Does a Manufacturer Benefit from Selling to aBetter-Forecasting Retailer?
Terry A. TaylorHaas School of Business, University of California, Berkeley, Berkeley, California 94720, [email protected]
Wenqiang XiaoStern School of Business, New York University, New York, New York 10012, [email protected]
This paper considers a manufacturer selling to a newsvendor retailer that possesses superior demand-forecastinformation. We show that the manufacturer’s expected profit is convex in the retailer’s forecasting accuracy:
The manufacturer benefits from selling to a better-forecasting retailer if and only if the retailer is already agood forecaster. If the retailer has poor forecasting capabilities, then the manufacturer is hurt as the retailer’sforecasting capability improves. More generally, the manufacturer tends to be hurt (benefit) by improved retailerforecasting capabilities if the product economics are lucrative (poor). Finally, the optimal procurement contractis a quantity discount contract.
Key words : supply chain contracting; asymmetric information; forecastingHistory : Received November 7, 2008; accepted April 14, 2010, by Aleda Roth, operations and supply chain
management. Published online in Articles in Advance July 21, 2010.
1. IntroductionAlthough a retailer’s skill at accurately forecastingmarket demand most obviously and directly impactsthe retailer, retailer forecasting accuracy impacts theentire supply chain including the manufacturer thatsupplies the retailer. The accuracy of a retailer’sforecast impacts how much she orders and her in-stock performance, which in turn impact manufac-turer profitability (Cederlund et al. 2007).1
Because they are closer to the end customer,retailers often have better information about marketdemand than the manufacturer. However, the degreeof superiority in forecasting demand varies (1) acrossretailers and (2) over time. First, some retailers areknown to be better forecasters than others. For exam-ple, in the retail consumer electronics industry, CircuitCity has been plagued by weak forecasting capabil-ities and trails behind best-in-class retailer Best Buy(Feldman and Cramer 2004, Widlitz 2005). How doesa retailer’s effectiveness in forecasting influence her
1 As evidence that manufacturers care about retailer forecastingaccuracy, a number of manufacturers have embarked on initiativesthat have improving the accuracy of their retailers’ forecasts as acentral objective. For example, Fraser (2003) reports on a survey of120 companies on Collaborative Planning, Forecasting and Replen-ishment initiatives and puts “improvements in trading partner fore-casting accuracy” (p. 75) at the top of the list of benefits anticipatedby survey respondents. The benefits listed subsequently flow fromthis improved accuracy: reduced out-of-stocks, improved servicelevels and increased sales.
attractiveness to a manufacturer that is selecting aretail partner? When faced with a pool of prospec-tive retailers, ceteris paribus, should a manufacturerselect a retailer that has strong, weak, or intermediateforecasting capabilities?Not only do forecasting capabilities vary across
retailers, they also vary over time at a single retailer.For example, a manufacturer’s retail partner mayinvest in forecasting capabilities (e.g., by purchasingrelevant software) with the intention of improving itsforecasting accuracy. Alternately, the retailer may dis-invest in forecasting (e.g., by laying off or redeploy-ing forecasting staff), understanding that this actionwill degrade its ability to accurately forecast demand.What impact should the manufacturer anticipate thatsuch changes by its retail partner will have on themanufacturer’s own performance? Should a manufac-turer relish and encourage either improved or wors-ened retailer forecasting accuracy?It may seem natural that a manufacturer would
benefit by selling to a better-forecasting retailer inthat by doing so, production decisions can be madewith more accurate information, reducing the costof supply/demand mismatch. However, selling toa better-informed retailer puts the manufacturer ata strategic disadvantage relative to the retailer. Theretailer may be able to use her informational advan-tage to extract a larger portion of the system profitfrom the manufacturer. The impact of improvedretailer forecasting accuracy on manufacturer profit
depends on the trade-off between these two factors,and it is this trade-off that we explore in this paper.This paper considers a manufacturer selling
to a newsvendor retailer that possesses superiordemand-forecast information. We show that themanufacturer’s expected profit is convex in theretailer’s forecasting accuracy: The manufacturer ben-efits from selling to a better-forecasting retailer if andonly if the retailer is already a good forecaster. Ifthe retailer has poor forecasting capabilities, then themanufacturer is hurt as the retailer’s forecasting capa-bility improves. More generally, the manufacturertends to be hurt (benefit) by improved retailer fore-casting capabilities if the product economics are lucra-tive (poor). Further, the optimal procurement contractis a quantity discount contract.The remainder of this paper is organized as follows.
Section 2 reviews the relevant literature. Section 3describes the model. Sections 4 and 5 contain theanalysis for the integrated system and the decentral-ized system, respectively. Section 6 provides numer-ical results. Section 7 provides concluding remarks.The proofs are in the appendix.
2. Literature ReviewThere is substantial literature that studies supplychain settings in which firms have distinct demandinformation. This literature can be classified into twostreams. One stream considers the impact of thetruthful sharing of private demand information (e.g.,Cachon and Fisher 2000, Lee et al. 2000, Aviv 2001).Demand-information sharing sometimes is not diffi-cult to achieve, e.g., when demand information onlyconsists of historical sales data that are readily veri-fiable from the information system. In this case, theinteresting questions include how to use the sharedinformation to improve supply chain performanceand what factors are crucial in affecting the magni-tude of the improved performance. However, whendemand information also involves the firms’ sub-jective assessment or private knowledge that is notverifiable by a third party, the credibility of truth-ful information sharing is in doubt because a firmmay have incentive to misrepresent its information.(For example, in our setting, the privately informedretailer has incentive to persuade the manufacturerthat market demand is weak (regardless of the actualmarket condition) so as to convince the manufacturerto offer more generous terms (e.g., a lower purchaseprice).) Indeed, in practice, the scope for opportunis-tic behavior and lack of trust have proven to be sub-stantial obstacles to demand-forecasting collaborationefforts (Fliedner 2003).A second stream of literature focuses on how self-
interested firms behave and interact in the face of pri-vate information, and in particular on how contracts
mediate these interactions (Cachon and Lariviere2001, Arya and Mittendorf 2004, Özer and Wei 2006,Burnetas et al. 2007, Mishra et al. 2007, Ren et al.2010). A typical theme in this stream of work is toexplore how contracts should be designed and then toevaluate the performance of optimal contracts and/orsimple and commonly used contracts. See Cachon(2003) and Chen (2003) for reviews. Our work fitswithin this stream. However, the focus of our work isdistinct: we concentrate on the impact of the retailer’sforecasting accuracy on the firms’ performance.The paper most closely related to our work is
Miyaoka and Hausman (2008). Similar to our work,they study a supply chain where the upstream firm(supplier) sells to a downstream newsvendor (man-ufacturer) who has private demand-forecast infor-mation. Unlike our work, the upstream firm mustmake a capacity decision. Even so, we share thesame objective, which is to evaluate the impact ofthe downstream firm’s forecasting accuracy on thefirms’ performance. However, they restrict attentionto the single wholesale price contract, whereas westudy the issue under both the wholesale price con-tract and the optimal procurement contract, with theemphasis on the latter. Furthermore, our results com-plement theirs in that they obtain analytical resultsfor the two extreme cases of forecasting accuracy (thedownstream firm is either perfectly informed or com-pletely uninformed) and provide numerical resultsfor intermediate cases, whereas we obtain analyticalresults for the full spectrum, albeit with a simplermodel. In particular, we provide a more completecharacterization as to how the upstream firm’s perfor-mance changes in the downstream firm’s forecastingaccuracy (e.g., the convexity property). The impact ofa privately informed firm’s forecasting accuracy onsupply chain performance has also been discussed,mainly through numerics, in Özer and Wei (2006)and Taylor (2006). In a setting where a manufacturerand retailer share the same demand-forecast informa-tion, Iyer et al. (2007) observe that the manufacturermay be better off when forecasting accuracy is poorbecause this mitigates double marginalization.Our work is also related to the economics and
accounting literature that studies the optimal level ofinformation asymmetry. Lewis and Sappington (1991)and Rajan and Saouma (2006) consider a principal-agent model where the agent privately exerts effortthat influences the output. The agent has privateinformation about his cost of effort. The authorsexamine the impact of the accuracy of the agent’s pri-vate information on the principal’s utility. They estab-lish an “all-or-nothing” result: the principal prefers todeal with either a completely uninformed agent ora perfectly informed agent. Even though our supplychain setting with asymmetric demand information
is quite different, it is interesting that we obtain aroughly parallel result, namely, that a manufacturerfacing a pool of retailers prefers to deal with eitherthe best or the worst forecaster.
3. ModelA manufacturer (he) produces a product at unit cost cand sells to a newsvendor retailer (she), who thensells at a fixed retail price p to a market with ran-dom demand D in a single selling season. The marketdemand is normally distributed, i.e., D ∼ N��0��0�.The salvage value of unsold inventory is normalizedto zero.Prior to the selling season, the retailer observes a
demand forecast S = D+�, where � ∼ N�0��1� is inde-pendent of D. Note that S is an unbiased estimatorof D, with the estimation error being normally dis-tributed. It follows from the conjugate property ofnormal distribution that the posterior demand dis-tribution under the forecast S is also normal (seeWinkler 1981), i.e.,
D�S ∼ N�a2�0 + �1− a2�S� a�0�� (1)
wherea ≡ �1√
�20 + �2
1
denotes the fraction of the original demand uncer-tainty, as measured by the standard deviation, thatremains after the forecast is observed. We refer to a asthe retailer’s forecasting accuracy parameter. The lowerthe value of a, the more accurate the retailer’s fore-cast. In the limiting case where a = 0, the forecastperfectly reveals the exact demand. In the oppositelimiting case where a = 1, the forecast contains novaluable information about demand and the poste-rior distribution is identical to the prior. For exposi-tional simplicity, we exclude these two extreme casesand restrict attention to a ∈ �0�1�. In considering theimpact of changes in forecasting accuracy, we assumethat the distribution of the underlying demand D (i.e.,the parameters �0 and �0) is fixed and only the levelof noise in the retailer’s forecast (as captured in �1)varies.The retailer privately observes her own forecast.
However, the distributions and all other parame-ters are common knowledge of both the manu-facturer and the retailer.2 Thus, the manufacturer
2 Typically, a manufacturer has some awareness of its (currentor prospective) retailer’s demand-forecasting investments in com-puter systems, software, and staff. Further, the manufacturermay have an understanding of the retailer’s historic forecastingperformance (either directly, or reflected in the retailer’s history ofstocking too much or too little) and the retailer’s familiarity withthe manufacturer’s product (based, for example, on what products
knows that the retailer’s forecast is normally dis-tributed, i.e., S ∼ N��0�
√�20 + �2
1 � , or equivalently,S ∼ N��0��0/
√1− a2�. It is convenient to rewrite S =
�0 + ��0/√1− a2��, where � ∼ N�0�1�. Because there
is a one-to-one mapping between S and �, we alsorefer to � as the retailer’s forecast. Given the retailer’sforecast � = , the posterior distribution in (1) is
D��= ∼ N��0 +√1− a2�0�a�0� (2)
Note that the retailer’s forecasting accuracy param-eter a impacts both the mean and standard devia-tion of posterior demand given a forecast: The moreaccurate the forecast (smaller a), the larger the weightof forecast (
√1− a2�0) in determining the posterior
mean, and the smaller the posterior standard devia-tion (a�0). (Although, for compactness, we representthe retailer’s forecast as a scalar � = , the retailer’sforecast of demand should be thought of as a distri-bution (2) rather than as a point forecast.)We assume both the manufacturer and the retailer
are risk neutral, maximizing their own expected prof-its. Because the retailer possesses superior informationabout the demand, the uninformed manufacturerfaces a typical adverse selection problem in contract-ing with the informed retailer. In such a situation, aprocurement contract can be represented by a trans-fer payment schedule T �q��, which specifies the pay-ment the retailer makes to the manufacturer when theretailer orders q units and reports observing forecast .
The sequence of events is as follows: First, themanufacturer specifies a transfer payment scheduleT �q�� (without knowing the retailer’s forecast), andthe retailer (privately) observes the forecast �. Sec-ond, the retailer orders q units and reports a fore-cast ; the manufacturer fulfills the retailer’s orderand receives the payment T �q� � from the retailer.Third, the market demand D is realized, and theretailer receives sales revenue pmin�D�q�. (Although,for generality, we describe the transfer paymentschedule as depending on the retailer’s reportedforecast, we subsequently will show that the opti-mal transfer payment schedule can be described asdepending only on the retailer’s order quantity. Thus,the sequence of events simplifies to the manufactureroffering a transfer payment schedule T �q� and theretailer responding by choosing an order quantity q�which more closely resembles managerial practice.)
the retailer has carried in the past). To the extent that this aware-ness and understanding is reasonably good, the assumption thatthe manufacturer knows or can infer the retailer’s forecasting accu-racy is reasonable, at least as an approximation. However, in somecases the manufacturer may lack an understanding of the retailer’sforecasting capabilities (e.g., if the manufacturer is unfamiliar withthe retailer and relevant public information is scarce); to addressthis case, a more complex model capturing this additional dimen-sion of private information is required.
Our primary goal is to examine the impact of theretailer’s forecasting accuracy on the manufacturer’sperformance under the optimal procurement contract,and we pursue this goal in §§5 and 6. Our main man-agerial insight is that improvement in the retailer’sforecasting accuracy hurts (benefits) the manufacturerwhen the retailer is a weak (strong) forecaster. At theconclusion of §5, we provide evidence that this mainmanagerial message is robust to the assumption ofnormally distributed demand, to the assumption ofthe nonlinear contract form, and to the assumptionthat the retail price is exogenous.
4. Integrated SystemAs a benchmark, we first briefly examine the impactof the retailer’s forecasting accuracy on the perfor-mance of the integrated system, where there is a sin-gle decision maker. After observing the demand fore-cast � = , the system faces the demand D ��= withdistribution (2). Letting D ≡ D ��=� we can write
D = �0 + √1− a2�0 + a�0X� (3)
where X ∼ N�0�1�. Let �� · � and �� · � denotethe standard normal density and distribution func-tion,respectively, and ��� · � ≡ 1 − �� · �. The systemdecides how much to produce by solving the follow-ing newsvendor problem:
maxq
ED pmin�D� q� − cq�� (4)
which can be rewritten as (see, e.g., Porteus 2002)
maxz
�p − c���0 + √1− a2�0� − a�0��z��� (5)
where z ≡ �q − �0 − √1− a2�0�/�a�0� and ��z� ≡
p���z�−z ���z��+cz. The first term in (5) is the profit onthe mean demand, and the second term is the expectedcost of supply/demand mismatch with order quantityq = �0 + √
1− a2�0 + a�0z�
a�0��z� = �p − c�ED�D − q�+ + cED
�q − D�+�
where x+ ≡ max�x�0� Because ��z� is strictly convexand is minimized at zI = ��−1�c/p�, given the forecast� = , the system’s optimal production quantity is
qI �� = �0 + √1− a2�0 + a�0z
I � (6)
and the system’s expected profit is
�I�� = �p − c���0 + √1− a2�0� − a�0��zI � (7)
Because � is standard normal, the system’s expectedprofit is
�I = E��I��� = �p − c��0 − a�0��zI �
Intuitively, the integrated system benefits fromimproved forecasting accuracy (i.e., smaller a): Withbetter forecast information, the system makes abetter-informed production quantity decision, whichreduces the cost of supply/demand mismatch. In ourcase this is manifest by the fact that �I decreasesin a� a consequence of the well-known result thatunder normal demand, a newsvendor’s expectedprofit decreases in the standard deviation of demand.Further, �d/da��I = −�0��zI � = −�0p��zI �� and p��zI �is increasing in p and increasing in c on c ∈ �0� p/2�and decreasing in c on c ∈ �p/2� p� (Qi and Zhu 2010).The implication is that improvement in forecastingaccuracy is of greater value to the integrated systemwhen the price p is high, the cost c is moderate (closeto p/2), and the underlying demand is volatile (�0 islarge). Because the impact of forecasting accuracy onan integrated system is well established, our contri-bution is in the analysis of the decentralized system,to which we turn in the next three sections.3
5. Decentralized SystemAlthough the integrated system always benefits fromimproved forecasting accuracy, it is not clear whether,in the decentralized system (under the optimal pro-curement contract), the manufacturer will always ben-efit from selling to a better-forecasting retailer. In thissection, first, we characterize the optimal procurementcontract (Propositions 1 and 2). Second, we character-ize the impact of the retailer’s forecasting accuracy onthe manufacturer’s profit under this optimal contract(Proposition 3).In our model setting with asymmetric information,
a general procurement contract can be representedby a transfer payment T �q��, which is a functionof the retailer’s order quantity q and reported fore-cast . It follows from the revelation principle thatin our model setting, finding the optimal transferpayment function is equivalent to finding the opti-mal menu of quantity-payment pairs q��� t��� thatinduces retailer truth-telling. The retailer selects fromthe menu by “reporting” a forecast , which corre-sponds to selecting the contract that stipulates q��units as the purchase quantity and t�� as the transferpayment. The menu induces truth-telling if it is in the
3 In the decentralized system, inefficiency arises because, althoughthe manufacturer offers the contract, it is the retailer’s right tochoose the order quantity. If this decision right of the retailer canbe transferred to the manufacturer, then the manufacturer is essen-tially transformed into a centralized decision maker; the manufac-turer achieves the integrated system profit by asking the retailer forher forecast and then dictating that the retailer order the integratedsystem optimal quantity; and the impact of forecasting accuracy onthe manufacturer’s expected profit is identical to its impact on theintegrated system’s.
retailer’s interest to report the forecast she actuallyobserved.The retailer observing a forecast � = is referred to
as the type- retailer. This retailer faces demand D. Ifthe type- retailer chooses the quantity-payment pair�q��� t���, then her expected profit is
��� � = pEDmin�D� q��� − t��
Let ��� ≡ ��� �. The optimal menu is the solu-tion to
max q� · �� t� · ��
E��t��� − cq���� �OBJ�
s.t. = argmax
��� �� �IC�
��� ≥ 0� for every �IR�
The incentive compatibility (IC) constraint ensuresthat it is in the best interest of the type- retailerto select the quantity-payment pair �q��� t���. Theindividual rationality (IR) constraint ensures that theretailer accepts the contract because her expectedprofit by choosing the intended contract is no lessthan her reservation profit, which without loss of gen-erality, is normalized to zero. We characterize thesolution in the following proposition.
Under the optimal menu, the type- retailer’s expectedprofit is
��� = p√1− a2�0
∫
−��z∗�x�� dx� (10)
and the manufacturer’s expected profit is
M = E���p − c��0 − a�0��z∗����
− p√1− a2�0��z∗���� �����/����� (11)
In what follows, we first interpret the optimalmenu (Proposition 2) and then turn to the impactof forecasting accuracy on the manufacturer’s profit(Proposition 3). To gain a better understanding ofthe optimal menu, we first note that q∗�� strictly
increases in (this follows from Lemma 2 inthe appendix). This is intuitive because it simplysays an order quantity intended for an optimistic-forecast-observing retailer is greater than that for apessimistic-forecast-observing retailer. The monotoneproperty of q∗�� implies the existence of its inversefunction, denoted by ∗�q�, i.e., ∗�q∗��� = . Con-sequently, the optimal menu is equivalent to thepayment schedule T ∗�q� ≡ t∗�∗�q��, which, by sim-ply specifying the transfer payment for any givenquantity, is a conceptually simpler way to imple-ment the optimal menu of quantity-payment pairs q∗��� t∗���Under payment schedule T ∗�q�, �d/dq�T ∗�q� can be
interpreted as the marginal wholesale price, because itis the price the retailer pays for the last unit. A pay-ment schedule in which the marginal wholesale priceis decreasing in the quantity purchased is a quantity-discount scheme, whereas a schedule in which themarginal wholesale price is increasing in the quan-tity is a quantity-premium scheme. In stochastic-demandsettings that are distinct from our own in that, interalia, common information is assumed, Tomlin (2003)and Cachon (2003) show that both quantity-discountschemes and quantity-premium schemes can be effec-tive tools in encouraging efficient quantity decisionsto the benefit of individual firms. In principle, in oursetting with asymmetric information about demand,it is an open question as to whether the optimalpayment scheme exhibits quantity discounts, quantitypremia, or a combination of the two.
Proposition 2. (a) The optimal payment scheduleT ∗�q� is a quantity-discount scheme:
�d2/dq2�T ∗�q� < 0
(b) The marginal wholesale price in the optimal paymentscheme, �d/dq�T ∗�q�� is strictly decreasing in the retailer’sforecasting accuracy parameter a.
Quantity discounts are commonly observed in prac-tice, and distinct explanations have been offeredfor their use. Quantity-discount schemes have beenshown to be effective tools in encouraging largerquantity decisions, to the benefit of firms, in set-tings with stochastic demand (Tomlin 2003, Cachon2003) and in settings with deterministic demand butfixed order costs (Weng 1995, Corbett and de Groote2000, Chen et al. 2001). A buyer that does not inter-nalize a supplier’s fixed order processing cost willorder frequently in small batches, and so quantitydiscounts are a natural mechanism to encourage thebuyer to order in a fashion that reflects the sup-plier’s economies of scale. Burnetas et al. (2007)show that quantity discounts can be effective in a
setting with asymmetric demand information. Propo-sition 2(a) provides a stronger result: Quantity dis-counts emerge endogenously as an optimal responseto private demand-forecast information; see Zhanget al. (2010) for a similar result in a considerably dif-ferent setting.To see the intuition as to why quantity discounts
are optimal, consider the manufacturer’s objectives inoffering a contract: differentiating among retailers thathave observed different signals, encouraging eachto purchase roughly the systemwide-efficient quan-tity (so as to maximize system profit), and extract-ing a large portion of the surplus from the retailer.The intuition for the optimality of quantity discountsis easiest to see when the retailer, after observingher forecast, still faces considerable uncertainty aboutdemand. In this case, her purchase quantity is sen-sitive to the marginal wholesale price. Virtually allretailers (all but those observing the most pessimisticforecasts), will anticipate being able to sell the firstfew units they acquire, so the marginal value of thesefirst units will be approximately the retail price. Themarginal value of additional units is decreasing, butthe extent of this decrease depends on the retailer’sprivately observed forecast. Charging a high marginalwholesale price for the first units (nearly the retailprice) and charging progressively smaller marginalwholesale prices for larger quantities accomplishestwo objectives: First, it makes the (low-quantity)contracts intended for retailers that observed unfa-vorable forecasts unattractive to retailers that haveobserved favorable forecasts, which limits the profitthe favorable-forecast-observing retailer can extract.More generally, making the marginal wholesale pricemove in tandem with the marginal value of units tothe retailer limits the surplus the retailer can capture.Second, it minimizes the quantity distortion (distor-tion in quantity away from the systemwide-efficientquantity) for the retailers that have observed favor-able forecasts, which is important because potentialsystem profits are the largest (and hence the impactsof quantity distortions most significant) under favor-able forecasts.Proposition 2(b) establishes that as the retailer’s
forecast accuracy worsens (a increases), the marginalwholesale price in the optimal procurement contractdecreases. To see the intuition, consider how theretailer’s price sensitivity is impacted by her forecast-ing accuracy. If the retailer has a very precise sense ofwhat demand will be, her purchase quantity will beinsensitive to the marginal wholesale price (so longas the marginal wholesale price is less than the retailprice, the retailer will purchase close to the level ofdemand she anticipates); consequently, it is optimalfor the manufacturer to charge a high marginal whole-sale price. Conversely, if the retailer has a poor sense
of demand, her purchase quantity will be sensitive tothe marginal wholesale price (because she is unsurewhether she will sell the units she purchases); conse-quently, it is optimal to charge a low marginal whole-sale price because the positive impact on the purchasequantity more than compensates for the lower per-unit revenue.The implication of Proposition 2(b) is that as the
retailer’s forecast accuracy deteriorates, the optimalpayment schedule T ∗�q� “flattens.” Figure 1 depictsthe optimal procurement contract as a function ofthe retailer’s forecasting accuracy. When the retaileris a weak forecaster (a = 095), the optimal contractexhibits substantial quantity discounts, for the rea-sons described above. In contrast, when the retaileris a strong forecaster (a = 005), the optimal con-tract exhibits little in the way of quantity discounts.Expanding on the intuition described above, when ais very small, the retailer that has observed forecast knows that demand will be very close to the posteriormean �0 + √
1− a2�0� and so will purchase almostprecisely this quantity so long as the marginal whole-sale price is less than the retail price. Accordingly, acontract which specifies a (constant marginal) whole-sale price that is slightly less than the retail price,differentiates among retailers that have observed dif-ferent forecasts, encourages them to purchase nearly-systemwide-efficient quantities, and allows the manu-facturer to appropriate nearly all of the system profit.Thus, a contract where the transfer payment is nearlylinear in the quantity purchased is optimal.However, as Figure 1 demonstrates, the optimal
contract T ∗�q� may exhibit significant nonlinearity. Ifa nonlinear contract is undesirable, the manufacturerwill achieve the same profit by instead offering amenu of linear contracts (or, equivalently, a menu oftwo-part tariffs):
T ∗�q��≡T ∗�q∗���+�d/dq�T ∗�q∗���·�q−q∗��� (12)
The linear contract intended for the type- retailer issimply the straight line that is tangent to the con-cave curve T ∗�q� at q = q∗��. Each correspondsto a particular linear contract, which is composedof a per-unit price �d/dq�T ∗�q∗��� and a fixed pay-ment T ∗�q∗��� − �d/dq�T ∗�q∗��� · q∗��. The retailercan select a contract with a low per-unit price and ahigh fixed payment (by “reporting” a large forecast )or a contract with a high per-unit price and a lowfixed payment (by reporting a low ).Before turning to our main focus—the manu-
facturer’s profit under the optimal procurementcontract—it is useful to briefly comment on theretailer’s profit. As is standard in adverse selec-tion models, the retailer’s informational advantageover the manufacturer translates into profit for the
Figure 1 Optimal Procurement Contract for Different Levels of Retailer Forecasting Accuracy
Optimal transfer payment T *(q) forweak-forecaster retailer (a = 0.95)
0 0.5 1.0 1.5 2.00
0.5
1.0
1.5
2.0
Quantity q
Optimal transfer payment T *(q) forstrong-forecaster retailer (a = 0.05)
Optimal transfer payment T *(q) formoderate-forecaster retailer (a = 0.50)
Note. Parameters are �0 = 1� �0 = 0�5� p = 1� and c = 0�2.
retailer (10). The source of this profit is that a type-retailer can threaten to select a contract intended fora retailer that has observed a less optimistic forecast,and discouraging the retailer from doing so requiresmaking the contract intended for the type- retailersufficiently attractive to her. The retailer’s expectedprofit is
R = E������� = E�
[p√1− a2�0
∫ �
−��z∗�x�� dx
]
= E��p√1− a2�0��z∗���� �����/����� (13)
This quantity is referred to as the retailer’s expectedinformation rent.We now turn to the second and main topic of
this section: how the retailer’s forecasting accu-racy impacts the manufacturer’s expected profit. Themanufacturer’s expected profit (see Equation (11))is equal to the expected profit from satisfying themean demand, �p − c��0� minus the expected costof supply/demand mismatch, E��a�0��z∗������ andminus the expected information rent captured by theretailer, (13). As the retailer’s forecasting accuracyimproves, the manufacturer can tailor its contract sothat the production quantity reflects this more pre-cise demand information, reducing the expected costof supply/demand mismatch. On the other hand, asthe retailer’s informational advantage over the man-ufacturer increases, it is natural that the informationrent captured by the retailer would increase. Whetheror not the manufacturer benefits from improvedretailer forecasting accuracy depends on the trade-offbetween the cost of supply/demand mismatch andthe retailer’s information rent. Because each of thesequantities depend on z∗��� to build understandingof the impact of forecasting accuracy parameter a onthe manufacturer’s profit, we first examine its impact
on z∗��. From (8), z∗�� is the number of standarddeviations of safety stock purchased by the -typeretailer in the optimal contract; we refer to z∗�� asthe safety stock factor.It is easy to check that z∗�+� = zI ; further, z∗�� is
strictly increasing in (see Lemma 2 in the appendix).The implication is that only the highest-type retailerorders the system-optimal safety stock and the othertypes always order less. This is because the manu-facturer distorts the quantities downward to limit theinformation rents earned by the retailer. The result ofno distortion for the highest type and downward dis-tortion for other types is a typical result in adverseselection.
Lemma 1. For every , the safety stock factor z∗��strictly increases in the retailer’s forecasting accuracyparameter a.
In other words, as the retailer’s forecasting accu-racy improves (a decreases), the manufacturer’s opti-mal contract lowers the safety stock factor for everytype retailer. The intuition is as follows. As theretailer’s informational advantage grows, she is ableto more accurately assess the value of various quan-tities of units to her; consequently, when faced witha fixed menu of contracts, the retailer is able to makea better-informed contract choice, which increasesher information rent. To recapture a portion of theretailer’s profit, it is optimal (see Proposition 2(b))for the manufacturer to increase the marginal whole-sale price �d/dq�T ∗�q� across the full range of quan-tities q. In response, the retailer selects a contractwith a smaller safety stock factor. Pushing down thesafety stock factor reduces the retailer’s informationrent (see (13)). As the retailer’s forecasting accuracyimproves (a decreases), the manufacturer worries lessabout distorting downward the safety stock factor
z∗�� (from the systemwide optimum) because doingso has less impact on the retailer’s order quantityq∗�� (see (8)) and, as a consequence, on the sys-temwide efficiency. Therefore, the more accurate theretailer’s forecast, the more willing the manufactureris to distort the safety stock factor to ameliorate theinformation rent paid to the retailer.We now turn to the paper’s main result.
Proposition 3. (a) The manufacturer’s expected profitunder the optimal procurement contract, M� is strictlyconvex in the retailer’s forecasting accuracy parameter a;there exists a threshold a ∈ �0�1� such that M is strictlydecreasing in a for a ∈ �0� a� and strictly increasing fora ∈ �a�1�.(b) The threshold a is solely determined by the ratio c/p
(i.e., a is independent of all other parameters).
Figure 2 provides a critical supplement to Proposi-tion 3(b): a is strictly increasing in c/p.Whether the manufacturer benefits from improved
retailer forecasting accuracy depends on the retailer’scurrent forecasting accuracy parameter a. There aretwo regimes, a ∈ �0� a� and a ∈ �a�1�. The manufac-turer benefits from the improved retailer forecastingaccuracy if a ∈ �0� a�, i.e., the retailer’s current fore-casting accuracy is already very good; whereas theopposite is true if a ∈ �a�1�, i.e., the retailer’s currentforecasting accuracy is very poor.First, the manufacturer benefits from improved
retailer forecasting accuracy if the retailer is alreadyvery good at forecasting, i.e., a ∈ �0� a�. The intuitionis as follows. When the retailer has strong forecast-ing capabilities, the optimal contract distorts the safetystock factor downward (Lemma 1), which causesE���0��z∗������ the normalized cost of supply/demand mismatch, to be relatively large. Therefore,
Figure 2 Threshold for Manufacturer a as Function of the ProductionCost to Retail Price Ratio c/p
0.7
0.8
1.0
0 0.2 0.4 0.6 0.8 1.0
Production cost to retail price ratio c /p
0.9
Threshold for manufacturer a
further improvement in the retailer’s forecasting accu-racy (reducing a) significantly reduces the total cost ofsupply/demand mismatch, E��a�0��z∗�����. Becausethe optimal contract is stingy (characterized by highmarginal wholesale prices and low safety stock fac-tors), improved retailer forecasting accuracy has arelatively minor impact on the retailer’s informa-tion rent. Therefore, the manufacturer benefits fromimproved retailer forecasting accuracy because thepositive impact on reduced supply/demand mis-match cost outweighs any potential negative impactfrom increased information rents.Second, the manufacturer is hurt by improved
retailer forecasting accuracy if the retailer is very poorat forecasting, i.e., a ∈ �a�1�. The intuition mirrorsthat of the strong-forecaster case. When the retailerhas weak forecasting capabilities, the optimal contractis generous (characterized by low marginal whole-sale prices and high safety stock factors). Under agenerous contract, increasing the retailer’s informa-tional advantage over the manufacturer translates intosubstantially larger retailer information rent. In con-trast, because the safety stock factors are close to thesystemwide optimal level, the normalized supply/demand mismatch cost is small, and consequently thesavings on the cost of supply/demand mismatch areminor. Consequently, when retailer forecasting accu-racy improves, the losses from larger information rentdominate, and the manufacturer is hurt.The size of the region in which the manufacturer is
hurt by selling to a better-forecasting retailer dependson the value of the threshold a. Because this ratio isrestricted to a limited range (c/p ∈ �0�1�), a can becompletely characterized for all problem parametersin a simple figure, Figure 2. Figure 2 shows that a isstrictly increasing in c/pSo, when should a manufacturer be especially con-
cerned that he will be hurt by selling to a better-forecasting retailer? The region in which this outcomeoccurs is larger (a is smaller) when the retail price pis high and the production cost c is low. The intuitionis that when the production cost is a small fractionof the retail price, it is optimal to offer a generouscontract so as to encourage the retailer to purchase alarge quantity (large safety stock factor). As describedimmediately above, when the contract is generous,the losses from larger information rent dominate thesavings from smaller cost supply/demand mismatch,and so the manufacturer is hurt by improved retailerforecasting.Thus, for manufacturers that sell high-margin prod-
ucts (e.g., innovative products (e.g., leading-edge elec-tronics), information goods (e.g., books), or goodswith strong brands (e.g., Apple, Nike, Polo), wherethe production cost is small relative to the retail
price), there is a wider range of retailer forecast-ing abilities for which the manufacturer is hurt bymarginal improvements. For manufacturers that selllow-margin products (e.g., mature computer hard-ware, where the production cost is high relative to theretail price), there is a smaller range of retailer fore-casting abilities for which the manufacturer is hurt bymarginal improvements.Proposition 3 also speaks to the setting in which
there is a discrete pool of prospective retailers, whichrequires an understanding that goes beyond theimpact of marginal changes in forecasting accuracy.Consider a manufacturer that is selecting a retailerto distribute his product from a pool of N retailers.Retailer i has forecasting accuracy parameter ai� and0 < a1 < a2 < · · · < aN < 1 Because the manufacturer’sexpected profit is convex in a, it is optimal for themanufacturer to select the strongest (a = a1) or weak-est (a = aN ) forecaster. It is straightforward to checkthat as a → 0, M → �p − c��0, which is equal to theintegrated system expected profit �I with a = 0. Thisis clearly the upper bound on the maximum expectedprofit that the manufacturer could possibly achieve.Therefore, if the pool includes a very strong forecaster(a1 is sufficiently close to zero), then the manufac-turer should select the strongest forecaster (a = a1). Ifthe retailers are all sufficiently weak forecasters (e.g.,a1 ≥ a), then the manufacturer should select the weak-est forecaster (a = aN ). All other things being equal(i.e., holding the forecasting accuracy of each retailerfixed), this scenario is more likely to occur when theproduction cost is small relative to the retail price.The implication is that manufacturers ought to
avoid blindly seeking out retailers with strong fore-casting capabilities. If the production cost is highor the pool of retailers contains a strong forecaster,then this naively appealing approach will serve themanufacturer well. However, if the production costis low and the pool of retailers is not as strong,the manufacturer may benefit by selling to a retailerwith inferior forecasting capabilities. We are not thefirst to point out that the manufacturer may bene-fit by selling to a retailer with inferior forecastingcapabilities. Taylor (2006) and Miyaoka and Hausman(2008) provide numerical examples in which the man-ufacturer’s expected profit is decreasing and thenincreasing in the retailer’s forecasting accuracy; wecomplement this work by establishing the convexityresult analytically.The convexity of the manufacturer’s profit func-
tion also has implications for the value of demand-forecast information to the manufacturer. In our basesetting, the retailer has more information about mar-ket demand than the manufacturer. However, in somesettings the manufacturer may be able, through itsown efforts, to obtain this additional information,
eliminating the retailer’s informational advantage(i.e., make the demand forecast � common knowl-edge). This may be the case, for example, when theadditional demand information is set of historicalsales data (which the manufacturer can piece togetherby working with its partners, or perhaps directlypurchase) or a third-party market demand analysis.Corollary 1 characterizes the value to the manufac-turer of acquiring the additional forecast informationpossessed by the retailer, as a function of the accuracyof that information.
Corollary 1. There exists a < a such that the manu-facturer’s gain in expected profit from observing the fore-cast � is strictly increasing in a for a ∈ �0�a� and isstrictly decreasing for a ∈ �a�1�
Acquiring forecast information to eliminate theretailer’s informational advantage is the most valu-able when the retailer’s informational advantage ismoderate. Intuitively, one might expect that the valueof eliminating the retailer’s informational advantagewould be increasing in the size of the informationaladvantage, because a large informational advantageshould translate into large informational rent for theretailer. This conjecture is incorrect because, as notedabove, when the retailer’s forecasting accuracy is veryhigh, the optimal contract allows the manufacturerto capture nearly the integrated system profit. Thatis, in this extreme case, the manufacturer eliminatesthe information rents almost completely even thoughthe retailer has superior information. This result sug-gests that it is not the retailer’s superior informa-tion that drives the information rents, but ratherthe extent to which the incentive compatibility (IC)constraint has bite, i.e., the extent to which achiev-ing efficient quantity self-selection requires distortingcontracts intended for pessimistic-forecast-observingretailers so that they are unappealing to optimistic-forecast-observing retailers. When the retailer is avery strong forecaster, very little of this distortionis required because a contract with a high marginalwholesale price and small quantity is naturally unap-pealing to an optimistic-forecasting retailer.An implication of Corollary 1 is that if forecasting
efforts (efforts that reveal forecast �) are costly to themanufacturer, it is optimal for the manufacturer toexert these efforts if and only if the retailer’s forecast-ing accuracy is moderate: a ∈ �al� ah�� where al andah satisfy 0 < al ≤ a ≤ ah < 1 (The precise value ofthe thresholds al and ah will depend on the manufac-turer’s cost of forecasting.)Stepping back, this paper’s main analytical result is
that under the optimal procurement contract, the man-ufacturer’s expected profit is convex in the retailer’sforecasting accuracy parameter a. This implies that the
manufacturer’s profit is decreasing and then increas-ing in any measure of forecasting accuracy that isa monotone function of a (or �1): Improvement inthe retailer’s forecasting accuracy hurts (benefits) themanufacturer when the retailer is a weak (strong)forecaster.In the online appendix (provided in the e-
companion),4 we provide evidence that this mainmanagerial message is robust to the assumption ofnormally distributed demand, to the assumption ofthe nonlinear contract form, and to the assumptionthat the retail price is exogenous. There is one caveatin each of the latter two cases. Regarding the normalassumption, in the online appendix, we provide analternative demand model in which the random vari-ables associated with demand are not required to benormally distributed, but only to have an increasingfailure rate. Our main result, Proposition 3(a), extendsto this alternative formulation. Regarding the contrac-tual form, under the optimal wholesale price contract(i.e., the contract in which the payment is linear inthe purchase quantity), the convexity result of Propo-sition 3(a) extends. The caveat for this case is thatfor reasonable parameters (e.g., coefficient of variation�0/�0 <
√�/2 125), the region in which the manu-
facturer is hurt by improved retailer forecasting doesnot exist. Regarding the retail price assumption, inthe online appendix, we generalize the model in §3to allow the retail price to be endogenous, assumingthat the demand curve is isoelastic. Parallel to Propo-sition 3(a), there exists a threshold a ∈ �0�1� such thatthe manufacturer’s expected profit under the optimalprocurement contract is strictly decreasing in a for a ∈�0� a� and strictly increasing for a ∈ �a�1� The caveatfor this case is that the region in which the manufac-turer benefits from improved retailer forecasting maynot exist (i.e., it may be that a = 0); however, in theonline appendix, we observe numerically that in thevast majority of cases, this region does exist.Having examined the impact of the retailer’s fore-
casting accuracy on the manufacturer analytically, wenext turn to a numerical study that examines theimpact of retailer forecasting accuracy on the supplychain in broader terms.
6. Numerical StudySo far, we have focused on the impact of the retailer’sforecasting accuracy on the manufacturer’s profit. Inthis section, we expand our study to examine theimpact of the retailer’s forecasting accuracy on theretailer’s profit and the decentralized total system’sprofit.
4 An electronic companion to this paper is available as part of the on-line version that can be found at http://mansci.journal.informs.org/.
At the outset, it is unclear whether the retailer ben-efits from having improved forecasting accuracy. Onone hand, having more accurate demand informationallows the retailer to make a better order-quantitydecision to alleviate the cost of supply/demandmismatch. On the other hand, a more accuratelyforecasting retailer faces less demand uncertainty, andconsequently her purchase quantity is less sensitive toher acquisition cost; consequently, the profit-seekingmanufacturer may respond by offering stingier con-tractual terms. Whether the retailer benefits fromimproved forecasting accuracy depends on the trade-off between these two factors. To investigate theimpact of the retailer’s forecasting accuracy on herexpected profit and the expected profit of the decen-tralized total system, we conducted a numerical study.We fixed p = 1 and �0 = 1, and varied the other threeparameters: �0 ∈ 010�012�014� �030�; c ∈ 020�025�030� �080�; a ∈ 001�003�005� �099�.Thus, we tested in total 7,150 instances. For eachinstance, we computed under the optimal procure-ment contract the retailer’s expected profit R andthe manufacturer’s expected profit M . Below, first,we summarize our numerical findings. Second, wediscuss the intuition for the impact of the retailer’sforecasting accuracy on the retailer’s and decentral-ized total system’s expected profits. Third, we discussthe managerial implications of the numerical findings.Our numerical observations for the impact of the
forecasting accuracy parameter a on the retailer’sand decentralized total system’s expected profit inseveral ways parallel those of our main analyticalresult, Proposition 3, which addresses the impact onthe manufacturer’s expected profit. In particular, weobserved that for every fixed value of �0 and c, thereexists a threshold ar ∈ �0�1� such that R is strictlyincreasing in a for a ∈ �0� ar � and strictly decreas-ing for a ∈ �ar�1�: as the retailer’s forecasting accu-racy improves, her expected profit increases and thendecreases. For every fixed value of �0 and c, thereexists a threshold at ∈ �0�1� such that the total sys-tem’s expected profit under the optimal procurementcontract M +R is strictly decreasing in a for a ∈ �0� at�and strictly increasing for a ∈ �at�1�: as forecastingaccuracy improves, system profit first decreases andthen increases. Under the assumption that R (M + R,respectively) is unimodal, which is the case in everyinstance in the numerical study, one can establishanalytically a result parallel to Proposition 3(b): thethreshold ar (at , respectively) is solely determined bythe ratio of the production cost to the retail pricec/p Figure 3 supplements Figure 2 by depicting thethresholds ar and at� in addition to a. There are tworegimes: if c/p < 04� then a < ar� otherwise, a > ar .Figure 4 illustrates the impact of the forecasting accu-racy parameter on the retailer’s, manufacturer’s, anddecentralized total system’s expected profits.
Figure 3 Thresholds for Manufacturer a� Retailer ar � andDecentralized Total System at as Functions of theProduction Cost to Retail Price Ratio c/p
0.5
0.7
0.9
1.0
0 0.2 0.4 0.6 0.8 1.0
Production cost to retail price ratio c /p
0.6
0.8
Threshold forretailer ar
Threshold formanufacturer a
Threshold fortotal system at
The retailer’s expected profit increases and thendecreases as her forecasting accuracy improves. Asnoted above, an improvement in the retailer’s fore-casting accuracy has two effects: First, the moreprecise forecast information allows the retailer tomake a better quantity choice. Second, the manufac-turer responds to the retailer’s improved forecastingaccuracy by offering a more stingy contract (Propo-sition 2(b)). The negative impact of the second effectoutweighs (is outweighed by) the positive impact of
Figure 4 Under the Optimal Procurement Contract, Expected Profits of the Retailer R� Manufacturer M� and the Total Decentralized System M +R asFunctions of the Retailer’s Forecasting Accuracy Parameter
Threshold for manufacturer a
Manufacturer profit M
Retailer profit R
Retailer’s forecasting accuracy parameter a
0.80
0.75
0.70
0.65
0.10
0.05
00 0.2 0.4 0.6 0.8
Reg
ion
1
Reg
ion
2
Reg
ion
3
1.0
Threshold for retailer ar
Total system profit M +R
–
Note. Parameters are p = 1� c = 0�2� � = 1� and �0 = 0�2.
the first effect when the retailer is a strong (weak)forecaster. To build intuition, it is helpful to con-sider the extreme cases. As noted previously, whenthe retailer is a very strong forecaster �a ≈ 0�� theoptimal contract achieves nearly the entire integratedsystem profit for the manufacturer, leaving very lit-tle profit for the retailer. When the retailer is a veryweak forecaster �a ≈ 1�� the integrated system opti-mal quantity does not depend on the retailer’s pri-vately observed forecast (from (6), qI �� ≈ �0 + �0z
I
for all ); consequently, the manufacturer can extractnearly the entire integrated system profit by offer-ing a contract in which the price of purchasing thisquantity is the expected revenue it generates, andother quantities are priced sufficiently high as to bemade unattractive to the retailer. Consequently, as astrong forecaster’s forecast accuracy improves (smalla decreases) or a weak forecaster’s accuracy wors-ens (large a increases), the retailer’s expected profitdecreases toward zero.As the retailer’s forecasting accuracy improves, sys-
tem profit first decreases and then increases. Thus,the intuitive result that the total system benefits fromimproved forecasting accuracy (which holds for theintegrated system, as discussed in §4) is reversed inthe decentralized system when the retailer is a poorforecaster. The impact of forecasting accuracy on thetotal system profit follows the pattern of its impact onthe manufacturer, not the retailer; this follows because
the manufacturer, as the Stackelberg leader, has a big-ger impact on system profit than the retailer.5
We now turn to the managerial implications of ournumerical findings. A retailer can improve her fore-casting accuracy by exerting effort to acquire andprocess demand-relevant data.6 The numerical resultssuggest that the retailer should be wary of improv-ing her forecast accuracy. The retailer should be par-ticularly concerned when she is already a strongforecaster; under such circumstances, even when thecost of improving her forecast accuracy is ignored,improved accuracy results in a loss to the retailer.(Taylor 2006 establishes similar results in a much sim-pler model.) To see this graphically, observe that inFigure 4 the retailer’s expected profit decreases as herforecasting accuracy improves (a decreases) when theretailer is already a strong forecaster (a < ar ). Figure 4also illustrates how the manufacturer’s and retailer’sinterests align or diverge with respect to improvedretailer forecasting accuracy. When the retailer isalready a very strong forecaster (Region 1, which cor-responds to a < a), improved retailer forecasting accu-racy has divergent impacts on the firms’ profits: theretailer is hurt, but the manufacturer benefits. Whenthe retailer is a weak forecaster (Region 3, which cor-responds to a > ar ), improved retailer forecasting hasthe opposite effect on the firms: the retailer benefits,but the manufacturer is hurt. When the retailer is amoderately skilled forecaster (Region 2, which corre-sponds to a ∈ �a� ar �), the interests of the firms arealigned: both firms are hurt by improved retailer fore-casting accuracy. Figure 4 depicts the case where theproduction cost is small (c/p < 04� so that a < ar ).When the production cost is large (c/p > 04� so thata > ar ), the results are identical with one exception.When the retailer is a moderately skilled forecaster(Region 2, which corresponds to a ∈ �ar� a� in the highproduction cost regime), again the interests of thefirms are aligned, but in this case both firms benefitby improved retailer forecasting.The numerical results suggest the following man-
agerial implications: When the retailer is a poor fore-caster, she has incentive to improve her forecastingaccuracy, but the manufacturer would be hurt by suchimprovements and so might be inclined to frustratethe retailer’s forecast improvement efforts. When theretailer is a strong forecaster, the manufacturer would
5 Further, we observed that for every fixed value of �0 and c�the difference between the integrated system profit and the decen-tralized total system profit �I − �M + R� is first increasing andthen decreasing as the retailer’s forecasting accuracy parameter aincreases in a ∈ �0�1� Thus, the gain in system profit from cen-tralization �I − �M + R� is largest when the retailer’s forecastingaccuracy is moderate.6 See Taylor and Xiao (2009) and Shin and Tunca (2010) for analysesthat formally model a retailer’s forecasting effort decision.
like the retailer to invest in improving her forecast-ing accuracy, but the retailer concerned only withher own profit lacks the incentive to do so. Onlywhen the production cost is large and the retaileris a moderately skilled forecaster, are the interestsof the two firms’ favorably aligned: both manufac-turer and retailer would like the retailer to improveher forecasting accuracy (provided that the cost ofdoing so is not too high). Perhaps our most surprisingnumerical observation is that improved retailer fore-casting accuracy can simultaneously hurt both firms.Our numerical results suggest that this occurs if andonly if the production cost is small (c/p < 04� so thata < ar ) and the retailer is a moderately skilled fore-caster (a ∈ �a� ar �).Before concluding, it is worth pointing out three
caveats. First, our results rely on the assumption thatthe manufacturer knows or can infer the retailer’sforecasting accuracy. If the retailer is able to improveher accuracy without the manufacturer’s knowledge,the results may differ. Second, in some contexts, theremay be a practical limit for how accurate a retailercan become in forecasting. In our setting this wouldcorrespond to imposing a lower limit on the retailer’sforecasting parameter, restricting a to a ∈ �a�1�� wherea > 0 If the level of inherent demand uncertainty thatcannot be resolved through retailer forecasting effortsis large (i.e., a is large), then our results may qual-itatively change. For example, if a > max�ar� a� at��then the effect of improved retailer forecasting accu-racy would always be to benefit the retailer andhurt the manufacturer and the total decentralized sys-tem. Third, our results rely on the assumption thatthe only contractual mechanism between the firmsis the quantity-based procurement contract. Whenthe retailer is a sufficiently strong forecaster (a < at),the positive impact of improved retailer forecastingaccuracy on the manufacturer outweighs the nega-tive impact on the retailer: the total system’s expectedprofit M + R increases. Consequently, if the firms canmake transfer payments that are contingent on theretailer’s forecasting accuracy (e.g., the manufacturerprovides direct support to help the retailer improveher forecasting capabilities), then the firms may beable to overcome the incentive misalignment that pre-vents the firms from capturing this gain when thefirms rely only on quantity-based contracts.
7. DiscussionOur main finding is that a manufacturer’s expectedprofit is convex in the forecasting accuracy of its retailpartner. This convexity result lends insight into twomanagerial questions. First, when faced with a poolof prospective retailers, ceteris paribus, should a man-ufacturer select a retailer that has strong, weak, or
intermediate forecasting capabilities? The convexityresult implies: if none of the retailers is sufficientlystrong, the manufacturer should choose the weakestforecaster; otherwise, the manufacturer should choosethe strongest forecaster.Second, does a manufacturer benefit when his retail
partner improves her forecasting capabilities? Ourconvexity result implies that a manufacturer bene-fits by improved forecasting at its retail partner ifand only if the retailer is already a good forecaster.To the extent that two retailers are quite distinct intheir forecasting capabilities, our model predicts thata marginal improvement in forecasting capabilitiesat these two retailers would have an opposite effecton their manufacturer-partners. Improved forecastingby a strong-forecaster retailer makes a “good” situa-tion better (for the manufacturer), whereas the sameimprovement by a weak-forecaster retailer makes a“bad” situation worse (for the manufacturer).We establish that the optimal procurement con-
tract exhibits quantity discounts: quantity discountsemerge endogenously as an optimal response to theretailer’s private demand-forecast information. Underthe optimal contract, the manufacturer tends to behurt by improved retailer forecasting when the prod-uct economics are lucrative. Conversely, the manufac-turer tends to benefit by improved retailer forecastingwhen the product economics are poor. We concludethat a manufacturer should be most concerned aboutimprovements in retailer forecasting accuracy whenthe retailer is a poor forecaster and the product eco-nomics are lucrative.
8. Electronic CompanionAn electronic companion to this paper is available aspart of the online version that can be found at http://mansci.journal.informs.org/.
AcknowledgmentsThe authors are grateful to the associate editor, referees, andseminar participants at the University of California, Berke-ley; University of California, San Diego; University of NorthCarolina, Chapel Hill; University of Pennsylvania; Univer-sity of Washington; and Washington University for helpfulcomments.
AppendixLemma 2 is useful in the proofs of Lemma 1 and Proposi-tions 1–3.
Lemma 2. z∗��� the solution to (9), is unique. z∗�� strictlyincreases in , a, and p, and strictly decreases in c
Proof of Lemma 2. By dividing the both sides of (9) byap ���z∗���, we have
Clearly, A�z� strictly increases in z. Because A�−� =c/p < 1 and A�+� = + > 1, there exists a unique solu-tion z∗�� that satisfies (14) for each . Because A�z� strictlydecreases in , a and p (strictly increases in c), the solutionz∗�� strictly increases in , a and p and strictly decreasesin c �
Proof of Proposition 1. The proof proceeds as follows.The bulk of the proof is devoted to identifying a solu-tion to the relaxed contract design problem in which the(IC) constraint is replaced by the corresponding first-ordernecessary condition. We then observe that the solution tothis relaxed problem satisfies the constraints of the originalproblem. Using (3), we can write
����
=pEDmin�D�q���−t��
=p��0+√1−a2�0�
+pa�0EXmin{
X�q��−�0−
√1−a2�0
a�0
}−t��
=p��0+√1−a2�0�
−pa�0
[�
(z��+
√1−a2
a�−�
)−(
z��+√1−a2
a�−�
)
× ��(
z��+√1−a2
a�−�
)]−t��� (15)
where z�� ≡ �q�� − �0 − √1− a2�0�/�a�0� and the last
equality follows from (3). It follows from (IC) and the Enve-lope Theorem that
�′�� = ���� �
�
∣∣∣∣=
= p√1− a2�0��z��� (by (15)),
which by integration, leads to ��� = ��−� + p√1− a2 ·
�0
∫
− ��z�x�� dx Note that ��� increases in . Clearly, atthe optimal solution, ��−� = 0. Hence we have
��� = p√1− a2�0
∫
−��z�x�� dx (16)
By definition of ���,
��� = ��� �
= p��0 +√1− a2�0�
− pa�0���z��� − z�� ���z���� − t�� (17)
From (16) and (17), we can express t�� by using z� · � asfollows:
which by pointwise optimization, maximized at z∗�� forevery � where z∗�� is uniquely (see Lemma 2) determinedfrom (9). The corresponding t∗�� can then be determinedfrom (18).
Clearly, the solution �z∗��� t∗��� constructed aboveyields an upper bound on the manufacturer’s expectedprofit, and also satisfies (IR) and the first-order necessarycondition of (IC). From Lemma 2, z∗�� increases in ,which is a sufficient condition to ensure that the solution�z∗��� t∗��� satisfies (IC). Therefore, �z∗��� t∗��� solves themanufacturer’s contract design problem (OBJ)–(IR). �
Proof of Proposition 2. (a) Note that
dt∗��
d= p�0
���z∗���
[√1− a2 + a
dz∗��
d
](19)
anddq∗��
d= �0
[√1− a2 + a
dz∗��
d
] (20)
By definition of T ∗�q�,
dT ∗�q�
dq
= dt∗��
d
/dq∗��
d
= p�0���z∗����
√1− a2 + a�d/d�z∗���
�0�√1− a2 + a�d/d�z∗���
(by (19) and (20))
= p ���z∗���� (21)
where is such that q = q∗��. We can also write�d/dq�T ∗�q� = p ���Z�q��, where Z�q� is the z value corre-sponding to q. Clearly, Z�q� is strictly increasing q. Hence,�d2/dq2�T ∗�q� = −p��Z�q��Z′�q� < 0.
(b) It follows from (21) that ��2/�q�a�T ∗�q� = −p��z∗��� ·��/�a�z∗��� where is such that q = q∗��. Because��/�a�z∗�� > 0 (from Lemma 2), ��2/�q�a�T ∗�q� < 0 �
Proof of Lemma 1. See Lemma 2. �
Proof of Proposition 3. (a) First we establish the con-vexity property. By the Envelope Theorem,
dM
da= E�
[− �0��z∗���� + pa�0√
1− a2��z∗����
�����
����
] (22)
Note that � ′�z� = −p ���z� + c. It follows from (9) thatp ���z∗��� − c > 0 for every . Thus
� ′�z∗��� < 0�
which together with the fact that z∗�� strictly increasesin a (see Lemma 2), implies that ��z∗��� strictly decreasesin a. Note that the second part of �d/da�M is clearly strictlyincreasing in a. Consequently, �d/da�M strictly increasesin a; therefore, M is strictly convex in a.
To show the existence of a ∈ �0�1�, it suffices to showthat �d/da�M �a→1−> 0 and �d/da�M �a→0+< 0. It followsfrom (9) that as a → 1−, z∗�� → zI for every . There-fore, by (22), we have �d/da�M �a→1−= + > 0. Similarly, asa → 0+, z∗�� → − for every . Therefore, by (22), we have�d/da�M �a→0+= − < 0.
(b) From (22), the definition of �� · �� and the fact thata is the unique solution to �d/da�M = 0� a is the uniquesolution to
E�
[− ��z∗���� + z∗��� ���z∗���� − c
pz∗���
+ a√1− a2
��z∗���������
����
]= 0 (23)
From (9), for every � z∗�� depends solely on the ratio c/pand a (i.e., z∗�� is independent of all other parameters).Therefore, in terms of the model primitives, the left-handside of (23) depends only on c/p and a Because for anyfixed c/p, the solution to (23), a� is unique, a is determinedsolely by the ratio c/p (i.e., a is independent of all otherparameters). �
Proof of Corollary 1. The optimization problem forthe manufacturer that observes forecast can be written as
max T �q�� q�
T �q� − cq�
s.t. q ∈ argmax�q
pEDmin�D� �q� − T � �q���
pEDmin�D� q� − T �q� ≥ 0
The retailer’s expected profit when she chooses quantity qis pED
min�D� q� − T �q�� the first constraint ensures that itis in the retailer’s best interest to select quantity q; and thesecond constraint ensures that the retailer accepts the con-tract because her expected profit by choosing the contract isno less than her reservation profit. Under the payment sched-ule T �q� = pED
�min�D� qI ���� for q ≤ qI �� and T �q� = pqfor q > qI ��� it is optimal for the retailer to order the inte-grated system quantity qI ��; the retailer cannot increaseher expected profit by selecting a distinct order quantity.Under this optimal order quantity qI ��, the manufacturerachieves the integrated system profit T �qI ��� − cqI �� =pED
�min�D� qI ���� − cqI �� Therefore, in expectation overthe forecast �� the manufacturer’s expected profit is the inte-grated system expected profit E��T �qI ��� − cqI ��� = �I
Consequently, the manufacturer’s gain in expected profitfrom observing the forecast is �I − M . Because M ≤ �I�
lima→0+ M = �I� M is strictly convex in a (from Propo-sition 3), and �d/da��I = −�0��zI �� lima→0+ �d/da�M <
−�0��zI � This together with the facts that �d/da�M �a=a= 0and M is strictly convex and continuous in a� im-plies the existence of a unique a ∈ �0� a� satisfying�d/da�M �a=a= −�0��zI � Therefore, �d/da���I − M� > 0 fora ∈ �0�a� and �d/da���I − M� < 0 for a ∈ �a�1� �
ReferencesArya, A., B. Mittendorf. 2004. Using returns policies to elicit retailer
information. RAND J. Econom. 35(3) 617–630.Aviv, Y. 2001. The effect of collaborative forecasting on supply chain
performance. Management Sci. 47(10) 1326–1343.Burnetas, A., S. M. Gilbert, C. E. Smith. 2007. Quantity discounts in
Cachon, G. P. 2003. Supply chain coordination with contracts. A. G.de Kok, S. C. Graves, eds. Handbooks in Operations Research andManagement Science: Supply Chain Management, Chap. 6. Else-vier B.V., Amsterdam.
Cachon, G., M. Fisher. 2000. Supply chain inventory managementand the value of shared information. Management Sci. 46(8)1032–1048.
Cachon, G. P., M. A. Lariviere. 2001. Contracting to assure supply:How to share demand forecasts in a supply chain. ManagementSci. 47(5) 629–646.
Cederlund, J. P., R. Kohli, S. A. Sherer, Y. Yao. 2007. How Motorolaput CPFR into action. Supply Chain Management Rev. 11(7)28–35.
Chen, F. 2003. Information sharing and supply chain coordina-tion. A. G. de Kok, S. C. Graves, eds. Handbooks in OperationsResearch and Management Science: Supply Chain Management,Chap. 7. Elsevier B.V., Amsterdam.
Chen, F., A. Federgruen, Y.-S. Zheng. 2001. Coordination mecha-nisms for a distribution system with one supplier and multipleretailers. Management Sci. 47(5) 693–708.
Corbett, C. J., X. de Groote. 2000. A supplier’s optimal quantity dis-count policy under asymmetric information. Management Sci.46(3) 444–450.
Feldman, J., J. Cramer. 2004. Hardlines retail. Report, April 26, SGCowen, New York.
Fliedner, G. 2003. CPFR: An emerging supply chain tool. Indust.Management Data Systems 103(1) 14–21.
Fraser, J. 2003. CPFR—Status and perspectives. D. Seifert, ed. Col-laborative Planning, Forecasting, and Replenishment: How to Createa Supply Chain Advantage. AMACON, New York, 70–93.
Iyer, G., C. Narasimhan, R. Niraj. 2007. Information and inventoryin distribution channels. Marketing Sci. 53(10) 1551–1561.
Lewis, T., D. Sappington. 1991. All-or-nothing information control.Econom. Lett. 37(2) 111–113.
Lee, H. L., K. C. So, C. S. Tang. 2000. The value of informationsharing in a two-level supply chain. Management Sci. 46(5)626–643.
Mishra, B. K., S. Raghunathan, X. Yue. 2007. Information sharing insupply chains: Incentives for information distortion. IIE Trans.39(9) 863–877.
Miyaoka, J., W. H. Hausman. 2008. How improved forecasts candegrade decentralized supply chains. Manufacturing ServiceOper. Management 10(3) 547–562.
Özer, Ö., W. Wei. 2006. Strategic commitment for optimal capacitydecision under asymmetric forecast information. ManagementSci. 52(8) 1238–1257.
Porteus, E. L. 2002. Foundations of Stochastic Inventory Theory. Stan-ford University Press, Stanford, CA.
Qi, F., K. Zhu. 2010. Endogenous information acquisition in supplychain management. Eur. J. Oper. Res. 201(2) 454–462.
Rajan, M., R. Saouma. 2006. Optimal information asymmetry.Accounting Rev. 81(3) 677–712.
Ren, Z. J., M. A. Cohen, T. H. Ho, C. Terwiesch. 2010. Informationsharing in a long-term supply chain relationship: The role ofcustomer review strategy. Oper. Res. 58(1) 81–93.
Shin, H., T. Tunca. 2010. The effect of competition on demandforecast investments and supply chain coordination. Oper. Res.Forthcoming.
Taylor, T. A. 2006. Sale timing in a supply chain: When to sell tothe retailer. Manufacturing Service Oper. Management 8(1) 23–42.
Taylor, T. A., W. Xiao. 2009. Incentives for retailer forecasting:Rebates vs. returns. Management Sci. 55(10) 1654–1669.
Tomlin, B. 2003. Capacity investments in supply chains: Sharing thegain rather than sharing the pain. Manufacturing Service Oper.Management 5(4) 317–333.
Weng, Z. K. 1995. Channel coordination and quantity discounts.Management Sci. 41(9) 1509–1521.
Widlitz, S. 2005. CC: SWAT teams and test catalog may be newcatalysts—neutral. Report, July 21, Fulcrum Global Partners,New York.
Winkler, R. L. 1981. Combining probability distributions fromdependent information sources. Management Sci. 27(4) 479–488.
Zhang, H., M. Nagarajan, G. Sošic. 2010. Dynamic supplier con-tracts under asymmetric inventory information. Oper. Res.,ePub ahead of print, August 17, http://or.journal.informs.org/cgi/content/abstract/opre.1100.0810v1.
