Do Lower Search Costs Benefit Intermediaries?

Abstract

Intermediaries—such as Amazon, Alibaba Group, and Walmart—play critical roles in digital markets: This paper proposes a model for analyzing how search cost affects their market outcomes. Distinguished from existing search literature, we introduce network effects to the market by allowing buyers to have positive preferences for product variety. In addition, we relax the exogenous demand and supply assumptions in the conventional search literature to study how search affects participation on both sides of the market. Through the model, we focus on studying and comparing two important types of intermediary: the two-sided platform; and the one-sided retailer. We find that reducing search costs does not necessarily increase the profit of both intermediary types. Furthermore, its influences are highly varied—depending on both the type of intermediary and its buyers’ preferences for variety.

1 Introduction

1.1 Motivation

The paper is mainly motivated by the increasing importance of online intermediaries and their search systems in digital markets. With advanced information technology, online intermediaries—such as Alibaba, Amazon, Priceline, Uber, iTunes, Walmart, and Airbnb—have flourished and become more critical in the current digital economy. Among them, two intermediary types have become ubiquitous in recent years: One is the “two-sided” platform that allows third-party sellers to make transactions with buyers on their platform. The other one is the “one-sided” retailer that resells to buyers the products that it acquires from suppliers.

To give a few examples: Most online marketplaces—such as Alibaba, eBay, and Airbnb—are mainly operated as two-sided platforms: These intermediaries do not claim legal possession of the products that are sold on their platform. They make profits by charging corresponding fees on sellers for each unit sold through the platform. In exchange, they provide third-party sellers with access to their consumers and the freedom to determine product prices. Besides online marketplaces, there are also numerous non-virtual two-sided platforms that function similarly. This includes flea markets that connect buyers and sellers, credit card companies that link merchants with consumers, and advertising-supported newspapers that operate as media for readers and advertisers.

In contrast, other retailers—such as Target, Walmart, Lowe’s, 7-Eleven, and Costco—function mainly as one-sided retailers: These intermediaries buy products from sellers first and then resell the products to buyers: The intermediaries retain full control over the market and are the owner of all of the products sold in the market. As a consequence, the intermediaries are able to determine the product prices directly and collect all of the profit that is derived from product selling.

Further, a significant number of intermediaries operate as a hybrid business mode of the two-sided platform and one-sided retailer: They not only connect and enable transactions among users—sellers and buyers—on two sides of the market, but they also sell their own products directly to consumers at the same time. One illustrating example is Amazon: According to a recent survery,¹ around 56 percent of sales on Amazon were sold by third-party sellers, where the intermediary operates as a two-sided platform; for the rest of the sales, Amazon sells directly to buyers and acts as a one-sided retailer.

Our paper focuses on studying and comparing the two-sided platform and one-sided retailer, respectively.

Past literature distinguished between the two-sided platform and the one-sided retailer mainly based on the allocation of control between the intermediary and third-party sellers (Hagiu and Wright, 2015, 2019). More precisely, the main difference lies in whether the intermediary or third-party sellers hold control over market activities. When the intermediary operates as a two-sided platform, it is individual sellers who independently control transactions—including product prices, shipping, and return. By contrast, when the intermediary operates as a one-sided retailer, it is the intermediary that takes complete control over all products. Because of the difference in selling mechanism, the market outcomes are highly varied in different market types and have been well illustrated in the past research.

This paper shows that the intermediary type also greatly affects the consequences of search cost reduction in digital markets. The search cost not only determines the search and purchase decision of buyers, but also the pricing strategies of sellers and the intermediary. With the rapid development in information technology, buyers’ search costs continue to decrease, and this plays an essential role in digital markets. Surprisingly, how search reduction affects the intermediary in different business modes has received scant attention.

When intermediaries operate in different market types, they face different goals. Since the two-sided platform makes profits by charging sellers commission fees, it focuses more on transaction volumes and the commission fees that are placed on sellers. However, the one-sided retailer sells products directly to buyers and collects all profit from product selling. Thus, it cares more about the product price and quantity sold in the market. This makes the influences of search cost on market outcomes vary depending on the intermediary type. According to the model, search cost reductions do not necessarily benefit intermediaries. Indeed, it depends both on the intermediary type and also on buyers’ preferences for variety.

1.2 Literature Review

Our work is related to several strands of economic research: First, it is built upon the literature on random search. In past decades, a considerable amount of search research (Wolinsky, 1986; Anderson and Renault, 1999) focused on analyzing the role of reducing search costs in the digital economy. For instance, Bakos (1997) showed that search cost reduction diminishes the inefficiencies in the market, depresses sellers’ ability to extract monopolistic profit, and optimizes productive resource allocation. In addition to the improved market efficiency, Kuksov (2004) found that diminishing search costs on the intermediary may also lead to greater product differentiation for consumers.

However, Dukes and Liu (2015) indicated that online intermediaries face two main conflicting goals: helping consumers to find desirable products and keeping sellers’ competition on track. Indeed, several studies have pointed out the intermediaries’ motivation to divert search. To give a few examples: Hagiu and Jullien (2011) indicated that information intermediaries may divert search to trade for higher revenue from per consumer visit or to influence sellers’ strategic variables. Casner (2020) showed that the intermediary strategically presents low-quality sellers, so as to soften competition among sellers. Heresi (2018) focused on platform competition and found strong obfuscation motivations for platforms as long as the degree of competition between platforms is sufficiently strong. Janssen and Shelegia (2015) and Janssen and Shelegia (2020) focused more on the vertical relations in a search market. Other research (Dinerstein et al., 2018; Mamadehussene, 2020) investigated the interplay between prominence and obfuscation on price comparison platforms.

In all of these studies, the demand and supply of the intermediary are taken as exogenously given and focus solely on search design. However, it has been well illustrated in the two-sided market literature (Rochet and Tirole, 2003, 2004, 2006) that one of the main roles of intermediaries is to coordinate the number of users on the two sides of the market. Furthermore, the cross-group network effect—which is an important characteristic of the economics of the intermediary—is largely ignored in the search literature.

Specifically, the importance of network effects—indicating that the surplus of a user depends on the number of users on the other side—is highlighted in past research (Yoo et al., 2002; Parker and Van Alstyne, 2005; Jullien, 2005; Hagiu, 2006). As pointed out by Rochet and Tirole (2006), not only the price level but also the price structure matters in the two-sided market due to the network effects. Armstrong (2006) studied competition in the two-sided market and emphasizes the importance of indirect network effects, multi-homing, and price discrimination.

To introduce the network effects to the search model, we allow buyers to have positive preferences for variety. In other words, buyers strictly prefer intermediaries with more available brands to choose from. Unlike previous research (Stahl, 1982; Bar-Isaac et al., 2012) that adjusts value distribution, our definition of the preference for variety emphasizes more the dependence of a user’s surplus on the number of users on the other side. Based on the setup, we further discuss how different intermediary types affect the influence of search cost.

Besides the two-sided platforms that connect users, we also investigate the traditional intermediaries where the intermediaries buy and resell the product to buyers. Consequently, it is also naturally related to the literature on the economics of intermediation (Dixit and Stiglitz, 1977; Li, 1998; Biglaiser and Friedman, 1994; Chu and Chu, 1994; Kirmani and Rao, 2000; Rysman, 2009). Biglaiser (1993) showed that intermediaries—acting as experts—can improve social welfare by minimizing inefficient searching between buyers and sellers. Other research (Garman, 1976; Rubinstein, 1982; Mortensen, 1982; Wolinsky, 1987; Gehrig, 1993; Yavas, 1994; Spulber, 1996) confirmed that intermediaries play a role in improving social welfare when searching is costly and inefficient in bilateral search markets. This type of intermediary acts more like a middleman or pure retailer who buys products from sellers and then resells them to consumers. In our study, we call those intermediaries one-sided retailers.

Compared with the previous literature, our article makes several contributions: To the best of our knowledge, our research is the first to investigate how intermediary type influences the search cost impact in digital markets. It adds to the relatively large search literature not only by endogenizing the demand and supply of an intermediary but also by introducing cross-group network effects through buyers’ preference for variety.

The remainder of the paper is organized as follows: We first present our two-sided platform model in Sect. 2. Then we proceed to investigate the equilibriums of the one-sided retailer in Sect. 3. Last, we compare the two market types in Sect. 4 and conclude in Sect. 5.

2 Two-Sided Platform

We first analyze the scenario in which the intermediary operates as a two-sided platform. As is shown in Fig. 1, the two-sided platform does not claim any legal possession of products sold in its market. Its main function is to connect sellers with buyers and facilitates transactions among them. Like most marketplaces, we assume that buyers do not incur any fees. For each transaction, the platform places corresponding charges only on sellers. In exchange, sellers gain access to buyers on the platform and control over their product prices.

Fig. 1

Two-Sided platform

Motivated by the fact that most platforms require sellers to pay commission fees, our model focuses on the scenario where the platform charges a fixed percentage d of sellers’ product revenue.² To give a few examples: Expedia charges travel companies up to a 25% commission fee for reservations that are made through its website; Newegg imposes on sellers varied commission fees between 8% and 13%, depending on their product types; and Orbitz collects an average of 25% as a commission fee for each transaction via its website.

The players in our game are buyers, sellers, and a monopoly platform. Specifically, there are n sellers on the platform: Each produces a unique product (brand). To have our model tractable, we assume that sellers all have zero marginal cost. Their products are horizontally differentiated and have no quality differences. To make transactions, sellers have to go through this monopoly platform, which has a continuum mass l of potential buyers on the other side of the market. Each buyer has a unit demand for the goods sold on the platform and is only partially informed about the product’s attributes: such as brand, color, design, and so on.

To determine the idiosyncratic value of a specific product, a buyer has to collect and evaluate the product’s information. The evaluation process is costly: A consumer has to pay a fixed cost each time she gets to know a brand. When the search cost decreases, buyers search more and expect higher matched value. Thus, a reduction in search cost is usually believed to make buyers better off.

In the meanwhile, a decrease in search cost makes sellers worse off, as they face more elastic demand and more competition when buyers search more. Since a change in search cost has conflicting effects on buyers and sellers, it is critical for the intermediary to seek a balance. To investigate, we set up a game with the following timelines: The platform first determines its commission fees. Based on that, sellers make entry decisions, and buyers enter afterward. Once entering the market, sellers determine product prices, and buyers optimize their searches and purchases.

2.1 Transaction

To derive the subgame-perfect Nash equilibrium, we use backward induction to analyze buyers’ search strategy. Specifically, the utility of buyer j for purchasing a unit of good i takes the form

$$\begin{aligned} x_j(i)=v_{ij}-p_i, \end{aligned}$$

(1)

where $v_{ij}$ is the matched value between buyer j and product i; and $p_i$ is the corresponding price of product i. Buyers do not know the idiosyncratic value of $v_{ij}$ before making searches. They know only that the realization of $v_{ij}$ is i.i.d. and uniformly distributed in the interval $[0,\bar{v}]$. To uncover the realized value of $v_{ij}$, a buyer has to pay a search cost: $s>0$. If the buyer finds a product with a sufficiently high matched value, she can stop searching and purchase the product immediately. Otherwise, the buyer may choose to make another search. If she does, the platform will assign a random product to her again, and the game repeats.

Following the conventional random search literature (Bakos, 1997), our study assumes that there is no replacement or recall in the search.³ At the symmetric equilibrium, a buyer expects brands that she has not yet visited all charge prices $p^\star$.

As a result, the buyer will form a stationary search strategy with a reservation value $w^{\star }$: The reservation product value $w^{\star } \in (0, \bar{v})$ is defined as

$$\begin{aligned} \int ^{\overline{v}}_{w^{\star }}(v-w^{\star }) dG(v)= & {} s, \end{aligned}$$

(2)

where $G(v)=v/\bar{v}$ is the distribution of $v_{ij}$: The left side of Eq. (2) measures the expected utility increment from one additional search when the buyer samples a brand with product value $w^{\star }$. The right side of Eq. (2) is the cost of an additional search.

According to the equation, the marginal benefit of an additional search is lower than the marginal cost if the buyer finds a product whose matched value is higher than $w^{\star }$. Consequently, it is optimal for the buyer to stop searching and purchase the first brand i whose product value is no less than $w^\star$. Note that when the product value is sufficiently small such that $\bar{v}<s/2$, there is no non-negative solution to Eq. (2). In this case, buyers have no motivation in searching and the market converges to the trivial monopoly outome, as shown in Diamond (1971). To focus on the more meaningful market equilibrium, we assume throughout the paper that $\bar{v}$ is always sufficiently large to rule out this degenerate case.

Given the stationary search strategy, the expected demand for a particular firm whose price is p while other firms charge $p^{\star }$ is

$$\begin{aligned} D(p,p^{\star },n)=Q\left\{ \frac{[1-G(w(p))]}{n} *\sum _{k=0}^\infty G^k(w^{\star })\right\} , \end{aligned}$$

where $w(p)=w^\star +p-p^\star$ is the reservation value of the focal firm. The right-hand side of the equation is the product of the number of buyers joining the platform Q and the probability that a randomly drawn buyer purchases from the focal firm: This probability consists of two terms as shown in the curly bracket: The first term captures the probability that the focal brand is chosen and purchased in the $(k+1)$th search by the buyer; the second term captures the probability that all of the brands in the previous k searches are not chosen by the buyer.

After simplification, the individual demand can be written as

$$\begin{aligned} D(p,p^{\star },n)=Q\frac{[1-G(w(p))]}{n[1-G(w^{\star })]}. \end{aligned}$$

(3)

As is shown in Eq. (3), an increase in the focal firm’s price p generates higher reservation value w(p). This indicates that buyers have a lower probability of stopping and purchasing the specific brand.

When the platform charges commission fees d, the expected profit of a seller whose price is p when all other firms charge $p^{\star }$ equals

$$\begin{aligned} \pi (p,p^{\star },n)= (1-d)p *D(p,p^{\star },n), \end{aligned}$$

where: the first component of the equation measures the markup that is received by sellers in each transaction; and the second component is the individual demand. Maximizing the third-party seller’s profit, we derive the product price at the symmetric equilibrium as

$$\begin{aligned} p^{\star }=\sqrt{2s\bar{v}}. \end{aligned}$$

(4)

As is shown in Eq. (4), the equilibrium product increases in the search cost. When the search cost increases, buyers form a lower reservation value and search less. As a consequence, each seller faces less elastic demand and has more monopolistic power.

2.2 Buyer Entry

We then move forward to investigate buyers’ entry decisions. Buyers are only partially informed about the market before joining the platform. Given the stationary search strategy, a buyer’s expected matched product value equals $(w^{\star }+\bar{v})/2$. Correspondingly, the net expected utility from product purchase can be written as $(w^{\star }+\bar{v})/2-p^{\star }$. Note that this utility is decreasing in buyers’ search costs. To guarantee positive value in product purchase $(w^{\star }+\bar{v})/2-p^{\star }$, we assume in the two-sided market that the search cost is not too high such that $s<2\bar{v}/9$.

In addition to the product value, we also let the utility of buyers be increasing in the number of sellers that join the market. This is motivated by the fact that a platform with thousands of brands is usually more attractive to buyers than a platform with only a few products. A large platform holding more brands usually represents positive signals to buyers, as it is usually more reliable than small firms in complementary services such as shipping, return, and customer services. Furthermore, even when the matched values of all products follow the same value distribution, having more brands reduces the variance of expected product value and decreases the buyers’ risk of matching to unsatisfying products.

This phenomenon such that a buyer’s surplus increases in the number of sellers that are available on the platform is the so-called cross-group network effect. As is well illustrated in Rochet and Tirole (2004, 2006), the network effect is extremely critical and prevalent in multi-sided markets, where an intermediary connects users and facilitates the transactions among them.

To model this effect, we follow the existing literature(Armstrong, 2006; Yoo et al., 2002; Jullien, 2005) in letting buyers’ utility in joining a market be linearly increasing in the number of sellers. Thus, a buyer’s expected surplus from joining the platform equals

$$\begin{aligned} U=\left( \frac{w^{\star }+\bar{v}}{2}-p^{\star }\right) +\mu *n, \end{aligned}$$

where $(w^{\star }+\bar{v})/2-p^{\star }$ measures the net utility that is derived from product purchase; n is the number of products that are available on the platform; and the last term – $\mu *N$—represents the network effect that is derived from product variety: When $\mu$ increases, buyers care more about the number of available brands that are on the platform and thus have stronger preferences for variety. Throughout the paper, we assume that buyers’ preferences for variety are positive such that $\mu >0$.⁴

Based on the expected utility, buyers make entry decisions. If the expected utility for joining the platform is low, buyers can choose to stay outside. Specifically, we let buyers’ outside option values follow a uniform distribution in [0, H]. A buyer will join the platform only when the expected surplus U is higher than her outside option: The equilibrium number of buyers on the platform equals $Q= (U *l)/H$, where l is the mass of potential buyers.

Since sellers join the platform prior to buyers, a seller’s expected profit from joining the platform depends on its expectation of the number of buyers. To simplify our model without loss of generality, we assume that all sellers share the same expectation such that there will be $Q^e=Exp[Q]$ buyers joining the platform. In this context, a seller’s net profit for joining the platform equals

$$\begin{aligned} \pi ^i=(1-d)\sqrt{2s\bar{v}} *\frac{Q^e}{n}-f, \end{aligned}$$

where $\sqrt{2s\bar{v}}$ is the markup of the product; f is the entry cost of the seller; and $Q^e/n$ measures the expected individual demand at the symmetric equilibrium. Sellers continue entering the market until the expected profit is zero: $\pi ^i=0$. Thus the number of sellers at equilibrium can be simplified as

$$\begin{aligned} n= \frac{(1-d)Q^e\sqrt{2s\bar{v}}}{f}. \end{aligned}$$

(5)

Given the commission fee that is charged by the platform, the number of sellers that join the platform increases in search costs. This is due to the fact that an increase in search cost reduces buyers’ motivation to search and softens the competition among sellers. As a consequence, each seller expects higher profits on the platform, and more sellers join the platform.

2.3 Platform Optimization

Last, we study the platform’s optimal pricing strategies and how the search cost influences market outcomes. To simplify the notation, we let $L=l/H$ and $F=f/Q^e$. When we plug in the equilibrium price $p^{\star }$, the reservation value $w^{\star }$, and the equilibrium number of sellers n, the aggregate demand for the platform can be simplified as

$$\begin{aligned} Q= L\left( \bar{v}-\frac{3\sqrt{2s\bar{v} }}{2} +\frac{\mu (1-d)\sqrt{2s\bar{v}}}{F}\right) . \end{aligned}$$

(6)

As is indicated in Eq. (6), reducing search costs does not necessarily attract more buyers to the platform. Although buyers enjoy higher matched product values and lower product prices when the search cost decreases, they also suffer from the loss of product variety on the platform.

Given the commission fees, the relative strengths of these two conflicting effects depend on buyers’ preference for variety $\mu$. When buyers have extremely high preferences for product variety, reducing buyers’ search cost will always decrease their surplus for using the platform. Although the low search cost endows consumers with better-matched products, the loss of product variety induced by greater competition outweighs the benefit from matches. However, the weight of variety in buyers’ utility drops as the preference for variety decreases. Consequently, consumers prefer lower search costs when their preferences for variety are low.

In this context, a two-sided platform’s profit equals

$$\begin{aligned} \pi _p=(d*p^{\star })*Q, \end{aligned}$$

where: $d*p^{\star }$ measures the revenue that is collected from each transaction; and Q is the total number of transactions on the platform. The platform maximizes its profit by determining the optimal commission fees. Solving the first-order condition, we derive the optimal pricing strategy as:

$$\begin{aligned} d^{\star }=\frac{F\sqrt{2\bar{v}} +2\mu \sqrt{s} -3F\sqrt{s}}{4\mu \sqrt{s}}. \end{aligned}$$

(7)

The resulting number of buyers at equilibrium is computed as

$$\begin{aligned} Q_t^{\star }=\frac{L\left( 2\mu \sqrt{2s\bar{v}}+2F\bar{v} -3F\sqrt{2s\bar{v}}\right) }{4F}\text {and }n_t^\star =\frac{3F\sqrt{2s\bar{v}}+2\mu \sqrt{2s\bar{v}}-2F\bar{v}}{4Fu}. \end{aligned}$$

(8)

When search costs decrease, buyers search more, and it becomes less profitable for sellers to join the market. When the search cost is too low such that $s<2F^2\bar{v}/(3F+2\mu )^2$, there is no seller on the platform so the market collapses. One hypothetical example would be when buyers have zero search costs. In this setting, the market becomes perfectly competitive, and the product price reduces to zero. Consequently, no seller will join the platform. To avoid the trivial market equilibria, we focus on the scenario where $s>2F^2\bar{v}/(3F+2\mu )^2$ in the two-sided market. In this context, the platform’s equilibrium profit equals:

$$\begin{aligned} \pi _p^{t{\star }}=\frac{L\left( 2\mu \sqrt{2s\bar{v}} +2F\bar{v}-3F\sqrt{2s\bar{v}}\right) \left( \sqrt{2}F\bar{v}+2\mu \sqrt{s\bar{v}} -3F\sqrt{s\bar{v}}\right) }{8\sqrt{2}\mu F}. \end{aligned}$$

(9)

We observe that the influence of search cost on the platform’s profit depends on buyers’ preferences for variety: We have established the following results:

Proposition 1

When the intermediary operates as a two-sided platform, reducing buyers’ search costs does not necessarily increase the platform’s profit:

• If consumers’ preferences for variety are low such that $\mu <\frac{3F}{2}$, the platform’s profit is convex and decreases in search cost.

• If consumers’ preferences for variety are high such that $\mu \ge \frac{3F}{2}$, then the platform’s profit is concave and increases in search cost.

Proof

See Appendix A.1$\square$

As is shown in Fig. 2, lower search costs enable the platform to achieve high profits when buyers have low preferences for variety. However, when the preference for variety is extremely high, we observe that the platform’s profit is lower when the search costs decrease.

Fig. 2

Two-sided platform profit. Note: The platform profits in this figure are based on parameter values $\bar{v}=100$, $F=1$, $L=1$, and $\mu =1 (2)$ for the low (high) preference variety case. The search costs are restricted to the corresponding interval where there are positive numbers of sellers and transactions

The underlying mechanisms of our results are as follows: When there is a reduction in search cost, there are multiple effects on the two-sided platform, and those effects are linked by network effects. On the one hand, a reduction in search cost provides better search services for buyers and enables the platform to charge possibly higher fees for each transaction. On the other hand, it also generates more competition among sellers and reduces the number of sellers that join the market. When buyers’ preference for variety is low, the buyers’ benefit from lower search costs is dominant thus the platform’s profit increases in search cost reduction. While when buyers have extremely strong preferences for variety, the marginal benefit from search reduction decreases due to the loss of variety. This makes the platform’s profit always decreases in search cost reduction.

3 One-Sided Retailer

In this section, we analyze the scenario where the intermediary operates as a one-sided retailer and sells directly to buyers in the market. As shown in Fig. 3, the one-sided retailer first purchases products from sellers at a wholesale price and then resells them to buyers. Different from the two-sided platform that charges sellers and enables them to make transactions with buyers, the one-sided retailer sells directly to buyers and collects all corresponding revenues.

Consequently, the goals of these two types of intermediaries are different. The two-sided platform gives market control to sellers and makes profits by facilitating transactions between the sellers and buyers, while the one-sided retailer controls the market directly and aims to maximize profit through its direct sales of the products to buyers.

Fig. 3

One-sided retailer

Specifically, the characteristics of a one-sided retailer can be summarized as follows: First, the one-sided retailer has the legal possessions of products sold in the market. This indicates that the intermediary can coordinate prices among the products in the one-sided market. Also, the one-sided retailer directly determines the number of brands that are available in its market while the two-sided platform usually achieves the goal indirectly through seller fees. Because of these unique characteristics, the influence of search cost on the market outcomes of a one-sided retailer is different from that of a two-sided platform. Note that a one-sided retailer could be either an online website or a brick-and-mortar store. As long as the intermediary has these characteristics of a one-sided market, it should have the corresponding attributes in our model.

To investigate, we set up a game for the one-sided retailer with the following timelines: The intermediary first determines the number of products that are sold in its market. Based on the product variety, buyers make entry decisions. Same as on the two-sided platform, we assume that buyers do not incur any fees to use the one-sided retailer. Once buyers enter the market, the intermediary determines product prices. Last, buyers optimize their searches and purchases.

When the intermediary operates as a one-sided retailer, sellers no longer participate directly in the market. Instead, the intermediary now determines product variety and prices directly. Without loss of generality, we assume that all of the bargaining power lies within the intermediary when the one-sided retailer makes wholesale offers to sellers: The one-sided retailer is able to purchase products at the sellers’ marginal cost. Introducing variable degrees of bargaining power just increases the intermediary’s marginal cost, but it should not affect our main conclusions in the study, as is shown in Appendix A.2.

Our paper focuses on the difference between two business modes induced by their varied control over product price and variety. However, the bargaining process makes it less obvious how varied controls affect market outcomes. For instance, when the marginal costs are different in the two markets, it becomes less transparent whether the higher product price in the one-sided market comes from its higher marginal cost, or from the fact that the one-sided retailer has more control over product price. Consequently, we adopt the assumption that the one-sided retailer has all the bargaining power to focus on more critical aspects.

3.1 Transaction

The type of intermediary does not affect buyers’ search or transaction strategies. Consequently, each buyer continues searching until she finds a product whose value is no less than the reservation value $w^\star$ defined by Eq. (2). Unlike the two-sided platform where individual sellers determine product prices to maximize individual profits, the one-sided retailer aims to maximize the total profit by optimizing over all product prices. Since the one-sided retailer is now the owner of all products, it can easily coordinate product prices and reduce competition among brands. Specifically, it is optimal for the platform to charge a product price $p^\star =w^\star$, such that the marginal buyer of each brand receives zero surplus.

This is because once the price is lower than $w^\star$, the intermediary can always raise all product prices simultaneously without losing any demand. However, if the price is higher than $w^\star$, the market will not be fully covered and the intermediary’s profit decreases.⁵ Consequently, the equilibrium product price of the one-sided retailer equals

$$\begin{aligned} p^\star =\bar{v}-\sqrt{2\bar{v}s}. \end{aligned}$$

(10)

We find that although an increase in search cost raises the product price on the two-sided platform, it reduces the product price of the one-sided retailer.

On the two-sided platform, the product price is increasing in search cost because a reduction in search increases buyers’ reservation value and motivates them to search more. This will generate more competition among brands and reduces their product prices. However, the intermediary is the common owner of all products in the one-sided market and thus can coordinate product prices: It is able to extract all of the surplus of the marginal buyer. When the search cost decreases, each buyer has a higher reservation value. As a consequence, the surplus of the marginal buyer increases, and the one-sided retailer is able to charge higher product prices in the market.

3.2 Buyer Entry

When the intermediary operates as a one-sided retailer, sellers no longer interact directly with buyers. The intermediary determines the number of brands and makes corresponding transaction decisions directly. As for the buyers, they will enter the market as long as they expect to derive higher utility from joining the market rather than staying outside. Specifically, a buyer’s expected surplus from entering the market still takes the format of $U=(w^\star +\bar{v})/2-p^{\star }+\mu n$, where the first term is the expected product value, and the last term measures consumers’ preference for variety. Plugging in the optimal pricing strategy of Eq. (10) and reservation value of Eq. (2), we are able to simplify the expected utility for entering the market as

$$\begin{aligned} U=\frac{\sqrt{2\bar{v}s}}{2}+\mu n. \end{aligned}$$

Given the product variety, a buyer’s surplus from using the one-sided retailer increases in search cost s: An increase in search cost reduces the reservation value $w^\star$, and thus decreases the expected matched product value $(w^\star +\bar{v})/2$ of the one-sided retailer. However, an increase in search cost also reduces the product prices of the one-sided retailer. Thus, an increase in search cost generates conflicting effects on buyers’ surplus from joining the one-sided retailer. In addition, the effect on the product price is stronger than the effect on the reservation value. Consequently, the expected surplus of product purchases is increasing in consumers’ search costs. This is exactly the opposite of the two-sided platform, where a buyer’s surplus decreases in search cost.

This different effect of search cost on the buyer’s surplus mainly comes from the diverse pricing mechanisms of the two intermediary types: On the two-sided platform, the product price is determined by individual sellers. An increase in search cost generates less competition among sellers and increases product prices. Thus, an increase in search cost reduces the expected product value while increasing product prices on the two-sided platform. Both effects decrease buyers’ surplus from entering the market. When the intermediary is a one-sided retailer, the product price is coordinated by the intermediary, which aims to maximize total profit. Thus an increase in search cost reduces the product prices and moves the buyers’ surplus in the opposite direction to that on the two-sided platform.

Following the same setup as on the two-sided platform, we assume that there is a continuum of l potential buyers in the market and that their outside option follows the uniform distribution in [0, H]. Remembering that L is used to denote l/H, we derive the aggregate demand for the platform as

$$\begin{aligned} Q= \frac{L[2 \mu n+ \sqrt{2\bar{v} s}]}{2}, \end{aligned}$$

(11)

where the number of buyers is increasing in search cost.

3.3 Platform Optimization

Last, the one-sided retailer maximizes its profit by choosing the optimal product variety: n. When the intermediary independently offers n brands and sells directly to buyers, we assume that it faces a corresponding cost of K(n): We let this operation cost take the format $K(n)=k n^2$, such that it is strictly increasing and convex in the number of brands n. When the parameter $k>0$ increases, it becomes more expensive for the one-sided retailer to offer additional brands. With this setup, the profit of the one-sided retailer equals

$$\begin{aligned} \pi _p=p *Q-k n^2, \end{aligned}$$

where the first term denotes the product revenue and the last term measures operation costs.

Solving the first-order condition, we derive the intermediary’s optimal strategy as

$$\begin{aligned} n_o^\star = \frac{\mu L(\bar{v}-\sqrt{2\bar{v}s})}{2k}. \end{aligned}$$

(12)

As is shown in Eq. (12), the optimal variety $n_o^\star$ of the one-sided retailer is increasing in buyers’ preference for variety. When buyers have strong preferences for variety, the intermediary is motivated to introduce more brands to its market. In addition, we find that the equilibrium product variety is decreasing in buyers’ search costs. This is due to the fact that an increase in search cost reduces the equilibrium product price, and thus decreases the additional benefit that accompanies additional brands.

The resulting number of buyers at equilibrium then equals

$$\begin{aligned} Q_o^\star = \frac{L(k\sqrt{2s\bar{v}}+L\mu ^2\bar{v}-L\mu ^2 \sqrt{2s\bar{v}})}{2k}. \end{aligned}$$

(13)

The influence of search cost on the number of buyers depends on buyers’ preference for variety. When buyers have strong preferences for variety, the equilibrium number of buyers is decreasing in buyers’ search costs. Otherwise, it is the opposite.

As is indicated in Eq. (11), an increase in search cost increases the utility that is derived from product purchase. However, it also decreases the product variety, as is shown in Eq. (12), and thus the utility derived from variety decreases. Furthermore, the strength of this product variety effect is determined by buyers’ preferences for variety. When buyers have strong preferences for variety, the influence of search cost on product variety is stronger, and thus the number of buyers decreases in search cost. Otherwise, the influence on utility in product purchase is more dominant, and the number of buyers increases in search cost.

Last, we further investigate the one-sided retailer’s profit: Plugging in the equilibrium product variety from Eq. (12), we are able to simplify the one-sided retailer’s profit as

$$\begin{aligned} \pi _p^o\star =\frac{L\bar{v} (\sqrt{\bar{v}}-\sqrt{2s}) (2k\sqrt{2s}-L\mu ^2\sqrt{2s}+L\mu ^2\sqrt{\bar{v}})}{4k}. \end{aligned}$$

As is shown in Fig. 4, the influence of search cost on the retailer’s profit critically depends on buyers’ preference for variety: When the preference for variety is low, the platform’s profit at first increases in search cost and then decreases; while when the preference for variety is sufficiently strong, lower search costs are always more profitable for the one-sided retailer.

Fig. 4

One-Sided Platform Profit. Note: The platform profits in this figure are based on parameter values $\bar{v}=100$, $k=1$, $L=1$, and $\mu =0.5 (3)$ for the low (high) preference variety case. The search costs are restricted to the corresponding interval where buyers search actively

On the one hand, a reduction in search cost enables the one-sided retailer to charge higher product prices. Each product becomes more profitable, and the product variety in the market also increases. On the other hand, buyers incur higher product prices, which reduces the aggregate demand. When buyers have high preferences for variety, the benefit from greater product variety offsets the demand loss from high product prices. Thus the one-sided retailer becomes better off when search cost decreases. However, when buyers’ preference for variety is low, the demand loss from high product prices becomes dominant. In this context, a decrease in search cost lowers the aggregate demand for the intermediary. This reduction in aggregate demand decreases the additional benefit induced by search cost reduction. As a consequence, the platform’s profit at first increases in search cost and then decreases.

Specifically, our finding can be summarized as:

Proposition 2

When the intermediary operates as a one-sided retailer, reducing buyers’ search costs does not necessarily increase the retailer’s profit:

• If consumers’ preferences for variety are low such that $\mu <\sqrt{\frac{k}{L}}$, then the platform’s profit is concave in search costs: There exists some $s^o\in (0, \frac{2\bar{v}}{9})$ such that the platform’s profit increases when $s<s^o$ and decreases otherwise.

• If consumers’ preferences for variety are high such that $\mu >\sqrt{\frac{k}{L}}$, then the platform’s profit is convex and always decreasing in search cost.

Proof

See Appendix A.3$\square$

According to our findings, the influence of search cost on the one-sided retailer can be differentiated from the effect on the two-sided platform: We observe a negative relation between the intermediary’s profit and search cost in the two-sided market when buyers’ preferences for variety are low; while in the one-sided market, the retailer’s profit exhibits an inverted U-shaped relationship with the search cost. Although the intermediary’s profit is monotonic in search cost in both markets when buyers’ preferences for variety are sufficiently strong, their shapes are different: The two-sided platform’s profit increases in search cost when buyers’ preferences for variety are high, while the one-sided retailer’s profit is the opposite.

4 Discussion and Comparison

In the previous sections, we analyze two intermediary types that are critical and prevalent in digital markets: One is the two-sided platform, whose main role is to connect sellers and buyers and facilitate transactions among them. For each transaction, the two-sided platform charges sellers corresponding commission fees. In exchange, it provides sellers with access to potential buyers, and the sellers retain control over the transactions. The other is the one-sided retailer that buys products from sellers first and then resells them to buyers. Different from the two-sided platform, the one-sided retailer is the owner of all of the products. Thus, the market remains in full control of the retailer, who directly collects all of the profits from product sales.

4.1 Market Comparison

Our paper finds that a reduction in search cost does not necessarily enable the intermediary to obtain higher profits. Indeed, the impact of search cost on the intermediary’s profit is jointly determined by both the intermediary type and its buyers’ preferences for variety. When the preferences for variety are high, the two-sided platform’s profit increases in search cost, while the one-sided retailer’s profit is the opposite. When the preferences for variety are low, the profit of the two-sided platform decreases in search cost while the one-sided retailer exhibit a non-monotonic relationship with the search cost: The one-sided retailer’s profit first increases then decreases in buyers’ search cost.

The varied influences of search cost on market outcomes mainly come from the highly diverse market controls for the two intermediary types. To better understand the results, let us compare how search costs affect the product prices and participation in those two market types respectively: When the intermediary operates as a two-sided platform, it gives market controls to third-party sellers: Each seller optimizes the product price to maximize its individual profit. A decrease in search cost motivates buyers to search more and generates more elastic demand. As a consequence, there is more competition among sellers which motivates them to decrease product prices. Given the commission fees, the reduction in search cost decreases sellers’ expected profit for joining the platform, and thus the number of sellers decreases.

When the intermediary operates as a one-sided retailer, the intermediary has full control over the market. Specifically, the intermediary directly determines all product prices and the number of brands in the market. Instead of charging sellers fees, now the intermediary focuses more on the total revenue collected in product selling. Compared with the two-sided platform where product prices are determined independently by individual sellers, the one-sided retailer has the advantage of coordinating the product prices in its market.

As a consequence, when there is a decrease in search cost, the expected product value of all buyers increases. As a consequence, it is optimal for the one-sided retailer to increase all product prices simultaneously. With higher product prices, the benefit to the retailer that accompanies an additional brand increases, and therefore a reduction in search cost also increases product variety. Specifically, our results can be summarized as follows.

Corollary 1

An increase in search cost moves the product variety of the two intermediary types in opposite directions:

• An increase in search cost increases the product variety on the two-sided platform.

• An increase in search cost decreases the product variety of the one-sided retailer.

A useful hypothetical example could be when buyers have no search costs: On the two-sided platforms, buyers would keep searching until the whole market is exhausted. Thus the market then switches to a perfectly competitive structure and there is no profit left for sellers. As a consequence, the market converges to the trivial case where no seller is willing to use the platform, as was indicated in Eq. (5). Unlike the two-sided platform where no seller will join the market, the one-sided retailer can successfully organize the market and reach a positive profit even when buyers have no search cost. Indeed, the retailer is able to achieve the highest product price $\bar{v}$ if $s=0$, as buyers find better-matched products and the retailer can extract all surplus from buyers. Consequently, the one-sided retailer is motivated to reach a sufficiently high product variety when buyers have no search costs.

In addition to the search cost, it is also interesting to examine the influence of the cross-group network effect on the platform. To investigate, we study how buyers’ preferences for variety affect the platform’s profit and find:

Corollary 2

The intermediary’s profit is always increasing in buyers’ preferences for variety in both market types.

Proof

See Appendix A.4$\square$

4.2 Welfare Analysis

In our model, sellers always have zero profit due to the free entry condition, and consumer surplus is increasing in the number of buyers that join the market. Since most policy makers rarely care about monopoly profit but focus more on buyer surplus, we first investigate how search costs affect the equilibrium number of buyers for the two intermediaries and then compare them.

When there is a decrease in search cost, a buyer’s surplus of joining the intermediary incurs multiple effects: First, a change in search cost affects the search strategy and product price. Therefore, it affects the utility that a buyer derives from product purchases. Second, changes in search cost potentially affect the number of brands in the market, and therefore, the utility that is derived via product variety changes. Furthermore, our previous analysis also shows that a change in search costs moves these two effects in the opposite directions for the different market types.

Our model indicates that a reduction in search cost does not always benefit buyers or increase consumer surpluses. The results can be easily derived from Eqs. (8) and (13) and summarized as follows:

Corollary 3

The influence of search costs on consumer surplus is determined by the intermediary type and the magnitude of buyers’ preferences for variety:

• If buyers’ preferences for variety are low such that $\mu <\frac{3F}{2}$, then the aggregate demand and the consumer surplus on the two-sided platform are strictly decreasing in search cost.

• If buyers’ preferences for variety are high such that $\mu >\frac{3F}{2}$, then the aggregate demand and the consumer surplus on the two-sided platform are strictly increasing in search cost.

• If buyers’ preferences for variety are low such that $\mu <\sqrt{\frac{k}{L}}$, then the aggregate demand and the consumer surplus in the one-sided market are strictly increasing in search cost.

• If buyers’ preferences for variety are high such that $\mu >\sqrt{\frac{k}{L}}$, then the aggregate demand and the consumer surplus in the one-sided market are strictly decreasing in search cost.

On the two-sided platform, a reduction in search cost increases expectation in product value, reduces product prices, and thus increases buyers’ utility in product purchase. It also reduces product variety and corresponding utility: The strengths of these two conflicting effects depend on buyers’ preferences for variety. When buyers’ preferences for variety are low, the effect on product purchases is dominant, and the number of buyers decreases in search cost; when buyers’ preferences for variety are high, the effect on product variety is stronger, and the number of buyers increases in search costs.

When the intermediary operates as a one-sided retailer, it is the opposite: A reduction in search cost increases product prices and consequently reduces buyers’ utility in product purchases. However, lower search cost also increases the one-sided retailer’s product variety and increases buyers’ utility. As is true for the two-sided platform, the relative strength of these two conflicting effects is determined by buyers’ preferences for variety. When the preferences for variety are strong, the variety effect is dominant, and the number of buyers decreases in search cost. Otherwise, it is the opposite.

We then further compare the consumer surplus in the two market types: Our comparison focuses on the scenario where $s\in (2F^2\bar{v}/(3F+2\mu )^2, 2\bar{v}/9)$: Thus both markets have positive numbers of sellers (brands) and transactions. Otherwise, it is clear that the one-sided retailer will generate more surplus because there are either no sellers or no transactions on the two-sided platform, as was shown in the previous analysis.

Proposition 3

Consumer surplus on the two-sided platform could be either higher or lower than that in the one-sided market, depending on the retailer’s costs, buyers’ preference for variety, and search costs:

• If buyers’ preferences for variety are low such that $\mu <\frac{F}{2}$, then consumer surplus for the one-sided retailer structure is always higher.

• If buyers’ preferences for variety are medium such that $\frac{F}{2}\le \mu <F$ and buyers’ search costs satisfy $s \ge \frac{2F^2\bar{v}}{(5F-2u)^2}$, the consumer surplus for the one-sided retailer structure is always higher than the two-sided platform. Otherwise, there exists a retailer cost cutoff $\bar{k}>0$ such that the consumer surplus in the one-sided retailer is higher than the two-sided platform when $k< \bar{k}$ and lower when $k> \bar{k}$.

• If buyers’ preferences for variety are high such that $\mu >F$, then the consumer surplus for the one-sided retailer structure is higher than the two-sided platform when $k< \bar{k}$. Otherwise, it is the opposite.

Proof

See Appendix A.5$\square$

Our results indicate that consumer surpluses in the two market types are jointly determined by buyers’ preferences for variety, their search cost, and the retailer’s operating cost. In the one-sided market, the consumer surplus is decreasing in the retailer’s operating cost k, and is always larger than that in the two-sided market when the operating cost approximates zero. However, what remains ambiguous is the relationship between the two market types in consumer surplus when the retailer’s operating cost becomes extremely high. If the number of buyers in this extreme case is still higher than that on the two-sided platform, the surplus in the one-sided market is always higher. Otherwise, there exists an operating consumer cost cutoff $\bar{k}$ such that the consumer surplus in the one-sided market is higher when the operating cost $k<\bar{k}$ and is lower otherwise.

To illustrate, let us check the number of buyers in the extreme case when the retailer’s operating cost equals infinity. In this scenario, we find that the number of buyers in the one-sided market equals $L\sqrt{s \bar{v}}/\sqrt{2}$, which is an increasing function in search cost s; while as was shown in Eq. (8), the number of buyers on the two-sided platform is decreasing in search costs when buyers’ preferences for variety are low such that $\mu <\frac{3F}{2}$. Consequently, given sufficiently high search costs, the consumer surplus in the one-sided market is always higher than that on the two-sided platform.⁶ In all other cases, the consumer surplus in the one-sided market is higher when the operating cost $k<\bar{k}$ and is lower when the operating cost $k>\bar{k}$.

5 Conclusion

This paper compares two critical intermediary types in digital markets. Specifically, they are the one-sided retailer that buys products from sellers and resells them to buyers, and the two-sided platform that enables transactions among users—sellers and buyers–and charges commission fees. The study is mainly motivated by the critical role of intermediaries in digital markets and their increasing adoption of search technology in the e-commerce market. Through the study, we are interested in knowing how a reduction in buyers’ search costs affects the market outcomes for different market types.

To investigate, we build our model upon the existing literature with respect to random search and the economics of intermediaries, with the following novelties: First, the paper contributes to the search literature by endogenizing the demand and supply of the market. Second, we introduce network effects—which are one of the most critical characteristics of digital markets—to the search model by allowing buyers to have positive preferences for variety. Last, we compare two critical types of intermediary in the digital market using an innovative search approach.

Our results show that a decrease in search cost does not necessarily always enable the intermediary to obtain higher profits. More precisely, the influence of search cost on the intermediary’s profit is jointly determined by buyers’ preferences for variety and the intermediary type.

Besides the innovative findings, our research also has limitations and could be extended in many promising directions. For instance, our study focuses on the monopoly case where buyers have to go through the intermediary to make transactions and the market is fully covered. As a consequence, the product variety of the intermediary is critical as consumers cannot get varieties from other places. Furthermore, we also endow the one-sided retailer with the ability to extract all surplus from buyers by assuming that buyers are no longer sensitive to price once entering the market. If the one-sided retailer faces a down-sloping demand, the model becomes more complex and requires further investigation.

To make the model tractable and simple, we also assume that consumers all have the same search cost and share the same belief about product value distribution between 0 and $\bar{v}$, while sellers all have zero marginal cost. However, current intermediaries are able to affect not only the search cost of buyers but also the distribution of products in search results. Thus additional adjustments could be made to investigate several current popular search technologies—including targeted search, machine learning, and personalized listing.

A Appendix

1.1 A.1 Proof of Proposition 1

As shown in Sect. 2, the equilibrium profit of a two-sided platform equals

$$\begin{aligned} \pi _p^{t\star }=\frac{L\left( 2\mu \sqrt{2s\bar{v}}+2F\bar{v} -3F\sqrt{2s\bar{v}}\right) \left( F\bar{v}\sqrt{2}+2\mu \sqrt{s\bar{v}}-3F\sqrt{s\bar{v}}\right) }{8\sqrt{2}\mu F}. \end{aligned}$$

(14)

Thus, the derivative of the two-sided platform’s profit with respect to consumers’ search costs can be simplified as

$$\begin{aligned} \frac{\partial \pi _p^{t\star }}{\partial s} =\frac{L\bar{v} (3F-2\mu )(3F\sqrt{2s}-2\mu \sqrt{2s} -2F\sqrt{\bar{v}})}{8F\mu \sqrt{2s}} \end{aligned}$$

Since the denominator is always positive, the sign of the derivative is determined by its numerator $Num_t(s)=L\bar{v} (3F-2\mu )(3F\sqrt{2\,s}-2\mu \sqrt{2\,s}-2F\sqrt{\bar{v}})$. Our paper focuses on the setting $2F^2\bar{v}/(3F+2\mu )^2<s<2\bar{v}/9$ such that there is positive participation on both the buyer and seller sides. In this context, the expression $3F\sqrt{2s}-2\mu \sqrt{2s}-2F\sqrt{\bar{v}}$ is always negative. Thus we conclude that the platform’s profit is increasing in search cost when $\mu \ge 3F/2$. Otherwise, it is the opposite

Furthermore, we also check the second order derivative to investigate the relationship:

$$\begin{aligned} \frac{\partial ^2\pi _p^{t\star }}{\partial s^2}=\frac{L(3F-2\mu ) \bar{v}^\frac{3}{2}}{8\sqrt{2}\mu s^\frac{3}{2}}. \end{aligned}$$

Thus the platform’s profit is convex and decreases in search cost when $\mu < 3F/2$, while it is concave and increases in search cost when $\mu \ge 3F/2$.

1.2 A.2 Robustness and Extensions

1.2.1 Per-Transaction Fee on Two-Sided Platform

We first investigate the digital market where the two-sided platform charges only a fixed per-transaction fee: t. One illustrating example could be Swappa,⁷ which is a popular platform for mobile device selling. Regardless of selling prices, the platform charges sellers a fixed amount fee for each item that is sold through the platform.

The change of pricing instruments should not influence consumers’ search or transaction decisions. Consequently, consumers follow the same search strategy such that they will stop searching and purchase the first product whose value is equal to or higher than the reservation value $w^\star$. Correspondingly, the individual demand of a particular firm whose price is p while other firms charge $p^{\star }$ still equals $D(p,p^{\star },n)=Q[1-G(w(p))]/ n[1-G(w^{\star })]$. When the platform charges a per-transaction fee t, it is equivalent as if the marginal costs of third-party sellers increase by t. Thus, the expected profit of a seller whose price is p when all other firms charge $p^{\star }$ equals

$$\begin{aligned} \pi (p,p^{\star },n)= (p-t)D(p,p^{\star },n). \end{aligned}$$

The maximization of the seller’s profit is equivalent as if he incurs a marginal cost of t. Thus, the optimal product price at the symmetric equilibrium is

$$\begin{aligned} p^{\star }= t+\sqrt{2s\bar{v}}, \end{aligned}$$

(15)

where the price is increasing in consumers’ search cost on the platform.

Given sellers’ pricing strategies at the symmetric equilibrium, we then move forward to investigate buyers’ entry decisions. Buyers join the platform only when they expect to obtain higher utility on the platform than the outside options. Specifically, a buyer’s expected surplus of using the platform equals $E[u]=(w^\star +\bar{v})/2-p^\star +\mu *n$, and thus the number of buyers joining the platform equals $Q=L((w^\star +\bar{v})/2-p^\star +\mu *n)$. When the platform changes pricing instruments, consumers expect the same matched value $(w^\star +\bar{v})/2$ between their taste and product. However, the product price and the number of sellers change correspondingly.

As for sellers, a third-party seller has to pay a fixed per-transaction fee t. Thus the markup of a seller equals $\sqrt{2s\bar{v}}$, and his expected net profit is

$$\begin{aligned} \pi ^i=\sqrt{2s\bar{v}} *\frac{Q^e}{n}-f. \end{aligned}$$

Sellers continue entering the market until the expected profit equals zero; thus the number of sellers at equilibrium can be simplified as

$$\begin{aligned} n=\frac{\sqrt{2s\bar{v}}}{F}. \end{aligned}$$

(16)

Given the per-transaction fee, the number of sellers is strictly increasing in buyers’ search costs. An increase in search cost motivates buyers to search less and generates less elastic demand. As a consequence, each seller has more monopolistic power and expects a higher profit from joining the platform.

Last, we study the platform’s optimal pricing strategies and how the search cost influences market outcomes. From the equilibrium number of sellers in Eq. (16) and the product price in Eq. (15), we are able to derive the aggregate demand at equilibrium as

$$\begin{aligned} Q= L\left[ \bar{v}-\frac{3\sqrt{2s\bar{v} }}{2}-t +\frac{\mu \sqrt{2s\bar{v} }}{F}\right] , \end{aligned}$$

and the platform’s profit equals $\pi _p=t*L[\bar{v} -\frac{3\sqrt{2\,s\bar{v} }}{2}-t+\frac{\mu \sqrt{2\,s\bar{v} }}{F}]$. Solving for the FOC, we derive the optimal pricing strategy as:

$$\begin{aligned} t^\star =\frac{\bar{v}}{2}+\frac{\sqrt{2s\bar{v} }(2\mu -3F)}{4F} \end{aligned}$$

and the resulting platform’s profit can be simplified as

$$\begin{aligned} \pi _p^\star =L*\left( \frac{\bar{v}}{2} +\frac{\sqrt{2s\bar{v} }(2\mu -3F)}{4F}\right) ^2. \end{aligned}$$

As is shown in the equation, how a reduction in search cost affects the aggregate demand and per-transaction fee remains uncertain. To investigate, we compute the derivative of the platform’s profit with respect to the search cost:

$$\begin{aligned} \frac{\partial \pi _p^\star }{\partial s}=\frac{L\bar{v}(2\mu -3F) (F\sqrt{2\bar{v}}+2\mu \sqrt{s}-3F\sqrt{s})}{8F^2\sqrt{s}}. \end{aligned}$$

We find that the derivative is an increasing function with respect to the search cost s. With easy calculation, we note that $\partial \pi _p^\star /\partial s$ is always positive when $\mu \ge 3F/2$; otherwise, it is the opposite. Thus when buyers’ preferences for variety are high such that $\mu \ge 3F/2$, the platform’s profit is always increasing in search cost s. When buyers’ preferences for variety are lower than 3F/2, the platform’s profit decreases in search cost s.

To summarize: The influence of search cost on the platform’s profit when the platform charges per-transaction fees is the same as when the platform charges commission fees.

1.2.2 Differential Commission Fees on Two-Sided Platform

We also study the setting where the two-sided platform charges differential commission fees to sellers based on their elasticities. Particularly, we assume there are two types of sellers in the market, with varied entry costs and thus different elasticities. Without loss of generality, we use $F_i$ to denote the “entry cost” for type $i\in \{1,2\}$ comparable to F in the benchmark, and let $F_1>F_2$. To keep the model tractable, we let the two types of sellers have mutually independent demander bases: A demander who considers one type of seller will not consider the other type of seller. The platform can perfectly tell the type of sellers and charge a commission fee $d_i$ for type i.

In this context, the transaction decisions are the same as the benchmark: Only the demander entry and platform optimization change. The number of sellers and the aggregate demand equal

$$\begin{aligned} n_i= \frac{(1-d_i)\sqrt{2s\bar{v}}}{F_i} \text { and } Q_i=L\left( \bar{v}-\frac{3\sqrt{2s\bar{v} }}{2} +\frac{\mu (1-d_i)\sqrt{2s\bar{v}}}{F_i}\right) \end{aligned}$$

for type i. Correspondingly, a two-sided platform’s profit is now:

$$\begin{aligned} \pi _p=(d_1*p^\star )*Q_1+(d_2*p^\star )*Q_2. \end{aligned}$$

Following the same procedure, we derive the optimal pricing strategy as:

$$\begin{aligned} d_i^{\star }=\frac{F_i\sqrt{2\bar{v}} +2\mu \sqrt{s}-3F_i \sqrt{s}}{4\mu \sqrt{s}} \end{aligned}$$

and the resulting equilibrium profit could be simplified as

$$\begin{aligned} \pi _p^{{\star }}= & {} \frac{L\bar{v}}{8 \sqrt{2} F_1 F_2 u}[-9 \sqrt{2}F_1^2 F_2 s - 9 \sqrt{2} F_1 F_2^2 s + 24 \sqrt{2} F_1 F_2 s u -4 \sqrt{2} F_1 s u^2 - 4 \sqrt{2} F_2 s u^2 \nonumber \\{} & {} + 12 F_1^2 F_2 \sqrt{s\bar{v}} + 12 F_1 F_2^2 \sqrt{s\bar{v}} -16 F_1 F_2 \sqrt{s \bar{v}} - 2 \sqrt{2} F_1^2 F_2 \bar{v} -2 \sqrt{2} F_1 F_2^2 \bar{v}]. \end{aligned}$$

(17)

To investigate the influence of search cost on the platform’s profit, we calculate the derivative of the platform’s profit with respect to the consumers’ search cost:

$$\begin{aligned} \frac{\partial \pi _p^\star }{\partial s}= & {} \frac{L\bar{v}}{8 \sqrt{s} F_1 F_2 u}\\{} & {} [\sqrt{s} (9 F_1 ^2 F_2 + 9 F_1 F_2^2 - 24 F_1 F_2 u\\{} & {} \qquad + 4 F_1 u^2 + 4 F_2 u^2) - \sqrt{2 \bar{v}} F_1 F_2 (3 F_1 + 3 F_2 - 4 u) ]. \end{aligned}$$

With easy calculation, we find that the derivative is an increasing function with respect to the search cost s. Thus, the derivative is always positive when $u>(3 F_1 + 3 F_2)/4$. Otherwise, we further check the value of $\partial \pi _p^\star / \partial s$ at two extreme points: When $s=2\bar{v}/9$, we have $\partial \pi _p^\star / \partial s=4 \sqrt{2 \bar{v}} uL (F_2 u + F_1 (-3 F_2 + u))/3$; when $s=(2 F_1^2 \bar{v})/(3 F_1 + 2 u)^2$, we have $\partial \pi _p^\star / \partial s=(2 \sqrt{2 \bar{v}} F_1 u L(-9 F_1 F_2 - 3 F_2^2 + 2 F_1 u + 6 F_2 u))/( 3 F_1 + 2 u)$.

Thus, we find that the platform’s profit is always decreasing when $u<3F_1F_2/(F_1+F_2)$. If the buyers’ preference for variety is medium such that $3F_1F_2/(F_1+F_2)<u<(3 F_2 (3 F_1+F_2))/(2 (F_1+3 F_2))$, the platform first decreases in s and then increases. When the preference for variety is relatively high –$u>(3 F_2 (3 F_1+F_2))/(2 (F_1+3 F_2))$—the platform’s profit becomes strictly increasing in search costs.

1.2.3 Prominent Search on Two-Sided Platform

Now we consider the case where one firm—a—is made prominent after buyers enter the two-sided platform. Consequently, the prominent firm is sampled first by all consumers, and all other firms are symmetrically distributed and randomly sampled afterward. Following the existing research, we focus on the equilibria where the prominent firm charges $p_a$ and all other non-prominent firms charge the same price $p_n$. In this context, buyers will behave as the random search case in the benchmark once they reject the prominent firm. Thus each non-prominent firm faces the same profit maximization problem, and the equilibrium price is the same as the benchmark: $p_n=p^{\star }=\sqrt{2s\bar{v}}$.

The expected surplus from purchasing from a non-prominent firm is $w^{\star }-p_n$ and the demand $D_a(p_a,p_n)$ for the prominent firm equals

$$\begin{aligned} D_a(p_a,p_n)=Q[1-G(w(p_a))], \end{aligned}$$

where $w(p_a)=w^{\star }-p_n+p_a$ is the reservation value of the prominent firm. If the product value of the prominent firm is higher than $w(p_a)$, its expected utility from purchases is higher than the non-prominent firms, and thus it is optimal for buyers to purchase from the prominent firm directly. Maximizing its profit, we find that the prominent firm’s pricing strategy is the same as the non-prominent firm: $p_a=p^{\star }$. However, its demand is much larger and equals $Q(1-G(w^{\star }))$. Since all other firms are symmetric, the non-prominent firms share the remaining demand equally. Clearly, it is much more profitable to be the prominent firm, and sellers bid for this prominent position until no one derives any profit from being prominent.⁸ Consequently, the expected profit of sellers is the same as the benchmark, and so is a buyer’s expected utility from joining the market.

The platform collects both the commission fees and the bid from the prominent firm and derives a profit

$$\begin{aligned} \pi _p^\star =d p^{\star } Q*G(w^{\star })+p^{\star }(1-G(w^{\star }))Q, \end{aligned}$$

where: the first term is the revenue from commission fees, and the second term is the additional revenue from selling the prominent position. Solving for the FOC, we derive the optimal pricing strategy as:

$$\begin{aligned} d^\star =\frac{4 \sqrt{2} s u - 2 u \sqrt{s\bar{v}} -F (3 \sqrt{2} s - 5 \sqrt{s\bar{v}} + \sqrt{2} v)}{4 \sqrt{2} s u - 4 u \sqrt{s\bar{v}}}, \end{aligned}$$

and the resulting platform’s profit can be simplified as

$$\begin{aligned} \pi _p^\star =L\frac{ \sqrt{\bar{v}} }{8 F u (\sqrt{\bar{v}} -\sqrt{2s} )} \left( 3 \sqrt{2} F s - 5 F \sqrt{\bar{v}} + 2 \sqrt{\bar{v}} +\sqrt{2} F v \right) ^2. \end{aligned}$$

To investigate the effects of search cost on the platform’s profit, we calculate the derivative of the platform’s profit with respect to the search cost:

$$\begin{aligned} \frac{\partial \pi _p^\star }{\partial s}= & {} \frac{L(3 \sqrt{2} F s - 5 F \sqrt{\bar{v}} + 2 \sqrt{\bar{v}} +\sqrt{2} F v)}{2 \sqrt{s} (\sqrt{2s} - \sqrt{\bar{v}})^2}\\{} & {} \left( -18 F s + \sqrt{s}(17 F \sqrt{2 \bar{v}} - 2 u \sqrt{2 \bar{v}} ) - 8 F v + 4 u v\right) . \end{aligned}$$

We find that the sign of the derivative is the same as the equation $Z(s)=-18 F s + \sqrt{s}(17 F \sqrt{2 \bar{v}} - 2 u \sqrt{2 \bar{v}} ) - 8 F v + 4 u v$. When $u>2F$, the derivative can easily be proved to be always positive in the interval $s\in (2F^2\bar{v}/(3F+2\mu )^2, 2\bar{v}/9)$. When $u<F/4$, we find that Z(s) increases in s and its value at the maximum $s=2\bar{v}/9$ is negative. Thus, the platform’s profit always decreases in search cost.

When $F/4<u<2F$, Z(s) first increases in s and then decreases. Furthermore, its value at $s=0$ is negative, and the value of Z(s) is always positive at $s=2\bar{v}/9$. We further check its value at $s=2F^2\bar{v}/(3F+2\mu )^2$. Specifically, we find that the sign depends on buyers’ preference for variety u: There exists a $u_p>0$ such that $Z(s)>0$ if $u>u_p$. Otherwise, it is the opposite. Thus, when $u>u_p$, the platform’s profit always increases in search cost, while when $F/4<u<u_p$, the platform’s profit first decreases and then increases in search cost.

In summary, the platform profit decrease in search cost when buyers’ preference for variety is sufficiently low: $u<F/4$. When buyers’ preference for variety is sufficiently high such that $u>u_p$, the platform profit always increases in search cost. When buyers’ preferences for variety are between these two extremes, the platform’s profit first decreases and then increases in search costs.

1.2.4 Robustness in One-Sided Retailer

When the one-sided retailer does not have all of the bargaining power, or the sellers anticipate the platform’s eventual retail prices on their products and adjust their wholesale prices accordingly, its marginal cost is no longer zero. Specifically, we assume that the retailer incurs a marginal cost of m for each product. The presence of marginal cost does not affect price coordination or buyer entry; thus the equilibrium product price and number of buyers in the one-sided market still equal to: $p^\star =\bar{v}-\sqrt{2\bar{v}s}$ and $Q=\frac{L[2 \mu n+ \sqrt{2\bar{v} s]}}{2}$.

In this context, the profit of the one-sided retailer now equals $\pi _p=(p-m) *Q-k n^2$, and the maximization derives the optimal number of products in the market:

$$\begin{aligned} n_o^\star = \frac{\mu L(\bar{v}-\sqrt{2\bar{v}s}-m)}{2k}, \end{aligned}$$

and the resulting profit can be simplified as

$$\begin{aligned} \pi _p^{o\star }=\frac{L(\bar{v}-\sqrt{2\bar{v}s}-m) (-L m \mu ^2 + 2 k \sqrt{2\bar{v}s} -L \mu ^2 \sqrt{2\bar{v}s} + L \mu ^2 \bar{v})}{4k}. \end{aligned}$$

To investigate the influence of search cost on the retailer’s profit, we calculate its derivative and derive:

$$\begin{aligned} \frac{\partial \pi _p^{o\star }}{\partial s} =\frac{-kLm\sqrt{2\bar{v}}+L^2m\mu ^2 \sqrt{2\bar{v}} -4kL\bar{v}\sqrt{s}+2 L^2 \mu ^2 \bar{v}\sqrt{s} +k L \sqrt{2\bar{v}^3} - L^2 \mu ^2 \sqrt{2\bar{v}^3}}{4k\sqrt{s}}. \end{aligned}$$

Based on the equation, we find that the value of $\partial \pi _p^{o\star }/\partial s$ at $s=0$ depends on the strength of consumers’ preference for variety $\mu$. Furthermore, we also find that the second derivative—$\partial ^2 \pi _p^{o\star }/\partial s^2=L\sqrt{\bar{v}}(L\mu ^2-k)(\bar{v}-m)/4k\sqrt{2\,s^3}$—depends on $\mu$. When $\mu >\sqrt{k/L}$, the value of $\partial \pi _p^{o\star }/\partial s$ at $s=0$ is negative, and the derivative is strictly increasing in s. However, when $\mu <\sqrt{k/L}$, the value of $\partial \pi _p^{o\star }/\partial s$ at $s=0$ is positive, and the derivative is strictly decreasing in s. As for its value at $s=\bar{v}/2$, it is always negative with a sufficiently high $\bar{v}$.

Thus, we are able to derive the same conclusion that the retailer’s profit is always decreasing in search cost when $\mu >\sqrt{k/L}$. Otherwise, there must exist some $s'_o\in (0, \bar{v}/2)$ such that platform’s profit increases when $s<s'_o$ and decreases when $s>s'_o$. The influence of search cost on the intermediary’s profit depends critically on buyers’ preference for variety.

We also consider the possibility that the one-sided retailer allows sellers to bid for a prominent position, such that it is sampled first by all consumers prior to other brands. Only this prominent seller can display its product in the market and sell to buyers, all other products remain to be sold by the retailer directly to buyers. As is true for the prominent search on the two-sided platform, the pricing strategy of non-prominent products sold by the retailer is not affected. Furthermore, the pricing strategy of the prominent seller is also the same as the non-prominent product determined by the retailer: The price is still equal to $p^\star =\bar{v}-\sqrt{2\bar{v}s}$ for all products after introducing the prominent product. However, the prominent product is able to attract $1-G(w^\star )$ portion of the aggregate demand, while the non-prominent products share evenly the rest.

With the same search strategy, product value, and price, a buyer’s expected utility in using the one-sided retailer is the same. Thus the aggregate demand for the retailer is also $Q= L[2 \mu n+ \sqrt{2\bar{v} s}]/2$. Sellers continue increasing their bids until the prominent position is no longer profitable. Moreover, they no longer need to pay entry costs in the one-sided retail market because the retailer will now be in charge of the market operation. In this context, the retailer is able to extract all profit $p^\star (1-G(w^\star ))Q$ from the prominent seller, and the prominent seller’s profit equals

$$\begin{aligned} \pi _p=p^\star G(w^\star ) Q+p^\star (1-G(w^\star ))Q-k n^2, \end{aligned}$$

where: the first term is its revenue from product selling; the second term is the fee from the prominent position; and the last term is the operating costs. With simple calculations, we can easily show that the platform’s profit maximization problem—$\max _n p^\star *Q-k n^2$—is the same as the benchmark, and we can derive the same conclusion.

1.3 A.3 Proof of Proposition 2

As was shown in Sect. 3, the equilibrium profit of a one-sided retailer equals

$$\begin{aligned} \pi _p^{o\star }=\frac{L\bar{v} (\sqrt{\bar{v}}-\sqrt{2s}) (2k\sqrt{2s}-L\mu ^2\sqrt{2s}+L\mu ^2\sqrt{\bar{v}})}{4k}. \end{aligned}$$

Thus, the derivative of the one-sided retailer’s profit with respect to consumers’ search cost equals

$$\begin{aligned} \frac{\partial \pi _p^{o\star }}{\partial s} =\frac{1}{4k\sqrt{s}}[\sqrt{2\bar{v}^3}L(k-L\mu ^2) +2\sqrt{s}(L^2\mu ^2\bar{v}-2kL\bar{v})]. \end{aligned}$$

To investigate, we first check the value of $\partial \pi _p^{o\star }/\partial s$ at $s=\bar{v}/2$. We find that the derivative at $s=\bar{v}/2$ is always negative. However, the value of $\partial \pi _p^{o\star }/\partial s$ at $s=0$ depends on the strength of consumers’ preferences for variety $\mu$. Furthermore, we also find that the second derivative—$\partial ^2 \pi _p^{o\star }/\partial s^2=L\sqrt{\bar{v}^3}(L\mu ^2-k)/ 4k\sqrt{2\,s^3}$—depends on $\mu$. When $\mu >\sqrt{k/L}$, the value of $\partial \pi _p^\star /\partial s$ at $s=0$ is negative and the derivative is strictly increasing in s. However, when $\mu <\sqrt{k/L}$, the value of $\partial \pi _p^\star /\partial s$ at $s=0$ is positive, and the derivative is strictly decreasing in s.

Thus, we are able to conclude that the retailer’s profit is always decreasing in search cost when $\mu >\sqrt{k/L}$. Otherwise, there must exist some $s_o\in (0, \bar{v}/2)$ such that the platform’s profit increases when $s<s_o$ and decreases when $s>s_o$. The influence of search cost on the intermediary’s profit depends critically on buyers’ preferences for variety.

1.4 A.4 Proof of Corollary 2

As was shown in Sect. 2, the equilibrium profit of a two-sided platform equals

$$\begin{aligned} \pi _p^{t\star }=\frac{L\left( 2\mu \sqrt{2s\bar{v}}+2F\bar{v} -3F\sqrt{2s\bar{v}}\right) \left( F\bar{v}\sqrt{2}+2\mu \sqrt{s\bar{v}}-3F\sqrt{s\bar{v}}\right) }{8\sqrt{2}\mu F}. \end{aligned}$$

(18)

Thus, the derivative of the two-sided platform’s profit with respect to consumers’ preference for variety $\mu$ can be simplified as

$$\begin{aligned} \frac{\partial \pi _p^{t\star }}{\partial \mu } =\frac{L\bar{v} (-9\sqrt{2}F^2s+4\sqrt{2}s\mu ^2 +12F^2\sqrt{s\bar{v}}-2\sqrt{2}F^2\bar{v})}{8\sqrt{2}F\mu ^2}. \end{aligned}$$

Since the denominator is always positive, the sign of the derivative is determined by the equation $N(\mu )=-9\sqrt{2}F^2\,s+4\sqrt{2}s\mu ^2 +12F^2\sqrt{s\bar{v}}-2\sqrt{2}F^2\bar{v}$, which is an increasing function with respect to buyers’ preference for variety $\mu$. Given $2F^2\bar{v}/(3F+2\mu )^2<s<2\bar{v}/9$, we also find N to be always positive at $\mu =0$. As a consequence, the platform’s profit always increases in buyers’ preference for variety $\mu$.

As was shown in Sect. 3, the equilibrium profit of a one-sided retailer equals

$$\begin{aligned} \pi _p^{o\star }=\frac{L\bar{v} (\sqrt{\bar{v}}-\sqrt{2s}) (2k\sqrt{2s}-L\mu ^2\sqrt{2s}+L\mu ^2\sqrt{\bar{v}})}{4k} \end{aligned}$$

It can be easily observed that the retailer’s profit increases in buyers’ preference for variety.

1.5 A.5 Proof of Proposition 3

As was analyzed in the previous sections, the equilibrium number of buyers in two markets is

$$\begin{aligned} Q_t^{\star }=\frac{L\left( 2\mu \sqrt{2s\bar{v}}+2F\bar{v} -3F\sqrt{2s\bar{v}}\right) }{4F} \text {and }Q_o^\star =\frac{L(k\sqrt{2s\bar{v}}+L\mu ^2\bar{v}-L\mu ^2 \sqrt{2s\bar{v}})}{2k} \end{aligned}$$

respectively. Thus, the difference between $Q_t^{\star }$ and $Q_o^\star$—$\Delta Q=Q_t^{\star }-Q_o^{\star }$—can be simplified as

$$\begin{aligned} \Delta Q=\frac{L\sqrt{\bar{v}}}{4Fk} [-2 F L u^2 (-\sqrt{2s} + \sqrt{\bar{v}}) +k (-5 \sqrt{2s} F + 2 u \sqrt{2s} + 2 F \sqrt{\bar{v}})]. \end{aligned}$$

The sign of $\Delta Q$ is determined by the equation $\Delta Q'=-2 F L u^2 (-\sqrt{2\,s} + \sqrt{\bar{v}}) + k (-5 \sqrt{2\,s} F + 2 u \sqrt{2\,s} + 2 F \sqrt{\bar{v}})$, whose value at $k=0$ must be negative. When $u>5F/2$, the equation $\Delta Q'$ is always increasing in k. Thus, there must exist $\bar{k}>0$ such that $\Delta Q>0$ when $k>\bar{k}$. Otherwise, it is the opposite.

When $u<5F/2$, the equation $\Delta Q'$ could either be increasing or decreasing in k, depending buyers’ search cost s. Specifically, $\partial Q'/\partial k=-5 \sqrt{2\,s} F + 2 u \sqrt{2\,s} + 2 F \sqrt{\bar{v}}$ is positive when $s<2F^2\bar{v}/(5F-2u)^2$, and is negative when $s>2F^2\bar{v}/(5F-2u)^2$. To guarantee a positive number of sellers and the expected utility from purchase to be positive for the two-sided market, we assume $2F^2\bar{v}/(3F+2\mu )^2<s<2 \bar{v}/9$ throughout the paper.

Comparing the relationship between $2F^2\bar{v}/(5F-2u)^2$ and the interval $(2F^2\bar{v}/(3F+2\mu )^2, 2 \bar{v}/9)$, we notice that $2F^2\bar{v}/(5F-2u)^2< 2F^2\bar{v}/(3F+2\mu )^2$ when $u<F/2$, $2F^2\bar{v}/(3F+2\mu )^2<2F^2\bar{v}/(5F-2u)^2<2 \bar{v}/9$ when $F/2<u<F$, and $2F^2\bar{v}/(5F-2u)^2>2 \bar{v}/9$ otherwise.

In the case $u<F/2$, the equation $\Delta Q'$ is always decreasing in k, and thus the consumer surplus in the one-sided market is always higher than that on the two-sided platform.

In the case $F/2<u<F$, the sign of $\partial Q'/\partial k$ depends on the search costs: The equation $\Delta Q'$ is increasing in k when $s<2F^2\bar{v}/(5F-2u)^2$: The consumer surplus in the one-sided market is higher than the consumer surplus on the two-sided platform if $k<\bar{k}$, and is lower than the consumer surplus on the two-sided platform otherwise. The equation $\Delta Q'$ is decreasing in k when $s>2F^2\bar{v}/(5F-2u)^2$: The consumer surplus in the one-sided market is always higher than the consumer surplus on the two-sided platform.

In the case $u>F$, the equation $\Delta Q'$ is always increasing in k: The consumer surplus in the one-sided market is higher than the consumer surplus on the two-sided platform if $k<\bar{k}$, and is lower than the consumer surplus on the two-sided platform otherwise.