How Often Should You Reward Your Salesforce? Multi-Period. Incentives and Effort Dynamics

Transcription

1 How Often Should You Reward Your Salesforce? Multi-Period Incentives and Effort Dynamics Kinshuk Jerath Fei Long Columbia Business School Columbia Business School 3022 Broadway 3022 Broadway New York, NY New York, NY Aril 2018

2 HOW OFTEN SHOULD YOU REWARD YOUR SALESFORCE? MULTI-PERIOD INCENTIVES AND EFFORT DYNAMICS ABSTRACT We study multi-eriod salesforce incentive contracting when sales agents can decide their effort levels dynamically. Focusing on a two-eriod scenario with a risk-neutral agent with limited liability, we analyze a central uestion faced by firms: Should saleseole be evaluated and granted comensation over a short or a long time horizon (in our case, er eriod or at the end of two eriods), and what should be the otimal comensation contract? While it is elementary that a fully flexible two-eriod contract will weakly dominate a eriod-by-eriod contract, we find that a two-eriod contract can strictly dominate a eriod-by-eriod contract, because it allows reward to be made contingent on a more extreme sales outcome, even though it allows the agent to game effort exertion. We show that this insight continues to hold when the agent can borrow or ostone sales between eriods. However, if the time eriods are deendent (e.g., a fixed amount of inventory has to be sold across the two time eriods), then a eriod-by-eriod contract can strictly dominate under certain conditions. We also derive imlications for the agent s multi-eriod effort rofile and show that various effort rofiles that may aear to be sub-otimal, e.g., delaying effort, may be induced otimally by the firm; i.e., the firm may not want to induce high effort in both eriods under all conditions. Throughout the aer, we show that two often ignored factors, namely the agent s outside otion and level of limited liability, are imortant determinants of the otimal time horizon and contract form, and of the agent s effort resonse. Keywords: Salesforce comensation; dynamic incentives; time horizon; limited liability; effort dynamics; inventory. INTRODUCTION Salesforce exenditures account for 10% 40% of the revenues of US firms (Albers and Mantrala 2008); this is of the order of hundreds of billions of dollars annually and is about three times the amount sent on advertising (Zoltners et al. 2008). Conseuently, how to best motivate saleseole is of rime imortance to firms, and salesforce comensation design roblems have drawn significant attention from economists and marketing researchers, rominent early aers in each area being Hölmstrom (1979) and Basu et al. (1985), resectively. In most cases, demand outcomes are uncertain and sales effort is not fully observable by the firm; this makes the determination of comensation, which is to reimburse for effort exerted, a difficult roblem. Tyical comensation contracts used by firms are comrised of a fixed art (e.g., base salary) and a variable art (e.g., commissions on sales or discrete bonuses awarded on achieving a uota of sales in a secified time eriod). According to Joseh and Kalwani (1998), who 1

3 conducted a survey of Fortune 500 firms, 58% of the firms use commissions and over 90% use uotabased contracts in their comensation lans. The advantage of uota-based lans is that they rovide stronger incentives to saleseole to reach a high level of sales, as variable comensation is rewarded only in that case. However, if the uota is too difficult or too easy to achieve, the effort exertion by the saleserson will be sub-otimal. This issue does not exist in a commission-based contract because every additional sales unit brings in the same additional reward. Determining whether uota-based bonuses or commissions should be used is an imortant roblem that firms face. Furthermore, firms emloy saleseole for extended eriods of time and they have to determine how freuently saleseole should be evaluated and rewarded, e.g., monthly, uarterly, semi-annually, annually, a combination of these, etc. Coughlan and Joseh (2012) refer to this as the time horizon over which rewards are offered to the saleserson, and state that essentially all firms face this roblem as well. As they discuss, when uota-based incentives are used in such a multi-eriod setting, the issue of dynamic gaming arises. This is because in a multi-eriod scenario the agents may strategically choose their effort exertion over time, based on how uncertain outcomes are realized and how the contract will determine reward in current and future eriods. As a canonical examle, consider a scenario in which the saleserson is aid a bonus if a articular sales uota is reached in six months. To reach his six-month uota with minimum effort, the agent may strategically shirk work in the first uarter hoing for a high demand outcome without much effort, and exert greater effort only in the second uarter in case of low demand realization in the first uarter. For similar strategic reasons, sales agents who have already achieved the uota in early sales cycles may not have the incentive to ut in extra effort later. This henomenon of agents dynamically adjusting their sales effort both ostonement of effort exertion and shirking after achieving the uota has been widely documented (e.g., Oyer 1998, Chen 2000, Steenburgh 2008, Misra and Nair 2011, Jain 2012, Kishore et al. 2013). Secifically, ostonement of effort exertion is considered an esecially undesirable outcome from the firm s oint of view because it involves not exerting effort before the sales uota is reached; it is sometimes known as the hockey stick attern (because effort exertion is flat in early eriods and increases sharly in later eriods, thus taking the shae of a hockey stick; Chen 2000). Such erverse gaming incentives are, however, not resent in commission-based contracts where every additional unit of demand brings in the same additional comensation. Clearly, the firm has multile otions for structuring the time horizon for the contract. The firm 2

4 could offer a short time horizon contract that evaluates and rewards the agent freuently (e.g., every uarter). Alternatively, the firm could offer a longer time horizon contract that evaluates and rewards the agent less freuently (e.g., six months); in this case, the firm would also have to determine the rewards for a larger number of ossible realizations. In each of these cases, the firm would have to determine the structure of the contract, for instance, should it include only commissions or should it include rewards for reaching uotas. If the firm chooses a longer time horizon with a uota-based contract, it would be more exosed to gaming. We therefore see that the two roblems of determining the time horizon of comensation and determining the otimal comensation structure are inter-related. This is an issue that essentially every comany that uses a salesforce must resolve non-linear uota-based incentive contracts lead to stronger incentives but invite gaming, while linear commission-based incentive contracts reduce gaming but also weaken incentives. A number of emirical aers have studied this tradeoff but have not reached a clear answer regarding which factor the incentive effect or the gaming effect dominates in a multi-eriod dynamic incentives scenario under which conditions. Oyer (1998) analyzes aggregate data sanning scores of industries (from the Survey of Income and Program Particiation (SIPP)) in which uotabased lans are used and detects dynamic gaming effects, and suggests that this gaming hurts more than the incentive effect hels. Steenburgh (2008) analyzes individual saleserson-level data from a Fortune 500 comany that sells durable office roducts and uses uota-based lans, and determines that stronger incentives dominate the downside from gaming (and also states that analyzing these data in aggregate would roduce results similar to those reorted in Oyer (1998)). Misra and Nair (2011) uses a dynamic structural model to analyze data from a Fortune 500 contact lens manufacturer and shows that a lan that uses only commissions erformed better than a uota-based lan (that was originally in use at the comany); using only commissions makes the time horizon decision irrelevant. Kishore et al. (2013) studies this uestion using data from a large harmaceutical firm in an emerging market and finds that commissions do better than uotas by reventing gaming, but this comes at the cost of neglecting nonincentivized tasks. Chung et al. (2014) uses a dynamic structural model to analyze data from a Fortune 500 office durable goods manufacturer and determines that uotas, through higher effort motivation, erform better than lans without uotas in site of gaming effects being resent; it also finds that both short term and long term uotas have roles to lay. Chung and Narayandas (2017) studies the effect of time horizon under a uota lan and finds that a shorter horizon generally ensures more consistent 3

5 effort but may not be more rofitable for the firm due to a change in the roduct mix being sold. Across these studies, choosing a better (even if not otimal ) comensation lan can lead to very significant increases in revenues and rofits, of the order of 5% to 20%. These aers also carefully document the effort exertion rofiles of agents induced by different tyes of contracts in a multi-eriod scenario. They consistently reort delaying effort and shirking after a good outcome as an issue of concern in long time horizon contracts. Overall, existing emirical research has found the roblem of determining the otimal time horizon (and contract form) to be highly relevant across a wide variety of scenarios but has reached mixed conclusions regarding this. In this aer, we conduct a theoretical investigation to shed light on this fundamental uestion that, arguably, any firm in any industry that emloys a salesforce faces (and, in a recent review artcile, Coughlan and Joseh (2012) list as a very imortant yet under-researched issue in salesforce management): What time horizon should the firm use to evaluate and comensate the saleserson, and what should be the associated contract? Should the firm offer multile seuential short time horizon contracts (which enables the firm to have more control over the effort exertion of the saleserson in every eriod) or should it offer a long time horizon contract (which allows the saleserson more freedom to adjust his effort rofile to game the system but also allows the firm to make variable comensation contingent on an outcome that is more difficult to achieve)? What are some key factors that influence this decision? Furthermore, what effort rofile(s) will be induced by the otimal incentive contract? To answer these uestions, we build a stylized rincial-agent model in which a firm interacts with a saleserson for two time eriods. In this context, using short time horizon for evaluation imlies offering two eriod-by-eriod contracts where each contract is determined at the start of a eriod and ays at the end of the eriod based on the outcome of the eriod. On the other hand, using long time horizon for evaluation imlies offering a two-eriod contract that is determined at the start of the first eriod and ays once at the end of the second eriod based on the outcomes of the two eriods. (In the rest of the aer, we will use long time horizon contracting interchangably with two-eriod contract, and short time horizon contracting interchangably with eriod-by-eriod contracts. ) We assume the demand outcome in each eriod to be stochastically deendent on the effort exerted in that eriod, and assume the demand outcomes in the two eriods to be indeendent of each other. In the two-eriod contract, the agent can dynamically adjust his effort level in the later eriod based on the early eriod s demand outcome which also influences his first-eriod effort exertion decision. We assume that the firm and the 4

6 saleserson are risk neutral, and that the agent has limited liability. Limited liability can be thought of as rotection from downside risk for the saleserson, i.e., he will be guaranteed a minimum ayment even in the case of an unfavorable market outcome (which is a robust feature of real-world comensation lans). We assume that the agent s limited liability can be lower or higher than his outside otion; the latter can haen, for instance, when the saleseole s skills are most valuable in a sales context and they cannot exect comarable comensation in other rofessions (Kim 1997, Oyer 2000). Our analysis shows that, for the firm, a fully flexible two-eriod contract weakly dominates a eriodby-eriod contract, as exected. Interestingly, however, we find that the two-eriod contract, even though it allows gaming of effort by the agent, strongly dominates the eriod-by-eriod contract under certain conditions. In the otimal two-eriod contract it is sufficient to determine comensation based on the cumulative sales for the two eriods and, under different conditions (discussed shortly), the otimal two-eriod contract is either an extreme contract that concentrates the reward only at the highest cumulative outut level, or a gradual contract with rewards at all (but the lowest) cumulative outut levels. In fact, similar to Hölmstrom and Milgrom (1987), the otimal gradual two-eriod contract can be interreted as identical to a commission contract. Furthermore, we obtain an interesting euivalence result that states that the otimal two-eriod gradual (commission) contract is identical in all ways (i.e., in terms of exected effort exertion, sales outcomes and total comensation) to a seuence of otimal eriod-by-eriod contracts; in other words, a long time horizon contract with commissions achieves the same outcomes as a seuence of otimal short time horizon contracts. Whether the extreme long time horizon contract or the gradual long time horizon contract (euivalently, a seuence of short time horizon contracts) is otimal can be exlained by understanding the two familiar countervailing effects at lay. The first is the beneficial incentive effect, which is that, given the agent s limited liability, an extreme lan rovides a larger incentive to work comared to a gradual lan because any outut lower than the highest ossible does not rovide any additional reward. However, the extreme lan also leads to a negative gaming effect, that is, dynamic gaming of effort in the second eriod based on the outcome of the first eriod hurts the rincial. The extreme contract is otimal when the incentive effect is stronger than the gaming effect, and this is the case when the effectiveness of the agent s effort is either low or high. This is because in the extreme contract the loss in demand due to the gaming effect is larger for higher effort effectiveness, but in the otimally designed contract the robability that this loss will haen is lower for higher effort effectiveness. Therefore, 5

7 the exected demand loss due to the gaming effect in the extreme contract is highest for intermediate effectiveness levels, and this loss is large enough to offset the incentive effect, so that in this region the extreme contract is not otimal. As limited liability decreases (fixing the agent s outside otion) the friction from moral hazard becomes smaller and the incentive effect becomes less imortant, so that the gradual contract becomes otimal in a larger arameter sace. In terms of the agent s effort exertion, we find that multile effort exertion rofiles are ossible under different conditions under the otimal contract effort exertion in both eriods; effort exertion in the first eriod and effort exertion in the second eriod conditional on high or low realized outcome in the first eriod; and no effort exertion in the first eriod and effort exertion in the second eriod conditional on high or low realized outcome in the first eriod. The last attern is esecially interesting as it imlies that in the otimal contract the firm induces delaying of effort (or hockey stick effort rofile). This effort ostonement is tyically interreted negatively (Chen 2000), and as something to avoid; our analysis shows that it indeed can be generated under an otimal contract even with indeendent eriods, and this haens when limited liability is intermediate. In other words, observing less than high effort exertion either early or late may not necessarily imly that the contract is not effective that may, in fact, be otimal for the firm and, indeed, induced by it through the contract. This imlies that one has to carefully understand and consider the setting and environmental factors when making inferences about contract efficiency from dynamic effort rofiles of agents. Next, we extend our basic model such that the two time eriods are not comletely indeendent. Secifically, we introduce the idea of an exogenous and limited amount of roduct inventory that has to be sold in the two eriods, such that the contract design decisions for the rincial in the two time eriods become deendent. (Note that demand outcomes in the two eriods are still indeendent.) We find that in this scenario the rincial may find it otimal to use a eriod-by-eriod contract in which the second-eriod contract is decided based on the outcome of the first eriod. Such a eriod-by-eriod contract can strongly dominate the two-eriod contract because it gives the rincial more flexibility in adjusting the contract. This cannot be reroduced by a two-eriod contract under the (reasonable) assumtion that total comensation cannot be decreasing in sales. Furthermore, with limited inventory, the rincial s incentive to induce effort in the first eriod is lesser, i.e., the rincial may otimally desire effort ostonement by the agent in a larger arameter sace. A number of aers, including Oyer (1998), Steenburgh (2008), Misra and Nair (2011), Jain (2012), 6

8 Chung et al. (2014) document another kind of gaming (in addition to effort gaming) in a dynamic incentives setting they show that in a multi-eriod setting with non-linear contracts, sales agents ull in orders from future eriods if they would otherwise fall short of a sales uota in one cycle, whereas they ush out orders to the future if uotas are either unattainable or have already been achieved. We extend our basic model to study such strategic sales ull in and ush out behavior, which also introduces deendence between the eriods. Allowing this affects short time horizon contracts because it gives the agent more freedom to game the system, but it does not affect long time horizon contracts. In accordance with this insight, we find that if sales ull in and ush out is ossible then a long time horizon contract becomes more attractive to the rincial, because it evaluates the agent only for the outut at the end of the two eriods. Our research is related to the body of work on dynamic incentives with reeated moral hazard. One stream of this work assumes the firm to be risk neutral but agents to be risk averse, which leads to contracting frictions. Under this aradigm, a seminal aer, Hölmstrom and Milgrom (1987) shows that a linear contract is otimal for the rincial when a number of other assumtions hold. We note that the gradual two-eriod contract that we derive as otimal for the firm under certain conditions can be interreted as a linear contract as well, but it is only otimal for intermediate values of effort effectiveness (recall that we assume the agent to be risk neutral). A number of aers in the risk aversion aradigm revisit the assumtions of Hölmstrom and Milgrom (1987) and show the otimality of non-linear contracts (Rogerson 1985, Sear and Srivastava 1987, Schättler and Sung 1993, Sung 1995, Hellwig and Schmidt 2002, Sannikov 2008, Rubel and Prasad 2015). A second stream of the work on dynamic incentives assumes agents to be risk neutral with limited liability, which leads to a different kind of contracting friction (our aer falls under this aradigm). Bierbaum (2002) studies how to induce high effort from the agent in each of two eriods (which may not be rofit maximizing for the rincial), while we allow different effort rofiles to be induced by the otimal contracts under different conditions. Kräkel and Schöttner (2016) study the firm s choice between commissions and bonuses and determines conditions under which one or the other (or a combination) is otimal when the reward must be aid at the end of multile eriods. Schöttner (2016) studies otimal contracting when the agent s effort costs change over time. None of these aers, however, consider whether long time horizon or short time horizon contracts are otimal. Relatedly, they also do not allow the agent to strategically borrow or ostone sales between eriods, neither do they consider the case of 7

9 limited inventory to be sold across two eriods which creates a articular form of deendence between eriods. In addition, these aers normalize the values of the outside otion and the limited liability and are unable to study comarative statics with resect to these uantities on the otimal contract and the effort rofile induced. More broadly, our research adds to the extensive literature on salesforce incentives in marketing which, in addition to the aers already cited, includes Raju and Srinivasan (1996), Godes (2004), Simester and Zhang (2010) and Zhang (2016), among many others. Our extension with limited inventory is related to the work on salesforce comensation when oerational considerations are imortant (Chen 2000, Plambeck and Zenios 2003, Dai and Jerath 2013, Saghafian and Chao 2014, Dai and Jerath 2016, Dai and Jerath 2018). The rest of the aer is organized as follows. In the following section, we describe the model. Next, we conduct the analysis and obtain our key insights regarding the different forces at lay, and the comarison between short and long time horizon contracts. Following this, we allow for eriods to be deendent by assuming that the rincial has limited inventory to be sold in the two eriods. In the final section, we conclude with a discussion. The roofs are rovided in an Aendix and an Online Aendix (indicated aroriately). MODEL We develo a simle agency theoretic model in which a firm (the rincial) hires a saleserson (the agent) to exert demand-enhancing effort. There are two time eriods denoted by t {1, 2}. Demand in both eriods is uncertain and indeendent. Let D t be the demand realization in eriod t, which can be either H or L with H > L > 0. The agent s effort increases the robability of realizing high demand levels. The effort level in eriod t, denoted by e t, can be either 1 or 0, i.e., the agent either works or shirks in each eriod; however, the rincial does not observe the effort level. We can think of effort level 0 as a saleserson making a client visit (which is observable and verifiable) and effort level 1 as the saleserson s additional effort sent in talking to and convincing the client to make the urchase (which the firm cannot observe or verify). Without effort exertion (e t = 0) demand is realized as H with a robability of, and with effort exertion (e t = 1) this robability increases to (0 < < < 1). A larger imlies greater effectiveness of the saleserson s effort, while can be interreted as the natural market outcome. We assume that all the demand created can be met and each unit sold gives a revenue 8

10 of 1 and has a marginal cost of zero. The cost of effort is given by > 0 for e t = 1 and is normalized to zero for e t = 0. We assume that both the firm and the saleserson are risk neutral. Unlike the firm, however, the saleserson has limited liability, imlying that he must be rotected from downside risk. Secifically, we assume that the saleserson has a limited liability of K in each eriod, i.e., to emloy the agent for one eriod, the rincial must guarantee a comensation of at least K under any demand outcome. Limited liability is a widely observed feature of salesforce contracts in the industry, and this assumtion is a standard one in the literature (Saington 1983, Park 1995, Kim 1997, Oyer 2000, Laffont and Martimort 2009, Simester and Zhang 2010, Dai and Jerath 2013). The limited liability assumtion also imlies the existence of a wage floor for the saleserson, which is aligned with industry ractice. We assume that the saleserson s reservation utility is U for each eriod, and that the limited liability can be either lower or higher relative to the agent s reservation utility. For instance, if the saleserson s alternative emloyment oortunities are attractive, then limited liability can be relatively low comared with reservation utility, but if saleseole s skills are most valuable in a sales context and they cannot exect comarable comensation in other rofessions, then limited liability can be relatively high comared with reservation utility (as also discussed in Kim 1997, Oyer 2000). The agent is reimbursed for effort using an incentive contract. Effort is unobservable to the firm and demand is random but can be influenced by effort, so the firm and the agent sign an outcome-based contract. The firm can contract using a short time horizon, i.e., two eriod-by-eriod contracts, where each contract is determined at the start of each eriod and ays at the end of the eriod based on the outcome of the eriod. Alternatively, the firm can roose a long time horizon, single two-eriod contract that is determined at the beginning of the first eriod and ays once at the end of the second eriod based on the outcomes of the two eriods. 1 The timings of the different games will be clarified in the relevant sections. We now roceed to the analysis. ANALYSIS First-Best Scenario We start by resenting the first-best solution (for instance, if the agent s effort is observable). In this 1 The discrete demand distribution that we have assumed ensures that effort will not change the suort of the demand distribution; otherwise, the rincial may be able to infer the agent s effort from the demand outcome and would induce the agent to work by imosing a large enalty for demand outcomes that cannot be obtained under euilibrium effort but can be obtained under off-euilibrium efforts, as argued in Mirrlees (1976). 9

11 case, the two eriods are indeendent and euivalent and it is sufficient to study just one eriod. The firm can imlement any effort level e t in either eriod, by reimbursing the agent a fixed salary s t which must be at least K while ensuring the agent s articiation. The rincial s roblem in each eriod is the following. max s t E[D t e t ] E[s t e t ] s.t. U A (e t ) U (P C t ) s t K (LL t ) Here, (P C t ) is the agent s articiation constraint, where U A (e t ) stands for the saleserson s exected net utility on exerting effort e t, which is eual to s t if the agent exerts effort and is eual to s t if the agent does not exert effort. It states that to emloy the sale agent, the rincial needs to rovide a fixed salary that makes the agent s exected net utility from exerting effort e t no less than his outside otion, which simlifies as s t U + if effort is exerted, and as s t U if effort is not exerted. (LL t ) stands for the agent s limited liability constraint, which ensures that the agent receives a fixed salary s t no less than his limited liability K. If the contract secifies effort exertion in eriod t {1, 2}, i.e., e t = 1, the rincial s exected rofit is eual to the exected market demand subject to the agent s effort exertion, H + (1 )L, minus the minimal salary to ensure effort exertion, max{u +, K}, i.e., H + (1 )L max{u +, K}. If e t = 0, the rincial gets the natural market outcome and ays the minimal salary to emloy the saleserson, i.e. H +(1 )L max{u, K}. This leads to the following first-best solution (the roof is in Section A1 in the Aendix). Proosition 1 (Otimal First-Best Solution) The first-best contract and outcomes are as er the following table. U K H L e F B s F B U K 0 H L U + U K < 0 H L + U K 1 U + U K < H L 0 K U K 0 H L < 0 U U K < 0 H L < + U K In the table in Proosition 1, the first column gives the condition on U K, the second column gives K 10

12 the condition on H L, the third column gives the effort exertion under the otimal salary, and the fourth column gives the otimal salary. Figure 1 deicts the first-best solution with resect to the range of the demand distribution (H L), the agent s effectiveness arameter ( ), and the agent s outside otion relative to his limited liability (U K). From Figure 1, we can infer that the rincial would like the agent to exert effort when the uside market otential is large, or when the effectiveness of the agent s effort is high, or when the agent s limited liability is large relative to his outside otion. Intuitively, the firm would like to direct the saleserson to work hard if and only if the increase in the exected demand subject to the agent s effort exertion (given by ( )(H L)) outweighs the marginal cost for soliciting effort (given by max{u +, K} max{u, K}). When limited liability is low relative to the agent s outside otion (given by K U), the rincial only needs to comensate the agent for his outside otion lus cost of effort. Therefore the additional cost for soliciting effort is, and the rincial solicits effort exertion if and only if H L. When limited liability is intermediate (i.e., U < K U + ), even if the rincial does not solicit effort, he still has to ay the agent his limited liability, so the additional cost for soliciting effort becomes + U K. In this case as K increases, the additional cost for soliciting effort decreases, thus the rincial solicits effort in a larger arameter sace. When limited liability increases beyond U +, the rincial ays the agent his limited liability regardless of effort levels and there is no additional cost for soliciting effort, therefore, the rincial instructs the agent to exert effort given any H L. The above arguments give the following counterintuitive result. Corollary 1 In the first-best scenario, as limited liability increases the rincial solicits effort in a weakly larger arameter sace. Period-by-Period (Short Time Horizon) Contract In this scenario, the rincial contracts over a short time horizon and secifies a one-eriod contract at the beginning of the first eriod, and then secifies another one-eriod contract at the beginning of the second eriod. The effort for each eriod is rewarded searately, and therefore we call this a disaggregate contract. As the two eriods are identical and indeendent, it is sufficient to study just one eriod. Consider the roblem for eriod t {1, 2}. Since demand follows a binomial distribution, the rincial offers uota-bonus contracts with uota levels χ t {H, L} and bonuses b χt,t 0, where the bonus b χt,t is aid to the saleserson if and only if the sales reach the uota χ t, together with a fixed salary of s t. Indeed, it suffices for the rincial to consider only two of the decision variables. Without 11

13 loss of generality, we normalize b L,t to 0 and simlify the notation of b H,t as b t, i.e., the rincial does not issue bonus when the demand outcome is L and issues bonus b t when the demand outcome is H. The rincial s roblem in each eriod is the following. max E[D t e t ] E[s t + b t e t ] s t,b t s.t. U A (e t ) > U A (ẽ t ) (IC t ) U A (e t ) U (P C t ) s t, s t + b t K (LL t ) The articiation constraint (P C t ) and the limited liability constraint (LL t ) can be interreted in a similar way as in the first-best scenario. In addition, the contract needs to satisfy an incentive comatibility constraint (IC t ), which states that to induce effort e t, the rincial needs to ensure that the agent gains a higher net utility by exerting effort e t comared with a different effort level ẽ t. Before solving the otimal contract for the rincial, we first derive the best contract for the rincial to induce any given effort level. To imlement e t = 1, from the incentive comatibility constraint (IC t ), the rincial needs to set b H,t satisfying s t + b t s t + b t, which simlifies into b t. The articiation constraint (P C t ) reuires that the agent s exected utility from exerting effort no lower than his reservation utility, that is, s t + U. To meet the limited liability constraint (LL t) we need the guaranteed salary no less than the agent s limited liability, i.e., s t K. The solution is that to imlement e t = 1, the rincial offers a fixed salary s t = max{k, U }, and a bonus b t = if the demand outcome is high. To imlement e t = 0, it is enough for the rincial to only offer the agent a fixed salary s t = max{k, U}. The overall solution to the otimal eriod-by-eriod contract is secified in the following roosition (the roof is in Section A2 in the Aendix). Proosition 2 (Otimal Period-by-Period Contract) The otimal eriod-by-eriod contract and outcomes are as er the following table. 12

14 U K H L e t s t b t U K H L U 0 U K < H L () 2 U K 1 K U K < 0 H L () 2 U K K H L < U 0 0 U K < H L < () 2 U K 0 U 0 U K < 0 H L < () 2 K 0 Figure 2 deicts the otimal eriod-by-eriod contract with resect to the range of the demand distribution (H L), the agent s effectiveness arameter ( ), and the agent s reservation utility relative to his limited liability (U K). In Region I, the rincial does not want to induce effort even in the first-best scenario. In Region II, the rincial wants to induce effort in the first-best scenario but not in the eriod-by-eriod contracting scenario. In Region III, the rincial wants to induce effort in the eriod-by-eriod scenario. Note that when limited liability is relatively small (K U ), even if effort is unobservable, the rincial can still achieve the first-best solution by enalizing the agent for low demand realization and rewarding the agent for high demand realization. As limited liability increases beyond U, the rincial cannot ay the agent less than his limited liability when demand realization is L, therefore the first-best solution is no longer achievable. This leads the rincial to induce effort in a smaller arameter sace as limited liability increases. When limited liability exceeds U, the agent needs to be guaranteed his limited liability, with or without a bonus to induce effort. Therefore, the rincial induces effort if and only if the extra cost for inducing effort () 2 is offset by the increase in exected demand from exerting effort (H L)( ). From Figure 2, we can see that as limited liability increases, while in the first-best scenario the rincial solicits effort in a weakly larger arameter sace (as er Corollary 1; reresented by the dashed line), with unobservable effort he will induce effort in a weakly smaller arameter sace (reresented by the solid line). Two-Period (Long Time Horizon) Contract In this scenario, the firm rooses a long time horizon two-eriod contract at the beginning of the first eriod and ays once at the end of the second eriod based on the outcomes of the two eriods. The timeline of the game is as follows. At the beginning of eriod 1, i.e., T = 1, the rincial rooses the contract and the agent decides whether or not to accet the offer. If acceted, the agent then decides 13

15 on his effort in the first eriod, e 1. At the end of T = 1, the agent and the rincial observe the demand outcome for the first eriod, D 1. The agent then chooses his second eriod effort e 2. At the end of T = 2, the agent and the rincial observe the second eriod demand outcome D 2. The agent then gets aid according to the contract. A key feature of this scenario introduced due to unobservability of effort and the contract aying at the end of two eriods is that the agent can game the system the agent can choose effort in eriod 2 based on the outcome of eriod 1 (and, realizing this, can also choose the effort in eriod 1 strategically). We denote the two-eriod effort rofile by (e 1, e H 2, el 2 ), where the second eriod s effort ed 1 2 is contingent on the first eriod s demand realization, D 1. In full generality, this contract involves a guaranteed salary for emloying the agent for two eriods, lus a bonus issued at the end of the two eriods that is contingent on the whole history of oututs. We denote the fixed salary as S, and denote the bonus aid at the end of T = 2 by b 2 (D 1, D 2 ). Such a contract thus stiulates four ossible bonuses, b 2 (L, L), b 2 (L, H), b 2 (H, L) and b 2 (H, H). To revent the agent from restricting sales to L when demand is H, we imose a constraint on the two-eriod contract given by b(h, H) max{b(h, L), b(l, H)}, i.e., the bonus aid when demand in both eriods is realized as H should be no lower than that aid when demand in only one of the eriods is realized as H. (Note that we do not need this assumtion to derive our result as this constraint does not bind in the otimal contract, but this is a natural assumtion and simlifies the analysis.) Under this constraint, we obtain the following lemma (the detailed roof is in Section A3.1 in the Aendix). Lemma 1 When the two eriods are indeendent of each other, in the weakly dominant two-eriod contract, b 2 (H, L) = b 2 (L, H). Lemma 1 imlies that it is sufficient for the rincial to ay the agent at the end of two eriods a bonus according to cumulative sales (which can be 2L, H + L or 2H) and indeendent of the sales history. 2,3 We denote the fixed salary by S, normalize the bonus ayment when the total sales are 2L as 0, denote the bonus ayment when the total sales across two eriods are H + L by B 1, and denote 2 Only the contract to induce (0, 1 ) is history-deendent, but we find it sub-otimal for the rincial when the two eriods are indeendent. However, later analysis, we will show that such a history-deendent contract can be otimal when the two eriods become deendent. 3 The lemma holds without discounting and with risk-neutral agents. As shown by Sear and Srivastava (1987) and Sannikov (2008), if agents discount their future utility, or if the agent is risk averse, a ath-deendent contract can be otimal. 14

16 the bonus ayment when the total sale are 2H by B 2. We formulate the rincial s roblem as follows. max E[D e 1, e H 2, e L 2 ] E[S + B 1 + B 2 e 1, e H 2, e L 2 ] B 1,B 2 s.t. U A (e H 2 ) > U A (ẽ H 2 ) (IC H 2 ) U A (e L 2 ) > U A (ẽ L 2 ) (IC L 2 ) U A (e 1 e H 2, e L 2 ) > U A (ẽ 1 e H 2, e L 2 ) (IC 1 ) U A (e 1, e H 2, e L 2 ) 2U (PC) S, S + B 1, S + B 2 2K (LL) (IC2 H) stands for the agent s incentive comatible constraint in the second eriod following D 1 = H, where U A (e H 2 ) reresents the agent s net ayoff in Period 2 uon exerting effort eh 2. If the agent exerts effort, he will get S + B 2 with robability and S + B 1 otherwise; without exerting effort, he will get S + B 2 with robability and S + B 1 otherwise. To induce e H 2, the rincial needs to ensure that the agent gets a higher ayoff uon exerting effort e H 2, comared with a different effort level ẽh 2. Similarly, (IC L 2 ) stands for the incentive comatible constraint for inducing effort level el 2 in the second eriod following D 1 = L. Then, (IC 1 ) reresents the incentive comatible constraint in the first eriod. U A (e 1 e H 2, el 2 ) denotes the agent s net ayoff across two eriods uon exerting e 1 in the first eriod, given that the agent is induced to exert effort (e H 2, el 2 ) in the second eriod. If e 1 = 1, his total net ayoff will be U A (e H 2 ) with robability and U A(e L 2 ) otherwise; if e 1 = 0, his total net ayoff will be U A (e H 2 ) with robability and U A (e L 2 ) otherwise. To induce e 1, the rincial needs to ensure that the agent gets a higher total net ayoff on exerting e 1, comared with a different effort level ẽ 1. The articiation constraint (P C) and the limited liability constraint (LL) are similar to that in the eriod-by-eriod case, excet for that we multily the right-hand sides by two under a two-eriod contracting. To arrive at an otimal contract for the rincial, it is crucial to understand how the agent s effort rofile in the two eriods changes with the bonuses B 1 (rovided for H +L) and B 2 (rovided for 2H) in the two-eriod contract. The following lemma describes this effort rofile (the roof is immediate from the roof of Lemma 1). 4 Note that since e 2 deends on D 1, which is random, we write e 2 in terms of its exectation value. For instance, if the agent exerts effort in eriod 1 and will exert effort in eriod 2 only if the outcome in eriod 1 is H, then e 2 = 1 with robability, so we write this effort rofile as 4 We make the assumtion that when the agent is indifferent between exerting effort or not, he will choose to exert effort. 15

17 (1, ). Lemma 2 (Agent s Resonse to Two-eriod Contract) Given B 1 and B 2, the agent s exected effort rofile (e 1, E[e 2 ]) is as er the following table. B 1 (B 1, B 2 ) (e 1, E[e 2 ]), B 2 B 1 (1, 1) 0 B 1 <, B 2 + (1 )B 1 + (1, ) B 2 B 1 < B 2 B 1 B 1, B 2 + (1 )B 1 (1, 1 ), B 2 + (1 )B 1 < + (0, ), B 2 + (1 )B 1 < (0, 1 ) 0 B 1 <, 0 B 2 B 1 < (0, 0) Figure 3 illustrates Lemma 2 grahically. The x-axis, B 1, is the incremental reward when total sales increase from 2L to H + L; the y-axis, B 2 B 1, is the incremental reward when total sales increase from H + L to 2H. If both rewards are small, there is no effort exertion in either eriod, denoted by e = (0, 0), which is Region I. If both rewards are large, the agent will ut in effort in both eriods, i.e., e = (1, 1), which is Region IV. For other regions, the effort exertion decisions are more involved. If the agent does not secure the bonus B 1 after eriod 1 with the demand outcome L, he will not exend additional effort if B 1. If the agent secures the bonus B 1 after eriod 1 with the demand outcome H, he will not exend additional effort if B 2 B 1. In other words, B 1 and B 2 B 1 motivate the agent to exert effort in the second eriod if demand in the first eriod turns out to be L and H, resectively. Furthermore, the agent s effort exertion at T = 1 deends on the valus of both B 1 and B 2 B 1. In Regions II and VI, the agent does not work in eriod 1 and chooses to ride his luck in eriod 1. However, in Region II, he works in eriod 2 if the demand outcome is unfavorable, i.e., L, in eriod 1, and in Region VI, he works in eriod 2 if the demand outcome is favorable, i.e., H, in eriod 1. In Regions III and V, the agent works in eriod 1. However, in Region III, he works in eriod 2 if the demand outcome is unfavorable, i.e., L, in eriod 1, and in Region V, he works in eriod 2 if the demand outcome is favorable, i.e., H, in eriod 1. We now determine the otimal comensation lan for the firm. We find the otimal contract by balancing the exected revenue E[D] less the exected comensation cost E[S + B 1 + B 2 ]. Proosition 3 characterizes the otimal two-eriod contract for the rincial (the detailed roof is in Section A3.2 in the Aendix). 16

18 Proosition 3 (Otimal Two-eriod Contract) The otimal two-eriod contract and outcomes are as er the following table. Region U K H L (e 1, E[e 2 ]) S B 1 B 2 I II III IV U K 2 2() H L < 2 2(1+)() U K < 2 2() H L < U K () 2 () (0, 0) 2U U K < 2 2(1+)() H L < 2 + (1+)() 2 U K < 0 H L < 2 + (1+)() 2 2 2() U K < 2 2(1+)() U K < 2() U K < 2 2() U K < 2 2(1+)() U K < 0 U K < 2() 2 2() 2() +() 2 2 2() 2 2(1+)() U K < 0 U K H L < +()2 (1+)() 2 () 2 U K () H L < H L < 2(U K) (1 +) (1 )() 2 (1+)() 2 (1+)() H L < 2(U K) (1+)() 2(U K) 2U 2U 2K (1+)() (0, ) 2U (1+ 2) (1+)() 2 2(U K) (1 )() (1+ 2) H L < (1 +), or, (1+)() 2 (1 )() 2 2(U K) 2 + (1+)() 2 (1+)() H L < (1 +), or, (1 )() 2 (1+) (1+)() 2 H L < H L or, 2() U K < H L (1 +) (1 )() 2 U K < 2K, or, 2U (1 +) (1 )() 2, or, (1, ) 2K 0 (1 +) (1 )() 2, 2(U K) 2() H L (1 +), (1 )() 2 2K 2K 2K 2 2U (1 )() or, (1, 1) 2K 1 2K 1+ () 2 We illustrate the result with the aid of Figure 4. The otimal contract is either a gradual contract (in which B 1 > 0, i.e., it rewards bonuses at both H + L and 2H) or an extreme contract (in which B 1 = 0, i.e., it rewards bonuses only at 2H). In Region I, the rincial does not want to motivate effort. In Region II, the rincial finds it otimal to use the extreme contract to motivate the effort rofile (0, ) by giving a bonus B 2 =. In Region III, the rincial finds it otimal to use the extreme contract to motivate the effort rofile (1, ) by giving a bonus B 2 = 1+ () (which is larger than ). In Region IV, the rincial finds it otimal to use the otimal gradual two-eriod contract to motivate the effort rofile (1, 1). To develo the intuition behind these results, we first focus on the case when limited liability is sufficiently high. Secifically, we assume K = U, in which case the rincial ays a fixed salary of S = 2K for inducing any effort rofile. From Figure 4, we can see that in this case the otimal contract is either the extreme two-eriod contract with B 2 = 1+ () to imlement e = (1, ), or the gradual two-eriod contract with bonuses B 1 =, B 2 = 2 to imlement e = (1, 1). To understand why, we discuss two effects that are oerative, namely the incentive effect and the gaming effect. First, we discuss the incentive effect. In Figure 5, we vary, the effectiveness of the agent s effort, keeing H L fixed. Generally seaking, more effective agents reuire lower incentives to work because the outcome is a better signal of effort exerted. In line with this, the exected bonus ayments under the extreme contract, ( 2 + ), and under the gradual contract, 2, both decrease with. 17

19 However, the difference between them, E[B] gradual E[B] extreme = ( 1 1), is always ositive, as shown by the solid line in Figure 5. This means that the rincial always ays a smaller exected bonus under the extreme contract than under the gradual contract. Therefore, on the ositive side, the extreme contract benefits from the incentive effect: it rovides more effective incentives for an agent with limited liability, thus saving on the bonus ayment for the rincial. The reason behind this is that under limited liability, the rincial concentrates comensation at a high level of sales. In a eriod-by-eriod contract the highest level of sales at which reward can be given is H while in a two-eriod contract this level is 2H; this can lead to higher incentive rovision in a two-eriod contract (even though the reward is given only once). Another interesting observation from Figure 5 is that the incentive effect, as measured by the solid line, shrinks as increases. This is because as moral hazard frictions decrease with more effective agents, so will the comarative advantage of the extreme contract on saving incentive costs. However, in a dynamic setting, such a non-linear reward structure will suffer from the agent s gaming. As a conseuence, on the negative side, the rincial obtains less demand under the extreme contract, as the dashed line in Figure 5 illustrates. Mathematically, E[D] gradual E[D] extreme = (1 )()(H L) is always ositive. As we have mentioned, due to the non-linear structure of the extreme contract, an agent will game the system by varying his effort in a dynamic setting. Secifically, the agent exerts effort in the first eriod, but if the first eriod outcome turns out to be L, the agent will give u on effort exertion in the second eriod, leading to a demand loss for the rincial. Interestingly, as gets larger, agents under both contracts generate higher sales, but the difference between the sales they generate, caused by the gaming effect, takes an inverse-u shae. This is because when increases, the demand loss, if it haens, ( )(H L), gets larger, but the robability of its haening, (1 ), decreases. Combining the incentive effect and the gaming effect, we can see from Figure 5 that if is small, the incentive effect dominates and the extreme lan outerforms the gradual lan the gaming loss under the extreme contract is relatively small comared with its advantage in roviding incentives. Above a threshold of, the gaming loss becomes dominant and the gradual contract is referred by the rincial. However, as continues to increase, the gaming loss begins to decline, rendering the extreme contract better again. Overall, when is either very small or very large, the incentive effect will be more significant than the gaming effect and the extreme lan outerforms the gradual lan. The above analysis is based on the remise that limited liability is sufficiently high. Now we discuss the otimal contract as limited liability decreases. We fix H L and at a low level so that when 18

20 limited liability is sufficiently high the rincial does not want to induce effort. As limited liability decreases, the friction due to moral hazard becomes smaller, and the rincial starts to motive effort using the extreme two-eriod contract, which rovides more effective incentives than the eriod-byeriod contract. Since limited liability is still relatively high in this scenario, the rincial only induces e H 2 = 1 through a low ultimate bonus and the full effort rofile is e = (0, ) that is, there is no early effort exertion in the first eriod, and there is effort exertion in the second eriod if the early eriod realizes as high. As limited liability continues to decreases further, the rincial imlements e 1 = 1 through a high ultimate bonus and the full effort rofile is e = (1, ) that is, the agent exerts effort in the first eriod, and he will continue exerting effort in the second eriod if the early eriod realizes as high. When limited liability becomes small enough, the rincial will imlement effort e = (1, 1) using the gradual two-eriod contract. Put together, the receding discussion exlains the atterns in Figure 4. When limited liability is not too small (relative to the agent s outside otion), in a market with small uside demand otential, and with either very inefficient or very efficient saleseole, the extreme contract erforms best for the rincial. In other circumstances, it is rofitable for firms to roose a gradual contract to motivate hard work in both eriods. Next, we state an interesting corollary. Corollary 2 Under a two-eriod contract, both e = (1, ) and e = (0, ) can be otimally induced exected effort rofiles by the firm. The corollary has interesting and imortant imlications for observed effort rofiles of agents. It states that a attern of giving u in the second eriod (exected effort rofile e = (1, ), i.e., effort is exerted in the first eriod but then not exerted in the second eriod if the first eriod demand is realized as low) is actually otimally induced by the firm under certain conditions. It also states that a attern of delaying effort or a hockey stick effort rofile (exected effort rofile e = (0, ), i.e., effort is not exerted in the first eriod but then exerted in the second eriod only if the first eriod demand is realized as high) is also otimally induced by the firm under certain conditions. To see why, note that effort ostonement is otimal for the firm when the limited liability, the demand uside otential, and agent s effort effectiveness are all at intermediate levels. In this case, the rincial would like to induce effort using an extreme two-eriod contract with a low ultimate bonus, which rovides the most effective incentives. In other arameter saces, early effort exertion is referred by the rincial under 19