Salesforce Contracting Under Uncertain Demand and Supply:Double Moral Hazard and Optimality of Smooth ContractsTinglong Dai,a Kinshuk Jerathb
aCarey Business School, Johns Hopkins University, Baltimore, Maryland 21202; bColumbia Business School, Columbia University,New York, New York 10027Contact: [email protected], https://orcid.org/0000-0001-9248-5153 (TD); [email protected], https://orcid.org/0000-0003-0732-5863 (KJ)
Received: March 8, 2018Revised: December 5, 2018; March 13, 2019Accepted: April 5, 2019Published Online in Articles in Advance:August 28, 2019
Abstract. We consider the compensation design problem of a firm that hires a salespersonto exert effort to increase demand. We assume both demand and supply to be uncertainwith sales being the smaller of demand and supply and assume that, if demand exceedssupply, then unmet demand is unobservable (demand censoring). Under single moralhazard (i.e., when the salesperson’s effort is unobservable to the firm), we show that theoptimal contract has an extreme convex form in which a bonus is provided only forachieving the highest sales outcome even if low realized sales are due to low realizedsupply (onwhich the salesperson has no influence). Under double moral hazard (i.e., whenthe firm can also take supply-related actions that are unobservable to the salesperson), weshow that the optimal contract is smoother as it involves positive compensation for in-termediate sales outcomes to assure the salesperson that the firm does not have an in-centive to deviate to an action that hurts the agent; in fact, under certain conditions, thecontract is concave in sales. We also determine conditions under which, if possible, thefirm should postpone contracting until after supply is realized.
History: Ganesh Iyer served as the senior editor and Dmitri Kuksov served as associate editor for thisarticle.
Keywords: salesforce compensation • yield uncertainty • demand censoring • double moral hazard • quota-bonus contract •early versus late contracting
1. IntroductionFirms engage salespersons to increase demand fortheir products. Salesforce compensation is a major ex-pense for firms, especially in business-to-business (B2B)settings, and the total spend of U.S. firms on salesforcecompensation is approximately three times their spendon advertising (Zoltners et al. 2008). There is a large lit-erature on salesforce compensation contracts rooted inagency theory (e.g., Holmstrom 1979, Basu et al. 1985,Holmstrom and Milgrom 1987, Lal and Srinivasan1993, Park 1995, Raju and Srinivasan 1996, Kim 1997,Oyer 2000, Herweg et al. 2010, and Simester andZhang 2014). This literature assumes that the agent’ssales effort is unobservable and there is demand un-certainty, which makes it difficult to infer salesforce ef-fort fromobserving realizeddemand, leading to the issueof moral hazard.
Realized demand can only be fulfilled if there issufficient supply, and virtually all of the work onsalesforce compensation has assumed that the supplyis unbounded and always assured (typically, supply-related assumptions are not even explicitly stated).This, however, may not always be the case as firms mayonly stock a limited amount of inventory to meet short-run demand. Indeed, how much inventory to stock
under uncertaindemand, often called the “newsvendorproblem,” is a primary focus of study of the field ofoperations management (Porteus 2002, Cachon andTerweisch 2012). Recent work has considered the im-portance of supply in determining salesforce compen-sation contracts. For example, Dai and Jerath (2013,2016) assume limited supply and show that, coun-terintuitively, this leads to higher powered contracts(in which “higher powered” means that the bonus islarger).Furthermore, in practice, there are many situations
in which a firm’s inventory level may not only belimited, but may also not be fully predictable, that is,supply may be random. (Note that we use the terms“supply,” “inventory,” and “yield” interchangeablythroughout the paper.) For example, in the case ofwine production, “Vineyards are variable. Growershave known this for as long as they have been grow-ing grapes” (Bramley and Hamilton 2004, p. 32). Theuncertain yield has implications for demand fulfill-ment. Random yield is, indeed, a widely observed phe-nomenon in myriad industries and scenarios, includ-ing electronic fabrication and assembly (Lee and Yano1988),mining (Kamrad and Ernst 2001), semiconductormanufacturing (Stapper and Rosner 1995), agriculture
(Kazaz 2004), refining and chemical manufacturing(Rajaram and Karmarkar 2002), vaccine and drugmanufacturing (Dai et al. 2016), andmultistage customproduction processes (Wein 1992). Yield uncertaintymay also play a significant role in the case of procur-ing from unreliable suppliers (Dada et al. 2007). Fur-thermore, Yano and Lee (1995) argue that randomyield is prevalent even outside of the aforementionedindustries/scenarios because of “many traditional dis-crete parts manufacturing processes that experiencerandom yields.” For example, a wide variety of con-sumer products (e.g., smartphones) are assembledfrom multiple parts with uncertain yields. Anotherfactor that may lead to supply uncertainty is inventoryshrinkage, for example, unforeseen loss of stocked in-ventory resulting from theft, mismanagement (e.g.,damage in the handling and storage of the product),expiration, etc., which is a common problem in ware-houses and retail stores (Raman et al. 2001, Liu et al.2010). In most, if not all, of these situations, the firmshire a salesforce to sell the products to other firms(B2B selling) or consumers (business-to-consumer orB2C selling), and it is important to understand theimpact of supply uncertainty on salesforce compen-sation contracts. However, to our knowledge, the im-pact of supply uncertainty on salesforce compensa-tion has not been studied, and this is a gap in theliterature that we make an effort to start to fill.1
An important consideration with random demandand limited (deterministic or random) supply is that,with positive probability, demand and supply do notmatch, and sales is the minimum of the two. We con-sider a setting in which the firm can only observe sales,which implies that, in the case when demand exceedssupply, the firm cannot observe the demand in excessof the supply. This is because it is typically not possibleto keep track of demand that was or could have beenrealized but was not fulfilled because of a lack of in-ventory, especially if customers choose not to order orto postpone their purchase rather than backorder theproduct. This is a widely observed phenomenon com-monly referred to as demand censoring. In recognition ofits real-world importance, a growing economics, mar-keting, and operations literature has studied the man-agerial implications of demand censoring (Braden andFreimer 1991; Anupindi et al. 1998; Downs et al. 2001;Ding et al. 2002; Chen and Plambeck 2008; Lu et al.2008; Besbes and Muharremoglu 2012; Conlon andMortimer 2013; Dai and Jerath 2013, 2016; Rudi andDrake 2014; Chen et al. 2017). Demand censoring maybe viewed as a specific form of information censoringand, in our setting, because of it, the firm cannot usethe realized demand as the basis for determining howmuch to pay a salesperson. In fact, the firm has to workwith realized sales, which is a worse signal than realized
demand (because of truncation at the inventory level) ofthe salesperson’s effort.The following is an example of a B2B situation in
which all the aspects that we highlight—namely sales-force, demand uncertainty, supply uncertainty, doublemoral hazard, and demand censoring—are opera-tive. Consider a company that sells office products,such as electronic equipment (e.g., projectors), furni-ture, and stationery. The firm typically orders prod-ucts and stocks them in a local warehouse with a leadtime of several weeks or even several months; thereis reasonable uncertainty about exactly how many ofand when the ordered units will reach because ofrandom supply disruptions and how many of thesupplied units will actually be available (for reasonssuch as shrinkage), but the firm can take certain costlyactions to increase the probability of high yield (i.e.,inventory yield is uncertain, and to improve this yield,the firm makes an inventory decision/action unob-served to the sales agent). On the other hand, salesagents go out in the field to describe these products toprospective customers in the hope that they will beconvinced of their benefits and will order these prod-ucts from the company if and when the need for thesearises for the customers (i.e., unobservable sales effortby the agent increases the level of demand, and realizeddemand is uncertain) with a promise that, after an orderis placed, delivery will be done in a few days (i.e., ifsupply is not available in the short term, then realizeddemand cannot be met in the short term). When thecustomer actually wants to order, the customer may usea website on which the products are displayed alongwith whether they are available for immediate deliveryor not, and sales from a geographical area are tied tothe salesperson serving that area. In this case, if a pro-spective customer sees that a product is not available forimmediate delivery, the customer may not even placethe order, or possibly, an unavailable product is noteven listed on the website, in which case, again, theconsumers cannot order it or indicate that they wantedit (i.e., lost demand is not observed). It may also bepossible that the customer calls the firm’s salespersonto place an order and is told that the particular productthe customer wanted is not available; the salespersonmay not record the missed order, and even if the firmdid claim to record orders, the firm may not fully be-lieve the lost demand number because the salesper-son would have an incentive to inflate this number toclaim that the salesperson generated high demand thatwas not fulfilled because of inventory issues (i.e., again,lost demand is not observed). Such a situation, whichhas all the essential components that we study, occursin many B2B settings.Another B2B situation in which this problem arises
is in the media ad sales context. We directly quote
Robert Dillon, whowas vice president of North Americasales strategy and operations at Yahoo! (Sales Lead-ership Forum 2010, pp. 27–28):
Yahoo! is in the media business. We create inventoryfor people looking at web pages. We call that supply.Our sales representatives are out there generatingdemand from our advertisers. Supply can move dra-matically up and down and it can move dramaticallyin various verticals. And, back to our market model,it’s difficult to predict those shifts and difficult toexplain those shifts when they happen. So a sales repmay have done everything right on the demand side,but because of some strange shift in inventory that’sdifficult to explain, they haven’t hit their number.That’s the challenge that we work through and try tocompensate for and plan for.
Clearly, in this case, there is supply and demandvariability, the sales agent can take actions to increaseoverall demand, the firm can take actions to increaseoverall supply, and unmet demand may go unexpressedby the buyer or unrecorded by the sales agent or thenumber may not be believed by the firm.
Such examples can also be readily provided for B2Csettings, for example, for the smartphone division ofa firm, such as Samsung. Briefly, the firm makes aninventory decision and can take actions to promotea high yield, but short-term yield is uncertain (thisproblem is especially acute in the smartphone in-dustry2), and short-term demand can only be metwithwhat is available. Customers go to off-line stores,such as a Samsung Experience Shop or a Samsungstore-within-a-store in BestBuy, to obtain informa-tion about the smartphones, where in-store associatesexert unobservable effort to convince customers of thebenefits of the product(s) offered. Customers later goonline to order the product they want, where they seeavailability status; typically, if they see a product asunavailable they are unable to place an order for italthough sometimes they might not even see it listedon the website, and so lost demand is not observed.On the other hand, if a customer orders a product in-store and is told that it is out of stock, this fact may notbe recorded (inwhich case lost sales are not recorded),and/or the firmmay not believe these numbers as thein-store associate would have the incentive to artifi-cially inflate the missed sales.
To study such situations, we construct a stylizedprincipal–agent model of a firm that hires a sales-person to market a product with uncertainty in bothsupply and demand. To keep the model simple whileconveying the main insights, we assume both supplyand demand to have discrete distributions with thesame support. The firm takes an inventory-relatedaction to influence the supply distribution (but doesnot influence the demand distribution); the sales-person’s effort influences the demand distribution
(but does not influence the supply distribution). Weassume that the firm contracts with the salespersonbefore yield uncertainty is resolved. We characterizethe firm’s optimal contracting decision under thestandard assumption in the contract theory literaturethat salesforce effort boosts demand in a way suchthat the demand distribution satisfies a monotonelikelihood ratio property (MLRP) (which essentiallyimplies that a higher demand level is a more reliableindicator that the salesperson has exerted effort);likewise, we model how the firm’s inventory-relatedaction influences supply by assuming the inventorydistribution satisfies MLRP. Because of demand cen-soring, the firm can only observe the realized salesand has to contract on this as the outcome metric.In a benchmark in which we fix the firm’s inventory-
related action, we show that the optimal compensa-tion contract provides a bonus only if observed salesare the highest possible (even if sales were limited by alow inventory realization on which the salesperson hasno control). This is a simple contract that is similar tothe optimal salesforce compensation contract withoutsupply limitations when the demand distribution sat-isfies MLRP, in which the salesperson is rewarded abonus only when the most desirable demand outcomeis achieved (see, e.g., Laffont and Martimort 2001).The aforementioned contract provides a bonus only
if observed sales are the highest possible even if saleswere more limited by low inventory realization thanby low demand realization. One may think of this asan overly extreme contract, especially because the sales-person’s effort does not influence the supply distri-bution. Echoing this point, we analyze our focal sce-nario in which the firm’s inventory-related action isendogenous, but there is randomness in the finalsupply that is available to meet demand. For instance,the firm may choose a high or low intensity of au-diting inventory, proactively addressing upstreamsupply issues or controlling inventory shrinkage; allthese activities influence the inventory available at thetime of meeting demand. The salesperson observesthe final inventory but does not observe the firm’sinventory-related action. Just as demand uncertaintyand effort unobservability imply a demand-relatedmoral hazard problem for the firm (i.e., the firmcannot verify the salesperson’s effort from the re-alized demand), if the firm’s inventory-related actionis unobservable to the salesperson, yield uncertaintyimplies a supply-related moral hazard problem forthe salesperson (i.e., the salesperson cannot verify thefirm’s original inventory action from the final availableinventory), and a double moral hazard problem arises.In this case, in which the salesperson does not have
transparency regarding the firm’s supply-related ac-tion, we show that the firm optimally offers the sales-person a smoother contract, that is, positive bonus
is provided for intermediate sales outcomes as well.This is because the firm is tempted to take a less ef-fective inventory-related action (without the sales-person’s knowledge) if this helps the firm to reducethe expected compensation for the salesperson, andto motivate the salesperson under this concern, thefirm may have to reward the salesperson even whenthe most desirable sales outcome is not achieved. Inother words, under double moral hazard, when thefirm’s inventory-related action is not observable, thecontract is lower powered; in fact, the contract mayeven be concave in realized sales. This is an inter-esting result because it shows that supply-side moralhazard can lead to concave contracts even when theagent is risk neutral; this is different from extant lit-erature that argues that risk-neutral agents are of-fered convex contracts (Laffont and Martimort 2001,Dai and Jerath 2013), and concavity in a contract istypically driven by risk aversion of the agent (Basuet al. 1985, Rubel and Prasad 2016). We note thatZoltners et al. (2006) report that concave compensa-tion plans (which they call “regressive” plans asopposed to convex compensation plans, which theycall “progressive” plans) are widely used by firms; inthis context, we show that regressive plans are pos-sible even if the salesperson is not risk averse andbecause there may be double moral hazard.
Finally, we examine the optimal timing of con-tracting, that is, if it is possible to contract with thesalesperson after yield uncertainty is resolved, shouldthe firm do so?3 When the firm contracts with thesalesperson after yield uncertainty is resolved, theinventory outcome is known, and the salesforcecompensation is only contingent on the demand out-come. However, a trade-off arises from the reduceduncertainty (Dai and Jerath 2013). On the one hand,when the yield is low, the sales outcome is con-strained by the low inventory level rather than a lowdemand outcome, and it is not worthwhile to en-gage the salesperson. In this case, the firm can avoidwasteful salesforce expenses (by not hiring the sales-person) in view of the inventory information. On theother hand, when the yield is relatively high but lessthan the highest possible realization of demand, thesales quantity is bounded above by the available in-ventory, and the firm faces the issue of demand cen-soring. Because of its limited observability of the salesoutcome, the firm has to share a higher rent with thesalesperson to induce the same demand inducing effort.Jointly, these two effects drive the optimal timingof salesforce contracting under yield uncertainty.We characterize the firm’s optimal contracting de-cision, which allows us to show a number of counter-intuitive results. For instance, we find that, as theprobability of high inventory outcome increases (i.e.,there is a lower chance that the inventory outcome is
low), the firm might be more inclined to wait to con-tract with the salesperson after observing the inven-tory outcome.Our paper contributes to the literature on double
moral hazard. To the best of our knowledge, the priordouble moral hazard literature with risk-neutral prin-cipals and agents (including, e.g., Cooper and Ross1985, Romano 1994, Bhattacharyya and Lafontaine1995, Kim and Wang 1998, and Roels et al. 2010)assumes that both parties have unlimited liability,whereas we assume that the agent has limited lia-bility. The feature that the agent has limited liability,albeit unique to the double moral hazard literature, isa standard assumption in the salesforce compensa-tion literature. This distinguishing feature in ourmodel implies that, even under single moral haz-ard, the moral hazard problem cannot be solved by“selling the firm,” and rent sharing is necessary.Double moral hazard further implies that the firm hasto share more rent and settle with a less efficientcontract that is smoother.In addition to the literatures mentioned earlier, our
paper contributes to the nascent literature on jointlymodeling incentive and operational issues. Chen (2005)focuses on designing sales compensation contractssuch that inventory can be managed more effec-tively through smoothing demand and eliciting moremarket information. Plambeck and Zenios (2003)derive an optimal incentive contract for a produc-tion manager for a specific production process. Daiand Jerath (2013, 2016) study salesforce compensa-tion contracts under limited inventory but do notallow for uncertainty in inventory. This is the novelangle that we add. We show that, if the inventorydecisions are exogenous or observable to the sales-person, then the contract form is similar to that withoutinventory considerations (an “extreme” contract that isconvex in sales wherein a bonus is rewarded only ifthe highest possible sales outcome is achieved), butif inventory decisions are unobservable to the sales-person, then the contract is a smoother one (and maybe concave in sales). Furthermore, supply uncertaintyleads to the question of the timing of contracting and,for the class of contracts that we consider, we deriveconditions for contracting before or after supply un-certainty is resolved.The rest of the paper is organized as follows. In Sec-
tion 2, we describe our model. In Section 3, we ana-lyze the firm’s optimal salesforce contract in a pre-liminary case with an exogenous inventory-relatedaction. In Section 4, we consider the focal case ofdouble moral hazard with endogenous, unobserv-able, inventory-related action by the firm followed by arandom supply shock. In Section 5, we consider thefirm’s optimal timing of offering the incentive con-tract. In Section 6, we conclude with a discussion.
2. ModelWemodel a firm thatmanufactures/stocks and sells aproduct. The demand for the product is uncertain.The firm employs a salesperson to exert sales effort toincrease the demand. We assume that the demand,denoted by D, can be high (H), medium (M), or low(L),H>M> L> 0. The salesperson’s effort, denoted bye, can be high (eH) or low (eL) and influences the de-mand according to the following probabilities:
Pr(D ! ξ|e) ! pξ if e ! eHqξ if e ! eL
for ξ ∈ {H,M,L}.{
(1)
This states that, if the salesperson exerts high effort,the probabilities of demand being H,M, and L arepH , pM, and pL, respectively, and if the salespersonexerts low effort, these probabilities are qH, qM, and qL,respectively. Consistent with the principal–agent liter-ature, we assume the MLRP such that
pHqH
>pMqM
>pLqL
> 0. (2)
The MLRP property essentially states that a higherdemand outcome is a more reliable indicator that thesalesperson exerted effort. Note that the MLRP di-rectly implies that pH > qH and pL < qL, but pM and qMcan have any relationship. We denote by ψ> 0 thesalesperson’s disutility of effort when the salespersonexerts high sales effort (i.e., e ! eH) and normalize thesalesperson’s disutility of effort to zero when thesalesperson exerts low sales effort (i.e., e ! eL).
We assume that the firm has limited inventory tosell. This inventory level, denoted by I, is uncertain,and can be high (H), medium (M), or low (L). The firmcan choose an inventory-related action, denoted by a,that influences the inventory according to the fol-lowing probabilities:
Pr(I ! ξ|a) ! rξ if a ! aHsξ if a ! aL
for ξ ∈ {H,M,L}.{
(3)
When a ! aH, the firm takes a highly effective action toensure ample inventory. Such an action may entail,for example, activities preventing inventory shrink-age, damage, and spoilage. When a ! aL, the firmtakes a less effective inventory-related action. Weassume that the inventory-related action is costless;this assumption is noncritical and helps us to un-derstand the impact of inventory uncertainty in aclearer and simpler manner. Similar to the demandside, we assume MLRP for the supply side such that4
rHsH
>rMsM
>rLsL
> 0, (4)
which implies rH > sH and rL < sL.
We assume that both the firm and the salespersonare risk neutral. Unlike the firm, however, the sales-person has limited liability, implying that the sales-person must be protected from downside risk. Spe-cifically,we normalize the salesperson’s limited liabilityto zero, that is, the salesperson’s salary must be non-negative under any outcome of demand. Limited li-ability is a widely observed feature of salesforcecontracts in the industry, and this assumption isa standard one in the literature (cf. Laffont andMartimort 2001; examples in the salesforce litera-ture include Sappington 1983; Park 1995; Kim 1997;Oyer 2000; and Dai and Jerath 2013, 2016). In oursetting, the limited liability assumption implies thatthe firm provides a nonnegative fixed wage to thesalesperson, which is aligned with industry practice.This directly implies that the firm cannot use a profit-sharing mechanism to achieve the first-best outcome.We also normalize the salesperson’s reservation util-ity to zero without loss of generality.The firm’s revenue comes from matching supply
with demand such that the actual selling quantity ismin{D, I}. Each unit of sales generates a revenue ofρ> 0. We assume that this per-unit price is exogenousand the salesperson does not adjust this price, anassumption that is uniformly made in the salesforcecompensation literature and has real-world support(Chung et al. 2014). We assume that the effort cost islow relative to the unit revenue such that it isworthwhile for the firm to induce a high sales effortexcept in the case in which the firm contracts with thesalesperson after the inventory is realized to be low.We also assume that inventory is costless (note thatwe have already assumed that the inventory-relatedaction is costless). These assumptions of no inventory-side costs, though nonstandard, allow us to focussharply on the effects of the uncertainty in inventoryrather than on the costs related to inventory. In-cluding the inventory-side costs will not lead to anyqualitative change in the insights that our analysisprovides.Throughout the paper, we consider the setting in
which the firm cannot verify the portion of realizeddemand D that is in excess of the stocked inventorylevel. For example, when the demand is H yet theinventory level is M, the realized sales areM, and thefirm only knows the demand is no less than M as itcannot observe the actual demand. This phenomenonis referred to as demand censoring. As discussed inthe introduction, a large number of papers in theliterature document this important phenomenon andstudy its implications (Braden and Freimer 1991;Anupindi et al. 1998; Downs et al. 2001; Ding et al.2002; Chen and Plambeck 2008; Lu et al. 2008; BesbesandMuharremoglu 2012; Conlon andMortimer 2013;Dai and Jerath 2013, 2016; Rudi and Drake 2014;
Chen et al. 2017). Because of demand censoring, thefirmhas to use the sales quantity rather than the actualdemand as the basis for determining the salesforcecompensation plan.
We summarize the notation that we use in Table 1.The timeline of the game, as illustrated in Figure 1,
is as follows. First, the firm offers the salesperson atake-it-or-leave-it compensation contract, which thesalesperson accepts or rejects. Second, the firm takesan inventory-related action to influence inventorydistribution. Third, if the salesperson accepts the con-tract, the salesperson exerts effort to boost demand.Fourth, the inventory is realized asH,M, or L. Fifth, thedemand is realized as H,M, or L, and sales are de-termined as the minimum of demand and inventory.Note that, in this formulation, the fourth and fifthstages can be merged into one stage.
3. Benchmark: Exogenous InventoryIn this section, we derive initial insights related to thesalesperson’s compensation contract under random
yield. For this purpose, we “switch off” the part of themodel in which the firm has an option to choose aninventory-related action. Instead, we fix the firm’sinventory-relation action at a ! aH such that inven-tory is random but its distribution is exogenous, whichimplies that, in this model, moral hazard exists onlyon the side of the salesperson. The insights obtainedin this section enable us to better understand theforces at play in the main model with double moralhazard in the next section.We note that the setup andthe results in this section are, nevertheless, inter-esting as well as novel to the literature for the fol-lowing reasons. First, this section allows for uncer-tainty in both supply and demand; as discussed earlier,previous literature on salesperson compensationeither assumes unlimited supply or limited but de-terministic supply as in Dai and Jerath (2013, 2016).Second, we show that, if MLRP holds on the de-mand side and supply is uncertain, then a propertysimilar to MLRP holds for realized sales (Lemma 1).Under this property, counterintuitively, it is optimalfor the firm to provide a bonus for only the highestsales outcome even when low realized sales are dueto low realized supply, which the salesperson cannotinfluence. Third, despite the presence of two sourcesof uncertainty (demand and supply), we show thefirm incurs the same expected cost of compensat-ing the salesperson as in the case with only demanduncertainty.There are three possible sales outcomes, that is, H,
M, and L. After the demandD and the inventory level Iare realized, the actual sales, denoted by the randomvariable Y, are equal to min{D, I}. Suppose that thesalesperson exerts an effort level of eH . Because we fixthe inventory-related action at aH , the probability ofeach possible sales outcome is
Pr{Y ! ξ|e ! eH} !rHpH if ξ ! HrHpM + rMpM + rMpH if ξ ! MrL + pL − rLpL if ξ ! L.
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
Likewise, when the salesperson exerts an effort levelof eL, the probability of each possible sales outcomecan be represented as
Pr{Y ! ξ|e ! eL} !rHqH if ξ ! HrHqM + rMqM + rMqH if ξ ! MrL + qL − rLqL if ξ ! L.
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
Define pYξ ! Pr{Y ! ξ|e ! eH}, ξ ∈ {H,M, L}; that is,pYξ is the probability that sales is equal to ξ under higheffort given the probability distribution of inven-tory realization. Similarly, define qYξ ! Pr{Y ! ξ|e ! eL},ξ ∈ {H,M,L}. We can think of these as the parametersof the sales distributions with high and low effort
Table 1. Notation
D Demand, which is subject to uncertainty and can beH (high), M (medium), or L (low)
I Inventory, which is subject to uncertainty and can beH (high), M (medium), or L (low)
e The salesperson’s effort, which can be high (eH) or low (eL)ψ The salesperson’s disutility from exerting a high effort level
(i.e., e ! eH)ψ The salesperson’s disutility from exerting a high effort level
(i.e., e ! eH) if the effort is exerted after inventory isrealized
pξ The probability that demand is ξ ∈ {H,M,L} when thesalesperson exerts high effort (i.e., e ! eH)
qξ The probability that demand is ξ ∈ {H,M,L} when thesalesperson exerts low effort (i.e., e ! eL)
a The firm’s inventory-related action, which can be eitherhighly effective (aH) or less effective (aL)
rξ The probability that inventory is ξ ∈ {H,M,L} when thefirm takes a highly effective inventory-related action(i.e., a ! aH)
sξ The probability that inventory is ξ ∈ {H,M,L} when thefirm takes a less effective inventory-related action(i.e., a ! aL)
ρ Unit revenueY Sales quantity, which is the minimum of demand and
inventory (i.e., Y ! min{D, I})pYξ The probability that sales quantity is equal to ξ ∈ {H,M, L}
under a high sales effort (i.e., e ! eH) and a highlyeffective inventory-related action (i.e., a ! aH)
qYξ The probability that sales quantity is equal to ξ ∈ {H,M, L}under a low sales effort (i.e., e ! eL) and a highly effectiveinventory-related action (i.e., a ! aH)
∆S The firm’s loss in its expected sales quantity under a highsales effort (i.e., e ! eH) when it switches from a highlyeffective inventory-related action (a ! aH) to a lesseffective one (i.e., a ! aL)
Bξ The firm’s bonus for the salesperson when the salesoutcome Y ! ξ for ξ ∈ {H,M,L}
given the parameters of the demand distributionswith high and low effort and the parameters of theinventory distribution. In the lemma, we establish animportant property.
Lemma 1. If pHqH
> pMqM
> pLqL
holds, then pYHqYH
> max pYLqYL,pYMqYM
{ }
holds.
In this scenario, the compensation contract of thesalesperson should be specified for every possiblerealization of sales, accounting for every possiblecombination of demand and inventory realizationthat can lead to that particular realization of sales.However, the result of Lemma 1 simplifies the anal-ysis of the scenario under consideration. The fol-lowing proposition shows that the optimal salesforcecompensation contract actually takes a very simpleform. (We assume that the unit revenue is highenough such that it is always worthwhile to inducethe salesperson to exert a high effort.)
Proposition 1. The optimal compensation plan is to pay thesalesperson a bonus of ψ
rH(pH−qH) if the sales are H units (i.e., ifI ! D ! H) and zero otherwise. The firm’s expected paymentfor motivating a high salesforce effort is ψ
1−qH/pH.This proposition states that, in the case of con-
tracting before inventory realization, without knowl-edge of the actual inventory outcome, it is optimal forthe firm to use an extreme, convex reward structure.Under this structure, the firm rewards the salespersonwith a bonus only when the highest possible sales levelis achieved and nothing for sales lower than this level(even if low saleswere due to low yield realization overwhich the salesperson has no control).5 The intuition ofthe result is that, because of demand censoring, thefirm cannot observe the true demand outcome andhas to determine the salesperson’s effort level byobserving the sales outcome. The distribution of thesales outcome is endogenous with the salesperson’seffort functioning as a key parameter. Given threepossible sales outcomes (H, M, and L), the firm seeksthe outcome that is most indicative of the fact thatthe salesperson has exerted a high effort. Mathemati-cally, this problem corresponds to finding the outcomewith the maximum likelihood ratio which, accordingto Lemma 1, is H.
Note that the only assumption needed for Propo-sition 1 to hold is MLRP in terms of demand distri-bution, which is a standard assumption in the contract-theory literature; no additional assumptions about
yield uncertainty are needed. Furthermore, Propo-sition 1 shows that, in the case of contracting beforeinventory realization, the amount of the bonus, givenby ψ/[rH(pH − qH)], depends on rH: the lower rH is, thehigher the bonus is. This makes intuitive sense; forinstance, if the probability that the salesperson ob-tains the bonus is low because the probability of highinventory realization is low, then the salespersonshould receive a larger bonus when the salespersonactually receives it. However, the firm’s expected pay-ment to the salesperson, ψ/(1 − qH/pH), is indepen-dent of rξ, ξ ∈ {H,M,L}. In other words, by con-tracting before inventory realization, the firm canfully address the risk resulting from yield uncertaintyand incurs the same expected cost of salesforce com-pensation as in the case without yield uncertainty.
4. Endogenous Inventory-Related Action:Double Moral Hazard
In the previous section, we analyze a benchmark inwhich the firm’s inventory-related action is fixed ata ! aH. In this section, we analyze our main model,allowing the firm to endogenously determine itsinventory-related action to influence the distributionof inventory. As discussed in Section 2, we assumethat the firm’s inventory-related action is costless,which helps us concisely and crisply characterize theimpact of inventory uncertainty; incorporating thecost of the inventory-related action will not qualita-tively alter our findings. The firm’s inventory-relatedaction may or may not be observable to the sales-person, and we analyze both of these cases. Herewe consider the case in which the firm’s inventory-related action is endogenous and not observed bythe agent (and, at the end of this section, we considerthe case in which the firm’s inventory-related actionis endogenous and observed by the agent).The analysis until now shows that a contract that
awards the salesperson a bonus only for the highestsales outcome is optimal. This is an extreme “bangbang” contract although, in reality, smoother con-tracts that reward a salesperson even for lower re-alized sales outcome are seen. In this section, we showthat, if the firm’s inventory-related action is not ob-servable to the agent, such a “smoother” or “lowerpowered” contract emerges (without adding anyadditional assumptions such as risk aversion, etc.). Inthis case, in the timeline in Figure 1, the second andthird stages can be merged into one stage.
Figure 1. Timeline of Contracting Under Supply Uncertainty
Note that this is the case of double moral hazard(see, e.g., Bhattacharyya and Lafontaine 1995), andthe previous analysis does not hold. To see this, notethat, under some conditions, once the salesperson hasaccepted the contract in Proposition 1, the firm mightbe tempted to deviate to an inventory-related actionof aL, under which the probability for the salespersonto earn a bonus decreases from rHpH to sHpH and thefirm’s expected payment decreases from ψ
1−qH/pH tosHrH· ψ1−qH/pH. (In an extreme case in which sH ! 0, that is,
a less effective inventory-related action leads to im-possibility of achieving a high inventory outcome, byswitching from aH to aL, the firm effectively voids thesalesperson’s likelihood of receiving a bonus.) Thus,by switching from aH to aL, the firm’s expected savingsfrom salesforce compensation is
1 − sHrH
( )· ψ1 − qH/pH
.
Switching from aH to aL, however, means a lowerexpected sales quantity and results in a loss of thefirm’s expected revenue. We define
as the absolute value of the firm’s expected loss ofsales by switching from aH to aL given that thesalesperson chooses e ! eH . The magnitude of thefirm’s expected revenue loss is, thus, ρ∆S.
For ease of exposition, we define the followingtwo constants:
τ1 ≜ (pH − qH)rM + (pM − qM)(rH + rM), (5)
τ2 ≜ (sM − rM)(pH + pM) − (rH − sH)pM. (6)
We focus on the case in which τ2 > 0 (i.e., sM − rM ispositive and sufficiently large, which is satisfiedunder the condition specified in endnote 4). We as-sume that B∗
H ≥ B∗M, which gives the parametric con-
dition ρ∆S ≥ 1 − sH+sMrH+rM
( )· ψ
1−qH+qMpH+pM
, which also immedi-
ately implies that the firm chooses an inventory-related action of aH . The following proposition pro-vides the optimal compensation contract and showsthat this is a smoother contract; that is, under someconditions, a bonus is awarded even for non-maximum sales outcomes. Note that the threshold1−sH
rH1−qH
pH
· ψ∆S in the proposition comes from the comparison
between 1 − sHrH
( )· ψ1−qH/pH and ρ∆S.
Proposition 2. If the salesperson cannot verify the firm’sinventory-related action, the optimal compensation contractis to pay the salesperson a bonus of B∗
i for a sales outcomei ∈ {H,M, L} such that
i. If ρ ≥ 1−sHrH
1−qHpH
· ψ∆S, B
∗H ! ψ
rH(pH−qH), and B∗M ! B∗
L ! 0.
ii. If ρ<1−sH
rH1−qH
pH
· ψ∆S,
B∗H ! τ1ρ∆S + τ2ψ
τ1(rH − sH)pH + τ2(pH − qH)rH, (7)
B∗M ! (rH − sH)pHψ − (pH − qH)rHρ∆S
τ1(rH − sH)pH + τ2(pH − qH)rH, and (8)
B∗L ! 0. (9)
In both cases, the firm chooses a supply-side action of aH.
Recall that, without double moral hazard, the op-timal contract is independent of the per-unit revenueρ. With double moral hazard, the optimal contractdepends on ρ. When ρ is large enough (the case ofProposition 2(i)), the contract stays the same as inProposition 1. However, when ρ is small enough (thecase of Proposition 2(ii)), the firm pays bonuses forsales equal to M and H even though the MLRP isassumed to hold (as one would expect, B∗
M ≤ B∗H).
Note that this is a fundamental change in the con-tractual form compared with the case of single-sidedmoral hazard, in which the firm paid a bonus just forsales equal toH. Specifically, the contract is smootherand not as high powered. A positive bonus for me-dium sales outcome is there to protect the salesper-son from receiving no compensation at all should thefirm, unobservable to the agent, choose an inventory-related action of aL. Under MLRP, any positive com-pensation for an outcome that is not the most desir-able causes a deadweight loss to the system, which isthe inefficiency introduced into the system becauseof the unobservability of the firm’s inventory-relatedaction.For the values of ρ in the case of Proposition 2(ii), as
ρ increases, the firm reduces B∗M, the bonus paid to the
salesperson for a medium sales outcome, but in-creases B∗
H, the bonus paid to the salesperson for ahigh sales outcome; at ρ ! 1−sH
rH1−qH
pH
· ψ∆S, the contract pa-
rameters {B∗H,B
∗M} are the same in cases (i) and (ii) of
the proposition. This is illustrated in Figure 2. In-tuitively, as ρ increases, everything else being thesame, the firm has a lower incentive to switch itsinventory-related action from aH to aL, making it lessnecessary to use B∗
Then we have the following corollary that is imme-diate from Proposition 2(ii).
Corollary 1. If ρ< ρ, the optimal compensation plan(B∗
L,B∗M,B
∗H) is concave in the sales outcome; otherwise, it is
convex.
The plots in Figure 3 illustrate the different types ofcontracts that are possible (in the plots, the dashedlines are for illustration only). In Figure 3(a), whereρ ≥ 1−sH
rH1−qH
pH
· ψ∆S, the contract takes an extreme convex form
with bonus awarded only when the realized sales areequal to H. In Figure 3(b), where ρ ≤ ρ<
1−sHrH
1−qHpH
· ψ∆S, the
contract takes a smoother yet convex form with arelatively small bonus awarded when the realizedsales are equal toM and a large bonus awarded whenthe realized sales are equal toH. In Figure 3(c), whereρ< ρ, the contract takes a smoother, concave formwith a relatively large bonus awarded when therealized sales are equal to M and a not much largerbonus awarded when the realized sales are equalto H.Corollary 1 states that the optimal compensation
plan may be concave when the per-unit revenue ρ issmall enough. The intuition behind this result is that,when ρ is sufficiently small, under the bang bangcontract as characterized in Proposition 1—a convexcompensation plan—the firm is tempted to choose aless effective inventory-related action to reduce itssalesforce compensation without significant impacton its expected revenue. Anticipating this possibility,the salesperson would not accept this bang bangcontract. Therefore, the firm must offer a contractwith a compensation level for a medium sales out-come. Furthermore, under certain conditions, thiscompensation level should be sufficiently close to that
Figure 2. (Color online) Effect of ρ on B∗H and B∗
M
Note. Parameters are {pH ,pM,pL}! {0.7,0.2,0.1}, {qH ,qM,qL}! {0.5,0.2,0.3}, {rH ,rM,rs}! {0.6,0.15,0.25}, {sH ,sM,sL}! {0.1,0.4,0.5}, {H,M,L}!{100,75,50}, and ψ!50.
Figure 3. (Color online) Different Contractual Forms Under Double Moral Hazard
Notes. In panels (a)–(c), the value of ρ is 12, 9, and 6, respectively. All the other parameters are the same as in Figure 2.
for a high sales outcome, leading to a concave com-pensation plan.
Basu et al. (1985) show that it is optimal for the firmto choose a concave compensation plan (i.e., de-creasing commission rates with outcome) only whenthe salesperson is risk averse (as characterized by,e.g., a constant risk-aversion utility function). Rubeland Prasad (2016) also show that, in their dynamicsetting, concavity in the compensation plan arisesfrom risk aversion. On the other hand, papers thatassume a risk-neutral salesperson with limited liabil-ity show that extreme, nonlinear, quota-bonus plansthat concentrate all variable payment at one outcomeof sales are optimal (Park 1995, Kim 1997, Oyer 2000).Recent work shows that adding inventory consider-ations to these settings maintains the extreme formof the optimal contract (Dai and Jerath 2013, 2016)though it has implications for the reward amount andthe inventory level.
In our case, the salesperson is risk neutral withlimited liability. Nevertheless, we show that a smoother(even concave) contract can be optimal when there isunobservability in the inventory-related actions. Inother words, we identify a force different from riskaversion—namely supply-related moral hazard—that can lead to a fundamentally different form of theincentive compensation plan.6 One can indeed expectsupply-side moral hazard to be operative in realitybecause of unobservability of the firm’s inventory-related action to a salesperson whose focus is onvisiting clients in the field to increase demand ratherthan closely monitoring the firm’s inventory-relatedaction that is often undertaken by a different silo inthe company.
We briefly discuss what happens if we assume thesalesperson to be risk averse (while maintaining theassumption of limited liability). Oyer (2000) showsthat, under single moral hazard, assuming limitedliability with risk neutrality leads to a quota-bonuscontract, that is, all marginal compensation is con-centrated at one point, and adding risk aversion tothat leads to marginal compensation being concen-trated on a range of critical points, that is, it leads to asmoother contract. If we add risk aversion to our set-ting, then there are two forces leading to the contractgetting smoother, namely double moral hazard andrisk aversion of the salesperson, but the key insightthat doublemoral hazard leads to a smoother contractis still operative.
We also note that our “discrete” model construc-tion with three states for demand and inventory leveland two effort levels, though certainly stylized, isnot a limiting model setup when compared with a“continuous” model construction with continuousdemand and effort. To see this, note that previous
research has already shown that, under limited lia-bility and single moral hazard, the result that a quota-bonus contract is optimal is obtained for both discrete(Laffont and Martimort 2001, Dai and Jerath 2013)and continuous models (Park 1995, Kim 1997, Oyer2000).7 Our main insight here is that, when there isdoublemoral hazard, that is, the agent cannot observea relevant action of the firm, the firm must make thiscontract smoother to assure the agent that the agentwill still obtain some marginal compensation forlower demand outcomes. It is straightforward to seethat this force should be operative equally in bothdiscrete and continuous models.
4.1. Observable Inventory-Related ActionFor completeness, we now briefly discuss the case inwhich the firm’s inventory-related action as aH oraL is observable to the salesperson. We obtain thefollowing proposition regarding the firm’s optimalinventory-related action.8
Proposition 3. If the firm’s inventory-related action isobservable, then the firm always sets this action as aH. Inaddition, the firm chooses the same compensation contractas that in Proposition 1.The reason for this result is that, compared with an
inventory-related action of aH, an inventory-relatedaction of aL implies the same expected payment to thesalesperson but a lower expected revenue (see prooffor details). Given that the inventory-related action isaH, the analysis in Section 3 applies, and the contractin Proposition 1 is optimal.
5. Timing of ContractingSo far, we have assumed that effort exertion by thesalesperson must happen before yield is realized,which implies that the firm must contract with thesalesforce before yield uncertainty is resolved. Insome situations, however, it may be possible to exertsales effort after yield is realized, and in these situ-ations, it may be beneficial for the firm to wait andcontract after yield certainty is resolved. In this sec-tion, we analyze the case of offering a compensationcontract after yield uncertainty is resolved and com-pare it with the case of offering a compensation con-tract before yield uncertainty is resolved (we callthese cases “late contracting” and “early contract-ing,” respectively). This provides insights related tothe optimal timing of contracting. (We assume thatadvanced contracts, such as those that include menusof contracts, cannot be used because of practical con-tracting frictions. We discuss this further at the end ofthis section.) We show that either early or late con-tracting may be optimal, depending on the interac-tions among yield uncertainty, demand censoring,
and moral hazard. The key trade-off is between waste-ful salesforce compensation expenses in early con-tracting and overcompensation because of demandcensoring in late contracting.
Before we proceed, we briefly note that there is aliterature on early versus late contracting in laborand product markets that studies matching betweenmarket participants (Roth and Xing 1994, Priest 2010).A critical difference between our work and the early-contracting literature is thatwe consider a contractingproblem in a single principal/single agent environ-ment, in which the issue of matching between marketparticipants is not relevant.
We assume the same supply-and-demand envi-ronment as specified in Section 2. In addition, we as-sume that the cost of effort exertion is higher if effortis exerted after inventory is realized. This assumptionis coming from the idea that, to effect the same changein the demand distribution, that is, change it from(qL, qM, qH) to (pL, pM, pH), in a shorter amount of time,the effort cost must be higher.9 We define the sales-person’s costs of effort as ψ and ψ, respectively, wheneffort exertion occurs before and after inventory re-alization, where ψ>ψ> 0.
If it is possible to delay effort exertion, then, just asthe firm has the flexibility of choosing the timing ofcontracting, in early contracting, the salesperson hasthe flexibility of choosing the timing of exerting effort.In other words, in early contracting, the salespersonmay choose to delay effort exertion until after theinventory is realized. To see the incentive behind this,recall from Section 3 that the optimal contract awardsa bonus to the salesperson only if sales equal H. Theimplication is that, if the inventory is M or L, theexpected sales outcome is always less than H andthe bonus is not awarded. The salesperson, by delay-ing effort exertion, may obtain inventory informa-tion and not necessarily choose a high effort level.Therefore, the firm would not offer this contract inequilibrium unless the firm expects the salespersonwill conformwith the firm’s contracting choice by notdelaying effort exertion. The following lemma spec-ifies the condition for this to happen.
Lemma 2. In the case of contracting before inventory re-alization, the salesperson chooses to exert effort before in-ventory realization if ψ ≥ ψ/rH.
Lemma 2 indicates that, if ψ ≥ ψ/rH, early con-tracting is viable because the salesperson is better off
exerting effort before the inventory is realized. In therest of this section, we focus on the interesting casein which ψ ≥ ψ/rH .
5.1. Contracting After Inventory RealizationIn this section, we consider the case of late contractingin which the firm contracts with the salesperson afterthe inventory is realized. The timeline of the game isas follows. First, the firm and the salesperson observethe realized inventory level as H,M, or L. Second, thefirm offers the salesperson a take-it-or-leave-it com-pensation contract that the salesperson accepts orrejects. Third, the salesperson determines the sales-person’s optimal effort level based on both thecompensation plan and the inventory level. Fourth,the demand is realized as H,M, or L, and sales aredetermined as the minimum of demand and in-ventory. Figure 4 illustrates the timeline.When inventory is realized, the firm observes this
realized level. There is the possibility that the firmstrategically does not disclose this inventory level tothe salesperson. (We assume that the firm does not lieabout the inventory level; that is, if it reports an in-ventory level to the agent, it reports truthfully. This isbecause, for any inventory level that the firm mis-reports, there is always a positive probability that thefirm’s lie is identified, in which case it can be taken tocourt. However, the firm can choose to not disclosethe realized inventory level.) We note that the firm, infact, always truthfully discloses the realized inventorylevel. To see this, consider that the realized inventorylevel is H. Then the firm discloses this information sothat the agent puts in the agent’s best effort. Next,assume that the realized inventory level is M. If thefirm does not disclose this, then the agent knows theinventory level must not be H (from the precedingargument); that is, it is either M or lower (i.e., L).Clearly, this only implies that the agent has reducedincentive to work hard than if the agent knows thatthe inventory level is M; therefore, once again, thefirm discloses the inventory level. Next, assume thatthe realized inventory level is L. In this case, it doesnot matter to the firm whether it discloses or not asminimum demand is always L; therefore, the firmagain discloses the inventory level (and, if it does notdisclose, then the agent can infer that it must not beHor M; that is, it must be L). In other words, the firmalways truthfully discloses the inventory level, andwe simply assume that,10 in the first stage of the game,
Figure 4. Timeline for Contracting After Inventory Realization
both the firm and the salesperson observe the realizedinventory level.11
Under late contracting, the optimal compensationplan for the salesperson is different based on therealized inventory level. The analysis of this case ison the lines of the analysis in Dai and Jerath (2013), sowe only provide a brief outline.
Case (i). I ! H: Because of MLRP, the salesperson ispaid a bonus only if the sales are H units, and thisbonus is equal to
ψpH − qH
.
The firm’s expected payment in this case is
ψ
1 − qHpH
.
Case (ii). I ! M: In this case, the maximum salesoutcome is capped by the inventory level, leading todemand censoring. Because of the MLRP assumption,the firm chooses to reward the salesperson when thesales are M units. Under the optimal contract, thesalesperson receives a positive bonus if the observedsales are M units and zero otherwise. The optimalbonus is
ψpH + pM − (qH + qM)
.
The firm’s expected payment in this case is
ψ
1 − qH+qMpH+pM
,
which, by MLRP, is higher than ψ/(1 − qH/pH) (thefirm’s expected payment in case (i)). Thus, because ofdemand censoring, the firm has to provide a higherexpected payment despite obtaining a lower expectedsales amount (because E[min{D,M}]<E[min{D,H}]).This scenario captures the drawback of contractingafter inventory is realized: when the inventory ismedium, both the firm and the salesperson un-derstand that the sales outcome can never be high.Thus, the firm has to provide a bonus for a mediumsales outcome which, according to Lemma 1, is notthe outcome most indicative of the salesperson’seffort choice, leading to overcompensation of thesalesforce.
Case (iii). I ! L: Because the inventory level is low,the sales would always be L units regardless of thedemand. The firm, thus, chooses not to induce anysalesforce effort, which is equivalent to offering asalesforce contract with a bonus of zero. Indeed, thefirm can avoid unnecessary salesforce expenses in thislow-yield scenario.
The following lemma holds for the case of latecontracting.
Lemma 3. In the case in which the firm contracts with thesalesperson late, that is, after the inventory is realized, thefollowing holds:i. If I ! H, the salesperson is offered a contract that pays
a bonus only if the sales are H units, and this bonus is equalto ψ
pH−qH.
ii. If I ! M, the salesperson is offered a contract that paysa bonus only if the observed sales are M units, and thisbonus is equal to ψ
pH+pM−(qH+qM).iii. If I ! L, the salesperson is not offered a contract of
employment.The firm’s expected payment for the salesperson is
rH · ψ
1 − qHpH
+ rM · ψ
1 − qH+qMpH+pM
. (10)
This analysis reveals that the key benefit of con-tracting after observing the inventory position is thatthe firm may avoid wasteful salesforce expenseswhen the inventory outcome is low. However, in thecase in which the inventory outcome is medium,because of demand censoring, the firm has to pay apremium to induce a high salesforce effort.
5.2. Optimal Timing of ContractingWe now determine the optimal timing of contractingfor the firm. Note that the firm has the same expectedrevenue in both early and late contracting (because,in both scenarios, in equilibrium, effort is exertedwhenever effort exertion can increase sales). Therefore,it is sufficient to compare the firm’s expected sales-force payment under the two scenarios (given inProposition 2 for early contracting and in Equation (10)for late contracting). The following proposition pres-ents the result for the case in which the per-unit rev-
enue is sufficiently high (i.e., ρ ≥ 1−sHrH
1−qHpH
· ψ∆S):
Proposition 4. In the case of ρ ≥ 1−sHrH
1−qHpH
· ψ∆S, if
1 − rHrM
≤ ψψ·
1 − qHpH
1 − qH+qMpH+pM
, (11)
then it is optimal for the firm to contract with the sales-person before the inventory is realized (and the contract isas per Proposition 2(i)). Otherwise, it is optimal for the firmto contract with the salesperson after the inventory isrealized (and the contract is as per Lemma 3).Note that the left-hand side of (11) has parameters
related to yield uncertainty, and the right-hand sideof (11) has parameters related to demand uncertaintyand effort cost. Intuitively, the firm’s optimal contract
timing depends on the trade-off between yield un-certainty and demand censoring. On the one hand, ahigh likelihood of the low inventory outcome makesit more appealing for the firm to observe the realizedinventory level first before committing to the con-tracting decision with the salesperson. On the otherhand, early commitment to the salesforce compen-sation plan helps to mitigate the effect of demand cen-soring because, without knowing the realized in-ventory level ex ante, the firm can hedge and specify apositive bonus awarded to the salesperson onlywhenthe sales are high. This is essentially transferring theburden of inventory uncertainty to the salesperson,and if the cost of delayed effort exertion is highenough, then the salesperson accepts this burden tobe able to exert effort early. By saying that the firmmay be better off by contracting before the inventoryis realized, we have the following interesting impli-cation: when choosing to contract before the in-ventory outcome is realized, the firm may benefitfrom its lack of inventory information because thebonus is paid to the salesperson only if the salesoutcome is H.
We now generate several insights regarding theeffect of supply-and-demand uncertainty on the firm’soptimal timing of contracting. First, consider the roleof the inventory distribution parameters. Specifi-cally, consider rH , which is the probability of the highinventory outcome. The following corollary showsthe impact of rH on the firm’s optimal contractingtiming.
Corollary 2. In the case of ρ ≥ 1−sHrH
1−qHpH
· ψ∆S, as rH increases,
i.Holding rM constant, the firm ismore inclined to contractwith the salesperson before the inventory is realized.
ii. Holding rL constant, the firm is more inclined tocontract with the salesperson after the inventory is realized.
As the probability of a high inventory outcome in-creases, the probability of yield loss (i.e., the realizedinventory is less than H) correspondingly decreases.As a result, inventory becomes a less restrictive factor,and one might intuit that the firm would be moreinclined to contract with the salesperson before ob-serving the realized inventory level, which is whatCorollary 2(i) states. However, Corollary 2(ii) statesthat the opposite could be true under certain condi-tions. To understand this, note that holding rL con-stant, as rH increases, rM decreases, reducing the is-sues that arise from demand censoring. Therefore,the advantages of late contracting dominate. Morespecifically, holding rL constant and decreasing rMlead to an increased ratio of rL over rM. Note that theleft-hand side of (11) can be rewritten as 1 + rL/rM,which captures the effect of yield uncertainty, whereasthe right-hand side of (11) captures the effect of
demand censoring. The first effect occurs because ofthe possibility that the supply can fall at the lowerbound of the support of the demand, whereas thesecond effect occurs because of the possibility thatthe supply is inadequate for fulfilling all the de-mand. The tension between these two effects drivesthe firm’s optimal timing of contracting. As rL/rMincreases, in the case of yield loss, the firm is morelikely to face a low inventory scenario than a mediuminventory scenario. Recall from our previous analysisthat contracting after inventory realization helps thefirm avoid unnecessary salesforce efforts when facedwith a low inventory outcome but might introducethe effect of demand censoring when faced with amedium inventory outcome. Therefore, as the prob-ability of low inventory increases relative to theprobability of medium inventory, the firm is moreinclined to contract after observing the inventoryoutcome.Second, we can consider the role of the demand
distribution parameters.We have the following lemmaregarding the monotonicity of the right-hand sideof (11).
Lemma 4. In the case of ρ ≥ 1−sHrH
1−qHpH
· ψ∆S, if pM ≤ qM or
pM > qM and pHqH
≥ pMqM
+pMqM
· pMqM
− 1( )
1 + qMqH
( )√, the right-
hand side of (11) decreases in pH.
This lemma immediately gives the followingcorollary.
Corollary 3. In the case of ρ ≥ 1−sHrH
1−qHpH
· ψ∆S, if pH is higher, that
is, the salesperson’s effort is more effective, the firm is morelikely to contract with the salesperson after observing therealized inventory level.
If the salesperson’s promoting effort is more ef-fective, all else equal, the firm has a better indicatorof effort exertion of the salesperson, reducing theissues resulting from demand censoring. Therefore,the advantages of late contracting dominate. In otherwords, for a higher pH, the agency cost would belower, and the firm would share a smaller rent withthe salesperson to motivate a high effort level. Thiswould make it more desirable to contract before in-ventory realization. On the other hand, given thesame yield uncertainty, if pH is higher, although thephenomenon of demand censoring becomes moresalient (because, given a medium inventory level, thedemand is more likely to exceed the inventory level,leading to unobserved demand), in the case of de-mand censoring, the firm can expect to share a lowerrent to induce salesforce effort as shown in the sec-ond part of (10) (rM · ψ/(1 − (qH + qM)/(pH + pM))). Thismakes it more desirable to contract before inventoryrealization. Corollary 3 states that the first effect may
dominate the second one; that is, the reduction in thecost associated with demand censoring may not be ashigh as the reduction in agency costs without ac-counting for the effect of inventory.
Our results in this section suggest that the firm mayprefer late contracting to early contracting under cer-tain conditions. The reason is that, under late con-tracting, there are scenarios in which the salespersondoes not exert effort; by comparison, under earlycontracting, the salesperson always exerts effort.This is a novel and nonobvious trade-off betweenyield uncertainty and demand censoring that ouranalysis reveals. Our comparison was between earlycontracting and late contracting, and we do notconsider more elaborate contracts in the form ofmenus or hybrids of early and late contracting be-cause, in practice, such contracts may be difficult tospecify and execute. That said, we anticipate ourresult that late contracting may be sometimes pre-ferred to hold qualitatively; whenever early con-tracting turns out to be not optimal, in the optimalcontract, there must be some contracting elementsthat allow the agent not to exert effort contingenton inventory information.
In contrast to Proposition 4, the next propositionprovides the optimal timing of contracting for the case
in which the per-unit revenue is low (i.e., ρ<1−sH
rH1−qH
pH
· ψ∆S):
Proposition 5. In the case of ρ<1−sH
rH1−qH
pH
· ψ∆S, if
pHrHB∗H + (pMrH + pHrM + pMrM)B∗
M
≤ rH · ψ
1 − qHpH
+ rM · ψ
1 − qH+qMpH+pM
. (12)
then it is optimal for the firm to contract with the sales-person before the inventory is realized (and the contract isas per Proposition 2(ii)). Otherwise, it is optimal for thefirm to contract with the salesperson after the inventory isrealized (and the contract is as per Lemma 3).
When the per-unit revenue is sufficiently low, thefirm has to offer a positive bonus for an intermedi-ate sales quantity (i.e., B∗
M > 0). As a result, the firmhas to incur a higher expected salesforce paymentthan the case in which the per-unit revenue is suffi-ciently high. Thus, compared with the case with ahigh per-unit revenue, the firm is less likely to con-tract with the salesperson before the inventory isrealized.
One notable difference between Propositions 4and 5 is that, in the former, the condition for earlycontracting to be optimal does not depend on the per-unit revenue, whereas, in the latter, it does (becauseboth B∗
H and B∗M depend on ρ). The following corollary
follows from Proposition 5:
Corollary 4. In the case of ρ<1−sH
rH1−qH
pH
· ψ∆S, as ρ decreases, the
firm is less likely to contract with the salesperson before theinventory is realized.
The intuition behind Corollary 4 is that, in the caseof a low per-unit revenue, as the per-unit revenuedecreases, the firm faces the pressure to choose a lessefficient contract to mitigate double moral hazard.Hence, early contracting becomes a less desirableoption.
6. Conclusions and DiscussionWe study salesforce compensation incentives in asetting in which both demand and supply are sto-chastic, and realized supply may be lower than re-alized demand (and unmet demand is not observ-able). Our research is the first to study salesforcecompensation under supply uncertainty, which is animportant real-world issue in many situations. Undermoral hazard (i.e., when the salesperson’s effort isunobservable to the firm), we characterize the opti-mal contract and show that it has an extreme convexform in which a bonus is provided only for achiev-ing the highest sales outcome even if low realizedsales are due to low realized supply over which thesalesperson has no influence (this result is driven byLemma 1, which we consider as one of our key re-sults). However, when the inventory decision is en-dogenous but unobservable by the salesperson, thatis, there is supply-relatedmoral hazard, doublemoralhazard arises. We characterize the optimal contractand show that it may be smoother as it may involvepositive compensation for intermediate sales out-comes; in fact, under certain conditions, the contract isconcave in sales (Corollary 1). This is an importantfinding because, although previous literature arguesthat it is optimal to provide a risk-neutral agent an ex-treme convex contract, this result shows that addingsupply-related moral hazard to the mix can lead tothe optimal contract being a concave contract (whichZoltners et al. 2006 report is used widely by firms).Our findings, therefore, shed light on how opera-tional considerations may drive the design of incen-tive contracts, especially when the salesperson does nothave complete transparency regarding the inventory-related actions of the firm. In addition, we studywhether the firm should contract with the salespersonbefore or after inventory realization (assuming it hasthe latter option at all) by characterizing the noveltrade-off between avoiding unnecessary marketingexpenses because of supply uncertainty (in the case oflow yield) and overcompensation because of demandcensoring (in the case of intermediate yield).Our results are of relevance to managers in a num-
ber of ways. First, we show that, if there is no moralhazard on the firm’s side, that is, the agent has full
transparency into the firm’s inventory-related actions,then, even if yield is random and even though thesalesperson’s effort has no impact on inventory yield,the firm can use performance-based compensationcontracts based on realized sales that are convex inshape. Second, we show that, if there is moral hazardon the firm’s side as well, that is, the agent does nothave full transparency into the firm’s inventory-related actions, then the firm has to make the con-tract smoother and, under certain conditions, mayeven have to make this contract concave; we specifythese conditions. Third, we show that, under randomyield, it may sometimes be beneficial for the firm towait until yield uncertainty is resolved to decide thecompensation contract; however, it may not always bebeneficial to do so, and we specify these conditions.
Our research can be extended in a number of di-rections. For instance, in the current model, we haveassumed that the salesperson obtains inventory in-formation exogenously. However, if the firm en-dogenously reveals inventory information, then it candetermine when and how much information to re-lease, and its revelation strategy itself may signal theinventory level. We have also assumed that the priceof the product is exogenous. One alternative sce-nario (that is beyond the scope of this study) is tomakethis price endogenous and empower the salespersonwith some ability to determine this price (Simester andZhang 2014). In such an effort, it would be importantto carefully determine the timing of the pricing de-cision as before or after yield uncertainty is resolved.Another interesting direction would be to study thecase of asymmetric information, in which the firmor the salesperson may have better information thanthe other party on inventory or demand. A streamof literature on the interfaces of operations andmarketing (e.g., Taylor 2006, Iyer et al. 2007, andBiyalogorsky and Koenigsberg 2010) has examinedthe cases in which the downstream player in a supplychain may own the inventory; although it might notbe reasonable in our case to assume that the salesper-son rather than the firm can own the inventory, it ispossible that the firm and the salesperson have dif-ferent information about the inventory yield distri-bution. It would also be interesting to analyze how thefirmmay want to alleviate some of these problems; forexample, should the firm invest in reducing the vari-ability of supply to reduce the firm’s moral hazardissue or should it invest in improving the observ-ability of lost demand? Finally, in many industries,firms may purchase insurance products (e.g., cropinsurance in the agricultural setting) to hedge yield-related risks. An interesting research problem wouldbe to study how yield-related insurance optionsimpact promotional effort and salesforce compensation.
Along these lines, future research may incorporate fi-nancial hedging strategies (e.g., futures and optionscontracts) that are commonly employed in the semi-conductor and related industries.
AcknowledgmentsThe authors are grateful to the senior editor, the associate editor,and two anonymous reviewers for the constructive re-view process. The authors thank Charles Angelucci, DilipMookherjee, and Raphael Thomadsen; as well as seminar par-ticipants at City University of Hong Kong, City University ofNew York, Columbia University, Duke University, Hong KongUniversity of Science and Technology, Washington Universityin St. Louis, University of Chicago, University of SouthernCalifornia, Yale University, the 37th ISMS Marketing ScienceConference (2015), INFORMS International Conference (2016),MSOM Conference (2016), and Zero-Decade Marketing The-ory Symposium (2019) for their helpful comments.
Appendix. Proofs
Proof of Lemma 1. We first prove that pYMqYM
<pYHqYH. ByMLRP, we
havepHqH
>pMqM
>pLqL
. (A.1)
Hence, we have
pYMqYM
! rHpM + rMpM + rMpHrHqM + rMqM + rMqH
! (rH + rM) · pM + rMpH(rH + rM) · qM + rMqH
!pMqM
+ rMrH+rM · pHqM
1 + rMrH+rM · qHqM
<
pHqH
+ rMrH+rM · pHqM
1 + rMrH+rM · qHqM
(by MLRP)
!pHqH
· 1 + rMrH+rM · qHqM
( )
1 + rMrH+rM · qHqM
! pHqH
! rHpHrHqH
! pYHqYH
.
Next, we prove that pYLqYL
<pYHqYH. Note from (A.1) that pH > qH
and pL < qL,12 which gives pL/qL < 1< pH/qH. Therefore,
pYLqYL
! rL + (1 − rL)pLrL + (1 − rL)qL
< 1 (by pL < qL)<pHqH
(by pH/pL > 1)
! rHpHrHqH
! pYHqYH
.
Therefore, we have max pYLqYL,pYMqYM
{ }<
pYHqYH. Q.E.D.
Proof of Proposition 1. Suppose that the firm pays thesalesperson a bonus, denoted by BH , when the sales outcome
is high and zero otherwise. The firm’s problem can be writtenas the following program:
maxBH
ρ · rHpHH + (rHpM + rMpM + rMpH)M[
+ (rL + pL − rLpL)L] − rHpHBH
(A.2)
s.t. rHpHBH − ψ ≥ rHqHBH (IC)rHpHBH − ψ ≥ 0. (IR)
Clearly, the individual rationality (IR) constraint follows fromthe incentive compatibility (IC) constraint and is, thus, redundant.By solving this problem, we have B∗
H ! ψrH(pH−qH). The firm’s ex-
pected payment to the salesperson is, thus, rHpHB∗H ! ψ
1−qH/pH.Next, we show that the firm only provides the salesperson
a bonus when the sales outcome isH. Suppose the firm uses adifferent compensation scheme in which the salesperson ispaid a bonus Bi when the sales outcome i ∈ {H,M, L}, and atleast one of {BM,BL} is positive. The firm’s problem can bewritten as the following program:
Again, the IR constraint follows from the IC constraint andis, thus, redundant. The IC constraint gives BH ≥ (ψ− ϵM − ϵL)/(rHpH − rHqH), where ϵM!BM[(rHpM+rMpM+rMpH)− (rHqM+rMqM+rMqH)] and ϵL ! BL[(rL + pL − rLpL)− (rL + qL− rLqL)].Therefore, the firm’s expected payment to the sales-person is
Therefore, the firm is better off by paying the salesperson apositive bonus only when the sales outcome is H. Q.E.D.
Proof of Proposition 2. When ρ ≥ 1−sHrH
1−qHpH
· ψ∆S, we have
ρ∆S ≥ ψ1 − qH
pH
− sHrH
· ψ1 − qH
pH
. (A.8)
Note that the left-hand side of (A.8) denotes the absolutevalue of the firm’s expected revenue loss by switching itsinventory-related action from aH to aL, whereas the right-hand side of (A.8) denotes the absolute value of the firm’sexpected saving from its compensation to the salespersonfor motivating an effort of eH. In other words, by choosing aless effective inventory-related action (aL), the firm’s reve-nue loss outweighs its savings from salesforce compensation.Thus, under the optimal salesforce compensation contract ascharacterized in Section 3, the firm does not have any in-centive to change its inventory-related action from aH to aL asdoing so would reduce its expected profit.
When ρ<1−sH
rH1−qH
pH
· ψ∆S, the optimal contract characterized in
Section 3 cannot sustain because, if the salesperson chooses ahigh effort level, the firm is better off choosing an inventory-related action of aL, which results in an expected reduction fromsalesforce force compensation, represented by ψ
1−qHpH
− sHrH· ψ1−qH
pH
,
that is greater than the expected revenue loss (∆S · ρ).Thus, the firmmust provide the salespersonwith a positive
bonus BM when the sales outcome is M. The firm’s objectivecan be formulated as
pHrHH + (pMrH + pHrM + pMrM)M[
+ (pL + rL − pLrL)L] · ρ − pHrHBH
− (pMrH + pHrM + pMrM)BM.
(A.9)
The bonus BM must be large enough such that the firm doesnot have any incentive to choose an inventory-related actionof aL (instead of aH):
By (A.12), we have ∆B(ρ)< 0 if ρ< ρ, and ∆B(ρ) ≥ 0 oth-erwise. Furthermore, we have from (A.13) that
ρ<(H − L)(rH − sH)pH(H − L)(pH − qH)rH
· ψ∆S
!1 − sH
rH
1 − qHpH
· ψ∆S
,
which completes the proof. Q.E.D.
Proof of Proposition 3. First, we show that an inventory-related action of aL is a dominated option. Note that, when thefirm chooses an inventory-related action of aL, because such anaction is observable to the salesperson, using an argumentsimilar to those in the proof of Proposition 1, in the optimalcontract, the firm pays the salesperson a bonus only forachieving sales equal toH, and the value of the bonus is ψ
sH (pH−qH).The firm’s expected payment to the salesperson is given by
sHpH · ψsH(pH − qH)
! ψ1 − qH
pH
,
which is equal to the firm’s expected payment to the sales-person when choosing an inventory-related action of aH. Inaddition, we have from ∆S> 0 that the firm’s expected rev-enue from an inventory-related action of aL is lower than thatfrom an inventory-related action of aH. Therefore, the firm isbetter off choosing an inventory-related action of aH insteadof aL.
Next, given that the inventory-related action is aH, theoptimal contract follows from Proposition 1. Q.E.D.
Proof of Lemma 2. Depending on the size of ρ, we have twocases:
i. ρ ≥ 1−sHrH
1−qHpH
· ψ∆S. Consider the case in which the firm offers a
contract before inventory realization, according to which abonus of B is provided when Y ! H. Under the contract, if thesalesperson chooses to exert effort before inventory re-alization, the salesperson’s expected utility is rHpHB − ψ. Ifthe salesperson waits until observing the realized inventory,then the salespersonwould only exert effort when the inventoryI ! H, incurring a cost of ψ>ψ. Thus, the salesperson’s expectedutility is rH(pHB − ψ). The condition for the salesperson to exerteffort before inventory realization is rHpHB − ψ ≥ rH(pHB − ψ),which is equivalent to ψ ≥ ψ/rH.
ii. ρ<1−sH
rH1−qH
pH
· ψ∆S. Note from Proposition 2(ii) that, in this case,
the salesperson may receive a positive bonus when the salesoutcome is M. Given a compensation contract withB∗H ,B
∗M > 0, by exerting effort before inventory realization, the
salesperson’s expected utility is
rHpHB∗H + (rHpM + rMpM + rMpH)B∗
M − ψ. (A.14)
If the salesperson waits until observing the realized in-ventory and the realized inventory isM, the salesperson mayor may not exert effort depending on the system parametervalues. We discuss both cases one by one.
a. If the salesperson is willing to exert effort even if therealized inventory is M, the salesperson’s expected utilityfrom waiting is
rH(pHB∗H + pMB∗
M − ψ) + rM[(pH + pM)B∗M − ψ]. (A.15)
By comparing (A.14) with (A.15), we find the condition forthe salesperson to exert effort before inventory realization isψ ≥ ψ/(rH + rM), which follows from ψ ≥ ψ/rH.
b. If the salesperson is willing to exert effort only if therealized inventory is H, the salesperson’s expected utilityfrom waiting is
rH(pHB∗H + pMB∗
M − ψ). (A.16)
By comparing (A.14) with (A.16), we identify the conditionfor the salesperson to exert effort before inventory realization:ψ> ψ
rH− rM(pH+pM)
rH· B∗
M, which, again, follows from ψ ≥ ψ/rH .Q.E.D.
Proof of Lemma 3. Similar to the argument in section 4 ofDai and Jerath (2013). Q.E.D.
Proof of Proposition 4. By comparing the firm’s expectedsalesforce compensation in the case of contracting after in-ventory is realized [see Equation (10)] against that in the caseof contracting before inventory is realized [see Proposition2(i)]. Q.E.D.
Proof of Corollary 2. Note that the left-hand side of (11),using rH + rM + rL ! 1, can be rewritten as 1 + rL/rM, whichincreases in rL and decreases in rM; the right-hand side of (11)is independent of ri, i ∈ {H,M, L}. The corollary thus followsfrom Proposition 4. Q.E.D.
Proof of Lemma 4. Let us write the right-hand side of (11) asa function of pH , that is, f (pH) ! ψ/ψ · (1 − qH/pH)/(1 − (qH + qM)/(pH + pM)). Its first-order derivative
f ′(pH) !ψψ·pMqH(pM−qM) − (pMq2H + p2HqM−2pHpMqH)
p2H(pH + pM − qH − qM)2
has a positive denominator. If pM ≤ qM, its numerator
Proof of Corollary 3. Follows from Lemma 4. Q.E.D.
Proof of Proposition 5. By comparing the firm’s expectedsalesforce compensation in the case of contracting after in-ventory is realized [see Equation (10)] against that inthe case of contracting before inventory is realized [seeProposition 2(ii)]. Q.E.D.
Proof of Corollary 4. It suffices to prove that the left-handside of (12), that is,
decreases in ρ.Note that the denominator of (A.18) is independent of ρ.
The first-order derivative of the numerator of (A.18) in termsof ρ is
[pHrHτ1 − (pMrH + pHrM + pMrM)(pH − qH)rH]∆S,which can be reorganized as
rHqHqM[rH + rM] · pMqM
− pHqH
( )∆S.
This quantity is negative because of MLRP. Hence, theproof is complete. Q.E.D.
Endnotes1There is a recent literature on robust contracts in which the principalevaluates possible contracts by their worst-case performance overunknown actions that the agent may take (Antic 2014, Carroll 2015,Carroll and Meng 2016, Yu and Kong 2017). Our work here is of adifferent flavor as it can be thought of as the traditional principal–agent problem of Holmstrom (1979) but with uncertain supply thatcan sometimes be less than demand.2 See https://bit.ly/2BIFWfB.3We assume for this extension that advanced contracts, such as thosethat include menus of contracts, cannot be used because of practicalcontracting frictions. Otherwise, a sufficiently complex early contractcan achieve any outcome that a late contract can achieve.4Note that MLRP for the supply side is neither necessary nor suf-ficient for our results to hold. We have assumed it as a parallel to theassumption of MLRP on the demand side, which makes it a palatableassumption. A sufficient condition for our analysis in Section 4 tohold is rH > sH and rM < sM − pM
pH+pM · (rH − sH) and rL < sL.5 It is well known that, in a situation without inventory consider-ations, the MLRP on the demand distribution implies that the firmonly needs to reward the salesperson when the highest demandoutcome is achieved (see, e.g., Laffont and Martimort 2001).6Recent work has shown that linear contracts can be optimal in asetting with risk neutrality with limited liability of the agent. Forinstance, Krakel and Schottner (2016) and Jerath and Long (2018)show this in a dynamic setting, and Antic (2014), Carroll (2015), andYu and Kong (2017) show this under a robust contracting paradigmwith uncertainty about the agent’s technology, the agent’s action set,and the agent’s effectiveness, respectively. In our case, the optimalcontractmay be linear (though a linear contract is not always optimal).7Note that, in addition to MLRP, the continuous formulation in Park(1995) and Kim (1997) needs the convexity of the distribution functioncondition to hold although Oyer (2000) needs that the participation
constraint of the agent should not be binding; in this sense, thecontinuous formulations may be considered more restrictive than thediscrete formulations. Furthermore, there are certain well-knowntechnical challenges associated with the continuous modelingframework (most notably, regarding the first-order approach typi-cally used under the assumptions of risk neutrality and limited lia-bility of the salesperson). As Laffont and Martimort (2001, pp.200–201) point out, “The first-order approach has been one of themostdebated issues in contract theory” because the validity of the approachhas not been well established, and “when the first-order approach isnot valid, using it can be very misleading.” As a result, “most of theapplied moral hazard literature” adopts a discrete formulation.8For conciseness of analysis, we consider a parametric space in whichthe firm prefers the salesperson to choose an effort of eH regardless ofwhether the firm takes an inventory-related action of aH or aL.9This is easy to generate using a convex cost function of within-dayeffort. Assume that the salesperson can effect the change from(qL, qM, qH) to (pL, pM, pH) by working for a total of D hours. Alsoassume that to work h hours in a day, the salesperson’s cost is h2, thatis, the marginal cost of effort of every additional hour worked in aday is higher. Lets say the salesperson has 10 days toworkD hours, sothe salesperson works D/10 hours per day at a total cost of10(D/10)2 ! D2/10. Now suppose the salesperson must work Dhours in five days, that is, the salesperson needs to work D/5 hoursper day at a total cost of 5(D/5)2 ! D2/5, which is higher.10We note that this holds even for a demand distribution with morethan three points of support.11A variation to the timeline in Figure 4 is that the inventory is re-alized after the sales contract is determined but before the salespersonexerts effort. In this case, the firm again has the choice of whether itshould disclose the inventory level to the salesperson (under therequirement that, if there is disclosure, it must be truthful). Followingthe same arguments as before, we can see that the firm discloses theinventory level.12We can show this by contradiction. Suppose pH < qH ; then (A.1)implies both pM < qM and pL < qL, which is impossible to hold becausepH + pM + pL ! qH + qM + qL ! 1. Likewise, suppose pL > qL; then(A.1) implies both pM > qM and pH > pH , which is impossible to holdbecause pH + pM + pL ! qH + qM + qL ! 1.
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