Optimal Sales Force Compensation

Transcription

1 Optimal Sales Fore Compensation Matthias Kräkel Anja Shöttner Abstrat We analyze a dynami moral-hazard model to derive optimal sales fore ompensation plans without imposing any ad ho restritions on the lass of feasible inentive ontrats. We explain when the ompensation plans that are most ommon in pratie fixed salaries, quota-based bonuses, ommissions, or a ombination thereof are optimal. Fixed salaries are optimal for small revenue-ost ratios. Quota-based bonuses (ommissions) should be used if the revenue-ost ratio takes intermediate (large) values. If firms fae demand unertainty, markets are rather thin, and the revenue-ost ratio large, firms should ombine a ommission with a quota-based bonus. If word-of-mouth advertising affets sales, a dynami ommission that inreases over time an be optimal. When entering a new market or launhing a new produt, firms should install long-term bonus plans. Key Words: sales fore ompensation, ommissions, quota-based bonuses University of Bonn, Institute for Applied Miroeonomis, Adenauerallee 24-42, D Bonn, Germany, tel: , fax: , [email protected]. Humboldt University Berlin, Shool of Business and Eonomis, Spandauer Str. 1, D Berlin, Germany, tel: , [email protected]. 1

2 1 Introdution When selling goods to ustomers, firms rely heavily on personal selling via sales fores. To ompensate their sales agents, U.S. firms spend about $800 billion eah year, whih is almost three times as muh as they spent on advertising in 2006 (Zoltners et al. 2008). Empirial studies doument a large variety of sales fore ompensation plans aross firms. Based on survey data, Joseph and Kalwani (1998) report that 5% of the 266 analyzed firms ompletely forgo monetary inentives and exlusively pay fixed salaries to their sales agents. The majority of the firms, however, use some form of inentive pay. Among the firms in the sample, 24% employ only ommissions, 37% employ only a bonus sheme, and 35% use both ommissions and bonus pay. For implementing bonus pay, firms typially ompare agents realized sales with a predetermined quota. All in all, ommissions and quota-based bonuses are the most frequently observed forms of sales fore ompensation. In this paper, we analyze an ageny-theoreti framework that an explain under whih onditions one form of ompensation plan dominates the other. We employ a dynami moral-hazard model that allows us to solve for the profit-maximizing ompensation plan without imposing any ad ho restritions on the lass of feasible ontrats. We find that those ompensation shemes that are ommon in pratie fixed salary, quota bonus, ommission, or a ombination thereof are indeed often optimal. In partiular, we an show under whih onditions ombining a ommission with a quota-based bonus maximizes firm profits. In our model, a sales agent has to sell a produt in eah of two periods. The agent an influene sales by two types of ativities. On the one hand, the agent has to perform some basi tasks that an be easily monitored by the firm. For example, a firm an easily hek whether a sales lerk is present to answer ustomer questions or to reload empty raks, or whether a sales representative ontats a ustomer. On the other hand, a sales agent an atively ommuniate the advantages of the firm s produt in fae-to-fae enounters and invest time and effort to learn a ustomer s speifi needs, whih is hard to monitor. The first type of ativities an be ostlessly enfored by the firm. However, it is prohibitively ostly for the firm to diretly enfore that the agent additionally engages in the seond type. As a onsequene, we an differentiate between two effort levels. If the agent does not engage in the seond type of ativities, he will only fulfill the basi tasks, whih orresponds to low effort. However, if the agent deides to engage also in the seond type of ativities to omplement the basi tasks, we will speak of high effort. As the firm annot observe whether the agent has hosen low or high effort, there is a moral-hazard problem. If the firm wants to motivate the agent to exert high effort, it has to design an adequate sales fore ompensation plan ontingent on realized sales. Sine sales 2

3 outomes ruially depend on the agent s efforts, they an serve as an effetive performane measure. We assume that both the agent and the firm observe sales at the end of eah period. The agent is proteted by limited liability so that his ompensation must be non-negative. 1 For our basi model, we obtain the following results. If the ratio between sales revenues and the agent s effort osts is suffiiently low, the firm pays only a fixed salary that indues low effort in both periods. For intermediate revenue-ost ratios, the firm wants to avoid that the agent always exerts low effort. Optimal ompensation is then given by the lass of utoff inentive plans that make the agent work hard in the first period, but lead to high effort in the seond period if and only if a first-period sale ourred. A entral feature of suh an inentive plan is that the agent must sell the produt in the first period to earn a reward. A quota-based bonus sheme whih offers a bonus to the agent if and only if he was suessful in both periods belongs to this lass of ontrats. For high revenue-ost ratios, optimal ompensation is desribed by the lass of permanent inentive plans that always indue high effort. The well-known ommission sheme, whih pays a ertain ommission rate per sale to the agent, belongs to this lass of inentives plans. Importantly, the quota-based bonus and the ommission are mutually exlusive as optimal sales fore ompensation. The advantage of the quota-based bonus over the ommission is that the former indues high effort in the first period at lower ost for the firm. However, this also implies that the agent might be demotivated in the seond period if it is lear that he annot make the quota. The firm aepts this drawbak only if the potential loss, indiated by the revenue-ost ratio, is not too large. Moreover, for a given revenue-ost ratio, a quota-based bonus is optimal if sales respond either suffiiently little or suffiiently strongly to the agent s effort. We also onsider two variants of our basi model demand unertainty and preferene unertainty. In the former ase, future demand annot be perfetly foreseen and may depend on previous sales. We show that, when the firm wishes to indue high effort in both periods, demand unertainty an imply the optimality of an inentive ompensation plan that ombines a ommission with a quota-based bonus. This is the ase when the produt market is rather thin, i.e., a first-period sale makes it harder to sell the produt in the seond period. By ontrast, if a first-period sale makes a seond-period sale more likely (e.g., due to word-of-mouth advertising), a dynami ommission that inreases over time is optimal. We further find that, under demand unertainty, a ommission sheme an exhibit another omparative advantage beause the firm might be able to implement the optimal ommission even if it has less knowledge on short-term demand developments than the sales agent. In suh a situation, the ommission is the uniquely 1 Limited liability is a ommon assumption in ageny models. See, e.g., Sappington (1983), Innes (1990), Demougin and Fluet (1998), Oyer (2000), Poblete and Spulber (2012), and Kishore et al. (2013). 3

4 optimal permanent inentive plan. Moreover, we show that the three pratially most relevant inentive plans ommission, bonus, and a ombination thereof are all unique solutions to the firm s optimization problem when the firm ollets sales figures less frequently and hene observes only total sales at the end of the seond period. Interestingly, sarer information on sales performane has no adverse effet on firm profits beause the aforementioned inentive plans do not require intermediate sales information. Preferene unertainty reflets a situation where it is unertain how ustomers respond to the sales agent s effort, e.g., beause the firm enters a new market or launhes a new produt. In this ase, a ommission is no longer optimal. Instead, the firm maximizes profits by either a fixed salary, a quota bonus, or an inentive sheme that fouses on seond-period sales. Finally, we disuss how advertising, the infeasibility of long-term ontrats, and limited liability of the firm affet the optimal ompensation plan. In the latter two situations, the quota bonus as an optimal sales fore ompensation turns out to be less robust. translated into lear empirial preditions, whih we disuss in the onlusion. Our results an be niely Our paper loses a gap in the theoretial literature on optimal sales fore ompensation by providing optimality onditions for the different types of inentive ompensation plans that are typially observed in pratie. The analysis of optimal sales fore ompensation under moral hazard goes bak to Basu et al. (1985). They onsider a single-period setting with a riskaverse agent and show that optimal inentive pay usually is a non-linear inreasing funtion of sales. It is argued that ommonly used ompensation plans an be seen as a pieewise linear approximation of their optimal ontrat. Raju and Srinivasan (1996) show that suh an approximation indeed only leads to a small loss relative to the optimal ontrat. While there has been extensive researh on optimal single-period sales fore ompensation (see Coughlan (1993) and Albers and Mantrala (2008) for an overview), the literature on multi-period sales fore ontrating is relatively sare. Dearden and Lilien (1990) and Lal and Srinivasan (1993) extend the framework by Basu et al. (1985) to dynami environments. Dearden and Lilien (1990) introdue prodution learning effets and show how the ompensation plan should be optimally adjusted. Lal and Srinivasan (1993) onsider a setting that satisfies the properties for the optimality of a linear inentive sheme as haraterized by Holmström and Milgrom (1987). By ontrast, we provide onditions under whih non-linear inentive shemes (e.g., quota-based bonuses) dominate linear ones (e.g., ommissions) in a dynami setting. 2 (2012) investigates a behavioral-eonomis approah and shows that multiperiod quotas an 2 Oyer (2000) shows that a bonus tied to a quota an be the uniquely optimal ontrat in a stati setting. Also employing a stati model, Bak and Klez-Simon (2013) find that introduing quota bonuses for sales agents engaging in Bertrand ompetition an inrease firm profit. Agents, however, inur a onstant ost per sale and do not hoose effort. Jain 4

5 solve an agent s self-ontrol problem. Our results offer an explanation for quotas to be optimal when sales agents behave rationally. The analysis of Kishore et al. (2013) shows that quotabased bonuses dominate ommissions in the presene of multitasking. However, the authors do not solve for the optimal ontrat. Complementary to the urrent paper, Shöttner (2015) analyzes optimal sales fore ompensation plans fousing on a situation where the firm always wants the agent to perform high effort in eah of n periods, but an only observe total sales at the end of the final period. In ontrast to the present paper, the agent s effort osts and the sales-effort responsiveness an vary over time. She shows that pure ommissions are optimal whenever the agent is hardest to motivate in the last period. This result is robust to different types of ost and demand externalities. Hene, our result on the optimality of ommissions in the lass of permanent inentive plans is also valid in alternative settings. The modelling approah used in this paper, where an agent s basi ativities an be easily enfored but a seond type of additional ativities has to be inentivized, is widely aepted in ageny theory (e.g., Che and Yoo (2001); Laffont and Martimort (2002); Bolton and Dewatripont (2005); Shmitz (2005, 2013); Simester and Zhang (2010); Dai and Jerath (2013); Kaya and Vereshhagina (2014); Shöttner (2015)). 3 Our paper is partiularly related to the dynami models with moral hazard and limited liability that analyzed by Bierbaum (2002), Shmitz (2005, 2013), and Shöttner (2015). In ontrast to our paper, these works fous on a situation where the prinipal wishes to indue high effort in eah period. 4 Thus, by assumption, a pure quota-based bonus annot be optimal beause it would lead to low effort when it turns out that the quota annot be made. 5 As another differene to our setting, Bierbaum (2002) assumes that the prinipal an extrat payments from the agent up to his first-period wage in the seond period, or dismiss the agent after the first period. In Shmitz (2005, 2013), the first-period outome affets the effetiveness of effort in the seond stage. By ontrast, there are no externalities aross periods in our model. The only exeption is that, under demand unertainty, the probability that there will be a ustomer in the seond period an depend on first-period sales. The remainder of the paper is organized as follows. Setion 2 introdues the basi model. In Setion 3, we solve this model and show that fixed salaries, quotas, and ommissions are optimal ontrats that are mutually exlusive. Setions 4 and 5 onsider demand unertainty 3 Continuous effort would not be analytially tratable in our dynami setting. 4 An exeption is Shmitz (2005). Similar to our paper, the prinipal implements low effort in the seond period when his return in ase of suess is low. 5 Our result that the prinipal may wish to indue high effort only after a first-period suess resembles a finding by Ohlendorf and Shmitz (2012), who analyze a dynami limited-liability model with ontinuous effort. In their framework, an optimal dynami ontrat always exhibits memory. 5

6 and preferene unertainty as extensions of the basi model and investigate the robustness of the previously optimal ontrats. Setion 6 disusses further variations of the basi model. Setion 7 onludes. 2 The Basi Model A firm hires a sales agent to sell its servie or produt in eah of two periods. In eah period i (i = 1, 2), the agent interats with one ustomer and hooses non-observable effort e i {0, 1}. Low effort (e i = 0) desribes a situation where the agent only fulfills some basi tasks (e.g., being present to answer the ustomer s questions or ontating the ustomer). High effort (e i = 1) orresponds to a situation where the agent additionally engages in ativities suh as atively ommuniating the advantages of the firm s produt or learning a ustomer s speifi needs. Sine the firm annot observe whether the agent engaged in more than just the basi ativities or not, it faes a typial moral-hazard problem. When the agent hooses low effort, the ustomer buys the produt with probability α. With high effort, the sale probability inreases to α + ρ, with α, ρ > 0 and α + ρ < 1. 6 The parameter ρ thus indiates the sales-effort responsiveness. The agent s effort osts are e i, where > 0. When a sale ours, the firm reeives the revenue R > 0. We assume that ρr > 0, i.e., high effort is effiient. Let x i {0, 1} indiate whether ustomer i bought the produt (x i = 1) or not (x i = 0). The total number of sales is x = x 1 + x 2, x {0, 1, 2}. At the end of period i, both the firm and the agent observe x i. 7 At the beginning of the first period, the firm offers the agent a ompensation sheme w that depends on sales, w = (w 00, w 10, w 01, w 11 ), where w x1 x 2 denotes the payment when sales x 1, x 2 {0, 1} ourred. The firm designs the ompensation sheme to maximize expeted profit, i.e., expeted revenues minus expeted ompensation. Note that our setting allows for a variety of ompensation plans, inluding fixed salaries, ommissions, and quota-based bonuses. We assume that the sales agent is risk neutral 8 and that the firm annot speify negative payments to fine the agent for poor performane. The orresponding limited-liability ondition w x1 x 2 0 for all x 1 and x 2 prevents the trivial outome that the firm always ahieves effiieny beause it does not fae a real ontratual frition. assumed to be zero. Finally, the agent s reservation value is The timeline of the game is as follows. First, the firm offers a ompensation ontrat. Then 6 We exlude α = 0 beause in this ase the prinipal an always indue the effiient (first-best) solution. 7 In Setion 6, we disuss a situation where the firm only observes total sales x. 8 Akerberg and Bottiini (2002), Hilt (2008), and Bellemare and Shearer (2010) empirially doument that less risk averse agents sort themselves into more risky jobs. Thus, in pratie, we an expet sales agents to have a high risk tolerane. 6

7 the agent aepts or rejets the ontrat offer. If he rejets, the game will end and the agent earns his reservation value. If he aepts, he will interat with the first-period ustomer and deide on effort e 1. After having observed the first-period sales outome, the agent deals with another ustomer in the seond period and hooses e 2. Finally, the sales agent is paid aording to the ompensation ontrat. 3 Optimal Compensation in the Basi Model Depending on the magnitude of sales revenues R, effort osts, basi suess probability α, and sales-effort responsiveness ρ, the firm may prefer low-powered, medium-sized, or highpowered inentives. Therefore, we first identify the relevant senarios haraterized by the implementation of different effort ombinations. Eah senario will turn out optimal for ertain parameter onstellations. In senario 1, the firm indues low effort in both periods (e 1 = e 2 = 0). The firm then optimally offers a fixed salary of zero beause both the agent s reservation utility and the loweffort osts are normalized to zero. Optimal ompensation thus is w F = (0, 0, 0, 0) and the firm s expeted profits are π F = 2αR. By ontrast, in senario 2, effort is high in both periods (e 1 = e 2 = 1). We label the orresponding lass of optimal ontrats as permanent inentive plans sine they provide high effort inentives independent of the sales history. The firm s ontrating problem is solved by bakward indution. To this end, onsider the agent s effort hoie with the seond ustomer. To implement e 2 = 1 for either first-period outome, the wages w x1 x 2 have to satisfy the seondperiod inentive onstraint (α + ρ)w x1 1 + (1 α ρ)w x1 0 αw x1 1 + (1 α)w x1 0 for x 1 = 0, 1, (1) whih requires that the agent s expeted net inome is greater or equal when hoosing high instead of low effort, irrespetive of the sales history x 1. The inentive onstraint is equivalent to w x1 1 ρ + w x 1 0 for x 1 = 0, 1. (2) Antiipating high effort with the seond ustomer, the agent also hooses high effort with the first ustomer (i.e., e 1 = 1) when this leads to a higher expeted net inome aross both periods 7

8 than low first-period effort. The orresponding first-period inentive onstraint is (α + ρ) [(α + ρ)w 11 + (1 α ρ)w 10 ] + (1 α ρ) [(α + ρ)w 01 + (1 α ρ)w 00 ] α [(α + ρ)w 11 + (1 α ρ)w 10 ] + (1 α) [(α + ρ)w 01 + (1 α ρ)w 00 ] ρ [(α + ρ)(w 11 w 01 ) + (1 α ρ)(w 10 w 00 )]. (3) The firm wants to indue permanent inentives at minimal osts. Hene it solves min w 11,w 10,w 01 w 00 0 (α+ρ)2 w 11 +(α+ρ)(1 α ρ)w 10 +(1 α ρ)(α+ρ)w 01 +(1 α ρ) 2 w 00 (4) subjet to the inentive onstraints (2) and (3) and the partiipation onstraint that guarantees the agent a non-negative expeted net inome. The partiipation onstraint, however, is implied by the limited-liability onstraint w 11, w 10, w 01, w 00 0 and the inentive onstraints and we an hene ignore it. 9 The optimal solution to the firm s problem omprises w 00 = 0, whih implies w 01 = ρ. From the first-period inentive onstraint (3), it follows that the firm annot do better than setting w 11 and w 10 suh that (α + ρ)w 11 + (1 α ρ)w 10 = ρ + (α + ρ)w 01 = (1 + α + ρ) ρ. (5) The lass of optimal permanent inentive plans is thus desribed by w 00 = 0, w 01 = ρ, and any ombination of w 11 and w 10 satisfying (5) and (2). One possibility is to set w 10 = ρ and w 11 = 2 ρ, whih desribes a ommission sheme where the sales agent reeives the ommission ( ) ρ per sale. The ompensation sheme wc = 0, ρ, ρ, 2 ρ is thus optimal and the firm s expeted profit is [ π C = 2(α + ρ)r (α + ρ) (1 + α + ρ) ρ + (1 α ρ) ] ( = 2(α + ρ) R ). (6) ρ ρ Beause the agent is proteted by limited liability, he earns a rent in this senario, i.e., the agent s expeted wage minus effort osts is stritly larger than his reservation utility. The rent ] amounts to 2 [(α + ρ) ρ = 2α ρ. In senario 3, the agent s seond-period inentives depend on the sales outome of the first period. The firm indues e 1 = 1 but implements e 2 = 1 if and only if x 1 = 1. That is, the agent should exert high effort with the seond ustomer if and only if he ould sell the produt to the first ustomer. We term the orresponding lass of optimal ontrats utoff inentive plans 9 This is true for all settings studied in this paper. 8

9 sine, as we will show below, the agent must be suessful with the first ustomer as a neessary ondition to earn a reward. From ondition (2), we an dedue the seond-period inentive onstraints w 11 ρ + w 10 and w 01 < ρ + w 00. (7) The first-stage inentive onstraint differs from onstraint (3) above, beause the agent antiipates that seond-period effort will not always be high: (α + ρ) [(α + ρ)w 11 + (1 α ρ)w 10 ] + (1 α ρ) [αw 01 + (1 α)w 00 ] α [(α + ρ)w 11 + (1 α ρ)w 10 ] + (1 α) [αw 01 + (1 α)w 00 ] ρ [(α + ρ)w 11 + (1 α ρ)w 10 αw 01 (1 α)w 00 ] (1 + ρ). (8) Again, the firm minimizes wage osts for implementing the given effort ombination. It now solves min w 11,w 10,w 01,w 00 0 (α+ρ)2 w 11 +(α+ρ)(1 α ρ)w 10 +(1 α ρ)αw 01 +(1 α ρ)(1 α)w 00 (9) subjet to (7) and (8). 10 As an immediate onsequene, we obtain w 00 = w 01 = 0. A ommission sheme an therefore not be optimal. By the inentive onstraints (7) and (8), all w 11 and w 10 satisfying (α + ρ)w 11 + (1 α ρ)w 10 = (1 + ρ) ρ and w 11 ρ + w 10 (10) minimize the firm s wage osts. Therefore, w 00 = w 01 = 0 and the onditions (10) haraterize the lass of optimal utoff inentive plans. One optimal plan is w 10 = 0 and w 11 = 1+ρ α+ρ ρ, whih orresponds to a quota-based bonus where the agent only earns a bonus if he sold the produt ( ) to both ustomers. Hene, the ompensation sheme w Q = 0, 0, 0, 1+ρ is optimal and the firm obtains the expeted profit α+ρ ρ π Q = (α + ρ)r + [ (α + ρ) 2 + (1 α ρ)α ] R (α + ρ)(1 + ρ) ρ ( = (α + ρ) R ) + [ (α + ρ) 2 + (1 α ρ)α ] R (α + ρ). (11) ρ The agent earns the rent α ρ. Besides the three senarios we have disussed so far, there are other effort ombinations that an be implemented. From the firm s point of view, however, none of the other ombinations is 10 The partiipation onstraint is again implied by the other onstraints. 9

10 R Ω H (α, ρ) fixed salary ommission quota Ω L (α, ρ) 0 1 ρ 1 α ρ Figure 1: Optimal ompensation plans optimal. 11 We thus have three andidate solutions that may lead to maximum firm profits a fixed-salary in senario 1, a ommission in senario 2, and a quota-based bonus in senario 3. In senarios 2 and 3, the optimal inentive ompensation plan is not unique. However, ommission and quota-based bonus are mutually exlusive: In senario 2 (senario 3), the uniquely optimal wage if only the seond ustomer buys the produt is w 01 = ρ (w 01 = 0). Thus, the firm will not use a quota sheme in senario 2 and it will not employ a ommission in senario 3. Comparing firm profits in the three senarios and defining the thresholds Ω L (α, ρ) := leads to the following result. α + ρ 1 + ρ 1 + α + ρ ρ 2 < α + ρ 1 ρ 1 (α + ρ) ρ 2 =: Ω H (α, ρ) (12) Proposition 1 The firm maximizes expeted profits by paying a fixed salary, or using a quotabased bonus, or paying a ommission. The fixed salary w F = (0, 0, 0, 0) is optimal if and ( ) only if R Ω L (α, ρ); the quota-based bonus w Q = 0, 0, 0, 1+ρ α+ρ ρ is optimal if and only if ( ) Ω L (α, ρ) < R < Ω H (α, ρ); and the ommission w C = 0, ρ, ρ, 2 ρ is optimal if and only if Ω H (α, ρ) R. The agent earns the rent α ρ under the ommission. under the quota-based bonus and the rent 2α ρ Proof. See Appendix. 11 See the Additional Material 1. 10

11 Proposition 1 shows that one of three pratially relevant ompensation shemes maximizes the firm s expeted profit. Figure 1 illustrates the firm s optimal hoie as a funtion of R/ and ρ. 12 Eliiting high effort from the agent in at least one period is only worthwhile if the revenue-ost ratio, R/, is suffiiently large, R/ > Ω L (α, ρ). This ondition is also satisfied when α is suffiiently low. Intuitively, the firm wants to provide inentives when a ustomer is unlikely to buy if the agent does not put forth an extra effort. Furthermore, the firm will implement inentive pay if ρ is suffiiently large, i.e., if sales are highly responsive to effort. The sales agent then responds more strongly to inentives, whih makes induing high effort less ostly for the firm beause the rent it has to leave to the agent dereases under either inentive sheme. To provide inentives, the firm an hoose between a ommission and a quota-based bonus. It uses a ommission if it wants to generate strong inentives, i.e., if effort should be high with every ustomer. In ontrast, the quota-based bonus reates only intermediate inentives. 13 Beause the agent is rewarded only when he sold the produt to both ustomers, he will no longer be motivated to exert high effort when he failed to make a sale in the first period. Our results show that suh a demotivation effet is frequently in the firm s interest. 14 The omparative advantage of the quota-based bonus is that the relatively high probability of no reward payment makes inentive provision less ostly for the firm: the agent s rent under the quota-based bonus is only half as large as his rent under the ommission. Still, the agent exerts high effort with probability α + ρ in the seond period under the quota-based bonus. On the downside, however, expeted revenues are lower under the bonus than under the ommission beause the hane of a sale in the seond period is lower under the bonus. Therefore, a bonus is optimal when R/ is suffiiently small, implying that the loss in revenues is dominated by the derease in ompensation. This is the ase if and only if R < Ω H (α, ρ) = α+ρ 1 ρ 1 (α+ρ). The ρ 2 α+ρ higher α + ρ and thus 1 (α+ρ), the more likely the first ustomer is to buy and, onsequently, the quota-based bonus does not lead to a loss in revenues beause effort will also be high in the seond period. A higher α thus makes a bonus more attrative ompared to a ommission. For ρ, however, there is a ounterating effet, refleted by 1 ρ. The larger ρ, the easier the agent is ρ 2 to motivate. Therefore, always induing high effort by a ommission beomes more attrative. However, if ρ is suffiiently large, high effort is almost ertain to result in a sale. This makes 12 Reall that, by assumption, high effort is effiient, i.e., ρr > 0. Beause this implies R/ > 1/ρ, only parameter onstellations above the hyperbola 1/ρ are feasible. 13 Kishore et al. (2013) analyze data from a pharmaeutial orporation that swithed from a bonus plan to a ommission plan to ompensate its sales fore. They find that the ommission plan led to more effetive inentives. Overall produtivity inreased by 24%. 14 Steenburgh (2008) empirially analyzes the effets of bonuses on sales fore performane. He onludes that [...] bonuses ause some salespeople, those who are unlikely to make quota, to redue effort, but this effet is more than ompensated for by produtive inreases in output by other salespeople. (Steenburgh 2008, p. 252) 11

12 a loss with the seond ustomer highly unlikely under a quota sheme and, thus, the bonus dominates the ommission: lim ρ 1 α Ω H (α, ρ) =. Consequently, firms with suffiiently low or suffiiently high effort-sales responsiveness should adopt a quota-based bonus. This nonmonotoniity result onerning the impat of ρ on the optimal inentive sheme is illustrated in Figure 1 and summarized in the following orollary. Corollary 1 Assume that the firm wants to provide effort inentives, i.e., R > Ω L (α, ρ). A quota-based bonus is optimal if sales respond either suffiiently little or suffiiently strongly to effort. The following two setions onsider more omplex senarios to hek the robustness of our previous findings. 4 Demand Unertainty In the basi model, we have assumed that the agent s opportunities to sell the produt are onstant over time, i.e., the agent an be ertain that there will be a ustomer also in the seond period. However, in pratie firms and their sales agents often fae demand unertainty, i.e., they annot perfetly foresee future market developments for their produt. Moreover, from working in the field the agent is often better informed about short-run demand developments than the firm. To apture this problem, we now assume that first-period demand is ertain but the seond ustomer only arrives with an exogenously given probability β x1 (0, 1) where x 1 {0, 1} again denotes first-period sales. There are three possibilities. We an have β 1 = β 0 so that seond-period demand is unertain but independent of whether the agent was suessful with the first ustomer or not. Alternatively, seond-period demand depends on whether there was a first-period sale. In that ase, we may either have β 1 > β 0 indiating that first-period suess inreases the probability of a seond-period ustomer ompared to a first-period failure, e.g., due to word-of-mouth advertising. Or we may have β 1 < β 0, e.g., beause the market has only few potential ustomers ( thin markets ) so that a suessful sale in the first period leads to a signifiant redution of the remaining market apaity (e.g., markets like real estate markets or the high-end art market where very expensive goods are traded). The time struture of the model hanges as follows: First, the firm deides on the ompensation plan. Then, the first ustomer arrives and the agent deides on exerting effort. Afterwards, the agent (but not the firm) observes whether there will be a seond ustomer (with probability 12

13 β x1 ) or not (with probability 1 β x1 ). If a ustomer arrives, the agent again deides on effort. The firm does not learn whether seond-period failure is the result of no demand. Define ˆΩ L := 1 + ρβ 1 1 αβ 0 + (2α + ρ) β 1 α + ρ ρ 2. Solving for the optimal ompensation plan leads to the following findings: Proposition 2 Suppose there is demand unertainty suh that the seond-period ustomer arrives with probability β x1 (0, 1). Assume further that β 0 2β 1, i.e., the market is not too thin. (i) If R ˆΩ L, the fixed salary w F is optimal. (ii) If ˆΩ L < R Ω H (α, ρ), the ) quota-based bonus w Q du (0, = 1+ρβ 0, 0, 1 (α+ρ)β 1 ρ is optimal. (iii) If R > Ω H (α, ρ) and β 0 β 1, ( ) the sheme wdu BC = 0, ρ, ρ, 2 ρ + α β 0 β 1 α+ρ β 1 ρ is optimal. That is, the firm maximizes its profits by using a ommission ombined with a bonus to be paid when the agent was suessful in both periods. If R > Ω H (α, ρ) and β 0 < β 1, the dynami ommission w SC du = (0, [1 + α(β 0 β 1 )] ρ, ρ, [2 + α(β 0 β 1 )] ρ ) is optimal. Proof. See Appendix. Proposition 2 onsiders a situation where markets are not too thin, i.e., β 0 is not too large relative to β 1. The proof of the proposition shows that we have a similar solution as in the basi model. The firm offers the fixed salary w F if the revenue-ost ratio, R/, is suffiiently small. If the revenue-ost ratio takes intermediate values, optimal sales fore ompensation is given by a lass of utoff inentive plans that never reward the agent if he failed in the first period (i.e., w 01 = 0). In partiular, the firm annot do better than using a quota-based bonus that is adapted to the given situation with demand unertainty. This bonus, w 11, dereases in β 1 beause a low probability β 1 makes it more diffiult for the sales agent to obtain the bonus, whih requires sales to two ustomers. This effet demotivates the agent. By hoosing a high bonus, the firm restores the agent s inentives. 15 For a suffiiently large revenue-ost ratio, the firm relies again on a lass of permanent inentive plans, i.e., it always indues high effort. However, in ontrast to the basi model, this lass of permanent inentive plans inludes a pure ommission only if there are no demand externalities aross periods (i.e., β 0 = β 1 ). If markets are thin (i.e., β 0 > β 1 ), the firm maximizes profits by ombining the ommission from the basi model, α ρ, with the bonus is paid when the agent sold the produt in both periods. 16 β 0 β 1 α+ρ β 1 ρ that If β 0 > β 1, a first-period failure is 15 Note that the bonus w 11 approahes infinity as β 1 goes to zero. A very large bonus may, thus, not be feasible if the firm is wealth onstrained. We disuss this problem in more detail in Setion In a reent empirial study, Chung et al. (2014) find that quota-based bonuses enhane sales agents performane in firms that also employ ommissions. 13

14 quite attrative for the agent sine it leads to a higher probability of earning a seond-period rent. To work against this detrimental inentive effet, the firm needs to attah an extra reward to two sales in order to inentivize the agent in both periods. By ontrast, if β 0 < β 1, the firm an employ a dynami ommission with ommission rates that inrease over time. 17 It is no longer optimal for the firm to pay the ommission ρ for a first-period sale. The agent s payoff from a first-period suess an be lowered beause the agent is also motivated by the prospet to earn a seond-period rent with a higher probability when he is suessful in the first period. The lower threshold for implementing a quota-based bonus, ˆΩL, inreases in β 0 and dereases in β 1. Intuitively, induing high effort in period 1 is less (more) worthwhile, if it lowers (inreases) the hane of having a seond-period ustomer relative to low effort. The higher threshold for implementing a bonus, Ω H (α, ρ), is independent of β 0 and β 1, and idential to the orresponding threshold in the basi model. The proof of Proposition 2 shows that the expeted returns and the expeted labor osts inrease in β 0 and β 1 under both lasses of inentive plans and that these effets anel eah other out when the respetive profits are ompared. For β := β 1 = β 0, the lass of permanent inentive plans from Proposition 2 (iii) ontains the ommission w C from the basi model, whih is independent of β. The ommission sheme thus is a quite robust inentive devie when there is demand unertainty that is independent of the sales history. Moreover, we obtain the following uniqueness result. Corollary 2 Assume that R > Ω H (α, ρ) and let β := β 1 = β 0. Suppose the firm does not observe β. The ommission w C then is the uniquely optimal sales fore ompensation beause it is the only optimal inentive sheme whose implementation does not require knowledge of β. If ˆΩ L < R Ω H (α, ρ), the firm always requires knowledge of β to determine the optimal ompensation. Proof. See Appendix. As in the basi model, the optimal inentive sheme is not unique. However, as the proof of orollary 2 shows, the ommission is the only inentive sheme that indues e 1 = e 2 = 1 and is independent of β. 18 Hene, if the firm always wants to indue high effort, it an implement a ommission even without knowing β. This is relevant beause in pratie the firm will often have less preise information about β than the agent. We now onsider the ase where markets are rather thin. In this ase, another effort profile 17 Note that w 11 = w 10 + w 01 as required for a pure ommission sheme. 18 There is no inentive sheme that is equivalent to the quota w Q du but independent of β. 14

15 an be optimal. Define { 1 (1 α) β0 + (ρ α) β 1 Ω (β 0, β 1 ) := min, 1 (ρ α) β } 0 + (ρ α)β 1 α + ρ 1 β 0 + (α + ρ) β 1 1 (α + ρ) β 0 + (α + ρ) β 1 ρ 2. Proposition 3 Assume that the market is rather thin, i.e., β 0 > 2β 1. If (α + ρ) /ρ 2 < R < Ω (β 0, β 1 ), the firm indues high effort only in the seond period and implements the ompensation plan wdu (0, F C = 0, ρ, ρ ). Otherwise, the ases (i)-(iii) in Proposition 2 apply. Proof. See the proof of Proposition 2 in the Appendix. When the firm faes a very thin market and the revenue-ost ratio is rather small, it is not optimal for the firm to indue high effort in the first period. The reason is the impat of the strong negative externality of high first-period effort on seond-period demand. The firm thus offers a fixed salary for the first period and fouses on the provision of seond-period inentives, paying the agent the ommission /ρ independent of his first-period performane. 5 Preferene Unertainty In this setion, we drop the assumption of the basi model that the firm and the agent exatly know how exerting high effort affets the probability of a sale. This simplifying assumption seems realisti if the firm and the agents know the ustomers preferenes very well and an thus antiipate how the ustomers respond to high effort. However, there also exist situations in whih the market s harateristis are not perfetly known. In partiular, how ustomers will respond to effort might not be preditable. For example, when launhing a new produt or entering a foreign market with a given produt, the firm and its agents do not exatly know whether it will be easy (i.e., high value of ρ) or diffiult (i.e., low value of ρ) to sell the produt. To analyze suh preferene unertainty, we assume that the market either onsists of type-1 ustomers being haraterized by ρ = ρ 1, or it onsists of type-2 ustomers being desribed by ρ = ρ 2. The ustomer type is distributed suh that ρ = ρ 1 ρ 2 with probability p with probability 1 p with ρ t + α < 1 (t = 1, 2). Hene, we also have E[ρ] + α < 1 with E[ρ] = pρ 1 + (1 p)ρ 2. The orresponding variane is given by V ar[ρ] = (ρ 1 E[ρ]) 2 p + (ρ 2 E[ρ]) 2 (1 p). As in the basi model, we assume that exerting high effort is effiient, i.e., E[ρ]R > 0. We further assume that observed output in period 1 is not a reliable signal to update beliefs over ρ (e.g., beause the market is large so that observing one ustomer s behavior is a negligible sample). 15

16 Hene, the probability distribution of ρ remains the same for the two periods. All remaining assumptions of the basi model (e.g., a seond ustomer that arrives for sure in period 2) are retained. We obtain the following result: Proposition 4 The optimal sales fore ompensation is unique. If R/ is suffiiently small, the firm will hoose the fixed salary w F. If R/ is suffiiently large, the firm will either use the utoff inentive plan w Q pu = (0, 0, 0, (1 + E [ρ]) /E[ρ(α + ρ)]) or the permanent inentive plan w L pu = (0, 0, /E[ρ], (/E[ρ]) + (/E[ρ(α + ρ)])). While w Q pu is a simple quota-based bonus, w L pu reflets a long-term bonus sheme that emphasizes seond-period suess. The firm prefers w Q pu to w L pu if and only if Proof. See Appendix. ( E[(α + ρ) 2 ) ] E [α + ρ] E [ρ] < ( E[ρ 2 ] (1 α) E[ρ] ) R. (13) E[ρ(α + ρ)] E [ρ] In ontrast to the basi model and the ase of demand unertainty, preferene unertainty always leads to a unique solution for the optimal inentive ontrat. As in the basi model, the firm indues zero inentives if the revenue-ost ratio, R/, is small. Otherwise, it either uses a quota-based bonus or a long-term bonus sheme. The latter leads to a positive payment only if the sales agent is suessful with the seond ustomer and rewards a first-period sale solely if it is followed by a seond-period sale. Hene, the optimal inentive shemes emphasize long-term sales beause they lead to zero payments in ase of seond-period failure. Rewarding short-term suess by paying a ommission per sale, whih belongs to the optimal forms of sales fore ompensation in the basi model, is no longer optimal under preferene unertainty. The proof of Proposition 4 gives a tehnial intuition for this result. The proof shows that, ontrary to the basi model, the firm s iso-ost urves for implementing effort are flatter than the straight line desribing the binding first-period inentive onstraint. 19 In other words, the introdution of preferene unertainty hanges the rate of substitution between w 11 and w 10 in favor of w 11 in the firm s objetive funtion. It is thus less ostly for the firm to implement high effort by using w 11 instead of w 10. This leads to a unique solution for the optimal sales fore ompensation, whih fouses on inentives for seond-period sales. Inspetion of (13) shows under whih onditions the firm prefers the quota-based bonus to the long-term bonus sheme and vie versa. The left-hand side of (13) is negative (see the Appendix for a proof). Hene, a suffiient ondition for (13) to be satisfied is that the righthand side of (13) is positive. The right-hand side is inreasing in α and, for α 1 E[ρ], it 19 In the basi model, V ar[ρ] = 0 so that both slopes are idential (see (32), (34) and (35)). 16

17 approahes (E[ρ 2 ] (1 (1 E[ρ])) E[ρ])R = ( E[ρ 2 ] E[ρ] 2) R = V ar[ρ]r, whih is stritly positive. Thus, there exists a ut-off value ᾱ (0, 1 E[ρ]) suh that ondition (13) is satisfied if α > ᾱ. Reall that in the basi model the quota-based bonus is also attrative for the firm when α is large. Then a sale is rather likely to our in the first period, implying that the disadvantage of the quota-based bonus low effort in the seond period after failure in the first is unlikely to take effet. Hene, our findings on the optimality of the quota sheme from the basi model are robust with respet to the impat of α. The findings of this setion indiate that bonus shemes are partiularly effetive inentive devies when firms launh new produts or enter new markets. This result is in line with empirial observations. As Joseph and Kalwani (1998, p. 151) report, 26% of all bonus-paying firms use bonuses to boost inentives for selling new produts. The authors onlude that bonus payments tied to speifi organizational goals suh as promoting new produt sales or sales to new ustomer groups help bring about goal ongruene between the interests of the salesperson and the long-term objetives of the firm (Joseph and Kalwani (1998), p. 158). 6 Disussion In Setions 4 and 5, we introdued two alternative forms of unertainty, resulting in a more omprehensive setup for the disussion of optimal sales fore ompensation. In the following, we will onsider less fundamental extensions of the basi model to further test the robustness of our results. 6.1 Non-Observability of the Sales Sequene So far we have assumed that the firm an ostlessly observe sales at the end of eah period. In pratie, however, measuring sales performane is usually ostly. Thus, eteris paribus, the firm prefers to ollet sales figures less frequently. To analyze the impat of sarer sales information on optimal ontrating, we now assume that the firm only observes total sales x at the end of the seond period, but not sales in eah period i, x i. This implies that the firm does not know the sequene of sales, i.e., when only one sale oured, the firm annot tell whether the agent was suessful in the first or seond period. By ontrast, the agent still observes sales in every period beause he knows whether a ustomer bought the produt after finishing the sales talk. In suh a situation, the firm annot base payments on the sequene of sales. Given our previous 17

18 analysis, solving the firm s new problem is straightforward. We just have to add the restrition w 01 = w 10 to the firm s problem in eah senario of the basi model from Setion 3. Sine the fixed wage w F, the quota-based bonus w Q, and the ommission w C desribed in Proposition 1 all satisfy this ondition, they are still feasible and hene optimal when the firm annot observe the sales sequene. Moreover, the quota-based bonus and the ommission are the only shemes from the previously optimal lass of utoff and permanent inentive plans, respetively, that satisfy w 01 = w 10. Thus, sarer sales information does not redue the firm s profit but implies unique optimality of the two pratially relevant inentive shemes. Proposition 5 Suppose the firm annot observe the sales sequene and thus w 01 = w 10 has to hold. The firm then still earns the same profits as in the basi model. The optimal ompensation plan, however, is unique: The firm either implements the fixed salary w F (if R Ω L (α, ρ)); or the quota-based bonus w Q (if Ω L (α, ρ) < R < Ω H (α, ρ)); or the ommission w C (if Ω H (α, ρ) R ). Besides pure bonuses and ommissions, the third inentive sheme of high pratial relevane is a ombination of the two. It turns out that this inentive sheme, whih belongs to the optimal lass of utoff inentive plans under demand unertainty, is also uniquely optimal when the firm annot observe the sales sequene. Proposition 6 Suppose there is demand unertainty as modeled in Setion 4 and the firm annot observe the sales sequene. Consider the ase where markets are thin but not too thin, i.e., β 1 β 0 2β 1. The firm then still earns the same profits as under an observable sales sequene, but the optimal ompensation plan is unique: The firm either implements the fixed salary w F (if R ˆΩ L ); or the quota-based bonus w Q du (if ˆΩ L < R Ω H (α, ρ)); or the ombination of ommission and bonus, w BC du (if Ω H (α, ρ) < R ). 6.2 Renegotiation-Proofness So far we have impliitly assumed that the firm an ommit to a long-term ontrat overing both periods. However, if suh ommitment is not possible, both the firm and the agent may prefer to renegotiate the ontrat after having observed the first-period outome. In that ase, the ontrat would not be renegotiation-proof. In this setion, we assume that the firm annot ommit to a long-term ontrat and hene has to derive the optimal renegotiation-proof ontrat. A ontrat overing both periods is renegotiation-proof if, at the beginning of the seond period, there does not exist a short-term ontrat for period 2 that makes no party worse off but at least one party stritly better off ompared to the given two-period ontrat. 18

19 First, onsider the firm s optimal behavior in the seond period, assuming for the moment that it is not restrited by any ontrat. The firm prefers either e 2 = 1 or e 2 = 0. To indue high effort e 2 = 1, the firm optimally offers the wage /ρ in ase of suess and zero payment in ase ( ) of failure, leading to the profit (α + ρ) R ρ. Low effort e 2 = 0 is optimally implemented by a fixed wage of zero and hene the firm s profit is αr. The firm thus prefers high effort to low effort if and only if R > α+ρ. Note that α+ρ (Ω ρ 2 ρ 2 L (α, ρ), Ω H (α, ρ)), where Ω L (α, ρ) and Ω H (α, ρ) denote the two ut-off values from Proposition 1. We an now solve for the optimal renegotiation-proof ontrat. If R > Ω H (α, ρ), the firm again maximizes profits by implementing the ommission sheme w C. Beause the ommission indues high effort in both periods by paying /ρ per sale, the firm does not want to renegotiate the ontrat after the first period. If R ( α+ρ, Ω ρ 2 H (α, ρ)), the firm will no longer implement the quota sheme w Q or any other utoff inentive plan beause it is not renegotiation-proof. To see this, suppose the firm offered a utoff inentive plan. If the agent fails in period 1, he will exert zero effort in period 2 beause he has no hane to realize two sales now. Hene, as R > α+ρ, the firm would prefer to renegotiate the old ontrat and replae it by a short-term ρ 2 ontrat that offers wage /ρ in ase of suess and zero payment in ase of a failure. agent would aept this ontrat beause he would then earn a rent. Thus, beause in the given situation the firm always wants to indue high effort, the ommission sheme w C is an optimal renegotiation-proof inentive sheme. If R The (Ω L (α, ρ), α+ρ ], however, the quota sheme w Q ρ 2 will be renegotiation-proof sine the firm has no inentive to offer a new ontrat after a firstperiod failure. If R Ω L (α, ρ), the firm is always interested in induing zero effort. Hene, the fixed salary w F is still optimal. The following proposition summarizes our findings: Proposition 7 Suppose the firm annot ommit to a two-period long-term ontrat so that optimal ontrats have to be renegotiation-proof. If R Ω L (α, ρ), the fixed salary w F is optimal. If R (Ω L (α, ρ), α+ρ ], the quota-based bonus w Q is optimal. If R ρ 2 > α+ρ, the ommission ρ 2 sheme w C is optimal. A omparison of Proposition 1 and Proposition 7 shows that, ompared to the basi model, the firm should replae a quota sheme with a ommission for a ertain set of parameters if it annot ommit to long-term ontrats. Intuitively, if the revenue-ost ratio is suffiiently large, the firm is always interested in induing high effort. Thus, maintaining the quota sheme w Q in ase of first-period failure is then not redible to the agent. Suh a problem does not our under a ommission or a fixed salary where inentives remain onstant over time. Note that the redibility problem of the quota-based bonus is also detrimental to first-period inentives. 19

20 Antiipating renegotiation of the quota sheme in ase of first-period failure, 20 the agent will exert high effort in period 1 if and only if ] (α + ρ) [(α + ρ)w Q 11 + (1 α ρ)wq 10 + (1 α ρ) [(α + ρ)w01 r + (1 α ρ)w00] r ] [(α + ρ)w Q 11 + (1 α ρ)wq 10 α ρ + (1 α) [(α + ρ)w01 r + (1 α ρ)w00] r ] (1 + ρ), [ (α + ρ)w Q 11 + (1 α ρ)wq 10 (α + ρ)wr 01 (1 α r)w r 00 where supersript r denotes the payments after renegotiation of the quota sheme, i.e., w01 r = ρ and w00 r = 0. It is easily verified that the ondition does not hold. Thus, first period inentives are ompletely erased. Negative inentive effets from renegotiating initial ontrats are also known from other long-term inentive shemes suh as stok options. 21 Suh shemes will beome ineffetive when a manager learns that it is impossible for him to reah the long-term goal. If the manager antiipates that the long-term sheme may be adjusted or omplemented by an optimal shortterm sheme later on, even early inentives an be ompletely erased. 6.3 Advertising Firms often invest in advertising to inrease sales. Our findings from the basi model allow us to disuss the impat of advertising on the optimal sales fore ompensation. In priniple, there are two possibilities how advertising and the agent s effort an interat to boost sales. Depending on the produt s harateristis, effort and advertising an be either substitutes or omplements. Consider, first, the ase of effort and advertising being substitutes, i.e., exerting high effort is less important to sell the produt with advertising than without. Advertising then inreases the basi probability of a sale, α. For example, this is the ase if advertising informs about a low produt prie and ustomers primarily deide on the basis of prie when hoosing between different produts. Seond, advertising and effort an be omplements, i.e., sales beome more responsive to effort under advertising. In this ase, advertising inreases ρ. For example, onsider the ase of highly omplex produts. Advertising an be used to all the ustomers attention to new innovative features of these produts, making the ustomers urious. But sales effort is still needed to explain to them the usefulness of the new features. In this situation, the advertised features help the agent during his sales talk. Beause both Ω L (α, ρ) and Ω H (α, ρ) inrease in α, advertising redues the provision of 20 The quota sheme would not be renegotiated after a first-period suess beause the agent would not agree to lowering w See, e.g., Aharya et al. (2000) on the inentive effets of resetting strike pries on stok options. 20

21 inentives and hene effort when advertising and effort are substitutes. This result niely orresponds to the observation by Basu et al. (1985, p. 269) that salaries are typial of industries like the pakaged goods industries where advertising already strongly affets sales. By ontrast, Ω L (α, ρ) dereases in ρ, implying that a fixed salary beomes less likely under advertising when effort and advertising are omplements. Despite this omplementarity, however, advertising may atually lead to lower effort beause the effet of ρ on Ω H (α, ρ) is ambiguous (omp. Fig. 1): if ρ is already quite large and is further inreased by advertising, the firm will replae a permanent inentive plan (e.g., ommission) with a utoff inentive plan (e.g., quota-based bonus). 6.4 Limited Liability of the Firm In our basi model, only the sales agent is proteted by limited liability or, equivalently, is wealth-onstrained. However, the firm may be wealth-onstrained as well, whih an make large inentive payments to the agent infeasible. Suppose the firm has zero initial wealth so that wage payments to the agent annot exeed total revenues xr with x {0, 1, 2}. In that ase, the optimal fixed salary in senario 1 is learly feasible. The optimal ommission in senario 2 is also feasible beause ρ R is implied by the assumption ρr 0. By ontrast, the optimal quota sheme in senario 3 may not be appliable any longer sine the wage w 11 = 1+ρ α+ρ ρ an be larger than 2R for Ω L (α, ρ) < R < Ω H (α, ρ). 22 Proposition 8 If the firm has zero initial wealth, the fixed salary and the ommission are still feasible, but the quota-based bonus is feasible if and only if The intuition is the following. 1+ρ 2ρ(α+ρ) < R. When an agent is risk neutral but proteted by limited liability, the optimal quota sheme typially ombines a large quota with a high bonus in ase of suess. 23 In our setting, the sales agent will only obtain a positive bonus if he is suessful in both periods, whih yields the maximum possible quota. The orresponding bonus payment w 11 an be so large that it exeeds total revenues 2R. The ondition in Proposition 8 shows that a wealth-onstrained firm an apply a quota-based bonus when α is suffiiently large beause then the bonus w 11 is rather small. 24 If the ondition does not hold, the firm will still employ a utoff inentive plan when Ω L (α, ρ) < R < Ω H (α, ρ). The firm then maximizes its profit by the inentive sheme (0, (1 α) ρ, 0, (2 α) ρ ), whih satisfies the onditions in (10) and never exeeds total revenues ρ Note that w 11 < 2R is equivalent to < R. Further, we have 1+ρ (ΩL (α, ρ), ΩH (α, ρ)) if, e.g., 2ρ(α+ρ) 2ρ(α+ρ) α = 0.1 and ρ = See also Demougin and Fluet (1998) and Oyer (2000) on a model with ontinuous effort ρ Note that lim α 1 ρ = 1+ρ < 1, whih together with the assumption 1 < R implies that the 2ρ(α+ρ) 2ρ ρ ρ ondition in Proposition 5 holds for suffiiently large α. 21

22 7 Conlusions We analyze a dynami moral-hazard model on optimal sales fore ompensation. Induing inentives is ostly for the firm sine sales agents earn rents. Our theoretial framework allows to design the optimal ompensation plan without imposing any ad ho restritions on the lass of feasible ontrats. Frequently observed inentive plans quota-based bonuses, ommissions, and a ombination thereof turn out to belong to the optimal ontrat lasses, but are mutually exlusive. It is therefore ruial for the firm to aurately investigate its speifi environment before deiding on sales fore inentives. We show that the aforementioned inentive plans an even be the unique solution to the firm s optimization problem when there is asymmetri information between the agent and the firm regarding the degree of demand unertainty or the sequene of sales. Our analysis leads to several testable preditions. First, firms with higher revenues are more likely to adapt inentive plans. Moreover, as revenue inrease, firms should favor ommissions over quota-based bonuses. For given revenues, firms with rather low or rather high effortsales responsiveness 25 should prefer bonuses to ommissions. If there is demand unertainty, firms with intermediate revenues ontinue to rely on quota-based bonuses. High-revenue firms, however, are likely to adopt more sophistiated ompensation plans. If markets are thin but not too thin, suh firms should supplement a ommission by a quota-based bonus. If present sales inrease future sales prospets (e.g., through word-of-mouth advertising), dynami ommissions that inrease over time should be observed. By ontrast, in very thin markets firms should use inentives autiously, e.g., ommission plans should be observed only toward the end of the produt-life yle. If a high-revenue firm enters a new market or launhes a new produt, it is likely to rely on long-term inentive shemes that partiularly emphasize suess in later periods. Low-revenue firms should still employ simple quota-based bonuses. Finally, ommissions tend to be more prevalent than quota-based bonuses in firms that fae a tight liquidity ondition, annot ommit to long-term ompensation plans, or find it hard to predit future demand developments. To keep the analysis tratable, we investigate dynami aspets of sales fore ompensation plans in a two-period framework. This is the simplest way to apture that intermediate information on sales performane an indue the agent to adapt his effort. In pratie, intermediate information may be available more often during the ontrat period. However, one also has to keep in mind that ontrat duration is restrited by limited ommitment to long-term ontrating and the fat that the firm annot postpone payments to agents for too long. A general 25 Effort-sales responsiveness an be approximated by sales response funtions. 22

23 multiperiod framework leads to a onsiderably larger set of feasible effort alloations and, thus, andidate solutions to the firm s ontrating problem. Nevertheless, without intermediate sales information on the side of the firm, a pure ommission remains an optimal permanent inentive plan in a broad range of settings if the firm always prefers high effort, as Shöttner (2015) demonstrates. 8 Appendix Proof of Proposition 1: The ommission sheme (senario 2) will dominate the quota sheme (senario 3) iff ( 2(α + ρ) R ) > R [(2 + ρ) α + ρ (1 + ρ)] (α + ρ)(1 + ρ) ρ ρ (14) R > 1 ρ α + ρ 1 (α + ρ) ρ 2 =: Ω H (α, ρ). The ommission will outperform the fixed salary (senario 1) iff ( 2(α + ρ) R ) > 2αR R ρ > α + ρ ρ 2. (15) Note that Ω H (α, ρ) > α+ρ. Hene, the ommission is the optimal inentive sheme if and only ρ 2 if R > Ω H (α, ρ). Otherwise, the firm prefers a quota-based bonus to the fixed wage if and only if R [(2 + ρ) α + ρ (1 + ρ)] (α + ρ)(1 + ρ) ρ > 2αR R > 1 + ρ α + ρ 1 + α + ρ ρ 2 =: Ω L (α, ρ). (16) Proof of Proposition 2: There are four andidate solutions for the optimal inentive ontrat, whereas the remaining possible ontrats an never be optimal (see the Additional Material 2). (I) If the firm wants to indue e 1 = e 2 = 0, it will offer w F and obtains expeted profits [1 + αβ 1 + (1 α) β 0 ]αr. (17) (II) Next, suppose the firm wishes to indue e 1 = e 2 = 1. If the agent has met the seond ustomer, the seond-period inentive onstraint (2) from the basi model will apply again. 23

24 However, the first-period inentive onstraint under demand unertainty now reads as follows: (α + ρ) [β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 ] +(1 α ρ) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 )w 00 ] α [β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 ] +(1 α) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 )w 00 ] (α + ρ) (β 1 w 11 β 0 w 01 ) + (1 α ρ) (β 1 w 10 β 0 w 00 ) +(1 β 1 )w 10 (1 β 0 )w 00 [1 ρ (β 0 β 1 )] ρ. (18) Hene, the firm solves min (α + ρ) [β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 ] w 11,w 10,w 01,w (1 α ρ) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 )w 00 ] subjet to (2) and (18), leading to optimal payments w 00 = 0 and w 01 = ρ. The firm s problem an, therefore, be simplified to [ ] min (α + ρ) β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 + (1 α ρ)β 0 w 11,w 10 0 ρ subjet to w 11 ρ + w 10 and β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 [1 ρ (β 0 β 1 ) + (α + ρ)β 0 ] ρ. The firm thus optimally hooses wages (w 10, w 11 ) that satisfy β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 = [1 ρ (β 0 β 1 ) + (α + ρ)β 0 ] ρ (19) and w 11 ρ + w 10 and w The set of optimal ontrats an also be rewritten as w 11 = 1 ρ (β 0 β 1 ) + (α + ρ)β 0 β 1 (α + ρ) ρ 1 β 1(α + ρ) w 10. (20) β 1 (α + ρ) and 0 w 10 [1 + α(β 0 β 1 )] ρ. Consequently, if β 0 β 1, the inentive sheme ( 0, ρ, ρ, αβ 0 + (α + 2ρ) β 1 β 1 (α + ρ) ) = (0, ρ ρ, ρ [, 2 + α α + ρ β 0 β 1 β 1 ] ) ρ (21) 24

25 implements high effort in both periods at minimal ost. If β 0 < β 1, however, the firm must ( ) set w 10 < w 01. The sheme 0, [1 + α(β 0 β 1 )] ρ, ρ, [2 + α(β 0 β 1 )] ρ then minimizes wage osts. The firm s expeted profit is { (α + ρ) [1 + (α + ρ)β 1 + (1 α ρ)β 0 ] R [1 + (1 ρ)β 0 + ρβ 1 ] }. (22) ρ (III) Now suppose the firm indues e 1 = 1 but e 2 = 1 if and only if x 1 = 1. The first-period inentive onstraint an be written as (α + ρ) [(α + ρ)β 1 w 11 + (1 β 1 (α + ρ)) w 10 β 1 ] + (1 α ρ) [αβ 0 w 01 + (1 αβ 0 ) w 00 ] α [(α + ρ)β 1 w 11 + (1 β 1 (α + ρ)) w 10 β 1 ] + (1 α) [αβ 0 w 01 + (1 αβ 0 ) w 00 ] (α + ρ)β 1 w 11 + (1 β 1 (α + ρ)) w 10 αβ 0 w 01 (1 αβ 0 ) w 00 (1 + ρβ 1). (23) ρ The firm thus minimizes (α + ρ)[β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 ] + (1 α ρ)[αβ 0 w 01 + (1 αβ 0 ) w 00 ] subjet to w 11 ρ + w 10, w 01 ρ + w 00, and (23). Therefore, w 00 = w 01 = 0, and the two other optimal wages are desribed by min (α + ρ)[β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 )w 10 ] subjet to w 11 ρ +w 10 and (23). Under the optimal ontrat, the latter onstraint boils down to (α + ρ)β 1 w 11 + (1 β 1 (α + ρ)) w 10 = (1 + ρβ 1). (24) ρ Optimal are, e.g., w 10 = 0 and w 11 = 1+ρβ 1 (α+ρ)β 1 ρ, whih together with w 00 = w 01 = 0 desribes a quota sheme. The firm s expeted profit is (α + ρ)[r + β 1 (α + ρ)r] + (1 α ρ)β 0 αr (α + ρ) (1 + ρβ 1). (25) ρ (IV) Consider the ase where effort is high only with ustomer 2: e 2 = 1 and e 1 = 0. The firm minimizes α [β 1 ((α + ρ)w 11 + (1 (α + ρ))w 10 ) + (1 β 1 ) w 10 ] + (1 α) [β 0 ((α + ρ)w 01 + (1 (α + ρ))w 00 ) + (1 β 0 ) w 00 ] 25

26 subjet to w 01 ρ + w 00 and w 11 ρ + w 10 as seond-period inentive onstraints, and α [β 1 ((α + ρ)w 11 + (1 (α + ρ))w 10 ) + (1 β 1 ) w 10 ] +(1 α) [β 0 ((α + ρ)w 01 + (1 (α + ρ))w 00 ) + (1 β 0 ) w 00 ] (α + ρ) [β 1 ((α + ρ)w 11 + (1 (α + ρ))w 10 ) + (1 β 1 ) w 10 ] +(1 (α + ρ)) [β 0 ((α + ρ)w 01 + (1 (α + ρ))w 00 ) + (1 β 0 ) w 00 ] (1 + ρ (β 1 β 0 )) ρ β 1(α + ρ)w 11 + (1 (α + ρ) β 1 ) w 10 (1 (α + ρ) β 0 ) w 00 β 0 (α + ρ)w 01 as first-period inentive onstraint. Thus, w 00 = w 10 = 0 and w 01 = w 11 = ρ ompensation leads to expeted profit are optimal. This (α + (1 α) (α + ρ) β 0 + α (α + ρ) β 1 ) R ((1 α) β 0 + αβ 1 ) (α + ρ) ρ. (26) We now ompare the profits in the different senarios. Profit in senario (IV) will be larger than profit in senario (I) iff α + ρ ρ 2 < R. (27) Moreover, profit in senario (IV) exeeds the profits in senarios (II) and (III), respetively, iff R < 1 (ρ α) β 0 + (ρ α)β 1 1 (α + ρ) β 0 + (α + ρ) β 1 α + ρ ρ 2 (28) and R < 1 (1 α) β 0 + (ρ α) β 1 1 β 0 + (α + ρ) β 1 α + ρ ρ 2. (29) For the onditions (27) (29) to hold at the same time, it is neessary that β 0 > 2β 1, whih guarantees that the right-hand sides of (28) and (29) are larger than α+ρ. If one of the three ρ 2 onditions (27) (29) is violated, the firm will prefer one of the senarios (I) (III). Expeted profits (22) in senario (II) exeed expeted profits (25) in senario (III) iff ondition (14) holds, i.e., iff R/ > Ω H (α, ρ). Comparing (17) and (25) yields that the fixed salary of senario (I) leads to higher profits then the quota sheme of senario (III) iff R < 1 + ρβ 1 1 αβ 0 + (2α + ρ) β 1 α + ρ ρ 2 := ˆΩ L < Ω H (α, ρ). Proof of Corollary 2: Beause R > Ω H (α, ρ), the firm optimally indues e 1 = e 2 = 1. By the proof of Proposition 26

27 2, an optimal ompensation sheme is desribed by (20) and 0 w 10 [1 + α(β 0 β 1 )] ρ. Inserting β 0 = β 1 = β leads to 0 w 10 ρ and w 11 = = 1 + (α + ρ) β 1 β(α + ρ) w 10. β (α + ρ) ρ β(α + ρ) 1 + (α + ρ) β 1 β(α + ρ) δ. β (α + ρ) β(α + ρ) Let w 10 = δ ρ with δ [0, 1]. Thus, w 11 = ( 1 δ β(α+ρ) + (1 + δ) ) ρ. For δ = 1, we obtain a ommission sheme that is independent of β. For δ [0, 1), the inentive sheme depends on β. Now suppose ˆΩ L < R Ω H (α, ρ) so that the firm indues e 1 = 1 but e 2 = 1 if and only if x 1 = 1. The orresponding optimal ompensation is desribed by (24) and w 11 ρ + w 10 (see the proof of Proposition 2). Rewriting the two optimality onditions and using β 1 = β leads to w 11 = 1 + ρβ β(α + ρ) ρ 1 β (ρ + α) w 10 and (1 αβ) β(α + ρ) ρ w 10. Assume w 10 = δ (1 αβ) ρ, δ [0, 1]. Thus, w 10 is independent of β iff δ = 0. However, in this β(α+ρ) ρ ase, w 11 = 1+ρβ know β. still depends on β, i.e., for the optimal inentive sheme the firm needs to Proof of Proposition 4: (I) If the firm wants to indue zero inentives, it will offer w F = (0, 0, 0, 0). Hene, the firm s expeted profit is 2αR. (II) Suppose that the firm wants to indue e 1 = e 2 = 1. To implement e 2 = 1 given the first-period outome x 1 {0, 1}, it must hold that: (α + E[ρ]) w x1 1 + (1 (α + E[ρ])) w x1 0 αw x1 1 + (1 α) w x1 0 w x1 1 E[ρ] + w x 1 0 (30) Analogously to the basi model, the first-stage inentive onstraint reads as E [ρ ((α + ρ) (w 11 w 01 ) + (1 α ρ) (w 10 w 00 ))]. (31) The firm thus solves min E[(α + w 11,w 10,w 01,w 00 0 ρ)2 w 11 + (α + ρ)(1 (α + ρ)) (w 10 + w 01 ) + (1 (α + ρ)) 2 w 00 ] s.t. (30), (31). 27

28 The firm optimally hooses w 00 = 0 and w 01 = /E[ρ]. Conerning the optimal hoie of w 11 and w 10, the firm minimizes total osts C haraterized by the iso-ost urves C A 2 E[ρ] A 1 w 11 = w 10, (32) A 2 where A 1 := E[(α + ρ) 2 ] and A 2 := E[(α + ρ)(1 (α + ρ))], subjet to the onstraints and A 4 + w 11 E[ρ] w 10 (33) A 3 A 3 w 11 w 10 E[ρ] A 4 A 4 (34) with A 3 := E[ρ(α + ρ)] and A 4 := E[ρ(1 (α + ρ))]. Graphially, the onstraints (33) and (34) define a feasible region within the (w 11, w 10 )-spae, and the firm needs to identify the lowest isoost urve (32) that stays in the feasible region. Condition (33) orresponds to the area below a straight line with positive slope that intersets with the horizontal axis at w 11 = /E[ρ]. Condition (34) desribes an area above a negatively sloped straight line that intersets with the horizontal axis at w 11 = (/E[ρ]) + (/A 3 ) > /E[ρ]. Thus, the two straight lines interset above the horizontal axis. Sine the absolute value of the slope of the iso-ost urves is smaller than the absolute value of the slope of the straight line haraterized by (34), i.e., the firm will optimally hoose A 1 A 2 < A 3 A 4 0 < V ar[ρ], (35) w 11 = E[ρ] + A 3 = E[ρ] + E[ρ(α + ρ)] and w 10 = 0, yielding expeted profits π (II) = 2E [α + ρ] R E [α + ρ] E [(α + ρ) 2] E [ρ] E [ρ (α + ρ)]. (III) Suppose the firm indues e 1 = 1 but e 2 = 1 if and only if x 1 = 1. Analogous to the basi model, the first-stage inentive onstraint is given by E [ρ(α + ρ)] (w 11 w 10 ) E [ρα] (w 01 w 00 ) + E [ρ] (w 10 w 00 ) (1 + E [ρ]). (36) 28

29 The firm thus minimizes E[(α + ρ) 2 w 11 + (α + ρ)(1 α ρ)w 10 + (1 α ρ)αw 01 + (1 α ρ)(1 α)w 00 ] subjet to (36) and the seond-period inentive onstraints w 11 E[ρ] + w 10, and w 01 < E[ρ] + w 00. Hene, w 00 = w 01 = 0 is optimal, and the firm minimizes osts C haraterized by the iso-ost urves subjet to the onstraints C A 2 A 1 A 2 w 11 = w 10 (37) and w 11 E[ρ] w 10 (38) (1 + E [ρ]) A 3 w 11 w 10. A 4 A 4 (39) Note that the zero of the straight line desribed by onstraint (39) (i.e., w 11 = (1 + E [ρ]) A 3 ) is larger than the zero of the straight line desribed (38) /E[ρ]: (1 + E [ρ]) A 3 > (1 + E [ρ]) E[ρ] > E[ρ(α + ρ)] E[ρ] is true sine E[ρ] > E[ρ(α + ρ)] pρ 1 + (1 p) ρ 2 > pρ 1 (α + ρ 1 ) + (1 p) ρ 2 (α + ρ 2 ) holds beause ρ t + α < 1 (t = 1, 2). Thus, we have qualitatively the same graphial solution as in ase (II) above. The absolute value of the slope of the iso-ost urves is smaller than the absolute value of the slope of the of the straight line desribed by (39), i.e., ondition (35) is satisfied. Hene, the firm optimally hooses w 11 = (1 + E [ρ]) = (1 + E [ρ]) A 3 E[ρ(α + ρ)] and w 10 = 0, whih haraterizes a quota-based bonus. Expeted profits are π (III) = E [α + ρ] R + {E[(α + ρ) 2 ] + E[(1 α ρ)α]}r E[(α + ρ)2 ] (1 + E [ρ]). E[ρ(α + ρ)] Besides the senarios (I) to (III), there are four additional ases that an be implemented. From the firm s point of view, however, none of these ases is optimal (see the Additional Material 3). 29

30 The firm will prefer senario (I) to senario (II) iff π (II) < 2αR R E [α + ρ] E [(α + ρ) 2] < 2E [ρ] 2 + 2E [ρ] E [ρ (α + ρ)], and senario (I) to senario (III) iff π (III) < 2αR R < E[(α + ρ) 2 ] (1 + E [ρ]) ((1 + α) E[ρ] + E[ρ 2 ]) E[ρ(α + ρ)]. It will prefer the quota sheme desribed by senario (III) to the other ompensation sheme desribed by (II) iff π (II) < π (III) ( E[(α + ρ) 2 ) ] E [α + ρ] E [ρ] < ( E[ρ 2 ] (1 α) E[ρ] ) R. E[ρ(α + ρ)] E [ρ] Proof that the left-hand side of (13) is negative: E[(α + ρ) 2 ] E [α + ρ] E [ρ] < E[ρ(α + ρ)] E [ρ] α2 + 2αE[ρ] + E[ρ 2 ] αe[ρ] + E[ρ 2 ] < 1 E [ρ] + α E [ρ] 2 α 2 + 2αE[ρ] + E[ρ 2 ] < α + E[ρ2 ] E [ρ] + 1 E [ρ] α2 + E[ρ2 ] E [ρ] 2 α E [ρ] 1 α 2 + (2E[ρ] 1) E [ρ]2 E[ρ 2 ] E [ρ] E [ρ] 2 α + E [ρ] 1 E[ρ 2 ] < 0 E [ρ] Dividing both sides by E[ρ] 1 E[ρ] < 0 leads to α 2 + E[ρ2 ] (2E[ρ] 1) E[ρ] 2 α + E[ρ 2 ] > 0. (40) (1 E[ρ]) E[ρ] Sine V ar[ρ] = E[ρ 2 ] E[ρ] 2 > 0 implies that E[ρ 2 ] > E[ρ] 2 and 2E[ρ] 1 < 1 E[ρ] < 1, ondition (40) is true. Referenes Aharya, V. V., K. John, and R. K. Sundaram (2000). On the optimality of resetting exeutive stok options. Journal of Finanial Eonomis 57, Akerberg, D. and M. Bottiini (2002). Endogenous mathing and the empirial determinants of ontrat form. Journal of Politial Eonomy 110,

31 Albers, S. and M. Mantrala (2008). Models for sales management deisions. In B. Wierenga (Ed.), Handbook of marketing deision models. Springer: Berlin. Bak, B. and A. Klez-Simon (2013). Quota bonuses with heterogeneous agents. Eonomis Letters 119, Basu, A., V. S. R. Lal, and R. Staelin (1985). Salesfore ompensation plans: An ageny theoreti perspetive. Marketing Siene 4, Bellemare, C. and B. Shearer (2010). Sorting, inentives and risk preferenes: Evidene from a field experiment. Eonomis Letters 108, Bierbaum, J. (2002). Repeated moral hazard under limited liability. Working Paper. Bolton, P. and M. Dewatripont (2005). Contrat Theory. MIT Press: Cambridge, MA. Che, Y.-K. and S.-W. Yoo (2001). Optimal inentives for teams. Amerian Eonomi Review 91, Chung, D. J., T. Steenburgh, and K. Sudhir (2014). Do bonuses enhane sales produtivity? a dynami strutural analysis of bonus-based ompensation plans. Marketing Siene 33, Coughlan, A. T. (1993). Salesfore ompensation: A review of MS/OR advanes. In J. Eliashberg and G. L. Lilien (Eds.), Handbooks in Operations Researh and Management Siene, Marketing, Volume 5, pp Amsterdam: North-Holland. Dai, T. and K. Jerath (2013). Salesfore ompensation with inventory onsiderations. Management Siene 59, Dearden, J. A. and G. L. Lilien (1990). On optimal salesfore ompensation in the presene of prodution learning effets. International Journal of Researh in Marketing 7, Demougin, D. and C. Fluet (1998). Mehanism suffiient statisti in the risk-neutral ageny problem. Journal of Institutional and Theoretial Eonomis 154, Hilt, E. (2008). The negative trade-off between risk and inentives: Evidene from the amerian whaling industry. Explorations in Eonomi History 45, Holmström, B. and P. Milgrom (1987). Aggregation and linearity in the provision of intertemporal inentives. Eonometria 55,

32 Innes, R. (1990). Limited liability and inentive ontrating with ex-ante ation hoies. Journal of Eonomi Theory 52, Jain, S. (2012). Self-ontrol and inentives: An analysis of multiperiod quota plans. Marketing Siene 31, Joseph, K. and M. Kalwani (1998). The role of bonus pay in salesfore ompensation plans. Industrial Marketing Management 27, Kaya, A. and G. Vereshhagina (2014). Partnerships versus orporations: Moral hazard, sorting, and ownership struture. Amerian Eonomi Review 104, Kishore, S., R. Rao, O. Narasimhan, and G. John (2013). Bonuses versus ommissions: A field study. Journal of Marketing Researh 50, Laffont, J.-J. and D. Martimort (2002). The Theory of Inentives: The Prinipal-Agent Model. Prineton University Press: Prineton and Oxford. Lal, R. and V. Srinivasan (1993). Compensation plans for single- and multi-produt salesfores: An appliation of the Holmstrom-Milgrom model. Management Siene 39, Ohlendorf, S. and P. Shmitz (2012). Repeated moral hazard and ontrats with memory: The ase of risk-neutrality. International Eonomi Review 53, Oyer, P. (2000). A theory of sales quotas with limited liability and rent sharing. Journal of Labor Eonomis 18, Poblete, J. and D. Spulber (2012). The form of inentive ontrats: Ageny with moral hazard, risk neutrality, and limited liability. RAND Journal of Eonomis 43, Raju, J. S. and V. Srinivasan (1996). Quota-based ompensation plans for multiterritory heterogeneous salesfores. Management Siene 42, Sappington, D. (1983). Limited liability ontrats between prinipal and agent. Journal of Eonomi Theory 29, Shmitz, P. (2005). Alloating ontrol in ageny problems with limited liability and sequential hidden ations. RAND Journal of Eonomis 36, 2, Shmitz, P. W. (2013). Job design with onfliting tasks reonsidered. European Eonomi Review 57,

33 Shöttner, A. (2015, aepted for publiaton). Optimal sales fore ompensation in dynami settings: Commissions versus bonuses. Management Siene. Simester, D. and J. Zhang (2010). Why are bad produts so hard to kill? Management Siene 56, Steenburgh, T. J. (2008). Effort or timing: The effet of lump-sum bonuses. Quantitative Marketing and Eonomis 6, Zoltners, A. A., P. Sinha, and S. E. Lorimer (2008). Sales fore effetiveness: A framework for researhers and pratitioners. Journal of Personal Selling and Sales Management 28,

34 Additional Material (not intended for publiation) Additional Material 1: The Suboptimal Cases in the Basi Model There exist four additional ases that an be hosen by the firm. First, onsider the ase where the firm indues e 1 = 1, but e 2 = 1 if and only if x 1 = 0. That is, the sales agent should exert high effort with the seond ustomer only if he ould not sell the produt to the first ustomer. The first-stage inentive onstraint thus beomes (α + ρ) [αw 11 + (1 α)w 10 ] + (1 α ρ) [(α + ρ)w 01 + (1 α ρ)w 00 ] α [αw 11 + (1 α)w 10 ] + (1 α) [(α + ρ)w 01 + (1 α ρ)w 00 ] ρ [αw 11 + (1 α)w 10 (α + ρ)w 01 (1 α ρ)w 00 ] (1 ρ). (41) The firm now solves min w 11,w 10,w 01,w 00 0 (α + ρ)αw 11 + (α + ρ)(1 α)w 10 + (1 α ρ)(α + ρ)w 01 + (1 α ρ) 2 w 00 subjet to w 11 < ρ + w 10, w 01 ρ + w 00, and (41). Consequently, w 00 = 0. Furthermore, w 01 should be as small as possible, w 01 = ρ. It follows that αw 11 + (1 α)w 10 should be minimal. Hene, by (41) and w 11 < ρ + w 10, all w 11 and w 10 satisfying αw 11 + (1 α)w 10 = 1 ρ ρ + (α + ρ) ρ = (1 + α) ρ and w 11 < ρ + w 10 1 α ρ are optimal (e.g., w 11 = 0 and w 10 = 1+α ). The firm s profit is [ (α + ρ)r + [(α + ρ)α + (1 α ρ)(α + ρ)] R (α + ρ)(1 + α) ρ + (1 α ρ)(α + ρ) ρ ( = (2 ρ)(α + ρ) R ), ρ ] whih is smaller than the profit in senario 2 desribed by (6). Seond, onsider the ase where effort is high only with one ustomer: e i = 1 and e j = 0 (i, j = 1, 2, i j). Let, w.l.o.g., i = 1. The firm minimizes (α + ρ)αw 11 + (α + ρ)(1 α)w 10 + (1 α ρ)αw 01 + (1 α ρ)(1 α)w 00 1

35 subjet to w 01 < ρ + w 00 and w 11 < ρ + w 10 as seond-stage inentive onstraints, and (α + ρ) [αw 11 + (1 α)w 10 ] + (1 α ρ) [αw 01 + (1 α)w 00 ] α [αw 11 + (1 α)w 10 ] + (1 α) [αw 01 + (1 α)w 00 ] αw 11 + (1 α)w 10 αw 01 (1 α)w 00 ρ. as first-stage inentive onstraint. Thus, w 01 = w 00 = 0, and optimal sales fore ompensation is desribed by αw 11 + (1 α)w 10 = ρ and w 11 < ρ + w 10, being satisfied, e.g., for w 11 = 0 and w 10 = 1 1 α ρ. The profit is [2(α + ρ)α + (α + ρ)(1 α) + (1 α ρ)α] R (α + ρ) ρ = [2α + ρ] R (α + ρ) ρ. (42) This profit is smaller than profit (6) under senario 2 if [2α + ρ] R (α + ρ) ( ρ < 2(α + ρ) R ) R ρ > α + ρ ρ 2. Profit (42) is smaller than the profit under senario 1 if [2α + ρ] R (α + ρ) ρ < 2αR R < α + ρ ρ 2. Thus, induing high effort only with one ustomer annot be optimal. Third, the firm may be interested in implementing low effort with ustomer 1 and high effort with ustomer 2 if and only if x 1 = 1, that is, e 1 = 0 and e 2 = 1 iff x 1 = 1. The first-stage inentive onstraint is α [(α + ρ)w 11 + (1 α ρ)w 10 ] + (1 α) [αw 01 + (1 α)w 00 ] (α + ρ) [(α + ρ)w 11 + (1 α ρ)w 10 ] + (1 α ρ) [αw 01 + (1 α)w 00 ] (1 + ρ) ρ[(α + ρ)w 11 + (1 α ρ)w 10 αw 01 (1 α)w 00 ]. (43) Hene, the firm s problem an be haraterized as follows: min w 11,w 10,w 01,w 00 0 α(α + ρ)w 11 + α(1 α ρ)w 10 + (1 α)αw 01 + (1 α) 2 w 00 subjet to w 01 < ρ + w 00, w 11 ρ + w 10, and (43). 2

36 It is optimal to have w 00 = w 01 = w 10 = 0 and w 11 = ρ. Expeted profit is αr + α(α + ρ)r + (1 α)αr α(α + ρ) ρ = (2 + ρ)αr α(α + ρ) ρ, (44) whih is smaller than (42) if (2 + ρ)αr α(α + ρ) ρ < [2α + ρ] R (α + ρ) ρ R > α + ρ ρ 2. On the other hand, (44) is smaller than profit under e 1 = e 2 = 0 if (2 + ρ)αr α(α + ρ) ρ < 2αR R < α + ρ ρ 2, implying that this third alternative senario annot be optimal for the firm. Fourth, the firm may wish to implement low effort with ustomer 1 and high effort with ustomer 2 if and only if x 1 = 0, that is, e 1 = 0 and e 2 = 1 iff x 1 = 0. The first-stage inentive onstraint reads as α [αw 11 + (1 α)w 10 ] + (1 α) [(α + ρ)w 01 + (1 α ρ)w 00 ] (α + ρ) [αw 11 + (1 α)w 10 ] + (1 α ρ) [(α + ρ)w 01 + (1 α ρ)w 00 ] (1 ρ) ρ[αw 11 + (1 α)w 10 (α + ρ)w 01 (1 α ρ)w 00 ]. (45) The firm s problem is min w 11,w 10,w 01,w 00 0 α2 w 11 + α(1 α)w 10 + (1 α)(α + ρ)w 01 + (1 α)(1 α ρ)w 00 subjet to w 01 ρ + w 00, w 11 < ρ + w 10, and (45). It is thus optimal to have w 00 = w 10 = w 11 = 0 and w 01 = ρ. Expeted profit is αr + α 2 R + (1 α)(α + ρ)r (1 α)(α + ρ) ρ = [2α + ρ αρ] R (1 α)(α + ρ) ρ, (46) whih is smaller than profit (42) if [2α + ρ αρ] R (1 α)(α + ρ) ρ < [2α + ρ] R (α + ρ) ρ R > α + ρ ρ 2. 3

37 On the other hand, profit (46) is smaller than profit under e 1 = e 2 = 0 if [2α + ρ αρ] R (1 α)(α + ρ) ρ < 2αR R < α + ρ ρ 2. Additional Material 2: The Suboptimal Cases Under Demand Unertainty As in the basi model, again the firm an hoose one of four additional ases. First, suppose the firm indues e 1 = 1, but e 2 = 1 if and only if x 1 = 0. That is, the sales agent should exert high effort with the seond ustomer only if he ould not sell the produt to the first ustomer. The first-stage inentive onstraint now beomes The firm solves (α + ρ) [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] +(1 α ρ) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 ) w 00 ] α [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] +(1 α) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 ) w 00 ] αβ 1 w 11 + w 10 (1 αβ 1 ) β 0 (α + ρ)w 01 (1 (α + ρ) β 0 ) w 00 1 ρβ 0. (47) ρ min (α + ρ) [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] w 11,w 10,w 01,w (1 α ρ) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 ) w 00 ] subjet to w 11 < ρ + w 10, w 01 ρ + w 00, and (47). Consequently, w 00 = 0. Furthermore, w 01 should be as small as possible, w 01 = ρ. It follows that β 1 (αw 11 + (1 α)w 10 )+(1 β 1 ) w 10 = αβ 1 w 11 +(1 αβ 1 ) w 10 should be minimal. Hene, by (47) and w 11 < ρ + w 10, all w 11 and w 10 satisfying αβ 1 w 11 + (1 αβ 1 ) w 10 = (1 + αβ 0 ) ρ and w 11 < ρ + w 10 are optimal. The firm s profit is [ (α + ρ) (1 + αβ 1 + (1 α ρ) β 0 ) R (1 + (1 ρ) β 0 ) ] ρ whih is smaller than profit (22). Seond, onsider the ase where effort is high only with ustomer 1: e 1 = 1 and e 2 = 0. The 4

38 firm minimizes (α + ρ) [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] + (1 α ρ) [β 0 (αw 01 + (1 α)w 00 ) + (1 β 0 ) w 00 ] subjet to w 01 < ρ + w 00 and w 11 < ρ + w 10 as seond-stage inentive onstraints, and (α + ρ) [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] +(1 α ρ) [β 0 (αw 01 + (1 α)w 00 ) + (1 β 0 ) w 00 ] α [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] +(1 α) [β 0 (αw 01 + (1 α)w 00 ) + (1 β 0 ) w 00 ] αβ 1 w 11 + (1 αβ 1 ) w 10 αβ 0 w 01 w 00 (1 αβ 0 ) ρ (48) as first-stage inentive onstraint. Thus, w 01 = w 00 = 0, and optimal sales fore ompensation is desribed by β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 )w 10 = ρ and w 11 < ρ + w 10. This ompensation leads to profit ((α + ρ) (1 + αβ 1 ) + (1 (α + ρ)) αβ 0 ) R (α + ρ) ρ, (49) whih is smaller than profit (25). Third, the firm may be interested in implementing low effort with ustomer 1 and high effort with ustomer 2 if and only if x 1 = 1, that is, e 1 = 0 and e 2 = 1 iff x 1 = 1. The first-stage inentive onstraint is α [β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 ) w 10 ] +(1 α) [β 0 (αw 01 + (1 α)w 00 ) + (1 β 0 ) w 00 ] (α + ρ) [β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 ) w 10 ] +(1 α ρ) [β 0 (αw 01 + (1 α)w 00 ) + (1 β 0 ) w 00 ] (1 + ρβ 1 ) ρ (α + ρ)β 1w 11 + (1 (α + ρ) β 1 ) w 10 αβ 0 w 01 (1 αβ 0 ) w 00. (50) Hene, the firm s problem an be haraterized as follows: min α [β 1 ((α + ρ)w 11 + (1 α ρ)w 10 ) + (1 β 1 ) w 10 ] w 11,w 10,w 01,w (1 α) [β 0 (αw 01 + (1 α)w 00 ) + (1 β 0 ) w 00 ] subjet to w 01 < ρ + w 00, w 11 ρ + w 10, and (50). 5

39 It is optimal to have w 00 = w 01 = w 10 = 0 and w 11 = ρ. Profit is given by (1 + (1 α) β 0 + (α + ρ) β 1 ) αr (α + ρ) αβ 1 ρ, whih is smaller than profit (17) iff Rρ 2 < (α + ρ), and smaller than profit (26) iff Rρ 2 > (α + ρ). Fourth, the firm may wish to implement low effort with ustomer 1 and high effort with ustomer 2 if and only if x 1 = 0, that is, e 1 = 0 and e 2 = 1 iff x 1 = 0. The first-stage inentive onstraint reads as α [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] +(1 α) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 ) w 00 ] (α + ρ) [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] +(1 α ρ) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 ) w 00 ] (1 ρβ 0 ) ρ αβ 1w 11 + (1 αβ 1 ) w 10 (α + ρ)β 0 w 01 (1 (α + ρ) β 0 ) w 00. (51) The firm s problem is min α [β 1 (αw 11 + (1 α)w 10 ) + (1 β 1 ) w 10 ] w 11,w 10,w 01,w (1 α) [β 0 ((α + ρ)w 01 + (1 α ρ)w 00 ) + (1 β 0 ) w 00 ] subjet to w 01 ρ + w 00, w 11 < ρ + w 10, and (51). It is thus optimal to have w 00 = w 10 = w 11 = 0 and w 01 = ρ, leading to profit ( α + ((1 (α + ρ)) α + ρ) β0 + α 2 β 1 ) R β0 (1 α) (α + ρ) ρ, whih is smaller than profit (17) iff Rρ 2 < (α + ρ), and smaller than profit (26) iff Rρ 2 > (α + ρ). Additional Material 3: The Suboptimal Cases Under Preferene Unertainty First, suppose the firm wants to indue e 1 = 1, but e 2 = 1 if and only if x 1 = 0. The first-stage inentive onstraint beomes E [ρα] (w 11 w 10 ) + E [ρ] (w 10 w 00 ) E [ρ(α + ρ)] (w 01 w 00 ) (1 E [ρ]) (52) 6

40 and the firm minimizes E [ (α + ρ)αw 11 + (α + ρ)(1 α)w 10 + (1 α ρ)(α + ρ)w 01 + (1 α ρ) 2 w 00 ] subjet to w 11 < E[ρ] + w 10, w 01 E[ρ] + w 00, and (52). Hene, w 00 = 0 and w 01 = /E[ρ]. The firm minimizes osts C haraterized by the iso-ost urves C A 2 B 2 B 2 E[ρ] B 1 w 11 = w 10 (53) B 2 subjet to the onstraints w 11 E[ρ] < w 10 (54) (1 E [ρ]) + A 3 E[ρ] E [ρα] E [ρ] E [ρα] E [ρ] E [ρα] w 11 w 10 (55) with B 1 := E [(α + ρ)α], B 2 := E [(α + ρ)(1 α)], and A 2 and A 3 as defined above. The LHS of (54) intersets with the horizontal axis at w 11 = /E[ρ]. The LHS of (55) intersets with the horizontal axis at 26 w11 = (1 E [ρ]) + A 3 E [ρα] E[ρ] The LHSs of both onstraints (54) and (55) interset at > E[ρ]. ŵ 11 := (1 p) p (ρ 1 ρ 2 ) 2 + 2E [ρ] E [ρ] 2 and ŵ 10 := (1 p) p (ρ 1 ρ 2 ) 2 + E [ρ] E [ρ] 2. Sine E [ρα] / (E [ρ] E [ρα]) = α/ (1 α) = B 1 /B 2, the slopes of the iso-ost urves and the LHS of (55) are idential. Therefore, eah wage ombination that lies on the LHS of (55) between the points (w 11, w 10 ) = (0, [(1 E [ρ]) + A 3 E[ρ] ]/[E [ρ] E [ρα]]) and (w 11, w 10 ) = (ŵ 11, ŵ 10 ) desribes an optimal solution, leading to profits π first = E [α + ρ] R + {E[(α + ρ) α] + E[(1 α ρ) (α + ρ)]}r E[(α + ρ)α] (1 p) p (ρ 1 ρ 2 ) 2 + 2E [ρ] E [ρ] 2 E[(α + ρ)(1 α)] (1 p) p (ρ 1 ρ 2 ) 2 + E [ρ] E [ρ] 2 E[(1 α ρ)(α + ρ)]. E[ρ] 26 Note that (1 E[ρ])+ A 3 E[ρ] > (1 E [ρ]) E[ρ] + E[ρ(α + ρ)] > E [ρα] is true sine E[ρ(α + ρ)] > E [ρα]. E[ρα] E[ρ] 7

41 We an show that π (II) > π first is always satisfied: π (II) > π first E [ ρ 2] E [α + ρ] E [(α + ρ) 2] R + E [αρ] R E [ρ] E [ρ (α + ρ)] > E[(α + ρ)α] (1 p) p (ρ 1 ρ 2 ) 2 + 2E [ρ] E [ρ] 2 E[(α + ρ)(1 α)] (1 p) p (ρ 1 ρ 2 ) 2 + E [ρ] E [ρ] 2 E[(1 α ρ)(α + ρ)]. E[ρ] Reall that exerting high effort is effiient, i.e., E [ρ] R >. Inserting R = /E [ρ] into the inequality yields p (ρ 1 ρ 2 ) 2 (1 p) Ω (pρ 1 + (1 p) ρ 2 ) 2 ( pρ (1 p) ρ2 2 + α (pρ 1 + (1 p) ρ 2 ) ) > 0 with Ω := α ( pρ 1 (1 + (1 + p) ρ 1 ) + ( p p ) ρ 2 2) +(1 p) 2 ρ 3 2 +p2 ρ 3 1 +α2 (pρ 1 +(1 p) ρ 2 )+ ((ρ 2 + ρ 1 + 2α) pρ 1 ρ 2 + αρ 2 ) (1 p), whih is true. Seond, onsider the ase where effort is high only with one ustomer: e i = 1 and e j = 0 (i, j = 1, 2, i j). Let, w.l.o.g., i = 1. The firm minimizes E[(α + ρ)αw 11 + (α + ρ)(1 α)w 10 + (1 α ρ)αw 01 + (1 α ρ)(1 α)w 00 ] subjet to w 01 < E[ρ] + w 00 and w 11 < E[ρ] + w 10 as seond-stage inentive onstraints, and αw 11 + (1 α)w 10 αw 01 (1 α)w 00 E [ρ]. as first-stage inentive onstraint. Thus, w 01 = w 00 = 0 is optimal. The firm minimizes osts C desribed by the iso-ost urves C E[(α + ρ)(1 α)] α 1 α w 11 = w 10 subjet to (1 α)e [ρ] α 1 α w 11 w 10 and w 11 E [ρ] < w 10. Sine the slopes of the iso-ost urves and the LHS of the first onstraint are idential, we obtain a similar solution as in the first additional ase above. For example, w 11 = 0 and 8

42 w 10 = /[(1 α)e [ρ]] is optimal. Eah solution leads to expeted profits We an show that π (III) > π seond : π seond = E[α + ρ]r + αr E[(α + ρ)(1 α)]. (1 α)e [ρ] π (III) > π seond ( pρ (1 p) ρ 2 ) E[(α + ρ) 2 ] (1 + E [ρ]) 2 + pαρ 1 + (1 p) αρ 2 R E[ρ(α + ρ)] E[(α + ρ)(1 α)] >. (1 α)e [ρ] Inserting R = /E [ρ] and rearranging gives whih is true. p (ρ 1 ρ 2 ) 2 (1 p) ( α + pρ (1 p) ρ α (pρ 1 + (1 p) ρ 2 ) ) (pρ 1 + (1 p) ρ 2 ) ( pρ (1 p) ρ2 2 + pαρ 1 + (1 p) αρ 2 ) > 0, Third, the firm may be interested in implementing low effort with ustomer 1 and high effort with ustomer 2 if and only if x 1 = 1, that is, e 1 = 0 and e 2 = 1 iff x 1 = 1. The first-stage inentive onstraint is (1 + E [ρ]) E (ρ [(α + ρ)w 11 + (1 α ρ)w 10 ] ρ [αw 01 + (1 α)w 00 ]). (56) Hene, the firm s problem an be haraterized as follows: min E[α(α + ρ)w 11 + α(1 α ρ)w 10 + (1 α)αw 01 + (1 α) 2 w 00 ] w 11,w 10,w 01,w 00 0 subjet to w 01 < It is optimal to have w 10 = 0 and w 11 = iso-ost urves ( ) E[ρ(α + ρ)] (1 + E [ρ]) E [ρ] E [ρ] + w 00, w 11 E [ρ] + w 10, and (56). E[ρ]. The firm minimizes osts C desribed by the C E[α(α + ρ)] (1 α) 2 (1 α) 2 E [ρ] α 1 α w 01 = w 00 subjet to E[ρ(1 α)] α 1 α w 01 w 00 and w 01 E [ρ] < w 00. (57) Note that, again, the iso-ost urves and the LHS of the first-stage inentive onstraint have idential slopes. However, now we have two possible solutions: (i) If the interept of the LHS 9

43 of the first-stage inentive onstraint is negative, i.e., if ( E[ρ(α + ρ)] E [ρ] ) (1 + E [ρ]) E[ρ(1 α)] < 0 (1 p) p (ρ 1 ρ 2 ) 2 < 1 α, (58) pρ 1 + (1 p) ρ 2 then w 01 = w 00 = 0 is optimal and expeted profits amount to π third,(i) = αr + {E[α(α + ρ)] + (1 α) α}r E[α(α + ρ)] E [ρ]. (ii) If (58) is not satisfied, then independent of the relation of the zeros of the two inentive onstraints in (57), the pair ( E[ρ(α + ρ)] w 01 = 0 and w 00 = E [ρ] desribes an optimal solution, leading to expeted profits ) (1 + E [ρ]) E[ρ(1 α)] π third,(ii) = αr + {E[α(α + ρ)] + (1 α) α}r ( ) E[ρ(α + ρ)] (1 α) 2 E[α(α + ρ)] E [ρ] (1 + E [ρ]) E [ρ] E[ρ(1 α)]. Sine π third,(i) > π third,(ii), the third additional ase annot be optimal if one of the profits of the other ases exeeds π third,(i). Induing no inentives at all is better for the firm than the third additional ase, iff π third,(i) = αr + {E[α(α + ρ)] + (1 α) α}r E[α(α + ρ)] E [ρ] < 2αR E[ρ]R < E[α + ρ] E [ρ]. The firm will prefer the seond additional ase to the third additional ase, iff π seond > π third,(i) E[ρ]R > E[α + ρ] E [ρ]. Thus, the firm does not prefer the third additional ase. Fourth, the firm may wish to implement low effort with ustomer 1 and high effort with ustomer 2 if and only if x 1 = 0, that is, e 1 = 0 and e 2 = 1 iff x 1 = 0. The first-stage inentive onstraint reads as (1 E[ρ]) E[ρ (αw 11 + (1 α)w 10 (α + ρ)w 01 (1 α ρ)w 00 )]. (59) 10

44 The firm solves min w 11,w 10,w 01,w 00 0 E[α2 w 11 + α(1 α)w 10 + (1 α)(α + ρ)w 01 + (1 α)(1 α ρ)w 00 ] subjet to w 01 E [ρ] + w 00, w 11 < E [ρ] + w 10, and (59). It is thus optimal to have w 10 = w 11 = w 00 = 0 and w 01 = /E [ρ]. Expeted profit is π fourth = αr + α 2 R + (1 α)e[α + ρ]r (1 α)e[α + ρ] E [ρ]. Sine π fourth < 2αR E[ρ]R < E[α + ρ]/e [ρ] and π fourth < π seond E[α + ρ]/e [ρ] < E[ρ]R, the firm does not prefer the fourth additional ase. 11