An Economic Analysis of Product Recommendation

in the Presence of Quality and Taste-Match Heterogeneity

Zhan (Michael) Shi T.S. Raghu

∗

Abstract

This paper investigates the strategy for product recommendation. Speciﬁcally, we analyze a platform-

based market where consumers search and purchase products that potentially differ in quality. In addi-

tion, consumers have idiosyncratic tastes for a product, and the extent of this heterogeneity may vary

from one product to another. In other words, there may be products with low taste dispersion (products

for which there is less heterogeneity among consumers) as well as products with high taste dispersion.

Our modeling framework elucidates how platform recommendation inﬂuences the market-level equilib-

rium outcomes, thereby informing the optimal recommendation strategy. We ﬁnd that the quality and

taste-dispersion dimensions can interact to affect the overall effectiveness of product recommendation

strategies. Conditioning on taste dispersion, recommending high quality products increases both pro-

ducer proﬁts and consumer surplus. Conditioning on quality, recommending high taste-dispersion prod-

ucts may, however, increase or decrease producer proﬁts depending on the joint effect of proﬁt margin

and purchase probability. The direction of change in consumer surplus is also uncertain—recommending

a high taste-dispersion product is more likely to increase (decrease) consumer surplus if the quality is

low (high). Importantly, we show that when the platform cannot discern product types, recommendation

strategies based on observed price or sales signals cannot guarantee the optimal outcome in the general

case.

Keywords: platform recommendation, consumer search, market equilibrium, recommendation strategy,

platform economics

∗

All authors: Department of Information Systems, W.P. Carey School of Business, Arizona State University. Main Campus, PO

BOX 874606, Tempe, AZ 85287, USA. Shi: zmshi@asu.edu. Raghu: raghu.santanam@asu.edu.

1

1 Introduction

Consumers search and evaluate a large number of alternatives in crowded markets to buy the products

that meet their speciﬁc needs. The success of products, especially new products, critically depends on

consumer discovery during the search process (e.g., Goeree 2008, Brynjolfsson et al. 2011). To reduce the

search burden, market platforms deploy product recommendations, which range from price- or sales-based

recommendations to human-curated recommendations. Amazon’s “Best Sellers” page, for instance, features

products with the highest sales in the past hour, while its “Interesting Finds”

1

site recommends product lists

selected by in-house curators. Another example is Apple’s iOS App Store, where the platform publishes

rankings of most-downloaded apps and highlights editor-curated lists such as “App of the Day,” “Game of

the Day,” and featured new apps.

2

The majority of the existing research on platform recommendation is focused on the accuracy of pref-

erence prediction (see survey in Adomavicius and Tuzhilin 2005, Ricci et al. 2011) or the effectiveness of

recommendation at the individual product level (e.g., Senecal and Nantel 2004, Lin 2014). While there are

several papers that have investigated some broader market impacts of platform recommendation (e.g., Fleder

and Hosanagar 2009 on sales diversity and Liang et al. forthcoming on spillover to related products), to the

best of our knowledge, there are no formal analyses of how the platform should select products for recom-

mendation. This is the fundamental gap that motivates our present study. Answering this question requires

an understanding of the equilibrium implications of platform recommendation at the market-level. When

the platform recommends a product, consumers are more likely to discover and purchase the recommended

product. Everything else equal, the products that are not recommended would have less exposure and sales.

Therefore, recommendation can be seen as a platform intervention that shifts (a portion of) consumer search

effort and demand from the rest of the market to the recommended products. Hence, in deciding its recom-

mendation strategy to optimize market-level outcomes, the platform should evaluate the tradeoff between the

potential gains for consumers and producers of the recommended products and the potential losses incurred

on the non-recommended products. This research focuses on analyzing the equilibrium market outcomes

1

See https://techcrunch.com/2016/11/28/amazon-expands-its-online-gift-shop-interesting-ﬁnds-adds-human-curation/.

2

See https://www.macrumors.com/2017/07/21/editorial-app-store-ios-11-beta/. Industry reports and practitioner-oriented publi-

cations suggest that these App Store recommendations require signiﬁcant editorial efforts (https://mashable.com/2017/09/23/inside-

the-new-apple-app-store/#N2aWCf3PPZqR) and have an outsized impact on the recommended apps’ commercial success

(https://sensortower.com/blog/ios-11-featuring-impact). Moreover, a recent change of the App Store reﬂects even greater em-

phasis on platform recommendations through editorial content, which has been observed as a strategic move by Apple to improve

the discovery of quality new apps and the long-term health of the store ecosystem (https://www.storemaven.com/ios-11-app-store-

updates-and-its-impact-on-app-discovery/).

2

and the entailed tradeoffs when the platform recommends different types of products. Speciﬁcally, we exam-

ine the following salient questions: (1) How will platform recommendation impact equilibrium consumer

surplus and producer proﬁts? (2) What type of products should the platform select for recommendation

under different objectives? (3) Are the commonly used recommendation strategies based on price and sales

optimal in terms of platform-level outcomes?

We develop a model of platform-based market to systematically analyze the impact of platform rec-

ommendation on market equilibrium. Speciﬁcally, we consider a heterogeneous product market where the

utility for the product is composed of a quality dimension that all consumers agree upon and a taste-match

dimension that varies across consumers (following the product differentiation literature, we henceforth use

the terms vertical and horizontal dimensions for the two components). Consumers have incomplete informa-

tion about the products, and they incur a search cost to learn the consumption utility and price for any given

product. During the search process, consumers sample products sequentially and adopt a rational stopping

rule. We progressively explore a set of scenarios where products differ in quality and/or taste match. In

each scenario, we ﬁrst consider the benchmark case where consumers sample products in a random order,

and derive the equilibrium condition. We then assume that platform recommendation exogenously alters the

consumer search sequence by placing the recommended products at the top of the list, and compare how

producer proﬁts and consumer surplus would change relative to the benchmark case when different types of

products are selected for recommendation. The analysis thus yields the theoretically optimal recommenda-

tion strategy for the platform, provided it is able to discern product heterogeneity in quality and taste match.

Using the results on optimal product selection as the basis for comparison, we then examine whether relying

on the observed price or sales signals can help the platform to achieve the optimum when it cannot determine

product types.

The main ﬁndings are as follows. First, if there is no systematic difference between products, platform

recommendation has no aggregate effect on total producer proﬁts and consumer surplus. Second, if prod-

ucts differ in their quality (i.e., when conditioning on taste dispersion), then recommending high quality

products can increase both producer proﬁts and consumer surplus. Third, if products differ in the disper-

sion of consumer taste match (i.e., when conditioning on quality), then recommending high taste-dispersion

products may increase or decrease producer proﬁts, depending on the joint effect of proﬁt margin and pur-

chase probability. The direction of change in consumer surplus is also uncertain—recommending a high

taste-dispersion product is more likely to increase (decrease) consumer surplus if the product quality is low

3

(high). Remarkably, recommendation strategies based on price or sales will not guarantee optimum in the

general case where heterogeneity exists in both the vertical and horizontal dimensions.

This research joins the literature that studies the market implications of platform recommendation.

Fleder and Hosanagar (2009) modeled how popularity-based platform recommendation would inﬂuence

consumer choice at the micro level and used simulation to demonstrate that the effect of recommendation

on product variety at the individual level and that at the platform level can be different. Based on the view-

rank data of the camcorder section on Amazon, Kim et al. (2010)’s numerical results suggested almost all

consumers beneﬁted from the platform’s collaborative-ﬁltering-based product recommendation. Recently,

Liang et al. (forthcoming) examined the spillover effects of platform-provided editorial recommendation

on related products. While these papers examined various aspects of the market implications of platform

recommendation, they did not focus on studying how the platform should choose products for recommenda-

tion. Our paper contributes to this emerging literature by providing a systematic analysis on the equilibrium

implications of platform recommendation. Our model yields predictions on what types of products should

be selected for optimizing speciﬁc market outcomes, and the analysis allows us to compare the different

implications of product type- and popularity-based recommendation strategies. Our analytical framework

draws upon the important prior work that models consumer search (e.g., Weitzman 1979, Perloff and Salop

1985, Wolinsky 1986, Anderson and Renault 1999, Armstrong et al. 2009). On top of the consumer-search

framework, we analyze the market implications of platform recommendation. By explicitly considering

product heterogeneity in the vertical and horizontal dimensions, we show that they have very different im-

plications on optimal platform recommendation strategy and demonstrate that optimality cannot always be

guaranteed using observed price and sales signals.

2 Basic Model Setup

In this section, we introduce our modeling framework. Our setup builds on the consumer search models of

Wolinsky (1986) and Anderson and Renault (1999). Using the basic consumer-search framework, we will

then in the following sections introduce platform recommendation into the model, analyze how it changes

the market equilibrium, and discuss the strategic implications.

The basic model setup is as follows. Suppose u

i j

is the utility that consumer i would get from consuming

4

product j, and it is composed of two parts:

u

i j

= v

j

+ h

i j

, (1)

where v

j

is product j’s quality that is valued in common by all consumers, and h

i j

, independently dis-

tributed across consumer-product pairs, captures the match between product j’s design and consumer i’s

idiosyncratic taste.

3

At price p

j

for product j, consumer i’s net utility of choosing j is u

i j

− p

j

. The range

of v

j

measures vertical heterogeneity across different products. For a particular j, the variance of h

i j

over i

measures the dispersion of product j’s consumer taste match (in other words, to what extent different con-

sumers’ preferences to product j agree with each other). Thus, as illustrated in Figure 1, the distribution of

consumer valuation of a product in the population is characterized by both its quality and the dispersion of its

consumer taste match. These two factors will determine the products types when we discuss differentiated

products in the following sections.

consumption

utility u

ij

range ofh

ij

captures

dispersion of taste match

v

j

quality

u

ij

=v

j

+h

ij

h

ij

Notes: The utility consumer i gets from consuming product j is composed of two parts: v

j

the quality of product j that is commonly

valued by all consumers and h

i j

the idiosyncratic taste match. The thick vertical line represents the range of consumption utility in

the consumer population. Quality v

j

determines the mean consumption utility for the product, and the range of possible h

i j

values

determines the dispersion of taste match, which is also the dispersion of consumption utility for the product.

Figure 1: Illustration of Product Quality and Taste Match

The product space on most market platforms is very large. For simplicity, we assume there is an inﬁnite

number of products, and each consumer is looking to purchase at most one product. Additionally, consumers

must incur a search cost of s to discover the consumption utility and price for any given product available

in the market. The search cost in the model encompasses the total time and cognitive cost incurred by

3

Without loss of generality, h

i j

is assumed to have a mean of zero over i since any nonzero mean can be absorbed into v

j

as part

of the quality.

5

consumers in ﬁnding and evaluating a product. Consumers are assumed to sample products sequentially: At

any given time, consumers can either purchase a product they have already learned about, continue searching

by sampling another product (and paying the search cost), or leave the market without purchase.

4

The model timing is as follows. The producers set the price given product quality. Then, consumers

search products to purchase. We assume consumers know the distributions of v

j

and h

i j

in the market but

not the speciﬁc v

j

and h

i j

for any product j.

5

Producers maximize their expected proﬁt, and consumers

maximize their expected surplus. Without loss of generality, we further assume the number of consumers is

one,

6

and the marginal cost of producing one unit of product is the same for all products which we normalize

to zero.

In Sections 3 to 5, we derive the model equilibrium and analyze the product-recommendation strategies

under three different assumptions regarding product heterogeneity in the market. We start from the setting

where the products are ex ante homogeneous in the sense that they have the same quality and taste-match

dispersion (Section 3), and then gradually increase the complexity to allow for product differentiation, ﬁrst

only in quality (Section 4) and then in both quality and taste-match dispersion (Section 5). Figure 2 shows

the roadmap. For each setting, we ﬁrst analyze the case where consumers sample products in a random order.

The equilibrium condition of this case serves as the benchmark for comparison. We subsequently consider

scenarios where the consumer search sequence is exogenously changed by different platform recommenda-

tion strategies, including product type-based recommendation and sales- or price-based recommendation,

and examine the resulting implications on market equilibrium. We particularly focus on characterizing the

effects on equilibrium producer proﬁts and consumer surplus. Producer proﬁt is deﬁned as producer rev-

enue minus production cost, and consumer surplus is deﬁned as consumption utility net of buying price and

search cost. Producer proﬁts and consumer surplus are the key equilibrium quantities of interest because, as

discussed in the literature on two-sided markets (e.g., Rochet and Tirole 2003, Armstrong 2006), they are

most likely to be the source of revenue for the platform. For instance, some commonly adopted business

models for market platforms include (1) charging producers per-sale royalties (e.g., Apple iOS App Store

and Google Play Store), (2) charging consumers a membership fee (e.g., Netﬂix), and (3) accruing direct

4

Sequential search is an assumption widely adopted in both theoretical search models (e.g., Anderson and Renault 1999) and

empirical studies of online settings (e.g., Kim et al. 2010). For more general search rules, refer to Morgan and Manning (1985).

5

That consumers know the distributions of v

j

and h

i j

can be justiﬁed by making the assumption that consumers search the market

repeatedly and they learn the distributions over time. Alternatively, we can assume consumers have beliefs about the distributions

and they behave according to their beliefs; as part of the equilibrium condition, the beliefs are required to be consistent with the

true distributions.

6

Thus, the probability that the consumer purchases a particular product can be interpreted as the market share of the product.

6

revenues from hardware sales (e.g., Apple iOS devices and Microsoft Xbox). Each of the business models

can be considered as extracting a fraction of the total producer proﬁts (1) or consumer surplus (2 and 3). For

ease of reference, we summarize the important notations in Table 1.

Setting 1:

Ex ante

homogeneous

products

(one type of product)

Setting 2:

Products differentiated

only

in quality

(two types of products)

Setting 3:

Products differentiated in

both quality and taste-

match dispersion

(four types of products)

Quality

Taste match

dispersion

Quality

Taste match

dispersion

Taste match

dispersion

Quality

Figure 2: Roadmap of Analyses in Sections 3, 4 and 5

Table 1: Notations

Model primitives

v

j

The quality of product j

h

i j

The idiosyncratic taste match of consumer i for product j

u

i j

(= v

j

+ h

i j

) The consumption utility of product j for consumer i

p

j

The price of product j

s The search cost

Distributions of quality and taste match

v The common quality in Setting 1

v, v The low and high quality in Settings 2 and 3

t The common taste dispersion in Setting 1

t, t The low and high taste dispersion in Setting 3

Equilibrium quantities

p

∗

()

Equilibrium price, subindex indicating product type

q

∗

()

Equilibrium conditional probability of purchase, subindex indicating product type

π

∗

, π

0∗

, π

00∗

Expected total industry (producer) proﬁts in Settings 1, 2 and 3 respectively

y

∗

, y

0∗

, y

00∗

Expected consumer surplus in Settings 1, 2 and 3 respectively

7

3 Setting 1: Ex Ante Homogeneous Products

We start by considering the simplest case where all products in the market are ex ante homogeneous from the

consumer’s perspective. We use this case to illustrate the derivation of market equilibrium with and without

platform recommendation. Speciﬁcally, we assume the vertical dimension (product quality, v

j

) is identical

for all products, and the horizontal dimension (idiosyncratic taste match, h

i j

) follows a uniform distribution.

As such, the products are ex ante homogeneous in the sense that the consumption utilities provided by

different products all have identical mean and variance. Note that the products are still heterogeneous ex

post since the consumer will draw different h

i j

values for different j. Formally, we introduce the following

assumption on the distributions of v

j

and h

i j

in the market.

Assumption (P1). For all j, v

j

= v ≥ 0, and h

i j

∼ Uniform[−t,t], t > 0.

3.1 Market Equilibrium without Recommendation under Setting 1

When there is no platform recommendation, we assume that the consumer searches for products in a com-

pletely random order. The equilibrium condition derived here will serve as the benchmark for evaluating the

effects of platform recommendation.

Assumption (C1 - without recommendation). The consumer samples products in a random order.

Under assumptions P1 and C1, a symmetric market equilibrium results where all producers charge the

same price. In such an equilibrium, the consumer faces a stationary environment, i.e., at each time point

during the search process the utility distribution of un-sampled products stays the same. The search theory

has established the following optimal stopping rule for consumer search in a stationary environment (which

also holds in the other settings we will consider later). The optimal stopping rule is given by a threshold

such that the consumer buys the ﬁrst product she ﬁnds where the net utility (consumption utility minus

price) exceeds the threshold (e.g., see MacQueen and Miller Jr 1960, McCall 1970, and Weitzman 1979).

The threshold can be interpreted as the consumer’s reservation utility. Using the result on optimal search

behavior, we establish the existence of market equilibrium and characterize the conditions that determine

the key equilibrium quantities in the following proposition.

Proposition 1. Under Assumptions P1 and C1, if the search cost is moderate (to ensure the consumer

participates in the market so there is positive demand), there exists a symmetric equilibrium characterized

8

by product price p

∗

and consumer reservation utility y

∗

≥ 0 such that

(1) given consumer reservation utility y

∗

, p

∗

maximizes each producer’s expected proﬁt, and

(2) given prices p

∗

, the search stopping rule given by reservation utility y

∗

maximizes the consumer’s

expected surplus. Speciﬁcally, p

∗

and y

∗

satisfy:

p

∗

= argmax

p

j

p

j

q

j

= argmax

p

j

p

j

Z

t

−t

I(v + h − p

j

− y

∗

≥ 0)

1

2t

dh, (2)

Z

t

−t

max(0,v + h − p

∗

− y

∗

)

1

2t

dh = s. (3)

Proof. All proofs are in the online appendix.

Equation 2 is the pricing equation and it sets the proﬁt-maximization condition for producer j. In the

equation, I() is the indicator function, and the integral gives the probability that the consumer buys product

j conditional on discovering the product, which per the optimal stopping rule is the probability that the net

utility, v + h − p

j

, is larger than the reservation utility, y

∗

. Note that when setting the price, each producer

takes the prices of other products as given. As a result, a product’s own price does not affect the probability

that the consumer ﬁnds the product. Therefore, the expected demand of product j is the integral in equation

2 multiplied by a constant. Equation 3 states the reservation utility the consumer holds in her stopping rule

is optimal. Intuitively, if the consumer has already found a product that gives her net utility y

∗

, then the

left hand side of equation 3 gives the expected gain (over y

∗

) to the consumer by sampling another product.

Since the right hand side of equation 3, s, is the cost of sampling another product, the two sides being equal

indicates the consumer is just indifferent between continuing searching and stopping. In other words, the

consumer should stop searching if she has found a product that provides a utility greater than y

∗

(hence, y

∗

is

the reservation utility). Note that y

∗

is also the expected consumer surplus, so y

∗

≥ 0 has to be met to ensure

the consumer participates in search at all. For a nonnegative y

∗

solution to exist for equations 2 and 3, the

search cost s cannot be too large. For instance, if the search cost is so large that it exceeds the best possible

consumption utility, then it is not rational for the consumer to search at all.

When a nonnegative y

∗

solution does exist, the ﬁrst order condition of equation 2 gives the equilibrium

price

p

∗

=

v +t − y

∗

2

. (4)

9

Ex ante, the expected total sales is 1 and industry proﬁt is p

∗

because with inﬁnite products, the consumer

will ﬁnd a match almost surely. But the expected proﬁt for each producer individually is zero because the

probability of being chosen is negligible. With zero marginal cost, the producer of the product that the

consumer purchases earns a proﬁt of p

∗

, which is also the proﬁt of the industry as a whole. The expected

consumer surplus is just the reservation utility y

∗

. We summarize the equilibrium quantities in the following

corollary.

Corollary 1. In the equilibrium deﬁned in Proposition 1, the expected industry proﬁt π

∗

= p

∗

, and the

expected consumer surplus is y

∗

.

3.2 Platform Recommendation under Setting 1

Now consider that the market platform intervenes in the search process by recommending an ordered list of l

products, which we index by j ∈ {1,2,...,l}. With the intervention in place, the consumer is more likely to

discover the recommended products. However, if the consumer does not purchase any of the recommended

products in the ordered list, she would have to continue sampling the remaining non-recommended products.

We therefore assume the consumer ﬁrst follows the platform recommendation by sequentially sampling from

product 1 to product l, and after that she searches the remaining products in a completely random fashion.

Assumption (C2 - with recommendation). The consumer ﬁrst samples the recommended products in the

predetermined order, and then randomly samples the remaining products.

We claim that in the new equilibrium, all producers, including those of the recommended products, still

charge the price p

∗

and the consumer still adopts the stopping rule that is characterized by reservation utility

y

∗

, where p

∗

and y

∗

are the same as given in Proposition 1.

Proposition 2. Under Assumptions P1 and C2, the product price p

∗

and consumer reservation utility y

∗

deﬁned by equations 2 and 3 still constitute a market equilibrium.

The equilibrium result follows by backward induction. First, note that after searching the l recommended

products without purchase, the market functions exactly as before so the producers of the non-recommended

products charge p

∗

and the consumer expects a surplus of y

∗

. Anticipating the remaining products charging

p

∗

, equation 3 shows the consumer will hold reservation utility y

∗

if she is currently considering the last

recommended product, l. Given the expected reservation utility, equation 2 still characterizes producer l’s

10

proﬁt-maximization problem so p

∗

l

= p

∗

. Expecting producer l will be charging p

∗

, the consumer will again

hold the reservation utility y

∗

at product l − 1 (equation 3).

Since products are ex ante homogeneous and the equilibrium prices and reservation utility are the same

as in the case of random search, the expected consumer surplus does not change with platform recommen-

dation (i.e. y

∗

). The expected total producer proﬁt is still π

∗

= p

∗

, which is ex post earned by the product

that is actually chosen.

Corollary 2. Under Assumption P1, platform recommendation does not change the expected industry proﬁt

and expected consumer surplus.

In contrast to the case of random search, the expected proﬁt does vary between producers due to platform

recommendation. In the following, we show how recommendation shifts proﬁts between producers (though

it does not alter industry proﬁt as a whole). Let q

∗

be the probability that the consumer buys a product

conditioning on discovering it (the integral in equation 2 evaluated at p

j

= p

∗

):

q

∗

=

Z

t

−t

I(v + h − p

∗

− y

∗

≥ 0)

1

2t

dh =

v +t − y

∗

4t

. (5)

Since the consumer will search the ﬁrst product in the recommendation list with certainty, the expected sales

for the product at the top of the list is q

∗

and the expected proﬁt is p

∗

q

∗

. The consumer will continue to

search the second product in the list if she does not purchase the ﬁrst product, so the expected sales for the

second product in the recommendation list is (1 − q

∗

)q

∗

and the expected proﬁt is p

∗

(1 − q

∗

)q

∗

. In general,

for the jth product, j ≤ l, the expected sales is (1 −q

∗

)

j−1

q

∗

and the expected proﬁt is p

∗

(1 −q

∗

)

j−1

q

∗

. The

expected total sales of the recommended products is 1 − (1−q

∗

)

l

, and that of the remaining products is (1−

q

∗

)

l

. For the non-recommended products, the expected sales and proﬁt are individually negligible. In other

words, the recommended products have larger sales and proﬁt than the non-recommended products. Among

the recommended products, the expected sales and proﬁt decrease exponentially in the recommendation

sequence.

In summary, for ex-ante homogenous products, no matter whether the platform maximizes total pro-

ducer proﬁts or consumer surplus, it would be indifferent between using and not using recommendation.

Recommendation would only shift proﬁts between producers: The recommended products beneﬁt from the

preferential exposure (which leads to higher sales), but the gain to the recommended products equals the loss

to the non-recommended products. Since the products do not have any systematic difference, on expectation

11

the consumer would not beneﬁt either. When recommendation does exist, the recommended products would

expect larger proﬁts. As such, producers have an incentive to compete for recommendation and they will

want their products to be listed high up in the recommendation sequence.

4 Setting 2: Products with Heterogeneous Quality and Identical Taste Dis-

persion

In this section, we allow products to have different quality (i.e., heterogenous in vertical dimension). For

ease of exposition, we assume that a half of the products is of low quality and the other half is of high

quality.

7

Formally, we replace Assumption P1 with Assumption P2 below.

Assumption (P2). For all j,

v

j

=

v, prob = 1/2

v, prob = 1/2

,

where 0 < v < v, and h

i j

∼ Uniform[−t,t], t > 0.

Similar to the approach in Setting 1, we ﬁrst characterize the market equilibrium under random search

and then analyze the implications of product recommendation.

4.1 Market Equilibrium without Recommendation under Setting 2

In the symmetric equilibrium under random search (Assumption C1), all producers with v

j

= v charge p

∗

v

,

all producers with v

j

= v charge p

∗

v

, and the consumer adopts a stopping rule with reservation utility y

0∗

.

Formally, we characterize the equilibrium conditions in the following proposition.

Proposition 3. Under Assumptions P2 and C1, if the search cost is moderate (to ensure the consumer partic-

ipates in the market and both types of products have positive demand), there exists a symmetric equilibrium

characterized by product prices (p

∗

v

, p

∗

v

) and consumer reservation utility y

0∗

≥ 0 such that

(1) given consumer reservation utility y

0∗

, p

∗

v

and p

∗

v

maximize expected proﬁts for type v and type v

producers respectively, and

7

Our results will not change qualitatively if we make the distribution continuous.

12

(2) given prices (p

∗

v

, p

∗

v

), the search stopping rule given by reservation utility y

0∗

maximizes the con-

sumer’s expected surplus. Speciﬁcally, (p

∗

v

, p

∗

v

) and y

0∗

satisfy:

p

∗

v

j

= argmax

p

j

p

j

q

j

= argmax

p

j

p

j

Z

t

−t

I(v

j

+ h − p

j

− y

0∗

≥ 0)

1

2t

dh, v

j

∈ {v, v}, (6)

1

2

Z

t

−t

max(0,v + h − p

∗

v

− y

0∗

)

1

2t

dh +

1

2

Z

t

−t

max(0,v + h − p

∗

v

− y

0∗

)

1

2t

dh = s. (7)

Similar to equation 2, equation 6 characterizes producer j’s proﬁt-maximization problem, given its prod-

uct quality v

j

and consumer reservation utility y

0∗

. Equation 7 states that y

0∗

is the optimal reservation utility

given p

∗

v

for low quality products and p

∗

v

for high quality products in the market. The expected increase

in utility from discovering one more product in this setting is the average of the expected gain from a low

quality product and that from a high quality product respectively. Under random search, the two product

types are encountered with equal probability.

The ﬁrst order condition gives

8

p

∗

v

=

v +t − y

0∗

2

, (8)

p

∗

v

=

v +t − y

0∗

2

. (9)

p

∗

v

> p

∗

v

, i.e., high quality products charge a higher price than low quality products. The realized industry

proﬁt is p

∗

v

or p

∗

v

, depending on whether the consumer buys a low quality or high quality product. The

expected proﬁt for each individual producer is negligible. To calculate the expected industry proﬁt, note that

the probability (conditional on discovery) of purchasing a low quality product and that for a high quality

product are

q

∗

v

=

v +t − y

0∗

4t

, (10)

q

∗

v

=

v +t − y

0∗

4t

(11)

respectively. Hence, the expected sales of low quality products is

q

∗

v

q

∗

v

+q

∗

v

and that for high quality products

is

q

∗

v

q

∗

v

+q

∗

v

. The expected total industry proﬁt is

q

∗

v

p

∗

v

+q

∗

v

p

∗

v

q

∗

v

+q

∗

v

. We summarize the equilibrium quantities in the

8

Note that if the reservation utility is higher than a threshold, i.e., v + t < y

0∗

, then the conditional probability of choosing a

low quality product is always zero regardless of its price. Only one kind of products—the high quality ones—would sell in such a

market. The analysis would reduce to the homogeneous-product case. To avoid discussing such corner solutions, we focus on the

scenario where both types of products have positive demand—the reservation utility is not too large, meaning the search cost is not

too low.

13

following corollary.

Corollary 3. In the equilibrium deﬁned in Proposition 3, the expected industry proﬁt π

0∗

=

q

∗

v

p

∗

v

+q

∗

v

p

∗

v

q

∗

v

+q

∗

v

, p

∗

v

<

π

0∗

< p

∗

v

, and the expected consumer surplus is y

0∗

.

4.2 Platform Recommendation under Setting 2

As in Setting 1, the platform recommends an ordered list of l products, and the consumer’s behavior changes

to ﬁrst search the recommended products in the determined order and then sample the remaining products

randomly (Assumption C2). We further assume that the consumer does not know which types of products

are being recommended. Following the same backward-induction logic as in the analysis of homogeneous

products, the recommended products will still charge p

∗

v

or p

∗

v

in the new equilibrium, depending only on

their quality. The consumer will still use a reservation utility of y

0∗

.

To analyze the impact of recommendation on producer proﬁts, suppose the platform can discern the

quality difference between products. The platform can increase the expected total producer proﬁts by rec-

ommending high quality products. To illustrate the idea, let l = 1. The consumer samples the recommended

product ﬁrst. If she buys the product, which happens with probability q

∗

v

or q

∗

v

, the proﬁt earned is p

∗

v

or

p

∗

v

, depending on whether the recommended product is of low or high quality. If the consumer is unsat-

isﬁed with the ﬁrst product, which happens with probability 1 − q

∗

v

or 1 − q

∗

v

, she then searches the other

products randomly so the expected industry proﬁt from the second product onward is π

0∗

(Corollary 3).

Thus, the new expected industry proﬁt when a low quality product is recommended is q

∗

v

p

∗

v

+ (1 − q

∗

v

)π

0∗

and q

∗

v

p

∗

v

+ (1 − q

∗

v

)π

0∗

when a high quality product is recommended. Observe that q

∗

v

p

∗

v

(or q

∗

v

p

∗

v

) is the

expected proﬁt gain on the recommended product, and −q

∗

v

π

0∗

(or −q

∗

v

π

0∗

) is the expected proﬁt loss on the

non-recommended products. Since p

∗

v

< π

0∗

< p

∗

v

and q

∗

v

< q

∗

v

, the expected gain outweighs the expected

loss when the recommended product is of high quality, and the reverse is true when the recommended prod-

uct is of low quality. More generally, if the platform recommends l high quality products, the expected

industry proﬁt becomes

q

∗

v

p

∗

v

+ q

∗

v

p

∗

v

(1 −q

∗

v

) +.. . + q

∗

v

p

∗

v

(1 −q

∗

v

)

l−1

+ (1 − q

∗

v

)

l

π

0∗

= (1 −(1 − q

∗

v

)

l

)p

∗

v

+ (1 − q

∗

v

)

l

π

0∗

,

which approaches the best possible industry proﬁt p

∗

v

as l → ∞—when a large number of high quality

14

products are recommended, the consumer will have to reject all of them before she encounters a low quality

product so she will almost surely purchase a high quality product. Therefore, it is in the platform’s best

interest to recommend high quality products if the goal is to maximize the total producer proﬁts.

To explain the impact on expected consumer surplus, we discuss the basic intuition using the case when

only a single product is recommended (l = 1). Under the equilibrium stopping rule, the consumer’s expected

surplus is the sum of three parts: the search cost incurred for sampling the recommended product (−s), the

expected surplus gain from the recommended product, and the option value of continued searching (the

expected surplus from randomly searching the remaining products, i.e., y

0∗

). Since the search cost and

option value do not depend on the recommended product, we determine whether recommending a high

(low) quality product will increase or decrease consumer surplus by comparing the expected utility gain from

ﬁnding a high quality product with that from ﬁnding a low quality product, i.e., the two integrals in equation

7. We ﬁnd that the expected net utility gain from ﬁnding a high quality product is larger than that from

ﬁnding a low quality product (the calculation is provided in the online appendix). Therefore, recommending

high quality products can increase the expected consumer surplus. We summarize the impacts of platform

recommendation on expected producer proﬁts and consumer surplus in the proposition below.

Proposition 4. Under Assumption P2, recommending a type v product generates a larger expected industry

proﬁt and a larger expected consumer surplus than recommending a type v product.

When conditioning on taste dispersion, i.e., considering products that are heterogeneous only in the

vertical quality dimension, the theoretically optimal recommendation strategy for the platform is clear and

also intuitive from Proposition 4—it should always recommend high quality products, regardless of whether

the platform’s incentive is to maximize total producer proﬁts or consumer surplus.

If the platform is unable to distinguish between high and low quality products, then the platform may

still use observed market signals for selecting recommended products. Two strategies commonly employed

by platforms are based on price and past sales respectively. Under the current setting, we know high quality

products will charge a higher price than low quality products (p

∗

v

> p

∗

v

, see equations 8 and 9) and high

quality products are also expected to have larger sales than low quality products (q

∗

v

> q

∗

v

, see equations 10

and 11). Thus, even if the platform does not observe product quality directly, it can still leverage the price

and sales signals to identify high quality products. As such, the platform can increase expected producer

proﬁts and consumer surplus by recommending high price products and/or high sales products. As we

15

will show in the next section, this convenient result will not hold when products differ in the horizontal

dimension.

5 Setting 3: Products with Heterogeneous Quality and Taste Dispersion

In this section, we relax assumption P2 by assuming the products differ not only in the vertical dimension

but also in the horizontal dimension.

Assumption (P3). For all j,

v

j

=

v, prob = 1/2

v, prob = 1/2

,

where 0 < v < v, and

h

i j

∼ Uniform[−t

j

,t

j

], t

j

=

t, prob = 1/2

t, prob = 1/2

,

where 0 < t < t. The draw of t

j

is independent of that of v

j

.

Under Assumption P3, there are four product types: (v

, t), (v, t), (v, t), and (v, t). Each type accounts

for one quarter of all products in the market.

9

The difference in the horizontal dimension (t versus t) does

not affect the mean consumption utility (determined by v

j

) but measures the dispersion of the taste match

distribution. A product with t

j

= t has a larger consumer taste dispersion than a product with t

j

= t. A simple

interpretation is that a product with t

j

= t has a more radical design than a product with t

j

= t—the consumer

either likes the high taste-dispersion product a lot or hates it. When producers set price, they know their own

product’s type. The consumer knows the distributions of v

j

and t

j

in the market but not any speciﬁc v

j

or t

j

value.

5.1 Market Equilibrium without Recommendation under Setting 3

In the symmetric equilibrium under random search (Assumption C1), producers of the same type charge the

same price, and the consumer adopts a stopping rule with reservation utility y

00∗

. Formally, we characterize

the equilibrium conditions in the following proposition.

9

The results will not change qualitatively if we assume a continuous distribution for t

j

.

16

Proposition 5. Under Assumptions P3 and C1, if the search cost is moderate (to ensure the consumer

participates in the market and all four types of products have positive demand), there exists a symmetric

equilibrium characterized by product prices (p

∗

v,t

, p

∗

v,t

, p

∗

v,t

, p

∗

v,t

) and consumer reservation utility y

00∗

≥ 0

such that

(1) given consumer reservation utility y

00∗

, p

∗

v,t

, p

∗

v,t

, p

∗

v,t

, and p

∗

v,t

respectively maximize expected proﬁts

for products of type (v, t), (v, t), (v, t), and (v, t) , and

(2) given prices (p

∗

v,t

, p

∗

v,t

, p

∗

v,t

, p

∗

v,t

), the search stopping rule given by the reservation utility y

00∗

maxi-

mizes the consumer’s expected surplus. Speciﬁcally, (p

∗

v,t

, p

∗

v,t

, p

∗

v,t

, p

∗

v,t

) and y

00∗

satisfy:

p

∗

v

j

,t

j

= argmax

p

j

p

j

q

j

= argmax

p

j

p

j

Z

t

j

−t

j

I(v

j

+ h − p

j

− y

00∗

≥ 0)

1

2t

j

dh, v

j

∈ {v, v} t

j

∈ {t,t}, (12)

1

4

Z

t

−t

max(0,v + h − p

∗

v,t

− y

00∗

)

1

2t

dh +

1

4

Z

t

−t

max(0,v + h − p

∗

v,t

− y

00∗

)

1

2t

dh +

1

4

Z

t

−t

max(0,v + h − p

∗

v,t

− y

00∗

)

1

2t

dh +

1

4

Z

t

−t

max(0,v + h − p

∗

v,t

− y

00∗

)

1

2t

dh = s. (13)

Similar to equations 2 and 6, given type (v

j

, t

j

) and consumer reservation utility y

00∗

, equation 12 char-

acterizes producer j’s proﬁt-maximization problem. The ﬁrst order condition gives

10

p

∗

v,t

=

v +t − y

00∗

2

, p

∗

v,t

=

v +t − y

00∗

2

, (14)

p

∗

v,t

=

v +t − y

00∗

2

, p

∗

v,t

=

v +t − y

00∗

2

. (15)

Given the prices in equations 14 and 15, equation 13 characterizes the optimal reservation utility. The

left-hand side of the equation is the expected increase in utility from sampling one more product—the four

integrals are the expected gains from discovering each of the four product types. Under random search, each

case happens with probability 1/4.

In the equilibrium, the probabilities of purchasing the four product types conditional on discovery are

q

∗

v,t

=

v +t − y

00∗

4t

, q

∗

v,t

=

v +t − y

00∗

4t

, (16)

10

Here we are assuming all four types of products have positive demand similar to footnote 8.

17

q

∗

v,t

=

v +t − y

00∗

4t

, q

∗

v,t

=

v +t − y

00∗

4t

(17)

respectively. Analogous to Corollary 3 in the previous section, we have the following result on the expected

industry proﬁt and consumer surplus.

Corollary 4. In the equilibrium deﬁned in Proposition 5, the expected total industry proﬁt

π

00∗

=

q

∗

v,t

p

∗

v,t

+ q

∗

v,t

p

∗

v,t

+ q

∗

v,t

p

∗

v,t

+ q

∗

v,t

p

∗

v,t

q

∗

v,t

+ q

∗

v,t

+ q

∗

v,t

+ q

∗

v,t

,

and the expected consumer surplus is y

00∗

.

5.2 Platform Recommendation under Setting 3

We now analyze the impact of platform recommendation. As before, we assume the consumer does not

know which types of products are being recommended, and when searching, she ﬁrst goes through the rec-

ommended products in the predetermined order and subsequently samples the remaining products randomly

(Assumption C2). Using the same backward-induction argument from the previous analyses, we know the

recommended products in the new equilibrium will still charge the same price given in equations 14 and 15.

When conditioning on taste dispersion, our analysis in Section 4.2 has provided results on optimal

product selection with respect to quality. As such, we now examine the difference between recommending

high taste-dispersion products and recommending low taste-dispersion products, conditioning on quality. In

other words, we compare recommending type (v

j

, t) products and recommending type (v

j

, t) products in

terms of the impact on total producer proﬁts and consumer surplus.

If the consumer purchases the recommended product, the transaction generates a proﬁt of p

∗

v

j

,t

(for

(v

j

, t)) or p

∗

v

j

,t

(for (v

j

, t)). From equations 14 and 15, we observe that p

∗

v

j

,t

> p

∗

v

j

,t

. That is, the high

taste-dispersion products have a larger proﬁt margin than the low taste-dispersion products. However, rec-

ommending a high proﬁt-margin product does not necessarily maximize industry proﬁt. The underlying

reason for the decoupling of producer proﬁt and industry proﬁt is the following. When product heterogene-

ity exists only in the vertical dimension, a high proﬁt-margin product (i.e., high quality product) is also more

likely to be accepted by the consumer than a low proﬁt-margin product (i.e., low quality product). When

taste dispersion in the horizontal dimension is introduced, whether recommending a high taste-dispersion

product would increase or decrease the expected industry proﬁt depends on the joint effect of proﬁt margin

18

(price) and purchase probability (proportional to sales). The probability that the consumer accepts a high

taste-dispersion product can actually be smaller under some conditions. To see when low taste-dispersion

products are more likely to be accepted than high taste-dispersion products, refer to equations 16 and 17.

Speciﬁcally, the equations show q

∗

v

j

,t

< q

∗

v

j

,t

if and only if v

j

> y

00∗

. The condition v

j

> y

00∗

is more likely

to hold when both product quality and search cost are simultaneously high (because the reservation utility

y

00∗

decreases with the search cost). As before, if the consumer rejects the recommended product (which

happens with probability 1 − q

∗

v

j

,t

or 1 − q

∗

v

j

,t

) she then searches the remaining products in a random order

so the expected industry proﬁt from the remaining products would be π

00∗

(Corollary 4). Calculating and

comparing the expectations yield the result on industry proﬁt in Proposition 6.

In general, neither the price signal nor the sales signal alone can help the platform select the recom-

mended product to maximize total producer proﬁts. An interesting implication from Proposition 6 is that it

is feasible to combine the observed price and sales signals to achieve the theoretical optimum. Note that π

00∗

is a function of equilibrium prices and sales. Therefore, even if the platform has no knowledge of product

types (v

j

, t

j

), it can nonetheless compute q

∗

v

j

,t

j

(p

∗

v

j

,t

j

− π

00∗

) for each product using the observed prices and

sales. Then, as the condition indicates, recommending product(s) where q

∗

v

j

,t

j

(p

∗

v

j

,t

j

− π

00∗

) is the largest

maximizes industry proﬁt.

Unlike the setting of Section 4.2 where recommending high quality products is always aligned with

maximizing consumer surplus, we ﬁnd here that the recommendation’s impact on consumer surplus is con-

ditional on a quality level threshold. Speciﬁcally, if v

j

> y

00∗

+

p

tt, recommending a low taste-dispersion

product will increase the expected consumer surplus, and if v

j

< y

00∗

+

p

tt, recommending a high taste-

dispersion product will increase the expected consumer surplus. Taking the search cost and product hori-

zontal heterogeneity as given, recommending a type t product is more likely to increase (decrease) expected

consumer surplus if product quality is low (high). Since high dispersion means larger probability mass on

extreme values of taste match, an intuitive understanding of the result would be the following: If the quality

level is low, then the upside potential is more important so it would be beneﬁcial to recommend a “riskier”

product (high dispersion); if the quality level is already high, then minimizing the downside risk is more

important so it would be beneﬁcial to recommend a “safer” product (low dispersion).

Relying on price or sales signals will not necessarily lead to optimal recommendation for maximizing

consumer surplus. Since the sign of the price difference between the high dispersion and low dispersion

products is independent of the level of quality (p

∗

v

j

,t

is larger than p

∗

v

j

.t

regardless of v

j

, see equations 14 and

19

15), price-based recommendation will not be optimal for the same reasons discussed above. For the sales

signal, observe that q

∗

v

j

,t

S q

∗

v

j

,t

when v

j

T y

00∗

(see equations 16 and 17). Thus, the product quality threshold

for sales signal reversal (due to taste dispersion) is lower than the threshold speciﬁed in the condition for

consumer surplus. Hence, the switch between high vs. low taste-dispersion product recommendation occurs

at a lower product quality threshold than needed for optimizing consumer surplus. Therefore, recommenda-

tion based on sales cannot guarantee the best result either.

Proposition 6. Under Assumption P3, recommending a type (v

j

, t) product generates a larger expected

industry proﬁt than recommending a type (v

j

, t) product if and only if q

∗

v

j

,t

(p

∗

v

j

,t

− π

00∗

) ≥ q

∗

v

j

,t

(p

∗

v

j

,t

− π

00∗

);

recommending a type (v

j

, t) product generates a larger expected consumer surplus than recommending a

type (v

j

, t) product if and only if v

j

≤ y

00∗

+

p

tt.

In summary, when products are differentiated by their consumer taste dispersion, the platform’s optimal

product recommendation strategy is more complicated. Speciﬁcally, when the goal is to maximize industry

proﬁt, the type of products the platform should recommend is determined by the interaction of price and

sales as speciﬁed in Proposition 6. The theoretical optimum can be achieved by combining the observed

price and sales signals. When the goal is to maximize consumer surplus, the platform should recommend

high (low) taste dispersion products to maximize consumer surplus if the product quality is low (high).

However, if the platform cannot discern product type, recommendation based on price or sales will not

guarantee optimal strategy for maximizing consumer surplus. In the online appendix, we provide numerical

examples to demonstrate the results on optimal recommendation as well as to show that recommendations

based on the price or sales signal can be suboptimal.

6 Discussion and Conclusion

There is a consensus among practitioners and academics that platform recommendation would in general

beneﬁt the selected products. As such, product recommendation has become an integral component of plat-

form strategy (Boudreau 2010, Tiwana et al. 2010, Qiu et al. 2017) to enhance market outcomes. Extant

research, however, has not systematically addressed the impact of recommendation on producer proﬁts and

consumer surplus at the market level. Consequently, there is very little theoretical guidance on how the plat-

form should select products for recommendation. This work develops an analytical model by adapting the

consumer-search framework of Wolinsky (1986) and Anderson and Renault (1999) to elucidate the tension

20

between the product- and market-level outcomes (Huber et al. 2017) induced by platform recommenda-

tion. Our model characterizes the equilibrium implications of platform recommendation, which entails the

tradeoff between the potential gains from the recommended products and the potential losses from the non-

recommended products. Our analysis regarding optimal product recommendation emphasizes the selection

of recommended products to balance this tradeoff.

We contribute to the platform recommendation literature by systematically analyzing changes in pro-

ducer proﬁts and consumer surplus under platform recommendation. We compare the promotion of the

recommended products in the consumer search sequence vis-à-vis the benchmark case where consumers

search products randomly. The analysis provides the theoretical basis on which types of products the

platform should select for recommendation to optimize platform-level outcomes. The key insights of our

ﬁndings come from the separate analyses regarding product heterogeneity in the vertical and horizontal

dimensions (i.e. product quality and idiosyncratic taste match). Our results indicate that recommending

high quality products will increase both industry proﬁt and consumer surplus. Recommending high taste-

dispersion products, however, may increase or decrease industry proﬁt and consumer surplus depending on

the interaction of price and sales as well as the relationship between product quality and taste heterogeneity.

Importantly, when the platform cannot discern product types, recommendation strategies based on observed

price or sales signals cannot guarantee optimality in the general case.

The analyses in our paper also provide several conceptual and practical implications beyond the immedi-

ate scope of product selection for recommendation. Our result on sales-based recommendation complements

the existing knowledge about the market effects of platform-published sales rankings such as bestseller lists.

The literature has documented that bestseller lists not only reﬂect consumers’ past purchases, but also often

times directly inﬂuence consumer behavior (Sorensen 2007, Hendricks and Sorensen 2009). In fact, con-

sumers use the bestseller lists in crowded markets as a product discovery channel (e.g., see Bresnahan et al.

2013 in the context of mobile app market). As a result, bestseller lists create a market environment that fa-

vors the already-successful products so they tend to hurt product variety, i.e., they reinforce the superstar or

rich-get-richer effect (Sorensen 2007). To the extent that bestseller lists are used by consumers as a product

discovery channel, we conceptualize that their role is essentially to promote the best selling products in the

consumer search sequence. In this sense, bestseller lists can be seen as sales-based recommendation within

our analytical framework. Our ﬁnding then shows that in addition to having drawbacks for product variety

and equality, the effects of bestseller lists can be suboptimal even in terms of pure market efﬁciency (since

21

sales-based recommendation cannot achieve either optimal total industry proﬁt or consumer surplus).

The optimal product recommendation that our theoretical model suggests is based on the assumption

that the platform is able to determine product types in terms of heterogeneity in the vertical and horizon-

tal dimensions. If the platform cannot distinguish different product types (for example in the case of a

nascent market platform that has not built the editorial and data-analytical capability for assessing product

heterogeneity), recommendations based on readily available price or sales signals can be suboptimal in the

general case. From this point of view, providing platform recommendation can be thought of as a way

of shifting the costs of product search and evaluation from individual consumers to the platform owner.

Presumably, the platform owner’s search costs would be considerably lower than the aggregate costs for in-

dividual consumers, so it should be beneﬁcial for the whole ecosystem to have the platform provide product

recommendation.

Our analytical results have useful implications for generating personalized recommendations through

algorithm-based recommender systems. Many algorithmic recommender systems predict to what extent a

given user would like a given product (often as a preference score), and also provide a precision estimate

associated with the prediction (e.g., in the form of conﬁdence interval). Then, for a particular user and

a number of candidate products, the recommender system would output a series of preference score and

conﬁdence interval pairs. If we interpret our model’s vertical utility component as the preference score and

the horizontal component as a measure of the conﬁdence interval, then our results can potentially be applied

to determine the optimal ranking of these personalized recommendations. Of course, the success of using

the method will depend on the reliability of the predictions generated by the recommender system.

We identify several directions to extend the current research. Future studies can explore ways to incor-

porate platform recommendation as part of the market outcome rather than as an exogenous intervention.

Given the product heterogeneity in the market, we have examined how the platform should select the type

of products for recommendation. Future research can build on our framework to analyze how the size of

market-level gains would change with the degree of product heterogeneity. One approach to endogenizing

the extensive and intensive margins of platform recommendation is through formally modeling the plat-

form’s revenue structure and the costs associated with conducting product search and evaluation. Such an

approach would then lead to a characterization of the threshold at which point making a recommendation

becomes unproﬁtable for the platform owner.

In the current setting of the model, the distribution of different types of products is given and ﬁxed in

22

the market. In other words, we implicitly assume the producers have designed and developed their products,

and the model only considers producers producing and selling copies of their existing products. Taking a

longer-term view, researchers could investigate whether and how platform recommendation will inﬂuence

producers’ strategic decisions such as entry, investment, and innovation. Answering these questions can

help us understand the long term implications of platform recommendation on product quality and variety in

the marketplace, and more broadly contribute to our knowledge about the inﬂuence of platform governance

on the coevolution of ecosystems and modules (Tiwana et al. 2010). The current model in the paper assumes

that consumers are uninformed about all products without recommendation. Future research may seek to

relax this assumption and re-analyze the implications on the platform recommendation strategy, for example

by assuming the coexistence of well-established incumbent products and lesser-known new entrants. From

a practical perspective, one can then study producers’ best responses when the platform recommends their

own products or their competitors’ products.

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25

A Proofs

Proof of Proposition 1

Proof. We present a constructive proof of the existence of such an equilibrium.

First, assume the consumer participates in search. The search literature has established that the optimal

stopping rule for the consumer is myopic—there is a reservation utility such that the consumer purchases the

ﬁrst product where the consumption utility net of price (v

j

+ h

i j

− p

j

) is larger than the reservation utility.

Now consider the producers’ proﬁt-maximization problem given the consumer reservation utility is y.

Since the consumer searches products randomly, the probability of the consumer ﬁnding a particular pro-

ducer’s product is exogenous from the producer’s perspective. Hence, a product’s own price only affects

the probability that the consumer buys the product conditional on sampling the product. Therefore, for any

producer j, the proﬁt-maximization problem can be written

max

p

j

p

j

Z

t

−t

I(v + h − p

j

− y ≥ 0)

1

2t

dh, (18)

where the integral gives the conditional probability of purchase. The ﬁrst order condition gives the pricing

function

p

j

(y) = p(y) =

v +t − y

2

. (19)

With t > 0, it is obvious that the second-order derivative is negative.

Then consider the consumer’s surplus-maximization problem given product prices p. With the myopic

stopping rule, the optimal reservation utility should make the consumer just (un)willing to sample another

product—the expected gain in surplus by sampling another product is zero. Suppose currently the best net

utility the consumer can get is y. Then her expected net utility after one additional search is

R

t

−t

max(v + h −

p,y)

1

2t

dh (the max function is used because the consumer can always come back to the current best). The

cost of one additional search is s, so the expected gain in surplus is

R

t

−t

max(v + h − p,y)

1

2t

dh − s − y. The

reservation utility should make the expected surplus gain zero. Hence, after rearranging the terms, we have

Z

t

−t

max(v + h − p − y,0)

1

2t

dh = s. (20)

26

Plugging equation 19 into equation 20 yields

Z

t

−t

max(

v −t − y

2

+ h,0)

1

2t

dh − s = 0. (21)

The existence problem thus boils down to whether there exists a solution y∗ to equation 21. Further, the

solution needs to be nonnegative to ensure the consumer participates in search. Consider the left-hand side

of 21 a function of y and denote it F(y). Note that, F(y) is a continuous and decreasing function of y,

and F(v + t) = −s < 0. Therefore, for a nonnegative solution y∗ ≥ 0 to exist, the necessary and sufﬁcient

condition is F(0) ≥ 0, or

s ≤

Z

t

−t

max(

v −t

2

+ h,0)

1

2t

dh. (22)

Proof of Corollary 1

Proof. Since all products are priced at p

∗

, and with an inﬁnite number of products the consumer will ﬁnd a

satisfying product with probability 1, the expected industry proﬁt π

∗

= 1 × (p

∗

− 0) = p

∗

.

Suppose the expected surplus for the consumer to participate in the market is x. x can be written as

x = −s +

Z

t

−v+p

∗

+y

∗

(v +h − p

∗

)

1

2t

dh + (1 − q

∗

)x. (23)

The intuition behind the equation can be understood by thinking one step ahead about the ﬁrst search. The

expected surplus is the sum of three parts: the cost paid for the ﬁrst search (−s), the net utility from the ﬁrst

sampled product (if the consumer accepts the ﬁrst product), and the option value of searching the remaining

products (if the consumer rejects the ﬁrst product, which happens with probability 1 − q

∗

). The option value

of continuing is also x because after the ﬁrst product there remains an inﬁnite number of products to search.

Plugging equation 5 into equation 23 and comparing the result with equation 3 yield x = y

∗

.

Proof of Proposition 2

Proof. First observe that if the consumer rejects all recommended products and proceeds to the remaining

products, the market then functions the same as under Assumption C1. Hence, it follows from Proposition

1 that the non-recommended products charge p

∗

and the consumer expects a surplus of y

∗

. We only need to

27

show the recommended products would also charge price p

∗

. We do so by backward induction. Anticipating

the remaining products charging p

∗

, equation 3 shows the consumer will hold reservation utility y

∗

if she is

currently considering the last recommended product, l. Correctly expecting the reservation utility, producer

l’s proﬁt-maximization problem is still characterized by equation 2 so p

∗

l

= p

∗

. Expecting producer l will

be charging p

∗

, the consumer will again hold the reservation utility y

∗

at product l − 1 (equation 3). Hence,

the equilibrium result follows by backward induction.

Proof of Corollary 2

Proof. It directly follows from that the assumption that products are ex ante homogeneous and the equilib-

rium product price and consumer reservation utility do not change under platform recommendation.

Proof of Proposition 3

The proof follows exactly the same steps as that for Proposition 1. The only differences are that (1) here the

two types of products have their own proﬁt-maximization equation, and (2) the search cost cannot be too

small for ensuring the solution of reservation utility is not too large. Speciﬁcally, the following condition

needs to hold:

y

0∗

≤ v +t, (24)

so that there is positive demand for the low quality products (it automatically follows then the high quality

products have positive demand since v < v).

Proof of Corollary 3

Proof. Selling a low quality product generates a proﬁt p

∗

v

and selling a high quality product generates a

proﬁt p

∗

v

. The probability of the consumer purchasing a low quality product conditional on discovery is q

∗

v

,

and that for a high quality product is q

∗

v

. Since there is an inﬁnite number of products and both types account

for one half of the market, the expected sales of the low quality type is

q

∗

v

q

∗

v

+q

∗

v

and that of the high quality

type is

q

∗

v

q

∗

v

+q

∗

v

. Hence, the expected industry proﬁt π

0∗

=

q

∗

v

p

∗

v

+q

∗

v

p

∗

v

q

∗

v

+q

∗

v

. p

∗

v

< π

0∗

< p

∗

v

follows from that π

0∗

is a

convex combination of p

∗

v

and p

∗

v

.

The proof of the expected consumer surplus is the same as in Corollary 1.

28

Proof of Proposition 4

Proof. First consider the expected industry proﬁt. If the consumer accepts the recommended product, which

happens with probability q

∗

v

or q

∗

v

, the recommended product earns a proﬁt p

∗

v

or p

∗

v

, depending on the rec-

ommended product’s quality type. It follows from Corollary 3 that, if the consumer rejects the recommended

product, which happens with probability 1 −q

∗

v

or 1 −q

∗

v

, the remaining products together earn an expected

proﬁt π

0∗

. Therefore, if a low quality product is selected for recommendation, the expected industry proﬁt

is q

∗

v

p

∗

v

+ (1 − q

∗

v

)π

0∗

; if a high quality product is selected for recommendation, the expected industry proﬁt

is q

∗

v

p

∗

v

+ (1 − q

∗

v

)π

0∗

. Given p

∗

v

> p

∗

v

> 0 and q

∗

v

> q

∗

v

> 0,

q

∗

v

p

∗

v

+ (1 − q

∗

v

)π

0∗

< π

0∗

< q

∗

v

p

∗

v

+ (1 − q

∗

v

)π

0∗

.

Then consider the expected consumer surplus. Similar to equation 23, if the platform recommends a

product with quality v

j

, the expected consumer surplus, denoted ∆y

∗

v

j

, can be written

∆y

∗

v

j

= −s +

Z

t

−v

j

+p

∗

v

j

+y

0∗

(v

j

+ h − p

∗

v

j

)

1

2t

dh + (1 − q

∗

v

j

)y

0∗

= −s +

Z

t

−v

j

+p

∗

v

j

+y

0∗

(v

j

+ h − p

∗

v

j

− y

0∗

)

1

2t

dh + y

0∗

, (25)

where the ﬁrst term −s is the search cost paid on the recommended product, the second term is the expected

surplus gain from the recommended product, and the third term is the option value of searching the remaining

products. Observe that the ﬁrst and third terms do not depend on v

j

. So it boils down to comparing the

values of the second term between when v

j

= v and when v

j

= v. Plugging the expressions for p

∗

v

and p

∗

v

into equation 25, we get

∆y

∗

v

=

1

4t

((v −y

0∗

)t + (

v −y

0∗

−t

2

)

2

) −s + y

0∗

, (26)

∆y

∗

v

=

1

4t

((v −y

0∗

)t + (

v −y

0∗

−t

2

)

2

) −s + y

0∗

. (27)

Comparing ∆y

∗

v

and ∆y

∗

v

yields

∆y

∗

v

− ∆y

∗

v

=

1

4t

(v −v)

[(v +t − y

0∗

) +(v +t − y

0∗

)]

4

> 0.

The inequality follows from v < v and equation 24.

29

Proof of Proposition 5

The proof follo