consumer surplus under demand uncertainty

Consumer Surplus Under Demand Uncertainty

Maxime C. Cohen*Desautels Faculty of Management, McGill University, Montreal, Quebec H3A 1G5, Canada, [email protected]

Georgia PerakisSloan School of Management, MIT, Cambridge, Massachusetts 02139, USA, [email protected]

Charles ThravesDepartment of Industrial Engineering, University of Chile, Santiago de Chile, 8370456, Chile, [email protected]

C onsumer Surplus is traditionally defined for the case where demand is a deterministic function of the price. How-ever, demand is usually stochastic and hence stock-outs can occur. Policy makers who consider the impact of differ-

ent regulations on Consumer Surplus often ignore the impact of demand uncertainty. We present a definition of theConsumer Surplus under stochastic demand. We then use this definition to study the impact of demand and supplyuncertainty on consumers in several cases (additive and multiplicative demand noise). We show that, in many cases,demand uncertainty hurts consumers. We also derive analytical bounds on the ratio of the Consumer Surplus relative tothe deterministic setting under linear demand. Our results suggest that ignoring uncertainty may severely impact theConsumer Surplus value.

Key words: consumer surplus; demand uncertainty; OM-economics interface; rationing capacityHistory: Received: July 2019; Accepted: July 2021 by Haresh Gurnani after 2 revisions.*Corresponding author.

1. Introduction

Customer welfare is often measured by using a metriccalled the Consumer Surplus. This metric was pro-posed in 1844 by Jules Dupuit who defined the Con-sumer Surplus as the difference between consumers’willingness to pay and market price. Subsequently,this concept was extensively studied in the economicsliterature and applied to a multitude of domains. Inthese applications, however, it is assumed that thedemand function is deterministic and that the prod-ucts are always available (i.e., no stock-outs).In real-world settings, demand is stochastic by nat-

ure. Data scientists and statisticians constantly aim toimprove demand prediction algorithms, but nomethod will consistently yield a perfect prediction.Consequently, firms will do their best to predictdemand but prediction errors remain inevitable. Theoperations management (OM) community has pro-posed to study such settings by considering a stochas-tic demand function and intends to make optimaldecisions under uncertainty. OM researchers havestudied various problems such as supply chain man-agement, revenue management, and queuing sys-tems. While the focus has mainly been on the firm’sperspective, several recent research trends (e.g., sus-tainable operations) study the problem from the

consumers’ perspective. In particular, policy makersand regulators are often interested in assessing theimpact of specific policies or interventions on con-sumers. In this context, the relevant question for apolicy maker is whether the standard (presumablydeterministic) approach to calculate the ConsumerSurplus ends up overstating its value and, hence,paint an inaccurate picture. To measure customerwelfare in such settings, one needs a general defini-tion of Consumer Surplus under demand uncertainty,while accounting for the events where products arenot available (i.e., stock-outs).As observed in Krishnan (2010), demand uncer-

tainty renders the welfare analysis more intricate.Including a stochastic term in the demand functionmay hinder the existence of an underlying representa-tive customer utility function and, thus, utility analy-sis may not be possible. Besides this concern, demanduncertainty may also lead to scarcity and stock-outs.Given a price, the demand function represents themaximum number of units to be consumed (or at leastdesired to be consumed). Such a demand functioncaptures multiple consumers, each endorsed with aparticular willingness to pay for the product. Whendemand exceeds supply, not all customers who arewilling to buy the product will be served.1 As a result,the Consumer Surplus will depend on the set of

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consumers who receive the product. We formalizethis intuition by using the notion of capacity alloca-tion rule (see Section 2.2.1).Several prior papers have studied the impact of

demand uncertainty but their focus was mainly onprices, production quantities, and firms’ profits. Aclassical example is the newsvendor problem inwhich the optimal production quantity is derivedwhen the firm only knows the demand distribution(as opposed to the exact realization). Other studies,such as Mills (1959) and Karlin and Carr (1962), com-pared the optimal prices and profits earned by firmsin the stochastic setting relative to their deterministiccounterparts for additive and multiplicative demandnoises. However, these studies do not address theimpact of demand uncertainty on consumers. Asmentioned, assessing the impact of policies or inter-ventions on consumers is of great interest to policymakers. Given the stochastic nature of most real-world settings, one needs a precise way to measurecustomer welfare when demand and supply areuncertain. This paper aims to bridge this gap byproposing a first step in answering this question.What happens if we ignore stock-outs and demand

uncertainty when computing the Consumer Surplus?Obviously, the answer will depend on which cus-tomers receive the limited supply. In this paper, wepresent a formal definition of the Consumer Surplusunder stochastic demand and stochastic supply. Wethen compare the Consumer Surplus relative to thedeterministic setting for various noise structures andcapacity allocation rules. Our model and analysesallow us to answer the question of whether moreuncertainty is good or bad for consumers—and towhat extent. It can thus provide guidelines to firmsand regulators on the potential need for investingefforts in collecting additional data and in developingmore sophisticated prediction methods to reducedemand uncertainty.

1.1. ContributionsIn economics, the concept of Consumer Surplus hasmainly focused on cases where demand is a determin-istic function of the price, so that products are alwaysavailable (see, e.g., Tirole 1988, Vives 2001). As men-tioned, many real-world settings are stochastic by nat-ure and modeled via a stochastic demand function.With the goal of accurately measuring customer wel-fare, this paper offers a first step in generalizing thenotion of Consumer Surplus and examining theimpact of demand and supply uncertainty on con-sumers. We next summarize our contributions.

• Presenting a generalization of the Consumer Sur-plus notion. As discussed, when products arenot necessarily available, the Consumer

Surplus will naturally depend on which cus-tomers receive the limited supply. We usecapacity allocation rules to model the wayavailable units are allocated to consumerswhen demand exceeds supply. We first pro-vide a rigorous definition of a capacity alloca-tion rule. We then present a general definitionof the Consumer Surplus for multiple productsunder stochastic demand. To our knowledge,this paper is the first to propose such a generalextension.

• Studying the impact of demand and supply uncer-tainty on consumers. Armed with our ConsumerSurplus definition, we study the impact ofdemand and supply uncertainty on consumersfor several special cases (multiplicative andadditive demand noises). Specifically, we com-pare the expected Consumer Surplus to thedeterministic setting, under the same prices.We also extend our results to the setting whereprices are endogenously determined. We showthat in many cases, demand uncertainty hurtsconsumers. Under a demand with multiplica-tive noise, consumers are always better off inthe deterministic setting. Interestingly, thisresult holds for any demand function, anynoise distribution, and any allocation rule.Under an additive demand noise, we showthat the impact of demand uncertaintydepends on the allocation rule and on the con-vexity properties of the demand. Finally, weshow that regardless of the type of noise, theexpected Consumer Surplus under stochasticsupply is always lower relative to its deter-ministic counterpart. Ultimately, our resultssuggest that ignoring uncertainty may severelyimpact the Consumer Surplus.

• Deriving analytical bounds on the Consumer Sur-plus ratio relative to the deterministic setting underlinear demand. Under additional assumptions,we derive bounds on the impact of demanduncertainty on the Consumer Surplus. Weshow that the expected Consumer Surplusunder stochastic demand can be as far as 50%relative to the deterministic setting.

1.2. Literature ReviewThe Consumer Surplus is typically defined as the dif-ference between consumers’ willingness to pay andmarket price. This definition was first introduced byJules Dupuit in 1844 as utilite relative. Later on, AlfredMarshall relabeled this concept as Consumer Surplusin the Principles of Economics in 1890, denoting it as theupper triangle of the inverse demand curve. Sincethen, hundreds of academic papers were published

Cohen, Perakis, and Thraves: Consumer Surplus Under Demand Uncertainty2 Production and Operations Management 0(0), pp. 1–17, © 2021 Production and Operations Management Society



on this topic (we cannot give proper credit to theextensive research conducted on this topic). Severalstudies have developed analytical frameworks (see,e.g., Takayama 1982, Willig 1976), whereas othershave focused on applications spanning a multitude ofcontexts.The basic definition of the Consumer Surplus relies

on computing the area between the market price pand the inverse demand curve (see, e.g., Pindyck andRubinfeld 2018, Tirole 1988, as well as section 2.1below). This definition rests on the assumption thatthe product is always available and that there are nostock-outs. Specifically, in most previous work, theframework developed to compute the Consumer Sur-plus focuses on the case where the demand curve is adeterministic function of the price. As mentioned,many real-world settings are modeled using astochastic demand. To measure customer welfare insuch settings, one needs an appropriate definition ofthe Consumer Surplus. Indeed, calculating the Con-sumer Surplus while ignoring stock-outs may lead toa severe overestimate. Furthermore, the correct esti-mate naturally depends on the set of customers whoreceive the limited supply. Extending the notion ofConsumer Surplus under stochastic demand and sup-ply is one of the main goals of this paper.Several papers on peak load pricing and capacity

investments by a power utility under stochasticdemand address partially this modeling issue (seeBrown and Johnson 1969, Crew et al. 1995). Neverthe-less, the models developed in this literature are notapplicable to settings where consumers arrive ran-domly and not according to their valuations. Specifi-cally, in Brown and Johnson (1969), the authorsassume that the utility power facility has access to thewillingness to pay of consumers, so that it can declinethe ones with lowest valuations. This assumption isnot justifiable in a setting where a first-come-first-serve logic with random arrivals is more suitable(e.g., the retail industry). In a similar spirit, Ha (1997)and Liu and Van Ryzin (2008) study optimal rationingpolicies for a firm that faces uncertain demand. In Razand Ovchinnikov (2015), the authors study a price-setting newsvendor model for public goods and con-sider the Consumer Surplus for a single product withlinear additive stochastic demand. In Cohen et al.(2015), the authors study the impact of demand uncer-tainty on consumer subsidies for green technologyadoption and propose a special case of the definitionpresented in this paper. To our knowledge, this paperis the first to provide a rigorous extension of the Con-sumer Surplus definition under stochastic demandfor multiple products and any capacity allocationrule.The second contribution of this paper is to use our

Consumer Surplus definition to study the impact of

demand and supply uncertainty on consumers.Examining the impact of demand uncertainty (orstock-outs) has been a common research topic in theOM literature. Examples include inventory and sup-ply chain (Gupta and Maranas 2003, Kaya and Ozer2011, Ozer and Wei 2004), capacity investment (Goyaland Netessine 2007), and subsidies for green technol-ogy adoption (Cohen et al. 2015). The results of thispaper can be of interest to policy makers when assess-ing the customer welfare in various settings whereuncertainty is inherent.Structure of the paper. In section 2, we present our

results for the single product model. We then extendthe treatment for multiple products in section 3. Insection 4, we consider the impact of supply uncer-tainty. Finally, our conclusions are reported in section5. Most of the proofs of the technical results are rele-gated to the appendix.

2. Single Product

In this section, we consider the case of a supplier (alsoreferred to as seller) selling a single product. Thesingle-product setting is an important building blockas it allows us to develop the intuitions and toolsneeded for the setting with multiple products consid-ered in section 3. We assume that the seller is facing astochastic demand curve d(p, ϵ). Specifically, weexplore two commonly-used noise dependences: (i)additive, that is, d(p, ϵ) = d(p)+ϵ, where ϵ is a randomvariable with mean 0; and (ii) multiplicative, that is, d(p, ϵ) = ϵd(p), where ϵ is a positive random variablewith mean 1.2 The function d(p) is called the nominaldemand function and corresponds to the setting withdeterministic demand (i.e., ϵ = 0 with probability 1 inthe case with additive noise and ϵ = 1 with probabil-ity 1 for the multiplicative noise). We assume that d(p)is strictly decreasing. We refer to d−1(q) as the nominalinverse demand. We assume that d−1(q,ϵ) exists and isdefined so that d(d−1(q,ϵ),ϵ) = q.As is common in the OM literature, we assume that

the nominal demand d(p) is only a function of theprice, omitting arguments such as income and utility(which are explicit arguments of the Marshallian andHicksian demands, respectively).3 If d(p) comes froma representative consumer solving the utility-maximization problem (UMP), then this utility func-tion has to be quasilinear (so that the welfare can beproperly measured), namely, the representativeconsumer utility function is such thatUðx0, x1Þ ¼ x0þuðx1Þ, where x0 is the consumption ofa numeraire good with price equal to 1 (for moredetails, see Varian 1992, Chapter 10). The UMP can bewritten as maxx0;x1 x0þuðx1Þ subject to the budget con-straint x0þpx1 ≤ I. Equivalently, one can writeIþmaxx1 uðx1Þ�px1, resulting in a demand that only

Cohen, Perakis, and Thraves: Consumer Surplus Under Demand UncertaintyProduction and Operations Management 0(0), pp. 1–17, © 2021 Production and Operations Management Society 3



depends on the price p. Alternatively, one can con-sider a representative consumer solving theexpenditure-minimization problem (EMP) given byminx0;x1 x0þpx1 subject to Uðx0, x1Þ ≥ u0. The lattercan be rewritten as u0 þminx1 px1�uðx1Þ. As discussedin Varian (1992), the income is assumed to be highenough so that the product consumption is indepen-dent of income. We thus assume throughout thepaper that d(p) (and d(p, ϵ) in the stochastic case) isonly a function of the price (and noise). Assumingthat demand integrability conditions are satisfied, thisis equivalent to require a quasilinear utility functionand that the representative consumer’s income is highenough. Our goal is to study the Consumer Surplusunder a general stochastic demand function. First, webriefly recall the definition under deterministicdemand.

2.1. Deterministic DemandConsumer Surplus is an economic measure of con-sumer net welfare to quantify product consumptiongiven the incurred expenditures. This can be com-puted as the area between the market price p and theinverse demand curve (see an illustration in Figure 1).For given values of p and q = d(p), the Consumer Sur-plus under deterministic demand, CSdet, can be com-puted as:4

CSdet ¼Z dðpÞ

0

½d�1ðwÞ�p�dw: (1)

Equivalently, the Consumer Surplus can be com-puted by integrating over the price space:

CSdet ¼Z pmax

pdðzÞdz,

where pmax is d−1(0), that is, the inverse demand atzero or +∞ when d−1(0) is not defined (e.g., for d(p) = 1/p); sometimes called “the null price.” In theremainder of this paper, we focus on the ConsumerSurplus expression in Equation (1), where the inte-gration is taken over the quantity space. Note thatsince d(z) ≥ 0 and p ≤ d−1(0), one can see that CSdetis non-increasing in p.

2.2. Stochastic DemandWhen demand is stochastic, defining the ConsumerSurplus is more subtle due to the possibility of stock-outs. More precisely, if the demand at price p under aparticular realization ϵ happens to be greater than theavailable supply q (i.e., d(p,ϵ)−q > 0), some consumerswho are willing to make a purchase will not be serveddue to the limited supply. This stock-out event clearlyaffects customer welfare and, hence, should beaccounted for in the Consumer Surplus definition. To

further motivate this issue, consider the following“discretized” example with two customers with valu-ations 8 and 6, and assume that there are two itemsavailable at price p = 3. In this case, both items areallocated, and the Consumer Surplus is CS =(8−3)+(6−3) = 8. We now introduce a positive shockthat doubles the demand while keeping the same pro-portion of customers at each valuation (such a shockcorresponds to a multiplicative demand noise withϵ = 2). What is the Consumer Surplus in this case? Itis clear that the answer will depend on how bothavailable items are allocated to the four customers.Our goal is to propose a method for computing cus-tomer welfare under any possible allocation. In theprevious example, one can interpret the four cus-tomers’ valuations as the marginal utility of a repre-sentative consumer who maximizes welfare subject toa maximum consumption of two items. In this case,the two customers with the highest valuation willreceive the item. However, from a modeling perspec-tive, this might be restrictive since it does not provideany flexibility on the allocation. For example, whathappens if the items are randomly allocated to thefour consumers? Our proposed Consumer Surplusdefinition will sustain any type of allocation.We note that for stochastic demand functions, the

Consumer Surplus CS(ϵ) will be a function of thenoise realization ϵ. An upper bound on the ConsumerSurplus can be obtained by considering the case withinfinite supply, that is,

R dðp;εÞ0 ½d�1ðw, εÞ�p�dw. In real-

ity, as we mentioned, we have to account for situa-tions where demand exceeds supply. In particular,the actual Consumer Surplus will be a fraction of thisupper bound, based on the proportion of customers

Figure 1 Illustration of the Consumer Surplus for a Single ProductUnder a Deterministic Linear Demand




who are served. The way of formalizing this preciseproportion of served customers depends on the speci-fic rationing capacity rule under consideration.

2.2.1. Rationing Capacity Rules. In practice, whensuppliers receive an unexpectedly large amount oforders, they need to decide how to allocate their sup-ply. Even if the supply allocation rule is not chosen bythe seller, it is important to account for the way prod-ucts are allocated when computing the Consumer Sur-plus. In this context, one can consider several rules forallocating available quantities. In most settings, suppli-ers do not have access to customer valuations and sim-ply assign supply to the first customers that show up totheir stores. Examples include car dealers, fashion, elec-tronic products, and online shopping. In addition, cus-tomers’ arrival times are often independent ofvaluations for the product and, hence, we label this rulewith the superscript R (Random allocation). In otherwords, all potential customers who are interested inpurchasing the product, have the same likelihood to beserved. Two additional common rules are H (Highestwillingness to pay) and L (Lowest willingness to pay).As their names indicate, available supply is allocated tothe consumers with the highest (resp. lowest) willing-ness to pay,5 and discard the consumers with the lowest(resp. highest) valuations. Note that these two alloca-tion rules are the best (resp. worst) in terms of the totalcustomer welfare. Besides these three allocation rules,one can consider alternative rules. More precisely, ageneral allocation rule A can be any allocation of avail-able capacity q at price p to the consumers for a givendemand realization d(p, ϵ). Mathematically, an alloca-tion A is defined as a function6A : ½0, dðp, εÞ� ! ½0, 1�so that for any p, q, and ϵ, we have:

Z dðp;εÞ

0

AðwÞdw¼minfdðp, εÞ, qg: (2)

Then, for any w ∈ [0, d(p, ϵ)], AðwÞ can be inter-preted as the likelihood that an infinitesimal con-sumer receives the product. Equation (2) ensuresthat the total supplied units under the allocationrule A are equal to the volume of sales (i.e., the min-imum between demand and supply). Thus, given p,q, and ϵ, the allocation rules functions AH, AL, AR

are AHðwÞ ¼ 1fw≤ qg, ALðwÞ¼ 1fw≥ dðp;εÞ�qg, and

ARðwÞ ¼ min 1, qdðp, εÞ

n orespectively. Consequently,

given an allocation rule, the Consumer Surplus canbe defined as the sum of the surplus over con-sumers who are willing to make a purchase,weighted by their likelihood of receiving the prod-uct. Although this definition holds for any allocationrule, our analysis will focus on the three most popu-lar rules H, L, and R.

2.2.2. Definition and Graphical Interpretation. Inthis section, we present the Consumer Surplus defi-nition under a general stochastic demand functiond(p, ϵ). We first present the expression for a generalallocation rule A and then for the three aforemen-tioned rules (H, L, and R). In addition, we providea graphical interpretation by illustrating the defini-tions for the case of a linear demand with additivenoise. A similar methodology and graphical intu-itions can be found in Visscher (1973) for the totalwelfare (the sum of consumer and supplier sur-pluses).For a general allocation rule A, the Consumer Sur-

plus for any given ϵ and (p,q) is given by:7

CSAðεÞ¼Z dðp;εÞ

0

½d�1ðw, εÞ�p�AðwÞdw: (3)

Under such a general allocation rule, it is oftenimpractical to measure the Consumer Surplus. Wethus consider the three special rules, H, L, and R. Asdiscussed, the H and L rules can be seen as best- andworst-cases, respectively in terms of the ConsumerSurplus value (this is formally shown in Observation1 below).For the H rule, the Consumer Surplus for any given

ϵ and (p, q) is given by:

CSHðεÞ¼Z dðp;εÞ

0

½d�1ðw, εÞ�p�1fw≤ qg dw:

This rule is the best possible situation for the poolof consumers, as customers with high valuationstypically want the product the most and are servedwith the highest priority. In practice, this setting canoccur when customers are passionate and eager tobuy the product (e.g., concert tickets, new genera-tion of technology products). In the case wheredemand exceeds supply (i.e., d(p, ϵ) > q), the d(p, ϵ)−q customers with the lowest valuations arenot served and, hence, the surplus of those con-sumers is 0 (an illustration is shown in the left panelof Figure 2).For the L rule, the Consumer Surplus for any given


CSLðεÞ¼Z dðp;εÞ

0

½d�1ðw, εÞ�p�1fw≥ dðp;εÞ�qg dw:

This rule is the worst possible situation for con-sumers, as customers with high valuations areserved with the lowest priority. In practice, this set-ting can correspond to the case where low-valuationcustomers are deal seekers and arrive first to thestore (e.g., promotional events or flash sales). In thiscase, when demand exceeds supply (i.e., d(p, ϵ) > q),




the d(p, ϵ)−q customers with the highest valuationsare not served (see the center panel of Figure 2).For the R rule, the Consumer Surplus for any given


CSRðεÞ¼Z dðp;εÞ

0

½d�1ðw, εÞ�p�min 1,q

dðp, εÞ� �

dw: (4)

Under the R rule, customers arrive at random irre-spective of their willingness to pay for the product.For certain demand realizations, some proportion ofthese customers will not be served due to stock-outs. The proportion of served customers is givenby the ratio of actual sales over potential demand,that is, minf1, q

dðp, εÞg. Thus, the Consumer Surpluscan be defined as the total available surplus timesthe proportion of served customers. In this case, theConsumer Surplus can be depicted as the grey areabetween the inverse demand and the price (see theright panel of Figure 2). Equivalently, when demandexceeds supply, each infinitesimal consumer has asurplus weighted by q/d(p, ϵ). We note that the caseof a random allocation rule with an additive noisefor a single product was already considered in theliterature (see, e.g., Cohen et al. 2015, Raz andOvchinnikov 2015).We highlight that all the above definitions coincide

with the deterministic definition in Equation (1) whenthe noise vanishes (i.e., ϵ = 0 and ϵ = 1 for additiveand multiplicative noises respectively) and whenq ¼ E½dðp, εÞ�. In section 2.3, we study the impact ofdemand uncertainty on consumers by comparingCSdet to the expected Consumer Surplus E½CSAðεÞ� forthe H, L, and R allocation rules. It allows us to inferwhat would happen if one is ignoring demand uncer-tainty and stock-outs when measuring customer wel-fare. Finally, note that for any allocation rule, thefollowing property holds.

OBSERVATION 1. For any A, p, q, and ϵ, the followingholds:

CSLðεÞ≤CSAðεÞ≤ CSHðεÞ:

The proof of Observation 1 can be found in theappendix. As a result, we also have E½CSLðεÞ� ≤E½CSAðεÞ� ≤ E½CSHðεÞ�. Consequently, by studying theH and L rules, we cover the best and worst cases forconsumers.As discussed, the demand function comes from the

result of a utility-maximization problem solved by arepresentative consumer. Consequently, the source ofuncertainty in the demand function also comes fromthe corresponding uncertainty in the utility function.We emphasize that the underlying utility model—along with its own source of uncertainty—gives riseto the stochastic demand function we consider.Another way to derive the Consumer Surplus is bydirectly considering the representative consumer util-ity function denoted u(v). Interestingly, one canderive the same expressions for the Consumer Sur-plus under stochastic demand from a utility perspec-tive (the details are omitted for conciseness). Insummary, we defined the Consumer Surplus for astochastic demand function with a general capacityrule. At the same time, it extends the current under-standing of consumer utility and its connection to theConsumer Surplus.

2.3. Impact of Demand Uncertainty on ConsumersOur goal is to investigate to what extent ignoringstock-outs and demand uncertainty will lead to a mis-calculation of the Consumer Surplus. As discussed,this question can be of interest to policy makers whenassessing the customer welfare of specific policies orpublic interventions. We consider both additive and

Figure 2 Left: High Willingness to Pay Allocation Rule. Center: Low Willingness to Pay. Right: Random Allocation Rule. The Shaded Region Repre-sents the Consumer Surplus for the Case When Demand Exceeds Supply




multiplicative noises and the three allocation rulesdiscussed in Section 2.2.1. To simplify the analysisand to isolate the effect of demand uncertainty, wefirst assume that the market price is the same in boththe deterministic and stochastic settings. This can bemotivated by the fact that the supplier is serving twodifferent markets (e.g., selling the same product intwo different countries) and must set the same sellingprice due to regulations or marketing considerations.For example, studying the impact of demand uncer-tainty can be helpful when the supplier plans to entera new market for which sales data are very limited.We then extend our results to the more general settingwhere prices are exogenously determined.

2.3.1. Exogenous Pricing. When demand is deter-ministic, the supplier naturally matches supply withdemand, namely, q0 ¼ dðp0Þ, where p0 denotes themarket price. When demand is stochastic, we assumethat the price is still equal to p0 but the quantity qsto

can differ from dðp0Þ. For instance, the quantity couldbe set according to the optimal newsvendor ordering(see, e.g., Petruzzi and Dada 1999, Porteus 1990) oraccording to an alternative ordering policy.We first consider the case of a multiplicative noise,

d(p, ϵ) = ϵd(p). The result on the impact of demanduncertainty on consumers is summarized in the fol-lowing proposition.

PROPOSITION 1. Consider a stochastic demand functionwith a multiplicative noise under any given allocationrule A. We then have

½CSAðεÞ� ≤CSdet:

Proof. We show the result for the H rule. Then, byrelying on Observation 1, we conclude that theresult also holds for any allocation rule. We have

½CSHðεÞ� ¼R dðp0 ;εÞ0 ðd�1ðw, εÞ�p0Þ1fw≤ q0g dw

h i

¼R εdðp0Þ0 d�1 w

ε

� ��p0

� �1fw≤ q0g dw

h i¼E

R dðp0Þ0 ðd�1ðvÞ�p0Þ1fvε ≤ q0gεdv

h i

≤ R dðp0Þ0 ðd�1ðvÞ�p0Þεdv

h i¼ R dðp0Þ

0 ðd�1ðvÞ�p0Þdv¼CSdet,

where the last equality follows from ε½ � ¼ 1.

Interestingly, the result holds for any qsto, any noisedistribution, and any allocation rule. Therefore, nomatter how sophisticated the supplier is in terms ofallocating available supply, the consumers are alwayshurt when demand is stochastic relative to the deter-ministic setting (for a multiplicative noise). Indeed,positive demand shocks (i.e., ϵ > 1) bring additionalconsumers but their valuations will not increase bymuch. This is especially true for customers who value

the product the most. In particular, the maximal valu-ation, d−1(0), remains the same regardless of the noiserealization, since the inverse demand function atq = 0 is not affected by the noise realization. Thisobservation does not hold for an additive noise.We next consider the model with an additive

noise, d(p, ϵ) = d(p)+ϵ. In this case, the impact ofdemand uncertainty on consumers depends onthree different factors: (i) the convexity propertiesof the nominal demand function d(p), (ii) the alloca-tion rule, and (iii) the relation between dðp0Þ andqsto. The results are summarized in the followingproposition.

PROPOSITION 2. Consider a stochastic demand functionwith an additive noise and the three allocation rules, H,L, and R. We then have the following results:

• H rule: If qsto ≥ dðp0Þ and d−1(�) is convex,

½CSHðεÞ� ≥CSdet:

• L rule: If qsto ≤ dðp0Þ,

½CSLðεÞ� ≤CSdet:

• R rule: If qsto ≤ dðp0Þ and d−1(�) is concave,

½CSRðεÞ� ≤CSdet:

Note that in each case, the conditions are only suffi-cient. In other words, if one of the conditions is notsatisfied, we can find examples in which the inequal-ity can be in either direction. It follows that the impactof demand uncertainty on consumers (under an addi-tive noise) is driven by both the allocation rule andthe demand convexity or concavity. An interestingspecial case is linear demand with qsto ¼ dðp0Þ (i.e., thesupplier produces according to the expected demandvalue). In this case, the impact of demand uncertaintyon consumers crucially depends on the ability of thesupplier to identify and discriminate consumers. Inparticular, under the H rule (best scenario for con-sumers), the consumers are better off when demandis stochastic, whereas this conclusion is reversedunder the R or L rule. Regarding the convexity prop-erties, a convex demand (and thus a convex inversedemand) will typically result in a higher surplus gainfor consumers. This follows from the fact that posi-tive demand shocks will induce additional customerswith much higher valuations relative to the deter-ministic nominal demand. We illustrate this effect inthe left panel of Figure 3, where the shaded area




represents the surplus of these additional customersassuming they get served. However, under a con-cave demand (and hence a concave inverse demand),positive demand shocks will introduce additionalconsumers with valuations only slightly higher rela-tive to the deterministic setting, see the right panelof Figure 3.In addition, if qsto is thought in terms of the optimal

newsvendor quantity, the condition qsto ≤ dðp0Þ trans-lates into F�1ðp0�c

p0 Þ ≤ 0 (under an additive noise),where F(�) is the CDF of the noise ϵ and c is the (con-stant) marginal production cost. This condition is thussatisfied for products with low profit margins. In par-ticular, if ϵ is symmetric, the condition reduces to c ≤p0 ≤ 2c.One can also consider an alternative setting where

the supplier produces q0 units in both the determinis-tic and stochastic cases, where q0 is not necessarilyequal to dðp0Þ. Such a setting may correspond to situa-tions where suppliers need to make capacity decisionsbefore knowing the demand realization. In this case,we can extend most of the results presented in thissection (the results are omitted for conciseness). Nev-ertheless, it seems reasonable to assume that whendemand is deterministic, we have q0 ¼ dðp0Þ as thesupplier can tailor its production to exactly matchdemand.

2.3.2. Endogenous Pricing. We next consider thecase where prices are endogenously determined. Inreality, market prices can be set by solving a revenue-maximization problem. Consequently, the equilib-rium prices in both settings (deterministic andstochastic) may possibly differ and, thus, the Con-sumer Surplus will also differ. To capture the endoge-nous nature of prices, we incorporate the seller’s

pricing and production strategy into the revenue-maximization problem. We consider a marginal pro-duction cost c > 0. For the deterministic setting, theseller solves the following problem:

maxp

ðp� cÞdðpÞ: (5)

We denote by pd the maximizer of problem (5) andqd = d(pd). Then, the Consumer Surplus can be sim-ply computed by using the expressionCSdet ¼

R dðpÞ0 ðd�1ðwÞ�pÞdw. For the stochastic setting,

the optimization problem becomes

maxp, q

pE minfdðp, εÞ, qg½ �� cq, (6)

which is the price-setting newsvendor problem. Wedenote (ps,qs) the optimal solution of problem (6).Before stating the relation between the ConsumerSurplus in both settings, we first state a well-knownresult on the relationship between the optimal pricesfrom problems (5) and (6) for additive and multi-plicative noises (Karlin and Carr 1962, Mills 1959,Salinger and Ampudia 2011).

LEMMA 1. Under an additive noise pd ≥ ps, whereasunder a multiplicative noise pd ≤ ps.

The next proposition compares the Consumer Sur-plus in the deterministic and stochastic settings undera multiplicative noise when prices are endogenous.

PROPOSITION 3. Under a stochastic demand functionwith a multiplicative noise and any capacity allocationrule A, we have

E½CSAðεÞ�≤ CSdet: (7)

Figure 3 Left: Convex Inverse Demand Curve. Right: Concave Inverse Demand Curve. The Solid Line Represents the Nominal Demand Function,whereas the Dashed Line Corresponds to the Demand Function with an Additive Positive Shock. The Shaded Area Represents the PotentialAdditional Consumer Surplus




Thus, consumers are always better off in the deter-ministic setting not only under the same (exogenous)prices, but also when the optimal prices and quanti-ties are endogenously determined. The next proposi-tion summarizes the results for an additive demandnoise.

PROPOSITION 4. Consider a stochastic demand functionwith an additive noise and the H allocation rule such thatqs ≥ d(ps) and d−1(�) is convex. We then have

½CSHðεÞ� ≥CSdet: (8)

Note that in the case of an additive demand noise,unlike the case of exogenous prices (i.e., Proposition2), consumers are not always worse off under the R orL rule when prices are endogenously determined (weformally identified counterexamples). The rationalebehind this result is that uncertainty under an addi-tive noise induces a lower price, hence offsetting theinequalities for the R and L rules outlined in Proposi-tion 2 for the case with exogenous prices.We highlight that there is no clear way to compare

the Consumer Surplus under exogenous and endoge-nous pricing. Indeed, the results will highly dependon the value of the exogenous price. Consider, forexample, the setting with a single product. Let p0denote the price in the exogenous setting. If p0 is veryhigh, it is clear that the Consumer Surplus will belower under exogenous pricing, whereas if p0 is verylow, then the Consumer Surplus will be higher underexogenous pricing. In the revised paper, we have nowexplicitly mentioned the fact that this comparisonhighly depends on the value of the exogenous price.

3. Multiple Products

In this section, we consider a setting with n ≥ 2 prod-ucts. The demand function, dðp, εÞ : n�n ! n

þ, isexpressed as a function of the price vector p∈n anda random vector ε∈n with support Ω⊂n. Weassume that d(p, ϵ) is continuous in p and ϵ and dif-ferentiable almost everywhere with respect to p.8 Asbefore, we consider the following two cases: (i) addi-tive noise: d(p, ϵ) = d(p)+ϵ and (ii) multiplicativenoise: dðp, εÞ¼DεdðpÞ, where Dε refers to the diago-nal matrix with the elements of ϵ in its diagonal. Asstated in Krishnan (2010), for a demand with multi-plicative noise, the Slutsky symmetry conditions (de-fined formally below) can only be met if therealizations of the random variables ϵi are identicalacross all products, that is, Dε ¼ ε1I where I∈n�n isthe identity matrix. Throughout this paper, whenreferring to the setting with multiple products undermultiplicative noise, we impose Dε ¼ ε1I. We assume

that E½εi� ¼ 0 and E½εi� ¼ 1 for all i ∈ {1, . . ., n} inthe additive and multiplicative cases, respectively.We also assume that the demand for each product isdecreasing in its own price and non-decreasing in theother prices (i.e., ∂di

∂pi< 0 for all i ∈ {1, . . ., n} and

∂di∂pj

≥ 0 for all i, j ∈ {1, . . ., n}, i ≠ j). This is a com-mon assumption that captures the fact that the prod-ucts are substitutable goods (e.g., two competingbrands in the same category). Modeling the substi-tutability behavior of consumers by using this type ofdemand models is common in the literature (see, e.g.,Cohen and Perakis 2020, Cohen et al. 2020, Pindyckand Rubinfeld 2018). We highlight that the substitu-tion behavior is captured by the cross-price effectspresent in the demand function (here, substitutionrefers to customers switching products based on pricevariation, as opposed to focusing on stock-outevents). We note that in a model where the demand ofeach product is a function of all products’ prices,stock outs are not explicitly captured in the demandmodel. However, we formally capture the stock-outevents by inputting the minimum between demandand supply, namely, min{di, qi} for each product i =-1, . . ., n. Thus, consumers are still substitutingamong the products based on pricing considerations;but the demand corresponds to the total number ofdemanded units, as opposed to the sales. To accountfor potential stock outs, we truncate the demand bytaking the minimum between demanded units andavailable supply. In the appendix, we consider andanalyze an alternative model that explicitly capturessubstitution into the consumer utility-maximizationproblem by accounting for inventory constraints. Thismodeling framework naturally leads to a model inwhich the outcome consumption results in a demandfunction that allows for substitution among productsin the events of stock-outs. Interestingly, this alterna-tive utility model leads to the same results andinsights presented in this section. In addition, weassume that the price change of a particular producthas a stronger effect on its own demand realtive to thesum of the price changes of the other products (i.e.,

∑ j≠i∂di∂pj

�� < ∂di∂pi

�� for all j ∈ {1, . . ., n}). This assumption

is called the strict diagonal dominance condition and iscommon in the literature (see, e.g., Arrow and Hahn1971). The above assumptions imply that the negativeof the demand Jacobian (with respect to prices) is anon-singular M-matrix and, hence, the Jacobian of theinverse demand is non-positive with strictly negativeelements in its diagonal.

3.1. Deterministic Demand and SlutskyConditionsIn the deterministic setting, the demand vector is afunction only of the price vector,9 that is,




dðpÞ : n ! nþ. In this case, we naturally have d

(p) = q. Note that for a given q, the inverse demandfunction d�1ðqÞ represents the maximal price vectorfor which consumers will demand q units (i.e., con-sumers’ willingness to pay). This interpretation isimportant as we will compute the Consumer Surplusby integrating consumers’ willingness to pay over thequantity space. Alternatively, one can integrate overthe price space, as in [1000] where the author com-putes the Consumer Surplus for multiple productsunder deterministic demand. In this case, the Con-sumer Surplus is expressed as the path integral overthe sum of the demand functions of each product.Thus, the value of the integral may be path dependentif the Slutsky conditions are not satisfied.10 To avoidthe undesired path-dependence property, we assumethat the Slutsky conditions are satisfied. We acknowl-edge that this condition is somewhat restrictive. How-ever, if we were to relax the Slutsky condition, then itwould not be possible to adequately measure theConsumer Surplus (for a setting where the demandfunction comes from a representative consumerutility-maximization problem). To our knowledge,the Slutsky condition has been imposed in all thestudies that consider a formal demand function thatcomes from a utility maximization problem faced by arepresentative consumer. As otherwise, measuringthe Consumer Surplus (even in a deterministic set-ting) is impossible. In this paper, we will compute theConsumer Surplus as the path integral over theinverse demand for cases where there may be a mis-match between demand and supply. As mentioned,when demand is stochastic, the produced units cansometimes be lower than demand. As in the single-product setting, we address this issue by introducingan n-dimensional allocation rule that assigns theavailable supply to customers (see section 3.2 formore details). We next write the Consumer Surplus asthe path integral over the inverse demand function:

CSdet ¼ZC½d�1ðrÞ�p� �dr, (9)

where C is an integration path from 0∈n to q = d(p) and defined by the parametric functionr : ½a, b� ! Qn

i¼1½0, qi� which is continuous and differ-entiable almost everywhere. The expression in Equa-tion (9) is uniquely determined (i.e., pathindependent), if the cross derivatives of the inversedemand function (or demand function) are equal,see Tirole (1988). Otherwise, the expression in Equa-tion (9) would depend on the path of integration.Note that in the single-product setting, such an issuedoes not exist, as the path moves along a uniquedirection on a segment.So far, we assumed that demand and supply are

matching (i.e., d(p) = q). Nevertheless, in several

applications, this may not be the case. This motivatesus to study the more general setting when supply anddemand do not necessarily match (i.e., d(p) ≠ q).

3.2. Stochastic DemandConsider vectors p and q. We do not necessarilyimpose q ¼ E½dðp, εÞ�. As discussed, when demand isstochastic, there may be cases where productionquantities are lower than the demand for each prod-uct. Our goal is to define the Consumer Surplus formultiple products, while accounting for potentialstock-outs. Since the multiple-product setting is moreintricate than the single-product setting, for ease ofexposition, we first present the definition for the Hand L rules from a utility perspective. We will thenwrite these expressions as a function of the inversedemand. As discussed, the H and L rules correspondto the best- and worst-case scenarios in terms of Con-sumer Surplus and, hence, can serve as identifyingperformance bounds. Finally, we will consider the Rrule and generalize to any allocation rule.Under the H allocation rule, if the available units q

are lower than the demand d(p,ϵ), a portion of the util-ity will not be captured by consumers. Following a simi-lar argument as in the single-product case, the utilityunder this allocation will be captured by the first con-sumers on each item and, thus, the Consumer Surplusis uðminfq, dðp, εÞgÞ�uð0Þ�pTmin fq, dðp, εÞg,where u(0) = 0. The H rule implies that among thediðp, εÞ consumers who are willing to purchase producti, only the consumers with the highest valuations willbe served (i.e., the first minfdiðp, εÞ, qig customers).For the L rule, as in the single-product case, the utilityof consumers is captured by the last units consumed,namely, the consumers from max{d(p,ϵ)−q,0} to d(p,ϵ)and, thus, the Consumer Surplus under the L ruleis uðdðp, εÞÞ�uðdðp, εÞ�maxfq, 0gÞ�pTminfq, dðp,εÞg. We illustrate these two definitions for a settingwith two products in Figure 4.We note that the utility-maximization problem

leads to d�1 ¼rvu, so that the Consumer Surpluscan be written as a function of the inverse demand.For the H rule, the integral of the inverse demandshould go from 0 to min{q, d(p, ϵ)}. The latter canbe expressed using an integration path from 0 to d(p,ϵ), while weighting the inverse demand with thecorresponding allocation. We denote by Cε the pathfrom 0 to d(p,ϵ) and by rε the corresponding para-metric function along the path.11 Then, the integralpath for the H rule goes from 0 to min{q,d(p,ϵ)}with a weight of 1, whereas the remaining path goesfrom min{q,d(p,ϵ)} to d(p,ϵ) with a weight of 0. Wethen have

CSHðεÞ¼ZCε∑n

i¼1

½d�1i ðrε, εÞ�pi�1frεi ≤ qigdr

εi , (10)




where Cε is any path consistent with the H rule inthe sense that it crosses the point min{q,d(p,ϵ)}. Notethat the expression in Equation (10) is path indepen-dent across all such paths. In addition, the Con-sumer Surplus can alternatively be written as thedifference between the utility evaluated at min{d(p, ϵ), q} and 0 (which is consistent with the tradi-tional deterministic interpretation).For the L rule, the integral of the inverse demand

should go from max{d(p, ϵ)−q, 0} to d(p,ϵ). Recall thatunder the L rule, items are allocated to customerswith the lowest valuations (among all customers witha valuation above price). Namely, among the diðp, εÞcustomers who demand product i, only theminfdiðp, εÞ, qig customers with the lowest valuationswill be served. Equivalently, we can consider anypath Cε from 0 to max{d(p, ϵ)−q, 0} with a weight of 0and, then, from max{d(p, ϵ)−q, 0} to d(p,ϵ) with aweight of 1. As a result, we can write

CSLðεÞ¼ZCε∑n

i¼1

½d�1i ðrε, εÞ�pi�1frεi ≥ diðp;εÞ�qigdr

εi : (11)

We note that the Consumer Surplus expressions forthe H and L rules, in Equations (10) and (11), have anindicator term inside the integral which assigns aweight to the marginal utility. This term varies depend-ing on the allocation. Under the R rule, all (infinitesi-mal) customers have the same likelihood of receivingthe item. This translates into having an allocation termof minf1, qi

diðp, εÞg for each product i. Then, the Con-sumer Surplus for the R rule can be expressed as

CSRðεÞ¼ZCε∑n

i¼1

½d�1i ðrε, εÞ�pi�min 1,

qidiðp, εÞ

� �drεi ,

(12)

where Cε is the path that follows a straight line from0 to d(p,ϵ). Intuitively, the allocation term inside theintegral in Equation (12) assigns a weight to themarginal utility according to the ratio of availablequantities and demand. This allocation can be seenas the limiting case of the H (or L) rule. A moredetailed explanation of this limit interpretation isprovided in the appendix.As in section 2, the treatment can be extended to a

general allocation rule A. To this end, we need tospecify: (i) an allocation functionA :

Qnj¼1½0, djðp, εÞ� ! ½0, 1�n and (ii) an integration

path Cε represented by a parametric function rε from0 to d(p,ϵ) that satisfies for each i:

ZCεAiðrεÞdrεi ¼min qi, diðp, εÞ

� �: (13)

Equation (13) implies that the total number of allo-cated units for each product is equal to the mini-mum between demand and supply. As a result, weobtain:12

CSAðεÞ¼ZCε∑n

i¼1

½d�1i ðrε, εÞ�pi�AiðrεÞdrεi : (14)

Finally, the expected Consumer Surplus can beobtained by taking the expectation over ϵ:

E½CSAðεÞ� ¼ZΩCSAðεÞdFðεÞ:

As in the single-product setting, we can also derivethe above Consumer Surplus definitions from a utilityperspective (the details are omitted). As before, theConsumer Surplus under the different allocation rulessatisfies the following ordering.

OBSERVATION 2. For any A, p, q, ϵ, and Cε defined byrε, the following holds:

CSLðεÞ≤CSAðεÞ≤CSHðεÞ:

The proof of Observation 2 can be found in theappendix. Consequently, we have E½CSLðεÞ�≤ E½CSAðεÞ�≤ E½CSHðεÞ�.As we can see, the Consumer Surplus for multiple

products under stochastic demand depends on thenoise realization and on the allocation rule. Whensupply exceeds demand, there is no stock-out and weare back to the deterministic case. However, whendemand exceeds supply, some consumers may not beserved. In this case, the allocation rule will determinewhich customers are served, hence allowing us toproperly compute the Consumer Surplus. As men-tioned, the allocation rule is related to the order in

Figure 4 Example with n = 2 on How to Compute the Consumer Sur-plus for the H and L Rules




which customers are served for a particular product.For example, the H rule translates to first serving thecustomers with the highest valuations.

3.2.1 Example. We next present a concrete exam-ple with two products and compute the ConsumerSurplus under the three different allocation rules. Weconsider a linear demand function with an additivenoise of the form d(p,ϵ)=d−Bp+ϵ that satisfies theconditions mentioned at the beginning of section 3,13

and we denote by p0 and q0 the given price and quan-tity vectors. Table 1 reports the Consumer Surplusvalues under a specific noise realization. As we cansee, the value of the Consumer Surplus highlydepends on the allocation rule.Having defined the Consumer Surplus for a general

allocation rule under stochastic demand for multipleproducts, we next compare the expected ConsumerSurplus to the deterministic setting. We show that theimpact of demand uncertainty on consumers dependson several factors such as the convexity properties ofthe demand, the noise structure, and the allocationrule.

3.3. Impact of Demand Uncertainty on ConsumersIn this section, we extend the analysis and results formultiple products. If the different products are inde-pendent (i.e., no cross-item effects), one can extendthe results from the single-product case by simplycomputing the Consumer Surplus for each productseparately and summing up the n terms. Neverthe-less, the most interesting case is when the demand ofeach product depends on both its own price and theprices of other products (i.e., the price of product ialso affects the demand of products j ≠ i).

3.3.1. Exogenous Pricing. We consider a settingwith n products and set the (exogenous) price vectorto p0. As before, when demand is deterministic, theproduction quantities are set to match the nominaldemand, namely, q0 ¼ dðp0Þ. Under stochasticdemand, the supplier produces quantities qsto whichdo not necessarily match dðp0Þ. As in the single-product case, we consider both multiplicative andadditive noises and start with the setting where theprices are the same.

PROPOSITION 5. Consider a stochastic demand functionwith a multiplicative noise and qsto ¼ dðp0Þ. Then, underany allocation rule A, we have

E½CSAðεÞ� ≤ CSdet:

We next consider a demand model with an additivenoise.

PROPOSITION 6. Consider n products and a stochasticdemand function with an additive noise. We then havethe following results:

• L rule: if qsto ≤ dðp0Þ

E½CSLðεÞ� ≤CSdet:

• R rule: If d�1i ð�Þ is concave for all i ∈ {1, . . ., n},

and if qsto ≤ dðp0Þ

E½CSRðεÞ� ≤CSdet:

• H rule: Consider a linear demand (i.e., d = d−Bp+ϵ) with independent and symmetric noises. Ifqsto ≥ dðp0Þ and 3DB�1 �B�1 is positive semi-definite, where D�1

B is the diagonal matrix with thediagonal elements of B�1, then

E½CSHðεÞ� ≥CSdet:

We note that for the H rule, the Consumer Surplusinequality obtained in the single-product settingholds only under more restricted conditions (lineardemand with independent and symmetric noises plusan additional technical condition). The technical con-dition outlined for the H rule in Proposition 6 isalways satisfied for n ≤ 3. To ensure that demand val-ues remain non-negative, we consider formallyimposing a non-negativity demand constraint for thecase of an additive noise under the L or R rule. Wethen show that the results still hold after imposingsuch a non-negativity constraint (see more details inthe proof of Proposition 6 in the appendix). In sum-mary, we have shown that in many cases, demanduncertainty hurts consumers. Our results also suggestthat ignoring uncertainty may severely impact theConsumer Surplus value. In section 3.4, we will showthat for a linear demand with an additive noise, underthe R rule, the expected Consumer Surplus can be asfar as 50% from the deterministic setting.

3.3.2. Endogenous Pricing. As in the single-product setting, we next consider the case with

Table 1 Consumer Surplus Values for the Different Allocation Rules.Parameters: �d1 ¼ 10, �d2 ¼ 7, B11 = B22 = 1,B12 = B21 = −0.25, ϵ1 = 1, ϵ2 = 0, p1 = 4, p2 = 2, d1 = 7.5,d2 = 6, q1 = 6.5, and q2 = 6

Rule CS

High 60.667Low 52.133Random 56.400




endogenous prices. In particular, we look at the casewhere n firms compete in a price-newsvendor setting.Note that the deterministic demand case can be seenas a special case when the quantities to produceexactly match the demand at the equilibrium price.Each firm i has a marginal cost ci and solves the fol-lowing problem:

maxpi , qi

pi minfdiðp, εÞ, qig � ciqi:

To the best of our knowledge, the counterpart ofLemma 1 for multiple products has not been previ-ously established. It seems that such a general resultcannot be derived due to the lack of tractability. Togain analytical tractability, we consider the case of alinear demand model with either an additive noise(i.e., d(p,ϵ) = d−Bp+ϵ with E½ε� ¼ 0) or a multiplica-tive noise (i.e., d(p,ϵ) = ϵ(d−Bp) with E½ε� ¼ 1) andderive this result in Lemma 2 below. The outcomeof this model with an additive noise has been stud-ied in Chen et al. (2004) and Zhao and Atkins(2008). 14 We denote ðps, qsÞ the equilibrium priceand quantity vectors. The first-order condition overthe quantities leads to qsi ¼ �di�∑n

j¼1BijpsjþF�1i ðpsi�ci

psiÞ,

where Fi(�) is the cdf of ϵi. Then, the first-order con-dition over the prices leads to the following fixed-point equation: ΨðpsÞþ �d�Bp�Dðps� cÞ¼ 0, whereΨiðpsÞ¼E½minfεi, F�1

i ðpsi�cipsiÞg� and D∈n�n is a diag-

onal matrix with the diagonal elements of B in itsdiagonal and zero elsewhere. Equivalently, ΨðpsÞcan be written as ΨðpsÞ¼E minfdðps, εÞ, qsg½ ��dðpsÞ,thus corresponding to the expected differencebetween sales and demand. In the special case ofa deterministic demand, the equilibrium prices canbe obtained in closed form as pd ¼cþðBþDÞ�1ð�d�BcÞ.

LEMMA 2. Consider a linear demand function. Under anadditive noise pd ≥ ps, whereas under a multiplicativenoise pd ≤ ps.

Thus, Lemma 2 extends the result of Lemma 1 formultiple products under linear demand. The next twopropositions state the results for multiplicative andadditive demand cases.

PROPOSITION 7. For a linear demand function with anmultiplicative noise, we have

½CSHðεÞ� ≤CSdet:

PROPOSITION 8. For a linear demand function with anadditive noise, if 3DB�1 �B�1 is positive semi-definite, wehave

E½CSHðεÞ� ≥CSdet:

As in the single-product setting, the inequalities forthe R and L rules presented for the case of endoge-nous prices (see Proposition 6) where consumers arebetter off in the deterministic setting, do not hold any-more under endogenous prices. Indeed, when pricesare endogenously determined, the firms will charge ahigher price in the deterministic case, as stated inLemma 2. It can thus offset the Consumer Surplusinequality. We conclude this section by presenting aplot of the Consumer Surplus ratio (stochastic dividedby deterministic) as a function of the demand uncer-tainty magnitude for the different settings (see Fig-ure 5). Ultimately, our model and results allow us toquantify the impact of the demand uncertainty mag-nitude on the Consumer Surplus. For example,depending on the extent of the Consumer Surplusloss, firms can invest efforts in collecting additionaldata and in developing more sophisticated demandprediction methods to ultimately reduce demanduncertainty.

3.4. Analytical Bounds for Linear DemandTo draw additional insights on the comparison of theConsumer Surplus under stochastic and deterministicdemand, we consider the special case of lineardemand with an additive noise. For simplicity, weassume that all n products are symmetric and thatq = d(p), that is, production quantities exactly matchthe nominal demand, and we consider the randomallocation rule. More precisely, the demand functionfor the n products is given by:

dðp, εÞ¼ �d�Bpþ ε: (15)

In this case, we arrive at the following expressions:

CSdet ¼ ½dðpÞ�TB�1

2dðpÞ,

CSRðεÞ ¼ ½minfq, dðp, εÞg�TB�1

2dðp, εÞ,

where the first (resp. second) equation correspondsto the Consumer Surplus for deterministic (resp.stochastic) demand. Recall that under stochasticdemand, the Consumer Surplus becomes a randomvariable, so we are interested in E½CSRðεÞ�.Since we assume that all the products are symmet-

ric, the parameters �di, pi, and qi are identical for i =1, . . ., n. In addition, the matrix B is such that its diag-onal elements are equal to b and its off-diagonal ele-ments (cross-price sensitivity) are equal to −δ < 0(while satisfying the diagonal dominance conditionb > (n−1)δ). Finally, we assume that the additive




noises, ϵi ∀ i = 1, . . ., n, are i.i.d with bounded sup-port. In particular, we require that ϵi > −qi with prob-ability 1. For example, ϵi can be uniformly distributedin [−qi,qi].To gain analytical tractability, we consider an addi-

tive i.i.d noise with a two-point distribution (i.e.,εi ∈ f�A, Ag each with probability 0.5, for some A ≤qi). In this case, we have

CSdet ¼ 1

2n~Bq2,

E½CSRðεÞ� ¼ 1

2n B q2�0:5AðB q�B�1

ii AÞh i

:

Here, ~B denotes the row sum of the matrix B�1 (un-der symmetric products, all rows have the samesum). Also, B�1

ii and B�1ij are the same for all i and j

≠ i. We next derive a bound on the ratio of theexpected Consumer Surplus under stochasticdemand relative to deterministic demand.

PROPOSITION 9. Assume that demand is linear with anadditive noise distributed according to a two-point distri-bution. Then, the following holds:

1. When the cross-price sensitivity δ = 0 (i.e., inde-pendent products), we have

E½CSRðεÞ�CSdet

≥7

8:

2. For any 0< δ< bn�1 (i.e., substitutable products),

we have

E½CSRðεÞ�CSdet

≥1

2þ 1

2n,

To illustrate Proposition 9, we consider the examplewith n = 2, �d ¼ 12, b = 1, δ = 0, p = 2, and q = 12,with noise realizations ϵ1 = −2 and ϵ2 = 2. In thisexample, we have CSdet = 2q2/2 = 100 andE½CSRðεÞ� ¼ 0:5minf10, 8g�8þ0:5min f10, 12g�12¼ 92, that is, a decrease of 8% relative to CSdet. Moregenerally, Proposition 9 shows that when demand islinear with an additive i.i.d noise (distributed accord-ing to a two-point distribution), the expected Con-sumer Surplus from the random allocation can be asfar as 50% from the deterministic setting.

4. Random Supply

We next examine the Consumer Surplus understochastic supply. We directly study the multiple-product setting (the single-product setting can beobtained as a special case).

4.1. Random Supply and Deterministic DemandWe consider a deterministic demand function d(p)with a fixed vector p. The supply is random and rep-resented by the function s(p,δ) such thatE½sðp, δÞ� ¼ dðpÞ. As in the case of random demand, anadditive supply uncertainty can be expressed as s(p,δ)=d(p)+δ with E½δ� ¼ 0, whereas a multiplicative sup-ply uncertainty follows siðp, δÞ ¼ δidiðpÞ with E½δi� ¼ 1

(a) (b)

Figure 5 Consumer Surplus ratio (stochastic divided by deterministic) as a Function of the Demand Uncertainty Magnitude for Linear Demand andn = 2 Items. Parameters: �d1 ¼ �d2 ¼ 5, B11 = B22 = 1, B12 = B21 = −0.4, c1 = c2 = 1, p1 = p2 = 3, and q1 = q1 = 3.2




and δi > 0. Unlike the stochastic demand setting,stochastic supply allows for multiple random vari-ables under a multiplicative uncertainty.

PROPOSITION 10. Under a stochastic supply function s(p, δ) so that E½sðp, δÞ� ¼ dðpÞ, we have

E½CSHðδÞ� ≤CSdet:

Interestingly, we show that regardless of the type ofnoise (additive or multiplicative), the expected Con-sumer Surplus under stochastic supply is alwayslower relative to its deterministic counterpart. Wenext consider the general setting where both demandand supply are stochastic.

4.2. Random Supply and Random DemandWhen both demand and supply are stochastic, theresults will depend on the noise structure.

PROPOSITION 11. Under multiplicative demand and sup-ply uncertainties, we have

Eε;δ½CSHðε, δÞ� ≤CSdet:

Here, the expectation is taken with respect to bothnoises’ distributions. Interestingly, the result ofProposition 11 allows for correlated noises. Asexpected, the uncertainty will reduce the value of theexpected Consumer Surplus for any allocation rule,when the noise is multiplicative.

PROPOSITION 12. Under additive demand and supplyuncertainties, if d�1

i is concave, we have

Eε;δ½CSLðε, δÞ� ≤CSdet,

Eε;δ½CSRðε, δÞ� ≤CSdet:

Proposition 12 shows that the expected ConsumerSurplus is lower under a stochastic setting for boththe L and R rules, if d�1

i is concave and the noises areadditive. For the H rule, we identified exampleswhere the inequality can go either way. Overall, thissection shows that having an uncertain supply willmost often hurt consumers in terms of ConsumerSurplus.

5. Conclusions

A well-known concept to measure customer welfareis the Consumer Surplus. This tool was primarilydeveloped assuming that the demand is deterministicand that products are always available. In most real-

world settings, however, demand is modeled as astochastic function of the price. While many tradi-tional OM studies have focused on firms, severalrecent lines of research also account for consumers.This is especially true in sustainable operations,where customer welfare is of primary importance(e.g., Avci et al. 2014, Chemama et al. 2019, Sunar andPlambeck 2016). In this context, policy makers areoften interested in assessing the impact of policies onconsumers. The relevant question is then whether thestandard (presumably deterministic) approach to cal-culate the Consumer Surplus may end up miscalculat-ing customer welfare.Demand uncertainty or errors in demand predic-

tion may lead to stock-outs. In such a case, consumerswho want to purchase the product may not be served,hence ultimately affecting customer welfare. We pro-pose an extension of the Consumer Surplus whichaccounts for demand uncertainty and stock-outs. Wefirst introduce a mathematical definition of an alloca-tion rule and then present a definition of the Con-sumer Surplus under stochastic demand for multipleproducts and any allocation rule. We next use thisdefinition to study the impact of demand and supplyuncertainty on consumers. We show that demanduncertainty often hurts consumers. For example,under a demand with a multiplicative noise, con-sumers are always better off in the deterministic set-ting. Interestingly, this result holds for any demandfunction, noise distribution, and allocation rule.Under an additive demand noise, we show that theimpact of uncertainty crucially depends on the alloca-tion rule and on the convexity properties of thedemand.In several settings, the most practical allocation

rule is the random rule, that is, consumers arriverandomly and are served independent of their val-uations. For this rule, we show that consumers aretypically worse off under stochastic demand. Weshow that when demand is linear with an additivei.i.d noise, the expected Consumer Surplus underthe random allocation can be as far as 50% relativeto the deterministic setting. One possible way tomitigate this Consumer Surplus loss is by using asharing mechanism. For example, a supplier canshare its excess capacity (or demand) for productsamong its different stores. Alternatively, competingfirms may engage in a sharing inventory agree-ment, which can ultimately benefit consumers byincreasing the Consumer Surplus under stochasticdemand.In this paper, we assumed that the substitution

among products is captured by the cross-price termsin the demand function. In the appendix, we considerand analyze an alternative model that explicitly incor-porates substitution into the consumer utility-




maximization problem by accounting for inventoryconstraints. This modeling framework naturally leadsto a model in which the outcome consumption resultsin a demand function that allows for substitutionamong products when stock-outs occur. Interestingly,this alternative utility model leads to the same resultsand insights regarding the impact of demand uncer-tainty on Consumer Surplus, hence supporting thatour results are robust to the specific way of capturingsubstitution among products.This paper is far from providing the last word on

the topic of Consumer Surplus in stochastic settingsand opens several opportunities for future research.For example, an interesting extension is to consider amodel where the stochastic demand term depends onthe price. Using the methodology developed in thiswork, one can compute the Consumer Surplus for set-tings with stochastic demand. Since customer welfareplays a growing role in many applications, one canuse the tools presented in this paper to study andquantify the potential impact of government policieson consumers.

ACKNOWLEDGMENTS

The authors thank the department editor (Haresh Gurnani),the associate editor, and the two anonymous referees fortheir valuable feedback that has helped us improve thispaper.

Notes

1 In this paper, we consider a setting where prices andquantities are set prior to the demand realization, as inthe newsvendor problem.2With a slight abuse of notation, we denote by d(p,ϵ) thestochastic demand function and by d(p) its deterministicpart. Namely, if the only argument is the price, we referto the latter, whereas if the argument is composed of aprice and a random variable ϵ, we refer to the former.3The Marshallian demand function explicitly depends onprice and income: it corresponds to the solution of theutility-maximization problem solved by a representativeconsumer who maximizes utility subject to a budget con-straint. The Hicksean demand depends on price and util-ity: it is obtained as the solution of the expenditure-minimization problem solved by a representative con-sumer subject to a constraint on the minimum utilitylevel.4We assume that the nominal demand function is strictlydecreasing so that its inverse exists. However, thisassumption can be relaxed without altering our results.Indeed, if there is a countable disjoint set of intervalswhere the demand has a zero slope, then the integral inEquation (1) will have zero measure over those points.5In our context, an allocation function refers to a way ofmathematically expressing how available units are dis-tributed to consumers. This is the reason why we usedemand (quantity) as the argument of the allocation

function. Given the monotonicity of the inverse demand,we can equivalently characterize an allocation by usingthe valuation (price) space, allowing us to map allocationfunctions to the willingness to pay of consumers.6More precisely, A is a family of allocation functions para-metrized by (p, q, ϵ). We omit these arguments to lightenour notation. However, our analysis carefully accounts forthis dependence.7Equivalently, the integral in Equation (3) can beexpressed by integrating over the price space, that is,CSAðεÞ¼ R p

pmax :¼d�1ð0;εÞ½v�p�Aðdðv, εÞÞddðv, εÞ. We choose topresent our analysis by integrating on the quantity spaceas it allows us to develop sharper insights given that (i)the allocation function is a mapping from the quantityspace and (ii) we can rely on the interpretation of valua-tions through the inverse demand.8As in section 2, we impose these assumptions for ease ofexposition but our results still hold under more generaldemand functions, such as discontinuity and lack of dif-ferentiability in countable disjoint sets.9As mentioned in section 2, the underlying utility functionis assumed to be quasilinear and, hence, the demand isonly a function of price (and ϵ in the stochastic case).10The Slutsky conditions are satisfied if the demand cross-derivatives are equal (i.e., ∂di

∂pj¼ ∂dj

∂pifor all i ≠ j).

11To simplify notation, we simply write rε omitting thearguments q, p, and ϵ.12As in section 2, the integral in Equation (14) can beequivalently expressed by integrating over the price space,namely, CSAðεÞ¼ R

C0∑i½sεi �pi�Aiðdðsε, εÞÞddiðsε, εÞ, wheresε ¼ d�1ðrε, εÞ is the parametric function that correspondsto C0 and goes from pmax :¼ d�1ð0, εÞ to p (this equivalenceholds for any allocation rule).13In this case, it suffices to require d > 0 and that B is asymmetric strictly diagonal dominant Z-matrix (i.e., theoff-diagonal entries are less or equal than zero).14To ensure that the equilibrium is unique, it is enough toassume that ∑n

j¼1Bij is greater than 1/[cifi(−Ai)], where fi(�)is the pdf of ϵi and Ai is the minimum value in the sup-port of ϵi.

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Supporting InformationAdditional supporting information may be found onlinein the Supporting Information section at the end of thearticle.

Online Appendix.




consumer surplus under demand uncertainty

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