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Welfare Analysis of Dynamic PricingNingyuan Chen, Guillermo Gallego

To cite this article:Ningyuan Chen, Guillermo Gallego (2018) Welfare Analysis of Dynamic Pricing. Management Science

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MANAGEMENT SCIENCEArticles in Advance, pp. 1–13

http://pubsonline.informs.org/journal/mnsc/ ISSN 0025-1909 (print), ISSN 1526-5501 (online)

Welfare Analysis of Dynamic PricingNingyuan Chen,a Guillermo Gallegoa

aDepartment of Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong KongContact: nychen@ust.hk, http://orcid.org/0000-0002-3948-1011 (NC); ggallego@ust.hk, http://orcid.org/0000-0002-9664-3750 (GG)

Received: January 12, 2016Revised: February 22, 2017; August 14, 2017Accepted: August 31, 2017Published Online in Articles in Advance:May 4, 2018

https://doi.org/10.1287/mnsc.2017.2943

Copyright: © 2018 INFORMS

Abstract. Dynamic pricing is designed to increase the revenues or profits of firms byadjusting prices in response to changes in the marginal value of capacity. We exam-ine the impact of dynamic pricing on social welfare and consumers’ surplus. Wepresent a dynamic pricing formulation designed to maximize welfare and show that thewelfare-maximizing dynamic pricing policy has the same structural properties as therevenue-maximizing policy. For systems with scarce capacity, we show that the revenue-maximizing dynamic pricing policy and the market-clearing price are both asymptoticallyoptimal for welfare. We also find in most cases that revenue-maximizing dynamic pric-ing improves consumers’ surplus compared to the revenue-maximizing static price. Ourfindings can potentially transform the public image of dynamic pricing and provide newmanagerial insights as well as policy implications: (1) in large-scale systems with scarcecapacity, a central planner would essentially implement the same pricing policy as a firmwith monopoly power; (2) the revenue-maximizing dynamic pricing policy can benefitcustomers when the demand elasticity is in a small bounded interval as is the case forseveral important demand functions.

History: Accepted by Serguei Netessine, operations management.Funding: The research of N. Chen is supported by R9382 start-up fund from HKUST. The research of

G. Gallego is funded by the Crown Worldwide Professorship at HKUST.Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2017.2943.

Keywords: dynamic pricing • revenue management • consumers’ surplus • social welfare • Pareto efficiency

1. IntroductionDynamic pricing and revenue management practiceshave proliferated since the deregulation of the UnitedStates airline industry in the late 1970s. Airlinesdesigned a variety of fares and restrictions to appealto broad market segments and dynamically managedthe set of offered fares to maximize expected revenuesfrom fixed capacities. These practices later expandedto hotels, car rentals, and the entertainment indus-try to the point where most firms in the travel andleisure industries arguably cannot survive withoutusing dynamic pricing. Dynamic price is also used infashion retailing, especially when capacity cannot beadjusted within the sales horizon.As dynamic pricing penetrates new industries, exec-

utives are raising the question of how dynamic pric-ing affects consumers and social welfare (firm’s prof-its plus consumers’ surplus). They are concerned thatdynamic pricing may help firms in the short run buthurt them in the long run if consumers find lessvalue in the companies’ products. Indeed, the pub-lic and the media appear to believe that firms earnmore revenues using dynamic pricing only by extract-ing from consumers’ surplus. As an example, after Dis-ney introduced a dynamic pricing scheme for theme

parks in 2016, Bloomberg published an article, titled“Surge Pricing at Disneyland Could Boost Ticket Costsby 20%” (Palmeri 2016), indicating that customers mayhave to paymore on average under the new policy. This“zero-sum” hypothesis, while disproved in other con-texts such as third-degree price discrimination (Varian1985, Cowan 2012), has not been fully studied in thecontext of dynamic pricing. Firms may therefore bereluctant to implement dynamic pricing for fear of con-sumer backlash and negative press coverage.

In addition to managers’ concerns about the publicperception of dynamic pricing, regulators may also beinterested in the question of whether dynamic pricingreduces social welfare. If it is the case, then the legit-imacy of the practice may be questioned by advocacygroups. Regulators may also be interested in devel-oping dynamic pricing policies with the explicit goalof maximizing social welfare instead of profits or rev-enues. This problem is relevant to the central allocationof scarce resources and to the determination of tollsfor infrastructure projects such as bridges and tunnels.They may also wish to compare policies designed tomaximize social welfare to those designed tomaximizeprofits, to make a decision to deter (resp., encourage)private infrastructure if the two policies are signifi-cantly different (resp., sufficiently similar).

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1.1. Contributions and Main ResultsWe consider a central planner’s problem of maximiz-ing the total social welfare that can be obtained from cunits of capacity over a finite sales horizon [0,T]. Thefirm’s problem of maximizing revenues has been stud-ied by Gallego and van Ryzin (1994); the difference ofour study is that central planners are interested inmax-imizing the sum of consumers’ surplus and firm’s rev-enues over the sales horizon.We show that the welfare-maximizing dynamic pricing policy shares the samestructural properties as the revenue-maximizing pric-ing policy derived by Gallego and van Ryzin (1994).More precisely, both pricing policies are increasing inthe time-to-go and decreasing in the current inventorylevel. Moreover, when capacity is sufficiently scarce,the market-clearing price asymptotically maximizesboth revenue and social welfare as demand and capac-ity are scaled in proportion. Under certain conditions,even consumers’ surplus is asymptotically maximizedby the market-clearing price. Thus, in scaled systems,the objectives of the firm, the customers, and the cen-tral planner are essentially aligned when capacity isrelatively scarce.In industries without central planners, it is unlikely

that firms will price their products to maximize socialwelfare. However, they may also be reluctant to usethe revenue-maximizing dynamic pricing policy andinstead use the revenue-maximizing static price, eitherbecause managers believe in the zero-sum hypothesisthemselves or out of fear that their consumers subscribeto this belief. While it is clear that dynamic pricing ben-efits firms compared to the static price, does it in factreduce consumers’ surplus as some fear? Our analy-sis indicates the contrary: in many cases, compared toan optimal static price, dynamic pricing is win–win formanydemandfunctions.Moreprecisely, it increases theexpected revenue for thefirmandbenefits customers byincreasing their expected total surplus. This is becausethe social welfare allocated to the firm and the con-sumer, conditioning on a sale, is often well aligned. Weprovide indications of how the elasticity of demandrelates to the alignment of these objectives.

We make three main contributions. First, we con-duct a welfare analysis on the classic dynamic pricingmodel developed in Gallego and van Ryzin (1994). Wefind pricing policies that are asymptotically optimalfor revenue, social welfare, and consumers’ surplus,respectively. When capacity is scarce, we show that themarket-clearing price performs well for both the firmand the central planner, and in some important cases,for the customers as well.

Second, we show that contrary to common belief, therevenue-maximizing dynamic pricing policy improvesboth the revenue and consumers’ surplus over therevenue-maximizing static pricing policy for many

common demand functions. The elasticity of the de-mand plays an important role in the impact of dynamicpricing on customers.

Third, our results may change the public imageof dynamic pricing and have policy implications. Weshow that the dynamic pricing policy that maximizesrevenues is asymptotically optimal for welfare whencapacity is scarce. Hence, the use of dynamic pricingshould be accepted, if not welcomed, by regulators andthe public. This finding also has implications for policymakers regulating private infrastructure as they can beassured that if capacity is sufficiently constrained, thenthe objectives of the firm are well aligned with those ofpolicy makers.

Section 2 conducts a welfare analysis on dynamicpricing. We review the basic concepts of consumers’surplus, and formulate the optimality equation forthe welfare function. We also study fluid approxima-tion, and show the relationship between the welfare-maximizing and revenue-maximizing pricing policies.In Section 3, we define the Pareto efficient frontierof the welfare allocation between the firm and cus-tomers over the sales horizon. In Section 4, we showthat dynamic pricing is win–win for the constant elas-ticity demand function analytically and for the expo-nential demand function numerically. For the lineardemand function, dynamic pricing is win–win exceptin extreme situations where there is sustained demandover an extremely large booking horizon and capacityis very low.We then provide conditions on the demandfunction under which dynamic pricing is likely to bewin–win. Section 5 extends the classic dynamic pricingto accommodate network revenue management andqueueing problems such as toll roads and bridges.

1.2. Literature ReviewThe first stream of literature relates to our paperthrough the concepts of consumers’ surplus and wel-fare. Consumer’s surplus is the benefit that a customergains from participating in market transactions. Whenaggregated over all of the customers in the market,consumers’ surplus equals to the area to the right ofthe price under the demand curve (see, e.g., Varian2010). The concept originates in Dupuit (1844) and isessential to consumer theory and welfare economics.In the operations management and marketing litera-ture, the consideration of consumers’ surplus is usu-ally made from the perspective of a single customer.The consumer decides whether or not to buy a singleunit of the good. In this case, consumer’s surplus isusually defined for a single customer as the differencebetween her willingness to pay (WtP) and the pricecharged by the firm. For example, in Dhebar and Oren(1985), the customer only purchases the product if herWtP is greater than or equal to the price. The sameconcept is used by later papers such as Cachon and

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Swinney (2009), Su (2007), and Levin et al. (2009) toderive customers’ optimal strategy. We refer the read-ers to Wilson (1993) and references therein for earlierexamples. In this paper, we use both interpretations ofconsumers’ surplus, which turn out to be consistent inour setting.There is a vast literature on dynamic pricing in the re-

venuemanagement community.Gallego andvanRyzin(1994) investigate the optimal pricing policy of a firmselling a given stock of items over a finite horizon. Sincethen, dynamic pricing has been analyzed in variousmarket environments including inventory considera-tions and competition. SeeElmaghraby andKeskinocak(2003) and Bitran and Caldentey (2003), and refer-ences therein, for (dynamic) pricing models in rev-enue management. Another stream of literature, suchas Su (2007), Aviv and Pazgal (2008), Su (2007), Liuand van Ryzin (2008), Gallego et al. (2008), Chen andFarias (2018), and Chen et al. (2018), and referencestherein, considers strategic customers. These papersassume that consumers are usually surplusmaximizersand focus on finding the optimal pricing policy ratherthan analyzing the welfare gain from dynamic pricing.

The study of the impact of pricing policies on con-sumers has a long history. Thewelfare analysis of third-degree price discrimination in the case of market seg-mentation has been documented by, e.g., Varian (1985),Cowan (2007), and Cowan (2012). In general, price dis-crimination hurts social welfare unless the total out-put increases after price discrimination (Varian 1985).However, there are exceptions and price discrimina-tion can be win–win (Cowan 2012). The pricing pol-icy we are investigating applies to a different settingfrom market segmentation, and we find that welfareis more likely to increase after the firm switches froma static pricing policy to dynamic pricing. In termsof the pricing of electricity, Borenstein and Holland(2003) and Borenstein (2005) show that the compet-itive electricity market is inefficient unless the retailprice can vary in time as frequently as the produc-tion/wholesale costs, which is referred to as real-timepricing. As real-time pricing shares a similar form todynamic pricing, their papers are in line with our con-clusion that dynamic pricing improves social welfarein general. For brick-and-mortar retail, Stamatopouloset al. (2017) show that dynamic pricing is win–win inthe EOQ setting because it reduces holding costs andpasses them through to benefit consumers. In the caseof platforms setting prices andwages tomatch demandand supply, Cachon et al. (2017) show that dynamicpricing and dynamic wages in response to differentdemand states can maximize the platform’s profit andincrease consumers’ surplus compared to a fixed con-tract. Feldman et al. (2016) conduct an empirical studyof congestion pricing for parking spaces in the city ofSan Francisco and show that the welfare is increased

in popular areas but reduced in less-congested areas.For airline tickets, the pricing practice can be classi-fied into dynamic pricing, which arises from demanduncertainty and is the main subject of our paper,and intertemporal price discrimination. Dana (1999)and Lazarev (2013) find that social welfare can eitherincrease or decrease with intertemporal price discrim-ination, and that business travelers (with a high disu-tility of waiting) can benefit from an increasing num-ber of tourists (with a low disutility of waiting). Dana(1998) investigates the effect of advanced-purchase dis-counts (which can be interpreted as a form of dynamicpricing) when demand is uncertain, and concludesthat welfare is decreased when high-valuation cus-tomers have more uncertain demand but unchangedwhen low-valuation customers have more uncertaindemand. Our result differs from his because (1) ourmodel does not consider the heterogeneity in demanduncertainty of customers having different valuations,and (2) the dynamic nature of our model mitigates the“misallocation effect” defined in Dana (1998) that iscaused by rationing.Williams (2013) empirically showsthat travelers benefit from dynamic pricing using adata set of daily fares at the flight level. McAfee andVelde (2008) investigate dynamic pricing with con-stant demand elasticity and prove that the revenue-maximizing pricing policy can also achieve efficiency.The insights drawn in this paper strengthen their pointby helping to explain why and for what demand func-tions this is likely to happen. Although the implica-tions of our results (that dynamic pricing can benefitconsumers) are similar to those of some of the abovepapers, the main differences are as follows: (1) Westudy the welfare-maximizing pricing policy based onthe model by Gallego and van Ryzin (1994) and showthatwhen the system is scaled, thewelfare-maximizingpricing policy is similar to the revenue-maximizingpricing policy. (2) We explain why dynamic pricing isusually win–win compared to the revenue-maximizingstatic price, and derive conditions of the demand func-tion under which dynamic pricing is likely to benefitconsumers.

2. Dynamic Pricing to MaximizeSocial Welfare

In this section, we review the concept of consumers’surplus and consider a central planner who attemptsto maximize social welfare when a firm has a finitehorizon to sell a given inventory of perishable, non-replenishable products. Structural properties of thewelfare-maximizing pricing policy are explored. Wethen conduct an asymptotic analysis in the fluid regimefollowing the methodology of Gallego and van Ryzin(1994). We show that when capacity is scarce, thesame market-clearing fixed-price heuristic that asymp-totically maximizes revenue also asymptotically maxi-mizes welfare.

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Table 1. Constant Elasticity Demand (CES), Exponential Demand, and Linear Demand

d(p) S (p) U(q) Parameters

CES ap−b ab − 1 p1−b b

b − 1 a1/b q1−1/b a > 0, b > 1

Exponential λ exp(−p/θ) λθ exp(−p/θ) θq(1+ lnλ− ln q)1q≤λ + θλ1q>λ θ, λ > 0Linear (b − p)+/2a ((b − p)+)2/4a (−aq2 + bq)1q≤b/2a + b2/4a1q>b/2a a , b > 0

2.1. Consumers’ SurplusSuppose the firm sets a price of p and there is a rep-resentative consumer (see, e.g., Varian 2010, Andersonet al. 1988) who acts on behalf of all consumers inthe market. For q units of the product, the represen-tative consumer’s monetary utility U(q) of the prod-uct is an increasing, concave, and twice differentiablefunction with U(0) 0. The maximum net utility isgiven by S (p) , maxq≥0U(q) − pq. To maximize hernet utility, the representative consumer chooses thequantity q d(p) , (U′)−1(p) obtained by the first-order necessary condition, which is also sufficient fromthe assumed concavity of U. Here, ( · )−1 denotes theinverse function. This optimization defines consumers’surplusS ( · ) andmarket demand d( · ). By the envelopetheorem, the surplus of the representative consumersatisfiesS ′(p)−d(p). Moreover, since consumers earnno surplus at p ∞, we have S (p) ∫∞p d(x) dx.In Table 1, we provide three instances of the util-

ity function U( · ), the demand function d( · ), and theconsumers’ surplusS ( · ) in closed forms. Another wayto interpret the relationship between d(p) and S (p)without invoking the representative consumer and theutility function is to assume a mass λ of nonatomicconsumers with random WtP Ω. The market demandis d(p) λP(Ω ≥ p), as only those customers with WtPhigher than p buy the product. Their surplus is thedifference, if positive, between the WtP and the priceλE[(Ω − p)+] λ ∫∞p P(Ω ≥ x) dx ∫∞p d(x) dx S (p).Both interpretations are consistent, and we will usethem interchangeably in the paper.

2.2. Optimal Social Welfare: Stochastic CaseThe issue of capacity constraints with random demandarises naturally for firms aiming to maximize revenue,such as airlines, hotels, concert venues, and car rentalagencies. In these settings, the firm has a fixed invest-ment in finite capacity c that is not replenished dur-ing the sales horizon [0,T]. For convenience, we denotet ∈ [0,T] as the time-to-go, and x as the remaininginventory. At time t T, the firm has x c unitsof capacity. Demand is assumed to be a Poisson pro-cess with intensity d(p) for price p. We assume with-out loss of generality that unsold items at t 0 havezero salvage value. The firm’s revenue-maximizingdynamic pricing in this setting is studied in Gallego

and van Ryzin (1994). See Elmaghraby and Keskinocak(2003) for a comprehensive survey.

To provide the intuition for the formulation of thewelfare optimization problem, we first summarize theresults for the revenue-maximizing dynamic pricingpolicy. Let V(t , x) be the optimal expected revenue ofthe firm for time-to-go t ≥ 0 and x units of inventory. Itcan be shown that V(t , x) satisfies

∂V(t , x)∂t

maxp≥0d(p)(p −∆V(t , x)),

V(0, x) V(t , 0) 0,(1)

where ∆V(t , x) , V(t , x) − V(t , x − 1) represents themarginal value of the xth unit of inventory. Therevenue-maximizing (dynamic) price for state (t , x) isdenoted p∗V(t , x) p(∆V(t , x)) , arg maxp≥0d(p)(p −∆V(t , x)). Under mild conditions,1 p∗V(t , x) exists andis increasing in t and decreasing in x.

Next, we consider the optimal expected social wel-fare W(t , x) generated over the sales horizon for anonanticipating pricing policy p(t , x). We will showthat the problem essentially adds the consumer’s sur-plus conditioning on sales to the price in the objectivefunction corresponding to the firm’s problem. Indeed,at price p, the firm earns revenue at rate pd(p), andconsumers earn surplus at rateS (p); hence, social wel-fare increases at rate (p +S (p)/d(p))d(p). To maximizesocial welfare W(t , x), the central planner uses p∗W (t , x)for state (t , x).Proposition 1. The optimal social welfare W(t , x) satisfies

∂W(t , x)∂t

maxpS (p)+ d(p)(p −∆W(t , x))

S (∆W(t , x)), x ≥ 1, t ≥ 0, (2)

W(0, x) W(t , 0) 0,

where ∆W(t , x) W(t , x) − W(t , x − 1) represents themarginal value of capacity for social welfare. Moreover,p∗W (t , x)∆W(t , x) is increasing in t and decreasing in x.Remark 1. (1) The key element of the equations forboth revenue and social welfare is the “shadow cost”of selling one unit of the product. For the revenue-maximizing problem, the shadow cost is the marginalvalue of capacity for revenue ∆V ; in the welfare prob-lem, it is the marginal value of capacity for wel-fare ∆W . The optimal revenue rate ∂V/∂t is therefore

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maxpd(p)(p − ∆V), as if the firm were maximizingthe profit of selling one unit of its product at variablecost∆V . The optimal social welfare rate ∂W/∂t is equalto S (∆W), as if the central planner were maximizingthe social welfare of selling one unit of the product atcost ∆W , S (p)+ d(p)(p −∆W).(2) Recall the static pricing problem: if the cost is z,

then a price p results in profit d(p)(p − z) and con-sumers’ surplusS (p). It can be shown that tomaximizewelfare in the static case, the price should be set equalto the marginal cost z, and all of the welfare is earnedby customers (see the proof of Proposition 1 for theproof for this claim). In contrast, in the dynamic case,the price is equal to the marginal value of capacity forwelfare, which is positive in general; thus, the firm cangain a positive fraction of the total social welfare.(3) The welfare-maximizing price is not necessarily

lower than the revenue-maximizing price (see Section 4for an example), although this is true in most cases.

(4) Although p∗W (t , x) is not equal to p∗V(t , x), theyshare the same structural properties—i.e., they bothincrease in t and decrease in x. This implies that tomaximize social welfare, the central planner shouldalso immediately increase the price when an itemis sold, and gradually decrease the price over time,exactly as the firm would do.

2.3. Optimal Social Welfare: Fluid ApproximationThe fluid approximation has been studied extensivelyin the literature of dynamic pricing. The Poisson arrivalprocess of rate d(p) is replaced by a determinis-tic demand flow equal to the mean rate: within aninfinitesimal time period dt, the size of the demand isd(p) dt for price p. The revenue rate is therefore pd(p),and the surplus rate is S (p). This approximation usu-ally provides tractable heuristic pricing policies thatperform well in large-scale systems (see Section 2.4 fordetails). In this section, we will demonstrate that thepricing policies that maximize welfare, revenue, andconsumers’ surplus, respectively, in the fluid approxi-mation actually coincide in many cases.First consider the following welfare-maximizing

problem:

W(T, c) maxpt , 0≤t≤T

∫ T

0(S (pt)+ pt d(pt)) dt

subject to∫ T

0d(pt) ≤ c.

(3)

The objective is the total welfare over the horizon ofthe fluid approximation. The constraint states that thetotal demand should be less than the initial capacity.Proposition 2 shows that when the initial capacity isscarce, then the market-clearing price pmc , d−1(c/T) iswelfare maximizing.

Proposition 2. If d(p) is continuous and decreasing, andc < Td(0), then the optimal solution to (3) is pt ≡ pmc for allt ∈ [0,T]. Moreover, W(T, c) ≥W(T, c).We now consider the fluid problems for the firm and

consumers. For the firm (see Gallego and van Ryzin1994 for details), the revenue-maximizing problem is

R(T, c) maxpt , 0≤t≤T

∫ T

0pt d(pt) dt

subject to∫ T

0d(pt) dt ≤ c.

(4)

The optimal solution is to set the price at pt ≡maxpmc , p for the entire horizon, where

p,argmaxp≥0

pd(p)

is the revenue-maximizing price when capacity isunconstrained. The solution is pt pmc when c≤Td(p).This is the case when capacity is insufficient to sat-isfydemandat theunconstrained revenuemaximizer p.It is the most relevant case in revenue managementapplications.

The surplus-maximizing fluid problem for con-sumers is given by

S(T, c) maxpt , 0≤t≤T

∫ T

0S (pt) dt

subject to∫ T

0d(pt) dt ≤ c.

(5)

When c ≥ Td(0), the price that maximizes surplus isp f ≡ 0. This is clear since p f 0 is a feasible solutionand maximizes the objective function pointwise.

The problem is more subtle when capacity is scarcec < Td(0), and some definitions are required beforetackling this problem. Assume that d( · ) is everywheredifferentiable and let h(p) , −d′(p)/d(p) be the haz-ard rate associated with the random variable Ω withtail probability P(Ω ≥ p) d(p)/d(0) when d(0) < ∞.When Ω is well defined, S (p)/d(p) E[Ω − p |Ω ≥ p]is the expected conditional mean residual life at p.Recall that the random variable Ω is NBUE (new bet-ter than used in expectation) if S (p)/d(p) is decreas-ing in p. Moreover, an increasing h(p) implies that theassociated random variable is NBUE (see Hollanderand Proschan 1972 and references therein). We findthat the distribution of consumers’ WtP Ω plays animportant role in determining the form of the surplus-maximizing pricing policy. We are now ready to ad-dress the surplus-maximizing problem when capacityis scarce.

Proposition 3. Suppose c < Td(0). If Ω is strictly NBUE,then the optimal solution to (5) is pt ≡ 0 until the inventoryis depleted. If h(p) is strictly decreasing, then pt ≡ pmc .

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Proposition 3 states that ifΩ is strictly NBUE, whichis guaranteed if h(p) is strictly increasing, then the firmprices its product at pt 0 until capacity is exhaustedto maximize consumers’ surplus. The firm then usesthe choke price, possibly infinity, to turn off demand,generating no additional surplus during the rest ofthe horizon. This scheme results in surplus cS (0)/d(0).In contrast, if h(p) is strictly decreasing, then themarket-clearing price maximizes consumers’ surplus,resulting in surplus TS (pmc) cS (pmc)/d(pmc). Wenow present three examples: the CES demand has adecreasing hazard rate, and pmc maximizes (5); theexponential demand has a constant hazard rate, andany price between [0, pmc] is surplus maximizing; thelinear demand function has an increasing hazard rate,and thus pt 0 maximizes consumers’ surplus.

One may be surprised that a low price does notalways benefit consumers. This is because the firm onlyhas a fixed initial capacity. When the price is low, thedemand is high and the inventory is depleted quickly.However, it is not guaranteed that the customers whobuy the product have high WtP. In some cases (whenthe hazard rate is decreasing), it is beneficial to raisethe price to “screen out” consumers with lowWtP; theincrease in the average WtP of customers who buy theproduct is higher than the increase in price. The totalconsumers’ surplus may increase as a result.

Combining the previous results with Proposition 3,we see that if c < Td(p), then the welfare W(T, c) isdecomposed into two components: the firm’s revenuecpmc and the consumers’ surplus TS (pmc), with themarket-clearing price maximizing both welfare andrevenue. In addition, if h(p) is decreasing, then themarket-clearing price also maximizes consumers’ sur-plus. On the other hand, if Ω is NBUE, then usingthe market-clearing price results in a loss to consumerscompared to the optimal consumers’ surplus. The gapis equal to cS (0)/d(0) − TS (pmc) c(E[Ω] − E[Ω −pmc |Ω > pmc]). These results for the fluid approxima-tion suggest that the market-clearing price may be agood heuristic for revenue and welfare when capacityis scarce, and also a reasonable heuristic for surplus.This finding is summarized below:Corollary 1. If c ≤ Td(p) and h(p) is decreasing, thenW(T, c) R(T, c) + S(T, c), and the market-clearing pricesimultaneously optimizes the firm’s revenues, social welfare,and consumers’ surplus in the fluid approximation.

2.4. Asymptotic Optimality of the Fluid SolutionThe fluid approximation is closely related to the follow-ing asymptotic regime of the stochastic system whenwe use the WtP interpretation d(p) λP(Ω ≥ p): if theinitial capacity c and the aggregate arrival rate λTtend to infinity while the ratio λT/c remains constant,then the Poisson arrival process can be approximatedusing the fluid approximation. Gallego and van Ryzin(1994) have shown that the fixed price p f maxpmc , p

is asymptotically optimal for the firm. More pre-cisely, these authors show that V(T, c | p f )/V(T, c)→ 1,where V(T, c | p f ) is the expected revenue when a fixedprice p f is used until the end of the sales horizon oruntil capacity is exhausted (in which case the chokeprice is used). This is a highly relevant regime becausedemand and capacity are both high inmany industries.

Next, we show that when capacity is scarce (i.e.,c ≤ Td(p)), the fixed price p f pmc is asymptoticallyoptimal for social welfare as well. We consider asequence of systems indexed by n: c(n) nc andλ(n)T(n) nλT. Note that the values of λ and T do notessentially change the pricing policy (up to a linearchange of state variables) or the value function, as longas their product λT is fixed.

Proposition 4. If c < Td(p), then the welfare gener-ated by the market-clearing price pmc is asymptoticallyoptimal—i.e.,

limn→∞

W(T(n) , c(n) | pmc)W(T(n) , c(n)) 1.

Proposition 4 states that for large-scale scarce-capac-ity systems, the central planner can simply use themarket-clearing price and generate welfare that is o(n)less than the optimal social welfare. Meanwhile, thefirm also accepts the policy because it allows it toachieve revenue that is only o(n) less than the optimalrevenue.

By Proposition 3, the market-clearing price also as-ymptotically maximizes consumers’ surplus when thehazard rate is decreasing. Therefore, for large-scale sys-tems, using the market-clearing price is win–win to alarge extent, because it pleases both the firm and cus-tomers, and thus also the central planner.

3. Pareto EfficiencyBoth the welfare-maximizing and revenue-maximizingpricing policies considered in Section 2 are “Pareto effi-cient” in the following sense:

Definition 1. Anonanticipating pricing policy is Paretoefficient if no other nonanticipating pricing policies canweakly improve both the expected revenue and con-sumers’ surplus and strictly improve at least one ofthem.

Indeed, under mild conditions, the welfare-maxi-mizing and revenue-maximizing pricing policies areunique, and no other policies can weakly improvesocial welfare or revenues, respectively.

In this section, we characterize all Pareto-efficientpricing policies and the corresponding welfare alloca-tions. Because of the Markov property of the system,the family of nonanticipating pricing policies can beexpressed in the form of p(t , x)—i.e., the price is only

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Figure 1. Pareto Frontier of the Allocation of Revenue andConsumers’ Surplus for a Given Initial State (T, c)

S

V

Maximal surplus (π = 0)

Maximal revenue (π = 1)

Maximal welfare (π = 0.5)

a function of the current time-to-go and the remain-ing inventory. For any π ∈ [0, 1], consider the problemof maximizing a convex combination of the objectivefunction of the firm and the objective function of theconsumers. More precisely, assume that the quantity ofinterest is π × revenue + (1 − π) × consumers’ surplus.Let Wπ(T, c) denote its optimal value, which satisfies asimilar differential equation to (2) whereby we replaced(p)(p + S (p)/d(p)) by d(p)(πp + (1 − π)S (p)/d(p)).The pricing policy that achieves the optimal value isdenoted as pπ(t , x). Under this policy, the expected rev-enue and consumers’ surplus are denoted asVπ and Sπ,respectively.

Proposition 5. A pricing policy is Pareto efficient if andonly if it is pπ for some π ∈ [0, 1].When π 1, the firm’s revenue is maximized; when

π 0.5, social welfare is maximized; when π 0, con-sumers’ surplus is maximized. As π ranges from 0to 1, (Vπ , Sπ) generates the Pareto frontier on the V − Splane, as illustrated in Figure 1. By definition, thePareto frontier is always concave; when a straight lineof slope −π/(1 − π) is drawn, its tangent point tothe frontier corresponds to Wπ(T, c). In particular, thetangent point of the vertical line maximizes revenue(i.e., π 1). For an arbitrary nonanticipating pricingpolicy, the generated consumers’ surplus and revenuecorrespond to a point inside the area enclosed by thePareto frontier, the dotted lines, and the axes.

3.1. Asymptotic Analysis of the Pareto FrontierWe already know that dynamic pricing p∗V(t , x) is theoptimal nonanticipating policy for the firm. We willnow show that it also asymptotically maximizes socialwelfare. In fact, we will present a stronger and deeperresult: the Pareto frontier “shrinks” relative to n inthe same regime as in Section 2.4. This result showsthat the zero-sum hypothesis is false: even when thefirm is only maximizing its revenue using dynamicpricing, the resulting social welfare is almost as sub-stantial as that generated by the welfare-maximizingdynamic pricing policy. This means that even in the

rare case that consumers are hurt by dynamic pricing,the firm’s gain exceeds the loss to consumers. Considerthe asymptotic regime c(n) nc and λ(n)T(n) nλT.

Theorem 1. If c ≤ Td(p) and g(x), xd−1(x) is concave forx ≤ d(p), then the length of the segment π ∈ [0.5, 1] dividedby n converges to 0 as n→∞.

Theorem 1 states that in the limit, the segment π ∈[0.5, 1] grows more slowly than n. Since both socialwelfare and revenues grow linearly in n, the distancebetween the two allocations π 1 and π 0.5 becomesnegligible when n is large. This implies that any Pareto-efficient pricing policy pπ for π ∈ [0.5, 1], includingpπ1 p∗V and pπ0.5 p∗W , is asymptotically optimalfor all π ∈ [0.5, 1]. Theorem 1 has significant policyand practice implications. For policymakers concernedabout the social welfare generated by firms that usedynamic pricing, our results show that dynamic pric-ing does not hurt welfare significantly, especially forindustries with large-scale demand and capacity. It isworth pointing out that when c > Td(p), the segmentπ ∈ [0.5, 1] grows linearly in n. Moreover, the otherhalf segment, π ∈ [0, 0.5), does not shrink in n whenthe hazard rate is increasing, as implied by Proposi-tion 3. Thus, neither the market-clearing price nor thedynamic policy p∗V(t , x) can asymptotically maximizeconsumers’ surplus in that case.

3.2. A Numerical ExampleIn this section, we provide numerical comparisons ofp∗V and p∗W . We showed in Section 2.4 that p∗V tendsto be close to p∗W when capacity is constrained inlarge-scale systems. The objective of this section is toillustrate that the revenue-maximizing and welfare-maximizing pricing policies are similar even for small-scale systems. Consider exponential demand d(p) exp(−p) and S (p) exp(−p). Table 2 shows the ratio ofp∗V(T, c)/p∗W (T, c) for different values of T and c. Whenc/T is small (the lower triangle), p∗V is already veryclose to p∗W even for small-scale systems such as c 1and T 20. As capacity becomes increasingly scarce,the revenue-maximizing price approaches the welfare-maximizing price. When c/T is fixed (on the diagonal),

Table 2. Ratio of p∗V (T, c)/p∗W (T, c) for Different T and c

c

T 1 5 10 20 50 100

4 1.18 4.56 18.79 1× 108 ∞ ∞20 1.02 1.10 1.42 5.78 1× 108 ∞40 1.01 1.03 1.07 1.41 53.00 1× 1015

80 1.00 1.01 1.01 1.04 2.00 249.99200 1.00 1.00 1.00 1.00 1.02 1.42400 1.00 1.00 1.00 1.00 1.00 1.01

Note. In the lower triangle, the ratio is very close to 1.

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the ratio converges to one as c increases. Because expo-nential demand functions have constant hazard rates,Corollary 1 implies that W R + S and p∗V also asymp-totically achieves the optimal consumers’ surplus.

4. When Is Dynamic Pricing Win–Win?For unregulated industries, firms with monopolypower usually use either the revenue-maximizingdynamic pricing policy p∗V(t , x) or the revenue-maximizing static price p∗f . If firms are concerned abouttheir reputation and future revenues given the negativepublic perception of dynamic pricing, then they maychoose p∗f and forgo the benefits of dynamic pricing.In this section, we will clarify this misconception andshow that dynamic pricing is in many cases win–wincompared to using the static price p∗f . Moreover, wewill identify conditions of the WtP distribution underwhich dynamic pricing is likely to be win–win.

4.1. Pricing and Consumers’ SurplusWe first derive the consumers’ surplus for an arbitrarystatic price p f and the revenue-maximizing dynamicpricing policy p∗V(t , x). We showed in Section 2.2 thatthe generated surplus conditioning on a sale givenprice p is S (p)/d(p). Thus, by replacing the revenue pwith S (p)/d(p) in (1), we obtain the differential equa-tion for the consumers’ surplus S(t , x | p) generated byany nonanticipating pricing policy p(t , x). In particular,for the revenue-maximizing pricing policy p∗V , the con-sumers’ surplus for time-to-go t and remaining inven-tory x is S(t , x | p∗V). For any static price p f , S(t , x | p f )E[S (p f )/d(p f ) × minc ,Pois(Td(p f ))], where Pois(λ)represents a Poisson random variable with mean λ.We will compare S(t , x | p∗V) and S(t , x | p∗f ), wherep∗f arg maxpp E[minc ,Pois(Td(p))] is the revenue-maximizing static price. It is worth mentioning that p∗fremains a function of the initial state (T, c), while beingfixed over the horizon.We first investigate the Pareto efficiency of the re-

venue-maximizing static price. We will show that astatic policy is rarely Pareto efficient—i.e., it usuallyresults in a welfare allocation that is strictly inside thePareto frontier. This implies that we can find a pricingpolicy (not necessarily p∗V ) that improves both revenueand consumers’ surplus over the revenue-maximizingstatic price.Proposition 6. The revenue-maximizing static price p∗fresults in a welfare allocation that lies on the Pareto frontieronly if both of the following conditions are satisfied:1. It equals to the unconstrained revenue maximizer p;2. pd(p) is not differentiable at p∗f —i.e.,

limp→(p∗f )−

p(d(p)) − p∗f d(p∗f )p − p∗f

> limp→(p∗f )+

p(d(p)) − p∗f d(p∗f )p − p∗f

.

The first condition only holds when the initial capac-ity is abundant relative to demand at p—i.e., c>Td(p).

However, it rarely holds in revenuemanagement appli-cations as the theory of dynamic pricing mainlyfocuses on scenarios of scarce capacity. The second con-dition implies that the function pd(p) “sticks out” at p∗f .Those two conditions are rarely satisfied in practice;hence, the revenue-maximizing static price is almostnever Pareto efficient.

4.2. Common Demand FunctionsNext, we study the expected consumers’ surplus gener-ated by the revenue-maximizing dynamic/static pric-ing policies for three common demand functions:constant elasticity demand, exponential demand, andlinear demand. We show that consumers are better offwith dynamic pricing in most cases.Constant elasticity demand: Consider d(p) ap−b andthus S (p) ap1−b/(b − 1), for a > 0 and b > 1. It is firststudied in Karlin and Carr (1962) and has become apopular model to characterize how demand reacts tochanges in price. The advantage of the constant elastic-ity demand is that it can be transformed into a linearfunction by taking logarithms, making the parametersrelatively easy to estimate. Its other advantages includeits ability to characterize the nonlinear effects arising inthe market and the possibility of deriving it from theCobb–Douglas production function.

Recall that V(t , x | p) denotes the revenue earnedby the firm for a nonanticipating pricing policy p.The elegant connection between the revenue and con-sumers’ surplus of the CES demand is first discoveredin McAfee and Velde (2008) and is stated in a moregeneral form here:Proposition 7. If d(p) ap−b for a > 0 and b > 1, then forany nonanticipating pricing policy (including static poli-cies), we have

V(t , x | p) (b − 1)S(t , x | p).

Proposition 7 states that for any initial states, thefirm’s revenue is always a constant multiple of the con-sumers’ surplus for any nonanticipating pricing poli-cies. This implies that

S(t , x | p∗V)S(t , x | p)

V(t , x | p∗V)V(t , x | p) ≥ 1,

because p∗V maximizes the firm’s revenue. In otherwords, the revenue-maximizing dynamic pricing pol-icy also maximizes consumers’ surplus. Moreover,compared to any static pricing policy, both the firmand consumers benefit from dynamic pricing at thesame rate. The Pareto frontier of the CES demandfunction collapses to a single point. All Pareto-efficientallocations correspond to the same pricing policy—i.e., p∗V(t , x), for all π ∈ [0, 1]. In particular, p∗V(t , x) p∗W (t , x).We conduct a numerical experiment to investigate

the percentage improvement in consumers’ surplus

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Figure 2. (Color online) Percentage Improvement Due toDynamic Pricing in the Expected Surplus (Revenue) Overthe Optimal Static Pricing Policy, for d(p) ap−b ,S (p) ap1−b/(b − 1), and a 1, b 2

using the dynamic pricing policy over the revenue-maximizing static price—i.e., (S(T, c | p∗V)−S(T, c | p∗f ))/S(T, c | p∗f ) or equivalently (V(T, c | p∗V) − V(T, c | p∗f ))/V(T, c | p∗f ). Figure 2 illustrates the results for differ-ent initial states (T, c), ranging from scarce to abun-dant capacity. It suggests that dynamic pricing is espe-cially effective when capacity is scarce (c is smallrelative to T).Exponential demand: Consider d(p) λ exp(−p/θ)for λ, θ > 0 and S (p) λθ exp(−p/θ). Exponentialdemand is used extensively in the literature onmarket-ing and revenue management, because it is well de-finedwhen p→ 0 and infinitely differentiable. It allowsfor an elegant and insightful solution to V(t , x | p∗V)(Gallego and van Ryzin 1994). This analytical tractabil-ity is preserved for consumers’ surplus.Proposition 8. Consider d(p) λ exp(−p/θ), S (p) λθ exp(−p/θ). Let Λ λt/e. We have

V(t , x | p∗V) θ(Λ+ log(P(Pois(Λ) ≤ x))),

p∗V(t , x) θ+ θ log(

P(Pois(Λ) ≤ x)P(Pois(Λ) ≤ x − 1)

),

S(t , x | p∗V) θE[Pois(Λ) | Pois(Λ) ≤ x],V(t , x | p f ) p f E[minPois(td(p f )), x],S(t , x | p f ) θE[minPois(λtd(p f )), x].

The solutions can be directly verified from the differ-ential equations.

Although S(t , x | p∗V) has a closed form, we can-not solve the revenue-maximizing static price explic-itly. Thus, we resort to the numerical computation ofp∗f (T, c). Given that the relevant quantities only dependonλt and x,wecansimply setθλ1, and thenumeri-cal result reflects all possible parameter values.As illus-trated in Figure 3, dynamic pricing always benefits con-sumers, especially when capacity is scarce.

Figure 3. (Color online) Percentage Improvements Due toDynamic Pricing in the Expected Revenue (Upper Panel)and Consumers’ Surplus (Lower Panel) Over the OptimalStatic Pricing Policy, for d(p) λ exp(−p/θ) andS (p) λθ exp(−p/θ)

Linear demand: d(p)λ(1− ap)+ is another commonlyuseddemand functionbecauseof its simplicityandana-lytical tractability. Consumers of linear demand havequadratic surplus S (p) λ((1 − ap)+)2/2a. Unfortu-nately, in the context of dynamic pricing, there are fewanalytical results for the revenue or consumers’ sur-plus. Thenumerical results are illustrated inFigure 4 forλ a 1. For many initial state (T, c), consumers ben-efit from dynamic pricing—i.e., S(T, c | p∗V) ≥ S(T, c | p∗f ).However, dynamic pricing hurts consumers whencapacity is extremely scarce—i.e., T is very large and cis very small.

To investigate this phenomenon, consider the asymp-totic regime c 1 and T→∞. For simplicity, let d(p)1−p. By solving the differential equations explicitly, forthe dynamic pricing policy, we have

p∗V(T, 1)2+T4+T

, S(T, 1 | p∗V)2T(T + 4)2 .

On the other hand, the optimal static price maxi-mizes pE[min1,Pois(T(1−p))] p(1−exp(−T(1−p)))

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Figure 4. (Color online) Percentage Improvements Due toDynamic Pricing in the Expected Revenue (Upper Panel)and Consumers’ Surplus (Lower Panel) Over the OptimalStatic Pricing Policy, for d(p) λ(1− ap)+ andS (p) λ((1− ap)+)2/2a

and thus p∗f 1 − O(log(T)/T). Hence, the con-sumers’ surplus for the optimal static price is approx-imately S(T,1 | p∗f ) (1 − p∗f )(1 − exp(−T(1 − p∗f )))/2

O(log(T)/T), dominating S(T,1 | p∗V) O(1/T). Never-theless, it can be shown that the social welfare gen-erated by p∗V , which is 1−O(1/T), is higher than thatof p∗f , which is 1−O(log(T)/T), in the same asymptoticregime.For general demand functions, the revenue-maximiz-

ing static/dynamic prices usually do not have explicitforms. Numerical computations may therefore be nec-essary.Nevertheless,weattempt topartially answer thisquestion for general demand functions in Section 4.4.

4.3. DiscussionAmong the three examples, revenue-maximizing dy-namic pricing is win–win for the CES and exponentialdemand functions. For CES demand, it even generatesmore consumers’ surplus than any static pricing pol-icy. For linear demand, it may hurt consumers’ surpluswhen the initial capacity is extremely scarce relativeto the sales horizon. Two natural questions arise: Whydoes dynamic pricing improve consumers’ surplus for

most distributions of customers’ WtP but not for oth-ers? Which characteristics of a demand function makedynamic pricing favorable to customers?

We provide a heuristic answer to the first questionhere while formalizing it and answering the secondquestion in Section 4.4. The intuition lies in how thewelfare is allocated between the firm and the customersconditioning on a sale—i.e., the tuple (p ,S (p)/d(p)).Recall that if the price is p, then on a sale, the firmearns revenue p and the expected surplus generatedby the sale is S (p)/d(p). For the CES demand, theexpected surplus conditioning on a sale S (p)/d(p) p/(b − 1) is perfectly aligned with the revenue p asb > 1. In this case, when the firm maximizes its totalexpected revenue using dynamic pricing, it also coin-cidentally maximizes the total consumers’ surplus.For exponential demand, the expected surplus on asale is S (p)/d(p) θ, which is independent of the rev-enue. Thus, consumers’ surplus is not very sensitiveto the pricing policy. It turns out that dynamic pricingstill increases consumers’ surplus by increasing sales,but not as much as the CES demand. For the lineardemand, S (p)/d(p) (1 − ap)+/2 is negatively alignedwith the firm’s revenue p earned from a sale. When thefirm maximizes its revenue using dynamic pricing, itmay hurt customers because their objectives are mis-aligned. Nevertheless, even in the extreme case of c 1and a large value of T, we see that the dynamic pricingpolicy p∗V generates more welfare than the fixed pricingpolicy p∗f , and thus even here, the zero-sum hypothesisfails to hold.

4.4. When Is Dynamic Pricing Win–Win?Wenow formalize the intuition stated in Section 4.3. Aswe focus on the revenue-maximizing dynamic pricingpolicy, the resulting allocation is the rightmost pointon the Pareto frontier. If the Pareto frontier is very long,then the two allocations of π 0 and π 1 are far aparton the V − S plane. In this case, the consumers’ sur-plus generated by p∗V(t , x) corresponding to π 1 maybe significantly lower than that generated by p∗S(t , x),which is the surplus-maximizing pricing policy andcorresponds to π 0. In particular, the allocation ofthe revenue-maximizing static policy, which is a pointinside the Pareto frontier, can be higher than the alloca-tion of p∗V , implying higher consumers’ surplus. In fact,no pricing policies can be near optimal for both the firmand customers. On the other hand, if the Pareto fron-tier is short, then the two allocations corresponding toπ 0 and π 1 are close. As a result, the revenue-maximizing dynamic pricing policy generates a levelof consumers’ surplus that is only slightly lower thanthe maximal surplus. It is unlikely that the allocationof p∗f is placed higher than p∗V on the plane. In theextreme case, the frontier collapses to a single point forthe CES demand function. The revenue-maximizing

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pricing policy also maximizes consumers’ surplus orany Wπ(T, c).To find demand functions whose Pareto-efficient

frontier is short, we resort to the intuition discussedin Section 4.3: if the welfare allocation conditioning ona sale (p ,S (p)/d(p)) is aligned, then the term πp +

(1− π)S (p)/d(p) does not change significantly for π ∈[0, 1], and maximizing Wπ yields similar allocations.

Theorem 2. If pd(p)/(S (p)) ∈ [a , b] for all prices p thatmay be used, then

S(T, c | p∗V) ≥ab

S(T, c | p∗S).

Theorem 2 implies that the revenue-maximizingpricing policy almost maximizes consumers’ surplus ifthe ratio a/b is close to one. In this case, it is unlikelythat a static price can achieve higher consumers’ sur-plus than the revenue-maximizing pricing policy. Infact, under the condition in Theorem 2, we can showthat V(T, c | p∗S) ≥ (a/b)V(T, c | p∗V) by the same argu-ment. That is, the surplus-maximizing pricing pol-icy also achieves a fraction a/b of the maximal rev-enue. Therefore, the Pareto frontier is contained ina rectangle of length (1 − a/b)V(T, c | p∗V) and height(1− a/b)S(T, c | p∗S). As a/b approaches 1, the rectangleshrinks and the Pareto frontier becomes shorter. As aresult, dynamic pricing is more likely to be win–win.For the linear demand function d(p) 1 − p, we havepd(p)/S (p) 2p/(1 − p). The dynamic price is alwaysgreater than p 1/2. If the price set by the firm rangesfrom 1/2 to 2/3, then pd(p)/S (p) ∈ (2, 4) and the ratioa/b 1/2, so the pricing policy generates at least a halfof the maximal consumers’ surplus.

It is natural to ask what demand functions have avalue of a/b close to one. By Proposition 7, a/b 1for the demand function with constant elasticity. Here,demand elasticity e(p) equals −pd′(p)/d(p), which isa standard definition in economics. This implies thatthe welfare allocation is related to the demand elastic-ity. Indeed, because (pd(p))′ pd′(p)+ d(p) andS ′(p)−d(p), the demand elasticity reflects the ratio of thederivatives of the revenue function pd(p) and the con-sumers’ surplus S (p), which is precisely e(p) − 1. Ifthe elasticity has small variations, so does pd(p)/S (p);consequently, the welfare allocated to the firm and theconsumers (p ,S (p)/d(p)) is well aligned. This intuitionis validated in Proposition 9.

Proposition 9. For p ≥ p, if e(p) ∈ [emin , emax], then

pd(p)S (p) ∈ [emin − 1, emax − 1].

We restrict our attention to p ≥ p because this is thedomain of the price that the firm may consider. It alsoguarantees that e(p) ≥ 1 so that the bounds are positive.

5. ExtensionsIn this section, we explore several extensions to ourbasic case, including network revenue managementand a dynamic pricing model for queueing systems.

5.1. Network Revenue ManagementGallego and van Ryzin (1997) consider a firm witha set of finite resources that are used to produce aset of products. The firm sets prices dynamically foreach product over a finite horizon to maximize itsexpected revenue. This setting has been studied exten-sively because of its wide applicability to many indus-tries. In this section, we investigate the welfare impli-cations of the network revenue management problem.

Consider a representative customer who gener-ates utility U(q) from bundle q ∈ n

+, where the

utility function U( · ): n+→ + is assumed to be

twice differentiable and concave (the Hessian matrixis negative semidefinite). Maximizing her net utilitymaxq U(q) −p′q, where p ∈ n

+is the price vector

of the bundle, gives the demand function q d(p) (∇U)−1(p). Consumer’s surplus is therefore S (p) U(d(p)) −p′d(p).

Next, we study fluid models to maximize socialwelfare for the central planner and revenue for thefirm. We will use the transformation p ∇U(q) tocast the problem in terms of q instead of p. The ini-tial capacity is c ∈ m

≥0 for m resources. The utilizationmatrix A ∈ m×n

+specifies the resources used by each

product—i.e., ai j is the amount of resource i used byone unit of product j. We again assume a sales horizonof length T. The fluid welfare and revenue optimiza-tion is given by

minqt≥0

∫ T

0[S (∇U(qt))+q′t∇U(qt)] dt

∫ T

0U(qt) dt ,

and minqt≥0

∫ T

0q′t∇U(qt) dt

subject to the same constraint ∫T0 Aqt dt ≤ c. We intro-

duce the shadow costs of the products in bothproblems as the dual variable of the constraint. Thefollowing proposition characterizes and compares theoptimal solutions to both problems.Proposition 10. If the number of resources is equal to thenumber of products, the utilization matrix is invertible, andthe shadow costs are positive under optimality in both prob-lems, then the welfare-maximizing and revenue-maximizingpricing policies are identical and time invariant.

The positivity of the shadow costs implies the scar-city of the resources. In this case, we can show (seethe proof of Proposition 10) that the optimal demandrate always clears the market in both problems—i.e.,the constraint is binding. However, unlike Section 2,the market-clearing demand rates in the two optimiza-tions are different in general, unless m n and A is aninvertible matrix.

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Chen and Gallego: Welfare Analysis of Dynamic Pricing12 Management Science, Articles in Advance, pp. 1–13, ©2018 INFORMS

5.2. A Queueing ModelOur original model in Section 2 is motivated by a sys-tem with nonreplenishable capacity. The intuition andinsight carry over to fluid queueing systems, which aremotivated by the pricing of toll roads and bridges. Cen-tral planners or private companies operating roads andbridges charge a price for the entrance of vehicles. Theformer may be interested in maximizing social welfare,while the latter focuses on the returns of the invest-ment. While the model and the analysis are presentedin the online appendix, our findings imply that whenthe capacity is constrained, the revenue-maximizingtoll policy is equivalent to the welfare-maximizing pol-icy. In addition, if the hazard rate of the demandfunction is decreasing, then the public interest is alsobest served under this policy. This conclusion providesvaluable insights into the ongoing controversies overprivatization of toll roads in the United States given thegrowing concern that road privatization could hurt thepublic interest (Baxandall et al. 2009).

6. Conclusion and Future ResearchThis paper analyzes the impact of dynamic pricingon social welfare and consumers’ surplus. We findthat the market-clearing price asymptotically maxi-mizes both social welfare and firm’s revenue, as wellas consumers’ surplus under certain conditions, whendemand and capacity are scaled in proportion andcapacity is relatively scarce. Moreover, the revenue-maximizing dynamic pricing policy asymptoticallymaximizes social welfare in the same regime. This isbecause the Pareto frontier shrinks relative to the scaleof the system. As a result, the objectives of the firmand the policy makers becomes aligned. Even with-out the explicit goal of improving social welfare, therevenue-maximizing dynamic pricing policy benefitsconsumers compared to the static pricing policy inmost cases. We find that this phenomenon dependson how welfare is allocated between the firm and thecustomer conditioning on a sale and, ultimately, thevariation of the demand elasticity.We briefly explore other settings including network

revenue management and pricing in queueing models.However, the following related questions remain to beanswered in future research.

• Strategic customers. When customers are surplusmaximizers, they may wait for lower prices, and thefirm must incorporate this strategic behavior to maxi-mize the revenue at equilibrium. Su (2007) shows thatthe revenue-maximizing pricing policy is socially effi-cient in the presence of strategic customers. This maybe due to the deterministic framework adopted in thatpaper, as all stocked products are guaranteed to besold. Cachon and Swinney (2009) argue that the quickresponse of retailers to account for strategic customersleads to more consumers purchasing at higher prices.The impact on social welfare is therefore ambiguous.

On the other hand, Chen and Farias (2018) and Chenet al. (2018) study firms’ revenue-maximization prob-lems in the presence of customer strategic behaviors.They establish that a simple fixed pricing policy candeter strategic behavior and is optimal when the mar-ket size is large. It somehow justifies the use of fixedpricing, not from the consumers’ but from the firm’spoint of view. In general, surplus-maximizing strate-gic consumers and revenue-maximizing firms do notnecessarily lead to more social welfare at equilibriumcompared to the myopic case, as demonstrated by theprisoner’s dilemma. Comparative statics are needed toexamine the welfare function as the level of customers’rationality varies. The dynamic price and static pricewhen consumers are strategic can also be compared.A static price induces consumers to behave myopi-cally and may eliminate inefficiency because of strate-gic waiting and discounting at the cost of lower rev-enues for the firm.

• Competition. The literature on revenue manage-ment (see, e.g., Gallego and Hu 2014, Levin et al.2009) has been focused on oligopolistic competitionand Nash equilibria. However, welfare implicationshave been scarcely discussed. We expect firms to earnlower profits and provide higher consumer surplusin a competitive setting. It is therefore plausible thatsocial welfare increases under competition. In addi-tion, if capacity is relatively scarce, we would expectthe pricing policies at a Nash equilibrium to be verysimilar to that resulting from a central planner man-aging the aggregate capacity. On the other hand, it isless clear whether a dynamic price competition resultsin increased social welfare compared to a static pricecompetition in which firms commit to a fixed priceover the horizon. Possible reasons for firms using staticprices include government regulations and agreementsbetween firms. Unlike the monopolistic setting, staticprices do not necessarily generate less revenues thandynamic prices if other competitors commit to staticprices as well. We therefore believe that social welfaremay either increase or decrease when all firms switchfrom static prices to dynamic prices.

• Other forms of dynamic pricing or price discrim-ination. For example, intertemporal price discrimina-tion is another scheme commonly used in the airlineindustry. As the demand function varies over time, afirm adjusts prices according to not only the remainingcapacity but also the customers’ time-varying WtP. Aspointed out in the literature review, the welfare effectsof intertemporal price discrimination remain unclear.

• Because customers have different arrival time andWtP, the firm or the central planner may be concernedabout who benefit from dynamic pricing. Accordingto Pang et al. (2015), the revenue-maximizing price isa submartingale before the stopping time when theinventory drops to 1, and then becomes a supermartin-gale. However, it is still unclear whether a customer

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Chen and Gallego: Welfare Analysis of Dynamic PricingManagement Science, Articles in Advance, pp. 1–13, ©2018 INFORMS 13

arriving early enjoys higher surplus than a latecomer.To shed light on this question, we conduct a numeri-cal example whose details are provided in the onlineappendix. The numerical example suggests that anearly customer tends to enjoy higher surplus. The samequestion is also relevant when there are strategic cus-tomers and/or the demand function is time varying.

AcknowledgmentsThe authors thank three anonymous reviewers, the associateeditor, and the department editor for their constructive feed-back that greatly improved the manuscript. The authors alsothank Richard Ratliff at Sabre and Robert Phillips at Uber forvaluable discussions and comments.

Endnote1For example, d(p) is upper semicontinuous and d(p) o(1/p) asp→∞, which is assumed to hold throughout the paper.

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