SEARCH FRICTIONS IN MANY-TO-ONE MARKETS∗

Menghan Xu †

March 2020

Abstract

This paper studies how search frictions affect competition and matching efficiency in

many-to-one loan markets where a borrower requires support from multiple investors and

coordination is desired but absent. We develop a dynamic search model and show that

borrowers employ mixed strategies in quoting interest rates. More importantly, we find

that in many-to-one markets, the rate dispersion caused by search frictions facilitates co-

ordination and hence improves allocation efficiency relative to a no friction environment.

Furthermore, we extend the model by allowing the borrowers to choose loan sizes and

show that both price dispersion and size dispersion arise. We further present stylized facts

that consistent with the theoretical predictions by documenting a novel dataset on a large

panel of fundraisers’ behaviors. By structurally estimating the model primitives, we find

that with dispersed rates, the coordination effect can improve the aggregate funding prob-

ability by 28% compared with a random matching context.

Key Words: Search Friction, Many-to-one matching, Coordination.

JEL Classification: C78, D43, D83, L86

∗The author is indebted to Hugo Hopenhayn, Moritz Meyer-ter-Vehn and Pierre-Olivier Weill for their guid-ance and encouragement. The author is also grateful to Simon Board, Ichiro Obara, Tomasz Sadzik, Zhuoran Lu,Yangbo Song, Semih Üslü, Ning Sun, Haojun Yu, and all seminar participants at UCLA, Xiamen University, Shang-hai University of Finance and Economics, and the 3rd Microeconomics Workshop. The author has no relevant ormaterial financial interests that relate to the research described in this paper. This research is supported by theNational Natural Science Foundation of China (grant number 71703137).

†School of Economics and Wang Yanan Institute for Studies of Economics, Xiamen University. (Email: xumh@xmu.edu.cn)

1 Introduction

Since the seminal work by Stigler (1961), search frictions have been used to explain many

economic phenomena. For instance, it has been observed that persistent price dispersion ex-

ists among otherwise homogeneous products in different markets (De los Santos et al., 2012;

Hortaçsu and Syverson, 2004; Kaplan and Menzio, 2015; Lach, 2002). Search frictions prevent

consumers from learning prices perfectly and thereby softens sellers’ competition. While the

majority of the existing research has focused on bilateral transactions, in which each trans-

action requires a match between one seller and one consumer, this paper attempts to study

search frictions in markets where each seller must be matched with many consumers.

The many-to-one matching structure applies to a variety of environments, especially emerg-

ing online financial markets, such as crowdfunding sites and peer-to-peer lending platforms.

Fundraisers in such markets specify financial goals that must be met by attracting sufficient

investors. In this context, the crowd of investors engages in a coordination game, as if the

fundraisers’ financial goal is not met, the funding procedure fails and all involved investors

must search again and reinvest. In this paper, we attempt to study the joint effect of search

and coordination frictions. We show that with search frictions, borrowers1 implement mixed

strategies and induce rate dispersion, which serves as a coordination mechanism for investors

and enhances allocation efficiency.

Specifically, we construct a dynamic many-to-one matching model embedded with fixed-

sample search technology à la Burdett and Judd (1983). The environment consists of borrow-

ers and investors who enter the market over time. Each borrower’s project requires multiple

units of funds, and the per-unit value of the project is the same across all. Upon entry, bor-

rowers post their loan requests and quote interest rates. Meanwhile, investors, each endowed

with just one unit of funding, visit the market and make investment decisions based on their

observations. Once a listed project is matched with a sufficient number of investors, the bor-

rowers and matched investors will leave the market with their respective payoffs. As borrowers

are waiting for investors, they may be thrown out of the market at a constant rate,2 and the

1To keep the paper coherent, we use the borrower and investor to represent the price setter and searcher,respectively. Therefore, a high interest rate is equivalent to a low price. Note that our framework also applies toconsumer-seller environments.

2In most online financial markets, a fixed deadline is required. Because this is not our main focus, we simplify

1

funding procedure fails. Regarding the search technology, each investor randomly encounters

a finite number of listings, observes their sizes and rates and selects one in which to invest.

From the borrowers’ perspective, they are visited by and matched with investors as Poisson

events with an intensity endogenously determined by factors such as market depth and the

listing’s ranking in the investor’s sample.

The model highlights several key features and imperfections in such markets. First, in-

vestors can only sample a limited number of listings. This property makes it possible for all

borrowers to fail to secure funding perfectly. Moreover, it provides the borrowers with some

market power to quote lower interest rates. Second, to study the mixed effect of frictional

search and an absence of coordination, we assume that investors make independent decisions

without observing the listings’ investment history or communicating with others. Instead, in-

vestors form beliefs on the sampled listings’ funding status by observing their interest rates.

We start in the model by letting all borrowers have identical loan sizes, so that they are,

hence, ex ante homogeneous. Accordingly, the only possible variation in an investor’s sample

is the potentially different interest rates. The model produces two main results. First, we show

that a stationary equilibrium exists and is unique. In that equilibrium, borrowers employ mixed

strategies in setting interest rates when investors’ search is imperfect. The result explains how

search frictions induce rate dispersion in the market even if loans are of the same quality. Note

that the magnitude of rate dispersion and search friction do not have a monotonic relationship.

At extremes, the interest rates will converge to zero-profit and monopoly levels, respectively.

Second, in terms of matching efficiency, we show that under the many-to-one matching

environment, when investors make their decisions independently, the rate dispersion caused

by search frictions endogenously creates ex post heterogeneity among the listings, which will

act as a mechanism facilitating investor efforts to sort observed listings. Comparatively, absent

frictional search, borrowers will engage in Bertrand competition such that the sorting mech-

anism disappears because interest rates are identical. Therefore, the proportion of successful

matches in the Bertrand competition scenario will be the same as that in a random matching

environment and lower than that with rate dispersion. In this sense, a certain level of search

the model by incorporating a Poisson death shock, which allows us to derive comparative statics in a stationaryenvironment.

2

friction is beneficial to the allocation efficiency of the market. The welfare implication for the

many-to-one environment departs from that for typical bilateral trading. The rate dispersion

in the one-to-one matching market, as a special case in our model, simply determines the di-

vision of surplus between buyers and sellers. The coordination effect only emerges when one

borrower requires several investors’ support.

Next, we extend the model such that borrowers can choose both interest rate and loan size

when posting listings, aiming to study rate distributions across different sizes of loans. We

first show that both rate dispersion and size dispersion emerge in equilibrium under search

frictions. That is, borrowers are indifferent between posting large and small financial requests

in equilibrium. Intuitively, the former could generate higher surplus if fully funded, while the

latter has a lower risk of funding failure. In addition, the model predicts that borrowers who

post larger listings will also quote higher interest rates, which implies that borrowers consider

the "crowd" effect and therefore provide a risk premium to investors to compensate for the

possibility of investment failure. Specifically, borrowers who issue small loans and those who

issue large loans randomize interest rates within two disjoint intervals. Meanwhile, we show

that such a risk premium is high enough that investors almost surely prefer larger loans to

smaller ones if their sample involves both types.

Our theoretical analysis produces several testable predictions regarding many-to-one mar-

kets. The main empirical challenge is to find ex ante homogeneous financial products and verify

that the rate dispersion is mainly caused by search frictions. We document a novel dataset in-

cluding information on approximately 500,000 listings from one of China’s peer-to-peer loan

platforms, whose unique structure allows us to overcome the difficulty. Briefly speaking, the

agents trading on the platform form a three-tier hierarchical structure. The middle tier agents

issue short-term listings with low interest rates and invest in long-term projects with high rates.

The dealership-like practice allows us to observe the value of borrowing for the short term list-

ings3.

Having unified the qualities of the loans in terms of their main attributes, such as the value

of borrowing, maturity and default risk, we present a series of stylized facts consistent with

model predictions. The first group of evidence concerns the existence of search frictions in

3We introduce the detailed institutional background in Section 6.

3

the market. We observed that persistent rate dispersion is always present and that a large

fraction of high-rate projects were not funded, while some low-rate loans posted during the

same period and exhibiting the same quality were successfully financed. The findings indicate

that the matching process between investors and borrowers is not perfect. We further examine

the detailed patterns of rate dispersion. Similar to Lach (2002), individual and aggregate level

rate dispersion are of similar magnitude. In other words, the main cause of the dispersion

is individual-level randomization rather than some other hidden source of heterogeneity. In

addition, while the average rate is decreasing as the market becomes more competitive, the

dispersion level and the competitiveness of the market demonstrate an inverse-U relationship.

A similar pattern is documented by works such as Pennerstorfer et al. (2020).

The next group of empirical evidence illustrates how loan sizes affect borrowers’ strategy

in quoting interest rates and investors’ decisions. First, loan size and invest rate are positively

correlated, which confirms that borrowers who issue larger loans tend to offer higher returns

to compensate for investor risk. In addition, we observe that the funding speed, as measured

by time between two successive bids, for larger loans is on average faster than that for smaller

loans. It suggests that in equilibrium, investors prefer larger listings since the rates are suffi-

ciently high that the expected return is higher than that of smaller listings.

The last set of facts attempts to reflect whether the investors in the platform are able to

coordinate among themselves without using the interest rate as a facilitator. We find that for

most of the listings, funding speed is negatively correlated with interest rate but independent

of funding progress. This observation is consistent with our theoretical prediction that when

investors are unable to observe a listing’s funding status, the interest rate becomes the main

reference for their decisions. The empirical patterns observed refer to the main finding of

the paper, namely, that many-to-one matching markets are subject to a coordination problem,

which rate dispersion helps to address.

To quantitatively measure the welfare impact due to search frictions compared with the

no-friction environment, we develop a nonparametric method to estimate the search friction

primitive, which is defined as the distribution of investors’ sample size in the model. We show

that the distribution can be uniquely identified and measured by the loans’ funding probabilities

with respect to their percentile rank in the market. Accordingly, we transfer the model from

4

the absolute terms of an interest rate level to the relative terms as a percentile. The benefit

of the transformation is that we need not identify and estimate the parameters regarding the

value of interest rates. The simplification allows us to utilize more observations to calculate

funding probabilities, which increases estimation efficiency. According to the estimation, the

welfare gain is approximately 28% compared with that when rate dispersion is eliminated.

Related Literature

The paper is related to several strands of the literature. First, the theoretical framework dates

back to seminal papers on consumer search models, such as Varian (1980), Burdett and Judd

(1983) and Stahl (1989). Our model shares the common setup in which price investigation is

costly or frictional for buyers, which gives sellers some market power to charge higher prices.

In the equilibria of such frameworks, sellers will employ mixed strategies, thereby generat-

ing price dispersion, even if goods are homogeneous. Dynamic versions of Burdett and Judd

(1983) include works such as the Burdett and Mortensen (1998) on-job-search model and Head

et al. (2012) monetary search model. The difference between dynamic and static models is

that there are frequent entry and exit events.4 In addition to single product search, there is

a strand of research discussing multiproduct search. Namely, consumers look for complemen-

tary products from different sellers (e.g., McAfee (1995) and Zhou (2014)). However, since

the trading volume is not affected, the main impact of the search is how the surplus is split

between the two sides of the market. My work alters the frameworks by assuming that a seller

needs many buyers.5 Under this environment, we have the novel finding that search friction

has a positive effect by facilitating coordination and hence improves allocation efficiency.

Second, the paper is related to empirical studies on price dispersion among homogeneous

goods. The empirical evidence presented in this paper is consistent with many other works.

For instance, Lach (2002) and Kaplan and Menzio (2015) study retail markets and show that

dispersion persistently prevails even after controlling for individual-level heterogeneity. They

investigate the price distribution evolution at the store and household level. We document

a similar fact that individual-level dispersion makes the majority contribution to that at the

4A comprehensive summary and review of the research can be found in Baye et al. (2006).5We can treat borrowers as loan sellers and investors as loan buyers.

5

aggregate level by observing dealers’ repetitive borrowing behaviors over time. In addition,

our theoretical works, as well as many others (e.g., Stahl (1989)), predict that the relationship

between search friction and dispersion is non-monotone. In principle, search friction deter-

mines the intensity of sellers’ competition. The price will converge to perfect competition and

a monopoly level when the search friction is extremely small and large, respectively. Penner-

storfer et al. (2020) studies the gasoline market and measures search frictions as the informa-

tion owned by consumers. Different from their measurement, our study uses market depth6

as the proxy of market competitiveness and shows that the non-monotone pattern also exists.

Meanwhile, there is a growing literature that concerns search behaviors in online markets. For

example, Brynjolfsson and Smith (2000), Clay et al. (2001), and Goolsbee (2001) attempt to

understand the nature of search costs by comparing the degree of price dispersion between

online and traditional retail sectors. Our particular interest lies in how the online market can

promote financial innovations and what role frictional search plays.

Our study is also related to structural estimation regarding search models. The most re-

lated work is Hong and Shum (2006), who estimates a Burdett and Judd (1983) search model

using price data for online book retailers. Other studies include De los Santos et al. (2012),

who use consumers’ web search histories to study their search behavior and conclude that a

non-sequential search model better describes buyers’ behavior in online markets. In the stud-

ies of financial markets, a similar methodology is applied by Hortaçsu and Syverson (2004),

who considers the rate dispersion in S&P500 funds. All the markets focused upon by the above

works feature long-lived sellers and bilateral trading, while my work focuses on a market with

frequent inflows and outflows with many-to-one transactions. In addition, for the purpose of

evaluating welfare impact, our estimation only requires the percentile distribution on prices in-

stead of absolute price levels. This method simplifies the identification procedure and enhances

the estimation efficiency.

Finally, this paper contributes to the growing literature on rapidly developing online fi-

nancial markets such as crowdfunding and peer-to-peer lending. Different from traditional

centralized markets, fundraisers directly attract financial support from investors through plat-

6This proxy is commonly used in dynamic search models such as labor search (Burdett and Mortensen, 1998)and OTC markets (Duffie et al., 2005)

6

forms, and hence, the many-to-one matching pattern emerges. A natural research question is

how the “many” side behaves in the market. For instance, Zhang and Liu (2012) and Freed-

man and Jin (2017) attempt to empirically understand investor dynamics over project funding

cycles and network effects. They conclude that contributions by peer investors will exacer-

bate network externalities and hence affect others’ decisions. In contrast, our work studies

the problem of investors having no information on their peer investors’ decisions. Instead, the

investors form beliefs on the magnitude of the network effects according to information on

prices. Moreover, our work considers not only the dynamics of a project’s funding process but

also the competition among fundraisers. Other papers on such markets include Wei and Lin

(2016), who studies auctions in peer-to-peer lending markets.7 Kawai et al. (2014), who iden-

tifies and estimates a signaling model using data from a US crowdfunding credit platform, and

Strausz (2017), who studies demand uncertainty and moral hazard problems in the crowd-

funding market from a mechanism design perspective. Deb et al. (2019) also considers the

coordination problem in crowdfunding markets and focuses on the social learning aspect. All

of these studies assume that a fundraiser is a monopolist on the borrower side.8 Our frame-

work provides a benchmark by studying competition and stressing the coordination problem

in the studies of such markets.

The remainder of the paper is organized as follows. Section 2 uses a simple 2×2 model to

illustrate the main insight of the joint effect of search and coordination effects. In Section 3,

we introduce the main model and characterize equilibrium. Section 4 conducts comparative

statics and welfare analysis, which provides testable predictions of the model. In Section 5,

we extend the model to study the size effect in the markets. In Section 6, we test the model

predictions by presenting a series of stylized facts and structurally estimate the model for a

counterfactual exercise. Section 7 concludes the paper. Additional empirical tests, technical

proofs, tables and figures are attached in the appendixes.

7All principal platforms have since abandoned this mechanism.8Some census data regarding the crowdfunding market are surveyed in Agrawal et al. (2014) and Kuppuswamy

and Bayus (2018).

7

2 Illustrative Example

We begin with an illustrative example to highlight the intuition of our result. In this simple

environment, there are only two anonymous borrowers, each endowed with a project with

expected value 2v and requiring two units of financial support. The other side has two anony-

mous investors, each having only 1 unit of funds to invest. The game consists of two stages.

At stage 1, the two borrowers quote interest rates simultaneously. At stage 2, the two investors

search and invest independently. Suppose the two investors invest their money in the same

project with rate r; then, the project realizes total surplus 2v. Each investor earns r, and the

borrower retains a profit of 2(v − r). Otherwise, suppose the two investors choose different

borrowers or someone decides not to invest; then, both projects fail. If so, investors realize

outside option r0 < v and borrowers earn zero.

Since there are only two units of funds on the supply side, the market can generate a posi-

tive surplus only if the two investors choose the same project. In the anonymous environment,

however, there exists no information to allow the agents to concentrate their investments ex-

cept for the quoted rates. We will show that search friction can turn interest rates into a

coordination device between investors and improve market efficiency.

Specifically, search frictions are presented in a form in which each investor has a probability

λ ∈ [0, 1] to meet both projects (searchers). With probability 1−λ, one can only randomly

observe one of them (non-searchers). At stage 2, when λ = 0, both investors are randomly

matched with one of the two projects. As long as the interest rates are no less than r0, the

investors will participate and each borrower has a probability of 1/4 of achieving financial

goals; the expected social surplus is v. When λ < 1, there is a positive chance that investors

will compare the two projects. Since all parties are anonymous, the investors tend to invest in

the project with a higher rate. If the two rates are identical, the investors are indifferent and

will choose one of the two projects with equal probability.

Given the investors’ strategy at stage 2, when search is perfect, λ = 1, the first stage be-

comes a Bertrand competition, and both borrowers will quote r = v. Facing the same interest

rates, investors will, in turn, randomly choose one of the projects for investment. Finally, each

project also has a 1/4 probability of being fully funded, and the expected social surplus is still

8

v. When the search is imperfect, λ ∈ (0, 1), then there exists no pure strategy Nash equilibrium

for the borrowers. The logic behind this is simple. Assume the strategy profile is pure with iden-

tical rate r; both borrowers have a 1/4 probability of achieving both investors’ support, and the

expected profit is (v− r)/2. In this case, anyone who raises the interest rate by a small amount

will enhance the probability of success to (1−λ)2× 14 +2λ(1−λ)× 1

2 +λ2 = 1

4(1+λ)2, which

is obviously a profitable deviation. In fact, the unique symmetric equilibrium is in the form of a

mixed strategy with CDF F(r) and r0 as the lower bound, satisfying the following indifference

condition:

2(v− r0)(1−λ)2 = 2(v− r)�

(1−λ)2 +λ(1−λ)F(r)+λ2F(r)�

.

for any r on the support of the strategy space. The equation pins down the closed form of the

CDF F(r) = (1−λ)2

λr−r0v−r . More importantly, in terms of expected social surplus, the probability

that both investors contribute to the same project is

λ2 +(1−λ2)×12=

12+λ2

2.

The term λ2 represents the probability that both investors are searchers. Since the two projects

always have different interest rates under a mixed strategy equilibrium, the investors will tacitly

coordinate and invest in the project with a higher rate. However, as long as one of the investors

is a non-searcher, whose probability is 1−λ2, there is only a 1/2 chance that the two investors

will contribute to the same projects.9 Therefore, the expected surplus is equal to (1+λ2)v,

which is greater than v in cases λ= 0 or 1. This simple illustrative example shows that when

there is a certain level of friction and investors cannot observe all rates on the market, the

dispersed rate can help them coordinate to the same project and hence increase the market’s

allocation efficiency.

9Since rates are drawn from the same distribution F(r), the probability that ri > r j is equal to

∫ r̄

r0

F(r)dF(r) =12

.

.

9

3 Model

In this section, we represent the general version of the model and explain the structure of the

equilibrium. In the following sections, we discuss the allocation efficiency in such a market

with the presence of both a many-to-one market structure and search friction.

Briefly, we consider a dynamic search model with a many-to-one matching environment,

assuming that each borrower requires many units of funding but each investor has only one unit

to provide. In essence, borrowers enter, post an interest rate and wait to be funded. Investors

visit the market, sample a finite number of listings and choose the preferred one in which to

invest. After a borrower and enough investors are matched, investors receive the committed

return while the borrower retains the profit, and both parties leave the market.10

3.1 Model Ingredients

Time is continuous, and the market consists of two sides: borrowers (who are referred to below

using the feminine pronoun) and investors (who are referred to below using the masculine

pronoun).

Borrowers At each instant, a measure B · d t of new borrowers enters the market. Each bor-

rower requires N ∈Z+ units of external financing, and the projects are of the same expected

value N v. Upon entry, each borrower posts an interest rate r and then waits to be funded.

During the funding process, a borrower may exit the market due to a Poisson death shock with

intensity δ > 0.11 Conditional on being fully funded, a borrower pays each investor r and

leaves the market with profit N(v− r).

Investors The instantaneous investor inflow is I · d t. Each investor has one unit of money

to invest. Assume that investors have the same reservation rate r0 < v. In terms of investors’

payoff, in the one-to-one market, an investor’s utility from investment is simply r, while in

the many-to-one environment, due to the risk of failure, his payoff could be represented as

10Consistent with the empirical environment that will be introduced in Section 6, we consider a borrow-and-lend environment. In fact, our framework applies to any many-to-one trading scenarios in which the party whoquotes the price requires multiple agents to back

11The death rate δ also plays a role as a discount factor in the model.

10

expected utility, which will be described shortly. For simplicity, we assume that investors are

short-lived and invest only once. If the listing encounters a death shock, all matched investors

will leave the market with r0 permanently.12

Search and Matching Since it takes time for a listed project to be fully funded, there always

exists a measure of listed projects in the market, denoted as Lt . Upon entry, an investor

randomly observes ` ∈Z+ listed projects fromLt and invests in, at maximum, one. Regarding

the information available to the investors, we impose the following assumption.

Assumption 1. Investors can observe the rate r and size N of the ` listings in the sample.

We have yet to impose any structure on the distribution of ` but just denote Pr(`) as its

probability. We will discuss how different search technologies affect market equilibrium in the

characterization section below. On the other side of the market, after being posted, a listing

waits passively for investments. Assume that each listed project can meet an investor with a

Poisson intensity ηt > 0, which is endogenously determined by

E(`) · I = ηt ·Lt .

The equation balances the total number of meetings from both the investors’ and the listings’

perspectives. In principle, a larger investor inflow I , a higher expected value of `, and a lower

the number of listed projectsLt will result in a higher exposure for a project, namely, a higher

value of ηt . In terms of matching, assume that conditional on being met, a listing with rate r

has a probability pt(r) of being matched. Therefore, ηt pt(r) is the effective matching intensity.

3.2 Strategy and Equilibrium

In this section, we formulate agents’ problems and describe aggregate variables. Then, we

define the equilibrium. We focus on a symmetric stationary equilibrium, in the sense that

investors and borrowers employ the same strategies for investment and quoting interest rates,

respectively, and all the aggregate variables, such as L ,η and p(r), are time-invariant t.

12Alternatively, the model can easily be extended by assuming that an investor can resample by paying a costc > 0. If so, the only difference is that the reservation rate r0 will be determined by the cost.

11

Agents’ Problem

To characterize the borrower’s problem, denote P(r) as the probability that a listing with rate

r is fully funded. Thus, the borrower’s problem is to choose the interest rate to maximize the

expected profit

π̂= maxσ∈∆(r)

P(r) ·N(v− r). (1)

Here, we allow for any form of borrower strategy, which could either be a pure or a mixed

strategy. In general, denote the strategy as a rate distribution {F(r), f (r)}.

The investor’s problem, on the other hand, is to choose the project with the largest expected

utility in his sample. Because the investors cannot coordinate with each other, and N is the

same across all listings, the observed interest rates are the only information available to an

investor. Therefore, he needs to form beliefs on the funding probability of a listing with rate

r conditional on his own contribution, denoted as Q̃(r). Accordingly, an investor’s expected

payoff can be represented as

u(r) = Q̃(r)r +(1− Q̃(r))r0.

Furthermore, denote Q(r) as the actual conditional funding probability. In equilibrium, the

belief has to be consistent for all r, i.e.,

Q(r) = Q̃(r). (2)

Aggregate Variables

For the aggregate variables in the stationary equilibrium, first, the number of meetings has to

be balanced such that

η ·L = E(`) · I . (3)

Next, the distribution of listings should be invariant with respect to time. Each listing has two

attributes - quoted rate r and progress n ∈ {0, 1,2...N − 1}, which represents the number of

investors who have been matched with the project. Denote the density as g(r, n); then, in

12

equilibrium, the distribution dynamics satisfy the following equations:

L g(r, n−1)ηp(r) =L g(r, n)(ηp(r)+δ);

B f (r) =L g(r, 0)(ηp(r)+δ). (4)

The equations represent that for each group of listings with the same attributes (r, n), the

inflow and outflow are identical at each instant. Specifically, for a listing with rate r, the in-

tensity of accumulating one more investment is ηp(r), while any project may leave the market

due to death shock with intensity δ. According to the expression, f (r) represents not only

the borrowers’ strategy but also the distribution of new entrants; hence, B f (r) represents the

measure of new entrants with rate r. In addition, define g(r) ≡∑N−1

n=0 g(r, n) as the density of

listings with rate r, with G(r) defined as the corresponding CDF. The last condition required

for a stationary equilibrium is that the number of matches is balanced. Because in equilibrium,

r ≥ r0 for all listings, and each investor invests in exactly one of his sample. However, for each

listing with rate r, its matching intensity is ηp(r); therefore,

I =L∫

rg(r)ηp(r)dr. (5)

The evolution of the market and its key component is illustrated in Figure 1. Thus far, we have

demonstrated all conditions required for a stationary equilibrium.

Definition 1 (Stationary Equilibrium). Given state variables {B, I , N}, parameters {δ, r0, v},

and search technology {Pr(`),` ∈Z+}, a stationary equilibrium consists of (i) borrower’s pric-

ing strategy F(r); (ii) investor’s decision rule and belief Q̃; (iii) intensity of meeting η; (iv)

total measure of listings L ; and (v) rate distribution G(r). such that

• Borrower profit is maximized Equation (1);

• Investors’ decision rule is belief-consistent Equation (2);

• Balance of meetings Equation (3);

• Rate distribution is time-invariant Equation (4);

13

• Balance of matches Equation (5).

3.3 Existence and Uniqueness

Because investors make decisions based on their belief Q̃(r), an equilibrium might not be

unique. For the present, suppose that all investors are endowed with the belief that Q̃(r) > 0

if and only if r = r∗ for some arbitrary r∗ > r0; then, there might exist equilibria in which

all borrowers will choose r∗ while investors invest up to r∗. In the remaining analysis of the

paper, we exclude such unrealistic equilibria and restrict attention to the monotonic decision

rule. Namely, when N is the same across all listings and other investors’ decisions cannot be

observed, an investor will choose the project with the largest r in his sample. We will show

below that the decision rule is belief-consistent.

Because the aim of the paper is to study the impacts of search frictions, the properties of

`’s distribution play a significant role in determining the equilibrium. For instance, there are

two special cases.

Proposition 1. When Pr(` > 1) = 1, r = v (Bertrand competition). When Pr(` = 1) = 1,

r = r0 (Monopoly).

According to the proposition, if investors can always sample more than one listing and com-

pare them, then borrowers are engaging in Bertrand competition. However, if each investor

can see only one listing in his sample, then borrowers have all of the market power, and the

interest rate will degenerate to r = r0, which is the famous result shown in Diamond (1971).

In the rest of the paper, we will maintain the following assumption.

Assumption 2. Pr(`= 1) ∈ (0,1).

There are several possible interpretations of Assumption 2. At the individual level, ran-

domness may emerge because the number of listings in the market is stochastic; although in a

stationary environment, it is constant on average. Therefore, some investors can observe more,

while others can only observe a few. From the perspective of the whole population, it might

be the case that some investors treat searching and sorting as a costly behavior and tend to

invest in the first project they meet in the market. Thus, the borrowers are facing a probability14

distribution of different types of investors. Therefore, we do not impose any specific structure

on the distribution of `, except for Assumption 2.13 Given Assumption 2, following Burdett

and Judd (1983) and Varian (1980), it is easy to show that in any symmetric equilibrium,

borrowers never choose pure strategy or any point of masses.

Lemma 1. Under Assumption 2, there is no symmetric equilibrium in which all borrowers set the

same rate or choose some rate with strictly positive probability.

Proof. The logic of the proof of Lemma 1 is simple. Suppose that borrowers implement pure

strategy r < v; then, a small upward deviation will significantly increase the probability of

success while the loss in payoff is small. If r = v, then according to Assumption 2, deviating

to r0 will generate strictly positive profits.

Based on Lemma 1, we focus on characterizing a mixed strategy equilibrium with a contin-

uous distribution function {F(r), f (r)} with support r = [r, r̄]. Then, we have an immediate

result as follows.

Lemma 2. In equilibrium, r = r0.

In words, the lower bound of rate dispersion is always the investors’ reservation rate in

equilibrium. The logic behind Lemma 2 is that if r > r0 for all investors, deviating to r0 will

maintain the same probability of success because the only way in which the listing is funded

is if the counterpart investor has `= 1. However, reducing the interest rate to r0 will increase

the return conditional on being funded. Hence, choosing r0 is a profitable deviation. In equi-

librium, by the indifference principle of mixed strategies, all rates r ∈ r will give borrowers the

same return, which can be represented as

π̂≡ P(r)(v− r) = P(r0)(v− r0). (6)

Note that P(r) is the probability that a listing generates contributions from N investors before

experiencing a death shock. Define V (r, n) as the value function of a listing with rate r and

13In the literature, there are many alternatives to define the distribution of `. For example, in Burdett andMenzio (2018), ` takes a value of 1 (captive) or 2 (non-captive). Alternatively, Hong and Shum (2006) assumethat investors are heterogeneous in unit search costs. The lower the cost is, the larger the sample one could draw.In equilibrium, the unit cost is equal to the marginal benefit from drawing one more sample.

15

progress n, which satisfies the following dynamics:

V (r, n) = (1−δd t −ηp(r)d t)V (r, n)+ηp(r)V (r, n+ 1),

or equivalently, as d t → 0, we have

V (r, n) = q(r)V (r, n+ 1),

where

q(r) ≡ηp(r)

ηp(r)+δ(7)

is the probability that a listing with rate r generates one more match without encountering a

death shock. Specifically, because V (r, N) = N(v− r) and V (r, 0) = π̂, we have

P(r) = q(r)N =

�

ηp(r)δ+ηp(r)

�N

. (8)

In terms of p(r), an investor will choose the borrower with r only if it offers the highest rate

in the sample. Therefore, suppose the sample has size `, the borrower will attract the investor

with probability G(r)`. Equation (9) expresses p(r) in terms of G(r) and Pr(`).

Lemma 3. In stationary equilibrium, given G(r), p(r) takes the form

p(r) =

∑

` ` ·Pr(`) ·G(r)`−1

E(`), (9)

which is increasing in r.

Based on the assumption that investors will choose the higher interest rate r, Lemma 3

shows that p(r) is indeed increasing in r, as is P(r) by Equation (8). To show that the mono-

tone decision rule is belief-consistent, in Proposition 2, we derive the analytic expression of

Q(r) in terms of P(r) and p(r) and show that it is also increasing in r. More importantly,

Q(r) is increasing in r, which confirms that it is belief-consistent for the investors to choose

the project with the highest rate in their samples.

16

Proposition 2. In stationary equilibrium, Q(r) takes the form

Q(r) =1

1− P(r)Nδ

δ+ηp(r)

�

ηp(r)δ+ηp(r)

�N−1

. (10)

In addition, investors who choose the highest rate in their sample are belief-consistent.

To demonstrate the existence and uniqueness of a stationary equilibrium with mixed strate-

gies under the belief-consistent assumption that investors always choose to invest in the listing

with the highest rate, several intermediate results need to be shown. First, because the match-

ing intensity ηp(r) is independent of progress n, the stationary distribution condition (charac-

terized by Equation (4)) can be simplified and summarized as the relationship among the rate

distribution of all the listings, the distribution of new entrants and the funding probability of

the projects of rate r.

Lemma 4. In equilibrium, g(r), f (r) and P(r) satisfy

δL g(r) = B f (r)(1− P(r)). (11)

In Equation (11), δL g(r) is the mass of exits in a unit of time, while B f (r)(1− P(r))

describes the expected number of new inflows that will ultimately fail. In a stationary equilib-

rium with constant L , the two measures must be equal for any r ∈ r. In addition, the above

equation also implies that g and f have a monotone likelihood ratio, since P(r) is increasing

in r. That is, for any distribution inflow f , listings with a higher interest rate will be funded

faster.

To show the existence and uniqueness of stationary equilibrium, we start by demonstrating

that the meeting intensity is uniquely pinned down.

Proposition 3. In stationary equilibrium, η satisfies

δ

η

∫ 1

0

1

1−�

ηp̃(x)ηp̃(x)+δ

�N d x =B

IE(`)(12)

17

where ∀x ∈ [0, 1],

p̃(x) ≡∑

` ` ·Pr(`) · x`−1

E(`)(13)

Moreover, η > 0 exists and is uniquely determined if and only if N · B > I .

The necessary and sufficient condition N · B > I indicates that the total fund demand is

less than the supply, which is consistent with the example illustrated in Section 2. Otherwise,

the measure of listing stock is simply zero. We will maintain the condition in the rest of the

discussion. Proposition 3 significantly simplifies our analysis because η is uniquely determined,

which allows us to straightforwardly prove the existence and uniqueness of the stationary

equilibrium in the many-to-one matching framework. More importantly, Proposition 3 will

also play a crucial role in the identification of the structural estimation, as will be discussed

in Section 6. In principle, Equation (13) shows that equilibrium η can be described by the

function p̃(x) with support [0, 1]. Comparing the function p(r) defined in Equation (9), p̃(x)

depends only on the exogenous search technology Pr(`).

Theorem 1. There is a unique stationary equilibrium in which investors choose the highest interest

rate in their sample, and borrowers implement identical mixed strategies.

Proof. The logic behind the proof is also the algorithm for solving the problem numerically.

Beginning from the unique η, we can solve all of the equilibrium elements according to the

following steps:

(η,L )⇒ π̂⇒ (P(r), p(r))⇒ {G(r), g(r)} ⇒ {F(r), f (r)}.

For the details of the proof, see the Appendix.

4 Comparative Statics and Welfare Analysis

Given existence and uniqueness, in this section, we discuss how the equilibrium conditions

respond to the change in the model primitives. We will report several testable results that can

be verified in our empirical analysis. More importantly, we present the main finding of the

18

paper, which concerns the efficiency problem in a many-to-one matching market caused by the

joint effect of coordination and search frictions.

4.1 Comparative Statics

We first investigate how meeting intensity η and market depth L are affected by the sup-

ply/demand ratio z ≡ B/I and the size of each project N .

Proposition 4. In stationary equilibrium, (i) η is decreasing in z and N while L is increasing in

z and N. (ii) η converges to δE(`)/z as N →∞.

Proposition 4 (i) shows that as the ratio z ≡ B/I increases, given the same death rate and

search technology, the market will stock more listings, namely, a larger L , which indicates a

lower meeting intensity η. Similarly, when N increases, more time is required for a project

to be funded. Therefore, L will increase in N , while η will decrease in N . (ii) implies that

when N is increasing, the market depth L will never diverge, and hence, η will not diminish

to zero. Based on Proposition 4, the next result reflects how rate distribution is affected by

market demand and supply.

Proposition 5. Denote the stationary distribution by G(r; z, N). For two pairs of market supply

and demand z1 = B1/I1, z2 = B2/I2, if z1 > z2, then

G(r; z1, N) �FOSD G(r; z2, N).

Given z, if N1 > N2, then

G(r; z, N1) �FOSD G(r; z, N2),

and r̄ is increasing in z and N.

Intuitively, as the demand/supply ratio or average loan size increases, borrowers face more

severe competition and hence will commit to overall higher interest rates, which gives us the

FOSD result. Correspondingly, r̄ will increase in z and N . Moreover, the proposition also

implies that, as a second order effect, when N · B/I is small, nearly all listings can be funded,

and hence, the interest rate will be close to r0. However, as the demand side expands, this rate19

will converge to v. Therefore, the standard deviation of the interest rate and demand/supply

ratio should have an inverse U-shaped relationship. A numerical computation is reported in

Figure 5.

In addition, an immediate result of Proposition 4 is that the death rate δ plays an non-

significant role in determining rate distribution, funding profitability and borrowers’ expected

payoff.

Corollary 1. G(r), P(r) and π̂ are independent of δ.

Intuitively, when δ is large, η will be increased by the same magnitude. Although each list-

ing’s life-span becomes shorter, as the market becomes less competitive, the funding likelihood

can compensate for the death risk and thereby the funding probability remains unchanged. In

a stationary equilibrium, the borrower’s strategy and profit will remain the same and free of δ.

4.2 Welfare Implication

To begin, let us consider the traditional one-to-one trading environments, namely, N = 1. In

such markets, each investor can finance a project alone. Therefore, the measure of successful

listings at each unit of time is simply I . The interest rate, regardless of being unified or dis-

persed, is only the determinant of monetary transfer between the two sides and has no effect

on welfare. However, when N > 1 such that one listing cannot be fully funded by a single in-

vestor, coordination friction emerges. Moreover, investors who have made an investment will

be locked in with the listing and face a risk of failure. Hence, the allocation efficiency of the

market has a significant impact on welfare. In this section, we will show that the interest rate

dispersion induced by search friction can mitigate the negative influence caused by the lack of

coordination.

We measure it as the expected number of projects that will be fully funded,

W ≡ B

∫

rP(r) f (r)dr. (14)

Since each project may leave the market with intensity δ, a larger W reflects the expectation

that each project will achieve its financial goal faster.

20

By Equation (11), welfare can also be expressed as

W = B−δL ,

which is the difference between new inflows and dead listings at each moment. Hence, compar-

ing efficiency levels across different search technologies is equivalent to comparing equilibrium

market sizes L .

Theorem 2 (Efficiency). Given (I , B, N), a market with Pr(` = 1) ∈ (0, 1) achieves higher

welfare than one with Pr(`= 1) ∈ {0,1}.

Theorem 2 implies that although a certain level of friction prevents investors from viewing

all listings in the market, from a social perspective, under a many-to-one matching framework,

friction improves welfare, measured by the expected number of successfully funded projects,

compared with the perfect competition case. The novel effect is induced by a mixture of dis-

persed prices generated by search frictions and the lack of a coordination mechanism in the

many-to-one environment. As discussed above, in a one-to-one matching market, there is no

problem regarding coordination; hence, interest rates simply determine how to divide surplus.

However, in the many-to-one environment, if investors cannot observe peers’ behavior and

make their decisions independently, the interest rate is the unique information that helps them

make decisions. If there is no differentiation among projects, investors tend to choose which

to invest in at random, which is no better than the case in which each investor can observe

only one listing (the case of Pr(`= 1) = 1).

However, under a certain level of search frictions, the most important feature generated

here is the persistent rate dispersion that induces a mechanism for investors to sort the listings

in their samples. In principle, listings with higher rates not only provide investors with a po-

tentially higher return but also serve as a signal that helps investors coordinate. Such biased

funding dynamics increase the overall number of fully funded projects, indicating enhanced

allocation efficiency and social welfare. The general implications of the analysis shed light on

mechanism design and information disclosure in such many-to-one matching environments in

which the side of “one” posts the price, while the “many” side searches. It seems that in such

a decentralized matching market, if there is no other effective coordination mechanism, price21

regulation or unifying the quality of the goods is harmful to market efficiency, and perfect in-

formation disclosure might also be inefficient. Recall the simple example mentioned in Section

2: suppose that there are two valuable projects, each of which requires two units of contribu-

tion. If there is a mechanism to differentiate between the two projects, it would be easier for

the two investors to contribute to the same one. In my model, rate dispersion represents such

an endogenous property that improves coordination.

In general, the model applies to any two-sided market with increasing returns to scale.

For instance, in the labor market for start-up companies, new firms enjoy increasing returns

to scale in labor during their early stages. However, in general, adjusting hiring information

such as wages is costly in the short run, and it is impossible for a firm to contract with workers

based on how many employees it has hired. Therefore, if employers have perfect competition,

then all of them will grow at the same pace and will reach optimal scale after a longer period.

However, when competition is imperfect and features wage dispersion, the endogenous sorting

mechanism will cause some firms to reach optimal size earlier, which from the social perspective

is beneficial.

5 Endogenous Sizes

In this section, we extend the model by allowing borrowers to choose both loan size and rate.

For simplicity and without loss of generality, we focus on the strategy space in which borrowers

can either choose to borrow less (N = 1) or borrow more (N = 2). Accordingly, denote ⟨r, N⟩

as a listing with rate r and size N .14

Except for endogenous loan sizes, all the other model ingredients, such as search technology

Pr(`), the death rate δ and measures of borrowers and investors (B, I), remain the same. In

terms of investors’ information set, assume that they are able to observe a project’s rate and

size. In the market, an investor is able to support one small project alone, while the large

projects are still facing coordination problems.

First, we consider the two extreme cases. When the investors can always compare rates

14We use ⟨r, N⟩ to represent a project with rate r and size N to differentiate the notation of open intervalsrepresented by parentheses (a, b).

22

with Pr(` > 1) = 1, it is straightforward to see that all borrowers choose r = v in equilibrium.

Meanwhile, the borrowers are indifferent in terms of loan size, since their profits are always

zero. In the other extreme case with Pr(` = 1) = 1, the interest rate is always equal to r0,

since the borrowers possess high market power. However, the choice of the size is crucially

dependent on the meeting intensity, which is determined by (B, I). Imaging a profile in which

all borrowers choose N = 2. The large demand and frictional search will diversify the investors’

contributions and reduce η. In this case, some borrowers may find it is profitable to choose N =

1 to increase the funding probability. Similarly, a profile with all N = 1 cannot be sustained,

either. In equilibrium, a mixture of loan sizes will emerge. In the rest of this section, we will

analyze the rate and size dispersion together by focusing on Pr(`= 1) ∈ (0, 1).

In the following analysis, we discuss investors’ and borrowers’ choices and leave the rest of

the equilibrium characterization to Appendix A.1.

5.1 Investor Choice

In Section 3, we have shown that investors always choose listings with higher rates is belief-

consistent under the case of unified loan size. When a sample could have listings of different

sizes, it is necessary to discuss investors’ preference toward them. Obviously, there exist some

non-interesting belief-consistent strategies. For instance, suppose all the investors hold the

belief that N = 2 listings have zero funding probability and hence only choose N = 1 listings;

then, very few N = 2 listings will ultimately be funded, and in equilibrium under such belief,

all borrowers choose N = 1.

Instead, we assume that all the investors follow the preference that is strictly increasing in

rate r among listings of the same size. In addition, define Q̃N (r) as the investors’ belief on the

funding probability of a listing ⟨r, N⟩, and the conditional expected return is

u(r, N) = Q̃N (r)r +(1− Q̃N (r))r0.

Obviously, for any ⟨r, 1⟩, we have Q̃1(r) = 1 for sure, since an investor can fully support a

small loan through her own investment without coordination, such that u(r, 1) = r.

23

Two listings with different sizes are return equivalent15 ⟨r1, 1⟩ ∼ ⟨r2, 2⟩ to investors if and

only if Q̃2(r2)r2 +(1− Q̃2(r2))r0 = r1. Effectively, suppose two listings are return equivalent;

they have the same conditional and unconditional matching probability, denoted as pN (r) and

qN (r), respectively. In equilibrium, we also have qN (r) =ηpN (r)ηpN (r)+δ . By Equation (10), an

N = 2 listing’s conditional funding probability is equal to

Q2(r2) =2q2(r2)

1+ q2(r2).

In equilibrium, it must be guaranteed that Q2(r) = Q̃2(r). In addition, we form a relationship

between two listings that are of different sizes but return equivalent to investors. Specifically,

⟨r1, 1⟩ ∼ ⟨r2, 2⟩ can be written as

r1(r2) =2q2(r2)

1+ q2(r2)r2 +

1− q2(r2)

1+ q2(r2)r0. (15a)

Since p1(r1) = p2(r2), Equation (15a) can be also written as

r2(r1) =1+ q1(r1)

2q1(r1)r1−

1− q1(r1)

2q1(r1)r0. (15b)

The equations immediately suggest that r1 ≤ r2 if ⟨r1, 1⟩ ∼ ⟨r2, 2⟩, since investors require a

premium from larger listings to compensate for the risk of investment failure. Moreover, the

equations present the investors’ decision rule given off-equilibrium path listings. For instance,

in an equilibrium with all N = 2 listings with r2 ∈ r2, suppose some borrower chooses ⟨r1, 1⟩;

the investors’ preference as well as the listing’s funding probability can be determined by Equa-

tion (15a), and vice versa.

5.2 Borrower Strategy

Based on the investors’ preference, we consider the borrowers’ strategy in choosing rate r and

size N . We consider all possible mixed strategies in space ∆(r, N). Denote the probabilities

of choosing loan size N as hN ∈ [0, 1] with h1 + h2 = 1. Equivalently, let BN ≡ B · hN be the

15We use the term “return equivalence” for investors’ preference to differentiate it from “indifference” whendiscussing borrowers’ mixed strategy conditions.

24

instantaneous inflow of new listings with size N . Another dimension of borrowers’ strategy, rate

distribution, is denoted as {FN (r), fN (r)}, with support rN ⊆ [r0, v], N = 1, 2. We impose that

the supports rN be closed sets, which rules out that the rate distributions have a finite number

of atom points with zero density but actually have no influence on the funding probability.

We first consider the model primitives that support equilibrium with identical loan sizes;

the borrower always chooses either N = 1 (h1 = 1) or N = 2 (h2 = 1). Lemma 5 describes

the necessary and sufficient conditions for each type of equilibrium to emerge.

Lemma 5. The market supports equilibrium with all N = 1 if and only if BI ≥ 1+Pr(`= 1)0.5 ≡

z̄. Meanwhile, equilibrium with all N = 2 exists if and only if BI ≤ z for some z < z̄.

Briefly, two market forces determine the size distribution in equilibrium. First, suppose the

ratio z = B/I is large enough such that the funding demand side is more competitive; then,

to increase the probability of being fully funded, the borrowers tend to request smaller loans,

and vice versa. Another force is the search technology. When search is frictional, (Pr(`= 1) is

large), the investors’ samples become more diversified, and each listing has more market power

and finds it easier to accumulate financial support. Thus, borrowers have a higher incentive

to choose N = 2. Moreover, z < z̄ indicates that the set of primitives that support all N = 1

equilibria is disjoint with those that support equilibrium with all N = 2, and the two thresholds

segment the borrowers’ strategies into three categories.

For the remainder of the section, we focus on the scenarios with B/I ∈ (z, z̄) and imperfect

search technology Pr(`= 1) ∈ (0, 1), under which borrowers will implement a mixed strategy

in choosing both N and r. Note that in any equilibrium, the lowest possible interest rate is

always r0. In addition, it is also trivial to see that, for both types of listings, the rate distributions

FN have no mass point.

In equilibrium, borrowers’ indifference requires that the two types of listings have equal

expected profits π̂ = π̂2 = π̂1. Based on the indifference condition, Lemma 6 shows that any

two listings with different sizes almost surely do not have equivalent returns in equilibrium.

Lemma 6. In any equilibrium with nonempty and closed support rN for both N = 1,2, there

25

exists one and only one pair of r1 ∈ r1, r2 ∈ r2 that satisfy the return equivalence condition, i.e.,

⟨r1, 1⟩ ∼ ⟨r2, 2⟩.

Lemma 6 suggests that the rate distributions F1 and F2 lie on two intervals and that investors

almost surely prefer one type of project over the other. It seems natural to ask next which type

of listings is relatively attractive to investors.

Theorem 3. In equilibrium, r1 = [r0, r̄1] and r2 = [r2, r̄2] such that

⟨r̄1, 1⟩ ∼ ⟨r2, 2⟩.

Intuitively, when loans could be of heterogeneous sizes, borrowers tend to offer higher in-

terest rates for larger listings to compensate investors for the risk of failure. Theorem 3 asserts

that when loans are of different sizes, the interest rates are distributed in disjoint ranges, such

that investors always prefer N = 2 listings compared with N = 1 listings in equilibrium. In

other words, an endogenous lexicographic preference emerges in the market with differenti-

ated loan sizes. Figure 2 illustrates the payoffs of borrowers and investors.

6 Empirical Analysis

In this section, we provide empirical evidence on pricing and matching in many-to-one trading

markets with homogeneous goods. We construct a novel dataset using an online peer-to-peer

platform for micro-loans. The empirical environment is consistent with the theoretical setup

based on two key features of the market.

First, similar to other markets such as crowdfunding and joint ventures, the matching be-

tween the supply and demand sides follows exactly a many-to-one pattern on the platform.

That is, one borrower’s loan request needs contributions from multiple investors. In our sam-

ple, one listing was, on average, matched with seven investors.

Second, the unique market structure allows us to identify homogeneity among financial

products. Briefly, the agents trading on the platform form a three-tier hierarchical structure.26

The middle tier agents (called dealers), on the one hand, post listings to absorb funds from

the bottom tier lenders (called basic investors). On the other hand, they invest in the loans

issued by the top tier fundraisers (called primary borrowers). We focus on the listings issued

by the dealers, since we can observe both the inflow and outflow of the fund. This helps us to

overcome an empirical challenge, in which the value of borrowing, represented by the param-

eter v in the model, is usually not observable. Under the many-to-one transaction mechanism,

we argue that dealers who are matched with the same primary borrower have the same value

of borrowing. According, by controlling other observable attributes, we are able to identify

homogeneous listings in our sample.

The empirical analysis is organized as follows. First, we provide more details on the in-

stitutional background and describe the data structure. Then, we present some stylized facts

that meet theoretical predictions. Finally, we structurally estimate the theoretical model and

conduct counterfactual analysis to evaluate the coordination effect on the many-to-one market

under search frictions.

6.1 Institutional Background and Data Description

We collected historical data from Hongling Capital, Inc. (or my089.com), one of China’s largest

online micro-loan platforms, based in Shenzhen. The dataset has a horizon of 4 months (Jan

1st - May 1st, 2015) and includes information on 508,888 listings, of which 216,402 are fully

funded. The average annualized interest rate is approximately 14.03%, and the average matu-

rity is approximately 130.747 days (weighted by loan size). The average daily trading volume

is 155 million Chinese yuan (CNY) (approximately 25 million US dollars (USDs)).

The life cycle of one individual listing is as follows. (1) A borrower enters the platform by

posting a listing with loan information such as interest rate, amount requested, duration of the

loan, purpose of borrowing and so on. (2) After the loan request is listed, the borrower starts

waiting for investors’ contributions. (3) If the requested amount is raised before the deadline,16

the borrower receives the money and repays the debt according to the terms committed. (4)

Otherwise, if the requested amount is not met, the loan will be canceled, and the money will

16At my089.com, the deadline was 2 days, but most of the projects were de-listed before then.

27

immediately be returned to the investors.17

In terms of borrowing purposes, there are several categories of loans listed on the platform.

To demonstrate that rate dispersion exists among nearly homogeneous loans, we devote par-

ticular emphasis to two type: those listed by primary borrowers and those listed by dealers. We

name the two types of listings “primary” and “derived” loans, respectively, and they contributed

to 99.57% of the total trading volume on the platform.

Primary Loans The primary fundraisers are external borrowers such as firms and entrepreneurs

who need financial support. In practice, the primary borrowers sign contracts with the plat-

form, and the platform lists their loan requests to investors. The primary loans account for

approximately 31% of the total trading volume. They are of large size and have high returns

and long maturities. In our sample, all primary loans have an annualized interest rate of 18%.

The average size is 3.8 million CNY, and the average duration is approximately 611 days. More-

over, all primary loans were fully funded in the sample.

Derived Loans In addition to primary loans, the platform allows its clients to issue short-term

loans using their assets such as cash deposits and account receivables within the platform as

collateral. The most commonly used collateral is unexpired primary loans. For instance, if an

investor lent 10,000 CNY to a primary loan, she is allowed to post a listing within the platform

with maximum amount 8,000 CNY. To some extent, such internal loans with collateral are a

financial derivative with primary loans as underlying assets.

Compared with primary loans, both the maturities and interest rates of derived loans are

smaller. On average, the maturity is 27.98 days, while the annualized interest rate is 12.81%.

Such loans represent approximately 68% percent of the market. In the sample, derived loans’

funding probability is approximately 40%. We summarize the main statistics of the two types

of listings in Table 1.

Dealership through derived loans The special rule of derived loans creates an opportunity

for the platform’s clients to make a profit from investing in long-run primary loans by issuing

17This trading mechanism is the same as that in most of the online financial platforms, such as LendingCluband Prosper.com.

28

short-term derived loans and borrowing from basic investments. Short-term derived loans

could exist, since they meet the demand of basic investors who were only interested in short-

term investment, as the maturity of primary loans was too long to them. As illustrated in

Table 1, there is an approximately 5% difference between the returns on primary and those on

derived loans.

The data reflect that the dealers were quite active on the platform. Figure 3 illustrates the

daily transaction volumes of primary loans and one-month derived loans, which are highly

correlated. In addition, as the user ID of each borrower and investor on a listing can be ob-

served, the data allow us to track every registered customer’s lending and borrowing behavior.

In our sample, 7,746 of a total of 109,828 users engaged in both borrowing and lending. Using

this information, we identified dealers’ behaviors within the market. As illustrated in Figure

4, overall, 63.04% of the primary loans are contributed to by dealers using money borrowed

through derived loans.

Market Risks Another important dimension to address is the risk of listings on the platform.

During the time horizon of the data, the platform monitored all its primary loans and guaran-

teed the return (including both principle and interest) of the investors. If default or delayed

repayment occurs, the platform would use its reserve fund to repay investors. Since primary

loans are return guaranteed, derived loans are also secured. Therefore, all of the listings in the

market share the same systematic risk on the platform.

Data Structure

The data consist of three parts, described as follows.

• Loan Information records basic information regarding each listing, such as the bor-

rower’s ID, loan size, interest rate, and duration.

• Bidding Record has information on a listing’s bidding history and records the time and

amount invested for each participating investor. One issue to note is that the information

only includes loans that were fully funded.

29

• Snapshot records all listings’ funding dynamics on the platform every 10 minutes. It

allows us to continuously track the platform’s characteristics, such as the pace of funding

and new listings.

The dealers’ behavior creates an ideal natural laboratory for studying price competition in

many-to-one markets, as it unifies the risk and value of each borrower’s project, which helps

us to rule out other unobservable heterogeneity that may also cause price dispersion. We focus

only on empirical patterns regarding 1-month derived loans made for the purpose of investing

in primary loans. The selection procedure generally ensures that the listings in our sample are

of the same quality in terms of their

(a) Duration: 1 month;

(b) Value: primary loans’ return;

(c) Risk: platform’s systematic risk.

Accordingly, our restricted sample has a total of 297,685 listings, of which 126,121 were fully

funded. The average size and rate are 31,090 CNY and 14.32%, respectively. On average,

the number of investors per listing is 6.36, conditional on the listing being fully funded. The

summarized statistics are reported in Table 2.

6.2 Stylized Facts

In this section, we report key empirical evidence that motivates the main questions of the paper.

Specifically, we present some stylized facts that are consistent with the fact that search frictions

exist in the market and cause persistent rate dispersion. In addition, we show that investors in

such a many-to-one market indeed face coordination problems.

Funding Probability

Our first evidence for the existence of market friction is the funding probability of the listings

in the market.

30

Fact 1. While funding probability is increasing in the interest rate, there is a significant failure

rate for listings of all rate ranges.

According to Table 3, funding probability is increasing in the interest rate. However, even

for the listings with the highest interest rate level (≥16%), the funding probability is only ap-

proximately 60%, while listings with a lower rate still have an approximately 20% chance of

being funded. Further, to avoid time-dependent variation such as market demand and supply,

we compute the funding probability by percentile during each hour and observe a similar pat-

tern. According to Table 4, for listings at the 80th percentile, the funding probability is only

approximately 65%, while listings at the 20th percentile have a success rate of approximately

30%. The fact seems to suggest that there exists certain frictions on the investor side in viewing

listings on the platform.

Rate Dispersion

The second set of evidence concerns interest rate dispersion. As shown in Table 3, the interest

rates for 1-month loans range from 10% to 17%. As predicted by our model as well as many

others (e.g., Burdett and Judd (1983) and Stahl (1989)), when search frictions play a role,

every seller employs identical mixed strategies such that the market will have a nondegener-

ate price distribution. Alternatively, price dispersion can simply be described by unobservable

individual-level heterogeneity. For instance, it is possible that fundraisers with high-quality

projects might be willing to offer higher interest rates. Although we have argued that the bor-

rowers have nearly identical returns (by investing in the same primary loans) on their projects,

to demonstrate that search friction is the main cause of rate dispersion, it is necessary to com-

pare individual-level and market-level rate dispersion. Suppose that at the individual level, the

dispersion level is significantly smaller; then, it might be the case that rate differentiation is in-

duced by some unobservable heterogeneity instead of by the mixed strategy induced by search

friction. However, in most empirical works (e.g., Hong and Shum (2006), and De los Santos

et al. (2012)), the price distribution is generated from a static snapshot of the market. Hence,

those works generally assume that price dispersion results from the use of mixed strategies.

Fortunately, in my data, we are able to observe repeated listing behavior by the same borrower,

which allows me to compare rate distributions at the individual and aggregate levels.31

To measure rate dispersion, we employ two measures for robustness. First, following

Jankowitsch et al. (2011) (JNS), we define rate dispersion as follows:

dt =

√

√

√

√

1∑Kt

k=1 wkt

Kt∑

k

(rkt − r̄t)2 ·wkt

In the formula, rkt is the interest rate for listing k in period t. r̄t is the weighted average

rate in period t. Furthermore, wkt represents the weight imposed on listing kt. If we weight

each listing equally, then wkt = 1. Alternatively, we can weight each listing by its size. In

brief, this measure represents the root mean squared difference between the traded rates and

the respective valuation based on a weighted calculation of the difference. In our sample, on

average, each borrower posts 4.29 listings per day. We compare dispersion at the individual

level with aggregate dispersion on daily and weekly bases. For comparison, we also compute

the daily and weekly dispersion for all listings in our sample.

According to Table 5, the daily and overall levels of dispersion are slightly smaller than those

at the aggregate level, while dispersion at the weekly level is slightly larger. This suggests

that individual-level dispersion plays a prominent role in market-level dispersion. Second,

we perform a simple comparison of the minimum and maximum rates quoted by different

borrowers. As indicated in Table 6, there is no significant difference between dispersion at the

aggregate and individual levels.18 In summary,

Fact 2. Individual-level interest rate dispersion is similar to aggregate level rate dispersion.

Facts 1 to 2 provide evidence that search frictions do play an important role in influenc-

ing investors’ information and hence affect borrowers’ pricing strategy. The next set of tests

attempts to compare the average rate and dispersion under different levels of demand and

supply, namely, z ≡ B/I .

According to Proposition 5, when B/I increases, the rate distribution will FOSD the distri-

bution with a lower B/I ratio. This implies that the mean of interest rates should be increasing

in B and decreasing in I . As reported in Table 9, we have

18For the min/max measure, as an individual’s number of listings is much smaller than the total level, I considerthe min/max for individuals but the 10th percentile/90th percentile for the aggregate-level data. Hence, inprinciple, aggregate-level statistics should be somewhat smaller.

32

Fact 3. The average rate level is positively correlated with funding demand and negatively corre-

lated with funding supply.

For rate dispersion, when B/I is small, the rate will degenerate to r0; when B/I increases,

the rate will converge to v as the market becomes more competitive. Therefore, the second

moment of interest rates should demonstrate an inverse U-shaped relationship with z = B/I .

Therefore, we run a regression w.r.t. the JNS measure of the rate dispersion level, again z

and z2, controlling for other observable variables. As reported in Table 10, under the hourly

measure, the pattern is significant. However, in the regression using daily averages, as demand

and supply fluctuate within a day in the market, the pattern disappears.

Fact 4. There is an inverse U-shaped relationship between rate dispersion and the supply/demand

ratio (as measured by z = B/I).

Facts 3 and 4 confirm the theoretical prediction that the average rate is positively correlated

with the competitiveness of the market, while the dispersion level has an inverse U-shaped

relationship with market competitiveness. This is also consistent with the findings in other

markets with homogeneous goods, such as gasoline (Pennerstorfer et al., 2020).

Coordination Effect

The third observation confirms the existence of coordination friction. According to the theo-

retical analysis, if investors are able to communicate or coordinate, a listing of who has accu-

mulated more contributions will be filled faster than one of those that have just been listed.

According to existing works such as Zhang and Liu (2012) and Kuppuswamy and Bayus (2018),

herding behavior and community effects play a significant role in listings’ funding dynamics.

We implement a similar empirical strategy to theirs but observe that this effect does not exist

in most of the listings in our sample. Specifically, to determine whether funding is accelerated

by funding progress, we regress the time interval between two successive bids on the listing’s

33

lagged progress, controlling for other variables.19

DTi t = αPROGRESS CUMULATEDi t−1 + γRAT Ei + X i tβ1 + X iβ2 + X tβ3 + εi t .

The model (Panel 1 in Table 8) predicts that when the progress is increased by 10%, the fund-

ing speed can be increased by approximately 37 seconds. The impact is very small compared

with the average funding duration, which is approximately 65 minutes in the sample. Then,

we separated the sample into two groups. In the first group, the funding duration is less than

90 minutes. This group contains 90% of the listings in the sample, while listings in the second

group have longer funding times. According to panel 2, the majority of the listings’ funding

processes are independent of the amount of funding secured to date. However, even for listings

that required more time to be funded, the increase in the funding rate attributable to funding

progress is quite limited (10% progress difference equates to an approximately 6-minute de-

crease in funding time). The results of the reduced-form analysis suggest that funding progress

plays a nonsignificant role in listings’ funding dynamics, especially for a trading environment

with relatively high frequency.

Fact 5. When trading frequency is high, investors’ main coordination device is the interest rate.

Loan size and rate

In Section 5, we discuss the relationship between loan sizes and interest rates. By Lemma 5,

suppose loan size is the borrower’s choice variable; it tends to be negatively correlated with

B/I . Consistent with the model prediction, Table 12 illustrates the fact as follows.

Fact 6. Average loan size is negatively correlated with the number of listings and positively corre-

lated with the number of investors.

We compare interest rates across listings of different sizes during the same period by es-

timating a random effects model controlling for aggregate variables during that period. As

19In their analysis, because both successful and failed listings’ funding dynamics are observable, they measurethe correlation between lagged progress and the probability of generating one more investor. However, in my data,because the platform erases bidding records for failed loans, we can only track funding dynamics for successfullistings. Therefore, we use funding speed as the proxy to test herding behavior

34

reported in Table 11, there is a significant and positive correlation between sizes and rates.

Consistent with our theoretical analysis, when a fundraiser decides to borrow more in the

market, they will offer a higher rate to compensate for the size of their requests.

Fact 7. Rate and size are positively correlated.

In addition to larger listings having higher rates, Theorem 3 asserts that investors should

always choose larger loans in their samples, since the borrowers’ strategy in turn endogenously

forms a lexicographic preference for consumers. The result provides a testable prediction that

the larger loans on average have a higher funding speed. The prediction is confirmed in Table

8. Namely, it can be observed that the time interval between two investments is shorter for

larger loans by controlling other important variables such as funding progress, interest rate,

market supply and demand.

Fact 8. Larger listings have higher funding speed.

In summary, the imperfect funding probability together with persistent rate dispersion at

both the aggregate and individual levels provide clear evidence of market friction and imperfect

competition among borrowers. Moreover, the relatively stable funding speed over a listing’s

lifespan suggests that investors lack an effective coordination device.

6.3 Estimation

Based on the evidence presented, we show that the market is affected by search frictions and

lacks effective coordination. More importantly, the theoretical analysis shows that in such

markets, price dispersion induced by search frictions could reduce welfare loss. In this section,

we attempt to quantitatively estimate the efficiency gain due to rate dispersion by introducing

a straightforward method to measure search cost primitives.

Identification and Estimation Strategy

Similar to Hong and Shum (2006) and De los Santos et al. (2012), the key element in the

identification and estimation is the price dispersion data. Our estimation method departs from

the literature such that we only use the information on percentile distributions instead of that35

for absolute price distributions. Specifically, our welfare measure is defined as the ratio of the

expected number of fully funded listings under random matching to that under dispersed rates

at each instant

w≡WRandom Matching

WDispersed Price,

where W is characterized by Equation (14). As shown in Theorem 2, the ratio has to be no

greater than one. To estimate w, define function H(x) : [0, 1]→R by

H(x) ≡ p̃(x) ·E(`) =∑

`

` ·Pr(`) · x`−1.

Recall that p̃(x) is the probability that a listing will be backed conditional on being viewed.

Thus, H(x) represents the expected number of listings in an investor’s sample conditional on

x ∈ [0, 1] being the highest percentile within the sample. In principle, our model primitives

include search technology Pr(`), state variables (B, I , N), investors’ outside option r0, the bor-

rowers’ value of borrowing v and the death rate δ. While the state variables (B, I , N) can be

directly measured, parameters v, r0 and δ are hard to observe or precisely measure. Proposi-

tion 6 shows that the estimation procedure can be simplified, and it is sufficient to identify and

estimate H(x).

Proposition 6. Given (B, I , N), H(x) is a sufficient statistic to estimate w and can be identified

and computed by

H(x) =P(x; z, N)1/N

1− P(x; z, N)1/N

z∫ 1

0d x

1−P(x ,z;N)

(16)

where z ≡ B/I and P(x; z, N) is the funding probability of the project at the x percentile.

Equation (16) provides the information with the need to estimate H(x). While z = L/I

and N are observable, the funding probability P(x; z, N) can be nonparametrically estimated

by computing the average probability of funding at each percentile. Moreover, the proposition

simplifies the estimation by avoiding the need to identify r0 and v, which might be determined

by some unobservable elements such as the change in the platform’s systematic risk and vari-

ables’ external elements to the market. We describe our estimation strategy of H(x) as follows.

36

1. Separate the data into four-hour intervals;20

2. Compute zt = Bt/It ;

3. Compute the average number of investments per listing, denoted as Nt ;

4. For each t with the same It , Bt and Nt , compute the average funding probability P(x; z, N)

by choosing the proper grid for x;21

5. Compute H(x) according to Equation (16).

Estimation Results

Table 13 and 14 report the estimated result of H(x) based on grids with the 10th and 25th per-

centiles, respectively. Figure 6 graphically illustrates the function. Note that in the expression

of H(x), H(0) is equal to Pr(` = 1), while H(1) is the estimate of E(`). According to the es-

timates, on average, approximately 35% of investors have only one listing in their sample. On

average, the number of listings observed by each investor is approximately 1.76. The reported

number implies that although the number of listings in the platform seems to be large, each

investor’s effective sample size is quite limited. In a behavioral interpretation, there could be a

significant level of idiosyncratic randomness associated with investors’ decision making, which

gives borrowers some market power and leads to dispersed interest rates.

Next, we use the estimated H(x) to compute the welfare ratio w under different combina-

tions of N and B/I , and the result is shown in Figure 7. According to the graph, when N ·B < I ,

there will be no welfare loss between the dispersed price and random matching cases because

L = 0 in either case. Moreover, when N = 1, the ratio is equal to 1. According to our data,

the N · B/I ratio is approximately 1.8 on average, while the average N is approximately equal

to 6. According to the calibration, w = 0.7807. That is, with a dispersed rate as a coordination

mechanism, the approximate welfare gain could be approximately 28.09%.

20The selection of 4 hours reflects a trade-off between sample size and market volatility. In principle, the longerthe time intervals are, the larger the sample in each group, meaning that we can compute P more precisely.However, as supply and demand may vary over time, we need to keep the time interval shorter to maintain thestationary assumption.

21Although we have approximately 250,000 observations in the sample, conducting kernel estimation requiresa high dimension of x; therefore, in our result, we only use the 10th and 25th percentiles.

37

7 Conclusion

This paper studies how search frictions affect price competition and allocation efficiency in a

many-to-one matching market. Our theoretical analysis shows that the rate dispersion induced

by search frictions will facilitate investors’ coordination and hence improve market welfare.

Our results are also supported by the empirical evidence that a significant mismatch and

persistent price dispersion are observed in a microloan market. We argue that search frictions

faced by investors play an important role in affecting borrowers’ pricing behavior.

The key feature that distinguishes my work from classical search models is that the bor-

rower must attract many investors, so investors must coordinate which projects to fund. The

surprising finding of the paper is that a certain level of search frictions can fix the coordination

problem by creating ex post heterogeneity. Hence, given a fixed inflow of external funding, the

number of funded projects will increase due to the joint effect of the two types of frictions.

Furthermore, we endogenize the choice of loan sizes such that the borrowers face a trade-

off on whether to form a many-to-one transaction to enlarge trading scale or a one-to-one

transaction to guarantee trading probability. We find that both rate and size dispersion will

emerge.

More generally, our study provides a tractable framework to investigate competition be-

haviors in many-to-one markets, especially online financial markets, while the majority of the

existing models restrain the borrower as a monopolist. There are several interesting ways to

extend the model to capture additional important aspects of the market. For instance, in this

paper, we only focus on credit markets. By modifying agents’ payoffs, one could apply the

model to other types of platforms, such as crowdfunding markets that are equity based and

presale based.

Moreover, we hope that our model can be extended and applied to other fields. In fact,

the many-to-one feature is an extreme case of an economy with increasing return to scale.

This property is shared by many other scenarios, such as forming social networks, producing

public goods, and recruiting employees at firms’ early stages. Our model provides two insights

into these markets. First, an increasing returns-to-scale market prefers unbalanced growth,

which can be facilitated by heterogeneity among competitors. Second, if there is no exogenous

38

heterogeneity, information imperfections such as search frictions can help to generate one.

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41

A Appendix

A.1 Equilibrium with Endogenous Sizes

This section is the continuation of Section 5. We first define the stationary equilibrium with

endogenous sizes and then show the existence and uniqueness of the stationary equilibrium

when the market consists of listings with differentiated sizes. We will focus on the model

primitives that satisfy z ≡ B/I ∈ (z, z̄), where z and z̄ are characterized in Lemma 5. Note that

when the demand/supply ratio z = B/I is smaller than z or larger than z̄, all the listings will

be of the same size and the analysis will be identical to that in Section 3.

Equilibrium Characterization

When loan size is another control variable, the borrowers’ optimization problem becomes

π̂= maxσ∈∆(r,N)

PN (r) ·N(v− r). (A.1)

where ∆(r, N) represents the set of possible mixed strategies on r ∈ [r0, v] and N ∈ {1,2}.

The borrowers’ strategy is denoted as {hN , FN (r), fN (r)}.

For investors’ decision, denote Q̃N (r) and QN (r) as the investors’ prospective and actual

funding probability, respectively. If an investor chooses ⟨r, 1⟩, she can earn the return r imme-

diately. Therefore, Q̃N (r) and QN (r) are both equal to 1. To guarantee belief consistency, we

only need to consider N = 2 listings, i.e.

Q2(r) = Q̃2(r). (A.2)

In terms of aggregate variables, let LN be the total measure of listings with size N and the

balance of meeting is characterized by

η · (L1 +L2) = E(`) · I , (A.3)

where η still represents the Poisson intensity that a listing is visited by an investor. Denote the

42

stationary rate distribution as

{GN (r), gN (r)}.

In addition, let gN (r, n) be the density of a listing ⟨r, N⟩ who has achieved n ≤ N − 1 units

of fund. Since we only restrict on N ∈ {1, 2}, the distribution dynamics only involve three

equations,

B1 f1(r) =L1g1(r)(ηp1(r)+δ),

B2 f2(r) =L2g2(r, 0)(ηpN (r)+δ), (A.4)

L2g2(r, 0)ηp2(r) =L2g2(r, 1)(ηp2(r)+δ).

The first equation characterizes the stationary distribution of N = 1 listings, and the latter two

represent that for N = 2 listings.22 As for the condition of balanced matching, the instanta-

neous investment I will flow to both types of listings.

I =L1

∫

r∈r1

g1(r)ηp1(r)dr +L2

∫

r∈r2

g2(r)ηp2(r)dr (A.5)

The last condition required is that the total number of new listings should be consistent with

the borrower’s strategy in choosing N , i.e. h1 + h2 = 1. Equivalently, we need that

B = B1 + B2. (A.6)

Note that it is the last condition that differentiates the environment with differentiated sizes

from the one with fixed N . Although borrowers choose N and r simultaneously, it is equivalent

that they first decide how much to borrow and then determine the interest rate. Hence, there

will be two types of new entrants whose measures are endogenously determined.

Accordingly, the stationary equilibrium is described as in Definition A.1.

Definition A.1 (SE with endogenous sizes). Given state variables {B, I}, parameters {δ, r0, v},

and search technology {Pr(`),` ∈ N+}, a stationary equilibrium consists of (i) borrower’s

pricing strategy {hN , FN (r)}; (ii) investor’s decision rule and belief Q̃N ; intensity of meeting η;

22They are actually the special cases of Equation (4).

43

total measure of listings LN ; and (v) rate distribution GN (r). such that

• Borrower’s profit is maximized Equation (A.1);

• Investors’ decision rule is belief-consistent Equation (A.2);

• Balance of meetings Equation (A.3);

• Rate distribution is invariant of time Equation (A.4);

• Balance of matches Equation (A.5);

• The total number of new listings is fixed Equation (A.6).

Existence and Uniqueness

To show the existence and uniqueness of the equilibrium, we first characterize the dynamics of

the two types of listings, given B1 and B2. According to Lemma 4, for both N = 1, 2, we have

δLN gN (r) = BN fN (r)(1− PN (r)),

where PN (r) is the funding probability of ⟨r, N⟩ and takes the form

PN (r) = qN (r)N =

�

ηpN (r)ηpN (r)+δ

�N

.

Therefore, we have

BN = δLN

∫

r∈rN

1

1−�

ηpN (r)ηpN (r)+δ

�N dGN (r).

In addition, by Theorem 3, we know that in any equilibrium, for any r1 ∈ [r0, r̄1], r2 ∈ [r2, r̄2],

we have

⟨r2, 2⟩¥ ⟨r1, 1⟩

44

Thus, we can express pi(ri), i = 1,2 as

p1(r1) =

∑

` Pr(`)`(λG1(r1))`−1

E(`);

p2(r2) =

∑

` Pr(`)`(λ+(1−λ)G2(r2))`−1

E(`).

where λ ≡ L1/L ∈ (0, 1) represents the proportion of small listings in the market. The

expression can be derived by the procedure of Lemma 3. The only difference is that we need

to take both rate r and size N into consideration. For an N = 1 listing, even if r1 = r̄1, it will

be dominated by any ⟨r, 2⟩ in equilibrium. Accordingly, we can form the bijective mapping

from the rate distribution to x ∈ [0, 1].

x =

λG1(r) if r ∈ [r0, r̄1];

λ+(1−λ)G2(r) if r ∈ [r2, r̄2].

Then, we have

B = B1 + B2

= δL∫ r̄1

r0

1

1−�

ηp1(r)ηp1(r)+δ

�dλG1(r)+δL∫ r̄2

r2

1

1−�

ηp2(r)ηp2(r)+δ

�2 d(λ+(1−λ)G2(r))

= δL∫ λ

0

1

1−�

ηp̃(x)ηp̃(x)+δ

�d x +δL∫ 1

λ

1

1−�

ηp̃(x)ηp(x)+δ

�2 d x .

where ∀x ∈ [0, 1],

p̃(x) ≡∑

` ` ·Pr(`) · x`−1

E(`)

as defined by Equation (13). Hence, the market share for small listings λ ∈ [0,1] and meeting

intensity η satisfy

∫ λ

0

1

1−�

ηp̃(x)ηp̃(x)+δ

�d x +

∫ 1

λ

1

1−�

ηp̃(x)ηp(x)+δ

�2 d x =η

δ

BIE(`)

. (A.7)

45

By implicit function theorem, it is easy to show that λ and η are positively correlated. Intu-

itively, when the market has more small listings, namely λ is larger, the instantaneous outflow

of fully funded listings increases since one small listing can be fully funded by one single in-

vestor. Therefore, the market sizeL gets smaller, and the meeting intensity η becomes higher.

Equation (A.7) describes how composition of different listings λ affects other equilibrium

variables, especially η. Next, we consider how the composition is formed. Namely, we discuss

the borrower’s incentive in determining loan size, which will provide us another relationship

betweenη andλ. For instance, at extreme, Lemma 5 shows thatλ= 1 when B/I ≥ z̄ whileλ=

0 when B/I ≤ z. Intuitively, λ tends to be higher when the borrower side is more competitive,

which can be described by a smaller η.

In equilibrium, borrowers are indifferent among all the listings and the unique indifferent

pair ⟨r̄1, 1⟩ ∼ ⟨r2, 2⟩ form the equation

π̂= q̂(v− r̄1) = 2q̂2(v− r2),

where q̂ = q1( r̄1) = q2(r2). By Equation (15a), we have

q1(r0)+ q̂ =ηp1(r0)

ηp1(r0)+δ+

ηp1( r̄1)

ηp1( r̄1)+δ= 1.

Since p1(r0) = p̃(0) = Pr(`=1)E(`)

and p1( r̄1) = p̃(λ) =∑

` `·Pr(`)·λ`−1

E(`), we form another relation-

ship between λ and η,ηp̃(0)

ηp̃(0)+δ+

ηp̃(λ)ηp̃(λ)+δ

= 1. (A.8)

It is obvious that η and λ are negatively correlated. Intuitively, given a larger η, each listings

have higher chance to be viewed. Such market with high meeting intensity will induce more

borrowers choose larger listings in equilibrium, which implies smaller λ.

Based on the discussion above, we summarize our result in Proposition A.1.

Proposition A.1. When B/I ∈ [z, z̄], there is a unique stationary equilibrium with endogenous

sizes. In the equilibrium, borrowers implement identical mixed strategies over sizes and rates, and

investors endogenously form a lexicographic preference on sizes over rates.

46

Proof. The proof of the existence and uniqueness consists of two parts. First, we will show

that Equation (A.7) and (A.8) provide a unique solution (η,λ). Then, we show that all rest of

equilibrium components are uniquely determined.

Equation (A.7) induces an increasing implicit function between λ and η, denoted as η1(λ).

Meanwhile, equation (A.8) characterizes a decreasing function between the two η2(λ). There-

fore, if the solution exists, it must be unique. To show the existence, we consider the two

extreme cases with λ= 0, 1, respectively.

(i) When λ= 0, Equation (A.8) implies that ηp̃(0)ηp̃(0)+δ = 1

2 and hence

η2(0)

δ=

E(`)

Pr(`= 1).

Also, we know that an necessary condition for the equilibrium without all N = 2 is thatη∗

δ <E(`)

Pr(`=1) . As for η1(0), it is the meeting intensity under which every borrower has to

choose N = 2. It cannot be the equilibrium since the meeting intensity is too small and

all borrowers tend to switch to N = 2. Therefore, η1(0) < η∗. Accordingly, we have

η1(0) < η2(0).

(ii) When λ= 1, Equation (A.7) becomes

∫ 1

0

1

1−�

ηp̃(x)ηp̃(x)+δ

�d x =η

δ

BIE(`)

.

By that∫ 1

0 p̃(x)d x = 1E(`)

, we have

η1(1)

δ=

IE(`)

B− I.

For Equation (A.8), p̃(0) = Pr(`=1)E(`)

and p̃(1) = 1, we have

η2(1)

δ=

�

E(`)

Pr(`= 1)

�0.5

.

47

Notice that we assume B/I < z̄ = 1+ Pr(`= 1)0.5. Therefore, we have

η1(1)

δ=

IE(`)

B− I>

E(`)

Pr(`= 1)0.5>

�

E(`)

Pr(`= 1)

�0.5

=η2(1)

δ,

or

η1(1) > η2(1),

By η1(λ) is increasing and η2(λ) is decreasing and η1(0) < η2(0),η1(1) > η2(1), we can

conclude that there exists unique η and λ that satisfy both Equation (A.7) and (A.8). Given

the unique η and λ, the equilibrium profit can be uniquely pinned down by

π̂=ηp̃(0)

ηp̃(0)+δ(v− r0).

Since λ is known, we can use the indifference condition to find r̄1 by π̂ =ηp̃(λ)ηp̃(λ)+δ (v − r̄1).

Correspondingly, p1(r1) can be computed for all r1 ∈ [r0, r̄1]. Similarly, we can solve for

[r2, r̄2] and p2(r2). Then, by that

x =

λG1(r) if r ∈ [r0, r̄1]

λ+(1−λ)G2(r) if r ∈ [r2, r̄2]

GN (r) can be determined. The last step is to find borrower’s strategy {hN , FN (r)} byδLN gN (r) =

BN fN (r)(1− PN (r)).

A.2 Technical Proofs

In this section, we provide all the technical proofs that omitted in the main context.

Proof of Proposition 1

The proof of the proposition is intuitive. When Pr(` > 1) = 1, all the investors can compare

at least two listings. Therefore, all borrowers quote the rate at competitive level r = v. If

Pr(`= 1) = 1, borrowers become local monopolist to the investors. Hence, the rate will be as

48

low as investors’ reservation value. �

Proof of Lemma 2

In a symmetric equilibrium with continuous mixed strategy {F(r), f (r)}, suppose r > r0.

When a borrower’s rate r = r, then the listing must be the least attractive one on any investor’s

sample. Hence, the only possibility of being invested is that the investor’s sample size `= 1. If

so, every borrowers has incentive to lower the r, which will not affect funding probability but

will increase profit conditional on fully funded. �

Proof of Lemma 3

In stationary equilibrium, the measure of projects that are matched should be balanced on both

sides for each rate r ∈ r, i.e.

I ·Pr(rinvest ≤ r) = ηL ·Pr(r f und ≤ r).

The left-hand side is the total measure of investors whose invested project is below r, while the

right-hand sidte represents the listings who make a unit of progress and have rates no larger

than r. Note that

I ·Pr(rinvest ≤ r) = I ·∑

`

Pr(`) ·G(r)`.

For an investor, for each realization of `, the probability that the best rate is less than r is

G(r)`. However, the measure of projects with rate less than r and make one step further can

be represented as

Pr(r f und ≤ r) =

∫ r

r0

g( r̃)p( r̃)d r̃.

Take the derivative of the above two expressions w.r.t. r, then we have

I · g(r)∑

`

` ·Pr(`) ·G(r)`−1 = ηL · g(r)p(r).

Moreover, by Equation (3), we have the desired result. �

49

Proof of Proposition 2

When an investor observes a project with rate r, he has to infer the distribution of the project’s

progress distribution. First, provided that the project is still listed, its age distribution is ex-

ponential with parameter δ. Hence, the probability that the progress is equal to n can be

expressed as∫ ∞

0δe−δt ·

e−ηp(r)t(ηp(r)t)n

n!d t,

which is equal to1

1− P(r)δ(ηp(r))n

(δ+ηp(r))n+1.

Given n, the probability that the project can reach N before death is

�

ηp(r)δ+ηp(r)

�N−n−1

.

Therefore, the expected probability is equal to

11− P(r)

N−1∑

n=0

δ(ηp(r))n

(δ+ηp(r))n+1

�

ηp(r)δ+ηp(r)

�N−n−1

,

which gives us

Q(r) =1

1− P(r)Nδ

δ+ηp(r)

�

ηp(r)δ+ηp(r)

�N−1

.

To show that Q(r) is increasing in r, by the expression of P(r), we have

Q(r) =

�N−1∑

n=0

�

ηp(r)δ+ηp(r)

�n�−1�ηp(r)

δ+ηp(r)

�N−1

=N−1∑

n=1

�

1q(r)n

�−1

.

As q(r) is increasing in r, Q(r) is also increasing. The belief-consistency is simply induced

from the monotonicity of Q(r) since the investor’s expected utility is

u(r) = Q(r)r +(1−Q(r))r0

. �

50

Proof of Lemma 4

By stationary conditions described by Equation (4), we have

g(r, n) = g(r, n−1) ·ηp(r)

ηp(r)+δ

for all n ∈ {1, 2...N −1}. In addition, we know that

g(r) =N−1∑

n=0

g(r, n),

hence we have

L g(r, N −1) =L g(r, 0) ·�

ηp(r)ηp(r)+δ

�N−1

=B f (r)

ηp(r)+δ·�

ηp(r)ηp(r)+δ

�N−1

.

And by

B f (r) =L · (δ · g(r)+ηp(r) · g(r, N −1))

and Equation (8), we have

B f (r) =L ·δg(r)+ B f (r)P(r).

�

Proof of Proposition 3

By Equation (8) and (11) , we have

B = δL∫

r∈r

1

1−�

ηp(r)ηp(r)+δ

�N dG(r).

Note that G(r) is continuous with domain [0, 1]. Moreover, by Lemma 3, p(r) can be repre-

sented as p(G(r)), which implies that p is a function of G. Therefore, we can define p̃(x) as

in Equation (13) and obtain Equation (12). To show that there exists a unique η that solves

51

the problem, define

F (ρ) ≡ ρ∫ 1

0

1

1−�

p̃(x)p̃(x)+ρ

�N d x −B

IE(`),

where ρ ≡ δ/η. First, we show that F is monotone in ρ.

∂F∂ ρ

=

∫ 1

0

1

1−�

p̃(x)p̃(x)+ρ

�N d x −ρ∫ 1

0

N�

p̃(x)p̃(x)+ρ

�N−1

�

1−�

p̃(x)p̃(x)+ρ

�N�2

p̃(x)(p̃(x)+ρ)2

d x .

For simplicity, denote q̃(x) ≡ p̃(x)p̃(x)+ρ , then the above partial derivative equals

∫ 1

0

11− q̃(x)N

d x −∫ 1

0

Nq̃(x)N−1

(1− q̃(x)N )2 q̃(x)(1− q̃(x))d x .

To show that the term is positive, it is sufficient to show that the integrand is always positive,

or equivalently

1− q̃(x)N −Nq̃(x)N (1− q̃(x)) > 0.

It is easy to show that, first, the term is decreasing in q̃(x); second, when q̃(x) = 0 and 1,

the above term equals to 1 and 0, respectively. As q̃(x) ∈ (0, 1), the above term is positive

universally. Therefore, F is increasing in ρ and hence is decreasing in η. To demonstrate the

existence and uniqueness of η, rewrite F (ρ) as

F (ρ) =

∫ 1

0

p̃(x)+ρ∑N−1

n=0

�

p̃(x)p̃(x)+ρ

�n d x −L

IE(`).

When ρ goes to infinity, it is obviously that limρ→+∞F (ρ) = +∞> 0. By the Intermediate

Value Theorem and that F is increasing in ρ, there is a unique ρ such that F (ρ) = 0 if and

only if F < 0.

When ρ = 0, the above equation becomes

F (0) =1

NE(`)−

BIE(`)

,

52

which is negative if and only if N · B < I . �

Proof of Theorem 1

Because η is uniquely determined,L is also uniquely determined. Because the lower bound of

the borrowers’ mixed strategy is always r0, in equilibrium, the indifference condition implies

that

π̂= P(r0) ·N(v− r0),

where

P(r0) =

�

ηp(r0)

δ+ηp(r0)

�N

.

Note that G(r0) = 0; therefore, p(r0) = p̃(0). Therefore, P(r0) is uniquely specified, as is π̂.

In the next step, provided that the equilibrium profit is uniquely determined, we have

P(r) =π̂

v− r.

So

p(r) =δ

η

�

π̂v−r

�1/N

1−�

π̂v−r

�1/N

is uniquely determined. By Lemma 3, we find unique G(r) and g(r). Finally, f (r) = δL g(r)B(1−P(r))

is unique. In summary, beginning from the existence and uniqueness of η, all equilibrium

elements of the equilibrium can be derived step by step. �

Proof of Proposition 4

By Proposition 3, we have

ρ

∫ 1

0

1

1−�

p̃(x)p̃(x)+ρ

�N d x =B

IE(`),

where ρ ≡ δ/η. Then, monotonicity can be proven by the Implicit Function Theorem. The

limit of η can be derived by applying Bernoulli’s Inequality and the Squeeze Theorem.

53

(i) Monotonicity

Define z ≡ B/I , and let

F (ρ, N , z) ≡ ρ∫ 1

0

1

1−�

p̃(x)p̃(x)+ρ

�N d x −z

E(`).

Obviously, both ∂F/∂ N and ∂F/∂ z are negative. For ∂ F/∂ ρ, we have shown in the

proof of Proposition 3 that ∂F/∂ ρ is positive. By the Implicit Function Theorem, we

havedρdN

= −∂F/∂ N∂F/∂ ρ

> 0,

anddρdz

= −∂F/∂ z∂F/∂ ρ

> 0.

Therefore, ρ is increasing in both N and z, which implies that η is decreasing in N and

z. Moreover, by Equation (3), L is increasing in N and z. By ρ = δ/η, we conclude

that η is decreasing in N and L is increasing in N .

(ii) Limit

To find the limit of η as N goes to infinity, first, it is obvious that

BIE(`)

≤ ρ.

However,�

p̃(x)+ρp(x)

�N

=

�

1+ρ

p̃(x)

�N

> 1+Nρp̃(x)

,

54

which is guaranteed by Bernoulli’s Inequality. Therefore,

BIE(`)

= ρ

∫ 1

0

1

1−�

p̃(x)p̃(x)+ρ

�N d x

< ρ

∫ 1

0

1Nρ

Nρ+p̃(x)

=1N

∫ 1

0p̃(x)d x +ρ.

Therefore, we haveB

IE(`)> ρ >

BIE(`)

−1N

∫ 1

0p̃(x)d x .

By the Squeeze Theorem, ρ converges to LIE(`)

as N goes to infinity. Thus, η converges

to δIE(`)/L.

�

Proof of Proposition 5

By Lemma 3, as p(r) and G(r) are bijective, comparing G(r; B, I , N) is equivalent to comparing

p(r) under different (B, I , N). Recall that

p(r) =δ

η

�

π̂v−r

�1/N

1−�

π̂v−r

�1/N.

Let ρ ≡ δ/η and write π̂= P(r0)(v− r0). Then, we have

p(r) = ρ ·

p(r0)ρ+p(r0)

� v−r0v−r

�1/N

1− p(r0)ρ+p(r0)

� v−r0v−r

�1/N.

By Proposition 4, it is easy to show that p(r) is decreasing in B/I and N , as is G(r). In terms

of r̄, because p(r) is decreasing in B/I and N , while p̃(x) is fixed with p̃(1) = 1, then we have

that r̄ is increasing in B/I and N .

�55

Proof of Corollary 1

By Equation (12), ρ ≡ δ/η is uniquely determined by (B, I , N) and search technology Pr(`).

In order to show that a variable is independent of δ, it is sufficient to show that it is only a

function of ρ.

By Equation (6) and (8), a borrower’s expected profit can be written as

π̂= P(r0) ·N(v− r0) =

�

p(r0)

ρ+ p(r0)

�N

N(v− r0),

where by Equation (9) and that G(r0) = 0,

p(r0) =Pr(`= 1)

E(`).

Thus, the expected profit π̂ is independent of δ. Further, since for any r ∈ r, π̂ = P(r) ·

N(v − r), P(r) is independent of δ, which indicates that p(r) is independent of δ according

to Equation (8). Finally, by Equation (9), the rate distribution G(r) is free of δ.

�

Proof of Theorem 2

The proof consists of two steps. In step one, we compare two cases by fixing the search tech-

nology. In the first case, we assume that fundraisers are free to set their rates. In the second

case, we assume that the market imposes a price regulation such that all listings have to set a

rate of r∗. In step two, we will show that for any search technology, a unique price will result

in same level of social welfare.

(i) Equilibrium with rate dispersion When borrowers are free to set their prices, when

Pr(`= 1) ∈ (0, 1), as mentioned in the proof of Proposition 3, we have

∫ 1

0

ρ

1−�

p̃(x)p̃(x)+ρ

�N d x =B

IE(`)

56

where ρ = δ/η. However, when the price regulation is imposed, then for each meeting, the

probability that the project will be invested in is identical to all the projects in the market,

namely

p∗ =

∑

` Pr(`) · ` · 1`

E(`)=

1E(`)

.

Further, all projects have the same funding probability

P∗ =�

p∗

p∗+ρ∗

�N

where ρ∗ = δ/η∗. Hence, we have

ρ∗

1−�

p∗p∗+ρ∗

�N =B

IE(`).

By Equation (9),∫ 1

0p̃(x)d x =

∫

r∈rp(r)dG(r) =

1E(`)

.

Moreover, for any ρ > 0, function

h(z,ρ) ≡ρ

1−�

zz+ρ

�N

is convex in z, and strictly convex when N > 1. Therefore, by Jensen’s Inequality, we have

∫ 1

0

ρ

1−�

p̃(x)p̃(x)+ρ

�N d x ≤ρ

1−�

p∗p∗+ρ

�N .

And the inequality is strict if and only if N > 1. As has already been shown, h(z,ρ) is increasing

in ρ, to have∫ 1

0

ρ

1−�

p̃(x)p̃(x)+ρ

�N d x =ρ∗

1−�

p∗p∗+ρ∗

�N =1

E(`),

57

it has to be the case that ρ∗ ≥ ρ or equivalently η∗ ≤ η, which indicates a market with unified

price will have slower funding speed, i.e.

L ∗ ≥L .

(ii) Constant market size without dispersion Now we show that for any search technology,

when rate dispersion disappears,L remains constant. Suppose that there is no rate dispersion,

we have

p∗ =1

E(`).

Moreover, ηL = E(`)I , a project’s success rate is always

P∗ =�

ηp∗

ηp∗+δ

�N

=

�

II +δL

�N.

In the market, we have

δL ∗ = B(1− P∗) = B− B ·�

II +δL ∗

�N,

which indicates that L ∗ is uniquely determined regardless of search technology.

�

Proof of Lemma 5

The proof consists of three steps. In the first two steps, we show that equilibria with all N = 1

and N = 2 emerge if and only if B/I > z̄ and B/I > z, respectively. Then, we show that z̄ > z.

(i) Equilibrium with all N = 1 Given an equilibrium in which all borrowers choose a loan

of N = 1, the rates are distributed on the support r1 = [r0, r̄1]. We need to show that N = 2

listings are never more profitable. Note that the increasing function r2(r1) in Equation (15b)

1-to-1 maps the set r1 to another set r′2 = [r0, r̄2]. For any r1 ∈ r1, borrower’s profit is

π̂= q1(r1)(v− r1) = q1(r0)(v− r0).

58

As for deviations, if r2 ∈ r′2, then p2(r2) = p1(r1(r2)) and q2(r2) = q1(r1(r2)). If r2 > r̄ ′2, we

have p2(r ′2) = 1, while r ′2 < r0 will induce p2(r ′2) = 0. Accordingly, we will focus on deviations

⟨r ′2, 2⟩ such that r ′2 ∈ r′2 and the borrower’s profit is equal to

π′2 = 2q2(r ′2)2(v− r ′2).

By Equation (15b) and indifference condition, the profit is equal to

π′2 = q1(r ′1)(1+ q1(r ′1))(v− r ′1)− q1(r ′1)(1− q1(r ′1))(v− r0)

=�

q1(r0)(1+ q1(r ′1))− q1(r ′1)(1− q1(r ′1))�

(v− r0),

where r ′1 = r1(r ′2). Hence, there exists no profitable deviation if and only if π′2 ≤ π̂ for any

r ′1 ∈ r1. Equivalently, it implies the following inequality holds for any r ′1 ∈ r1,

q1(r0)(1+ q1(r ′1))− q1(r ′1)(1− q1(r ′1)) ≤ q1(r0)⇔ q1(r ′1)+ q1(r0) ≤ 1.

Since q1(r) is increasing in r, it is necessary and sufficient to have

q1( r̄1)+ q1(r0) ≤ 1

The equilibrium with all N = 1, the total measure L1 satisfies δL1 + I = B ⇔ L1 = B−Iδ .

Also by Equation (3), η=δIE(`)

B−I . In addition, by that p1(r0) =Pr(`=1)

E(`)and p1( r̄1) = 1, we can

solve for closed form q1( r̄1) and q1(r0) in terms of model primitives.

q1(r0) =I ·Pr(`= 1)

B− I + I ·Pr(`= 1); q1( r̄1) =

IB

.

Therefore, we have

q1( r̄1)+ q1(r0) ≤ 1⇔BI≥ 1+ Pr(`= 1)0.5 ≡ z̄.

(ii) Equilibrium with all N = 2 Under the equilibrium that all listings are of size N = 2,

the rates are distributed on the support r2 ∈ [r0, r̄2]. By Equation (15a), function r1(r2) 1-to-1

59

maps the interval to r′1 = [r0, r̄1].

The equilibrium profit is represented by

π̂= 2q2(r2)2(v− r2) = 2q2(r0)

2(v− r0).

If some borrower deviates to ⟨r ′1, 1⟩, by the same argument, the most profitable small project

must have r ′1 ∈ r′1, so that

π′1 = q1(r ′2)(v− r ′1).

By Equation (15a) and indifference condition, π′1 can be expressed as a function of r ′2

π′1 =2q2(r ′2)

2

1+ q2(r ′2)(v− r ′2)+

q2(r ′2)(1− q2(r ′2))

1+ q2(r ′2)(v− r0)

=

�

2q2(r0)2

1+ q2(r ′2)+

q2(r ′2)(1− q2(r ′2))

1+ q2(r ′2)

�

(v− r0),

where r ′2 = r2(r ′1). Hence, there exists no profitable deviation if and only if π′1 ≤ π̂ for any

r ′2 ∈ r2. Equivalently, it implies the following inequality holds for any r ′2 ∈ r2,

2q2(r0)2

1+ q2(r ′2)+

q2(r ′2)(1− q2(r ′2))

1+ q2(r ′2)≥ 2q2(r0)

2⇔ 2q2(r0)2 + q2(r ′2) ≥ 1.

for all r ′2 ∈ r2. Since q2(r) is increasing in r, it is necessary and sufficient to have

2q2(r0)2 + q2(r0) ≥ 1⇔ q2(r0) ≥

12

.

By that q2(r0) =ηp2(r0)ηp2(r0)+δ

and p2(r0) =Pr(`=1)

E(`), the inequality is equivalent to

δ

η

E(`)

Pr(`= 1)≤ 1.

By Equation (9), it is easy to show that η is decreasing in B/I . In other words, the inequality

hows if and only if B/I ≤ z, where z is determined by other model primitives.

60

(iii) Comparison of z̄ and z Finally, we show that z < z̄. In other words, the set of model

primitives that support N = 1 equilibrium is disjoint with whose support N = 2 equilibrium.

Assume not and suppose that z ≥ z̄, then for all primitives that satisfy B/I = z ∈ [z̄, z],

both types of equilibria with N = 1 and 2 can hold. Denote η1,η2 are the equilibrium meeting

intensities in the two equilibria, respectively. When N = 1, by replacing q1(r0) =ηp1(r0)ηp1(r0)+δ

,

q1( r̄) = η1η1+δ

and p1(r0) =Pr(`=1)

E(`), the inequality q1( r̄1)+ q1(r0) ≤ 1 is equivalent to

E(`)

Pr(`= 1)≥�η1

δ

�2.

For N = 2, we have shown thatE(`)

Pr(`= 1)≤η2

δ.

However, under the same technology and identical (B, I), it must be the case that η1 > η2

since L1 < L2 under the two equilibria. Thus, the above two inequalities cannot be satisfied

simultaneously since E(`)Pr(`=1) > 1 for sure.

�

Proof of Lemma 6

We first show the uniqueness of such return equivalent pair ⟨r1, 1⟩ ∼ ⟨r2, 2⟩ if it exists. By

Equation (15a), r1 and r2 satisfy

r1 =2q2(r2)

1+ q2(r2)r2 +

1− q2(r2)

1+ q2(r2)r0

and also by

q2(r) =ηp2(r)

ηp2(r)+δ

we have

p2(r2) =δ

η

r1− r0

2(r2− r1)(A.9)

Since the probability is also the same as p1(r1), define p∗(r1, r2) ≡ p1(r1) = p2(r2).

Moreover, the two projects have to generate the same expected profit, hence, for both

61

n = 1, 2, we have

π̂= NqN (rN )n(v− rN ) =

�

ηpN (rN )

ηpN (rN )+δ

�N

N(v− rN )

By eliminating p∗(r1, r2), π̂ and rN satisfy

2(v− r2)π̂= (v− r1)2 +(v− r1)(π̂− v + r0)+ (v− r0)π̂

and

2(v− r2)π̂= (v− r1)2.

Equating them by eliminating r2, we have

(v− r1)2 = (v− r1)

2 +(π̂1− v + r0)(v− r1)+ π̂(v− r0)

Obviously, the equation is linear in r1 and thus has unique solution given π̂,

r1 = v−π̂(v− r0)

v− r0 + π̂∈ (π̂, v).

And r2 is also determined by

r2 = v−(v− r1)

2

2π̂.

To show existence, suppose there is no such return equivalent pair. Define rN as the lower

bound of rN . Since the lowest rate has to be r0, if r1 = r0, then r2 > r0 for sure and vice versa,

since ⟨r0, 1⟩ and ⟨r0, 2⟩ are always return equivalent to the investors. W.l.o.g., let’s assume

r2 > r0. In this case, by choosing a project ⟨r ′, 2⟩ = ⟨r2− ε, 2⟩ for sufficiently small ε > 0, we

have p2(r ′) = p2(r2), since there exists no return equivalent project and both r1 and r2 are

closed, which leads to the contradiction.

Given uniqueness and existence of the return equivalent projects, we finally show that both

r1 and r2 are simply closed intervals. Assume not and w.l.o.g. suppose r2 consists of more than

one interval. Denote r̄ ′2 < r ′2 be the upper bound and lower bound of two adjacent intervals,

respectively. Now we show that it will lead to the contradiction that there exists two return

62

equivalent pairs r ′1 = r1( r̄ ′2), r̄ ′1 = r1(r ′2) such that [r ′1, r̄1] ∈ r1. The reasoning is as follows.

1. There exists some r1 ∈ r1 such that

⟨r ′2, 2⟩ � ⟨r1, 1⟩ � ⟨r̄ ′2, 2⟩

Otherwise, choosing any ⟨r2, 2⟩ with r2 ∈ [r ′1, r̄ ′1] will be a profitable deviation for the

borrower.

2. Let r′1 ⊆ r1 be the set of all rates that satisfy the preference relationship above. Then

r′1 is an open interval r′1 = (r ′1, r̄ ′1) ∈ r1. Otherwise, since r1 is closed, there exists an

open interval (r ′′1 , r̄ ′′1 )∩ r′1 = ;. Hence, it is a profitable deviation to choose ⟨r1, 2⟩ where

r1 ∈ (r ′′1 , r̄ ′′1 ).

3. Finally, it must be the case that

⟨r ′1, 1⟩ ∼ ⟨r̄ ′2, 2⟩; (r ′2, 2⟩ ∼ ⟨r̄ ′1, 1⟩

which contradicts the fact that the return equivalent pair is unique.

�

Proof of Theorem 3

By Lemma 6, there are only two possible cases in equilibrium.

(i) Investors prefer large listings. That is, for any r1 ∈ r1 and r2 ∈ r2, we have

p2(r2) ≥ p1(r1)

Hence, r1 takes the form [r0, r̄1] and r2 is represented by [r2, r̄2] with p1( r̄1) = p2(r2).

(ii) Investors prefer small listings. That is, for any r1 ∈ r1 and r2 ∈ r2, we have

p2(r2) ≤ p1(r1)

63

Hence, r2 takes the form [r0, r̄2] and r1 is represented by [r1, r̄1] with p1(r1) = p2( r̄2).

We aim to show that any profile under case (ii) will induce profitable deviation.

Assume the opposite and suppose there exists an equilibrium with r2 ∈ [r0, r̄2] and r1 ∈

[r1, r̄1]. No profitable deviation requires that no one will choose ⟨r0, 1⟩. Equivalently, we have

π̂= 2q2(r0)2(v− r0) ≥ q1(r0)(v− r0),

which indicates that q2(r0) = q1(r0) ≥ 1/2. Now, consider a borrower who deviates to ⟨r ′2, 2⟩

with r ′2 > r̄2, then

π′2 = (1+ q1(r ′1))π̂− (1− q1(r ′1))q1(r ′1)(v− r0),

where ⟨r ′1, 1⟩ ∼ ⟨r ′1, 2⟩. Obviously, the deviation is profitable, since

π′2− π̂= q1(r ′1)(2q1(r0)2 + q1(r ′1)−1)(v− r0).

and q1(r ′1) > q1(r0) > 1/2. By finding the profitable deviation, we show that such profile can

never be supported in equilibrium.

�

Proof of Proposition 6

First, by Equation (3),

W = B−δL = B−δ

ηE(`)I .

Therefore,

w≡Wrandom matching

WDispersed Price=

B− δη∗E(`)I

B− δηE(`)I,

where η and η∗ are the meeting intensity under dispersed rates and uniform rate, respectively.

Since (B, I) can be directly observed in the data, it is sufficient for us to identify δηE(`) and

δη∗E(`). By that H(x) ≡ p̃(x)E(`) and p̃(1) = 1, we can identify E(`) since H(1) = E(`).

64

By Equation (12), we have

δ

η

∫ 1

0

1

1−�

ηp̃(x)η̃p(x)+δ

�N d x =B

IE(`),

which implies that p̃(x) is sufficient to compute δ/η as long as N is observable. Given H(x),

p̃(x) can be computed simply by p̃(x) = H(x)/H(1).

Similarly, η∗ satisfiesδ

η∗1

1−�

η∗p∗η∗p∗+δ

�N =B

IE(`),

where p∗ = 1E(`)

is identifiable.

To derive Equation (16), since23

P(x) =

�

ηp̃(x)ηp̃(x)+δ

�N

,

we have

p̃(x) =δ

η

P(x)1/N

1− P(x)1/N

By Equation (3), we have

H(x) ≡ p̃(x) ·E(`) =δ

η

P(x)1/N

1− P(x)1/N·ηL

I=

P(x)1/N

1− P(x)1/N·δL

I

Finally, for the term δL , recall that by Equation (8) and (11) , we have

B = δL∫

r∈r

1

1−�

ηp(r)ηp(r)+δ

�N dG(r) = δL∫ 1

0

1

1−�

ηp̃(x)ηp̃(x)+δ

�N d x .

Thus,

δL =B

∫ 10 1−

�

ηp̃(x)ηp̃(x)+δ

�N =B

∫ 10

d x1−P(x)

,

which gives us the desired result.

�23In the proof, we simplify the notation, and use P(x) and P(x , z; N) interchangeably.

65

B Tables and Figures

B.1 Tables

Variable Mean Std. dev. Min Max

All ListingsLoan Size 37,489.39 199,656 400 20,000,000

Interest Rates 0.126417 0.0334 10% 20%Durations 29.31826 40.41698 1 810

Progress 49.07% 0.4448 0 100%Number of Observations 508,888

Primary LoansLoan Size 3,770,128 2,442,229 1,000,000 20,000,000

Interest Rates 18% 0 18% 18%Durations 612.787 227.4511 180 810

Progress 100% 0 100% 100%Number of Observations 939

Derived LoansLoan Size 30,442.37 48,530.61 400 5,000,000

Interest Rates 12.81% 0.0296 10% 20%Durations 27.98 30.59 1 360

Progress 47.58% 0.4451 0 100%Number of Observations 464,718

Table 1: Summary Statistics of All Listings

Note: The data for this table summarize information on 508,888 listings. Moreover, we report the

summary statistics for primary Loans and derived Loans. The time horizon of the data is from January

2015 to May 2015. All interest rates are annualized; loan sizes are measured by CNY(1 USD = 6.9

CNY); duration is measured by day.

66

Variable Mean Std. dev. Min Max

1-Month Derived LoansLoan Size 310,90.98 50,245.85 400 4,370,000

Interest Rates 14.32% .0076 10% 20%Number of Bidders 6.36 6.46 0 342

Progress 51.79% .4270 0 100%Number of Observations 297,685

Number of Successful Loans 126,121

Table 2: Summary Statistics of 1-month Derived Loans

Rate Listings Funded Listings Funding Probability Funding Probability(Weighted by Size)

≥10% 789 65 8.24% 4.77%≥11% 1,216 67 5.51% 8.84%≥12% 7,508 1,582 21.07% 43.69%≥13% 29,774 9,579 32.17% 47.27%≥14% 204,445 88,942 43.50% 59.56%≥15% 47,556 21,958 46.17% 54.54%≥16% 6,397 3,928 61.40% 65.24%Total 297,685 126,121 42.37% 57.34%

Table 3: Funding Probability by Interest Rate

Percentile Funding Probability Funding Probability(Weighted by Size)

10 7.75% 10.89%20 32.87% 30.87%30 36.81% 28.84%40 40.88% 32.81%50 49.95% 44.68%60 50.99% 43.68%70 62.79% 61.44%80 66.95% 67.62%90 75.42% 78.92%

Total 42.37% 57.34%

Table 4: Funding Probability by Rate

Note: The funding probability by percentile is computed by the average of hourly

statistics.

67

Daily Weekly Overall

Individual 0.3390% 0.5692% 0.5889%Aggregate 0.4042% 0.4870% 0.6756%

Table 5: Individual-level Rate Dispersion

Aggregate Individual

10 Percentile 90 Percentile Min MaxDaily 13.78% 14.67% 13.93% 14.66%

Weekly 13.71% 14.84% 13.65% 14.89%Overall 13.71% 15.21% 12.44% 15.88%

Table 6: Individual Rate Dispersion (min/max)

Rate Dispersion Hourly Daily Weekly Overall

All Listings 0.3403% 0.4042% 0.4870% 0.6756%Successful Listings 0.1268% 0.2186% 0.3498% 0.6218%

Table 7: Rate Dispersion over Time

68

Whole Sample Group 1 Group 2

DTi t DTi t DTi t

PROGRESS CUMULATEDi t−1 -373.1*** 1.260 -4045.1***(-52.25) (1.57) (-50.82)

RATEi -8898.7*** -3280.3*** -90183.8***(-21.32) (-69.09) (-21.20)

SIZEi 9.10e-4*** -1.89e-4*** -7.12e-3***(-31.33) (-53.41) (-25.58)

WEBSITE PAGEi t -6.316*** 29.18*** 60.47***(-9.36) (297.67) (14.99)

LISTING AGEi t 16.80*** 1.209*** 17.63***(555.25) (99.88) (154.25)

Bt 1.18e-5*** -6.02e-6*** 4.46e-5*(6.40) (-29.46) (2.09)

It -4.05e-6 -6.97e-6*** -6.85e-5(-1.38) (-21.40) (-1.92)

Vprimary -9.04e-7*** 1.15e-08 -6.68e-6***(-9.78) (1.12) (-6.69)

Tprimary -0.150*** 0.0106*** -1.602***(-11.93) (7.54) (-11.80)

Daytime 28.71*** 5.271*** 133.4*(5.90) (9.74) (2.39)

Weekend 0.446 4.120*** 366.3***(0.09) (7.24) (7.01)

Night -22.08** 33.53*** 171.6*(-2.83) (37.45) (2.39)

cons 1620.1*** 516.7*** 15753.1***(25.70) (72.29) (24.57)

Biddings 900,405 815,518 84,842Listings 126,149 113,349 12,800R-sq 0.3235 0.1742 0.3300

t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 8: Effect of Funding Progress

Note: The funding speed is computed only for fully funded listings. WEBSITE PAGEi t is listing i’s default pagenumber at the time t; LISTING AGEi t is listing i’s age at time t measured in minutes; SIZEi and Ratei are listingi’s size measured in CNY and annualized interest rate, respectively. Bt represents the amount of listing inflow attime t measure in CNY; It represents total amount of funding inflow at time t. Other control variables includeVprimary and Tprimary measures, which measure the total volume of primary loans and their respective durationmeasured by day; daytime represents the dummy if the time is between 9am and 5pm; night is the dummy if tis between 1am and 9am; weekday is the dummy if the time is from Monday to Friday. In the rest of the tables,the variables will be identically defined unless stipulated otherwise.

69

Hourly Daily

rt rt

Vt 1.42e-09*** 2.33e-10***(16.29) (9.66)

It -7.09e-10*** -2.75e-10***(-4.98) (-6.47)

Vprimary -3.20e-11*** -4.89e-11***(-9.55) (-4.27)

Tprimary -4.70e-6*** -1.68e-6(-8.87) (-0.98)

Daytime 1.75e-3***(8.48)

Night 4.65e-3***(18.83)

Weekday -7.69e-5 6.96e-5(-1.82) (0.52)

cons 0.144*** 0.143***(323.70) (104.66)

N 2,234 94R-sq 0.2851 0.6685t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 9: Average Interest Rate and Market Supply/Demand

Note: rt is the average rate at time t. In the first panel, t is measured in hours. In

the second panel, t is measured in days.

70

Hourly Daily

dt dt

zt 2.61e-4** -0.0154(3.28) (-1.05)

z2t -2.65e-6** 6.50e-3

(-2.82) (1.45)

rt 0.508*** 0.476***(24.63) (6.28)

Vprimary 1.06e-11** 1.66e-11(3.05) (1.80)

Tprimary -3.76e-6*** -1.10e-6(-6.32) (-0.70)

Daytime -8.31e-4***(-3.74)

Night -7.44e-4***(-3.34)

Weekday -4.97e-5 -1.45e-4(-1.07) (-1.18)

cons -0.0638*** -0.0512***(-20.49) (-3.78)

N 2,231 94R-sq 0.2709 0.6454t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 10: Rate Dispersion and Market Supply/Demand

Note: dt is the rate dispersion measured by JNS statistics. In the first panel, t is

measured in hours. In the second panel, t is measured in days.

71

Hourly Daily

ri t ri t

Sizei t 1.64e-08*** 1.70e-08***(77.77) (75.35)

Bt 6.20e-4*** 8.21e-6***(8.38) (3.99)

Vt 5.99e-08*** 1.40e-08***(13.23) (8.61)

It -6.05e-08*** -1.42e-08***(-13.25) (-8.65)

Vprimar y -2.40e-11*** -2.41e-11(-6.69) (-1.81)

Tprimar y -7.09e-6*** -2.88e-6(-12.30) (-1.51)

Weekday -5.94e-4 9.62e-5(-1.31) (0.64)

Daytime 1.28e-3***(5.99)

Night 4.66e-3***(16.65)

cons 0.145*** 0.139***(288.60) (81.55)

N 106,237 106,237R-sq 0.1505 0.1301

t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 11: Correlation between Sizes and Rates

72

Daily Hourly

Nt Nt

Lt -4.402*** -80.73***(-814.56) (-427.79)

It 2.66e-4*** 4.68e-3***(551.48) (360.14)

Vprimar y 2.95e-5** 3.41e-5***(113.59) (64.13)

Vprimar y 0.847*** 1.551***(20.40) (17.39)

Night -2757.5***(-58.77)

Daytime -3147.5***(-106.39)

Weekday -413.9*** -407.7***(-123.44) (-54.65)

cons 29308.5*** 30883.6***(840.52) (410.81)

N 116 2,696R-sq 0.8031 0.5316t statistics in parentheses∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

Table 12: Average Loan Size

Note: Nt measures average number of investors for each successful listings. In the

first panel, t is measured in days. In the second panel, t is measured in hours.

73

Mean SD 95% C.I.

0 0.3580 0.0828 [0.1923, 0.5236]10% 0.4674 0.1020 [0.2633, 0.6715]20% 0.6311 0.1615 [0.3080, 0.9542]30% 0.7286 0.1971 [0.3343, 1.1229]40% 0.8333 0.2144 [0.4045, 1.2622]50% 0.9772 0.2420 [0.4932, 1.4612]60% 1.2539 0.3183 [0.6172, 1.8906]70% 1.2995 0.2788 [0.7417, 1.8572]80% 1.2999 0.2636 [0.7725, 1.8272]90% 1.6737 0.3112 [1.0513, 2.2962]100% 1.7649 0.2851 [1.1946, 2.3351]

Table 13: Estimate of H(x) with 10th Percentile

Mean SD 95% C.I.

0.3844 0.0860 [0.2124, 0.5565]0.6152 0.1629 [0.2893, 0.9410]1.0356 0.3664 [0.3027, 1.7684]1.4727 0.5697 [0.3332, 2.6122]1.8020 0.5495 [0.7031, 2.9010]

Table 14: Estimate of H(x) with 25th Percentile

74

B.2 Figures

New BorrowersB, f (r)

New Investors ILisingsL , g(r, n)

Dead listingsδL

SuccessfulListings

L g(r, N −1)ηp(r)Borrowers exit with N(v− r)Investors exit with r

Borrowers exit with 0Investors exit with r0

Figure 1: Market Structure

Note: The graph illustrates the flow of the model. At each instance, a measure of I investors andB borrowers enter the market. The rate distribution of new listings is f (r). In the market, the totalmeasure of listings is L , with rate/progress distribution g(r, n). As shown in the figure, there are twopossible channels for a listing to leave the market. First, a listing may encounter a death shock withintensity δ; second, a fully funded project will leave the market and both parties earn positive payoffs.

75

π̂

r0 r̄1 r2 r̄2 v

Borrower’s payoff

r0

r1

v

r0 r̄1 r2 r̄2 v

Investor’s payoff

Figure 2: Payoffs with differentiated sizes

Note: (1) The left and right panels show the payoff of borrowers and investors, respectively. The solid and

dashed curves shows the payoffs on and off equilibrium path, respectively. (2) Left panel shows that borrowers

are indifferent among ⟨r, 1⟩with r ∈ [r0, r̄1] and ⟨r, 2⟩with r ∈ [r2, r̄2]. (3) Right panel shows that in equilibrium,

investors always prefer larger listings to smaller ones. There exists a unique pair of return equivalent listings

⟨r̄1, 1⟩ ∼ ⟨r2, 2⟩.

Figure 3: Dealer’s Contribution to Primary loans

Note: The graph illustrates the trading volume of different types of loans. The red line represents theloans issued by the primary fundraisers, while the blue dashed line is for one-month derived loans. Thepattern shows that they are highly correlated.

76

Figure 4: Volume of Different Types of Loans

Note: The graph illustrates the dealers’ contribution to primary loans. The red line represents the loansissued by the primary fundraisers, while the blue dashed line represents the fraction of primary loansthat are funded by dealers.

77

0.1 0.12 0.14 0.16r

0

0.2

0.4

0.6

0.8

1

G(r

)

Rate Distribution (L/I = 0.75)

N = 1N = 3N = 5

0 2 4 6z=B/I

0.1

0.12

0.14

0.16

0.18

Ave

rage

Rat

e

Average Rate

N = 1N = 3N = 5

0 2 4 6z=B/I

0

0.005

0.01

0.015

0.02

Sta

ndar

d D

evia

tion

Rate Dispersion

N = 1N = 3N = 5

0.1 0.12 0.14 0.16r

0

0.2

0.4

0.6

0.8

1

G(r

)

Rate Distribution (N = 3)

z = 0.35z = 0.75z = 1.75z = 3.75

2 4 6N

0.1

0.12

0.14

0.16

0.18A

vera

ge R

ate

Average Rate

z = 0.35z = 0.75z = 1.75z = 3.75

2 4 6N

0

0.005

0.01

0.015

0.02

Sta

ndar

d D

evia

tion

Rate Dispersion

z = 0.35z = 0.75z = 1.75z = 3.75

Figure 5: Rate Distribution vs Supply and Demand

Note: First row reports the change of rate distribution, average rate and rate dispersion w.r.t. differentloan size N ; Second row reports the change of rate distribution, average rate and rate dispersion w.r.t.different supply demand ratio B/I ; For parameters, let δ = 0.9, r0 = 0.1, v = 0.18. For the distributionof `, we apply distribution used in Burdett and Menzio (2018), letting Pr(`= 1) = 1−λ; Pr(`= 2) = λand choose λ= 0.7. As shown in Figure 6, it is closed to my estimated result.

78

Figure 6: Estimates of H(x)

N

z = B/I

Market average

Figure 7: Calibrated Wrandom matchWdispersed price

79