjusted to equal payout with bonus)No bonus (only commissions + −800
overachievement commission)No bonus (commissions adjusted to −206
equal payout with bonus)No overachievement commissions −1007Cumulative annual quota replaced with −107
quarterly quotaAnnual bonus split into quarterly and −005
5.4.2. Value of Overachievement Compensation.When overachievement commissions are eliminated,not surprisingly, overall revenues drop by 10.7%.Even accounting for the additional commission costs,profits are lower by about 2% (assuming a gross mar-gin of 33%).
To gain insight into how overachievement commis-sions impact sales, Figure 8(a) compares the effortlevel of sales agents who eventually meet or do notmeet the annual quota. For those who met the annualquota, their effort level does not decline even whenclose to the quota because of the overachievementcommission. In contrast, those who did not meet theannual quota decrease effort toward the end of theyear because they are unlikely to meet the quota,and therefore the overachievement commission hasno impact on their earnings. Thus the overachieve-ment commission provides incentives for the mostproductive salespeople even if they have already mettheir quota (or are likely to meet it). Table 10 indi-cates, not surprisingly, that overachievement commis-sions have the greatest impact on Segment 3, the mostproductive segment. Revenues drop by about 15%for this segment, whereas the effect on the least pro-ductive segment is substantially smaller, at 2%. Over-all, we conclude that overachievement commissionrates motivate high-performing salespeople to con-tinue exerting high levels of effort when they are closeto and exceed the quota, but these rates have a limitedimpact on low-performing salespeople.
5.4.3. Value of a Cumulative Annual Quota.Next we investigate the effect of cumulative annual
Figure 8(a) Effort: Overachievement Commission
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quota by replacing the cumulative annual quota withjust a fourth-quarter quota. That is, we set the fourth-quarter quota as the annual quota minus the sum ofthree previous quarters’ quotas 4QQ4 = AQ − QQ1 −
QQ2 − QQ35 and split the total bonus payments(annual bonus + quarterly bonuses) equally across allfour quarters.23 Furthermore, to isolate the effect ofbonuses, we also remove the overachievement com-mission for reaching the annual quota in both sce-narios. Overall, revenues drop by 12.0% when thecumulative annual quota and overachievement com-missions are removed. This decrease is greater thanthe 10.7% we obtained when we just dropped theoverachievement commission. Thus the cumulativeannual quota induces sales agents to exert greatereffort and raise revenues by 1.3%.
How does the cumulative annual quota work?On the negative side, the cumulative annual quotareduces the incentive of salespeople who have nothad success in the first three quarters to continue toput in effort in the fourth quarter. But on the positiveside, the cumulative annual quota allows salespeoplewho reach their quarterly quota in earlier quarters tocontinue to extend themselves in order to get themcloser to the annual target—making this additionaleffort early on helps sales agents buffer against nega-tive shocks in future quarters and allows them to bewithin striking distance of the annual quota. On bal-ance, the positive effect of the latter outweighs thenegative effect of the former.
We also consider the case where we split the cur-rent annual bonus into a fourth-quarter bonus (thesame amount as other quarterly bonuses) and anannual bonus so that salespeople do not give up inthe last quarter when they are far away from quota.Specifically, we split the $4,000 annual bonus intoa $1,500 fourth-quarter bonus and a $2,500 annualbonus. Although this did increase the effort in thelast quarter, it reduced revenues overall because sales-people did not put in as much effort earlier in theyear to be within striking distance of the annual quotabecause it is not as large. Total revenues droppedby 0.5%.
23 Specifically, the quarterly bonus is set to be (annual bonus + 3 ×
Overall, these results show that the cumulativeannual quota has a significant impact on salespeo-ple by inducing agents to work harder even whenthey reach their quarterly targets, again motivatingthe high-performing salespeople who reach quarterlyquotas to continue to remain productive and bringadditional sales.
5.4.4. Quota-Bonus Frequency. We next investi-gate the value of quarterly bonuses relative to annualbonuses. Figure 8(b) shows the comparison of effortbetween the current plan and when quarterly bonusesare eliminated and only the annual bonus is left.Effort drops consistently across the year when thereare no quarterly quotas. Overall revenues fall by 4%.Even in December, when the annual bonus is on thetable, revenue falls by 1% and effort falls by 2%.Thus annual bonus and overachievement commis-sions have less of an impact on year-end performancewithout quarterly bonuses. Why?
The quarterly bonus induces sales agents to workharder in a given quarter. But it also helps themachieve the annual quota by helping them stay ontrack of their annual goal. Without a quarterly bonus,sales agents do not have much incentive to workhard early on. This lack of incentive leads them tobe farther away from the annual quota by Decem-ber. Annual bonuses and overachievement commis-sion have little impact on effort as sales agents aremore likely to give up meeting the annual quota.
The impact of quarterly bonuses also differs acrossthe three segments of consumers. Table 10 indicatesthat quarterly bonuses have a relatively minimalimpact on Segment 3, the most productive segment,but they almost triple the impact of Segment 3 onSegment 1. In effect, quarterly bonuses are needed aspacers for the less productive salespeople than for themost productive salespeople.
To the best of our knowledge, there has been noanalysis to date on what the appropriate frequency ofquota and bonuses is. There has been some descrip-tive work in the education literature on how frequenttesting affects academic performance (for an exten-sive survey, see Bangert-Drowns et al. 1991) and some
experimental work in behavioral psychology (Heathet al. 1999). The basic idea is that achieving short-termgoals makes achieving long-term goals more feasi-ble. Our analysis shows that the short-term goals aremore valuable to the least productive segment; i.e., ineducation terms, it implies that weaker students gainmore by periodic testing relative to stronger studentswho would study regardless of exams.
5.4.5. Discussion. Overall, our results provideempirically grounded substantive insight on the roleof various elements of nonlinear compensation toimprove productivity. The numerical example in §2.2shows that quotas and bonuses serve as importantgoals for the average performers, inducing them toincrease their effort. Our empirical analysis showssupport for the role of quotas and bonuses as motiva-tional goals and stretch incentives, respectively. How-ever, it is important to recognize that other elementsof the compensation plan are critical in improvingthe performance of both high performers and lowperformance. First, in this setting, managers recog-nize that quota updating based on the salesperson’sown performance in the previous year can induceratcheting effects; such effects can reduce productiv-ity among the highest-performing salespeople, whodo not book sales much higher than their quota so asto avoid higher quotas the following year. By updat-ing quotas based not on an individual’s performancebut on the performance of a larger group of sales-people, managers can minimize ratcheting effects.In fact, we found empirically little statistical evidenceof effort shading as a result of ratcheting effects. Sec-ond, the high performers who are most likely to reachtheir quotas will reduce effort upon reaching them ifthere are no additional incentives offered. The firmused overachievement commissions and cumulativeannual quotas to induce the high-performing salesagents to continue to put in effort and excel in theirsales performance. These incentives, along with theavoidance of ratcheting effects, enabled the firm tokeep its best sales agents motivated to continue per-forming at a high level. Finally, quarterly bonusesserved as pacers to help keep the least productivesegment motivated so that it can be within strikingdistance of its long-term goals. By providing empir-ically grounded insight into the differential motiva-tional roles of different incentive components of anonlinear compensation plan for high-, average-, andlow-performing sales agents, this paper contributessubstantively to literature on the role of nonlineari-ties in quota-bonus plans and explains its widespreadpopularity in practice (Joseph and Kalwani 1998).
6. ConclusionPersonal selling is a primary marketing mix tool formost B2B firms to generate sales, yet there is little
research on how the compensation plan motivatesthe sales force and affects performance. This paperdevelops and estimates a dynamic structural modelof sales force response to a compensation plan withvarious components: salary, commissions, lump-sumbonuses for achieving quotas, and different commis-sion rates beyond achieving quotas. Our analysishelps us assess the impact of (1) different componentsof compensation and (2) the differential importanceof periodic bonuses on the performance of differenttypes of salespeople.
We find that the quota-bonus scheme used by thisfirm increases the sales force’s performance by serv-ing as an intermediary goal and pushing employ-ees to meet targets. Features such as overachieve-ment compensation reduce the problems associatedwith sales agents slacking off when they get close toachieving their quota. Furthermore, quarterly bonusesserve as a continuous evaluation scheme to keepsales agents within striking distance of their annualquotas. In the absence of quarterly bonuses, failurein the early periods to meet targets cause agents tofall behind more often than in the presence of quar-terly bonuses. Thus, the quarterly bonus serves as avaluable subgoal that helps the sales force stay ontrack in achieving its overall goal; such incentivesare especially valuable to low performers. In contrast,overachievement commissions increase performanceamong the highest performers.
In this empirical application, we introduce twoimportant methodological innovations to the market-ing literature. First, to the best of our knowledge,we provide the first empirical implementation of theArcidiacono and Miller (2011) approach to accommo-date unobserved heterogeneity within a two-step esti-mation framework. Second, we demonstrate that dis-count factors can be estimated in naturally occurringfield data using appropriate exclusion restrictions—and we overturn the conventional wisdom that esti-mation of discount factors requires augmentationwith survey data.
We now discuss the limitations of our paper, whichprovide promising avenues for future research. First,effort tends to be multidimensional, and one possibil-ity is that quotas and bonuses force people to focuson the effort that leads to final sales in bonus peri-ods, whereas agents may focus on the earlier stagesof the selling process in nonbonus periods. Such amultidimensional effort cannot be identified merelyfrom sales data. We hope data from customer rela-tionship management databases that track customerstages through the selling process can help shed lighton this issue. We believe this is an exciting area forfuture research.
Second, compensation contracts can serve to selectthe right type of salespeople. We do not address
selection issues. One possibility is to use a longerpanel of salespeople’s performance that includes attri-tion information. If there were variation in contractsthat affected employee retention, that could also helpaddress this problem. More work on scenarios withricher contracts needs to be done. For example, onecould study peer effects on sales performance andselection effects when firms shift from individual- toteam-based compensation (Chan et al. 2014).
In summary, this paper provides a rigorous frame-work to empirically understand how the sales forceresponds to a very rich compensation structure involv-ing many components of compensation: salaries,commissions, quotas, and bonuses at quarterly andannual frequencies. Our analysis helps obtain a num-ber of useful substantive insights. Nevertheless, theissues raised above provide an interesting agenda forfuture work.
AcknowledgmentsThis paper is the first essay of the first author’s doctoraldissertation at Yale University. The authors thank the editor,associate editor, two anonymous reviewers, Lanier Benkard,Adam Copeland, David Godes, Ahmed Khwaja, SubrataSen, as well as the seminar participants at Carnegie MellonUniversity, Cornell University, Emory University, the Geor-gia Institute of Technology, Harvard University, the HongKong University of Science and Technology, the IndianInstitute of Management, the Indian School of Business,New York University, Northwestern University, Pennsylva-nia State University, Stanford University, the University ofArizona, the University of British Columbia, the Universityof California at Berkeley, the University of Chicago, the Uni-versity of Houston, the University of Maryland, the Univer-sity of North Carolina, the University of Rochester, Wash-ington University, Western University, and Yale University,in addition to the conference participants at the 2009 UTDallas FORMS Conference, the 2010 Choice Conference inKey Largo, the 2010 Sales Conference in Kansas, the 2012Harvard Thought Leadership on the Sales Profession Con-ference, and the 2012 INFORMS Marketing Science Confer-ence for their comments and suggestions. The authors alsothank ISBM for providing financial support.
Appendix. Simulation Study to IdentifyDiscount Factors
Let realized sales be a function of effort, a function ofdemand shifter zt , and a normal random sales shock �itsuch that
yit = h4zt5+ eit + �it1 �it ∼ N401�250
To illustrate identification, we consider a simulation sce-nario with seasonality and quarterly quotas. The demandshifter zt represents the seasonality index comparable tothe ISR index in our empirical setting. For the purposeof the simulation, we assume the function h4 · 5 is lin-ear in the seasonality index; i.e., h4zt5 = �zt . Let thesalesperson’s utility be given as uit = −de2
it + Wit , whereWit = BI8t=316191129I8sit+yit>Q9 is the wealth earned from bonus
payment B for achieving quarterly quota Q. The cumulativesales state within quarter evolves as follows:
sit =
{
0 if t is the start of quota period1si4t−15 + yit otherwise0
We report the results of the simulation with seasonalityand quarterly quotas using three years of data. Let the vec-tor of the seasonality index for each month throughout theyear be given by (21315111213131417121315). We set thebonus at B = 60 and quarterly quota at Q = 30. Consistentwith the model in the paper, we estimate the first-stage esti-mates using our specification in Equation (1).
We varied the simulated number of individuals from 50to 300. The true values and the estimates and standarderrors for each simulation are reported in Table A.1. Evenwith 100 individuals (relative to our sample size of 350), weare able to recover the disutility parameter and the discountfactor with reasonable precision, lending confidence to theidentification arguments in §4.3.
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