Online Grocery Retail: Revenue Models and Environmental …...of appropriate revenue models and...

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Online Grocery Retail: Revenue Models and EnvironmentalImpactElena Belavina, Karan Girotra, Ashish Kabra

To cite this article:Elena Belavina, Karan Girotra, Ashish Kabra (2016) Online Grocery Retail: Revenue Models and Environmental Impact.Management Science

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MANAGEMENT SCIENCEArticles in Advance, pp. 1–19ISSN 0025-1909 (print) � ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2016.2430

© 2016 INFORMS

Online Grocery Retail: Revenue Models andEnvironmental Impact

Elena BelavinaBooth School of Business, University of Chicago, Chicago, Illinois 60637, [email protected]

Karan Girotra, Ashish KabraTechnology and Operations Management, INSEAD, 77305 Fontainebleau Cedex, France

{[email protected], [email protected]}

This paper compares the financial and environmental performance of two revenue models for the onlineretailing of groceries: the per-order model, where customers pay for each delivery, and the subscription

model, where customers pay a set fee and receive free deliveries. We build a stylized model that incorporates(i) customers with ongoing uncertain grocery needs and who choose between shopping offline or online and(ii) an online retailer that makes deliveries through a proprietary distribution network. We find that subscriptionincentivizes smaller and more frequent grocery orders, which reduces food waste and creates more value for thecustomer; the result is higher retailer revenues, lower grocery costs, and potentially higher adoption rates. Theseadvantages are countered by greater delivery-related travel and expenses, which are moderated by area geographyand routing-related scale economies. Subscription also leads to lower food waste–related emissions but to higherdelivery-related emissions. Ceteris paribus, the per-order model is preferable for higher-margin retailers withhigher-consumption product assortments that are sold in sparsely populated markets spread over large, irregularareas with high delivery costs. Geographic and demographic data indicate that the subscription model is almostalways environmentally preferable because lower food waste emissions dominate higher delivery emissions.

Keywords : online retail; food waste; environment; sustainabilityHistory : Received November 2, 2014; accepted December 19, 2015, by Vishal Gaur, operations management.

Published online in Articles in Advance.

1. IntroductionMore than a decade has passed since the spectacularfailure of Webvan, the heavily funded online groceryretailer. Today, retail-savvy tech companies, ambitiousstart-ups, and deep-pocketed investors (Instacart, Ama-zon Fresh, etc.) are again betting on the online groceryopportunity (Wohlsen 2014, Mitchell 2014).

How customers buy groceries is closely tied to twomajor causes of greenhouse gas emissions. First, drivingto buy groceries is—after driving to work—the mostcommon reason for using passenger vehicles; onlinegrocery shopping has the potential to reduce some ofthat driving by replacing individual trips to grocerystores with a more efficient delivery truck route. Second,the mode of buying fresh groceries also affects theamount of food wasted by consumers, which is a lesswell-known but arguably more important contributorto emissions. In the United Kingdom, eliminating foodscraps from landfills would be equivalent to takinga fifth of all cars off the roads (Gunders 2012); U.S.consumption and dietary habits make food waste aneven more significant climate pollutant (Food andAgriculture Organization of the United Nations 2013).By making grocery shopping more convenient, online

grocery retail could change customer buying patternsand thereby reduce food waste and its environmentalconsequences.

Despite the financial and environmental promiseof online grocery retailing, most early attempts wereunsuccessful. Since the failure of Webvan in 2001,customers have become more accustomed to onlineshopping, and contemporary retailers are more cautiousin designing their expansion strategies. Yet nearly15 years later, there is still only a limited understandingof appropriate revenue models and their operationalconsequences. The current generation of online groceryretailers is experimenting with different revenue models.Amazon charges a $7.99 per-order delivery fee in theSeattle area; a subscription model (Amazon PrimeFresh) is offered in the Los Angeles and San Franciscoareas, for which customers pay $299/year for theprivilege of unlimited free deliveries (McCarthy 2014).

This study is the first to present a (stylized) modelthat compares the two revenue approaches most com-monly used by online fresh grocery retailers: the per-order model, where customers pay for each delivery,and the subscription model, where customers pay onceeach year and subsequently receive unlimited free

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Belavina, Girotra, and Kabra: Online Grocery Retail2 Management Science, Articles in Advance, pp. 1–19, © 2016 INFORMS

deliveries. On the customer side, our analysis includesstochastic processes that model a customer’s ongo-ing grocery needs; the evolution of her perishablegrocery inventory; and her choice of offline/onlinestore, basket size, and order frequency in responseto the retailer’s delivery pricing. For the retailer, wedevelop a detailed delivery cost model, identifying thecost-minimizing delivery routing and relating customerorder streams to driving costs, delivery area geography,and demographics.

The subscription model’s lower ordering costs incen-tivize more frequent orders. It is worth noting that,in the context of perishable groceries, this greater fre-quency results not only in lower grocery sales but alsoin higher supplemental customer value vis-à-vis offlineshopping. When deciding how many groceries to buyin one order (the so-called basket size), customerstrade off the risk of having to place additional ordersagainst the risk of buying more groceries than can beconsumed within their shelf life. The former risk ismore consequential in the per-order model, leadingto larger basket sizes and greater total grocery sales.Thus, an online grocery retailer that offers subscriptionpricing creates extra value for customers by allowingthem to reduce their “safety stock” and their wasteof perishable groceries. From a financial standpoint,the retailer can extract this extra value by setting asufficiently high subscription price and thus earninghigher total revenues (grocery + subscription). We referto this as the revenue disadvantage of the per-ordermodel.

It is interesting that the extra revenues in the sub-scription model are earned while selling fewer perish-able groceries—in particular, fewer wasted groceries.Financially and environmentally, this means that theper-order model also has a food waste disadvantage.Yet the per-order model has the advantage that itsless frequent orders have the effect of reducing deliv-ery costs. The delivery area’s specific geography anddemographics moderate the extent to which theseeconomies play out. Overall, the subscription modelleads to more frequent orders and to higher deliv-ery costs and emissions, although the difference (incomparison with the per-order model) is diminishingin the adoption rate and certain delivery area char-acteristics. Thus, the per-order model has a financialand environmental order frequency advantage over thesubscription model. Finally, these advantages and dis-advantages result in different per-customer marginalcosts and revenues, which implies that the retailer’sprofit-maximizing prices lead to different rates ofadoption.

This analysis allows us to characterize the financiallyand environmentally preferred revenue model given theretailer’s product assortment (i.e., the products’ averagemargins and rates of consumption), the delivery area’s

characteristics (size, shape, and per-mile driving costs),the delivery area’s population demographics (popula-tion density, distribution of store visit costs, and comfortwith ordering online), and the economics of a retailer’sdelivery operations (per-mile driving costs and numberof orders a delivery vehicle can carry). We find that, allelse being equal, online retailers should prefer the per-order model (over the subscription model) for productswith higher margins and in markets that are spreadover a large and less regularly shaped area, are associ-ated with higher per-mile delivery costs, are sparselypopulated, or are served with faster delivery promises(which require that vehicles carry fewer deliveries ata time).

Finally, we calibrate the models with real demo-graphic and geographic data from major cities aroundthe world. We find that, for reasonable estimates ofinput parameters and for a wide range of cities andproduct categories, the financial consequences of thefood waste disadvantage and the order frequencyadvantage are comparable in magnitude and that thetrade-off at the heart of our analysis is of practical rele-vance. Moreover, which model yields greater revenuesdepends strongly on product margins and city charac-teristics. For a city such as Los Angeles and a retailerselling fresh product assortments with average grossmargins below approximately 12%, the subscriptionmodel is preferred; the per-order model is preferredfor higher margins. The critical gross margin is higherfrom about 25% to 35% for denser delivery regions(e.g., Manhattan) and for cities with lower per-miledelivery costs (e.g., Beijing).

As for the environmental consequences, the resultsof our calibrated estimates are even more interesting.The setup we employ predicts a trade-off between thetwo models, yet the actual estimates suggest that—foralmost all realistic delivery geographies, product assort-ments, and customer characteristics—the subscriptionmodel is environmentally preferable even though itentails more driving. Environmental costs associatedwith the per-order model’s food waste are much greaterthan the environmental costs of extra driving, so thefinancially relevant trade-off is of no practical concernwhen the environment is considered. Although thesubscription model’s additional driving makes it lessecofriendly for durable products, the food waste effectreverses that prescription when the delivered productis fresh groceries.

This paper makes three contributions. First, ouranalysis provides important insights and prescriptionsregarding the design of viable revenue models for theonline retailing of fresh groceries, which is arguablythe most lucrative and exciting open opportunity inonline retail. Second, our analysis and calibrated numer-ical study of the grocery value chain’s environmentalimpact show that food waste—an often overlooked

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Belavina, Girotra, and Kabra: Online Grocery RetailManagement Science, Articles in Advance, pp. 1–19, © 2016 INFORMS 3

contributor to emissions—is more important than theemissions associated with transportation, which isthe main focus of past work (e.g., Cachon 2014). Theimplication is that future work on the environmentalimpact of grocery supply chains should specificallyincorporate a consumer inventory model that accuratelyaccounts for consumer food waste. Third, our modelcombines different economic and operational compo-nents that have heretofore been studied independently(i.e., optimal delivery routing, perishable inventories,and choice of revenue model); hence our predictionsare more precise—and sometimes contradict—thosederived from piecemeal models. Thus the approachdescribed here provides a template for a more holisticanalysis of business models.

2. Related LiteratureOur paper is related to previous research on subscrip-tion versus usage-based pricing models, inventoryplanning for perishables, and delivery networks. Wealso contribute to the active literature on the environ-mental impact of operational choices.

Subscription vs. Per-Order Revenue Models. Past workhas considered the choice between the two revenuemodels in the context of consumer services such ascellphones, DVD rentals, etc. (see Cachon and Feldman2011 and the references therein). Our paper extends thisliterature by examining the choice of revenue model inthe rich new context of grocery delivery. We comparethe models not only from a financial but also from anenvironmental standpoint. The subsequent analysiswill show that our model exhibits positive externalities(scale economies) rather than the congestion effectsfound in existing work.

Inventory Management of Perishable Products. Ourmodel of consumer grocery ordering decisions buildsdirectly on the vast literature addressing firm-levelinventory models. In contrast to our setting, mostinventory management theory focuses either on single-period decision making or on nonperishable products.The literature analyzing perishable products is lessextensive (for an up-to-date review, see Nahmias 2011).

Delivery Networks. Our model includes the retailer’soptimal choice of delivery routing, combining the well-known traveling salesman problem (see, e.g., Brameland Simchi-Levi 1997) and an optimal division of thedelivery area into sectors (Daganzo 1984a, b). We drawextensively on this literature.

Environmental Impact of Operational Decisions. A recenthigh-impact body of literature has studied the environ-mental effects of operational decisions. Among otherauthors, Agrawal et al. (2012) compare the revenuemodels of leasing and selling. Akkas et al. (2014) usedata from a large grocery retailer to identify the driversof in-store product expiration. Cachon (2014) compares

the environmental performance of different supplychains in offline retailing. That paper is probably theclosest to ours, although we consider online groceryretail (while incorporating the role of consumer foodwaste) as well as the retailer’s choice of revenue model.

3. Model Setup3.1. The Grocery CustomerConsider an online grocery retailer that serves a market¡ of size A (see Figure 1). Potential customers aredistributed with uniform density � in this area. Thegrocery consumption of customers is random as a resultof unpredictable demand shocks (e.g., unexpectedguests, last-minute plans to dine out) and consists ofboth perishable and nonperishable items. Formally,each customer’s grocery consumption is generatedaccording to a Poisson process whereby instancesof consumption occur at a rate of � per year. Eachconsumption instance corresponds to a consumption of1 (normalized) monetary unit’s worth of perishablesand � monetary units’ worth of nonperishables.

Customers choose when to buy groceries and alsochoose their preferred channel: visiting an offline store,ordering at an online store that offers home delivery,or a combination of these channels. In addition, foreach purchase of groceries, they choose what amountto buy (i.e., the basket size).

A customer who buys groceries offline incurs anidiosyncratic and time-varying cost for each store visit.Customers differ in terms of the inconvenience theyassociate with visiting the offline store owing to differ-ent opportunity costs of time, distances from the grocerystore, fondness for shopping, and so forth. Further-more, a given customer may value this inconveniencedifferently at different times because of unpredictablechanges—for instance, on some days she might passby a grocery store, whereas a store visit might be lessconvenient on other days.

We model these store visit costs with a randomvariable �. At the time of each store visit, � takeseither the value x with probability � or the value 0with probability 1 −�;1 the latter represents convenientstore visits. The inconvenient (high) store visit cost xitself differs for each customer; this captures the time-invariant heterogeneity among customers. We assumethat x is distributed exponentially in the customerpopulation: x ∼ exp4�5.2 We use G4x5 to denote the

1 Section 4.4 of the electronic companion (available as supplementalmaterial at http://dx.doi.org/10.1287/mnsc.2016.2430) provides athree-level model for the intertemporal variation in store visit costs.2 Brown and Borisova (2007) provide empirical support for thisassumption; they find that the major component of the store visitcost is time. The value of time can be approximated by income,whose distribution is known to be roughly exponential (Wikipedia,The Free Encyclopedia, s.v. “Household income in the United States,”http://bit.ly/HHIIncome, accessed May 9, 2016).

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http://bit.ly/HHIIncome


Figure 1 The Grocery Delivery Business Model

Customer

Grocery consumption � Poisson (�)

Grocery shelf life, T

Market area �, size A

Firm

Avg distance per order, D

Per-mile delivery cost, �

Grocery margins, 1 – �

–

Subscription/Per-order (�)?S O

Effective density, � = ��Cost of storevisit � = {0, x}Pr(� = x) = �

x � exp()

plus a frictionalDelivery charge

cost of ordering,

––

––

When to buy groceries?

Amount of groceries to buy?

Shop offline or online?

cumulative distribution function of x; g4x5 is the densityfunction, and G4x5≡ 1 −G4x5 is the survival function.

A customer who instead orders from the onlineretailer incurs—in addition to the delivery charges—anordering cost �. This term captures the inconvenienceof going to the website, selecting groceries, placingthe order, and receiving the delivery. We assume thatgroceries ordered online are delivered almost instantly(same-day delivery is typically offered by prominentonline grocery retailers). Finally, the perishable groceriespurchased by the customer have a limited shelf life; arepresentative basket expires T days after purchase.

3.2. The Online Grocery RetailerThe online retailer of fresh groceries builds a web-basedstorefront and a proprietary distribution network tomake deliveries.3 This network consists of a warehousealong with a fleet of vehicles, each of which can makeas many as K deliveries per run.4 Product diffusion,market penetration, and Internet access constraintslimit the online retailer to serving only a fraction � ofthe population, so the effective density of potentialcustomers for that retailer is �= ��.

Revenues. The retailer generates revenue from thegrocery sales and also charges for delivery. It canchoose between two revenue models for its deliveryservice: the subscription model (S) or the per-ordermodel (O). In the former, customers pay a subscriptionfee s each year and enjoy free delivery for all ordersplaced during that year; in the latter, the customerpays a delivery fee o for each order.

3 Using third-party logistics providers is seldom viable in this contextbecause of the short delivery times, special transit requirements, andperishable nature of the products.4 The number of deliveries (K) is determined by the delivery offer. Inparticular, K is smaller for a faster delivery offer, for tighter deliverytime windows, for a longer time required to reach customers, andfor smaller delivery vehicles.

Costs. The retailer has two main variable costs: thecost of sourcing groceries and the cost of deliveringthem. It sources groceries at a cost of � times the saleprice, where � < 1 and 1 −� captures the gross margin.For every order delivered, the retailer incurs an averagedirect delivery cost of �D; here, � is the per-miledelivery cost (which subsumes the costs of fuel, labor,truck purchase, licensing, depreciation, etc.), and D isthe average distance traveled to deliver an order. It isdetermined by the lowest-cost feasible routing schemethat can fulfill all delivery commitments (we derive theexpression for this distance in Section 5.1).

An online retailer also incurs “picking” costs as wellas the costs of building a warehouse, buying vehicles,training employees, and so forth. These costs havethree potential components: one that depends on theamount of groceries sold, one that depends on thenumber of orders serviced by the online warehouse,and finally some fixed costs. The first component issubsumed by the sourcing cost (�), and the second iscaptured by a picking cost cp that the retailer incursfor each order. The third component (fixed costs) doesnot change under the two revenue models in question,so it has been excluded from all model comparisons.

3.3. Sequence of EventsThe sequence of events is illustrated in Figure 2. First,nature draws each customer’s type x, which capturesthe inconvenient store visit costs. This cost is knownonly to the customer, although its distribution is com-mon knowledge. Then the retailer chooses a subscrip-tion (S) or per-order (O) revenue model, � ∈ 8S1O9,and the relevant prices: the yearly subscription fee s orthe per-order fee o.

Next, the customer decides on her channel strat-egy w ∈ 8off1on-�1mix-�9. The customer may plan to(i) always shop offline (off), (ii) always shop online(on-�), or (iii) choose between online and offline shop-ping as a function of the store visit costs when apurchase is contemplated (mix).

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Figure 2 Sequence of Events

� ∈ � ∈

� �� ∈ � � �

Finally, the customer reviews her grocery inven-tory and, based on the realized consumption andspoilage of groceries, decides when to replenish them.At each replenishment time, the customer realizes hercurrent store visit cost, visits the chosen store (fol-lowing her channel strategy and the cost realization),and decides on her basket size—that is, how manyperishable and nonperishable groceries to buy in antici-pation of future consumption, spoilage, and store visitcosts. This replenishment–consumption cycle continuesindefinitely.

4. Customer BehaviorA customer’s decisions include choosing a channelstrategy w ∈ 8off1on-�1mix-�9, the timing 8Ti9 of ithreplenishment, and—at each replenishment time—thebasket size. We start by describing the timing andbasket size choices for any given channel strategy.

4.1. Order TimingThe customer continuously reviews her inventory leveland decides whether or not to place a new orderwhile bearing in mind the costs of replenishment,grocery spoilage, and the constraint that her grocerydemand must be met. Our analysis reveals that thereplenishment timing choice is driven by perishablegrocery: it is optimal to make the purchase of durablegoods coincide with the purchase of perishables (seeLemma 5 in the appendix).

The customer’s perishable grocery replenishment pro-cess can be viewed as a continuous-review inventorymanagement problem with state-dependent replenish-ment costs. The optimal replenishment policy in suchsettings is likely to be state dependent: the reorderpoint might depend on the time remaining beforeinventory expiration and the current realization ofstore visit costs. Berk and Gurler (2008) show thatpolicies with stationary reorder points are very close tothe state-dependent policies. Thus, we develop ouranalysis using a stationary reorder point. Groceries areperishable, there is a cost of replenishment, and thereis no lead time. Therefore the customer will replenishgroceries only when she runs out (through depletion

or expiration)—that is, replenish as seldom as possibleand no sooner than necessary.

4.2. Basket SizeThe optimal perishable grocery replenishment quantitymust be such that it appropriately trades off the cus-tomer’s costs of replenishing groceries against foodwaste. On the one hand, if the customer buys toomany groceries, then some of them will be wastedand thus lead to unnecessary grocery expenses. Onthe other hand, if she buys too few groceries, thenall items will be consumed before expiration; thatwould lead to an extra order/store trip and hence toextra future replenishment costs. These depend on thechannel strategy but not on the current realization ofstore visit costs, and so, for any given channel strat-egy, the consequences of buying too few groceries arethe same at every replenishment time. Thus, at eachreplenishment, the system returns to the same state.Consequently, the basket sizes are the same at eachreplenishment, QwTi

=Qw, and are a function of theexpected replenishment costs.

Since the customer buys the same quantity of per-ishables each time, successive cycles are indepen-dent and identical. The length of the ith cycle, CTi,depends on the basket size Q. The expected cyclelength is E6CTi4Q57 = �−14Q −

∑Qj=04Q − j5 · pj4�T 55,

pj4�T 5= e−�T 4�T 5j/j! (Lemma 5). The term in largeparentheses is the expected consumption of perishablegroceries per cycle, and � is the average consump-tion rate of perishables. Of Q units ordered, it isexpected that

∑Qj=04Q− j5 · pj4�T 5 will be wasted; pj

is the probability that consumption during the shelflife T is equal to j . Furthermore, the expected num-ber of orders in a year induced by ordering Q unitsat a time (Lemma 5): N4Q5 = 1 year/4E6CTi4Q575 =

�4Q−∑Q

j=04Q− j5 · pj4�T 55−1. We consider all costs on

an annual basis, so that annual subscription costs canbe incorporated accurately.

For nonperishable items, the replenishment policy issimply an order-up-to policy. At each replenishmentpoint, the customer buys enough nonperishables to

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bring her inventory level up to � times the perishableinventory Qw—that is, up to level �Qw (Lemma 5).

Now, for a size Q basket of perishable groceries andreplenishment costs a, we can write the customer’sexpected yearly cost as aN4Q5+QN4Q5+��. Here,�� captures the expected nonperishable purchasesand QN4Q5 the expected cost of perishable groceries—namely, the direct costs of purchasing groceries, Q,multiplied by the number of ordering cycles in a year,N4Q5. Finally, aN4Q5 captures the total replenishmentcosts. The optimal basket size is given by Q∗4a5 ≡

arg minQ4aN 4Q5+QN4Q5+��5. Lemma 5 provides theimplicit definition of the function Q∗4a5.

Lemma 1. Higher replenishment costs a lead to largerperishable grocery basket sizes, ¡aQ∗ > 0; fewer annual orders,¡aN4Q∗5 < 0; a higher annual volume of perishable groceriespurchased, ¡a4N 4Q∗5 ·Q∗5 > 0; and higher customer’s optimalgrocery cost, ¡a4aN4Q∗5+Q∗N4Q∗5+��5 > 0.

Proofs for all results are given in the appendix. Theoptimal basket size for perishables trades off the risk ofbuying too many groceries (which would lead to waste)against the risk of buying too few (which would triggeradditional replenishments and increase replenishmentcosts). Thus, a higher replenishment cost a inducescustomers to buy larger baskets, but less frequently.Larger baskets increase the likelihood that some ofthe groceries expire before they are consumed, thusincreasing the average waste. Given that annual groceryconsumption is independent of replenishment costs,higher waste implies that the customer purchases ahigher annual quantity N4Q∗5Q∗ of groceries. Finally,the “larger basket size” effect dominates the “lessfrequent replenishment” effect and so the customer’sgrocery costs are increasing in a.

4.3. Channel StrategyThe customer’s channel strategy w ∈ 8off1on-�1mix-�9determines her replenishment costs a. If the strategy isto always choose the offline store, w = off, then thereplenishment costs are simply the expected store visitcosts: aoff = E6�7=�x. If the strategy is to always choosethe online store, w = on-�, then the replenishmentcosts are the online ordering costs � plus the per-orderdelivery fee (when the per-order model is used).

Finally, suppose the customer chooses the contingentcombination strategy, w = mix-� . Of the various possi-ble combinations, the best one is to shop online whenthe store visit cost is high (�= x) and to shop offlinewhenever that cost is low (�= 0). Then the orderingcost is amix−O =�4�+ o5 under the per-order revenuemodel and amix−S =�� under the subscription revenuemodel.

The contingent combination strategy dominates thealways online strategy, but preference for the alwaysoffline strategy versus the combination strategy depends

on the customer’s type and the retailer’s specific offer.The customer’s optimal annual costs under the alwaysoffline strategy are

Coff ≡ 4�x+Q∗4�x55 ·N4Q∗4�x55+��1 (1)

while the costs with combination strategy are as follows:

Cmix-� =

4�o+��+Q∗4�o+��55

·N4Q∗4�o+��55+�� if � =O1

s + 4��+Q∗4��55 ·N4Q∗4��55+��

if � = S0

(2)

The customer’s choice between the always offline strat-egy and the combination strategy simply boils down tominimizing the yearly cost: w∗ = arg minw∈8off1mix-�9Cw.

We simplify the notation for the optimal order size bydenoting Qoff ≡Qx ≡Q∗4�x5 as the optimal perishablebasket size under the always offline strategy for atype x customer, Qo ≡ Q∗4�o + ��5 as the optimalbasket size when using the combination strategy andthe per-order revenue model, and Qs ≡ Q∗4��5 asthe optimal basket size when using the combinationstrategy and the subscription revenue model. Similarly,for the number of orders per year, we set (respectively)Noff ≡Nx ≡N4Qx5, No ≡N4Qo5, and Ns ≡N4Qs5.

5. Retailer DecisionsThe online grocery retailer builds a distribution networkto deliver groceries, chooses either the subscription orthe per-order model, and determines the relevant price.We start by analyzing the design of the distributionnetwork that meets the delivery requirements at thelowest cost.

5.1. The Proprietary Distribution NetworkThe retailer’s distribution network consists of a ware-house located in area ¡ and a fleet of delivery vehicles.We assume that the number of orders delivered by onevehicle in one delivery period K is significantly smallerthan the number to be delivered in the entire market atthe same time, K �A�d. We have �d = �N�−1; here, �is the density of the population that adopts the onlineservice (in Section 5.2 we define this term explicitly), Nis the annual number of orders per customer, and � isa coefficient that converts the annual number of ordersinto the number of orders per delivery period. Forexample, if the retailer delivers once a day, then �= 365.On the basis of the specific orders to be delivered in aperiod, the retailer devises the following distributionplan. First, it optimally partitions area ¡ into sectors ³i,i ∈ 811 0 0 0 1 I9, so that each sector has K customers. Eachsector is then assigned a vehicle that, following anoptimal route, visits each of the K customers. A detailedversion of this analysis is provided in Section 3 of theelectronic companion; the key results follow.

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Lemma 2. (i) When K orders are delivered by one vehicle,the average distance traveled to deliver an order in a regionof size A1 of uniform customer population density �1 andwith customer yearly order frequency N is

D4�1N1A1K5 û2�

√A

K+å4K5

√

�

�N1

with ¡Kå4K5 < 00 (3)

(ii) The per-order average distance is decreasing in thenumber of orders 4¡N D < 05 and in the adopting density4¡�D < 05, but it is increasing in the order costs 4¡aD > 05.The total annual distance traveled per customer, ND1 isincreasing in the number of orders 4¡NND > 05 but isdecreasing in the order costs 4¡aND < 05.

The first component of Equation (3), �√A, constitutes

the average “line-haul” distance that has to be traveledto reach a delivery sector, which must be covered twice(to get to the sector and back) and is distributed overthe K deliveries. The coefficient � is determined by theregion’s shape and the warehouse’s location. Shapeparameter � is greater when markets are more irregular:it is smallest for the circular regions, followed by squareand rectangular regions (�ci < �sq ≤ �re5. Increasingthe rectangle’s length-to-height ratio � (a higher �corresponds to the more elongated rectangle) alsoincreases the shape parameter ¡��re > 0 and so does anoncentral warehouse location (�re ≤ �nc

re ). The exactexpressions for �ci1 �sq1 �re, and �nc

re are provided inSection 3 of the electronic companion for the interestedreader.

The second component of Equation (3) is the travelingsalesman tour that visits each of K customers within thesector. The proof of Lemma 2 identifies an expressionfor the coefficient å4K5. It captures the economies ofscale in the traveling salesman tour: the per-orderaverage length of the optimal tour decreases as weadd more points (orders) to the tour.

Part (ii) shows that there are economies of scale indelivery. If there are more customers (i.e., if � increases)and/or if customers order more often (N increases),then the per-order delivery distance decreases. Alongthe same lines, a higher ordering cost a will lead bothto fewer orders and to fewer customers and thus to ahigher average distance traveled. Finally, with respectto total annual distance traveled per customer, theorder frequency effect dominates the average deliverydistance effect; hence a higher frequency leads to moreannual travel.

5.2. Choice of Revenue ModelSuppose the retailer chooses the per-order pricing model.Then customers whose type x is above a certain thresh-old will choose the combination channel strategy,whereas those of the type lower than this thresholdwill choose to always shop offline (Lemma 6).

The retailer’s profits (at the profit-maximizing per-order price) can then be written as

�O = maxo

4��41 −�5+ 4o+Qo5No

−8� · D4�G4x51�No1A1K5+�Qo + cp9 ·No5

·A��G4xo51 (4)

where xo = min8x s.t. Coff ≥ Cmix-O9. The first term inthis formulation is the per-customer profit from the saleof nonperishable items, and the second term is the per-customer delivery and perishable grocery revenues. Thethird term includes the variable cost components, theper customer-order costs of delivery, and the costs ofsourcing perishable groceries (and picking) multipliedby the number of customer orders per year. Finally, themultiplicative term represents the number of customersbuying from the online grocery retailer; that is, �G4xo5captures the fraction of potential customers who buyfrom the online retailer on any given day.

If instead the retailer chooses the subscription model,then again, the customer’s choice of channel strategy isa threshold choice in the customer’s type x. So the max-imum expected profit under subscription pricing (at theprofit-maximizing subscription price) is, analogously,

�S = maxs

(

��41 −�5+ 8s +Qs�Ns9

− 8� · D4�G4xs51 �Ns1A1K5+�Qs + cp9

·�Ns

)

·A�G4xs51 (5)

where xs = min8x s.t. Coff ≥Cmix-S9. As before, the firstterm is the nonperishable grocery profit, the second isthe per-customer subscription and perishable groceryrevenues, the next terms include the delivery andsourcing costs, and the multiplicative term capturesthe fraction of customers who buy online. Finally, theretailer will choose the pricing scheme that maximizesits profit: � = arg max�∈8O1S9�� .

6. Equilibrium Outcomes6.1. Per-Order Revenue Model

Lemma 3 (Equilibrium Outcome Under the Per-Order Revenue Model). (i) The online retailer charges adelivery fee o=x∗

o −�, where the optimal market coverage x∗o is

the unique solution to 4Nx+41−�5x¡xNx−¡xho4x55G4x5=4Cx−ho4x55g4x5. Here, ho4x5≡ 8� ·D4�G4x51�Nx1A1K5+�Q∗4�x5+�+cp9Nx−�41−�5�; Cx ≡ 4x+Qx5Nx.

(ii) Customers of type x > x∗o choose the contingent

combination channel strategy: shop online when store visitcosts are high 4�= x5 and shop offline when store visit costsare low 4� = 05. These customers all purchase the samebasket size Q∗4�x∗

o 5. Customers of type x < x∗o always shop

offline, and each such customer’s idiosyncratic basket sizeis Q∗4�x5.

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The equilibrium is best understood by examining theeffect of a price change on each term in the retailer’sprofits (Equation (4)) while keeping in mind customerresponse to this price (i.e., her ordering costs; seeLemma 1), delivery costs (Lemma 2), and the adoptionrate.

Recall that an increase in ordering costs increasesthe customer’s basket size, reduces the annual numberof orders per customer, and increases the amountof perishable groceries purchased by each customer(and hence the amount of food waste); see parts (ii)and (iii) of Lemma 1. Thus an increase in the per-orderprice increases the direct perishable grocery profits41 −�5QoNo but also increases a customer’s cost ofusing the online channel—both directly (owing tohigher annual delivery costs oNo) and indirectly (owingto higher perishable grocery expenses QoNo). As aresult, customer adoption G4xo5 declines and so doesthe amount ��G4xo5 of nonperishable groceries sold.With regard to delivery, an increase in the per-orderprice increases per-order delivery costs �D becauseof a lower order frequency and lower adoption rate;however, the annual per-customer delivery costs DNo

actually decrease because the order frequency effectdominates (Lemma 2(ii)). Hence the delivery profits4o−�D5No are increasing in o. Overall, the deliverycosts and perishable grocery profits increase whenthe retailer sets higher per-order prices—althoughdoing so reduces adoption by customers. A per-orderdelivery price of o = x∗

o − � (with x∗o defined as before)

optimally trades off the per-customer profit effectagainst the adoption effect or the market size effect.Given an optimal delivery fee, customers of type x ≥ x∗

o

employ the combination channel strategy while allother customers always use the offline channel. Theresulting yearly per-customer delivery and grocerypurchase cost is captured by ho4x

∗o 5.

6.2. Subscription Pricing

Lemma 4 (Equilibrium Outcome Under the Sub-scription Pricing Model). (i) The online retailer chargesa yearly subscription fee s∗ = 4�x∗

s +Q∗4�x∗s 55N 4Q∗4�x∗

s 55−4��+Qs5Ns , where the optimal market coverage x∗

s is a uniquesolution to 4Nx − ¡xhs4x55G4x5 = g4x54Cx + âx − hs4x55;here, hs4x5≡ 8�+� · D4�G4x51�Ns1A1K5+�Qs + cp9 ·Ns −�41 −�5� and âx ≡ 441 −�5/�56QxNx −QsNs7.

(ii) Customers of type x ≥ x∗s choose the contingent

combination channel strategy: shop online when store visitcosts are high 4�= x5 and shop offline when store visit costsare low 4�= 05. These customers all order the same basketsize Qs on an ongoing basis. Customers of type x < x∗

s

always shop offline; each such customer’s idiosyncratic basketsize is Q∗4�x5.

Subscription price affects the direct delivery revenuesand market adoption (which in turn affect delivery

costs) but has no effect on either the amount of groceriespurchased or the order frequency. Thus the trade-offdriving the choice of subscription price is simplerthan that for the choice of per-order price: highersubscription prices increase the retailer’s revenue butalso reduce adoption rates. Lower levels of adoptionentail higher delivery costs because in that case thereare fewer economies of scale. As before, our marginalcustomer is the one with high store visit cost x∗

s . Theresulting yearly per-customer delivery and grocerypurchase cost for online customers is captured by hs4x

∗s 5.

7. Comparing Revenue Models:Subscription vs. Per Order

7.1. Comparing Equilibrium Customer BehaviorIrrespective of the revenue model employed, there aretwo kinds of customers: adopting customers—those whofollow the contingent combination strategy of sometimesbuying offline and sometimes online—and customerswho always buy offline. Not surprisingly, the lattergroup’s behavior is not affected by the revenue modelemployed, whereas the former group’s behavior is. Evenwhen adopting customers buy offline, they considerthe expected costs of future grocery orders becausesuch future purchases include the possibility of onlinepurchases whose costs depend on the online retailer’srevenue model. This section examines the equilibriumbehavior of adopting customers, which has importantramifications for the retailer’s choice of revenue modeland for the environmental impact of that choice.

Theorem 1. Adopting customers order larger grocerybasket sizes and order less frequently in the per-order modelthan in the subscription model. Overall, the total offlineand online purchases of perishable groceries is higher in theper-order model (i.e., larger basket size dominates the lowerfrequency), and greater delivery distances are traveled in thesubscription model to serve an adopting customer. Offline oronline, adopting customers purchase the same amount ofnonperishable groceries in the two revenue models.

Corollary. Adopting customers’ expected food waste—that is, the amount of perishable groceries that expire beforebeing used—is higher in the per-order model than in thesubscription model.

At equilibrium subscription and per-order prices,for adopting customers, the expected ordering costsare higher in the per-order model. By Lemma 1, itfollows that the per-order model is characterized bylarger perishable grocery basket sizes, fewer orders,and more perishable groceries purchased annually.Expected perishable grocery consumption is simplythe mean demand rate, which is the same for the twomodels. Since more perishable groceries are purchasedin the per-order model, waste is higher.

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A higher annual grocery volume with fewer orderssuggests that the per-order retailer sells more percustomer while spending less delivering the groceriesand hence that the per-order model should dominate.This analysis is incomplete, however, because of anotherfactor distinguishing the two models. Since they lead todifferent customer behavior and grocery waste, the twomodels also create different supplemental value (overoffline shopping) for customers. This factor affects theprices (subscription or per order, as applies) that can beset and thus how much of that value can be extractedby the retailer. Furthermore, if different revenue modelscreate different levels of supplemental value, then theimplication is that the two models will exhibit differentadoption rates and hence different economies of scalein delivery. Together these effects lead to a drasticdeparture from the naïve analysis.

7.2. Comparing Equilibrium OutcomesThe equilibrium profits can be rewritten as maximiza-tion problems in which the retailer—rather than choos-ing the delivery price, which determines the rate ofmarket adoption—directly chooses an optimal adoptionlevel via the critical store visit costs x∗:

�s = maxx

�s4x5

≡ maxx

�4�41−�5�+Cx+âx−hs4x55A�G4x53 (6)

�o = maxx

�o4x5

≡ maxx

�4�41−�5�+Cx−ho4x55A�G4x50 (7)

Here, âx ≡ 441 − �5/�56QxNx − QsNs7. This formula-tion subsumes customer behavior and allows for anintuitive decomposition of model differences into(a) per-customer revenues, (b) per-customer costs, and(c) market adoption rates. Note that, regardless ofthe adoption rate, the nonperishable grocery profits (i.e.,��41 − �5) are the same in the two revenue mod-els; hence we can exclude them when comparingper-customer revenues and costs. The comparison ofrevenues and costs for per-customer delivery and per-ishable grocery is more involved. We start with therevenues.

From Theorem 1, it follows that the retailer’s per-customer perishable grocery revenues are higher in theper-order case because the customer buys more (shebuilds higher safety stock) in anticipation of higherordering costs. However, this extra grocery, in expecta-tion, ends up as waste and so does not create additionalvalue for the customer. That a given customer type(as signified by x) spends less on groceries in thesubscription model has important consequences forhow the retailer sets its subscription fee s. It can set sto extract additional value created by lower grocerypurchases, in which case the grocery revenue losses

of the subscription model are compensated by theincreased subscription fee; hence the term Cx is thesame in both models.

It is noteworthy that, under the subscription model,offline purchases are also reduced because expectedreplenishment costs are lower (since costs are expectedto be lower when shopping online). This allows theretailer to increase the subscription fee still more,an advantage that is captured by the term âx. Inessence, with a subscription fee the retailer can recoverany grocery revenue losses and can also extractthe customers’ gains as a result of their savingscompared with the offline shopping, as they stockand waste fewer perishable groceries. We call thisthe revenue disadvantage of the per-order model (seeFigure 3(a)).

Next we compare per-customer costs, which arecaptured by the following two equations:

hs4x5 = �QsNs + 4�+ cp5Ns +� ·2�

√A

KNs

+� ·å4K5

√

�

�G4x5�

√

Ns3

ho4x5 = �QxNx + 4�+ cp5Nx +� ·2�

√A

KNx

+� ·å4K5

√

�

�G4x5�

√

Nx0

In both models, these costs have four components.The first is the per-customer grocery sourcing cost,§· = �Q·N·, which is always higher for the per-ordermodel. As discussed previously, the per-order modelsells more groceries (see Theorem 1); so all else equal,the subscription model surprisingly outperforms fromthe standpoint of grocery cost. This is the second fun-damental difference between the two revenue models:the per-order model has a food waste disadvantage (seeFigure 3(b)).

The second component combines the retailer’s per-order cost cpN· and the customer’s ordering costs �N·

(in both models, the firm compensates the consumerfor her ordering inconvenience � by lowering prices).The third component is the per-customer annual line-haul delivery costs �442�

√A5/K5N·. Both of these cost

components are increasing in the order frequency andare therefore higher under the subscription model,Ns > Nx. The fourth and final component is the per-customer annual traveling salesman tour cost: theaverage per-order traveling salesman tour costs � ·

å4K5√

�/4�G4x5�N·5 multiplied by the order frequencyN·; thus, � ·å4K5

√

�/4�G4x5�5√

N·. Note that this cost isalso increasing in the order frequency but at a slowerthan linear rate, since there are two kinds of scaleeconomies: the order frequency itself (

√

N·) and the

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Figure 3 Per-Order vs. Subscription Revenue Model: Comparison of Per-Customer Revenue, Grocery, and Delivery Costs and of Overall Profits

Per-

orde

r an

d T

SP c

ost S

O

Order frequency

advantage

Prof

its �

s(x)

, �o(

x)

S

O

Rev

enue

–Adoption, x –Adoption, x –Adoption, x –Adoption, x

O

S

Revenuedisadvantage

Gro

cery

cos

tS

O

Food waste disadvantage

b c d

Note. TSP, traveling salesman problem.

adoption level G−1/24x5. For the same level of adoption,the traveling salesman tour costs will be higher in thesubscription model. Although different equilibriumadoption rates might change this, they are usually higherin the subscription model. We refer to the combinedeffect of these delivery costs (sum of the second, third,and fourth components) as the per-order model’s orderfrequency advantage (see Figure 3(c)).

In sum, the per-order model enjoys the order fre-quency advantage but suffers from the food waste andrevenue disadvantages. All three effects are on a per-customer basis, and the fourth and final effect concernsoptimal adoption levels. Because adoption influencesthe magnitude of the first three advantages by scalingthem upward, it determines which of them dominates.It turns out that all three differences are decreasingin the adoption level, as evidenced in panels (a), (b),and (c) of Figure 3. Figure 3(d) depicts the retailer’sprofit curves (as a function of the adoption level x,where all customers of type x ≥ x will adopt) under thetwo models. We can show that if in both models theadoption levels are above (respectively, below) the inter-section point of the two profit curves, then the financialimpact of the order frequency advantage dominates(respectively, is dominated by) the impact of the foodwaste and revenue disadvantages. We refer to the roleof adoption as the adoption effect.

7.3. Which Revenue Model Yields Higher Profits?The financial and environmental comparison betweenthe two revenue models is driven by a nonlinear interac-tion involving the revenues (the revenue disadvantage),the grocery costs (the food waste disadvantage), thedelivery costs (the order frequency advantage), and theadoption effect. However, further analysis shows thatan overall comparison of the two revenue models canbe expressed using a single metric—one that allows usto identify the preferred revenue model as a function ofthe market area’s spatial and demographic properties,the product’s characteristics, and the economics ofdelivery. Analyzing the constituent marginals allows

us to demonstrate a single-crossing property in theshape parameter.5

Theorem 2. If

¡2x∗oho4x

∗

o 5≥14�å4K5�2

√

�

��

Nx∗o

G4x∗o 51

then the subscription model is preferred when the shapeparameter of the delivery region is below a threshold level,� < � ; otherwise, the per-order model is preferred.6

The profit difference takes the form shown in Figure 4.As the shape parameter � increases (i.e., as the deliveryregion becomes more irregular/elongated), there aretwo associated changes: delivery costs increase (becauseof longer line-haul distances) and the adoption ratedeclines (in response to higher prices—recall that highercosts make higher prices optimal from the retailer’sperspective).

The increase in delivery costs increases the relativevalue of the order frequency advantage, since each tripbecomes more costly; this is the main effect that pro-gressively favors the per-order model. Higher deliverycosts mean that the optimal adoption rate decreasesin both models; yet despite the higher per-customerrevenues, in the subscription model this decline ismore rapid because there are more deliveries to makethan in the per-order model. Hence we observe second-order effects: lower adoption rates for the subscriptionmodel imply a scaling down of its advantages. Thisdynamic also favors the per-order model. However,reduced adoption also implies an increase in the orderfrequency advantage and in both the food waste and

5 The single-crossing property that drives Theorem 2 holds also forthe market size A, population density �, per-mile delivery cost �, andnumber K of deliveries per truck. The intuition is the same as thatfor the shape parameter (explained next), and both Theorem 2 andthe financially preferred model could be equivalently characterizedusing any of these parameters.6 Condition ¡2

x∗oho4x

∗

o 5≥14�å4K5�2

√

4�/4��554Nx∗o/G4x∗

o 55 is technical;based on our extensive numerical experimentation, the thresholdresult holds for all reasonable parameters.

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Figure 4 Retailer Profit: Subscription Model vs. Per-Order Model

Prof

it di

ffer

ence

, �s –

�o Order frequency advangage (OFA)

Food waste disadvangage (FWD)

Equilibrium adoption(faster for subscription)

OFA >

Delivery costs

As �

�

As �

Revenue disadvangage (RD)

–

FWD, RD

�

revenue disadvantages. This increase in order frequencyadvantage—when combined with the direct first-ordereffect through an increase in �—outpaces the increasein the food waste and revenue disadvantages (second-order effects), favoring once again the per-order model.Eventually, in both models the adoption level becomeslow enough to reduce not only absolute profits but alsothe difference between the models’ respective profits.Hence the profit lines cross only once, which allows usto characterize the preferred revenue model in termsof the threshold shape parameter alone.

The main advantage of an intuitive statement ofTheorem 2 is that it allows one to characterize the area,product, and operational characteristics that are bestsuited for each model.

Theorem 3. Ceteris paribus, the per-order model is pre-ferred over the subscription model if the population is sparse,the delivery area is either large or irregular/elongated, and/orretailer delivery commitments require that delivery vehiclescarry fewer orders. Formally, the threshold shape parameter� is decreasing in the delivery area A and in the per-miledelivery costs �, but it is increasing in �, the populationdensity, and K, the number of deliveries per truck.

Theorem 3 follows from our previous discussionabout the effect of increasing per-mile distances.A sparsely populated area (lower �), a large deliveryarea (greater A), and high delivery costs (higher �)lead to the same first- and second-order effects as anincrease in the shape parameter—namely, an increase inthe order frequency advantage; a lower optimal adop-tion rate; and a consequent increase in the food waste,revenue, and order frequency effects. The only differ-ence is that the order frequency–related phenomena inthe main and second-order effect appear through thetraveling salesman component of the delivery distancefor population density (and not through the line-haulcomponent, as in the case of the shape parameter); forthe per-mile cost �, they appear through both compo-nents. Finally, less batching of deliveries (reduced K)results in the same operating phenomena and, likethe per-mile delivery cost, operates through both theline-haul and traveling salesman distances. Lower K

implies that more trips are needed to the sectors andthat the tours within a sector are longer on a per-orderbasis, which increases the order frequency advantageand lowers the adoption rate.

7.4. Environmental Impact of Revenue ModelsWe compare the population’s driving and food waste–related carbon emissions from the two revenue models.To understand this comparison, it is useful to start bysimply comparing adopting and nonadopting customers.Regardless of the revenue model, customers that adoptthe online channel do so in order to reduce their groceryordering costs; for these customers, then, the onlinechannel is associated with less food waste and its relatedemissions. Moreover, orders are now pooled in deliveryand so the per-order travel is also reduced. But onlinecustomers will shop more frequently and thus inducemore trips (especially in the subscription model), whichcould cancel out the benefits from their lower foodwaste and per-order travel.

The total emissions for the delivery area’s pop-ulation of potential customers in the subscriptionmodel are written as Es = A�4Emix−S4Qs1Ns1x

∗s 5 +

Eoff4x∗s 55 and those in the per-order model as Eo =

A�8Emix−O4Qx∗o1Nx∗

o1 x∗

o 5+Eoff4x∗o 55. The first part of each

expression Emix-� captures the emissions resulting fromadopting customers who use a combination of the twomodes:

Emix-�4Q1N1 x5

= ef 4QN −�5 · G4x5+ ed ·�ND4�G4x51 �N1 ¡1 K5

· G4x5+ ep41 −�5N∫ �

xd0xg4x5dx0

Here, ed and ep are the CO2-equivalent per-mile emis-sions for a delivery truck and a passenger vehicle,respectively, and ef signifies the CO2-equivalent emis-sions for every dollar’s worth of food wasted. The firstcomponent is food waste emissions, which are calcu-lated as the carbon load of all food purchased (efQN )minus the carbon load of the food consumed 4ef�5. Thesecond component is emissions from driving, whichare generated by driving of the delivery truck (when

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the consumer is shopping online) or by the consumer’sown driving (when shopping offline); dx denotes thedistance traveled by consumers of type x when shop-ping offline has a high cost, and d0

x is the correspondingdistance when store visits happen to be cheap. Thesecond part Eoff is emissions resulting from customerswho employ the always offline channel strategy:

Eoff4x5 =

∫ x

04ef 4NxQx −�5

+ ep4�dx + 41 −�5d0x5Nx5g4x5dx0

Comparing the two models in terms of their envi-ronmental impact is analogous to comparing themin terms of profits. On the one hand, the per-ordermodel leads to more food waste and so has a higherenvironmental cost; hence the per-order model suffersfrom a food waste disadvantage. On the other hand,that model requires fewer deliveries, which translatesinto less driving and fewer carbon emissions. Finally,adoption rates, which reflect the number of customerswho actually use the online channel, differ across thetwo models, and thus there is also an adoption effect.Although the preferred model is determined by a non-linear interaction of these effects, it can be more easilycharacterized by using a single metric: a thresholdlevel of food waste emissions.

Theorem 4. If x∗s − x∗

o < ¯x∗, the per-order model is moreecofriendly than the subscription model when the emissionsef for every dollar’s worth of food wasted are less than athreshold level ef .

This threshold result is easier to establish than is thethreshold result for our profit comparisons. An increasein emissions from a dollar of wasted food increasesthe relative environmental consequences of the foodwaste disadvantage while leaving the other effectsunchanged. Increased unit emissions from food wasterender the per-order model progressively less “green,”so at some point the subscription model becomesgreener despite involving more travel.7 The resultnow follows. This analysis suggests that one revenuemodel’s being greener than the other depends on theparameters. However, in the next section we calibrateour model using realistic parameters and find that thesubscription revenue model almost always dominates.

8. Managerial Implications:Cities, Product Categories,and Revenue Models

The foregoing analysis can be used by an online groceryprovider to choose the revenue model that yields the

7 Condition x∗

s − x∗

o < ¯x∗ ensures that more food is wasted under theper-order model, i.e., its food waste disadvantage (see the proof ofTheorem 4 for details; ¯x∗ is defined therein). Based on our extensivenumerical experimentation, condition x∗

s − x∗

o < ¯x∗ holds for allreasonable parameters.

highest profits and the best chance of financial viability.We now illustrate this choice by calibrating our modelusing realistic values of the input parameters.

8.1. Financial EffectsWe proceed by analyzing four different delivery areas:Manhattan, Los Angeles, Paris, and Beijing (old city).Table 1 lists our parameter estimates for these cities,8

which vary widely in terms of geography. Paris, Beijing,and Manhattan are relatively small in area; Los Angelesis about 10 times their size. Paris and Beijing are roughlycircular/oval cities, whereas Los Angeles and Manhat-tan are rectangular. Note that Manhattan is extremelyelongated and so its shape parameter is much higher.The cities also vary widely with respect to householddensity, ranging from fewer than 3,000 households persquare mile in Los Angeles to about eight times thatdensity in Manhattan. Finally, the cities have differentlabor markets. The per-mile delivery costs in China areestimated to be less than a third of those in Westerncities. We estimate other relevant parameters usingcensus data and industry sources (see Table 2). In addi-tion to the estimates in Tables 1 and 2, we comparedthe two revenue models for many other cities andeven more parameter values; the full ranges examinedare reported in the bottom row of Table 1 and therightmost column of Table 2.

Figure 5 shows the prescribed revenue model strategyfor the different cities. The horizontal axes of eachpanel are the product gross margins. Gross margins(1 −�) vary considerably among different fresh grocerycategories and between premium and basic products.Products in the bulk section generally earn highermargins (around 40%); by contrast, margins in thebread aisle are only about 20%.9 The vertical axesrepresent the weekly consumption of groceries byconsumers. The average U.S. household consumptionof groceries is $86/week,10 but the exact value depends

8 Each city’s area is obtained from its Wikipedia entry. Density isobtained by dividing the population density by an average householdsize of 2.9 (Consumer Expenditure Survey of the Bureau of LaborStatistics, Table 1500: Composition of consumer units, 2012Q3–2013Q2(http://www.bls.gov/cex/, accessed May 9, 2016)). Shape parameter� is computed based on the following assumptions: Manhattancan be represented by a rectangle whose sides are proportionedas 1:5; Los Angeles, by a rectangle of proportion 2:3; Paris, byapproximately a circle; and Beijing, by an oval whose minor andmajor axes are proportioned as 5:6. Delivery cost � is computed asthe cost of operating a truck, including labor and delivery costs. ForWestern cities, the operating cost is $1.38 per mile (“The Real Cost ofTrucking,” http://bit.ly/1ojPJaF, accessed May 9, 2016) plus about$0.12 per mile for other items; for Beijing, costs are estimated to beabout 27% of Western costs (Foley 2014).9 According to Reinvestment Fund (2011).10 Consumer Expenditure Survey of the Bureau of Labor Statis-tics, Table 1500: Composition of consumer units, 2012Q3–2013Q2(http://www.bls.gov/cex/, accessed May 9, 2016).

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Table 1 City Characteristics

Area, A Density, � Shape Delivery cost,(sq. mi) (households per sq. mi) parameter, � � ($/mi)

Manhattan 23 241137 1016 105Los Angeles 500 21836 008 105Paris 40 181930 004 105Beijing 33.63 221168 0061 00413Range examined A ∈ 61016007 � ∈ 6212010007 � ∈ 6004137 � ∈ 600512057

on the market segment. Large households and high-net worth households naturally consume more, as dohouseholds that purchase premium, imported, organic,and/or luxury items.

Note that in each of the cities, retailers that sellhigher-margin product assortments are better off takingthe per-order approach to pricing. In fact, for all realisticparameter values, the per-order model is preferred aslong as the margin is roughly above a certain threshold(about 30% for Beijing, Paris, and Manhattan and about12% for Los Angeles).

The differences between cities are more interesting.Manhattan, Paris, and Beijing (panel (a) in Figure 5)are small and densely populated delivery areas. Ofthe three, Beijing and Manhattan are the most denselypopulated; however, Beijing has two advantages vis-à-vis Manhattan. First, in contrast to Manhattan’shighly skewed rectangle, Beijing has an oval shape; thisdifference in shape reduces travel distances. Second,because of Beijing’s lower labor costs, its per-miledelivery costs are only about 30% of those in Manhattan.Hence Beijing is the city in which deliveries are the

Table 2 Retailer and Product Parameters

Parameter Value Sources and comments

Cost of ordering � = 5$/order Hann and Terwiesch (2003) estimate the mean cost of online ordering to be approximately $5. Rangeexamined: � ∈ 601107 $/order.

Perishable/nonperish.mix

� = 005 Consumer Expenditure Survey of the Bureau of Labor Statistics, Table 1500: Composition of consumer units,2012Q3–2013Q2. Range examined: � ∈ 60127.

High store visitcost

x ∼ exp4�5E6x7= 30$ perround-trip

The cost of a store visit is estimated as the value of time spent shopping for groceries. Brown and Borisova(2007) report that the average consumer spends about 140 minutes each week shopping for groceries,including travel time. If we value the customer’s time at $13/hour, then the mean store visit costs $30 perround-trip. Range examined: E6x7∈ 6101507$.

Probability of highcost

�= 0042 Estimates in Chintagunta et al. (2012). Range examined: � ∈ 60117.

Product life T = 1 week Author estimates; we examine product lives between 3.5 days and 2 weeks, T ∈ 6005127.Number of

deliveriesK = 15/truck Peapod routing data (Peapod corporate fact sheet, http://bit.ly/peepod (accessed May 9, 2016)). Range

examined: K ∈ 651507 (per truck).Days in operation �= 365 Customers can usually place orders seven days a week (Peapod corporate fact sheet, http://bit.ly/peepod

(accessed May 9, 2016)).Picking costs cp = $2 Author estimates. Range examined: cp ∈ 60157.Market penetration � = 1% Author estimates. We examine penetration between 0.1% and 81%. In the United States, 81% of the

population uses the Internet but only about a third have fixed-line broadband ((Wikipedia, The FreeEncyclopedia, s.v. “Internet in the United States,” http://bit.ly/HHIIncome, accessed May 9, 2016). Effectivepenetration is capped by Internet access and varies as a function of, inter alia, marketing efforts;� ∈ 60011817 (percent).

easiest to make, so the subscription model—with itsfrequent deliveries and low levels of wasted food—is preferred for the most margins and consumptionrates. Comparing Paris and Manhattan, we see thatManhattan’s density is higher but that Paris has a morefavorable shape. The shape effect is more significant,so the subscription model should be preferred more inParis than in Manhattan.

Finally, Los Angeles (panel (b) of Figure 5) hasa much lower density of population and a muchlarger area. Here, the subscription model’s higher-orderfrequency constitutes more of a disadvantage thandoes the per-order model’s food waste; as a result, theper-order model is preferred more there.

These comparisons illustrate that the trade-offs onwhich we focus are not only theoretically but alsopractically relevant, leading to different choices invarious realistic contexts. Prospective online retailersshould employ the type of formal analysis described inthis paper in order to give themselves the best chanceof succeeding in the online grocery market.

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Figure 5 Product Characteristics and Preferred Revenue Models by City

O S

� �

O

S

�

�

Notes. The per-order revenue model is preferred to the right of the indifference curve (regions marked O); the subscription model is preferred to the left of theindifference curve (regions marked S). The warehouse is assumed to be centrally located.

8.2. Environmental EffectsWe also calibrate our results on the environmentalimpact of different revenue models. This analysisrequires estimates for the carbon emissions of travelfor passenger vehicles ep and delivery vehicles ed. Weuse estimates from Cachon (2014): ep = 00417 kg ofCO2 equivalents per mile and ed = 10683 kg of CO2equivalents per mile. We also require estimates of thefood waste emissions coefficient ef . There are a varietyof different sources with estimates that differ as areflection of the composition of consumers’ diet, consid-eration of food consumed or food wasted (wasted foodis more carbon polluting than consumed food), and theinclusion or exclusion of different stages of production,transport, storage, and consumption (since we areexamining food wasted by the consumer, we need toinclude the carbon load of all stages). We provide asummary of the most relevant sources as well as theirestimation techniques and potential biases in Section 1of the electronic companion. For purposes of numeri-cal analysis, we verify the results by considering thewidest plausible range of estimates—ef ∈ 60036110677 kgof CO2 equivalents per dollar of food wasted—builtby considering the most conservative lower bound(i.e., only production-related emissions and low-carbondiets). All results are presented for three representativevalues from that range. For the distance traveled byoffline consumers when their cost is high dx, we usean indicative average round-trip distance to the nearestgrocery store, dx = 1 miles. Likewise, for distance trav-eled when store visits happen to be cheap (consumeris passing by the store), we use d0

x = 005 miles.The results of our analysis for environmental impact

are more stark than those for financial impact; seeFigure 6. We find that, for all reasonable parametervalues, emissions under the subscription model arelower than those under the per-order model. The carbonimpact of extra food waste turns out to be muchhigher than that of extra driving. In other words,the negative effects of wasted groceries (in the per-order model) far outweigh the negative effects of extra

miles driven when delivering the groceries (in thesubscription model). This result holds for all plausibleranges of parameters for the cities of Paris, Beijing, andManhattan.

In practical terms, the subscription model’s emis-sions advantage over the per-order model amounts tobetween 5% and 10% of the food waste emissions cre-ated by an average citizen of the Western world. Thismeans that the combination of subscription pricing andonline grocery retailing can significantly reduce emissionsfrom the food supply chain. Finally, our findings indicatethat food waste plays a much more important rolein the emissions impact of grocery retail than do thetravel-related emissions upon which previous studieshave focused (e.g., Cachon 2014). Most previous analy-sis fails to account for emissions from food waste andso would wrongly condemn the subscription modelby focusing only on its driving-related downsides.Although this conclusion may well apply to nonperish-able products, the prescription should be reversed forperishable grocery items.

Our results are a bit moderated for the somewhatatypical city of Los Angeles—which has a slightlyelongated shape and (compared with the other twocities) a much larger area and much lower populationdensity—we find that, for the lower bound on thefood emissions coefficient (i.e., ef = 0036) and for awarehouse located far from the delivery area’s center,the subscription and the per-order model becomecomparable environmentally; furthermore, for someparameter values, the per-order model (marginally)dominates. For example, our finding has potential to bereversed for a vegan grocery store selling high-margin,fast-to-expire products, delivered from a suburbanwarehouse in a vast urban area such as Los Angeles.Section 2 of the electronic companion provides detailedestimates for such extreme cases.

Figure 6 illustrates how the environmental advantageof the subscription model depends on select param-eters. For all cities considered, the advantage of the

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Figure 6 Annual Per-Household Reduction in Emissions Under Subscription Pricing for Paris

Product life, T (weeks)

(b) (c)(a)

Mean demand rate, � ($/wk) Cost of ordering, � ($) Mean high store visit cost, 1/� ($)

(e) (f)(d)

Δ E

mis

sion

s(k

g of

CO

2 eq

.)

ef = 0.36

e f = 1

e f = 1.67

e f = 1 e f =

1.67

ef = 0.36

ef = 0.36

ef = 1

ef = 1.67

ef = 1

ef = 1.67

ef = 0.36

ef = 1

ef = 1.67

ef = 0.36

ef = 1

ef = 1.67

ef = 0.36

100

0

Δ E

mis

sion

s(k

g of

CO

2 eq

.)

150

0

70 100 2 10 10 50

Δ E

mis

sion

s(k

g of

CO

2 eq

.)150

0

Δ E

mis

sion

s(k

g of

CO

2 eq

.)

100

0

Δ E

mis

sion

s(k

g of

CO

2 eq

.)

100

0 Δ E

mis

sion

s(k

g of

CO

2 eq

.)

200

0

0.6 2.0

High-cost probability �0.1 1.0

Gross margins, 1– � (%)10 45

Notes. Unless stated otherwise, the city parameters used are those given in Table 1 for Paris and other parameters are from the second column of Table 2. Thewarehouse is assumed to be centrally located. Estimates are very similar for Beijing and Manhattan.

subscription model is higher (i) in small cities, wherethe driving disadvantage is small; (ii) for low margins,high delivery costs, and high store visit costs (all ofwhich increase waste because of lower adoption rates);and (iii) for low mean demand and product life, whichincrease waste in the per-order model. The advantageis higher for moderate costs of ordering (in the sub-scription model, waste is increasing in �; eventually,this increase approaches and exceeds the relativelymodest effect of � under the per-order model). Finally(as expected), more frequent customer use of the onlinechannel in comparison with offline shopping—as cap-tured by higher �—enhances the subscription model’sadvantage.

9. DiscussionIn our base model, it is never optimal for the retailerto offer customers a choice between subscription andper-order pricing. The reason is that there is only onedimension of customer heterogeneity and so one of thetwo models will yield higher retailer profits; offeringtwo options will lead to one revenue model cannibal-izing the other revenue model’s profits. That beingsaid, it could be optimal for the retailer to offer twopricing schemes if there were two (or more) dimen-sions of customer heterogeneity—for instance, if offlineshopping costs had three levels instead of two. In thiscase, the per-order model and the subscription modelwould be able to capture different market segments,and offering both models might allow the retailer to

screen customers in different segments and therebyimprove its profits. This possibility is an intriguingprospect for future study.

Our analysis assumed that delivery trucks carry thesame number of orders irrespective of the basket size.However, it is possible that if delivery batch sizes—asdetermined by delivery window promises—are large,then the truck’s size does become binding and thenumber of orders carried in a truck is a function ofthe basket size. Analysis of this scenario (providedin Section 4.1 of the electronic companion) showsthat this eventuality increases both the financial andenvironmental advantages of the subscription model,reinforcing our finding.

We did not consider any customer storage spaceconstraints. Incorporating such constraints (discussedin Section 4.2 of the electronic companion) does nothave any practical effect on our results (the constraintsare not binding for any reasonable settings). Anotherinteresting facet is the dependence of store visit costson the physical location of the customer. Interestingly,the geometric properties of the area serviced andadopting customer locations ensure that incorporatingthis dependence also does not change any of theexpressions provided and requires minor adjustmentsin the parameter values used (see Section 4.3 of theelectronic companion).

The costs of online or offline shopping may varyseasonally (in addition to the random variations alreadyconsidered); in effect, such variation shifts grocery

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purchase volume to days characterized by lower incon-venience costs. For our model, that would be equivalentto reducing the effective replenishment costs (� and �)—in which case our original analysis still applies. Wehave also assumed the inconvenience cost of onlineshopping to be the same throughout the population, yetcustomers might be heterogeneous in that respect. Prin-cipally, only the difference between online and offlinecosts matters; since the store visit cost is already hetero-geneous, this second degree of customer heterogeneityyields few (if any) new conceptual insights but makesall the mathematical expressions far more cumbersome.

In our model, there is no lead time between placingand receiving an order. Positive lead times will resultin more waste. A random shelf life of the grocerybasket and the fact that it consists of multiple itemsthat have different perishability also increase foodwaste, increasing the food waste disadvantage of theper-order model and, thus, making the subscriptionmodel more desirable.

Although our analysis assumed that the market areais exogenously specified, it could instead be chosenendogenously in concert with the service model. Ingeneral, the choice of service area (both its size andshape) is driven by several factors that we did notconsider. These factors include population density andthe rate of its decay from the center of a city, regulationand licensing, natural geographical factors (bodiesof water, etc.), the growth in delivery costs, and therate of increase in line-haul distances, etc., over andabove its dependence on per-customer revenues thatdepend on the revenue model. Our analysis could bemodified to include endogenous area choices, in whichcase the per-order (respectively, subscription) modelwould likely correspond to larger (respectively, smaller)service areas.

We did not explicitly consider the retailer’s manage-ment of inventory and fleet size. The pricing modelmay influence day-to-day variability in orders to bedelivered, which in turn would affect costs relatedto inventory management. Note also that reducedintertemporal variability at the warehouse wouldrequire fewer vehicles to provide the same service levelfor on-time order deliveries. We also assumed that thefixed costs of building and servicing the per-order andsubscription models are the same—that is, costs thatdepend neither on the volume of groceries sold nor onthe number of orders serviced are the same in eachmodel. Incorporating such costs would be but a minorextension.

Behavioral phenomena can also alter customer prefer-ences for different pricing schemes. Empirical analysisby Danaher (2002) shows that customers vary in theirsensitivity to subscription and per-order fees. Theremight be other behavioral effects—for example, theability to select appropriate groceries, impulse buying,

the physical exercise benefits of shopping offline, andthe effect on consumer behavior of marketing andpricing strategies. Many of these factors could be easilyincorporated into our model via adjustment of thestore visit costs.

We also assumed that the per-unit prices of groceriesare the same in both revenue models and in the offlinestore. Obviously, that may not be the case. Lower pricescan increase adoption, enhancing the relevant model’seffects. It is also worth mentioning that customers mayhave different grocery demand rates. Yet because theretailer cannot discriminate ex ante among differentcustomers—and must therefore offer different mar-ket segments the same pricing scheme—our analysisprovides a close first-order approximation. If, indeed,there are widely different customer segments, thenoffering a menu of contracts (and/or two-part tariffs)might help the retailer select which segments to serveand help it steer different segments toward differentrevenue models.

Finally, we considered an online retailer that com-petes with offline stores. This is a fair description ofalmost all major markets, but an extended model mightconsider the competition between different online retail-ers. Our model captures competitive aspects throughthe customer’s outside option cost of buying offline.Under online competition, this cost could be substi-tuted by the customer’s outside online option. Furtheranalysis of these issues remains an open question anda promising avenue for future work.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/mnsc.2016.2430.

AcknowledgmentsThe authors thank Dan Adelman, Ekaterina Astashkina, JohnBirge, Gérard Cachon, René Caldentey, Steve Chick, LaurensDebo, Donald Eisenstein, Vishal Gaur, Ruslan Momot, SergueiNetessine, seminar participants at the Tuck School of Businessat Dartmouth, and the anonymous review team for commentsand discussions. This work was supported by generousfunding from the Jane and Basil Vasiliou Faculty ResearchFund and the Neubauer Faculty Fellows program at theUniversity of Chicago Booth School of Business.

Appendix. ProofsAll statements are given in their order of appearance in themain text.

Lemma 5. (i) An optimal order-up-to policy is one where,at each perishable grocery replenishment point, the customerbuys enough nonperishables to bring the nonperishable inven-tory to � times the perishable inventory basket size—that is, upto level �Qw. (ii) E6CTi4Q57=�−14Q−

∑Qj=04Q− j5 ·pj4�T 55,

N4Q5 = 1 year/4E6CTi4Q575 and Q∗4a5 is a solution toQ−

∑Qj=04Q− j5pj 4�T 5≈ 4a+Q541−Pj 4Q1�T 551 Pj 4Q1�T 5=

∑Qj=0 pj4�T 5.

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Proof. (i) For nonperishable items, the customer has fullcontrol over when and how much to buy (because theseitems do not expire). Since there is no holding cost and sinceitems do not perish, the customer’s cost of shopping doesnot change provided no additional orders or shopping tripsare triggered in addition to the those required to restockperishable items. By ordering up to level �Qw , the customercan ensure that no additional orders are triggered owing toher consumption of nonperishables.

(ii) Adopting the analysis of Weiss (1980) to our setting,we obtain expressions for E6CTi4Q57 and N4Q5 and that thecustomer’s expected average long-run cost, 4a + Q54Q −∑Q

j=04Q− j5pj 4�T 55−1, is first decreasing and then increasing

in Q. Hence the optimal quantity is either the highest Q suchthat

℘ ≡a+Q+1

Q+1−∑Q+1

j=0 4Q+1−j5pj4�T 5−

a+Q

Q−∑Q

j=04Q−j5pj4�T 5

≤ 0

or the lowest Q such that ℘≥ 0. �

Proof of Lemma 1For high enough �T (�T > 10), Poisson4�T 5 random variablemay be considered approximately normal. We can thereforerewrite the expected waste during a cycle

∑Qj=04Q− j5pj 4�T 5≈

∫ Q

−�4Q−x5f 4x5dx, f 4x5≡ 41/�541/

√2�5e−41/254x−�T 52/�2 , and

� =√

�T . Thus, from Lemma 5, after a few simplifica-tions, Q∗ is given as a solution to

∫ Q

−�xf 4x5dx = aF 4Q5,

where F 4Q5≡ 1 −∫ Q

−�f 4x5dx; ¡aQ = F 4Q5/4f 4Q54Q+ a55 > 0.

Similarly, N4Q5 = �4Q −∑Q

n=04Q − n5pn4�T 55−1 ≈ �6Q −

∫ Q

−�4Q−x5f 4x5dx7−1 and ¡QN4Q5= −41/�5N4Q52F 4Q5 < 0

and so ¡aN4Q5 = ¡QN4Q5¡aQ < 0. Finally, ¡a4N4Q∗5Q∗5 =

41/�56N 4Q572aF 4Q5¡aQ> 0. Also, ¡a4a+Q∗5N 4Q∗5=N4Q∗5+¡aQ

∗¡Q44a+Q5N4Q55=N4Q∗5 > 0 because, at Q∗, we have¡Q44a+Q5N4Q55= 0. Now, note that ¡a4aNa5=Na + a¡aN

∗=

Na41 − aF 4Q5/4f 4Q54Q+ a5255 and that Q is implicitly de-fined by

∫ Q

−�xf 4x5 dx = aF 4Q5; therefore, aF 4Q5/4f 4Q54Q+a525

≈∫ Q

−�x4f 4x5/f 4Q55dx/4Q + a52. For �T sufficiently high,

∫ Q

−�xf 4x5dx ≈

∫ Q

0 xf 4x5dx (this must be true for the approx-imation of Poisson to be reasonable). Now, if Q < �T ,then

∫ Q

−�x4f 4x5/f 4Q55 dx ≈

∫ Q

0 x4f 4x5/f 4Q55 dx ≤∫ Q

0 x dx = 12Q

2;thus we obtain

aF 4Q5

f 4Q54Q+ a52≤

41/25Q2

4Q+ a52< 10

Furthermore, if Q>�T , then it follows (from the “increasingfailure rate” property of a normal distribution) that

f 4Q5

F 4Q5>

f 4�T 5

F 4�T 5=

2√

2��T0

That is, we need to show that

1√

4�/25�T>

a

Q+ a

1Q+ a

or 1 >a

Q+ a

√

4�/25�TQ+ a

0

These inequalities hold if �/2 <�T , which holds per ourassumption �T > 10.

Proof of Lemma 2(i) This result follows from adapting the analysis of Daganzo(1984a, b) to our setting (see Section 3 of the electronic compan-ion for details). The factor å4K5 is given as å4K5= 44K − 25+/4K + 1551/

√

�∗K + 4K − 1/K5�4�∗K5 for K≤4 and å4K5=�4�∗K5−1/

√

�∗K for K>4; �∗ =1 for K∈ 61177 and �∗ =607/Kfor K≥7. (ii) We have ¡N D=−å4K541/42N55

√

�/4�N 5<0 and¡�D=−å4K541/42�55

√

�/4�N 5<0. Now ¡aD= 4¡N D5¡aN >0,since ¡aN <0 (by Lemma 1); therefore, ¡NND=2�

√A/K+

å4K541/25√

�/4�N 5>0 and ¡aND= 4¡NND5¡aN <0.

Lemma 6. If and only if the customer with store visit cost xchooses the online retailer, then customers with store visit costsx ≥ x will also choose to shop online.

Proof. Since Coff is an increasing function of x, it followsthat (a) all customers for whom x ≥ x will purchase onlineand (b) all other customers will purchase offline. �

Proof of Lemma 3The firm seeks to maximize its profit: �O = maxo4��41 −�5+4o+Qo5No − 8�D4�G4x51�No1A1K5+�Qo + cp9No5A��G4xo5,where xo = min8x s.t. Coff ≥ Cmix-O9; Coff ≥ Cmix-O is equiv-alent to 4�x + Q∗4�x55N4Q∗4�x55 ≥ 4�o + ��5 + Q∗4�o +

��55N4Q∗4�o+��55. The constraint Coff ≥Cmix-O is alwaysbinding, and the firm will charge o+ � = xo; hence we canreformulate the problem as

max0≤x≤�

4Cx +�41 −�5�−ho4x55�A�G4x51

Cx ≡ 4x+Q∗4�x55N 4Q∗4�x550(8)

Since Coff is an increasing function of x, it follows that allcustomers for whom x ≥ x∗

o will purchase online and thatall other customers will purchase offline. The equation fordetermining the optimal solution follows from taking thederivative of (8), and that solution’s uniqueness follows fromthe concavity of (8). We have ¡xCx =N4Q∗4�x55+ 41−�5 ·x¡xN4Q∗4�x55, as N4Q∗4�x55+4�x+Q∗4�x55¡QN4Q∗4�x55= 0,since Q∗4�x5 is determined from ¡Q4�x+Q5N4Q5=N4Q5+4�x+Q5¡QN4Q5= 0.

Proof of Lemma 4The firm seeks to maximize its profit: �S = maxs4��41−�5+8s+Qs�Ns9−8�D4�G4xs51�Ns1A1K5+�Qs +cp9�Ns5A�G4xs51where xs = min8x s.t. Coff ≥Cmix-S9; Coff ≥Cmix-S is expressedas 4�x+Q∗4�x55N4Q∗4�x55≥ s+4��+Q∗4��55N4Q∗4��55.At xs , then, 4�xs + Q∗4�xs55N4Q∗4�xs55 = s + 4�� +

Q∗4��55N 4Q∗4��55, and thus s = 4�xs +Q∗4�xs55N 4Q∗4�xs55−4�� + Q∗4��55N4Q∗4��55 = 4�xs + Q∗4�xs55N4Q∗4�xs55 −

4��+Qs5Ns . Hence, �S = maxx4Cx +�41 −�5�+ âx −hs4x55� ·

A�G4x5. Now ¡x4Cx + âx5=Nx, as Q∗4�x5 is determined by¡Q4�x+Q5N4Q5=N4Q5+ 4�x+Q5¡QN4Q5= 0.

Proof of Theorem 1The theorem’s statements follow from Lemmas 1, 3, and 4combined with the higher cost of ordering under the per-order model: � ≤ x∗

o = o+ � since o ≥ 0; that is, Q∗4�x∗o 5 >Qs ,

N4Q∗4�x∗o 55 <Ns , and Q∗4�x∗

o 5N 4Q∗4�x∗o 55≥QsNs .

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Proof of Corollary to Theorem 1The yearly expected amount of groceries that is spoiled andhence becomes food waste can be written as 4Q·5N· − �.Theorem 1 establishes that 4Q·5N· is higher in the per-ordermodel, so the annual amount of wasted groceries is higher inthat model.

Lemma 7. Optimal x∗s is increasing in � ; i.e., ¡�x∗

s > 0.

Proof. We first establish that x∗s is increasing in � ; that is,

¡�x∗s > 0. To find the optimal x∗

s , the firm solves maxx4Cx +

�41 −�5�+ âx − hs4x55G4x5. Since the firm can guaranteea zero profit by choosing x = �, it follows for any x∗

s <�

that Cx∗s+�41 −�5�+ âx∗

s− hs4x

∗s 5 > 0 (A.2). By Lemma 4,

x∗s is implicitly defined by (using g4x5/G4x5= �> 0) Nx∗

s−

¡x∗shs4x

∗s 5= �4Cx∗

s+�41 −�5�+ âx∗

s−hs4x

∗s 55. The next step is

determining the behavior of x∗s with regard to � . For this, we

can use the implicit function theorem and our optimalitycondition to obtain

¡�x∗

s =��4

√A/K5Ns

�24Cx∗s+�41−�5�+âx∗

s−hs4x

∗s 55−¡x∗

sNx∗

s+¡2

x∗shs4x

∗s 50

The numerator is positive, so it remains only to estab-lish that the denominator is also positive. In the proof ofLemma 1 we showed that ¡Na/¡a < 0, from which it followsthat ¡Nx/¡x < 0; furthermore, the first summand is positivebecause Cx∗

s+�41 −�5�+ âx∗

s−hs4x

∗s 5 > 0 (see A.2). Finally,

the last summand is positive because

¡x4¡xhs5 �x∗s

= ¡x

(

12�z

√

Ns

G4x5

∣

∣

∣

∣

x∗s

)

=14�2z

√

Ns

G4x5

∣

∣

∣

∣

x∗s

> 01

z = �å4K5√

�/4��50

Similarly, x∗o is ¡�x

∗o = 4��4

√A/K5Nx∗

o5/4�24Cx∗

o+�41−�5�−

ho4x∗o 55− 42 −�5¡xNx − 41 −�5x¡2

xNx + ¡2x∗oho4x

∗o 55. �

Proof of Theorem 2First, observe that 4¡/¡�54�∗

o −�∗s 5=�42

√A/K58NsG4x

∗s 5−

Nx∗oG4x∗

o 59.Next we show that if NsG4x∗

s 5=Nx∗oG4x∗

o 5 for some � (whichentails that x∗

s > x∗o ) and if �∗

o >�∗s , then NsG4x∗

s 5 < Nx∗oG4x∗

o 5

for all � > � . To establish these conditions, we need only showthat ¡�x∗

s > ¡�x∗o for � > � . In Lemma 7 we established that

¡�x∗s > 0; now if ¡�x∗

o < 0, we directly obtain desired ¡�x∗s >

¡�x∗o . Thus, in what follows we consider ¡�x

∗o > 0. We can

rewrite the expression for ¡�x∗o (derived in Lemma 7) to obtain

¡�x∗o = 4�41/25�42

√A/K5Ns5/44Ns/Nx∗

o54�24Cx∗

o− ho4x

∗o 55 −

42 −�5¡xNx − 41 −�5x¡2xNx + ¡2

x∗oho4x

∗o 555. The numerators of

¡�x∗s and ¡�x

∗o are now equal, and we use dens and deno

to denote the respective denominators. Next we show thatdeno > dens for � > � :

deno − dens = �2{4Ns/Nx∗o5Cx∗

o− 4Ns/Nx∗

o5ho4x

∗

o 5

− 4Cx∗s+ âx∗

s−hs4x

∗

s 55}

+{

¡x∗sNx∗

s− 4Ns/Nx∗

o5¡x∗

oNx∗

o

− 41 −�54Ns/Nx∗o54x∗

o¡2x∗oNx∗

o+ ¡x∗

oNx∗

o5}

+ 84Ns/Nx∗o5¡2

x∗oho4x

∗

o 5− ¡2x∗shs4x

∗

s 590

We start by establishing that the second summand is posi-tive. We have already proved that ¡aNa = −Na41/4a+Qa5

25 ·4F 4Qa5/f 4Qa55, so

¡x∗sNx∗

s−

Ns

Nx∗o

¡x∗oNx∗

o

=Ns

4x∗o +Qx∗

o52

F 4Qx∗o5

f 4Qx∗o5−

Nx∗s

4x∗s +Qx∗

s52

F 4Qx∗s5

f 4Qx∗s5> 00

The third summand is also positive; ¡2x∗oho4x

∗o 5− 4Nx∗

o/Ns5 ·

¡2x∗shs4x

∗s 5 > 0, since

Nx∗o

Ns

¡2x∗shs4x

∗

s 5=z

4�2Nx∗

o

√

1

NsG4x∗s 51

and when ¡2x∗oho4x

∗o 5≥ 4z/45�2

√

Nx∗o/G4x∗

o 5, we have

¡2x∗oho4x

∗

o 5− 4Nx∗o/Ns5¡

2x∗shs4x

∗

s 5

≥ �4z/45�2Nx∗o

(√

1/4Nx∗oG4x∗

o 55−√

1/4NsG4x∗

s 55)

> 00

Finally, the first summand is positive, 4Ns/Nx∗o54Cx∗

o−ho4x

∗o 55−

4Cx∗s+ âx∗

s−hs4x

∗s 55 > 0, and can be rewritten as follows:

4x∗

o + 41 −�5Qx∗o5Ns − 4x∗

s +Qx∗s5Nx∗

s+Ns�Qs

+ zNs

(√

1/4NsG4x∗

s 55−√

1/4Nx∗oG4x∗

o 55)

0

We can now show that 4x∗o +41−�5Qx∗

o5Ns−4x∗

s +Qx∗s5Nx∗

s+

Ns�Qs is increasing in � : ¡� 4x∗o +41−�5Qx∗

o5Ns −4x∗

s +Qx∗s5Nx∗

s=

4¡�x∗o +41−�5¡�Qx∗

o5Ns −¡�Nx∗

s¡�x

∗s >0. Furthermore, at � we

have 4Ns/Nx∗o54Cx∗

o−ho4x

∗o 55−4Cx∗

s+âx∗

s−hs4x

∗s 55>0 if �∗

o >�∗s ;

hence 4Ns/Nx∗o54Cx∗

o−ho4x

∗o 55−4Cx∗

s+âx∗

s−hs4x

∗s 55>0 for all

� >� . This means that, for all �>� , we have ¡�x∗s > ¡�x

∗s , and

so x∗s > x∗

o . That is, 4¡/¡�54�∗o −�∗

s 5 < 0 for �>� .

Proof of Theorem 3By definition, � is such that �∗

s =�∗o . That is,

4�41 −�5�+Cx∗s+ âx∗

s− hs4x

∗

s 55G4x∗

s 5− z√Ns

√

G4x∗

s 5

= 4�41 −�5�+Cx∗o− ho4x

∗

o 55G4x∗

o 5− z√Nx∗

o

√

G4x∗

o 53

here hs4x5= 4�+ cp5Ns + �yNs +�QsNs ; ho4x5= 4�+ cp5Nx +

�yNx + �QxNx, z ≡ �å4K5√

�/� and y ≡ 42�√A5/K. Now

using the implicit function theorem yields

¡�

¡A= −�

¡y

¡A

4G4x∗o 5Nx∗

o−NsG4x∗

s 55

y4G4x∗o 5Nx∗

o− G4x∗

s 5Ns5< 0

because ¡y/¡A> 0; also,

¡�

¡�= −4�4¡y/¡�54G4x∗

o 5Nx∗o− G4x∗

s 5Ns5

+¡z

¡�

√

Nx∗oG4x∗

o 5−√

NsG4x∗s 5

y4G4x∗o 5Nx∗

o− G4x∗

s 5Ns5< 0

since ¡y/¡�1¡z/¡� > 0 and since 4G4x∗o 5Nx∗

o− NsG4x

∗s 55

and 4√

Nx∗oG4x∗

o 5 −

√

NsG4x∗s 55 always have the same sign.

Moreover,

¡�

¡�= −�

¡z

¡�

√

Nx∗oG4x∗

o 5−√

NsG4x∗s 5

y4G4x∗o 5Nx∗

o− G4x∗

s 5Ns5> 0

because ¡z/¡� < 0.

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Finally, ¡�/¡K = −44¡z/¡K54√

Nx∗oG4x∗

o 5−√

NsG4x∗s 55+ � ·

4¡y/¡K54G4x∗o 5Nx∗

o−NsG4x

∗s 555/4y4G4x

∗o 5Nx∗

o− G4x∗

s 5Ns55 > 0since ¡z/¡K1¡y/¡K < 0.

Proof of Theorem 4We can express the emission difference as follows: (z ≡

�å4K5√

�/�, y ≡ 42�√A5/K and e ≡ ep/ed): 41/4A�554Es −Eo5

= ed4†5 + ef 4∗5, where 4†5 ≡ �8G4x∗s 5yNs + z

√

G4x∗s 5Ns −

G4x∗o 5yNx∗

o− z

√

G4x∗o 5Nx∗

o9 + e41 − �58Ns

∫ �

x∗sd0xg4x5dx − No ·

∫ �

x∗od0xg4x5dx +

∫ x∗s

x∗o4�dx + 41 − �5d0

x5Nxg4x5dx9 and 4∗5 ≡

G4x∗s 5QsNs − G4x∗

o 5Qx∗oNx∗

o+∫ x∗

sx∗oNxQxg4x5dx. Both 4†5 and

4∗5 are independent of ef and ed. There are three pos-sible combinations: (i) 4∗51 4†5 < 0, in which case ef = 0;(ii) 4∗5 < 01 4†5 > 0, in which case ef = −ed44†5/4∗55; and(iii) 4∗5 > 0. We next show that case (iii) is ruled out bycondition x∗

s − x∗o < ¯x∗, where ¯x∗ ≡ max801 x∗9 and x∗ is such

that G4x∗o + x∗5QsNs +

∫ x∗o+x∗

x∗o

NxQxg4x5dx = G4x∗o 5Qx∗

oNx∗

o.

We first consider x∗s − x∗

o ≤ 0, which implies G4x∗s 5≥ G4x∗

o 5.We have G4x∗

o 54QsNs −Qx∗oNx∗

o5≤ 0 because QsNs <Qx∗

oNx∗

o(by

Theorem 1). Thus, we need only show that 4∗∗5≡ 4G4x∗s 5−

G4x∗o 55QsNs −

∫ x∗o

x∗sNxQxg4x5dx < 0. Since

∫ x∗o

x∗sNxQxg4x5dx >

Nx∗sQx∗

s

∫ x∗o

x∗sg4x5dx = Nx∗

sQx∗

s4G4x∗

o 5−G4x∗s 55, it follows that

4∗∗5 < 4G4x∗s 5− G4x∗

o 554QsNs −Nx∗sQx∗

s5 < 0. This inequality

allows us to obtain the desired result, 4∗5 < 0, which rulesout case (iii).

Next consider x∗s − x∗

o > 0, which implies G4x∗o 5≥ G4x∗

s 5.This means that, under the subscription revenue model, theretailer’s market coverage will be smaller, and so customerswith x ∈ 4x∗

o1x∗s 7 would waste more under the subscription

model (as they do not adopt) then under the per-order model(where such customers adopt online shopping). Conditionx∗s − x∗

o < ¯x∗ ensures that additional waste stemming fromnonadopters does not overcome the reduction in waste of theadopting customers, i.e., 4∗5 < 0, which rules out case (iii).This completes the proof. �

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http://bit.ly/23DpWj5

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http://bit.ly/GroceryDisruption

http://bit.ly/GroceryIndustry

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