Beating the Accuracy Trap:

Overinvestment in Demand Forecasting and

Supply Chain Coordination

under Downstream Competition

Hyoduk Shin and Tunay I. Tunca∗

Graduate School of BusinessStanford University

August 2007

Abstract

We study supply chain contracting with investment in demand forecasting under downstream compe-

tition. Supporting some recent industry observations, we show that under common pricing schemes,

such as wholesale price or two-part tariff, downstream firms overinvest in demand forecasting. Ana-

lyzing the bounds and determinants of overinvestment, we demonstrate that the wholesale price and

two-part tariff schemes can result in overinvestment up to twice the optimal level. As a result, the

supply chain surplus can also suffer substantially as the wholesale price and two-part tariff schemes

can result in nearly a full loss of supply chain efficiency. Further the losses with the best contracting

scheme in the class of quadratic contracts, which also include the wholesale price and two-part tariff

contracts, can amount to as much as half the first-best supply chain surplus. Examining the factors

that determine the severity of the efficiency loss, we show that an increased number of competing

retailers and uncertainty in consumer demand tend to increase inefficiency, whereas increased con-

sumer market size, consumer price sensitivity and demand forecast costs reduce the loss in supply

chain surplus. Finally, we propose a “market-based” contracting scheme that fully coordinates the

supply chain in quantities and demand forecast investment, and demonstrate its desirable properties

for implementability.

∗Graduate School of Business, Stanford University, Stanford CA 94305-5015. e-mails: hshin@stanford.edu,tunca tunay@stanford.edu. We are thankful to Hau Lee, Haim Mendelson, Evan Porteus, Jin Whang, Bob Wilson andseminar participants at Stanford University for valuable comments.

1 Introduction

Demand forecasting plays an important role in supply chain management. Every year companies spend

billions of dollars on software, personnel and consulting fees in an attempt to achieve accurate demand

forecasts (Ladesma 2004). The large investment in demand forecasting encompasses many dimensions,

such as collecting consumer preferences and demand-shaping, as well as predicting sales volume. Fore-

casting demand is crucial not only for the company itself, but also for the supply chain due to the fact

that the way in which the quality of forecasts is managed often affects the performance of the entire

supply chain (cf. Lee and Whang 2000). Because of the importance of demand forecasting on prof-

itability and the availability of advanced software and tools, in recent years there has been a substantial

growth of interest in demand forecasting in industry as well as in the academic literature. Within the

context of supply chains, where the payoff for each firm is affected by the actions of both vertical supply

chain partners and competitors, the role of demand forecasting is critical. Many questions regarding

the role of demand forecasting in supply chains arise: What is the right amount to invest in demand

forecasting? How should demand forecasts be shared, if at all, across the supply chain? If information

is shared within the supply chain, how can supply chain partners be coaxed into truthfully sharing their

private forecasts?

Among the questions that surround demand forecasting in supply chains, a very important one that

has not yet been adequately studied is the optimal level of forecast investment. The dramatic growth

in investment in demand forecasting in recent years not only resulted in increased profitability, but also

generated instances of substantial failures and, consequently, growing doubts about the value of investing

large amounts in forecasting (Whorten 2003). To achieve a high accuracy level in forecasts, companies

have to invest and risk substantial resources; yet it is easy to spend a large sum of money and still

not be able to achieve a satisfactory level of accuracy (see, e.g., Sullivan 2001 and Reese 2004). Some

experts have even expressed concerns about the increasing obsession and overinvestment in demand

forecasting in the industry. In a recent article, referring to the industry’s tendency to overinvest in

demand forecasting as the “accuracy trap,” Laucka (2005) argues that although accurate forecasting

is important, firms should consider moderating their investments in the area. She further argues that

supply chain objectives, in many cases, would be better served if companies directed their excess money

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and resources away from demand forecasting to other areas, such as process improvement, to adapt

efficiently to market changes.

When one considers the reasons for overinvestment in demand forecasting in supply chains, two ma-

jor factors stand out. First, in the presence of vertical disintegration and decentralized decision making,

supply chain coordination is already a challenging task (see, e.g., Cachon 2003 and Chen 2003). The

issue becomes more complicated when downstream parties invest in information acquisition; in this

case, the equilibrium depends not only on the quantities ordered by downstream parties, but also on

the investment each one makes in forecasting. As a result, the contract structures that must be designed

to coordinate the supply chain can become complicated. Second, horizontal competition among retail-

ers can create significant incentives for the competing firms to deviate from supply chain coordinating

production behavior and overinvest in demand forecasting. Combining these factors, simple, common

contract structures may, in fact, induce retailers to invest in demand forecasting at substantially sub-

optimal levels from the supply chain point of view.

Designing and implementing efficient contracts pose further challenges to supply chains in the pres-

ence of private information and costly information acquisition. First, the contract mechanism employed

must ensure that each downstream retailer reports his demand forecast truthfully. On the other hand,

it is very difficult in practice to convince firms to share their information, especially considering they

know that their competitors can benefit from that information. Thus, the mechanism employed should

address the incentives of the firms appropriately so that they will truthfully communicate their demand

forecasts. Second, in an environment with multiple downstream retailers having private information, a

contract agreed upon between the supplier and a retailer can reflect some information that the supplier

has obtained from that retailer’s competitors. Hence, after engaging in the contract with the supplier,

each retailer may be able to infer his competitors’ private information from the contracting outcome.

Consequently, after contracting occurs and before the consumer market clears, a retailer can update

his information and may find that his previously submitted order quantity is suboptimal, i.e., may

“regret” his contracted quantity. This issue can create problems with the implementation of a con-

tracting scheme and the realization of its outcome. Therefore, designing contracting schemes that are

“regret-free” in this sense requires close attention; when agents have correlated private signals, many

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contracting schemes that aim to achieve efficient outcomes fail to have the regret-freeness property.

In this paper, our purpose is three-fold. First, we demonstrate that under common contracting

schemes, such as wholesale price contracts and two-part tariffs, downstream competition indeed causes

overinvestment in demand forecasting, reducing the efficiency of the entire supply chain. Second, we find

the bounds of inefficiency that results from the employment of these contracting schemes on demand

forecast investment and supply chain surplus, showing that the degree of overinvestment as well as the

resulting supply chain losses can be very severe. We also study the factors that affect the severity of

this inefficiency. Third, we propose a mechanism that utilizes a “market index” for pricing and can fully

coordinate the supply chain, including the investment in demand forecasting by the downstream parties.

Further, we show that this scheme induces the retailers to fully and truthfully share their information.

In addition, even though information leaks through contracting, this mechanism is regret-free, i.e., no

retailer wants to change his order quantity even after observing the contracting outcome. Therefore

our proposed mechanism also satisfies important but hard-to-achieve implementability properties.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section

3 presents the model. Section 4 demonstrates the emergence of overinvestment in demand forecasting

with commonly employed contracting schemes. Section 5 analyzes the bounds and determinants of

overinvestment and supply chain surplus inefficiency. Section 6 discusses the sources of inefficiencies in

the setting, and presents the market-based contracting scheme that fully coordinates the supply chain

and studies its properties. Section 7 offers our concluding remarks. Mathematical preliminaries and

proofs for propositions are given in the online supplement.

2 Literature Review

Vertical disintegration and inefficiencies due to decentralized decision-making in supply chains have

been explored in many studies. The extensive literature on supply chain coordination examines mech-

anisms that can resolve the misalignment of incentives by making different parties act according to the

way a centralized decision-maker would behave in various settings. (See Cachon 2003 and Chen 2003

for comprehensive surveys.) Some examples of contracting schemes are revenue sharing (Cachon and

Lariviere 2005), channel rebates (Taylor 2002) and quantity flexibility contracts (Tsay 1999). There is

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also a large literature in economics on double-marginalization and vertical restraints. This literature

deals with the losses associated with double-marginalization resulting from wholesale price contracting

(Spengler 1950), and application of various additional contractual conditions to remedy this problem

and related issues. Such conditions, among others, include franchise fees, or two-part tariffs (see, e.g.,

Schmalensee 1981), resale price maintenance (see, e.g., Deneckere et al. 1997), product tie-ins (Burstein

1960), exclusive dealing to control inter-brand competition (Marvel 1982), and exclusive downstream

territories (Mathewson and Winter 1984). As in the supply chain coordination literature, these con-

tractual schemes are offered to provide solutions to incentive misalignments connected to various issues

that arise from decentralized decision making in industries. Considering the point of view of these two

main branches of literature, in this paper, we explore an important type of incentive misalignment. In

particular, we examine distortions resulting from private information and the incentives to invest in

demand forecasting under downstream competition.

Studying channel coordination with quadratic contracts, Ingene and Parry (1995) consider a two

layer supply chain with one upstream manufacturer and two downstream competing retailers. In their

setting there is no uncertainty, no private information and no investment in information acquisition.

They show that in such a setting, with two downstream retailers having possibly different demand

curves, by just providing sufficient degrees of freedom, a quadratic pricing scheme can coordinate the

supply chain. Our model explores a setting with much different incentive issues, namely contracting

under uncertainty in demand, and misalignment in incentives for investment in demand forecasting.

We show that, in our setting, a simple quadratic contracting scheme is not sufficient to coordinate the

supply chain. Instead, we propose a contracting scheme that achieves full supply chain coordination

in the presence of private information and investment in information acquisition by utilizing implicit

dissemination of retailers’ private information through the contract to coordinate the supply chain.

Further, we show that, in addition to achieving full supply chain coordination, our proposed structure

provides truthful and incentive-compatible information sharing and regret-freeness of the contract, which

are necessary and desirable properties for the implementability of the contracting scheme.

Various roles demand forecasting plays in supply chain management have been studied in the liter-

ature (see, e.g., Fisher and Raman 1996, Cachon and Lariviere 2001 and Terwiesch et al. 2004, among

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others). Aviv (2001) explores the benefits of the vertical sharing of demand forecast information by

comparing scenarios with and without demand information sharing in a collaboratively-managed sup-

ply chain with a single supplier and a single retailer. In a non-cooperative setting, Lariviere (2002)

studies the contracting schemes between a single supplier and a single retailer, in which the retailer’s

cost-effectiveness in forecasting is private information. He explores buy-back and quantity flexibility

contracts to simultaneously induce truthful revelation and optimal investment in forecasts, showing that

the latter type of contract can coordinate the supply chain and screen retailers who are efficient fore-

casters. In our research we consider a non-cooperative supply chain with multiple competing retailers

and costly information acquisition. Our results point to a type of inefficiency that has not yet been

explored in the literature, namely, the overinvestment in demand forecasting under common contracting

schemes and downstream competition.

Our paper is also a part of the literature on private information in oligopoly. The classic literature in

this area demonstrates the difficulties of inducing competing oligopolists to share private information (cf.

Novshek and Sonnenschein 1982, Vives 1984, Gal-Or 1985, 1986, Li 1985, Shapiro 1986, Raith 1996,

and Jin 2000). One of the primary conclusions derived from this literature is that when competing

firms have private information on a common uncertain variable, they do not want to share this private

information with their competitors. Li (2002) analyzes information sharing in a one-to-many supply

chain and concludes that, due to information leakage, competing downstream firms refuse to share

demand information not only with other downstream firms, but also with the supplier. We suggest

a mechanism that induces demand information sharing as equilibrium behavior. Furthermore, our

proposed contracting scheme guarantees coordination in investment in demand forecasting.

As a part of the information in oligopoly literature, Li et al. (1987) examine the welfare consequences

of investment in demand forecasting under a single-layer oligopoly with linear investment costs. In

their model, Cournot oligopolists acquire information about uncertain demand before competing in

the consumer market. The investment level of each competitor is observable to others. They consider

social welfare, i.e., the sum of the oligopolists’ profits and the consumer surplus as their performance

measure and explore the equilibrium investment levels in demand forecasting compared to the centralized

socially optimal solution. Their results on the comparison of the two investment levels are mixed:

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when the cost of demand forecasting is high, there is underinvestment in forecasting compared to the

welfare-maximizing level, whereas for low demand forecasting costs, there is overinvestment in demand

forecasting. In comparison, we explore a two-layer supply chain where the retailers’ investment levels

are unobservable to their competitors as well as the upstream supplier under general convex investment

costs. Vertical disintegration introduces contract alignment issues which, unlike Li et al. (1987), is

the focus of our study. We study the supply chain surplus, demonstrate that there is overinvestment

with common contracting schemes, find the bounds of the losses incurred, and explore the determinants

of overinvestment and inefficiencies in supply chain surplus. As a result of these differences in the

setup and the performance measures considered (e.g., supply chain surplus versus social welfare), our

results on information acquisition are different from those of Li et al. (1987). We show that under

common contracting schemes, there is overinvestment in demand forecasting compared to the first-best

level for the supply chain, no matter what the magnitude of forecasting costs is, as long as the first-

best investment level is non-zero. We also suggest a contracting scheme that aligns the equilibrium

investment level in demand forecasting with the first-best investment.

One important assumption made in the studies in information sharing in the oligopoly literature (as

cited above) is that the parties report their signals truthfully. However, unless they are compelled by

a mechanism that ties incentives to truthful reporting, competing oligopolists have strong incentives to

distort information, especially when sharing it with competitors. Studying the impact of strategic spot

trading in supply chains, Mendelson and Tunca (2007) demonstrate that partial truthful information

sharing in the supply chain can be achieved in a decentralized spot market. Our proposed contracting

scheme in this paper reveals that the upstream firm can offer a contract to implement the supply

chain surplus-maximizing contract by inducing full truthful information revelation and aggregation in

an incentive-compatible way as well as achieving coordination in demand forecast investments.

A number of researchers in economics literature have studied efficient mechanism design with infor-

mation acquisition. The papers in this area regard information acquisition as hidden action, implying

that the level of effort (investment) is unobservable to other players. Bergemann and Valimaki (2002)

show that when agents have private valuations of a good and acquire costly independent information,

a Vickrey-Clarke-Groves mechanism (VCG, see Clarke 1971 and Groves 1973) achieves regret-free (or

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ex-post) efficient allocation and an ex-ante efficient level of information acquisition. Under common

valuations and independent signals, efficient mechanism design is also examined in a group of studies,

including Dasgupta and Maskin (2000), Jehiel and Moldovanu (2001), Perry and Reny (2002). With

uncorrelated signals and under the assumption that deviations from equilibrium can be detected with

positive probability, Mezzetti (2002) demonstrates that a mechanism that achieves ex-post efficient

allocation with efficient ex-ante information acquisition exists.1

When the signals are correlated, Cremer and McLean (1985 and 1988) establish the existence of

regret-free efficient mechanisms with full surplus extraction, but with no information acquisition and

interdependence among the payoffs of the agents. Following Cremer and McLean, Obara (2003) studies

efficient allocation in perfect Bayesian equilibrium with information acquisition for correlated signals

and shows that there is no mechanism that guarantees full efficiency in a Bayesian equilibrium, even if

the objective of a regret-free implementation is relaxed. In our paper, in an environment of common

values and correlated signals, in which the quantity decision of each retailer affects the payoffs of the

other retailers through downstream competition, we present a contracting scheme that achieves efficient

production quantities and investment in information acquisition with full surplus extraction in a regret-

free way, as we demonstrate in Section 6.

3 The Model

A single supplier sells a good to n retailers who compete as a Cournot oligopoly in the consumer market.2

The (inverse) consumer demand curve is given by pc = K − γ∑n

i=1 qi, where pc is the clearing price in

the consumer market, and qi is the quantity that retailer i, 1 ≤ i ≤ n, orders and sells in the consumer

market.3 For simplicity in exposition, we normalize each retailer’s reservation value to zero. In the

demand curve, K is uncertain with mean K0 and variance σ20. The supplier’s unit production cost is c0.

1Externality-based coordinating schemes are also used in the supply chain management literature. Bernstein et al.(2003) demonstrate that a simple externality pricing scheme with Vendor Managed Inventory (VMI) can achieve supplychain coordination with no investment in demand forecasting or private demand information. Our proposed market-basedscheme also addresses externalities in coordinating the supply chain, but when there is private demand information andinvestment in demand forecasting, the issues for supply chain coordination are very different. In this case the utilization ofincorporating externalities into pricing must address the alignment of investment in demand forecasting and should takethe leakage of private information in the contracting process into account.

2We refer to the upstream firm as female and the downstream firms as male.3Inverting pc = K−γ

Pni=1 qi to get the demand curve with the quantity on the left-hand side, we obtain Q = K/γ−pc/γ,

where Q =Pn

i=1 qi. That is, forecasting K is, indeed, forecasting the consumers’ demanded quantity (at any given pricelevel). Presenting oligopolists’ demand forecasts in inverse demand form is the standard way of representing demandforecasts in similar models in the literature (see e.g., Li 2002 and the references therein), which we will follow in this paper.

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There are three time periods indexed by t = 1, 2, 3. At time t = 1, there is no information asymmetry

among the participants. The supplier offers retailers a contract, which specifies a price function P (q),

where q = (q1, . . . , qn). Denote q−i = (q1, . . . , qi−1, qi+1, . . . , qn) for i = 1, . . . , n, and Q =∑n

j=1 qj , the

total quantity ordered. At t = 2, each retailer i invests in demand forecasting and obtains a private

signal (si) about the state of the demand (i.e., K). Demand forecasting is costly; therefore, before

making the forecast, each retailer decides how much to invest. The more a retailer invests, the more

accurate the signal he obtains about the state of the demand. Upon receiving his signal, each retailer

i places his order qi to the supplier.4 At t = 3, supplier delivers the good, consumer market demand

is realized, and competing as a Cournot oligopoly, retailers sell the good in the consumer market. We

denote retailer i’s profit as Πi, and the total supply chain profit as ΠSC .

We assume unbiased signals and affine conditional expectations for the information structure of the

signal (see, e.g., Ericson 1969). That is, E[si|K] = K, and E[K|s] is affine in s, for all s = (s1, . . . , sn).

There are many conjugate pair distributions for the demand intercept and signals that satisfy these

assumptions, such as Normal/Multivariate Normal, Beta/Binomial and Gamma/Poisson, respectively

(see Ericson 1969 for a detailed discussion). Define νi , σ20/σ2

i , where σ2i , E [Var[si|K]], i = 1, . . . , n.

νi is the expected precision of retailer i’s demand signal, si, relative to the precision of K. As the

expected precision of the demand signal, νi, increases, σ2i decreases. Denote the cost function for

demand forecasting by C. That is, to have an expected forecast precision νi, retailer i must invest

C(νi). Clearly, to achieve higher precision, one needs to invest more, i.e., C is non-decreasing. We also

assume that C is convex, non-identically zero, and twice differentiable with C(0) = 0. The investment

level of each retailer is unobservable by other parties.

The definition of the equilibrium for our analysis is given in Appendix A. We focus on linear Bayesian

equilibria, in which qi has the form, qi(si) = α0i + αsi (si −K0). Equilibrium in linear order strategies

is a common feature of the models in the class we study, namely those that explore correlated private

information among oligopolists in an industry (cf. Vives 1984 and the follow-up literature). Note that

we do not impose the linearity constraint as a restriction. Rather, we will conjecture such equilibria4For notational convenience, if retailer i decides not to contract at all, we set qi = 0 by default. Note that for the

wholesale price contract a retailer’s ordering a zero quantity at this stage essentially means he is refusing the contract.However, for two-part tariff and general quadratic contracting schemes we will study in Section 4, a retailer’s participatingin the contract requires a fixed payment. Also note that in equilibrium, order quantities are non-zero almost surely.

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and verify these conjectures. For any given retailer, given the other retailers’ linear strategies, the

optimality of a linear order quantity strategy will follow endogeneously. In short, the equilibria we

study throughout the paper will be fully unrestricted Bayesian equilibria.

4 Overinvestment in Demand Forecasting

In order to understand the effect of competition on investment in demand forecasting and supply chain

efficiency, we need to derive the centralized first-best benchmark outcome for the supply chain. The

first-best benchmark (denoted by the superscript FB), as usual, assumes that the supply chain is fully

coordinated, i.e., all decisions are made in a centralized manner, and all information in the supply chain

is available to the decision maker. The analysis and the derivation of the first-best optimal total pro-

duction quantity for any given signal realization s (QFB(s)), optimal demand forecast investment levels

(νFB), and first-best channel profit (ΠFB) are given in Appendix A. Note that the first-best benchmark

is the idealized best and achieves the fully-coordinated outcome by utilizing normally nonexistent advan-

tages, such as centralized decision-making (no incentive issues) and the pooling of the dispersed private

information of multiple agents (no informational asymmetry).5 Throughout the paper, we will compare

the outcomes of the contracting schemes we examine to this fully coordinated first-best benchmark.

4.1 Overinvestment under common pricing schemes

In this section, our goal is to show that, in confirmation of recent industry observations, common

contracting schemes, such as simple wholesale pricing and two-part tariff can result in overinvestment

in demand forecasting. For ease of exposition, we start by giving the equilibrium solution for the general

standard quadratic contracting form Pi(qi) = w0 + w1qi + w2q2i , which we denote by the superscript q.

The common wholesale price contract is the case where w0 = w2 = 0. We denote the wholesale pricing

scheme with the superscript ws. In a similar manner, the common two-part tariff scheme corresponds

to the case with w2 = 0, and we denote this scheme with the superscript tpt.5The benefits of pooling information in the supply chain has also been said to be attained in certain inventory man-

agement schemes, such as Vendor Managed Inventory (VMI). See, e.g., Aviv and Federgruen (1998) and Chen (2003) formore details. In Appendix C, we demonstrate that, in a decentralized setting with downstream competition and privatecorrelated signals, simple information pooling cannot achieve coordination. On the other hand, in Section 6, we showthat a pooling effect as well as full coordination can be achieved in this setting without exogenous pooling assumptions byutilizing our market-based coordinating scheme.

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We start our analysis by presenting the equilibrium outcome for the general quadratic contracting

form. The following proposition presents this result.

Proposition 1 Given the pricing scheme Pi(q) = w0 + w1 qi + w2 q2i , provided that the second order

condition γ + w2 > 0 is satisfied, there exists a unique linear equilibrium. In equilibrium qi(si) =

αq0 + αq

s (si−K0) and νqi = νq, for all i, where αq

0 = (K0−w1)/(γ(n + 1) + 2w2), αqs = νq/(2(γ + w2) +

(2w2 + (n + 1)γ)νq), νq = ν∗ · 1{σ20≥4(γ+w2) C′(0)}, and ν∗ is the unique solution to the equation

(γ + w2)σ20

(2(γ + w2) + (2w2 + (n + 1)γ)ν)2− C ′(ν) = 0 . (1)

Proposition 1 presents the unique linear equilibrium in order quantities. The equilibrium is symmetric

in investment levels for the retailers (νq). Utilizing this equilibrium solution for the general class of

quadratic contracting schemes, we can deduce the equilibrium for the commonly used contract subclasses

of interest. For the wholesale pricing scheme, the supplier sets a constant price wws in order to maximize

her profit, i.e., the pricing scheme she offers is Pi(q) = wws qi. For two-part tariff scheme, the supplier

announces the pricing scheme as Pi(q) = wtpt0 + wtpt

1 qi and chooses wtpt0 and wtpt

1 to maximize her

expected profit while ensuring the participation of the retailers.

We can now explore the equilibrium and demand forecast investment behavior in the supply chain

under the wholesale price and two-part tariff schemes. The following proposition presents the optimal

wholesale price and the optimal two-part tariff schemes and compares the equilibrium investment levels

with first-best investment levels.

Proposition 2

(i) The optimal wholesale price contract for the supplier is specified by the wholesale price wws =

(K0 + c0)/2, and the optimal two-part tariff contract for the supplier is specified by the two para-

meters

wtpt0 =

14γn

((K0 − c0)2

n+

4nνtpt(1 + νtpt)σ20

(2 + (n + 1)νtpt)2

)− C(νtpt) , (2)

wtpt1 =

(n− 1)K0 + (n + 1)c0

2n, (3)

where νtpt = νws, and is as given in νq in Proposition 1 with w2 = 0.

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(ii) For n ≥ 2, under simple wholesale pricing and two-part tariff schemes, each retailer overinvests

in demand forecasting in equilibrium, i.e., νws = νtpt ≥ νFB. The inequality is strict when

σ20 > 4γ C ′(0).

Proposition 2 states the first main result of our paper: It is indeed the case that under downstream

competition, common contracting schemes such as the wholesale price and two-part tariff contracts

are prone to yielding overinvestment in demand forecasting. This becomes more apparent when one

pays attention to the case with no downstream competition, i.e., the case with n = 1. Notice that for

n = 1, substituting w2 = 0 in (1), the equilibrium demand forecast investment level for wholesale price

and two-part tariff contracts is the same as the first-best level. That is, when there is no downstream

competition, even when there is vertical disintegration, the wholesale price and two-part tariff contracts

do not result in overinvestment in demand forecasting. Further, plugging n = 1 in Proposition 2, and

comparing to the first-best outcome, one can see that, when there is only one retailer, by the virtue of

having coordinated the level of investment in demand forecasting, the two-part tariff scheme can even

achieve full supply chain coordination.

However, for n ≥ 2, the competition among retailers brings three additional layers of inefficiency

on top of vertical disintegration.6 First, the marginal value of each additional unit ordered by each

retailer is enhanced for that particular retailer compared to the corresponding value for the supply

chain. Second, each retailer’s optimal action in order quantities with respect to his private signal is not

aligned with the optimal reaction to that signal from the supply chain point of view. This deviation

from the supply chain optimal by competing agents creates an inefficiency in the supply chain in that

the supply chain coordinating contract has to take into account how each retailer would react to the

contract given each realization of his signal.

Third, because of downstream competition, the marginal value of precision of the signal for each

retailer is higher than the corresponding marginal value for the supply chain. Each retailer uses his

demand forecast for two purposes. First, his forecast signal provides him information about the realiza-

tion of the consumer demand. Second, he uses his forecast to estimate his competitors’ order quantities,

as their orders are related to their forecasts, which are in turn correlated to his forecast. Hence, for6We further discuss and analyze these sources of inefficiencies in Section 6.1.

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each retailer, a higher precision in his signal means not only a more accurate forecast for the demand

realization but also a better prediction of the order quantities of the competitors, which will determine

the consumer market clearing price. Therefore, increasing the marginal precision of his signal is more

valuable to him than it is to the supply chain, and consequently, in equilibrium, each retailer ends up

overinvesting in demand forecasting.

Notice that, the double-marginalization caused by vertical disintegration will result in underproduc-

tion in expectation for the wholesale price contract. With the added degree of freedom, plugging (3)

in the expression for αq0 as given in Proposition 1, the expected quantities will be coordinated under

the two-part tariff scheme, i.e., E[Qtpt(s)

]= E

[QFB(s)

], while, unlike the case for n = 1, for n ≥ 2,

statewise supply chain production quantity coordination cannot be achieved. That is, as a result of the

misalignment in production and demand forecasting caused by downstream competition, for a given

realization of signals, in equilibrium, there is a mismatch between the quantity produced under the

two-part tariff contracting scheme and the first-best production level. This results in a loss of supply

chain surplus for the two-part tariff scheme as the supply chain is unable to make use of the available

information efficiently. Following this result, one could suppose that perhaps, as it would be in settings

with no demand forecasting and no investment in information acquisition, a simple (ex-ante) quadratic

contracting scheme could solve the coordination problem by adding an additional degree of freedom to

contract design. We explore the question of adequacy of such schemes in our case in the next section.

4.2 Lack of Coordination with General Standard Quadratic Contracting Schemes

In this section, we demonstrate that with demand uncertainty and forecast investments under down-

stream competition, full coordination is impossible with standard quadratic contracting schemes.7 The

next proposition presents this result.

Proposition 3 When n ≥ 2 and there is positive investment in demand forecasting in the first-best

solution, i.e., when σ20 > 4γC ′(0), it is impossible to simultaneously achieve full coordination in statewise

total production quantity and the investment level using a standard quadratic pricing scheme. That is,

standard quadratic pricing schemes fail to coordinate the supply chain.7This is in contrast to some simpler settings, such as that of Ingene and Parry (1995), where simple quadratic contracting

schemes are sufficient to coordinate the supply chain.

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Proposition 3 states that in the presence of private demand forecast information and investment

in information acquisition, even the globally optimal quadratic scheme is not sufficient to fully coordi-

nate the supply chain. With the dependence of the retailers’ payoffs on each others’ quantities due to

downstream competition, and private forecast information, even if the supplier can coordinate the pro-

duction quantities statewise with a quadratic contracting scheme, she will still be unable to coordinate

the investment level in demand forecasting in the supply chain.

This observation points to a certain fundamental inefficiency in the class of standard quadratic

contracting schemes we analyze in a setting with demand uncertainty and forecasting. This class that

includes very commonly used supply chain contracting schemes in fact consists of ex-ante contracts, in

which the supplier makes a contract offer that ties the price each retailer pays only to the order quantity

of that retailer. As a result these schemes separate the pricing, and consequently, the order quantity

decision of each retailer from those of the other retailers, and therefore fail to make efficient use of

dispersed private demand information of the retailers. However, this poses the challenge of utilizing

the information contained in retailers’ demand signals and incorporating it into the supplier’s contract

offer without observing those signals. In addition, this indirect information sharing must be achieved

while ensuring truth-telling, as the retailers would have incentives to distort their information if asked

to share. We will discuss this issue further in Section 6 when we propose a contracting scheme that

achieves supply chain coordination by satisfying these required properties.

5 Determinants and Bounds of Overinvestment and Efficiency

In Section 4 we have seen that under downstream competition, the common wholesale price and two-part

tariff contracting schemes induce overinvestment in demand forecasting. Further, the analysis in Section

4.2 demonstrates that it is not possible to fully coordinate the supply chain using a general standard

quadratic pricing scheme. This raises important questions about supply chain efficiency: What are

the bounds of overinvestment? If inefficient investments in demand forecasting occur under common

contracting schemes, what is the extent that this gets reflected on the supply chain surplus? How

large can the supply chain surplus loss get? What are the determinants of the loss of efficiency in

investment in demand forecasting and supply chain surplus? In this section, we study the bounds for

13

overinvestment under these contracting schemes and explore the factors that determine the magnitude

of deviation from the first-best investment levels. We first find the general bounds on overinvestment

and supply chain surplus loss, and then examine the determinants of efficiency loss.

We start by investigating the upper and lower bounds of overinvestment in demand forecasting.

Note that the equilibrium investment levels of demand forecasting for the wholesale price and two-part

tariff schemes are identical. Further, by Proposition 1 and the analysis of the first-best case, when

σ20 ≤ 4γ C ′(0), we have νFB = νws = νtpt = 0. That is, in that case, the first-best and equilibrium

investment levels are trivially coordinated at zero. Similarly, as we discussed in Section 4.1, without

competition, i.e., when n = 1, supply chain can be coordinated in both investment and production

quantities under the two-part tariff and the quadratic pricing schemes. Thus, to avoid trivialities,

throughout this section we focus on the case where forecast investment coordination is relevant, i.e.,

σ20 > 4γ C ′(0), and n ≥ 2. The following proposition provides the bounds of overinvestment.

Proposition 4 For n ≥ 2, considering all possible increasing and convex demand forecast investment

cost functions,

(i)

1 <νws

νFB=

νtpt

νFB≤ 2n

n + 1. (4)

Furthermore, all bounds in (4) are tight. The lower bound can be achieved in the limit for any

parameter combination as C ′(ν) is large enough for ν → νFB from the right, and the upper bound

can be achieved for linear demand forecast cost functions, i.e., when C(ν) = cf ·ν for cf > 0.

(ii) For the optimal quadratic pricing scheme, equilibrium overinvestment (νq/νFB) can be unbounded.

This can occur in the limit, for C(ν) = cf ·ν as n →∞, and σ20 →∞ or cf → 0 or γ → 0.

Proposition 4 states that the equilibrium investment level under the common wholesale and two-part

tariff contracts can be as close to the first-best investment level as possible for certain parameter com-

binations and cost functions. That is, under best-case conditions, there is almost no supply chain

overinvestment. Such scenarios usually involve cost functions that become steep after a certain specific

threshold as part (i) of Proposition 4 indicates. For general cost functions, on the other hand, the out-

come can be very inefficient: Specifically, commonly used wholesale price and two-part tariff schemes

14

can result in an overinvestment in demand forecasting up to 2n/(n + 1) times the first-best level νFB.

Further, this inefficiency ratio increases as n increases and as n → ∞, the maximum inefficiency con-

verges to 2, i.e., with downstream competition, common contracting schemes can result in twice as

much investment as it is optimal for the supply chain in equilibrium. As can be seen from part (ii) of

Proposition 4, the inefficiency in forecast investment is not mitigated even when the contracting scheme

is relaxed to general quadratic contracting schemes. In fact, with the general quadratic contracting

scheme, the inefficiency in investment can become unbounded, when the number of competing retailers

is large and the demand uncertainty is large or the cost of investment or the sensitivity of consumer de-

mand to quantity produced is low. This is because, downstream competition can increase the intensity

of the overinvestment and a high demand uncertainty, low investment cost, or low demand sensitivity

can fuel this competition by increasing the retailers’ incentives to invest to very high levels as stated in

part (ii) of Proposition 4.

The negative effect of overinvestment in demand forecasting on the supply chain is not limited to the

cost of overinvestment. In fact, considering the ripple effects of overinvestment in demand forecasting

on contracted quantities, in equilibrium, the losses in supply chain profits with common contracting

schemes can be very substantial. The following proposition presents the effect of the demand forecast

overinvestment caused by downstream competition on supply chain profits.

Proposition 5 For n ≥ 2, considering all possible increasing and convex demand forecast investment

cost functions,

(i)

0 <E[Πws

SC ]E[ΠFB]

< 1, and 0 <E[Πtpt

SC ]E[ΠFB]

< 1 , (5)

with all bounds in (5) being tight. The lower bounds can be achieved in the limit for C(ν) = cf ·ν

and as n →∞ and σ20 →∞, and the upper bounds can be achieved for all convex increasing cost

functions in the limit as σ20 → (4γC ′(0))+, with the additional condition n → ∞ for the case of

wholesale price contracting.

(ii) With the optimal general quadratic contracting scheme, supply chain surplus loss can reach at least

as high as 50% of the first-best level, which can occur in the limit, for C(ν) = cf ·ν as n → ∞,

15

and σ20 →∞.

Proposition 5 states that the misalignment of incentives stemming from downstream competition and

investment in demand forecasting can reduce supply chain surplus tremendously to a near full inefficiency

level for the wholesale price and two-part tariff cases. While in certain cases, the two contracting

schemes can achieve nearly efficient results, part (i) states that, such cases occur when the uncertainty

in consumer demand curve is near levels that do not justify positive investments in demand forecasting as

specified by Proposition 1. This means that relatively efficient cases can arise when parties have limited

private demand information even if there is uncertainty in demand. However, in most cases, demand

uncertainty is large enough to induce substantial investment in forecasting by retailers. Proposition 5

shows that in such cases, especially when downstream competition is intense and uncertainty is high,

supply chain losses induced by overinvestment in forecasting and acquisition of private information can

reach very high levels, and in the limit the channel can even suffer nearly full loss of efficiency. Part

(ii) of Proposition 5 states that even when one generalizes contracting to full quadratic contracting

schemes, given that downstream fragmentation and demand uncertainty are large, the losses can be

very significant, reaching up to 50% of full supply chain surplus levels under the globally optimal

contract in this class.

The large degree of potential inefficiency noted in Proposition 5 highlights the importance of the

roles of private information and investment in demand forecasting, and suggests that in analyzing

supply chain coordination, demand forecasting should be carefully considered. This result is especially

significant when one considers that, ignoring investment in demand forecasting and private information

about market demand in the analysis, the two-part tariff and the quadratic contracting schemes are

traditionally postulated to perfectly coordinate the supply chain and yield full first-best profits. Our

analysis shows that ignoring demand forecasting when analyzing supply chain contracts can result in

highly inefficient recommendations and outcomes.

Given the substantial consequences of contracting with demand forecasting on supply chain surplus,

one question that immediately arises is what are the factors that determine the degree of inefficiency

resulting from simple contracts. We explore answers to that question next. To illustrate the effects of the

factors that determine the level of inefficiency, we analyze the comparative statics for overinvestment

16

and supply chain surplus loss for the case of linear investment cost in demand forecasting, which is

commonly used in the literature (see, e.g., Li et al. 1987). The following proposition presents this

result.

Proposition 6 Consider the case C(ν) = cf · ν and n ≥ 2.

(i) For the wholesale price and the two-part tariff schemes,

(a)

νws

νFB=

νtpt

νFB=

2n

n + 1. (6)

(b) E[ΠwsSC ]/E[ΠFB] and E[Πtpt

SC ]/E[ΠFB] are increasing in γ, cf and K0, and decreasing in σ20

and c0. Further, E[ΠtptSC ]/E[ΠFB] is decreasing in n and E[Πws

SC ]/E[ΠFB] is increasing in n if

2 ≤ n ≤ 1 + (K0 − c0)2/2(σ20 − 4

√γcfσ2

0 + 4γcf ), and decreasing in n, otherwise.

(ii) For the optimal quadratic pricing scheme, there is always overinvestment in demand forecasting,

i.e., νq/νFB ≥ 1. Further,

(a) νq/νFB is increasing in n and σ20, and decreasing in γ and cf .

(b) E[Πq

SC

]/E

[ΠFB

]is increasing in K0, and decreasing in n and c0. Moreover, there exists an

n ∈ IN, such that if n > n, E[Πq

SC

]/E

[ΠFB

]is increasing in γ and cf and decreasing in σ2

0.

Figures 1 and 2 demonstrate the effects of parameters on inefficiency in demand forecast investment

and supply chain surplus. As the number of competing retailers increases, the competition among them

intensifies, increasing the misalignment of incentives. As a result, as stated in part (i)(a) of Proposition

6, in wholesale price and two-part tariff schemes, the overinvestment in demand forecasting increases

in n asymptoting at 2. The same effect can also be observed for the optimal quadratic contracting

scheme, in panel (a) of Figure 1 as the overinvestment ratio in demand forecasting increases with n.

As can also be seen in the figure, the globally optimal quadratic contracting scheme can, in fact, hold

the overinvestment under control better than the wholesale price and the two-part tariff schemes for

small n. However, it cannot do much better than these commonly used contracting schemes in this

dimension, yielding a worse overinvestment ratio as n increases and eventually moving towards the

asymptotic level labeled LQP in the figure. The effect of the number of competing retailers on profits is

17

0 10 20 30 40 501

1.5

2

2.5

(a) ν/νF B vs. Number of Retailers

n

QPWS-TPTLQP

0 10 20 30 40 500.7

0.75

0.8

0.85

0.9

0.95

1(b) E[ΠSC]/E[ΠF B] vs. Number of Retailers

n

QPTPTWSLQP

0 5 10 151

2

3

4

5(c) ν/νF B vs. Market Price Sensitivity

γ

QPWS-TPTLQP

0 10 20 30 40 500.7

0.75

0.8

0.85

0.9

0.95

1

(d) E[ΠSC]/E[ΠF B] vs. Market Size

K0 − c0

QPTPTWSLQP

Figure 1: The effect of the number of competing retailers, consumer market size and price sensitivity ondemand forecast overinvestment and supply chain surplus. Panel (a) plots demand forecast investmentratio (ν/νFB) and panel (b) plots supply chain profit ratio (E [ΠSC ] /E

[ΠFB

]) with respect to the

number of retailers (n). Panel (c) plots demand forecast investment ratio (ν/νFB) with respect toconsumer market price sensitivity (γ). Panel (d) plots supply chain profit ratio (E [ΠSC ] /E

[ΠFB

])

with respect to expected market size (K0− c0). In the legends, WS indicates wholesale pricing scheme,TPT indicates two-part tariff scheme, QP indicates quadratic pricing scheme and LQP indicates theasymptotic limit of quadratic pricing scheme as n → ∞. The remaining parameters for all panels areσ2

0 =30, γ =1, c0 =0.5, cf = 0.5, n =3 and K20 = 50, where applicable.

subtle. For wholesale price contracting, as the number of competing retailers increases, the supplier can

have better control of quantities. That is, the increase in number of retailers reduces the efficiency loss

due to double-marginalization, which consequently facilitates coordination in quantities and increases

supply chain profits. On the other hand, as we have discussed above, an increase in the number of

retailers increases overinvestment in demand forecasting. As a result, under wholesale pricing, there

can be a threshold level of downstream competition, before which the supply chain surplus efficiency is

increasing in n, but after which the negative effect of overinvestment becomes dominant to decrease the

supply chain profits relative to the first-best level as can be seen in panel (b) of Figure 1. For two-part

tariff and the optimal quadratic contracting schemes, since the production quantities are coordinated

in expectation, only the overinvestment effect is present, decreasing the surplus efficiency ratios as the

18

number of competing retailers increases. As can also be seen from this figure, as the number of retailers

increases, the supply chain surplus with the wholesale price and the two-part tariff schemes converge,

while there is a substantial gap between the surplus these two common contracting schemes yield and

the supply chain optimum, due to overinvestment in demand forecasting.

As the consumer market price sensitivity (γ) decreases, the market becomes more attractive for

the retailers, which increases the magnitude of misalignment of incentives in demand forecasting. As

a result, as can be seen in panel (c) of Figure 1, the overinvestment ratio increases. Although the

globally optimal quadratic contracting scheme improves upon controlling overinvestment compared to

the common contracting schemes for low n and high γ values, as n becomes large, overinvestment

can become very significant as the limiting curve that corresponds to overinvestment with quadratic

contracting (labeled LQP) demonstrates. An increase in market potential, on the other hand, reduces

the relative effect of the inefficiency resulting from overinvestment as stated in part (ii)(b) of Proposition

6 and demonstrated in panel (d) of Figure 1.

When demand uncertainty increases, the marginal value of each unit of investment in demand

forecasting increases for each retailer. Consequently, an increase in market demand uncertainty increases

the relative overinvestment in demand forecasting as can be seen in panel (a) of Figure 2. Hence, the

surplus efficiency of the supply chain for the common contracting schemes decreases with increased σ20

from a level of near perfect efficiency for small values of σ20 to significantly lower levels as panel (b) of the

figure demonstrates. Notice that, increased downstream competition intensifies the surplus efficiency

reduction effect for quadratic contracting scheme as can be observed by the steepness of the asymptotic

efficiency ratio curves (LQP) in both panels. For large n values, as stated in part (ii)(b) of Proposition

6, surplus ratio for the globally optimal quadratic contract is decreasing with σ20. However, for smaller

n values, as can be seen in panel (b) of Figure 2, the best surplus ratio one can get with any quadratic

scheme is non-monotonic: It decreases with increased σ20 for small values of σ2

0, as the overinvestment

for the optimal contract sharply increases in this range. In contrast, when σ20 is larger, the effect of

σ20 on the overinvestment flattens out. At the same time, both the optimal quadratic and the first-

best profits increase at similar rates, making the ratio increase towards one. As can be seen from the

figure, for larger σ20 values, the supply chain efficiency gap between the common contracting schemes

19

0 50 100 150 200 250 3001

2

3

4

5

6(a) ν/νF B vs. Demand Uncertainty

σ2

0

QPWS-TPTLQP

0 100 200 3000.6

0.8

1(b) E[ΠSC]/E[ΠF B] vs. Demand Uncertainty

σ2

0

WS,T

PT

and

LQ

P

0 100 200 3000.98

0.99

1

QP

0 1 2 3 4 5 6 71

2

3

4

5(c) ν/νF B vs. Cost of Demand Forecast

cf

QPWS-TPTLQP

0 2 4 60.8

0.9

1

(d) E[ΠSC]/E[ΠF B] vs. Cost of Demand Forecast

cf

WS,T

PT

and

LQ

P

0 2 4 60.98

0.99

1

QP

QPTPTWSLQP

QPTPTWSLQP

Figure 2: The effect of consumer market demand uncertainty and the cost of investment in demandforecasting on overinvestment and supply chain surplus. Panel (a) plots demand forecast investmentratio (ν/νFB) and panel (b) plots supply chain profit ratio (E [ΠSC ] /E

[ΠFB

]) with respect to the

level of demand uncertainty (σ20). Panel (c) plots demand forecast investment ratio (ν/νFB) and panel

(d) plots supply chain profit ratio (E [ΠSC ] /E[ΠFB

]) with respect to cost of investment in demand

forecasting (cf ). The remaining parameters for all panels and the legends are as given in Figure 1.

and the globally optimal scheme in the general class of quadratic contracts also widens. Finally, when

the marginal cost of investment in precision of demand forecasting (cf ) increases, investment in demand

forecasting decreases in equilibrium for the common contracting schemes as well as the first-best. As a

result, the profit losses due to demand uncertainty and acquisition of private information become less

significant, and the surplus efficiency ratios increase, as can be seen in panel (d) of Figure 2. This effect

is preserved for the globally optimal quadratic scheme with sufficiently intense downstream competition,

as it is also stated in part (ii)(b) of Proposition 6. However, when the number of downstream retailers

is relatively small, the surplus ratio is not necessarily increasing in cf . For lower cf values, the losses

due to overinvestment is low. As a result, the quadratic contracting scheme achieves near first-best

optimal supply chain surplus. As cf increases, the optimal quadratic scheme loses efficiency relative

to the first-best and consequently, the surplus relative to the first-best can decrease. For larger cf

20

values, investment in demand forecasting and the corresponding surplus loss with quadratic contracting

diminish. Therefore the surplus ratio for the globally optimal quadratic contracting scheme can be

nonmonotonic with cf as can be seen in panel (d) of Figure 2.8

6 A Market-Based Contracting Scheme for Full Coordination

In Section 4, we have observed that the recently suggested phenomenon of overinvestment in demand

forecasting can occur under common contracting schemes with downstream competition in supply

chains. In addition, we have discovered that overinvestment can cause significant losses in supply chain

surplus in Section 5. These results call for the question how a contracting scheme can fully coordinate

the supply chain in both production quantities and investment in demand forecasting. In Section 4.2, we

discussed the fact that such a contracting scheme should make use of retailers’ private information and

the correlation among such information, when determining the payments before observing the retailers’

private information. In this section, we propose a contracting scheme that can achieve full supply chain

coordination including production quantities and investments in demand forecasting. Before we present

this solution, in Section 6.1, we first take a closer look into the sources of inefficiency in the setting we

analyze. This discussion clarifies why standard contracting schemes fail to coordinate the supply chain,

and provides insights into the issues that the coordinating contracting scheme should resolve.

6.1 Sources of Inefficiency

As we discussed in Section 4.1, there are two main sources of inefficiency in supply chain contracting

with investment in private demand forecasting.

The first main source of inefficiency, as with all models that deal with decentralized decision mak-

ing in supply chains and coordination, is vertical disintegration. The supplier and the retailers make

decisions to optimize their own disparate profit functions. This results in decisions that misalign the

consumer prices and the quantities produced in the supply chain. A usual way that this misalignment

demonstrates itself is through the common double-marginalization effect: Instead of a central decision

maker determining a single margin for the product, both the buyers and the seller independently add

margins on the product, which reduces the production quantities and the supply chain surplus sig-8Note that the corresponding graph and the underlying intuition for the parameter γ would be similar to that for cf .

21

nificantly. The double-marginalization effect is usually eliminated by removing the margins between

the supplier and the retailer by setting the wholesale price to c0. This corresponds to the case with

w0 =w2 =0 and w1 = c0 in our setting. For n = 1, as we discussed above in Section 4.1, the investment

in demand forecasting in equilibrium is the same as the first-best. Therefore, by Proposition 1, for

n = 1 and all s1, Q(s1)|Pi(q)=c0·qi= QFB(s1). That is, for n = 1, simply eliminating supplier marginal-

ization fully coordinates the supply chain. However, for n ≥ 2, it is not only the case that there is

overinvestment in demand forecasting as stated in Proposition 2, but also, by plugging (3) in αq0,

E[Q]∣∣Pi(q)=c0·qi

=n(K0 − c0)(n + 1)γ

>E[QFB] =E[Qtpt] =K0 − c0

2γ>E[Qws] =

n(K0 − c0)2(n + 1)γ

. (7)

That is, when there is downstream competition, eliminating double-marginalization by making the

wholesale price equal to the marginal production cost c0 does not solve the problem of coordination. In

fact, it leads to overproduction in expectation as (7) indicates, which brings us to the second source of

inefficiency.

The second main source of inefficiency in the market is the downstream fragmentation. This has

three consequences on misalignment:

The first consequence is the misalignment of production quantity decisions among the competing

downstream retailers. Given that a retailer is facing competition from other retailers, the value of the

marginal unit he sells is higher for him compared to the value of that unit for the supply chain, as

he keeps the revenue he makes from that unit while its price impact reduces the revenues of all other

retailers. As a result, in equilibrium, eliminating supplier margins leads to overproduction and the

supply chain surplus suffers. A successful coordinating contracting scheme should address this source

of misalignment simultaneously while resolving the misalignments caused by uncertainty in the market

and demand forecast investments, as we discuss below.

The second misalignment of incentives caused by downstream competition stems from uncertainty

in the market and decision making with demand forecast signals. Having a signal to predict demand

and facing competition from the other retailers in the market who also act utilizing their own signals,

each retailer reacts to his signal in a way that differs from the supply chain best action. As a result,

22

a successful contracting scheme should take into account each retailer’s reaction to his private demand

signal considering the equilibrium behavior under competition. Here, the fact that retailers have disperse

private signals about the realization of the demand curve and hence the order and production decisions

are made based on partial and fragmented information may suggest that pooling information and making

the information available centrally to all retailers can solve the coordination problem. However, with

downstream competition among retailers, pooling information is not sufficient to coordinate the supply

chain. We provide the formal analysis for this case in Appendix C. Miscoordination persists because

the competition among the retailers still distorts their reactions to the pooled signal compared to the

supply chain optimal behavior. Further, pooling information also dramatically changes the incentive

structure for investment in demand forecasting, and in fact results in underinvestment (see Proposition

C.1 in Appendix C). This is because pooling leads to a free-riding effect: If information is pooled, each

retailer bears the cost of his investment fully while the investment benefits the entire supply chain. As a

result, each retailer’s incentive to invest in information acquisition is lower compared to the supply chain

optimum. It is also essential to notice that pooling information would bring about important problems

with implementability. Pooling of private demand signals would require the retailers to truthfully report

their private demand forecasts. However, in such a case, the retailers would strongly prefer to distort

information when reporting their signals to the pool. In addition, even if truthfulness in reporting were

to be exogenously assumed, competing retailers would prefer not to share their private demand signals

with other retailers (see, e.g., Li 2002). These problems are important for the implementation of a

mechanism that coordinates the supply chain under demand uncertainty and private signals. In Section

6.2 we will show how these problems can be solved by the contracting mechanism we propose.

Finally, the third misalignment that downstream competition causes, namely misalignment in in-

centives in demand forecast investment, emerges because the competition among retailers makes the

precision of individual signals more valuable to individual retailers compared to the value of that preci-

sion for the supply chain, as we discussed in Section 4.1. This distortion also needs to be resolved with

the other sources of misalignment simultaneously in the coordinating contracting mechanism. In the

next section, we will show how this can be achieved by employing a “market-based” mechanism that

implicitly utilizes the existing information in the channel through contracting.

23

6.2 The Market-Based Contracting Scheme

Consider the pricing scheme offered by the supplier to retailer i, Pi(q) = w0 + (w1 + w2∑

j 6=i qj) qi.

In this scheme, the price that each retailer pays depends not only on the quantity he orders from the

supplier, but also on the total quantity ordered. In that sense, this pricing scheme contains a “market-

like” flavor, since it ties the price to the eventual total consumer demand for the product, and adjusts

the price accordingly. Note that Pi(q) can be rewritten as

Pi(q) = w0 + p(q)qi − w2 q2i , (8)

where p(q) = w1 + w2∑n

j=1 qj . That is, the coordinating pricing scheme can be viewed as a quadratic

contract, with a market outcome based linear price index. When w2 > 0, the price index is increasing

in total production quantity. In this sense, this contracting scheme incorporates the market’s opinion

into pricing. If the retailers receive high demand signals, their order quantities will be higher, which

will, in turn, push the price up. That is, higher signals about the demand will mean higher contracting

prices and vice-versa. We call this contracting scheme the market-based contracting and denote it with

superscript m.

Notice that from another point of view, this pricing scheme utilizes externality pricing. Externality

pricing schemes, such as VCG mechanisms, are known to effectively implement efficient outcomes in

the case of private valuations and signals (see, e.g., Bergemann and Valimaki 2002). However, there

is no guarantee that such mechanisms can achieve efficient allocations in Bayesian equilibrium when

the valuations are common and signals are correlated even without information acquisition by agents.9

In addition, there is no guarantee that such a mechanism will implement an efficient outcome for the

case of correlated signals or independent signals conditional on the state of the world (K) even with

no downstream competition.10 In our case, the uncertainty of each retailer’s profit depends not only

on his signal (si) but also on the other retailers’ signals (s−i = (s1, . . . , si−1, si+1, . . . , sn)). Moreover,

each retailer’s signal is correlated with other retailers’ signals. Finally, due to downstream competition,9See e.g., Example 3 in Dasgupta and Maskin (2000).

10See Cremer and McLean (1985, 1988) and Obara (2003), who show that even with no interdependence of utilitiesfor the agents (i.e., with no downstream competition in a single market for our case), it is still impossible to achieve fullcoordination even when one considers only a Bayesian equilibrium which is not necessarily “regret-free” (or ex-post). Wewill discuss the concept of regret-freeness below and demonstrate that our contracting scheme, in addition to achievingfull coordination, also satisfies this desirable property.

24

the expected profit for each retailer depends on both the quantity he orders and the quantities that

the other retailers order. Under these conditions, the following proposition states that a market-based

contracting scheme given in (8) can coordinate the supply chain fully, i.e., both in production quantities

statewise and in investment in demand forecasting.

Proposition 7 Consider the pricing scheme in (8). For any given w0, w1 and w2, there exists a

unique linear equilibrium. In equilibrium, qi(si) = αm0 + αm

s (si − K0), and νi = νm for all i, where

αm0 = (K0−w1)/(2γ+(n−1)(γ+w2)), αm

s = νm/(2γ+((n+1)γ+(n−1)w2)νm), νm = ν∗ · 1{σ20≥4γ C′(0)},

and ν∗ is the unique solution to the equation

γ σ20

(2γ + ((n + 1)γ + (n− 1)w2)ν)2− C ′(ν) = 0 . (9)

Further, there is a unique contract in the class defined by (8) that in equilibrium achieves the full supply

chain coordination in both statewise production quantities and in investment in demand forecasting and

in which the supplier extracts all supply chain surplus. In this contract wm1 = c0, wm

2 = γ, and

wm0 =

(K0 − c0)2

4γn2+

σ20 νFB

4γ (1 + νFB)·(

2 + (n + 1)νFB

2(1 + nνFB)

)2

− C(νFB) . (10)

An immediate observation from Proposition 7 is that, since w2 = γ > 0, writing the coordinating

contract in the market price index form given in (8), we can see that the coordinating contract offers

quantity discounts to the retailers. That is, it can be viewed as combination of a linear price that depends

on a market index and quantity discounts. Further, notice that for the contract suggested in Proposition

7, νm = νFB. Given this observation, for any s, the equilibrium total production quantity as a function

of the signals received by the retailers is identical to QFB(s), i.e., full statewise production quantity

coordination is achieved. In addition, the supplier maximizes her own profit by using this pricing

scheme having all retailers engage in the contract voluntarily and act in a decentralized manner. One

important point is that the market-based contracting achieves coordination with truthful information

sharing by the retailers in an incentive compatible way. That is, in this scheme each oligopolist retailer

indirectly reports his signal as an optimal action without the need for an exogenous assumption of

truth-telling. This is in contrast to the existing models in the literature (see, e.g., Gal-Or 1985 and

25

the follow-up studies), which artificially assume that competing oligopolists would report their signals

truthfully, which is a problematic assumption, given that the competing retailers would obviously have

incentives to distort their forecasts when sharing it with their competitors.

A further important point is on the actual incentives to share private demand forecasts. As the lit-

erature on information sharing in oligopoly demonstrates, even if one exogenously assumes truth-telling

by the oligopolists, when asked to share their demand signals, they choose not to do so (see, e.g., Li

2002). With the market-based contracting scheme as given in Proposition 7, each retailer willingly and

endogenously reveals his demand signal through his order quantity, and this information gets incorpo-

rated in the supply chain outcome. In short, by utilizing implicit dissemination of information through

order quantities, the contracting scheme we present not only addresses the demand forecast sharing

problem posed in the literature by inducing retailers to reveal their demand signals, but also induces

them to report their signals truthfully in an endogenous way, without relying on artificial truth-telling

assumptions.

Another important issue here is implicit dissemination of information from the implementation of

this contract. Given that a retailer’s paid price depends on the quantities and hence the forecasts of

competing retailers, each retailer’s optimal quantity is endogenously tied to his competitors’ demand

forecasts. That is, when pricing of a contract is sensitive to the market outcome, each retailer can infer

to other retailers’ signals from the contract outcome. This can in many cases create implementation

problems. For a contracting scheme to be implementable, it is important that the outcome of the

contracting scheme be “regret-free” in the sense that the quantity each retailer would have preferred to

order does not change after he observes the outcome of the contract and as a result draws inferences

about other retailers’ signals. That is, given the mechanism and after observing the outcome, no retailer

should want to change his equilibrium order quantity given what he knew after the contract outcome

was revealed.11 Formally, the equilibrium order quantities are regret-free if

E [Πi(qi,q−i, νi)|si, Pi(q)] ≥ E[Πi(q′i,q−i, νi)|si, Pi(q)

], (11)

for any alternative order quantities q′i, for 1 ≤ i ≤ n. After each retailer submits his order quantity, the

11See, e.g., Cremer and McLean (1985) and Klemperer and Meyer (1989) for extensive discussions of this issue.

26

supplier informs each retailer of his payment, Pi(q), according to the announced contracting scheme.

Then each retailer is able to infer the other retailers’ signals as reflected on the price he should pay,

which, in turn, reflects the quantities that the retailer’s competitors ordered. As a result, each retailer

updates his information about the realization of the demand due to the correlation of the demand signals.

In addition, he obtains information about the order quantities of his competitors. If the equilibrium

strategy profile also satisfies (11), then each retailer, despite having made his decision of order quantity

solely based on his own signal, does not want to change his decision even after having obtained the

additional information from the other retailers’ signals by inference based on the contracting outcome.

That is, he can submit his order quantity based only on the information he has, knowing that it will be

optimal for each realization of the contracting outcome. We subsequently show that in the market-based

contracting scheme, given this updated information, each retailer’s order quantity is regret-free.

Proposition 8 The equilibrium under the market-based pricing scheme is regret-free, i.e., equilibrium

q as given in Proposition 7 satisfies (11).

Proposition 8 states that the market-based contracting scheme manages to keep the incentives aligned,

in the sense that all retailers are satisfied with the quantities, that they ordered based only on their own

signals, even after they obtain additional information leaked through the contracting mechanism. In

fact, this contracting scheme facilitates horizontal information sharing among the competing oligopolist

retailers in the supply chain, as well as the information sharing between the retailers and the upstream

supplier. In this sense, the market-based contracting scheme achieves information pooling without

explicitly pooling the signals, but rather making efficient use of dispersed information in the supply

chain and inducing the retailers to make the decisions that are optimal for the supply chain in two

steps: First, the contracting structure ensures that each agent takes the first-best actions had they

possessed the full information. Second, the regret-freeness property ensures that in each realization of

the signals, the decision that each retailer makes is the same as the one that he would have made given

the information of all the other retailers. As a result, full and efficient utilization of the entire information

in the supply chain is achieved. Furthermore, this contracting scheme is easily implementable through

aggregation of orders by a central off-line or online system. That is, the market-based contracting scheme

we present achieves and implements full supply chain coordination as well as full information sharing,

27

while preventing the information leakage in the process from destroying the alignment of incentives in

decentralized decision making.

7 Concluding Remarks

In this paper we analyzed a phenomenon that can significantly undermine supply chain performance.

Specifically, we showed that in a decentralized supply chain, when there is downstream competition

and the competing retailers invest in demand forecasting, under common contracting schemes, such

as wholesale price contracting and two-part tariffs, the retailers overinvest in demand forecasting. We

also studied the extent of inefficiencies that can occur under such contracting schemes, showing that

in the wholesale price and two-part tariff schemes, total investment in demand forecasting can amount

to as much as twice the efficient level. Further, the resulting supply chain inefficiency can reach near-

full loss levels. Exploring the bounds of inefficiency on the general quadratic contracting schemes, of

which the common wholesale and two-part tariff schemes are subsets, we also find that even with the

optimal contracts in this class, losses in surplus can amount to 50% of the surplus of the first-best case.

Consequently, we suggested an implementable contracting scheme that fully coordinates the supply chain

in both production quantities and investment in demand forecasting. Our scheme relies on charging

the retailers in a manner that combines quantity discounts with a linear price index based on the total

orders of the retailers; in this sense, it is a “market-based” contracting scheme. We demonstrated

that this scheme achieves coordination with truthful information sharing and is also “regret-free,” in

the sense that although the retailers make their order decisions based only on their private signal and

before observing the outcome of the contract mechanism, their equilibrium orders will be optimal even

after they observe the contracting outcome.

In this paper, as it is the case with models that examine oligopolistic competition with correlated

information (see the papers we cite in Section 2 on the subject), we assumed that the retailers were

ex-ante symmetric. In our case, this means that the retailers are symmetric in their demand forecast

investment cost functions. As it is the case with all related models, such symmetry of the retailers are

needed for tractability, since without the assumption of symmetric cost functions, the analysis would be

intractable. It should also be noted that the equilibria we study in the paper is not restricted, and the

28

symmetry in the strategies of the players arise endogenously in fully unrestricted Bayesian equilibria.

In this study, one of the main issues we are interested in is the ability of contract structures to

coordinate the supply chain. Hence, as in many other studies in the literature that explore channel

coordination (see Cachon 2003 and Chen 2003), we assumed the supplier has “full bargaining power”

and makes a take-it-or-leave-it offer to the retailers. This enables the supplier to maximize the supply

chain profit and extract the surplus from the retailers, leaving them with their reservation profits. Under

other bargaining power structures it may not be possible to assure that the supplier will optimize the

supply chain profit in her optimal contracts. As in all supply chain models, in certain cases, such lack

of full bargaining power may inhibit supply chain surplus maximization. However, the results on the

existence of supply chain coordinating contracts, or lack there of, remain intact.

Every year, companies invest millions of dollars in software, personnel and resources to accurately

predict demand. Unfortunately, in numerous cases such efforts result in disappointments that negatively

affect the entire supply chain performance. Our analysis demonstrates that the significant overinvest-

ment in demand forecasting often occurs under common contracting schemes, contributing to substantial

supply chain inefficiency. In addition, our results suggest that contracting schemes designed to be more

sensitive to the market’s pulse can efficiently facilitate improvements on the solution of this problem.

Consideration of such schemes by supply chain partners can ultimately help reduce inefficiencies in the

supply chain and significantly contribute to improve savings and supply chain performance.

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32

Online Supplement

Beating the Accuracy Trap: Overinvestment in Demand Forecasting and Supply

Chain Coordination under Downstream Competition

Hyoduk Shin and Tunay I. Tunca

Graduate School of Business

Stanford University

A Mathematical Preliminaries

In this section, we present the formal mathematical notions for the equilibrium and the first-best

benchmark, and provide the analysis and the derivation for the first-best benchmark. Note that retailer

i’s profit is

Πi(qi,q−i, νi) , qi · (K − γQ)− Pi (qi,q−i)− C(νi) , (A.1)

the supplier’s profit is

ΠS(q) ,n∑

i=1

Pi(qi,q−i)− c0Q , (A.2)

and the total supply chain profit is

ΠSC(q,v) , ΠS(q) +n∑

i=1

Πi(qi,q−i, νi) = Q (K − γQ− c0)−n∑

i=1

C(νi) , (A.3)

where v = (ν1, . . . , νn).

A. 1 Definition of Equilibrium

For all contracting schemes examined in this paper, we explore the Bayesian Nash equilibrium of the

game. In equilibrium, retailer i, 1 ≤ i ≤ n, selects his profit maximizing order quantity qi and signal

precision νi, for i = 1, . . . , n, given the contracting scheme offered by the supplier. That is,

E [Πi(qi,q−i, νi)|si] ≥ E[Πi(q′i,q−i, ν

′i)|si

], (A.4)

for any alternative order quantities q′i and for any alternative investment levels ν ′i, given the other re-

tailers’ equilibrium investment levels, for 1 ≤ i ≤ n.

A. 2 The First-Best Benchmark

The first-best benchmark is, as usual, the case in the supply chain where the channel is integrated and

all decision making is centralized. Given this, the first-best problem can be formulated as

maxQ(s),v

{E [Q (K − γQ− c0)]−

n∑

i=1

C(νi)

}. (A.5)

EC.1

We denote this first-best case with superscript FB. The following proposition provides the solution to

this problem:

Proposition A.1 The first-best total production quantities and investment level in demand forecasting

are given by

QFB(s) =K0 − c0

2γ+

∑ni=1 νFB

i (si −K0)2γ(1 +

∑ni=1 νFB

i ), (A.6)

νFBi = νFB = ν∗ · 1{σ2

0≥4γ C′(0)} , (A.7)

for i = 1, . . . , n, where 1{·} is the indicator function, and ν∗ is the unique solution to the equation

1(1 + nν)2

· σ20

4γ− C ′(ν) = 0 . (A.8)

The first-best expected total supply chain profit is

E[ΠFB

]=

14γ

((K0 − c0)2 +

nνFBσ20

1 + nνFB

)− nC(νFB) . (A.9)

Proof: First, given s = (s1, . . . , sn) and v = (ν1, . . . , νn), by (A.3),

E[ΠFB|s,v]

= (E [K|s,v]− γQ− c0) Q−n∑

i=1

C(νi). (A.10)

Since K and s satisfy affine conditional expectations property, by Ericson (1969),

E [K|s,v] = K0 +∑n

i=1 νi(si −K0)1 +

∑ni=1 νi

. (A.11)

Taking the first derivative with respect to Q in (A.10), we have Q = (E [K|s,v]− c0) /2γ. Plugging in

(A.11) we obtain (A.6). Second, substituting (A.6) into (A.10), we have

E[ΠFB

]= E

[E

[ΠFB|s]] =

14γ

((K0 − c0)2 +

σ20

∑ni=1 νi

1 +∑n

i=1 νi

)−

n∑

i=1

C(νi) . (A.12)

Now, observe that the first term in (A.12) only depends on the sum of νi’s. Suppose that there exists

i 6=j, such that in the optimum νi 6= νj . Then replacing νi and νj with ν ′i and ν ′j , respectively, where

ν ′i = ν ′j = ν ′ = (νi + νj)/2 and keeping νk constant for all k 6=i, j,∑n

i=1 νi remains unchanged. If C(·)is strictly convex, then C(ν ′i) + C(ν ′j) < C(νi) + C(νj). Therefore in the optimum, νi = νj for all i, j.

If C(·) is linear, then there is a continuum of optima and the symmetric solution is one of them. Thus,

taking the first derivative of (A.12) and plugging in the symmetry, we obtain (A.8). Now, since C is

convex, non-decreasing and non-identically zero, the left hand side of (A.8) is decreasing in ν, and will

be strictly negative as ν →∞. Hence if σ20 ≥ 4γC ′(0), there exists a unique ν ≥ 0 that satisfies (A.8).

If σ20 < 4γC ′(0) on the other hand, ν = 0 will be the optimal ν. Thus (A.7) holds and the concavity of

EC.2

the objective function guarantees the optimality. This completes the proof. ¥Note that if C(·) is strictly convex, the optimal solution is unique. If C(·) is linear, the expected total

supply chain profit is the same as long as the total investment level is the same, given the optimal total

production function. Hence, there is a continuum of optimal solutions, among which the symmetric

optimal solution is one. This continuum includes the solution where the entire supply chain invests in

only one signal as the central planner is indifferent between investing in one signal or multiple ones.

However, when C(·) is strictly convex, although investing in a single cost function to get a single signal

is also in the feasible set of the optimization problem, the planner would not choose that option, since

there are decreasing returns to investment and it is optimal to “spread” the cost equally among as many

cost functions as possible.

B Proofs of Propositions

Proof of Proposition 1: Let qj(sj) = α0j + αsj (sj −K0), α0j, αsj ∈ IR and νj ∈ IR+, for all

j 6= i. Expected profit for retailer i after observing si is

E [Πqi |si] = qiE[K − γ

∑

j 6=i

qj(sj)− γ qi − w1 − w2 qi|si]− C(νi)− w0

= qi

(K0 − w1 +

νi (si −K0)

1 + νi

− (γ + w2)qi

−γ∑

j 6=i

(α0j + αsjE [sj −K0|si]))− C(νi)− w0 . (B.1)

Note that (B.1) is concave in qi if and only if γ +w2 ≥ 0. Since E [sj −K0|si] = νi/(1+ νi) (si−K0), the first order condition for qi from (B.1) is written as

qi =1

2(γ + w2)

(K0 − w1 − γ

∑

j 6=i

α0j +νi

1 + νi

(1− γ

∑

j 6=i

αsj

)(si −K0)

). (B.2)

Observe that qi is linear in si −K0. Substituting (B.2) into (B.1), we have

E [Πqi ] = E [E [Πq

i |si]]

=1

4(γ + w2)

(K0 − w1 − γ

∑

j 6=i

α0j

)2

+σ2

0 νi

4(γ + w2) (1 + νi)

(1− γ

∑

j 6=i

αsj

)2

−C(νi)− w0 . (B.3)

The first order condition for νi from (B.3) is

σ20

4(γ + w2) (1 + νi)2

(1− γ

∑

j 6=i

αsj

)2

− C ′(νi) = 0 , (B.4)

EC.3

and the second order condition,

− σ20

2(γ + w2) (1 + νi)3

(1− γ

∑

j 6=i

αsj

)2

− C ′′(νi) < 0 , (B.5)

is satisfied if γ + w2 > 0, i.e., if γ + w2 > 0, the objective function, (B.3), is strictly concave in νi

and there exists unique maximizer. From (B.2) and (B.4), we have

α0i =1

2(γ + w2)

(K0 − w1 − γ

∑

j 6=i

α0j

), (B.6)

αsi =1

2(γ + w2)

(νi

1 + νi

(1− γ

∑

j 6=i

αsj

)), (B.7)

C ′(νi) =σ2

0

4(γ + w2) (1 + νi)2

(1− γ

∑

j 6=i

αsj

)2

. (B.8)

Summing up (B.6) for all i, we obtain

n∑i=1

α0i =n(K0 − w1)

γ(n + 1) + 2w2

. (B.9)

Substituting (B.9) into (B.6) and simplifying, we obtain αq0. Note that (B.7) can be rewritten as

αsi =νi(1− γ

∑nj=1 αsj)

(γ + 2w2)νi + 2(γ + w2). (B.10)

Plugging (B.10) into (B.8), we have

C ′(νi) =σ2

0(γ + w2)(1− γ

∑nj=1 αsj

)2

((γ + 2w2)νi + 2(γ + w2))2 , (B.11)

for all i. Observe that since (B.11) holds for all i, the first order condition for all νi is identical.

Further, since the difference between the two sides of (B.11) is strictly monotonic for νi > 0, it

can have at most one solution in νi. It follows that νi = ν, for some ν ≥ 0, for all i. Adding up

(B.7) for all i, and plugging in νi = ν, we then have

n∑i=1

αsi =nν

2(γ + w2) + (2w2 + (n + 1)γ)ν. (B.12)

Substituting (B.12) into (B.7) and simplifying, we obtain αqs. Finally, substituting (B.12) into

(B.11), we obtain (1). Since C is convex, non-decreasing and non-identically zero, the left hand

side of (1) is decreasing in ν, and will be strictly negative as ν →∞. Consequently there exists

EC.4

a unique ν ≥ 0 that satisfies (1) if and only if σ20 ≥ 4(γ + w2)C

′(0), with νq = 0 otherwise.

This confirms νq and completes the proof. ¥

Proof of Proposition 2: Under a wholesale price contract, w0 = w2 = 0 and w1 = wws. Then,

by Proposition 1, plugging in αq0 and αq

s, and summing up over all i, the equilibrium total order

quantity is

Qws(s) =n(K0 − wws)

γ(n + 1)+

νws

γ(2 + (n + 1)νws)

n∑i=1

(si −K0) . (B.13)

Plugging (B.13) in, the supplier’s expected profit is

E [ΠwsS ] = E [(wws − c0) Qws] =

n(wws − c0) (K0 − wws)

γ(n + 1). (B.14)

It follows that wws = (K0 + c0)/2 maximizes the supplier’s expected profit. For a two-part tariff

contract, w0 = wtpt0 , w1 = wtpt

1 and w2 = 0. Noticing that the participation constraint for each

retailer must be binding, supplier’s optimal w0 has to equal the expected profit for each retailer.

Then by Proposition 1, plugging in w1 = wtpt1 and w2 = 0 in αq

0 and αqs, calculating the expected

retailer profit and plugging back in E[ΠtptS ], we obtain expected supplier profit as a function of

wtpt1 as

E[Πtpt

S

]=

n (K0 − wtpt1 )

γ(n + 1)

((K0 − c0)− n (K0 − wtpt

1 )

n + 1

)+

nνtpt(1 + νtpt) σ20

γ(2 + (n + 1)νtpt)2−nC(νtpt) . (B.15)

The first order condition for wtpt1 gives (3) and is sufficient for optimality since (B.15) is concave

in wtpt1 . (2) follows by plugging (3) back in αq

0 and αqs and equating expected retailer profit to

the reservation value zero. This proves part (i).

For part (ii), plugging w2 = 0 in (1), notice that since

(1

1 + n+12

ν

)2

>

(1

1 + nν

)2

, (B.16)

for all ν > 0, the left hand side of (1) is always greater than the left hand side of (A.8). Further,

the left hand side of both equations are strictly decreasing in ν. It follows that if νFB > 0, i.e.,

when σ20 > 4γC ′(0), then νws = νtpt > νFB. In addition, by (A.7) and νq, νws = 0 if and only if

νFB = 0. This completes the proof. ¥

EC.5

Proof of Proposition 3: Consider any quadratic pricing scheme that coordinates the produc-

tion quantities statewise. By (A.6), to achieve statewise production quantity, we need

νq

2(γ + w2) + (2w2 + (n + 1)γ)νq=

νFB

2γ(1 + nνFB), (B.17)

K0 − w1

(n + 1)γ + 2w2

=K0 − c0

2γn. (B.18)

For νFB > 0 and n ≥ 2, from (B.17), we obtain

νq =2(γ + w2)ν

FB

2γ + ((n− 1)γ − 2w2) νFB. (B.19)

Substituting (B.19) into (1), using (A.8) and simplifying, we then have

4γ(w2 + γ)

C ′(νFB)C ′

(2(γ + w2)ν

FB

2γ + ((n− 1)γ − 2w2) νFB

)

− (4(νFB)2w2

2 − 4γνFB(2 + (n− 1)νFB)w2 + γ2(2 + (n− 1)νFB)2)

= 0 . (B.20)

Since C is convex, the first term on the left hand side of (B.20) is increasing in w2. Now for

w2 > γ/νFB + (n − 1)γ/2, (B.19) can not be satisfied due to the non-negativity of νq and

therefore we can restrict our attention to the region where w2 ≤ γ/νFB +(n− 1)γ/2. By taking

the first derivative of the second term in the left hand side of (B.20) and plugging in this upper

bound for w2 it then follows that this term is non-decreasing in w2. Therefore the left hand side

of (B.20) is monotonically increasing in w2 for w2 ≤ γ/νFB + (n − 1)γ/2. When w2 = 0, left

hand side of (B.20) is negative. Let

w2 , γ

2(νFB)2

(νFB(2 + (n− 1)νFB) + 1−

√2νFB(2 + (n + 1)νFB) + 1

)∈

(0,

γ

νFB+

(n− 1)γ

2

).

(B.21)

We then have

2(γ + w2)νFB

2γ + ((n− 1)γ − 2w2) νFB

∣∣∣∣w2=w2

=1

2

(√2νFB(2 + (n + 1)νFB) + 1− 1

)> νFB , (B.22)

and hence, again by convexity of C,

4γ(w2 + γ)

C ′(νFB)C ′

(2(γ + w2)ν

FB

2γ + ((n− 1)γ − 2w2) νFB

)∣∣∣∣w2=w2

> 4γ(w2 + γ) . (B.23)

Further,

4(νFB)2w22 − 4γνFB(2 + (n− 1)νFB)w2 + γ2(2 + (n− 1)νFB)2

∣∣w2=w2

= 4γ(w2 + γ) . (B.24)

EC.6

Plugging (B.23) and (B.24) in, it then follows that at w2 = w2, the left hand side of (B.20) is posi-

tive. Hence there exists a unique wq2 ∈ (0, γ/νFB+(n−1)γ/2) that satisfies (B.17). Given this wq

2,

wq1 is uniquely determined from (B.18). Further, setting wq

0 = (E [ΠqSC ]− E [

∑i(w

q1qi + wq

2q2i )]) /n >

0 assures that the supplier extracts all expected supply chain surplus. Finally, note that since

wq2 > 0, the second order conditions for the optimality of qi and νi, i.e., γ + w2 ≥ 0 and (B.5),

are satisfied. Now,2(γ + w2)ν

FB

2γ + ((n− 1)γ − 2w2) νFB

∣∣∣∣w2=

γ(n−1)νFB

2(1+νFB)

= νFB . (B.25)

By (B.25) and plugging in and simplifying, it follows that the left hand side of (B.20) is negative at

w2 = γ(n− 1)νFB/2(1 + νFB). Then the monotonicity of (B.20) implies wq2 > γ(n−1)νFB/2(1+

νFB). Combining this with the monotonicity of (B.19) yields νq > νFB. Therefore the unique

quadratic pricing scheme that achieves statewise supply chain production coordination results in

overinvestment in demand forecasting. That is, simultaneous coordination of quantities in each

state and investments by the full quadratic contracting scheme is impossible. ¥

Proof of Proposition 4: For part (i), since, by Proposition 1, νws = νtpt, we will only give

the proof for νws. Let G(νws) =√

C ′(νFB)/C ′(νws). From (A.8) and (1), we have

νws

νFB=

2

n + 1

(nG(νws) +

G(νws)− 1

νFB

). (B.26)

Since (B.26) is increasing in G(νws), and since by Proposition 2, 0 ≤ G(νws) ≤ 1, νws/νFB

is maximized at G(νws) = 1, which is attainable for C(ν) = cfν, cf > 0. Plugging in (B.26),

we obtain the upper bound given in (4). For the lower bound, first by Proposition 2, we know

that νws > νFB for 4γC ′(0) < σ20. Let ε > 0 be given. Let C : IR+ → IR+ be an increasing,

convex, twice differentiable cost function with 4γC ′(0) < σ20 and define νFB > 0 as the first

best investment level for this function. Let Crε : IR+ → IR+ be an increasing, convex, twice

differentiable function such that

(i) Crε(νFB) = C(νFB), and C ′

rε(νFB) = C ′(νFB); (B.27)

(ii) C ′rε

(νFB + ε

)>

σ20

γ(2 + (n + 1) νFB )2. (B.28)

Define Cε : IR+ → IR+ as

Cε(ν) = C(ν) · 1{0≤ ν≤ νFB} + Crε(ν) · 1{ν>νFB} , (B.29)

Notice that Cε is increasing, convex and twice differentiable. Further, the first best investment

level for Cε is νFB. Define the equilibrium investment level for Cε under wholesale contracting

as νwsε . Then, by (A.8), (1) and (B.28) and the continuity of C ′

ε, we have νFB < νwsε < νFB + ε.

EC.7

It follows that the lower bound in (4) is tight.

To see part (ii), consider linear investment cost case, C(ν) = cfν. In the supplier’s profit

maximizing quadratic scheme, the retailers’ participation constraint will be binding. Therefore,

plugging αq0 and αq

s in (B.1), equating it to zero to solve for wq0, and plugging it in E[Πq

S], we

obtain

E [ΠqS] =

n(K0 − w1) ((γ + 2w2)K0 + nγw1 − (γ(n + 1) + 2w2)c0)

(γ(n + 1) + 2w2)2

+n(γ + 2w2)ν(1 + ν)σ2

0

(2(γ + w2) + (2w2 + (n + 1)γ)ν)2 − ncfν . (B.30)

Note that for any w2, (B.30) is concave and quadratic in w1. Hence for a given w2, the w1 that

maximizes (B.30) is

w1(w2) =(γ(n− 1)− 2w2)K0 + (γ(n + 1) + 2w2)c0

2nγ. (B.31)

Using (B.31) and (1), (B.30) can be simplified to

E [ΠqS] =

(K0 − c0)2

4γ+ ncfν

(1− γ

γ + w2

+

(2− γ

γ + w2

)ν

). (B.32)

Solving (A.8) and (1) for ν, with C(ν) = cf · ν, we obtain

νFB =1

n

(√σ2

0

4γcf

− 1

). (B.33)

and

νq =1

2w2 + (n + 1)γ

(√(γ + w2)σ2

0

cf

− 2(γ + w2)

). (B.34)

Denote y =√

(γ + w2)/γ and z =√

σ20/4γcf . Note that if y > z, then σ2

0 < 4(γ + w2)cf , and

hence νq = 0. Thus it is sufficient to focus on the region where y≤ z. By (B.33) and (B.34), we

then have

limn→∞

νq

νFB=

2(yz − y2)

z − 1. (B.35)

Now, substituting (B.34) into (B.32), we have

E[ΠqS] =

(K0 − c0)2

4γ+

ncf (z − y)

(y2 + (n− 1)/2)2

(2zy2 + y(n− 1)/2− z − (n− 1)/2

y− y3

). (B.36)

Taking the first derivative of (B.36) with respect to y, we see that as n→∞, the first order

EC.8

condition for optimality converges to

2y3 − zy2 − z = 0 , (B.37)

and the second order condition tends to −4cf (y3 + z)/y3 < 0. That is, for sufficiently large n,

(B.36) is strictly concave in y, and the optimal y converges to the solution of (B.37). Note that,

for all z > 0, there exists the unique solution y ∈ (z/2, z) for (B.37). Moreover there exists z

such that for all z≥ z, y ∈ (z/2, 3z/4). Observing that (B.35) is monotonically decreasing in y

for y ≥ z/2, and substituting these bounds in (B.35), we then conclude that for all z≥ z,

limn→∞

νq

νFB∈

(3z2

8(z − 1),

z2

2(z − 1)

). (B.38)

Note that z → ∞ as σ20 → ∞ or cf → 0 or γ → 0. Therefore, for any M > 0, for sufficiently

large n, and sufficiently large σ20 or sufficiently small cf or γ, νq/νFB > M , which proves the

desired statement. ¥

Proof of Proposition 5: Let cf > 0 be given and let C(ν) = cf · ν. Then by (1), we obtain

νws = νtpt =2

n + 1

(√σ2

0

4γcf

− 1

). (B.39)

By (B.33), (B.39), Propositions 1 and 2, and taking expectations we obtain

E [ΠwsSC ]

E [ΠFB]=

n((n + 2) (K0 − c0)

2 + 4(σ2

0 + 4γcf − 4√

γcfσ20

))

(n + 1)2((K0 − c0)2 + σ2

0 + 4γcf − 4√

γcfσ20

) , (B.40)

and

E[ΠtptSC ]

E [ΠFB]=

(n + 1)2 (K0 − c0)2 + 4n

(σ2

0 + 4γcf − 4√

γcfσ20

)

(n + 1)2((K0 − c0)2 + σ2

0 + 4γcf − 4√

γcfσ20

) . (B.41)

Let {vk} ⊂ IN+ × IR4+ be a sequence of parameter vectors such that limk→∞ n(k) = ∞, and

limk→∞

K0(k)− c0(k)√σ2

0(k)− 2√

γ(k)cf

= 0 . (B.42)

Then by (B.40) and (B.41), we have

limk→∞

E [ΠwsSC ]

E [ΠFB]= lim

k→∞E[Πtpt

SC ]

E [ΠFB]= 0 . (B.43)

EC.9

For the upper bound of profit ratio under two-part tariff scheme, note that as σ20 → (4γC ′(0))+,

νFB → 0 from (A.7) and (A.8). Further, plugging in (3), we can see that the optimal two-part tar-

iff scheme achieves the first-best supply chain profit in the limit as νFB → 0. For the upper bound

for the wholesale price scheme, observe that as n→∞, wtpt1 = ((n− 1)K0 + (n + 1)c0) /2n→ (K0+

c0)/2 = wws. Hence as n→∞, E[ΠtptSC ] = E[Πws

SC ], and the upper bound result for the wholesale

price scheme follows similar to that for the two-part tariff scheme. This completes the proof of

part (i).

For part (ii), again let C(ν) = cf · ν. From (A.9) and (B.36), we obtain

Rqπ,∞ , lim

n→∞E[Πq

SC ]

E[ΠFB]=

(K0 − c0)2 + 8γcf (z − y)(y − 1/y)

(K0 − c0)2 + σ20 + 4γcf − 4

√γcfσ2

0

, (B.44)

where y and z are as defined in the proof of Proposition 4. Note that from (B.37), limz→∞ y/z =

1/2. Plugging this, and the fact that z/σ20 = 1/(2

√σ2

0γcf ) in (B.44) and taking the limit as

σ20 →∞, we obtain

limσ20→∞

limn→∞

E [ΠqSC ]

E [ΠFB]=

1

2. (B.45)

That is, the efficiency loss can reach at least half of the first-best supply chain profit as stated.

This completes the proof. ¥

Proof of Proposition 6: Part (i)(a) follows from (B.33) and (B.39), and part (i)(b) follows

from taking partial derivatives in (B.40) and (B.41) with respect to each parameter stated. For

part (ii)(a), from (B.33) and (B.34), we obtain

Rqν , νq

νFB=

2ny(z − y)

(2y2 + n− 1)(z − 1), (B.46)

where y and z are as defined in Proposition 4, with y maximizing (B.36). Note that y > 0 since

γ + w2 > 0. Further, if y≥ z, νq = 0 by νq. Thus for comparative statics of investment ratio, Rqν ,

it is sufficient to consider the case in which y ∈ (0, z), i.e., the case in which optimal y satisfies

the first order condition for (B.36), given as

12zy6 − 4(4z2 − 3(n− 1))y5 − 12z(2n− 1)y4

+ 2(4(n + 1)z2 + (n− 1)(n + 3))y3 + (n− 1)(n + 11)zy2 + (n− 1)2z = 0 . (B.47)

Taking the partial derivatives of Rqν in (B.46) with respect to y and z, we have

∂ Rqν

∂ y=

2n (2(n− 1)y + 2y2z − z(n− 1))

(2y2 + n− 1)2(z − 1)(B.48)

∂ Rqν

∂ z=

2ny(y − 1)

(2y2 + n− 1)(z − 1)2, . (B.49)

EC.10

From (B.47), one can show that y≥ 1, which, by (B.49) implies ∂ Rqν/∂ z≥ 0. Further, applying

the implicit function theorem to (B.47), and since y < z and z > 1, we obtain 0≤ dy/dz≤ 1/2. In

addition, note that that 2·(∂ Rqν/∂ z)≥ |∂ Rq

ν/∂ y|. Since dRqν/dz = ∂ Rq

ν/∂ z+(∂ Rqν/∂ y)·(dy/dz),

we obtain dRqν/dz≥ 0, from which the comparative statics results of part (ii)(a) with respect to

σ20, γ and cf follow. Further,

∂ Rqν

∂ n=

2y(z − y)(2y2 − 1)

(z − 1)(2y2 + n− 1)2> 0 . (B.50)

Further, applying the implicit function theorem to (B.47) and simplifying we can show that

dy/dn ≥ 0. Using (B.47), (B.48) and (B.50), and simplifying, it follows that dRqν/dn = ∂ Rq

ν/∂ n+

(∂ Rqν/∂ y) · (dy/dn) ≥ 0.

For part (ii)(b), using C(ν) = cfν, (A.9) and (B.32), and simplifying, we obtain

Rqπ , E[Πq

SC ]

E[ΠFB]=

(K0 − c0)2 + 4γn(2− 1/y2)νq(1 + νq)cf − 4γncfν

q

(K0 − c0)2 + nνFBσ20/(1 + nνFB)− 4γncfνFB

. (B.51)

Since E[ΠFB]≥E[ΠqSC ], we have nνFBσ2

0/(1+nνFB)−4γncf ≥ 4γn(2−1/y2)νq(1+νq)cf−4γncf .

Also, since y, νq and νFB do not depend on K0 or c0, it then follows that Rqπ is increasing in K0

and decreasing in c0. Now for n, note that from (B.33), nνFB = z − 1. Then, by (A.7), E[ΠFB]

depends neither on n nor on y, which implies that y is the maximizer of Rqπ. By the envelope

theorem, dRqπ/dn = ∂ Rq

π/∂ n. Further, sign(∂ Rqπ/∂ n) = sign(∂ E[Πq

SC ]/∂ n). Using (B.47), we

obtain

∂ E[ΠqSC ]

∂ n= −2cf (2y

2 − 1)(z − y) ((2y2 − 1)(y2 − 2zy + 1) + n(2zy − 3y2 + 1))

y(2y2 + n− 1)3≤ 0 , (B.52)

which proves ∂ Rqπ/∂ n ≤ 0.

Now consider the asymptotic profit ratio Rqπ in (B.44), in which y satisfies (B.37). First, we

can rewrite (B.44) as

Rqπ,∞ =

(K0 − c0)2 + 4

√γcf (σ0 − 2y

√γcf )(y − 1/y)

(K0 − c0)2 + (σ0 − 2√

γcf )2. (B.53)

Applying the implicit function theorem to (B.37), we have

dy

dσ0

=dy

dz· dz

dσ0

=y2 + 1

4y(3y − z)√

γcf

. (B.54)

By (B.54) and (B.37), we then have

dRqπ,∞

dσ0

=∂Rq

π,∞∂σ0

+∂Rq

π,∞∂y

· dy

dσ0

EC.11

=4√

γcf (4γcf (y2 − 1)(z − 1)(2y − z − 1) + (y(y − z + 1)− 1)(K0 − c0)

2)

y ((K0 − c0)2 + 4γcf (z − 1)2)2 ≤ 0 . (B.55)

That is, for sufficiently large n, dRqπ,∞/dσ0 ≤ 0 as stated. Applying the implicit function

theorem to (B.37) once again, we obtain

dy

dγ=

dy

dz· dz

dγ= − σ0(y

2 + 1)

8yγ(3y − z)√

γcf

. (B.56)

It then follows from (B.37) and (B.56) that

dRqπ,∞

dγ=

∂Rqπ,∞

∂γ+

∂Rqπ,∞

∂y· dy

dγ

= −σ3/2 ((z − 1)(y2 − 1)(2y − z − 1)σ20 + z(2y3 − zy2 − (z + 1)y + z)(K0 − c0)

2)

2γy√

γcf (z2(K0 − c0)2 + (z − 1)2σ20)

2 ≥ 0 . (B.57)

Therefore, Rqπ is increasing in γ for sufficiently large n. The proof for cf follows similarly. ¥

Proof of Proposition 7: Let qj(sj) = α0j + αsj (sj −K0), for α0j, αsj ∈ IR and νj ∈ IR+, for

all j 6=i. Expected profit for retailer i after observing si under the given pricing scheme is

E [Πmi |si] = qiE[K − γ

∑

j 6=i

qj(sj)− γ qi − w1 − w2

∑

j 6=i

qj(sj)|si]− C(νi)− w0

= qi

(K0 − w1 +

νi (si −K0)

1 + νi

− γ qi

−(γ + w2)∑

j 6=i

(α0j + αsjE [sj −K0|si]))− C(νi)− w0 . (B.58)

Note that since γ > 0, (B.58) is concave in qi. Using the fact that E[sj −K0|si] = (νi/(1 + νi)) ·(si −K0), the first order condition for qi from (B.58) is written as

qi =1

2γ

(K0 − w1 − (γ + w2)

∑

j 6=i

α0j +νi

1 + νi

(1− (γ + w2)

∑

j 6=i

αsj

)(si −K0)

). (B.59)

Observe that qi is linear in si − K0. Substituting (B.59) into (B.58) and again plugging in

E[sj −K0|si] = (νi/(1 + νi)) · (si −K0), we have

E [Πmi ] = E [E [Πm

i |si]]

=1

4γ

(K0 − w1 − (γ + w2)

∑

j 6=i

α0j

)2

+σ2

0νi

4γ(1 + νi)

(1− γ

∑

j 6=i

αsj

)2

− C(νi)− w0. (B.60)

EC.12

The first order condition for νi from (B.60) is

σ20

4γ (1 + νi)2

(1− (γ + w2)

∑

j 6=i

αsj

)2

− C ′(νi) = 0 . (B.61)

And the second order condition,

−σ20

2γ (1 + νi)3

(1− (γ + w2)

∑

j 6=i

αsj

)2

− C ′′(νi) < 0 , (B.62)

is satisfied for all νi ≥ 0. From (B.59) and (B.61), we have

α0i =1

2γ

(K0 − w1 − (γ + w2)

∑

j 6=i

α0j

), (B.63)

αsi =νi

2γ (1 + νi)·(

1− (γ + w2)∑

j 6=i

αsj

), (B.64)

C ′(νi) =σ2

0

4(γ + w2)·(

1− (γ + w2)∑

j 6=i αsj

1 + νi

)2

. (B.65)

Summing (B.63) over all i, we obtain

n∑i=1

α0i =n(K0 − w1)

2γ + (n− 1)(γ + w2). (B.66)

Plugging (B.66) back into (B.63) and simplifying, we have αm0 . Solving αsi in (B.64), we obtain

αsi =νi

(1− (γ + w2)

∑nj=1 αsj

)

2γ + (γ − w2)νi

. (B.67)

Substituting (B.67) into (B.65), we have

C ′(νi) =γ2σ2

0

γ + w2

(1− (γ + w2)

∑nj=1 αsj

2γ + (γ − w2)νi

)2

, (B.68)

for all i. Since (B.68) holds for all i, νi satisfies the same equation for all i. Further, (B.68) can

not have multiple solutions for νi since the left hand side is increasing whereas the right hand

side is strictly decreasing in νi. Hence νi = ν, for ν ≥ 0. Aggregating (B.64) over all i and

substituting νi = ν, we obtain

n∑i=1

αsi =nν

2γ + ((n + 1)γ + (n− 1)w2) ν. (B.69)

EC.13

Plugging (B.69) back into (B.64) and simplifying, we have αms . Substituting νi = ν, for all i, and

simplifying (B.65), we obtain (9) for each retailer i, 1 ≤ i ≤ n. The existence and uniqueness

of the solution to (9) and the validity of νm can be shown as in the proof of Proposition 1. Now,

(9) is identical to (A.7) if and only if wm2 = γ. Further, given wm

2 = γ and νmi = νFB

i and by

(A.6), αm0 and αm

s , the total supply chain production quantity is the same as the first-best total

supply chain production quantity for any s, i.e., the statewise production quantity coordination

is achieved, if and only if wm1 = c0. This completes the proof. ¥

Proof of Proposition 8: When retailer i observes the price the supplier asks him to pay, i.e.,

Pi(qi,q−i) = wm0 + c0 qi + γ (

∑j 6=i qj) qi, he can refer to

∑j 6=i qj as

∑

j 6=i

qj =Pi(qi,q−i)− wm

0

γqi

− c0

γ. (B.70)

By (B.70) and Proposition 7, it then follows that

∑

j 6=i

sj = (n− 1)K0 +1

αms

(Pi(qi,q−i)− wm

0

γqi

− c0

γ− (n− 1)αm

0

). (B.71)

That is, in equilibrium, each retailer i can infer the sum of the remaining retailer’s demand

signals after observing the price the supplier asks him to pay, Pi(q). Now, since by Proposition

7, qj(sj) = (K0−c0)/2nγ+νm/2γ(1+nνm) (sj−K0), and since E[K|si,∑

j 6=i sj] = K0+(νm/(1+

n νm))∑n

j=1(sj −K0), we have

E[Πmi |si,

∑

j 6=i

sj] = qi E[K − γ∑

j 6=i

qj − γqi − c0 − γ∑

j 6=i

qj |si,∑

j 6=i

sj]− wm0 − C(νi)

= qi

(K0 − c0

n+

νm (si −K0)

1 + nνm− γ qi

)− wm

0 − C(νi) . (B.72)

Since γ > 0, (B.72) is concave in qi and taking the first order condition we obtain the optimal

order quantity function identical to the one derived in Proposition 7. Therefore (11) is satisfied.

This completes the proof. ¥

C The Analysis of the Pooled Information Case

In this section, providing the formal analysis for the pooled information case we discussed in Section

6.1, we explore the setting where the signals of all retailers are shared or visible to all other retailers.

That is, we study the case in which the signals for all retailers are collected in an information pool in

the style of the earlier literature (see, e.g., Li 2002 and the papers cited therein). We demonstrate that,

pooling information, even when one assumes that the retailers would share their information truthfully,

does not solve the investment coordination problem. Further, for the common contracting schemes, it

EC.14

leads to underinvestment. The following proposition states this result.

Proposition C.1 Assume that the retailers share their demand signals truthfully. It is not possible

to coordinate the production quantities statewise and investment in demand forecasting simultaneously

under any contracting scheme in the class of quadratic pricing schemes, Pi(qi) = w0 + w1qi + w2q2i .

Further, for the wholesale price and the two-part tariff contracting schemes, under any equilibrium

where retailers make positive investment in demand forecasting, they underinvest compared to the first-

best case.

Proof: Suppose that retailers pool their signals on market demand. We conjecture that qj(s) =

α0j +∑n

k =1 αsjk(sk −K0), for α0j , αsjk ∈ IR and νj ∈ IR+, for all j 6=i. Expected profit for retailer i

after observing the pooled information s under any quadratic pricing scheme can then be written as

E [Πqi |s] = qiE[K − γ

∑

j 6=i

qj(sj)− γ qi − w1 − w2 qi|s]− C(νi)− w0

= qi

{K0 − w1 +

∑nk =1 νk(sk −K0)1 +

∑nk =1 νk

− (γ + w2)qi

−γ∑

j 6=i

(α0j +

n∑

k =1

αsjk(sk −K0)

)}− C(νi)− w0 . (C.1)

As before, the retailer objective is concave in qi if and only if γ + w2 > 0. From (C.1), we can obtain

the first order condition for qi as

qi =1

2(γ + w2)

{K0 − w1 − γ

∑

j 6=i

α0j +n∑

k =1

νk

1 +∑n

m =1 νm− γ

∑

j 6=i

αsjk

(sk −K0)

}. (C.2)

Note that qi is linear in sk −K0. Substituting (C.2) into (C.1), we have

E [Πi] = E [E [Πi|s]]

=1

4(γ + w2)

{K0 − w1 − γ

∑

j 6=i

α0j

2

+n∑

k =1

σ20(1 + νk)

νk

νk

1 +∑n

m=1 νm− γ

∑

j 6=i

αsjk

2 }

+σ2

0

4(γ + w2)

{n∑

k =1

∑

l 6=k

νk

1 +∑n

m=1 νm− γ

∑

j 6=i

αsjk

νl

1 +∑n

m=1 νm− γ

∑

j 6=i

αsjl

}

−C(νi)− w0 . (C.3)

Taking the derivative of (C.3) with respect to νi and simplifying, we obtain

σ20

4(γ + w2)

1

(1 +∑n

k=1 νk)2 −

γ2

ν2i

∑

j 6=i

αsji

2− C ′(νi) = 0 . (C.4)

EC.15

Since qi = α0i +∑n

k =1 αsik(sk −K0), from (C.2), we obtain the following equations for all i = 1, . . . , n:

2(γ + w2)α0i = K0 − w1 − γ∑

j 6=i

α0j , (C.5)

2(γ + w2)αsij =νj

1 +∑n

m=1 νm− γ

∑

k 6=i

αskj , ∀ j . (C.6)

Summing up (C.5) over i, we obtain

n∑

i=1

α0i =n(K0 − w2)

(n + 1)γ + 2w2. (C.7)

By substituting (C.7) into (C.5), we have

α0i =K0 − w1

(n + 1)γ + 2w2. (C.8)

From (C.6), it follows that

αsji =1

γ + 2w2

(νi

1 +∑n

m=1 νm− γ

n∑

m=1

αsmi

), (C.9)

for all i, j. By (C.9), note that αsji =αski, for all j and k, and hence we obtain

αsji =νi

((n + 1)γ + 2w2) (1 +∑n

k=1 νk), (C.10)

for all i and j. Substituting (C.10) into (C.4), we have

σ20(nγ + w2)

(1 +∑n

k=1 νk)2 ((n + 1)γ + 2w2)2

− C ′(νi) = 0 , (C.11)

for i. Since the first term in (C.11) is the same for all i, we have νi = ν. Further there exists at most

one solution of νi to (C.11). Hence it follows that

νp = ν∗ · 1{(nγ+w2)σ20≥((n+1)γ+2w2)2C′(0)} , (C.12)

where ν∗ is the unique solution to the equation:

σ20(nγ + w2)

(1 + nν)2((n + 1)γ + 2w2)2− C ′(ν) = 0 . (C.13)

Plugging w2 = 0 in (C.13), notice that since

σ20

4γ(1 + nν)2>

nσ20

(n + 1)2γ(1 + nν)2, (C.14)

for all ν, i.e., the left hand side of (C.13) is always less than the left hand side of (A.8). Further, the

EC.16

left hand side of both equations are strictly decreasing in ν. It follows that νwsp = νtpt

p < νFB.

To coordinate the investment level, w2 should be set such that (C.13) matches with (A.8). This

implies that w2 has to solve

4w22 + 4nw2 + (n− 1)2γ2 = 0 . (C.15)

subject to γ + w2 > 0. The smaller of the two roots to (C.15) fails to satisfy the constraint. The larger

root is

w2 =γ

2(√

2n− 1− n)

, (C.16)

which yields γ+w2≤ 0 for n≥ 5. When n≤ 4, to coordinate production quantities statewise, from (A.6)

and (C.10), and plugging in (C.16), we have

n√2n− 1 + 1

=12

, (C.17)

which holds only when n=1. Hence, we conclude that it is impossible to coordinate the supply chain

fully using any quadratic pricing schemes under pooled information when n≥ 2. ¥

EC.17