Multi-Seller Access Pricing

for Shopping Platforms∗

Ming Gao†

May 6, 2020

Abstract

Through charging or subsidizing shoppers, a platform participates in the

pricing game played by n hosted third-party sellers, who sell different products

and share profits or revenues with the platform. The platform’s equilibrium

access price for shoppers is determined by bilateral effects between the plat-

form and each seller. A “sharing effect”is represented by the profit or revenue

share taken from a seller. A “buzz effect”embodies a seller’s ability to attract

shopper traffi c to the platform through adjusting her own price, as measured

by the platform’s cross-price elasticity with respect to the seller’s price. These

effects are aggregated across all sellers, and deducted from the platform’s stan-

dard markup. A positive access fee is optimal if the sum of profit shares taken

from sellers does not exceed 1, or if the revenue-share sum does not exceed

the per-shopper cost. When either sum exceeds a higher threshold, an access

subsidy emerges.

Key Words: multiproduct pricing game, agency selling, subsidy.

JEL Codes: D21, D42, L11, L81.

∗I thank Jean-Pierre Benoît, Mark Armstrong, Simon Cowan, Vincent Crawford, JeanineMiklos-Thal, David Myatt and two referees for very valuable comments. I also thank MasakiAoyagi, Yongmin Chen, Zhiqi Chen, Jay Pil Choi, Anastasios Dosis, Yuk-fai Fong, Hikmet Gu-nay, Ole Jann, Justin Johnson, Jianpei Li, George Mailath, Alexei Parakhonyak, David Ronayne,Greg Taylor, Julian Wright, Lixin Ye and other participants in seminars at Oxford, Fudan, SJTU,Peking and UIBE, and in the 2019 APIOC (Tokyo), 2019 AETW (Sydney), 2017 RES (Bristol),2016 IOMS (Hong Kong) and 2016 IIOC (Philadelphia) conferences.†School of Economics and Management, Tsinghua University, Beijing 100084, China. Email:

gaom@sem.tsinghua.edu.cn.

1 Introduction

Shopping malls and Amazon Marketplace are common examples of shopping plat-

forms that host third-party sellers and provide infrastructure to facilitate sellers’

transactions with shoppers. They usually adopt an “agency selling”business model,

where they take a share of sellers’profits or revenues1 but do not interfere in sellers’

price setting.2 Online or off-line, these platforms play an increasingly important role

in the retail industry. For instance, from 2015 onwards, more than half of all the

items on Amazon have been sold by third-party sellers annually.3 In this article,

we seek to understand some of the pricing patterns that emerge in the trilateral

business relationships among these platforms, sellers and shoppers. Our particular

focus is on how a platform should choose its optimal price towards shoppers, and

what distinctive role each seller may play in this choice.

We observe that, although it is clearly costly for the platforms to serve shop-

pers, a puzzling feature of this industry is the coexistence of platforms which charge

shoppers, those which provide free services, and those which even pay shoppers. For

instance, although most shopping malls provide parking space for shoppers, some

charge a fee for using it whereas others offer it for free. When parking is free, it

often requires purchase so that only actual shoppers benefit.4 Quite similar pricing

patterns exist in shipping for online shopping. Some platforms provide shipping

services at a fixed fee (e.g., an Amazon Prime membership covers all standard ship-

ping), whereas others waive this fee, usually for orders exceeding a certain check-out

threshold.5 To customers who require parking or shipping, such a fee acts as a fixed

price for access to the sellers on the platform.6 Free parking or shipping therefore

1In practice, profit sharing may take the form of the platform’s partial or full ownership of aseller, such as own-brand stores or restaurants owned by a shopping mall; revenue sharing neednot involve ownership, and may be easier to implement for typical independent sellers, or in theonline context generally.

2This distinguishes platforms from retailers, such as most supermarkets and Amazon’s ownretail segment.

3Source: Bezos’s letter to shareholders. See https://www.sec.gov/Archives/edgar/data/1018724/000119312519103013/d727605dex991.htm.4When malls offer free parking, shoppers are usually required to use their receipt to

redeem parking tickets before exiting. Normal fees apply otherwise. In Manhattan, forinstance, free parking can save shoppers up to $20 per hour in metered fees. (Source:http://www.nyc.gov/html/dot/html/motorist/parking-rates.shtml.)

5Here the parking or shipping is provided by the platform, not by individual sellers. Forinstance, JD.com, the world’s third largest Internet company by revenue in 2017 (Source:https://en.wikipedia.org/wiki/List_of_largest_Internet_companies), waives shipping fees on or-ders exceeding $15 (wich used to be $6 before 2014). An order may consist of products fromdifferent sellers.

6The actual name of such a price does not matter. The platforms that do not provide parking

1

corresponds to a zero access fee, and most likely constitutes subsidization towards

shoppers when considering the opportunity cost of parking or shipping facilities. In

fact, it is not rare for both online and off-line platforms to go further than provid-

ing free access, and literally subsidize shoppers by cash back coupons or gift cards

which directly or indirectly reduce total check-out prices,7 so that the effective access

price becomes negative. Our first research question naturally arises: When should a

platform charge shoppers a fee, and when should it offer a subsidy?

Moreover, we note two features on the seller side of the retail market that may

complicate a platform’s pricing choice. First, the number of sellers on a platform can

be quite large, and there exists tremendous heterogeneity among sellers. Consumers’

one-stop shopping needs usually require that the platforms attract various categories

of sellers, not just providers of differentiated products. Large shopping centers can

host hundreds of stores spanning the whole spectrum of products and services,8

whereas online platforms usually have even more sellers and variety.

Second, the sharing arrangement between each seller and the platform is given

before the seller’s own pricing decision, and the specific sharing percentage varies

significantly across different sellers. For instance, Amazon offers standardized take-

it-or-leave-it contracts to sellers, with “referral fee”percentages ranging from 6% to

45% across different product categories.9 Gould, Pashigian and Prendergast (2005)

report that 70% of “anchor”stores in shopping malls across the US pay nothing at

all to their malls by contract, whereas 99% of nonanchor stores’contracts require

sharing. The average sharing percentage is a mere 0.47% for anchors, compared

to a much higher 6.27% for nonanchors. They argue (on page 413) that “anchors

with well-established local market reputations are often in a commanding bargaining

situation (relative to the mall)”.

or shipping could simply charge a “membership fee”per shopper or a “transaction fee”per order,etc.

7Such coupons or cards typically apply as redeemable credit towards purchases or di-rect check-out price deductions. For instance, Macy’s offer free Visitor Savings Passes thatentitle shoppers to 10% off merchandise purchases when redeemed in person. (Source:http://www.visitmacysusa.com/sites/default/master/files/Macys_Discount_Voucher_Dom.pdf.)And the Fashion Outlets Chicago offers visitors a Green Savings Card, priced at$5 but actually often handed out for free from their Concierge Services or byapp download, that entitles shoppers to various redeemable discounts. (Source:http://www.fashionoutletsofchicago.com/Visitors#greensavingscard.)

8These include, for instance, fashion clothing, electronic devices, groceries, furniture, toys,restaurants, health/beauty salons, gyms, and cinemas, etc. The Mall of America, in Bloom-ington, Minnesota hosts more than 500 stores and restaurants, among other tenants. Source:https://www.travelchannel.com/interests/shopping/articles/top-10-us-shopping-malls.

9Source: https://sellercentral.amazon.com.

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Our second research question is motivated by these observations: When a large

group of sellers - who sell different products and have diverse sharing contracts - each

choose their own retail prices, what impact do they have on the platform’s optimal

access price?

Summary of Key Findings

We answer these questions in a general multiproduct pricing game played by a

platform and n sellers, where potential shoppers need to pay the platform to gain

access to these sellers. If the platform only received revenues from charging shoppers,

the standard revenue-demand trade-off would imply a positive markup on cost, and

hence a positive access price. Now suppose each seller also shares some revenue with

the platform, then it clearly creates an incentive for the platform to cut price. As a

lower access price attracts more shoppers to visit the platform, the demand for each

seller also increases. Other things equal, the more the platform internalizes seller 1’s

revenue, say, the more it also benefits from the price cut through sharing. Call this

the sharing effect due to seller 1, which should increase in the revenue share that

the platform takes from seller 1. Thus the equilibrium access price depends on the

relative strengths of i) the platform’s standard markup incentive (which decreases in

its price elasticity of demand), and ii) its price-reducing incentive from the sharing

effect of all sellers. When the latter is suffi ciently strong, an access subsidy may

emerge.

Whereas this intuitive rationale is formalized in our analysis, we also show that

it is coarse and incomplete. In particular, it ignores the impact that each seller’s

pricing has on the platform’s choice. In fact, whenever the access price is not zero,

all the sellers’and the platform’s pricing problems will generally be inter-dependent,

even if different sellers supply totally unrelated products (i.e., neither complements

nor substitutes - we formalize this point in Lemma 1).

We show that, besides the previous sharing effect, each seller also imposes a buzz

effect, which plays an equally important role in the platform’s price-reducing incen-

tive. This effect represents a seller’s ability to attract shopper traffi c to the platform

- or to create a “buzz”for the platform - through adjusting her own product’s price.

That is, it can be precisely measured by the platform’s cross-price elasticity with

respect to this seller’s price.

For illustration, consider a shopping mall which, in addition to a group of stores,

can also host either a Walmart convenience store or a generic grocery store of similar

size and style. Suppose further that all stores would share 1% of revenue with the

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mall so that the sharing effect does not vary. Other things equal, would the mall’s

access price (e.g., its parking tariff or cash back policy for shoppers) be different in

the two cases? Our analysis shows that, the access price would be lower if the mall

chooses the candidate store with more buzz (i.e., a higher cross-price elasticity) -

perhaps Walmart.

One tempting explanation might be that because Walmart will likely make more

sales due to a higher “brand value” than the alternative store, the platform (af-

ter receiving its share of revenue) can afford to lower the access price for shoppers.

However this intuition is only apparent. The platform need not afford a lower access

price unless it is profitable. The real reason is that, if the platform’s cross-price elas-

ticity is higher when measured by Walmart’s price, Walmart’s demand will increase

more as a result of the same reduction in access price by the platform,10 which in

turn will also benefit the platform more given the same sharing arrangement. In

other words, a seller with more buzz can translate the same reduction in access price

into a larger increase in shopper traffi c (to both this seller and the platform), and

this higher marginal benefit incentivizes the platform to cut price further.

It is useful to clarify that the buzz effect is not generally reflected in the absolute

scale of demand. A seller with more buzz is not necessarily one that constantly brings

high traffi c to the platform, but one that attracts proportionally higher incremental

traffi c to the platform when her price lowers by one percent. Walmart may perform

well on both fronts, but it is the latter that directly affects access pricing.11

As a result, the platform follows a new pricing rule, where each seller plays a

role through her sharing and buzz effects. The number and composition of sellers

affect the optimal access price through an aggregation of these seller-specific effects

across all sellers, which are jointly deducted from the platform’s standard markup.

(Propositions 1 and 3 summarize this pricing rule for different sharing scenarios.)

The aforementioned mechanisms are similar under profit or revenue sharing.

The only difference lies in whether or not the platform internalizes sellers’ costs.

Under profit sharing, it does care about each seller’s price-cost markup, and thus

10A technical detail behind this rationale, as discussed in part i) of Lemma 1, is the Slutskysymmetry: In this general multiproduct demand system, the cross-price demand derivatives aresymmetric. That is, the platform’s demand gain when a seller lowers her price by one unit is thesame as that seller’s demand gain when the platform lowers its access price by one unit.

11As discussed previously, Gould, Pashigian and Prendergast (2005) attribute the difference insellers’ sharing percentages to their bargaining power relative to the platform. A related open(empirical) question is whether the scale of a seller’s demand is reflected in her bargaining power,and hence in the sharing percentage. If so, the demand scale will indirectly affect access pricingthrough the sharing effect.

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each seller’s own-price elasticity also enters the platform’s pricing trade-offs. Other

things equal, a less elastic demand for an individual seller will induce the platform

to lower its price to attract more shoppers to this seller, because this seller will have

a higher markup in equilibrium, and the platform will gain more in shared profit.

Under revenue sharing, on the other hand, the sellers’own elasticities do not matter.

One strikingly simple implication of the pricing formula we find is: Under profit

sharing, a necessary condition for a negative access price is that the sum of all sellers’

profit shares taken by the platform must exceed 1. It means that a positive access

price is optimal whenever this condition does not hold. Moreover, because each

profit share is at most 1, this condition requires at least two sellers, and equilibrium

subsidy is therefore a multi-seller phenomenon. (Corollary 1 presents this result.)

To our best knowledge, our theory and results are new to the literature (refer-

enced in Section 7), and have the potential to be more convenient for applications

or empirical tests. Our general framework allows for virtually any form of “asym-

metries”and “correlations”across sellers. That is, i) different sellers’products may

follow different valuation distributions, and ii) any pair of products’valuations may

have any degree of positive or negative correlation, or neither, except only for per-

fect correlation. These properties may be especially useful in the context of retail

markets with numerous sellers and intricate cross-product relationships.

In the remaining parts of this article, Section 2 presents the basic model under

profit sharing; Section 3 derives the equilibrium access price; Section 4 presents

conditions for the choice between an access fee and a subsidy; Section 5 analyzes

the pricing game under revenue sharing; Section 6 discusses the welfare-maximizing

strategy; Section 7 reviews the related literature; and Section 8 concludes.

2 Modelling Framework

There is a group of n ∈ N sellers who each sell one product. Denote N ≡ {1, 2, ..., n}the set of all sellers, and let j ∈ N represent a seller and the product she12 provides,

at a marginal cost cj ≥ 0. Each seller individually chooses a price pj ∈ R for herproduct. There is a platform that serves these sellers and acts as their “portal”to

shoppers. The platform has two potential revenue sources: (i) it takes a share of

each seller’s profit or revenue; (ii) it charges an access price to each shopper. In the

baseline model, we study the more intuitive case when sellers share profits with the

12In this article, “she”refers to a seller, “he”refers to a shopper, and “it”refers to the platform.Throughout our research papers we use feminine and masculine pronouns approximately equally.

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platform, and the revenue-sharing scenario is analyzed in Section 5.

Each seller and the platform bilaterally decide how to share the seller’s profit,

if any, depending on their relative bargaining power. To model potential hetero-

geneity among sellers, assume that relative to seller j, the platform has bargaining

power αj ∈ (0, 1), which is exogenously given based on seller j’s idiosyncratic char-

acteristics.13 (1 − αj) then represents seller j’s bargaining power. Denote βj the

share of seller j’s profit that the platform ends up extracting, and assume that their

contracting process can be represented by an exogenous function gj(·), such thatβj = gj(αj), g′j ≥ 0, gj(0) = 0, and gj(1) = 1 for all j ∈ N . The remaining (1− βj)profit share is kept by seller j.

Some explanations are necessary. The mechanisms through which the bargain-

ing power parameters {αj}j∈N affect sharing between sellers and the platform are

parsimoniously modeled as a set of exogenous functions {gj(·)}j∈N . There are twokey assumptions here. Firstly, each gj(·) is (weakly) increasing, such that largerbargaining power enables the platform to extract a larger share of a seller’s profit.

This is consistent with the empirical evidence discussed in Section 1, and is one key

feature we want the current set-up to exhibit. Secondly, in any seller j’s bilateral

contracting process with the platform, besides αj, the effects of all other possible

factors on βj are incorporated in gj(·), where the subscript j indicates possible selleridiosyncrasy. In reality, the platform may bargain sequentially with different sellers

(e.g., in the shopping mall example), or offer sellers take-it-or-leave-it contracts (e.g.,

Amazon). The difference between gj(·)’s therefore accommodates these exogenousfactors that may not be reflected in bargaining power.14 For instance, in the case

of the non-negotiable sharing percentages offered by Amazon, each gj(·) is a con-stant that depends only on the product category to which j belongs. However, the

general “bargaining game”played by all the sellers and the platform is not formally

modeled.15 Importantly, this rules out inter-dependencies among different sellers’

contracting processes with the platform, such as endogenous sequentiality.16 This

13These include, e.g., whether the seller is an anchor store or a nonanchor. See Section 1 for adiscussion of the empirical findings by Gould, Pashigian and Prendergast (2005) that support thisassumption.

14All gj(·)’s may also be the same. Suppose each seller and the platform share the seller’s profitthrough asymmetric Nash bargaining (Osborne and Rubinstein, 1994), then in the solution eachseller’s share should be proportional to her bargaining power relative to the platform. Thus wehave gj(x) = x, which implies βj = αj for all j.

15Such a general bargaining game with n sellers and the platform can be very complicated,especially for a large n, and is beyond the scope of this article.

16For instance, if the platform bargains sequentially with different sellers, and each seller canchoose when to bargain, then each seller’s sharing arrangement may also depend directly on other

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modeling choice allows us to focus on the analysis of the “pricing game” among

sellers and the platform. A discussion of the ways to extend it is postponed till

Section 8.

The platform sets an access price m ∈ R (which may be negative as a subsidy)that applies to each shopper who purchases from any seller(s), and the platform does

not coordinate with any seller.17 The cost for the platform to serve each shopper is

c ≥ 0. There are no other costs.18

There is a continuum of potential shoppers of mass 1, each of whom is represented

by an n-dimensional real-valued vector x ≡ (x1, ..., xn), where xj is his private

valuation of (or the gross utility he derives from) product j. Each shopper demands

0 or 1 unit of each product. The net utility a shopper derives from purchasing a set of

products is simply the sum of his valuations for these products, minus all the prices

he pays. Therefore shopper valuation is additive and there exists no complementarity

or substitutability between different products in individual consumption. (This

assumption is relaxed in an online appendix which shows that it does not affect

our main findings and their intuition.19) However, as we will show shortly, different

products may still exhibit complementarity or substitutability on the aggregate level

as a result of the platform’s pricing strategy. No purchase results in zero utility.

Shoppers are heterogeneous such that x varies across them following a joint

distribution F with density f . Denote Fj and fj the marginal distribution and

density of xj. The sellers and the platform know these distributions but not the

exact value of each shopper’s type. The support of f is assumed to be bounded and

is normalized to [0, 1]n ≡ In.20

Assumption 1 f is continuous and f(x) > 0 if and only if x ∈ In.

sellers’bargaining power. The current model does not apply in such contexts.17In an online appendix we extend the analysis to the case when the platform can use multi-tier

fees that are contingent on purchase(s) from certain sellers.18All fixed costs of the platform or the sellers are ignored, as they do not affect their pricing

decisions. These include any lump-sum charge that the platform may impose on a seller (e.g., afixed rent), which in practice may affect the seller’s entry/exit decision, and hence the total numberof sellers on the platform. We have taken this number as given, which may be interpreted as a kindof capacity constraint of the platform (e.g., limited space in a shopping mall), or the equilibriumoutcome of a larger game which considers sellers’participation choice.

19The online appendix extends our model to the case of non-additive valuations with multi-unit demand. The assumption of additive valuations for different products, and the earlier oneon unit demand for each product per consumer, are both invoked only to reduce the burden oncharacterizing consumer demand later.

20Using a unit hypercube as support simplifies expressions but is not crucial. Our main resultsstill hold when we extend the support of f to a weakly convex and bounded subset of R+n withfull dimension, as shown in the online appendix.

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Note that there is no symmetry or independence assumption: each product may

have a different distribution Fj, and shoppers’valuations for different products may

be correlated. To make it possible for each seller and the platform to make some

profit, we further assume cj ∈ [0, 1) and c ∈ [0, n).

Timing Shoppers maximize their own utility, and sellers and the platform maxi-

mize their own profit. The timing of the game that they all play is as follows.

Stage 1. When the platform opens business, the sellers and the platform simul-

taneously announce their prices.21

Stage 2. Shoppers observe prices and decide whether to visit the platform, paying

the relevant access fee or receiving the subsidy upon entry. When there is an access

subsidy, we assume that a shopper must purchase at least one product to qualify for

the subsidy (e.g., suppose the platform can check receipts or implement subsidies

through non-transferable gift codes or coupons, which are worthless unless redeemed

upon purchasing something).22

Stage 3. Shoppers who visit the platform choose which product(s) to purchase,

if any.

Our analysis follows backward induction. We begin by characterizing shoppers’

demand, given the prices set by sellers and the platform. Then we find the platform’s

and each seller’s optimal price.

Demand First note that when m = 0, the additivity of valuations implies that

a shopper buys from seller j if and only if xj − pj ≥ 0. Therefore the demand for

seller j is simply 1− Fj(pj), independent of any other sellers’prices. Each shopperthen visits the platform if and only if they expect at least one product, say i, with

xi ≥ pi, so the total number of shoppers who visit the platform, or the platform’s

demand, is Pr[max(xi − pi,∀i ∈ N) ≥ 0] in this case.

Now consider the case with an access fee (m ≥ 0). Once a shopper has paid

this fee to visit the platform, he will buy from seller j if and only if xj − pj ≥ 0.

Thus a shopper will visit the platform if and only if his total expected surplus from

all purchases at least covers the fee. Let p ≡ (p1, p2, ..., pn). Denote D0(m,p) the

21It is worth noting that, real-life shopping malls and its sellers can usually freely change theirprices at any time, which indicates a lack of commitment power. Therefore we model the pricingstage as a static game.

22If no purchase was required to obtain a subsidy, all potential consumers in our context wouldvisit the platform simply for the subsidy, and the platform would incur a pure loss from the subsidywhich has no impact on the demand for any seller whatsoever. The platform then has no incentiveto offer such a subsidy. The assumption of contingent subsidy here rules out this case.

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platform’s demand, and we have

D0(m,p) = Pr[∑i∈N

max(xi − pi, 0) ≥ m], if m ≥ 0. (1)

The demand for seller j ∈ N is then

Dj(m,p) = Pr[xj ≥ pj and∑i∈N

max(xi − pi, 0) ≥ m], if m ≥ 0. (2)

Figure 1 shows the demand patterns for two sellers. The left panel corresponds

to m > 0, where the shoppers on the upper-right side of the line segments ab, bc,

and cd visit the platform. Those on the right side of be, bc, and cd buy from seller

1, and those above ab, bc, and cf buy from seller 2.

Figure 1: Shoppers’Purchase Choice when n = 2

The case with a subsidy (m < 0) is different because receiving the subsidy is

contingent on purchasing from at least one seller. Therefore a shopper visits the

platform if and only if there exists some seller i ∈ N who provides him with an

expected net surplus xi − pi −m ≥ 0. This means the platform’s demand is now

D0(m,p) = Pr[max(xi − pi −m,∀i ∈ N) ≥ 0], if m < 0. (3)

Now consider seller j’s demand. A shopper will clearly buy from her if xj−pj ≥ 0.

But this condition is no longer necessary when m < 0, because even if xj < pj, it

is still possible that buying product j will qualify the shopper for the subsidy from

the platform, which lowers the threshold for xj. Clearly, this latter case happens

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if and only if no other products are bought, and product j alone is worth the visit

to the platform, i.e., pj + m ≤ xj(< pj). (Note m is negative here.) This in

turn means that j must provide the highest net surplus among all products,23 i.e.,

xj − pj = max(xi − pi, ∀i ∈ N). In summary, the demand for seller j when m < 0

incorporates both cases:

Dj(m,p) = Pr[pj +m ≤ xj < pj and xj − pj = max(xi − pi,∀i ∈ N)]

+ Pr[xj ≥ pj], if m < 0.(4)

The right panel of Figure 1 shows the demand patterns when m < 0, where the

shoppers on the upper-right side of the line segments rs and st visit the platform.

Those on the right side of yw, sw, and st buy from seller 1, and those above rs, sw,

and wz buy from seller 2.

We have therefore found the demand functions D0(m,p) and Dj(m,p). Note

that when m = 0, (1) and (3) coincide, and so do (2) and (4). For expository

simplicity, their functional forms are provided in the appendix, where we also show

that they are differentiable on the appropriate domains (see the proof of Lemma 1).

Let K ⊆ N denote a general combination of products, then∑k∈K

(xk − pk)−m is

the utility derived by shopper x if he visits the platform and buys the products in

set K. Thus the maximized aggregate consumer surplus is

V (m,p) ≡ Ex[max{0,∑k∈K

(xk − pk)−m,K ⊆ N,K 6= ∅}]. (5)

By an envelope argument, we necessarily have,24 for any j ∈ N ,

Dj = −∂V∂pj

, and D0 = −∂V∂m

,

and therefore the next result follows. (All omitted proofs are provided in the Ap-

pendix.)

Lemma 1 When n ≥ 2, given (m,p), suppose there exist two sellers j, k ∈ N , suchthat j 6= k, Dj > 0, and Dk > 0. Then we have

i) ∂Dj∂m

= ∂D0∂pj;

ii) When m > 0, ∂Dj∂pk

< 0 (i.e., an access fee creates complementarity);

23Otherwise the consumer would buy another better product and already qualify for the subsidy,so the decision to buy j would not be pivotal for receiving the subsidy.

24Because V (m,p) is essentially the indirect utility function of the “representative”consumerin the market, the equations that follow can also be derived from Roy’s identity with no incomeeffect.

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iii) When m < 0, ∂Dj∂pk

> 0 (i.e., an access subsidy creates substitutability).

Point i) follows from the demand definitions and the Slutsky symmetry of V in

(5). In Figure 1, both ∂D1∂m

and ∂D0∂p1

correspond to the shoppers on segments bc and

cd in the left panel, and those on segment st in the right one.

Points ii) and iii) show that different choices of pricing strategy by the platform

can make the products of any two sellers complements or substitutes on the ag-

gregate level, even though each individual shopper has additive valuations of any

combination of products. The intuition of these results are as follows.

A positive access fee creates complementarity between any two products, because

of marginal multi-seller shoppers who buy both products (with or without other

products) and obtain just enough surplus to justify their visit to the platform (i.e.,

their participation constraints are binding). For these shoppers, a higher price of

either product will dissuade them from visiting the platform altogether, and by doing

so they also forego the other product. In the left panel of Figure 1, ∂D2∂p1

corresponds

to the shoppers on segment bc. As p1 rises, segment bc moves to the right, and the

shoppers originally on bc will stop buying product 2 (as well as product 1) and leave

the platform. Therefore seller 2’s demand decreases.

An access subsidy, on the other hand, induces some marginal shoppers to visit

the platform just to buy from a single seller so that they qualify for the subsidy. It

creates substitutability between any two sellers, because they now compete for those

single-seller shoppers who are lured in by the subsidy but are indifferent between

their products. Therefore a higher price of either product will send such shoppers

straight to the competitor next door. In the right panel of Figure 1, ∂D2∂p1

corresponds

to the shoppers on segment sw. As p1 rises, segment sw moves to the right, and

the shoppers originally on sw will switch from being indifferent to buying product

2 alone. Therefore seller 2’s demand increases.25

However, under a subsidy the complementarity described previously no longer

exists. When the price of a product increases, no multi-seller shoppers who are

indifferent between buying and not buying this product will leave the platform,

because their participation is “cushioned”by the subsidy. They simply stop buying

this more expensive product while keeping whichever other product(s) they have

already bought.

25Whenever indifferent among several subsets of products, we assume that a shopper eitherchooses the largest subset (if any), or randomizes with equal probabilities among all equal-sizedsubsets. More details are provided in the demand analysis from the product bundle perspective inthe online appendix.

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Nor does the substitutability under a subsidy described previously exist under

an access fee. When there is a positive fee to visit the platform, if a shopper is

indifferent between buying two stand-alone products, then buying a bundle of both

will be an even better choice. This is because each shopper only needs to pay the

access fee once. If he is willing to pay this fee for either single product, buying

both is strictly better because he will save one fee. If neither product alone is good

enough to attract him to visit the platform, it is still possible that buying both

will be worthwhile due to a “de facto discount”(equal to the access fee) enjoyed by

multi-seller shoppers. Either way, indifference between any two separate products

will not be a binding constraint on his choice among different product bundles, and

therefore there is no substitutability under an access fee.

3 Analysis of the Pricing Game26

3.1 Optimal seller pricing

Because all sellers and the platform set their prices at the same time, seller j chooses

her optimal price p∗j taking as given the access price of the platformm and the prices

of all other sellers, denoted p−j . Because seller j retains a share (1−βj) of her ownprofit, her optimal price is

p∗j = arg maxpj

(1− βj) · (pj − cj) ·Dj(m, pj,p−j).

The first-order condition gives

Dj(m, p∗j ,p−j) + (p∗j − cj) ·

∂

∂pjDj(m, p

∗j ,p−j) = 0,

which defines seller j’s best response to m and p−j as an implicit function. Denote

εj ≡ −∂Dj

∂pj· pjDj

, seller j’s price elasticity of demand, and

σj ≡ −∂Dj

∂pj· 1

Dj

=εjpj

(> 0), seller j’s “semi-elasticity”of demand.

Then we have the following familiar markup formula.27

26The properties of the best-response functions that are not immediately useful to understandthe equilibrium prices are not discussed in the main text, but provided in the online appendix.

27The term and definition of “semi-elasticity”here follow Rochet and Tirole (2006). In typicalsingle-product demand systems, the semi-elasticity as a function of price can be shown to equal

12

Lemma 2 Given m and p−j, seller j’s optimal price p∗j satisfies

p∗j − cjp∗j

=1

εj, or equivalently, p∗j = cj +

1

σj. (6)

Therefore, each seller marks up on her cost according to her (semi-)elasticity.

Clearly p∗j > cj ≥ 0, for any j ∈ N . Without loss of generality, from now on we

focus on such positive prices set by all sellers.

3.2 Optimal platform pricing

Given sellers’prices p, when the platform takes a βj share of seller j’s profit, and

sets a price m per shopper, its profit is given by

π(m,p) ≡∑j∈N

βj(pj − cj)Dj(m,p) + (m− c)D0(m,p),

and therefore its optimal price m∗ satisfies the following first-order condition:

(m∗ − c) · (−∂D0

∂m)︸ ︷︷ ︸

loss from reduced

demand for platform

+∑j∈N

βj(pj − cj) · (−∂Dj

∂m)︸ ︷︷ ︸

loss from reduced

demand for seller j

= D0.︸︷︷︸gain from

additional fees

(7)

Equation (7) defines the platform’s best response to sellers’prices p as an implicit

function. Denote

η0 ≡ −∂D0

∂m· mD0

, the platform’s price elasticity of demand,

σ0 ≡ −∂D0

∂m· 1

D0

=η0m

(> 0), the platform’s semi-elasticity of demand, and

ηj ≡ −∂D0

∂pj· pjD0

, the platform’s cross-price elasticity of demand with respect to pj.

When the market reaches equilibrium, that is, when both the platform and all

the sellers are setting prices optimally, we add superscript ∗ to the various elastic-ities defined previously to denote their equilibrium values. Therefore, the sellers’

equilibrium prices p∗ ≡ (p∗1, p∗2, ..., p

∗n) and price elasticities {ε∗j}j∈N satisfy (6) in

the hazard rate. In our multi-product setting, σj may be interpreted as the “attrition rate” ofseller j when it raises its price, i.e. the proportion of seller j’s lost demand due to a one-unit pricerise. Similarly, the platform’s semi-elasticity σ0 to be defined later also represents the platform’sattrition rate when it raises the access fee (or lowers a access subsidy) by one unit.

13

Lemma 2. Divide both sides of (7) by D0, and by part i) of Lemma 1 we have:

(m∗ − c) · σ∗0︸ ︷︷ ︸Standard trade-off:

loss from reduced demand for platform

as a fraction of revenue gain

+∑j∈N

βj ·η∗jε∗j︸ ︷︷ ︸

Additional loss-gain ratio due to seller j:

loss from reduced demand for

seller j as a fraction of revenue gain

= 1.

(8)

In equilibrium, when the platform increases the access fee (or lowers an access

subsidy) by one unit, the direct gain in revenue is equal to D0, the total number

of shoppers visiting the platform. The loss comes from reduced demand for the

platform as a whole and for different sellers. The first term on the left-hand side

of condition (8) measures the loss in the platform’s profit due to the decrease in its

own demand, (m∗ − c) · (−∂D0∂m

), as a fraction of the total revenue gain D0. This

loss-gain ratio exactly represents the platform’s standard revenue-demand trade-off.

The second term on the left-hand side of (8) represents the new trade-offs brought

about by profit sharing between sellers and the platform. Because ∂Dj∂m

= ∂D0∂pj, we

can use η∗j - the platform’s cross-price elasticity of demand with respect to pj -

to measure the loss in seller j’s revenue due to the decrease in her demand, as a

fraction of the total revenue gain D0. This loss in revenue is then converted into a

loss in profit by multiplying seller j’s optimal profit margin 1ε∗j, and only a βj share

of this loss is passed through to the platform. Therefore, each βj ·η∗jε∗jmeasures the

platform’s lost profit due to the decrease in seller j’s demand, as a fraction of the

platform’s total revenue gain D0. Clearly, this term only exists because of profit

sharing, and it amplifies the losses that the platform suffers from raising its fee. As

it will emerge repeatedly in our following results, we call it the platform’s additional

loss-gain ratio due to seller j.

Finally, when the original access fee is optimal, the sum of all these loss-gain

ratios should be exactly equal to 1.

The equilibrium prices of the sellers and the platform are given by the system of

n+ 1 simultaneous equations in (6) and (8). By (8) we immediately have:

Proposition 1 (Optimal Access Pricing) Given sellers’ equilibrium prices p∗

and price elasticities of demand {ε∗j}j∈N in (6), the platform’s equilibrium access

14

price m∗ is given by

m∗ = c+1

σ∗0︸︷︷︸ ·Standard

markup

(1−∑j∈N

βj︸︷︷︸Sharing

effect

· η∗j︸︷︷︸Buzz

effect

· 1

ε∗j︸︷︷︸Seller’s own

elasticity

). (9)

In a concise and unified manner, the analytical solution (9) shows that the equi-

librium access price is found by a reduction from the platform’s standard markup,

and the magnitude of this reduction depends on an aggregation of bilateral effects

between the platform and each seller.

Role of each seller in access pricing

1. Sharing effect (βj): Everything else equal, when the platform takes a larger

share of a particular seller’s profit, it will have an additional incentive to lower its

access price. This is because the benefit of the seller’s demand gain, as a result of a

lower access price, will now be passed through to the platform at a higher rate. Thus

the platform’s marginal benefit from a price cut is higher when any βj is larger.

2. Buzz effect (η∗j): When the total number of visitors to the platform is more

responsive to a particular seller’s price adjustment, it means this seller’s pricing

decision is more important in drawing shopper traffi c to the platform. Equivalently,

by ∂Dj∂m

= ∂D0∂pj

we know, a seller with more buzz can translate the same reduction

in access price into a larger increase in shopper traffi c, which in turn raises the

platform’s marginal benefit from a price cut.

Therefore, taking a larger profit share or having a higher cross-price elasticity

vis-a-vis an individual seller has substitutable positive effects on the platform’s in-

centive to lower its access price. Moreover, these two effects amplify each other as

multipliers, and neither one works without the other. That is, if some seller shares

nothing with the platform, or the platform’s cross-price elasticity measured by this

seller’s price is zero (i.e., her price does not affect total traffi c to the platform), then

she does not affect the access price. These findings imply that a small seller (in

terms of demand scale) may well have a large impact on the access price, and vice

versa.

3. Own-price elasticity (ε∗j): A less elastic demand for a particular seller also

favors a lower access price, because it implies that this seller has a higher profit

15

margin in equilibrium (according to Lemma 2). As a result, when the platform

lowers its price, its gain in shared profit from this seller will also be higher.

Alleviation of double marginalization If βj = 0 for all j, such that the

platform receives no shared profits from sellers, then (9) would define exactly the

monopoly pricing formula m∗ = c + 1σ∗0, with a standard markup. Because each

seller also marks her cost up according to (6), there exists “double marginalization”.

(However, here the sellers and the platform all directly charge shoppers by imposing

a markup on their own costs, and thus the situation is different from the conven-

tional case where a retail price is found by applying a markup on top of a wholesale

price.)

Equation (9) shows how each seller contributes individually towards alleviation of

the double marginalization problem in this market, and how they can jointly reverse

it. In the case with only one seller, whose optimal price would be p∗1 = c1+ 1σ∗0(by (6)

and the fact that η∗1 = ε∗1), by Proposition 1 we would have m∗ = c+ 1−β1

σ∗0, such that

a β1 share of the platform’s standard markup1σ∗0is deducted due to profit sharing.

Clearly m∗ decreases in β1 (and in turn also decreases in α1). Therefore, more

profit sharing from the seller (or more bargaining power of the platform) reduces

the platform’s markup and access price. It is also clear that it requires more than

one seller to make m∗ negative. In the next section we find the precise requirement.

4 Access Fee or Subsidy?

4.1 Necessary and suffi cient condition

Equation (9) immediately implies the following criterion.

Proposition 2 (Choice between Access Fee and Subsidy) In equilibrium, theplatform offers an access subsidy (m∗ < 0) if and only if∑

j∈Nβj ·

η∗jε∗j> 1 + cσ∗0. (10)

Intuitively, this condition compares the platform’s aggregated incentive to reduce

access price due to each seller (on the left-hand side), with its incentive to raise price

to gain revenue and save cost (on the right). The sign of its equilibrium price is

completely determined by the equilibrium values of (semi-)elasticities η∗j , ε∗j , σ

∗0, and

16

exogenous parameters c and αj (which determines βj). (10) is more likely to hold

when βj is larger, or when c is smaller, because in either case, the platform cares

more about boosting the sellers’demand so that they make more profits, and worries

less about spending money on attracting more shoppers. Now we derive suffi cient

conditions for an equilibrium access fee and a subsidy, respectively.

4.2 Suffi cient condition for an access fee

The next property will help us simplify (10).

Lemma 3 When n ≥ 2, given (m,p), suppose there exist at least two sellers with

positive demand. Then the platform’s demand is always less elastic than any such

seller’s demand when elasticities are measured by the seller’s price, i.e., for any

j ∈ N such that Dj > 0, we have

ηj < εj.

This property is quite intuitive. When n = 1, by definition we know η1 = ε1.

Whenever the platform serves multiple sellers, however, the change of one seller’s

price will always have a larger impact on that seller’s own demand than on the

platform’s demand. Lemma 3 implies∑j∈N

βjη∗jε∗j<∑j∈N

βj, and the next result follows

immediately from Proposition 2.

Corollary 1 (Equilibrium Access Fee) In equilibrium, the platform charges an

access fee (m∗ > c ≥ 0) if ∑j∈N

βj ≤ 1. (11)

Corollary 1 provides a simple suffi cient condition that depends only on the un-

weighted sum of the profit shares taken from different sellers, or equivalently, on the

exogenous parameters of the platform’s bargaining power (the α’s).

The intuition of (11) is the following. Starting from a situation with no access

fee, because the platform’s demand is always less elastic than that of each seller’s

demand (measured by the seller’s price), when the platform imposes a one-unit

access fee, the profit share βj also represents an upper bound of the platform’s

loss-gain ratio due to seller j (i.e., the second term in (8)), and therefore the total

loss-gain ratio due to all sellers will not exceed the sum of all sellers’shares,∑j∈N

βj.

When∑j∈N

βj ≤ 1, we are sure that the platform will lose less from the access fee

than it gains.

17

It is useful to clarify that the threshold “1” in condition (11) simply balances

the total losses and gains from an access fee, and therefore does not rely on the

normalization of the support of valuations or costs. Moreover, from Proposition

1 and Lemma 3 we know, when (11) holds, the equilibrium access fee is actually

higher than the cost to serve each shopper c, such that the platform directly profits

from the fee. It is also clear that, if n = 1, the platform should definitely charge a

positive fee, as β1 ≤ 1 always holds (this fee should equal c + 1−β1σ∗0

by Proposition

1, as discussed previously).

Note that Corollary 1 requires no information whatsoever about shopper valua-

tions or demand, nor revenue or cost of any seller or the platform. Therefore, Corol-

lary 1 holds for any general joint valuation distribution f that satisfies Assumption

1. In particular, shoppers’valuations of any pair of different sellers’products can

have any degree of positive or negative correlation, or neither (except only for perfect

correlation, which is ruled out by the requirement of a full-dimension support).

Corollary 1 is immediately applicable to real-life multi-seller platforms, and it

suggests very practical advice. The sum of profit shares in (11) should be easy to

calculate by simply checking the platform’s contracts with sellers. If we denote the

average profit share that the platform takes from all sellers as β̄ ≡ 1n

∑j∈N βj, then

an equivalent condition to (11) is β̄ · n ≤ 1. This implies that a platform serving a

small number of sellers and/or taking a small average profit share is more likely to

charge shoppers an access fee.

4.3 Suffi cient condition for an access subsidy

Let λj ≡η∗jε∗j. As mentioned in the discussion of equation (8), if the platform raises

its equilibrium price by one unit, the elasticity ratio λj precisely measures the loss

in seller j’s profit as a fraction of the platform’s revenue gain. Select the seller

whose profit experiences the smallest such loss, and denote her elasticity ratio as

λ̂ ≡ min{λj, j ∈ N}, then we must have∑j∈N

βjλj ≥∑j∈N

βjλ̂. Therefore,∑j∈N

βjλ̂

represents a lower bound on the platform’s total loss in shared profits from all sellers

when it imposes a one-unit access fee, as a fraction of the direct revenue gain D0.

Proposition 2 then immediately implies the following conclusion.

Corollary 2 (Equilibrium Access Subsidy) In equilibrium, the platform offers

18

an access subsidy (m∗ < 0) if ∑j∈N

βj >1

λ̂(1 + cσ∗0).

This result formalizes the intuition that, when the sellers’ combined sharing

effects are suffi ciently strong, an access subsidy becomes optimal. The threshold on

the right-hand side is larger than 1 because λ̂ ≤ η∗jε∗j≤ 1. It incorporates a trade-off

between the platform’s incentive to raise price to gain revenue and save cost (as

represented by 1 + cσ∗0), and its incentive to cut price due to factors beyond the

sharing effect (as represented by λ̂).

Sometimes, even if the platform charges a positive access price, it may not neces-

sarily cover its cost to serve each shopper. If this is the case, the platform essentially

still subsidizes access. The necessary and suffi cient condition for this to happen in

equilibrium follows directly from Proposition 1.

Corollary 3 In equilibrium, the platform directly makes a loss on access price

(m∗ < c) if and only if∑j∈N

βj ·η∗jε∗j> 1.

5 Revenue Sharing

In this section alone, we suppose sellers each share their revenues, instead of prof-

its, with the platform. Denote βRj ∈ (0, 1) the revenue share that the platform

takes from seller j. For expository simplicity, continue to assume that all {βRj }j∈Nare determined prior to the pricing game by the same bargaining power parame-

ters {αj}j∈N and exogenous functions {gj(·)}j∈N as before. Note that the demandfunctions are not affected.

5.1 Optimal seller pricing

Compared to profit sharing, revenue sharing leaves the sellers to bear all their own

costs, and therefore their optimal prices under revenue sharing reflect higher “vir-

tual”marginal costs. To see this, note that seller j retains a share (1 − βRj ) of her

revenue, and given the platform’s access price m and all the other sellers’prices p−j,

her optimal price is given by

pRj ≡ arg maxpj

(1− βRj ) · pj ·Dj(m, pj,p−j)− cj ·Dj(m, pj,p−j)

= arg maxpj

(1− βRj ) · (pj −cj

1− βRj) ·Dj(m, pj,p−j).

19

The first order condition implies the following result.

Lemma 4 Under revenue sharing, given m and p−j, seller j’s optimal price pRjsatisfies

pRj =cj

1− βRj+

1

σj. (12)

Under revenue sharing, each seller reacts as if her marginal cost is raised from

cj tocj

1−βRj, and she continues to follow the familiar markup rule to set her optimal

price, though σj is now evaluated at (m, pRj ,p−j).

Clearly, whenever cj > 0, the virtual marginal cost cj1−βRj

increases in βRj ; as βRj

approaches 1, cj1−βRj

approaches infinity. To focus on the more interesting outcomes

where each seller still makes some profits, we assume cj1−βRj

< 1 for each j ∈ N ,

which implies that pRj ∈ (cj

1−βRj, 1). This further means that βRj should not exceed

1− cj.

5.2 Optimal platform pricing

Under revenue sharing, given all sellers’ prices p, the platform’s profit when it

charges a price m is given by

πR(m,p) ≡∑j∈N

βRj · pj ·Dj(m,p) + (m− c) ·D0(m,p).

Note that all sellers’costs disappear from πR because the platform now only di-

rectly cares about their revenues. By the first order condition, we have the following

conclusion. (All equilibrium values are denoted with superscript R.)

Proposition 3 (Optimal Access Pricing under Revenue Sharing) Under rev-enue sharing, given sellers’ equilibrium prices pR ≡ (pR1 , ..., p

Rn ) in (12), the plat-

form’s equilibrium access price mR is given by

mR = c+1

σR0︸︷︷︸ ·Standard

markup

(1−∑j∈N

βRj︸︷︷︸Sharing

effect

· ηRj︸︷︷︸Buzz

effect

), (13)

where all (semi-)elasticities are evaluated at (mR,pR).

Under revenue sharing, the role that each seller plays in the platform’s pricing

choice is exactly represented by her sharing and buzz effects. The more an individual

20

seller shares revenues with the platform (i.e., larger βRj ), or the more important that

her pricing decision is in bringing shopper traffi c to the platform (i.e., larger ηRj ),

the more incentive that the platform has to lower its access price.

A comparison between Propositions 1 and 3 reveals that, under revenue sharing,

sellers’own-price elasticities (εj) no longer play a role in the platform’s access pricing.

This is because the platform no longer cares about their cost-based profit margins,

which depend on their individual price elasticities.

Proposition 3 immediately implies the following criterion.

Corollary 4 Under revenue sharing, the platform offers an equilibrium access sub-

sidy (mR < 0) if and only if ∑j∈N

βRj ηRj > 1 + cσR0 .

Like (10), this condition compares the platform’s incentive to lower its price (on

the left-hand side), to its incentive to raise it (on the right). We now further simplify

this condition.

Lemma 5 When n ≥ 2, given (m,p), suppose there exist at least two sellers with

positive demand, then for any such seller j, we have

ηj < σ0.

Intuitively, this means that, whenever there are multiple sellers, the percentage

change in the platform’s demand will always be larger if such a change is caused

by the platform raising its own access fee by one unit (inducing a change of σ0),

than when it is caused by any seller raising her individual price by a unit (inducing

a change of ηj/pj > ηj). Lemma 5 therefore echoes the intuition of Lemma 3 that

own-price demand effects are generally stronger than cross-price demand effects in

our multi-seller demand system.

Corollary 5 Under revenue sharing, the platform charges an equilibrium access fee(mR > 0) if ∑

j∈NβRj ≤ c. (14)

Condition (14) intuitively means that the platform’s benefits from reducing the

access fee, due to the combined sharing effects of all sellers, is not suffi cient to cover

the cost it incurs from serving shoppers. Putting (14) together with Lemma 5, we

21

have∑j∈N

βRj ·ηRjσR0

< c, which in turn implies the necessary and suffi cient condition for

a fee in Corollary 4. Therefore, the platform’s incentive to charge a fee dominates

whenever∑j∈N

βRj ≤ c holds.28

Like Corollary 1, Corollary 5 provides a simple condition which is based only

on exogenous parameters of the model and accommodates virtually any joint dis-

tribution of shoppers’valuations. Moreover, it also confirms the intuition that less

sharing from sellers - of either profits or revenues - favors an access fee.

Now select the seller with the weakest buzz effect in equilibrium, and let η̂R ≡min{ηRj , j ∈ N}. Then we have the following suffi cient condition for an equilibriumaccess subsidy.

Corollary 6 Under revenue sharing, the platform offers an equilibrium access sub-

sidy (mR < 0) if ∑j∈N

βRj >1

η̂R(1 + cσR0 ).

The intuition is similar to that of Corollary 2. Again, an equilibrium access

subsidy emerges when the sellers’combined sharing effects are suffi ciently strong.

Such a numerical example is provided in the appendix (after proving Corollary 6).

6 Welfare Maximization

The welfare in this market, as measured by the sum of consumer surplus (5), all

sellers’profits, and the platform’s profit, is given by

W (m,p) ≡ V (m,p)+∑j∈N

(pj − cj)Dj(m,p) + (m− c)D0(m,p).

6.1 Welfare under profit sharing

Suppose temporarily that the platform wants to maximize welfare by access price

mW (e.g., because it is owned by the government). Note that sellers are still maxi-

mizing their own profits. Denote pWj seller j’s optimal price in the current context,

which is derived from (6) given mW . The first-order condition gives the following

result, where equilibrium values are denoted with superscript W .

28Unlike (11), condition (14) does depend on the normalization assumptions on the support ofvaluations and costs, as each seller needs to obtain a non-negative profit under revenue sharing.See the discussion after (12) for more details.

22

Proposition 4 (Welfare under Profit Sharing) The platform’s welfare-maximizingaccess price under profit sharing is given by

mW = c− 1

σW0·∑j∈N

ηWjεWj

, (15)

where (semi-)elasticities are evaluated at (mW ,pW ).

We first note that the welfare-maximizing price marks down on the platform’s

cost to serve each shopper, in order to offset the markups by sellers. This lowers the

final prices that shoppers pay, and therefore increases equilibrium consumption and

welfare. If there is only one seller, whose optimal price would be pW1 = c1+ 1σW0, then

by (15) we would have mW = c− 1σW0, which completely offsets the seller’s markup.

Second, when there are multiple sellers, Proposition 4 shows that the platform’s

socially optimal “markdown”must account for: i) each seller’s buzz effect (ηWj ),

which translates an access price reduction into shopper traffi c generation, and ii)

each seller’s equilibrium price-cost margin ( 1εWj), which converts her revenue gains

from higher shopper traffi c into profit gains. Because any gain in seller profit is an

increase in social surplus, how it is split between a seller and the platform no longer

matters for welfare maximization.

Third, when the platform’s cost c is suffi ciently small, mW will be negative ac-

cording to (15), such that welfare maximization requires that the platform subsidize

shoppers, regardless of the sign of its profit-maximizing access price m∗ in (9).

6.2 Welfare under revenue sharing

Under revenue sharing, denote mWR the welfare-maximizing access price. The wel-

fare function W (m,p) remains the same. However, seller j now sets her optimal

price, denoted pWRj , following (12) instead of (6). Use superscript WR for all equi-

librium values, and we have the following result.

Proposition 5 (Welfare under Revenue Sharing) The platform’s welfare-maximizingaccess price under revenue sharing is given by

mWR = c− 1

σWR0

·∑j∈N

[βRj ηWRj + (1− βRj )

ηWRj

εWRj

],

where (semi-)elasticities are evaluated at (mWR,pWR).

23

Unlike (15) under profit sharing, here we observe that the revenue shares between

sellers and the platform do affect the welfare-maximizing access price. Intuitively,

when βRj is larger, seller j faces a higher virtual cost and thus raises her price, which

reduces equilibrium consumption and welfare. The platform then has more incentive

to subsidize shopper access in order to counter this reduction.

7 Review of Relevant Literature

Our research is mainly related to two streams of literature: (i) two-sided platforms

and markets, and (ii) multiproduct pricing.

First, the recent literature on two-sided markets (e.g., Caillaud and Jullien 2003,

Rochet and Tirole 2003 and 2006, Parker and Van Alstyne 2005, Armstrong 2006,

Hagiu 2009, among others) studies pricing strategies of two-sided platforms (e.g.,

credit cards, video game consoles, media, etc) towards buyers and sellers (as the two

sides), and provides an insightful intuition for the emergence of negative prices driven

by exogenous cross-side network effects. Some of these theories have also mentioned

shopping malls as applications. For instance, Armstrong (2006) shows that, when

buyers and sellers have utility functions that are directly increasing in the number

of participants on the opposite side, the platform may have an incentive to subsidize

the side that is either less dependent on the platform, or imposes a stronger positive

network effect on the opposite side. Either way, network effects imply that expanding

the user base on such a side will make the platform much more attractive to the other

side, and therefore can recoup the subsidy by charging the other side a high price.

Hagiu (2009) shows that, in addition to the previous motivation, a stronger intensity

of buyers’preference for seller variety (i.e., more sellers) shifts a monopoly platform’s

optimal pricing structure towards making a larger share of profits on sellers relative

to buyers. The reason is that sellers become less substitutable when buyers demand

more variety, and are therefore able to extract a larger share of the joint surplus

created by the interaction of the two sides. These rationales crucially depend on

the assumption that the numbers of buyers and sellers can both increase freely, and

each additional participant brings an exogenous positive value to everyone on the

opposite side.

We however take a different perspective and do not focus on the effect when

the number of sellers (or buyers) change. Instead, we model a platform with an

arbitrarily given number of participating sellers.29 This perspective is especially29Sellers’strategic entry and exit decisions are not modeled. The effect of changing the number

24

pertinent for brick-and-mortar malls where the number of sellers are usually fixed,

possibly due to the platform’s capacity constraint (e.g., limited space) or other

contractual constraints (e.g., long contract terms within which the stores do not

change). Once the seller number is fixed, the platform cannot attract more buyers

through attracting more sellers, or vice versa. Therefore, there is no need to invoke

network effects.30 Absent such exogenous effects, we show that a platform may still

choose to subsidize shoppers, driven by the new effects and channels we find under

profit or revenue sharing between sellers and the platform.

Moreover, through modeling the platform’s and all sellers’price setting in one

unified pricing game, our approach reveals the distinctive role that each seller plays

in platform pricing, which is especially relevant to the retail market. The two-

sided market literature, on the other hand, focuses more on explaining the common

equilibrium pricing patterns across different markets, and ignores several defining

features of the shopping applications we study, such as the rich heterogeneity in

sellers’product categories and sharing agreements with the platform, and sometimes

even the sellers’freedom to choose retail prices. Our model incorporates them all

and shows their importance.

Second, our demand modeling techniques draw from the vast literature on mul-

tiproduct pricing. For instance, Calem and Spulber (1984), Long (1984), McAfee,

McMillan and Whinston (1989) and Armstrong (2013), among others, study mixed

bundling (and/or two-part tariffs) of two products, and provide conditions for the

optimality of offering a bundle discount. Bliss (1988) studies the pricing behavior

of a retailer of multiple products, in an environment with shopping cost and retailer

competition. Armstrong (1999) shows that, in quite general circumstances, sim-

ple cost-based multiproduct two-part tariffs can be asymptotically optimal as the

number of products goes to infinity. Manelli and Vincent (2006) find conditions for

optimal mixed bundling of two or more products that are expressed as constraints

on product valuation distributions. Rhodes (2015) studies a monopolist multiprod-

uct retailer’s pricing behavior, when products are symmetric and independent (in

distribution) and consumers must pay a search cost to learn their prices. The main

difference between these works and ours is that, the individual product prices in our

model are chosen by separate profit-maximizing sellers respectively, whereas they

of sellers can be found through a comparative-statics analysis, although that is not our focus.30If shoppers directly prefer more sellers, such that a positive network effect does exist, its

impact on the shoppers’utility would still be a constant when the number of sellers is fixed, andit therefore won’t affect the shoppers’demand for the platform or for the sellers.

25

are all set by an integrated seller in their models.31 The intention and application of

these models are therefore different from ours. In terms of demand characterization

alone, the approach of Manelli and Vincent (2006) is closest to ours. However, we

do not assume independence across product distributions, which is required in much

of their analysis in order to derive desirable properties of the optimal pricing tariff.

In management research, Foros, Hagen and Kind (2009) model a mobile network

as a distribution channel for a fixed number of substitutable content commodities,

where the content providers share profits with the network. The network in their

model does not charge consumers directly, and therefore there is no access fee or

subsidy. Their focus is on how different profit-sharing contracts with the network

affect the content providers’pricing behavior. Our work is complementary to theirs

in the sense that, when the platform can also directly charge or subsidize consumers,

we show how the sharing and other effects also affect the platform’s optimal pricing

strategy towards consumers.

8 Conclusion

Given the close link between the retail market and many consumers’everyday life,

and the trend that it is increasingly shifting towards a platform-based business

format, it is important to understand how shopping platforms charge or subsidize

shoppers. Through modeling the profit maximization problems of a platform and

the multiple third-party sellers that it hosts in one unified multiproduct pricing

game, we derive an analytical solution of the platform’s equilibrium access price,

and provide conditions for it to be either a fee or a subsidy. Some of these conditions

require minimum information about the market, and provide applicable advice for

practitioners.

The key message of this research is that in a platform’s access pricing choice,

every one of the potentially numerous hosted sellers plays a distinctive role, based

on their sharing arrangement with the platform, their ability to attract shopper

traffi c through price adjustments, and their own profitability (under profit sharing).

A minor contribution of our work is introducing a general demand system that

accommodates any number of sellers, without assuming symmetry or independence

across product valuations. We hope that this generality improves the potential

31To be more precise, Armstrong (2013) also allows two sellers to jointly sell bundles of theirproducts, but one of the sellers - instead of a third party - takes the responsibility to set a bundlediscount.

26

applicability of our model and results.

We still have many significant simplifications of the real-life business examples in

order to facilitate a better illustration of our main messages. Firstly, the assumptions

of unit demand for each product per shopper and additive valuations for different

products are invoked to reduce the burden on characterizing demand. In an online

appendix we extend the model to the case of multi-unit demand with non-additive

valuations, without affecting our main findings and their intuition. That appendix

also includes an analysis of multi-tier access prices which implement price discrimi-

nation and can explain discounts or waivers of parking and shipping fees contingent

on order size. Although more complicated analyses are involved, the main insights

and intuition remain similar.

Secondly, we have assumed exogenous and independent bilateral contracting be-

tween the platform and each seller, and taken the platform’s bargaining power rela-

tive to each seller as exogenous. This is a compromise between our wish to generate

general heterogeneity in all sellers’sharing arrangements with the platform, and the

diffi culty in building a model that incorporates in a tractable way the full bargain-

ing game played by all sellers and the platform. Resolving the latter problem would

clearly make a natural and highly desirable generalization.

Thirdly, our results are derived from a monopoly model, and do not address

platform competition. When applicable, it would be interesting to study how com-

petitive effects among different platforms interact with our findings, and what new

insights emerge about equilibrium access pricing strategies.

Besides, it appears that our model and some results could be relevant to some

other industries where platforms also adopt the agency selling model, such as smart

phone app platforms, news/content aggregator platforms, etc.

27

9 Acknowledgement

I acknowledge the financial assistance of the Tsinghua University Initiative Scien-

tific Research Program (Projects 20151080389 and 20205080014), and the Research

Foundation of Humanities and Social Sciences, Ministry of Education of China

(Project No. 18YJC790032).

10 Appendix A - Supplementary material

An online appendix of this article can be found at

http://mis.sem.tsinghua.edu.cn/ueditor/jsp/upload/file/20200505/1588665394168014188.pdf.

11 Appendix B - Proofs

Lemma 1 By (1) through (4), the demand functions are:

Dj(m,p) =

∫ 1pjfj(xj) · Pr[

∑i∈N

max(xi − pi, 0) ≥ m|xj]dxj, if m ≥ 0;

1− Fj(pj) +∫ pjpj+m

fj(xj) · Pr[xj − pj ≥ max(xi − pi,∀i 6= j)|xj]dxj, if m < 0.

(16)

D0(m,p) =

∫ pj0fj(xj) · Pr[

∑i∈N,i6=j

max(xi − pi, 0) ≥ m|xj]dxj +Dj(m,p), if m ≥ 0;

1−∫ p1+m0

∫ p2+m0

· · ·∫ pn+m0

f(x)dxn · · · dx2dx1, if m < 0.

(17)

Assumption 1 implies that D0 and Dj are differentiable with respect to m, pjand pk, for any j, k ∈ N . In particular, we have

limm→0+

∂

∂mDj(m,p) = lim

m→0−∂

∂mDj(m,p) = −fj(pj) · Pr[xi < pi,∀i 6= j|xj = pj], and

limm→0+

∂

∂mD0(m,p) = lim

m→0−∂

∂mD0(m,p) = −

∑j∈N

fj(pj) · Pr[xi < pi, ∀i 6= j|xj = pj].

Note that in the following proof we focus on the domains of (m,p) where

pj, pk, pj +m, pk +m ∈ [0, 1) such that Dj > 0 and Dk > 0.

i) This is implied directly by the symmetry of second-order derivatives of V in

(5).

ii) When m > 0, ∂Dj∂pk

=∫ 1pjfj(xj) · ∂

∂pkPr[∑i∈N

max(xi − pi, 0) ≥ m|xj]dxj. Given

xj, the event∑i∈N

max(xi − pi, 0) ≥ m is affected (in a set-theoretic sense) by pk

28

only through the term max(xk − pk, 0) which clearly decreases in pk. Therefore,∂∂pk

Pr[∑i∈N

max(xi − pi, 0) ≥ m|xj] < 0, which implies ∂Dj∂pk

< 0.

iii) Whenm < 0, ∂Dj∂pk

=∫ pjpj+m

fj(xj)· ∂∂pk Pr[xj − pj ≥ max(xi − pi, ∀i 6= j)|xj]dxj.Given xj, the event xj− pj ≥ max(xi− pi,∀i 6= j) is affected by pk only through the

term max(xi − pi,∀i 6= j) which clearly decreases in pk (because k 6= j). Therefore,∂∂pk

Pr[xj − pj ≥ max(xi − pi,∀i 6= j)|xj] > 0, which implies ∂Dj∂pk

> 0.�

Lemma 2 Apply εj ≡ −∂Dj∂pj· pjDjin the first-order condition to derive (6).�

Proposition 1 Apply σ∗0 =η∗0m∗ in (8) and rearrange to get (9).�

Proposition 2 Implied by (9).�

Lemma 3 Use (16) and (17) from the proof of Lemma 1.

i) Whenm ≥ 0, becauseD0(m,p)−Dj(m,p) =∫ pj0fj(xj)·Pr[

∑i∈N,i6=j

max(xi − pi, 0) ≥ m|xj]dxj,

we have ∂D0∂pj− ∂Dj

∂pj= fj(pj) · Pr[

∑i∈N,i6=j

max(xi − pi, 0) ≥ m|xj = pj] > 0.

ii) When m < 0, we have ∂D0∂pj

= −∫ p−j+m0

f(x−j, pj +m)dx−j, and

∂Dj

∂pj=

∫ pj

pj+m

fj(xj) ·∂

∂pjPr[xj − pj ≥ max(xi − pi, ∀i 6= j)|xj]dxj

−fj(pj) · {1− Pr[0 ≥ max(xi − pi,∀i 6= j)|xj = pj]}−fj(pj +m) · Pr[m ≥ max(xi − pi,∀i 6= j)|xj = pj +m],

where the first term on the right-hand side is negative because

∂∂pj

Pr[xj − pj ≥ max(xi − pi,∀i 6= j)|xj] < 0, the second term is clearly also neg-

ative, and the third term is equal to ∂D0∂pj.

Therefore we have ∂Dj∂pj

< ∂D0∂pj

(< 0) regardless of the sign of m. And because

Dj < D0, we have εj = −∂Dj∂pj· 1Dj

> −∂D0∂pj· 1D0

= ηj.�

Corollary 1 Lemma 3 implies∑j∈N

βjη∗jε∗j<∑j∈N

βj, and (11) follows from Proposi-

tion 2.�

29

Corollary 2 By definition λ̂ ≤ η∗jε∗j,∀j ∈ N , and thus

∑j∈N

βjλ̂ > 1 + cσ∗0 implies the

condition in Proposition 2.�

Corollary 3 This follows directly from (9).�

Lemma 4 This is immediately implied by applying σj = −∂Dj∂pj· 1Djin the first

order condition of seller j’s profit maximization problem under revenue sharing.

Proposition 3 Given pR, the first order condition is

(mR − c) · (−∂D0

∂m) +

∑j∈N

βRj pRj · (−

∂Dj

∂m) = D0.

Divide both sides by D0, and use the definitions of σ0 and ηj, as well as the fact

that ∂Dj∂m

= ∂D0∂pj

from part i) of Lemma 1, we derive the desired condition.

Corollary 4 Implied by Proposition 3.�

Lemma 5 Any seller j with a positive demand must have pj < 1, which implies

ηj <ηjpj

= −∂D0∂pj· 1D0. By (1) through (4), similar to the proof of Lemma 3, we can

prove −∂Dj∂m

< −∂D0∂m. From part i) of Lemma 1 we know −∂D0

∂pj= −∂Dj

∂m, which then

impliesηjpj< −∂D0

∂m· 1D0

= σ0.�

Corollary 5 By Lemma 5 and (14), we have∑j∈N

βRj ηRj <

∑j∈N

βRj σR0 ≤ cσR0 < 1 + cσR0 ,

which then implies mR > 0 by Corollary 4.�

Corollary 6 By definition ηRj ≥ η̂R for any j ∈ N . Therefore∑j∈N

βRj > 1 + cσR0

implies the condition in Corollary 4.�

Numerical Example This is an example of equilibrium access subsidy under

revenue sharing. Let n = 1 and suppose x1 follows the uniform distribution on

[0, 1]. The following table shows the other parameter values and the equilibrium

30

outcomes, where mR < 0. All calculations are done via Scientific WorkPlace 5.

c1 0.7 pR1 0.946 Seller 1 Profit 0.004

c 0.1 mR −0.018 Platform Profit 0.005

βR1 20% ηR1 13.25 1σR1

= 1σR0

0.071

Propositions 4 and 5 They follow directly from the first-order condition∑j∈N

(pj − cj)∂

∂mDj(m,p) + (m− c) ∂

∂mD0(m,p) = 0,

evaluated at either (mW ,pW ) or (mWR,pWR), where seller prices follow either (6)

or (12), respectively.�

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