A Macroeconomic Theory of Banking Oligopoly∗

Mei Dong†

University of Melbourne

Stella Huangfu‡

University of Sydney

Hongfei Sun§

Queen’s University

Chenggang ZhouCanada Mortgage and Housing Corporation¶

June 2, 2021

Abstract

We study the behavior and macroeconomic impact of oligopolistic banks in a tractable

environment with micro-foundations for money and banking. In our model, large banks

interact strategically as they compete against each other to make loans. Banks face potential

liquidity issues as they try to secure deposits as a source of funds. We find that it is welfare-

maximizing to have the banking sector as oligopolistic, i.e., to have a small number of large

banks. In addition, inflation stimulates bank entry but always reduces welfare.

JEL Categories: D83, E43, E44, E51, E52, G21, G32, L26Keywords: banking, oligopoly, liquidity, market frictions.

∗We are grateful for feedback from George Alessandria, Yan Bai, Francesco Carli, Allen Head, Charlie Kahn,Thor Koeppl, Jim McGee, Stephen Williamson, Randall Wright, and many other participants of seminars andconferences for comments. The authors also thank the editor and two anonymous reviewers for their constructivecomments, which helped us to improve the manuscript. Dong acknowledges financial support under AustralianResearch Council DECRA scheme (DE120102589). Dong, Huangfu and Sun acknowledge support from AustralianResearch Council Discovery Project Grant (DP210101688). Sun acknowledges support from Social Sciences andHumanities Research Council (SSHRC).†Email address: mei.dong@unimelb.edu.au. Mailing Address: Department of Economics, Faculty of Business

and Economics, Carlton, Victoria, Australia, 3053.‡Email address: stella.huangfu@sydney.edu.au. Mailing Address: School of Economics, University of Sydney,

NSW, Australia, 2006.§Email address: hfsun@econ.queensu.ca. Mailing Address: Department of Economics, Queen’s University,

Dunning Hall, Room 309, 94 University Avenue, Kingston, ON, Canada, K7L 3N6.¶Email address: czhou@cmhc-schl.gc.ca. Mailing Address: Canada Mortgage and Housing Corporation, 700

Montreal Rd, Ottawa, Ontario, Canada, K1A 0P7.

1

1 Introduction

We study the behavior and the macroeconomic impact of oligopolistic banks in a tractable

environment with micro-foundations for money and banking. Nowadays in many coun-

tries, the banking sector is clearly an oligopoly in that it consists of a few large banks

who control a significant proportion of the banking business across the country. This is

demonstrated by Figures 1(a) and 1(b), which respectively report the concentration of

the banking sector measured by the fraction of total bank assets controlled by the top

3- or top 5-largest banks in ten largest advanced economies. For instance, the top five

banks in the U.S. accounted for about 46 per cent of total bank assets in 2017.

Data source: World Bank Global Financial Development Database

0

20

40

60

80

100

United States Japan Germany UnitedKingdom

France Italy Canada Australia Spain Netherlands

%

1(a): 3-bank Asset Concentration for 10 largest Developed Countries1997 2007 2017

0

20

40

60

80

100

United States Japan Germany United Kingdom France Italy Canada Australia Spain Netherlands

%

1(b): 5-bank Asset Concentration for 10 Largest Developed Countries1997 2007 2017

Figure 1: Concentration of the Banking Sector

Since the 2008 Global Financial Crisis, the importance of the banking sector has

prompted new research interests in macroeconomic models of banking. However, exist-

ing literature on banking often ignore the fact that the banking sector consists of a few

large banks and treats the banking sector as being composed of either one monopoly

bank or a continuum of competitive banks with no market power. As is well known

from the Cournot competition model, oligopolistic firms could behave differently from

monopolistic or competitive firms, which would further lead to different economic out-

comes. To overcome this gap between theory and evidence, it is necessary to model banks

as oligopolists and have the market structure of the banking sector (i.e., the number of2

banks) endogenously determined.

In this paper, we construct a dynamic general equilibrium model to study how the

structure of the banking sector affects the macroeconomy and welfare. In our model,

banks accept deposits from households and compete for shares in the credit market by

issuing loans to firms. There are search and matching frictions in the credit market.

Banks are large in the sense that each bank has a large number of loan offi cers working

to provide loans. We calibrate our model to the U.S. economy and obtain two sets of

results:

First, we hold the number of banks exogenous and examine how it affects welfare. We

find that welfare responds to the number of banks non-monotonically in a hump shape.

It is optimal to have an oligopolistic banking sector because welfare is maximized at a

small, yet greater than one, number of banks. The intuition is given by the following: As

the number of banks rises, intensified oligopolistic competition induces aggregate lending

to increase. All else equal, this improves welfare as more firms obtain funding and more

workers find jobs. Nevertheless, an increase in the number of banks also has two negative

welfare effects, through a market congestion channel and a liquidity channel.

In the market congestion channel, oligopolistic banks compete in making loans without

fully internalizing the negative search externality of it sending more offi cers to crowd the

credit market. As a result, the equilibrium matching probability for each offi cer to meet

a fund-seeking firm becomes ineffi ciently low, when the number of banks is above one.

In reality, this corresponds to banks opening up more branches and hiring more bankers

to gain better access to clients, as they compete for a better market share. A direct

consequence of this is to amplify the effective cost of making loans for each bank. In the

liquidity channel, banks face an increasing need for funds as the presence of more banks

induces boosted aggregate lending. Therefore, the banks find themselves in an intensified

bid for deposits as they compete with more banks in the credit market. This causes the

deposit rate to rise, which immediately raises the cost of funds for banks. In sum, both

channels place constraints on lending activities, which ends up diminishing welfare.

The overall welfare effect of having more banks in the economy is a combination

of the positive effect through lending competition and the two negative effects through

market congestion and liquidity constraints. As a result, competition in the banking3

sector improves welfare only to a certain extent. It is welfare-maximizing to have the

banking sector as oligopolistic, i.e., to have a small number of large banks.

The second set of results are obtained by endogenizing the number of banks to study

the relationship among inflation, bank entry, and welfare. We find that inflation stim-

ulates bank entry. On one hand, inflation worsens bank liquidity conditions and thus

causes the real deposit rate to rise, which increases the banking cost of raising funds.

On the other hand, aggregate lending declines with inflation and fewer loan offi cers oper-

ate in the credit market, which reduces the effective cost of making loans by raising the

probability for an offi cer to meet with a client. Overall, the latter tends to dominate the

former and thus bank profit rises with inflation. Finally, inflation monotonically decreases

welfare, as the negative effect of worsened liquidity conditions dominates.

Our theoretical framework is partly built upon Wasmer and Weil (2004) (henceforth

WW) and Berentsen, Menzio and Wright (2011) (henceforth BMW). In particular, WW

has search frictions in the credit and labor markets, while BMW features frictional labor

and goods markets (without a credit market). Although WW focuses on how credit

market conditions affect labor market outcomes, there is an important piece missing in

the model in that the source of lending by creditors is unaccounted for. In contrast,

we specifically fill in this missing piece of the puzzle by modeling a full-fledged banking

sector who gathers funds from workers before lending to firms. Moreover, this banking

sector is the main focus of our paper as we aim to study how decisions of oligopolistic

banks influence the aggregate economy.

Our model has two key features: (i) banks as oligopolists in a frictional credit market,

and (ii) liquidity constraints for banks that arise from timing mismatch of cashflows.

The former gives rise to the market congestion channel of which individual banks fail to

fully internalize the crowding-out externality of them sending more loan offi cers to the

credit market. The latter feature augments that a key function of banking is using liquid

demandable deposits to service illiquid long-term investments. The maturity mismatch

of assets and liabilities allows banks to earn a premium between the long- and short-term

rates, but exposes them to interest and liquidity risks. Excessive maturity mismatch

can not only leave banks vulnerable to runs but also threaten the stability of the entire

financial system, as evidenced by the Global Financial Crisis. Therefore, it is important

4

to take the liquidity mismatch problem into account when modelling the banking sector.1

The two features together allow us to identify the positive effect and the two negative

effects on welfare associated with having more banks. As the exogenous number of

banks rises, we find the equilibrium switches types from (i) banks not being liquidity

constrained, to (ii) banks being constrained on lending but need not retain earnings to

improve liquidity conditions, to (iii) banks being constrained on lending and resorting

to retained earnings, and finally, to (iv) banks being constrained on both lending and

meeting withdraw demands while having to retain earnings. When calibrated to the

U.S. economy, inflation causes the equilibrium to switch types consecutively in the order

from the first to the third type described above, as the number of banks held fixed.

Nevertheless, when bank entry is allowed, the equilibrium consistently rests on the third

type.

Our paper brings a unique contribution to the vast literature on banking. The theo-

retical banking literature mainly consists of microeconomic models of banking. However,

macroeconomic models of banking have attracted much interest in recent years espe-

cially since the 2008 Global Financial Crisis.2 Compared to the standard macroeconomic

banking models, our paper focuses on the long-run effect of having a oligopolistic banking

sector. It is closest in relation to the theories of money, credit and banking with explicit

micro-foundations for money and banking, and focuses particularly on bank lending ac-

tivities. In this regard, the literature narrows down to a few papers such as Berentsen

et al. (2007) and Sun (2007, 2011).3 Among these papers, only Sun (2007) considers1There are various ways to measure banks’ maturity mismatch. English et al. (2018) measure maturity

mismatch as the gap between the repricing time or maturity of bank assets and liabilities. During 1997 to 2007,the median maturity gap for 355 U.S. commercial banks has fluctuated in a relatively narrow range of 3 to 5years. Haddad and Sraer (2019) measure interest risk exposure using the normalized income gap that accountsfor maturities up to one year. A high income gap corresponds to a low exposure to long-term fixed-rate assetsand therefore less maturity mismatch. During 1986 to 2014, the average income gap of U.S. banks with morethan $1B in consolidated assets has a mean of 0.128 and exhibits pro-cyclical variations: peaked at 0.2 in 1994and reached the lowest level of 0.05 in 2009 after the financial crisis.Maturity mismatch can also be measured from the liquidity perspective. Bai et al. (2018) implement a liquidity

measure proposed by Brunnermeier et al. (2012, 2013), the Liquidity Mismatch Index (LMI), to measure themismatch between the market liquidity of assets and the funding liquidity of liabilities over the period 2002 —2013. They find that the aggregate LMI of U.S. bank holding companies worsens from less than -$1 trillion in2002 to -$3.3 trillion in 2008, before reversing back to pre-crisis level in 2009.

2See Gorton and Winton (2003) and Freixas and Rochet (2008) for surveys of microeconomics models ofbanking. Important contributions to macroeconomic models of banking include Gertler and Kiyotaki (2010,2015) and Gertler et al. (2016). These papers study potential banking crisis in the spirit of Diamond and Dybvig(1983).

3Other papers with micro-foundations of money and banking include Cavalcanti and Wallace (1999),Williamson (2004), He et al. (2005, 2008), Williamson and Wright (2010a, 2010b), Gu et al. (2013). Nev-

5

banks as oligopolists. In Sun (2007), banks serve as delegated monitors and there arises

a micro-foundation for money due to lack of double coincidence of wants. In the model,

the number of banks is exogenous. Competition among banks strictly improves wel-

fare by stimulating aggregate lending. In contrast, our model allows for an endogenous

number of banks. Moreover, we examine the consequence of liquidity problem faced by

oligopolist banks and find that oligopolistic competition among banks does not always

improve welfare.

More recently, Corbae and D’Erasmo (2013) endogenize the bank size distribution us-

ing a quantitative model of banking. They model banking competition as a Stackleberg

game and focus on the dynamics of bank size distribution during business cycles. Huang

(2019) develops a model of banking where an optimal number of banks arises due to

trade-off between monitoring costs and rewards to banks. The decentralized allocation

features excess entry of banks, which can be corrected by appropriately-designed banking

policies. While we argue that regulating the number of banks can improve welfare, our

model features a frictional goods market that allows us to examine how inflation interacts

with the structure of the banking sector and welfare. Chiu et al. (2020) consider mo-

nopolistic banks with inside money creation to study the impact of a central bank digital

currency on intermediation of commercial banks. Their model features Cournot quantity

competition in the deposit market and perfect competition in the loan market. Head et al.

(2020) propose a model with imperfect competition in the consumer-loan market based

on the noisy-search model as in Burdett and Judd (1983). Their model generates endoge-

nous concentration of banking sector and heterogenous loan rates among consumers. In

contrast, we model imperfect competition among banks as Cournot competition. Ahnert

and Martinez-Miera (2020) find that an interior number of banks is optimal in a model

with imperfect bank competition for deposits, bank runs and endogenous bank opacity.

The remainder of this paper is organized as follows. Section 2 describes the economic

environment. We provide value functions and derive the optimal decisions made by

workers, firms and banks in Section 3. Section 4 characterizes the banking equilibrium.

Section 5 explores the quantitative implications of our model and considers extension.

The final section offers some concluding remarks.

ertheless, these models do not have oligopolistic banks.

6

2 Model Environment

Time is discrete and has infinite horizons. Each time period t consists of three subperiods.

As shown in Figure 2, a centralized goods market (GM) and a credit market (CM) are

simultaneously active in the first subperiod. A frictional labor market (LM) is active in

the second subperiod. A decentralized goods market (DM) is active in the third subperiod.

All goods are perishable across periods. All agents discount each period at rate β ∈ (0, 1).

Discounting occurs following DM activities and preceding GM/CM activities.

Sub-period 1 Sub-period 2 Sub-period 3

Centralized Market (GM) Labour Market (LM) Decentralized Market (DM) t N (U,V) M (1, S) t+1

π, 𝜋𝜋𝑏𝑏, T, b, wage Withdraw D, Arrival of loan repayment Credit Market (CM)

L (B, E) 𝑔𝑔1,𝑔𝑔2 Interest, new deposit, Funding ϒ

Figure 2: Timing of Events

The economy is populated by three types of agents: workers, firms and banks, which

are indexed by h, f and b, respectively. The measure of infinitely-lived workers is one.

Workers supply labor and consume in both GM and DM. They also own all the firms

in the economy. The measure of firms is suffi ciently large, although not all firms will be

active at any point in time. Firms hire workers to produce output, and then pay wages

and dividends to workers. Posting a vacancy is costly and thus firms require funding

from banks before they can hire workers and start production. There is also a banking

sector composed of N banks, where N ≥ 1 is finite. Banks take in demand deposits from

workers and lend to firms.

Due to anonymity and the lack of double coincidence of wants, agents require a means

of payment in DM. An intrinsically worthless, perfectly divisible and storable object, fiat7

money, serves this purpose. The quantity of money at the end of subperiod 1 of period t

is Mt. We assume that Mt = (1 + π)Mt−1, where π is the growth rate of money supply.

New money is injected by lump-sum transfers to workers at the beginning of subperiod

1 in GM. The nominal price of GM goods is pt. We restrict our attention to stationary

equilibria in which the real value of aggregate money balances Mt/pt is constant. It

implies that pt = (1+π)pt−1 and π is the net inflation rate. To facilitate exposition, from

now on we adopt the following notation: M denotes money balances in period t, and M

represents money balances in period t+ 1, similarly for other variables. Throughout the

paper, we use GM goods as the numeraire. We denote the respective values of entering

GM, CM, LM and DM as W j, Ij, U j, and V j, where j ∈ {h, f, b} is the agent type.

Subperiod 1: In subperiod 1 of every period t, a frictionless GM and a frictional

CM are simultaneously active. Each bank in CM has a large number of loan offi cers who

assist in lending. At the beginning of this subperiod, each bank selects from a pool of

idle loan offi cers. The selected loan offi cers receive instructions on how to negotiate with

matched firms over loan terms, before being sent into CM. New firms can enter CM at

a sweat cost k > 0. Loan offi cers and new firms are matched randomly and bilaterally.

Matching is governed by the function L (B,E), where B =∑N

n=1Bn is the total measure

of loan offi cers from all banks, Bn is the measure of loan offi cers from bank n, and E is

the measure of new firms in CM.

A new firm faces a cost of γ > 0 units of GM goods every time it posts vacancies in

LM. As such, the loan contract requires the bank to lend γ to the firm in every subperiod 1

until the firm successfully hires a worker. From that point on, the firm will start repaying

a real amount a every period until the job terminates, which occurs with probability s

every period. The job separation shock occurs at the onset of each subperiod 2.

Subperiod 2: In subperiod 2, firms (those with funding from banks) and unemployed

workers are randomly and bilaterally matched in LM. Upon meeting, the two parties

bargain over the real wage, and production takes place immediately afterwards. Matching

is governed by the function N (u, v), where u is the measure of unemployed workers and

v is the measure of vacancy. The output in a match is y > 0, which is exogenous.

Subperiod 3: In subperiod 3, all workers and firms with output y are randomly8

matched in DM. Again, matching is bilateral and is governed by the functionM (1, 1− u),

where the measure of workers is 1 and the measure of firms with output y is 1− u. Once

matched, the parties engage in Kalai bargaining over the terms of trade (q, d), where a

real money balance d is used to purchase an amount q of DM goods.4 The DM goods

can be paid with only fiat money. Banks are not allowed to issue inside money, but

workers can withdraw their deposits to obtain fiat money.5 We assume that all matching

functions have constant returns to scale and satisfy the usual assumptions such as being

strictly increasing in both arguments.

3 Values and Decisions

In this section, we describe the value functions for each type of agents and solve the

bargaining problems in each market. The value functions for workers and firms are

standard in the models with frictional labor and goods market. The decisions faced by

banks, however, are unique to our framework and thus we provide a detailed description

of the banking problem.

3.1 Workers

Consider a worker with employment status e ∈ {0, 1} entering GM with real deposit

balances z, where e = 1 indicates that the worker is currently employed, and e = 0

means unemployed. The worker maximizes her utility by choosing the amounts of GM

consumption, X, and new demand deposits to hold in the current subperiod, z. Note

that workers will choose asset portfolios consisting of only demand deposits as long as

the nominal interest rate on demand deposits id > 0. The worker’s value of entering GM

is given by

W he (z) = max

X,z{X + (1− e)`+ Uh

e (z)} (1)

st. X + z = ew + (1− e)b+ Π + ρΠb,−1 − T + (1 + id) z,

4We choose Kalai bargaining according to Kalai (1977) because it helps to simplify the theoretical resultswithout loss of generality.

5Later we discuss the implications of allowing for inside money.

9

where ` is the exogenous utility from leisure, w is the real wage, b is unemployment

insurance (UI), Π and Πb,−1 are respectively dividends from firms and banks, and T is the

lump-sum transfer due to taxation and inflationary purposes. Note that dividends from

banks (Πb,−1) are paid to workers as money balances and thus are subject to inflation

with ρ = 1/(1 + π). As is standard with the Lagos-Wright type of environment, the

quasi-linear preference renders the model tractability. Notice that W he is linear in z:

dW he /dz = 1 + id. Moreover, from (1), the optimal choice of z is independent of z. As

will be shown later, the portfolio of the worker is independent of her employment status

e. As a result, every worker has the same z when exiting GM.

In LM, the value function for a worker is constructed according to her employment

status,

Uh1 (z) = sV h

0 (z) + (1− s)V h1 (z) , (2)

Uh0 (z) = λhV h

1 (z) + (1− λh)V h0 (z) , (3)

where s is the job separation rate, and λh = N (u, v) /u = N (1, v/u) is the probability

for an unemployed worker to find a job. Without loss of generality, we assume that if a

match is destroyed, a worker cannot gain employment until the next LM.

In DM, the worker meets a firm with probability αh, where αh =M(1, 1−u). If they

trade, the worker pays a real balance d for q units of goods and thus the value function

is given by

V he (z) = αh[u (q) + βW h

e (ρz − ρd)] + (1− αh)βW he (ρz) . (4)

Note that real balances that are carried into the next period are inflation-adjusted with

ρ. The function u(q) is the utility that the worker obtains by consuming q units of DM

goods, with usual properties of u(0) = 0, u′ > 0 and u′′ < 0.

For a worker, the above value functions of the three respective subperiods can collapse

into one Bellman equation of the entire time period. After repeated substitution, we

obtain the following recursive problem:

W he (z) = Ie + (1 + id)z + βEW h

e (0, 0)

+ maxz{αh[u(q)− β(1 + id)ρd]− z[1− β(1 + id)ρ]}, (5)

10

where the expectation is with respect to next period’s employment status e, I1 = w +

Π + ρΠb,−1 − T and I0 = b+ `+ Π + ρΠb,−1 − T . Note that the optimal choice of z does

not depend on the employment status of a worker. An interior choice of z > 0 satisfies

αh

[u′(q)

∂q

∂z− βρ(1 + id)

∂d

∂z

]= 1− βρ(1 + id). (6)

3.2 Firms

There are three types of firms: firms not associated with any bank or worker (type 0),

firms with a bank loan but not a worker (type 1) and firms who have both a bank loan

and a worker (type 2). We first consider the value of a type-0 firm at the beginning

of subperiod 1, represented by If0 . A type-0 firm can enter CM at a sweat cost k > 0

to search for a loan offi cer for funding. The firm will be matched with a loan offi cer

with probability φf = L (B,E) /E = L (B/E, 1), and enter LM as a type-1 firm. If not

matched, the firm remains type 0. Thus,

If0 = max{0, φfU f1 (a) + (1− φf )βIf0 − k}. (7)

Consider a type-1 firm with a loan contract of periodic repayment a (for short, con-

tract a). With probability λf = N (u, v) /v = N (u/v, 1), the firm finds a worker and

immediately produces output y. The firm then enters DM as a type-2 firm. Otherwise,

it remains type 1. Thus,

U f1 (a) = λfV f

2 (a) + (1− λf )βU f1 (a) . (8)

Only type-2 firms with output y enter DM to trade. In a match, the firm sells q units

of DM goods, which costs the firm c (q) in terms of GM goods and we have c(0) = 0,

c′ (q) > 0 and c′′ (q) ≥ 0. The residual y− c (q) is transformed into GM goods and sold in

the subsequent GM. The cost function here captures the variable costs that firms incur

in production. Correspondingly,

V f2 (a) = αfβW f

2 (y − c(q), ρd, a) + (1− αf )βW f2 (y, 0, a), (9)

where αf = M (1/(1− u), 1) is the probability of a firm meeting and trading with a

11

worker in DM and W f2 (·, ·, ·) is the value of a type-2 firm at the beginning of subperiod

1 of t+ 1.

In GM, a type-2 firm clears inventory x, makes loan repayment a, and pays wage w.

The profit x+ z−w− a aggregating across all type-2 firms constitutes dividends Π paid

to workers. Thus a type-2 firm’s value function is given by

W f2 (x, z, a) = x+ z − w − a+ sβIf0 + (1− s)V f

2 (a) . (10)

We assume that the lending relationship ends once the employment relationship ends due

to the separation shock. With probability s, the firm becomes type 0 again. Otherwise,

the firm remains type 2.

3.3 Banks

Banks issue demand deposits to workers in each subperiod 1. The demand deposit is a

contract between a bank and a worker, defined as follows.

Definition 1 The demand deposit contract states that: (i) for a real balance z deposited

in subperiod 1 of t, the worker is free to withdraw any amount less than or equal to z by

the end of t; (ii) The bank is to pay a net nominal interest rate, id > 0, on the worker’s

remaining real bank balance at the beginning of subperiod 1 of t+ 1.

Consider any bank n ∈ {1, 2, ..., N} in CM. The bank sends a measure Bn of loan

offi cers to CM. Sending an offi cer incurs a real cost κ > 0. Let A = {a : a ∈ [0,∞)} be

the set of all possible contracts and a ∈ A represent a contract with a particular periodic

repayment level, a. Moreover, let A represent the contract negotiated between loan

offi cers and firms in the current CM. Now define gni (a) as the density of the distribution

of all type-i contracts owned by bank n at the beginning of subperiod 1 of t, with i = 1, 2.

In other words, gni (a) represents the measure of type-i contracts owned by bank n that

require a periodic repayment of a. Moreover, type-1 contracts refer to those that are in

the funding stage (i.e., firms searching in LM) and the type-2 are those in the repayment

stage (i.e., firms making repayments to the bank).

Given that demand deposits are costly, the bank may have the incentive to retain

some of its earnings to help alleviate the cost of funds in the upcoming periods. We call12

the retained earnings “bank savings”. Let Kn denote the real balance of bank savings at

the end of subperiod 3 of t − 1, and Zn be the amount of demand deposits accepted by

bank n in subperiod 1 of period t. Moreover, Dn ∈[0, Zn

]is the expected amount of

withdrawal in DM coming up in subperiod 3 of t. Finally, let B−n be the total measure

of loan offi cers sent by all banks except n. That is, B−n =∑N

n6=k=1Bk. The probability

for a loan offi cer to meet a firm in CM is φb(Bn, B−n, E

)= L (B,E) /B = L (1, E/B),

where B = Bn + B−n. The periodic profit of bank n is given by

Πbn = ρKn +

∫a∈A

agn2 (a) da+ (1− κd) Zn − κBn − (1 + id) ρZnD

−γ[∫

a∈Agn1 (a) da+Bnφ

b(Bn, B−n, E

)]−Dn − Kn − κf , (11)

where κd ∈ (0, 1) is a variable cost associated with handling demand deposits and κf > 0

is a fixed cost. Note that κd is a sweat cost of the bank and does not affect the two liquidity

constraints specified next. Moreover,∫a∈A gni (a) da is the stock of type-i contracts held

by bank n, and Bnφb(Bn, B−n, E

)is the measure of new type-1 contracts created in the

current subperiod 1. The bank’s profit Πbn is to be rebated to workers as dividends at

the beginning of subperiod 1 of t+ 1.

Between the current subperiod 1 and the next, the bank has three sources of funds:

its own savings, ρKn, repayments from all type-2 contracts,∫a∈A agn2 (a) da, and newly

accepted demand deposits, Zn. Note that the bank’s own saving Kn is subject to the

inflation tax because this is the balance at the end of subperiod 3 of t− 1 and can only

take the form of fiat money.

Banking costs include the cost of handling demand deposits, κdZn, the cost of send-

ing loan offi cers to CM, κBn, the cost of funding new and existing type-1 contracts,

γ[∫a∈A gn1 (a) da+Bnφ

b(Bn, B−n, E

)], the cost of paying principal and interests on

previously-issued demand deposits, (1 + id) ρZnD, the cost of meeting demand of with-

drawals, Dn, and finally, the fixed cost, κf . In addition, the bank will choose a new

amount of own savings Kn to put aside by the end of subperiod 3 of t.

We now proceed to describe the bank’s decision problem. First, when making de-

cisions, bank n takes A, the current market loan term, as given because it is to be

13

determined through negotiation between loan offi cers and fund-seeking firms.6 When

sending loan offi cers to CM, bank n informs them that the marginal value of creating a

new loan to the bank is Vn1 (a) for any given a. The offi cers will carry out negotiations

according to this instruction. Moreover, the bank also takes the competitive deposit rate,

id, withdrawal demand in real terms, Dn, measure of new firms in CM, E, and all other

banks’decisions as given. Second, there are four state variables for bank n’s decision

problem. They are the stocks of type-i contracts, gni with i = {1, 2}, the total balance of

demand deposits at the beginning of subperiod 1, ZnD, and the savings from subperiod

3 of t− 1, Kn. Finally, the bank makes decisions on the measure of loan offi cers to send

to CM, Bn, the amount of new demand deposits to take in, Zn, and the real balance to

save by the end of subperiod 3, Kn.

We assume that all loan repayments submitted by firms in subperiod 1 of t will arrive

at the corresponding banks in subperiod 3 of t. This assumption is used to augment a

potential liquidity problem for banks, resulting from mismatched timing of payments. In

reality banks need to constantly manage the flow of funds by keeping track of the timing

of payments. We interpret the late arrival of repayments as a result of some exogenous

reasons such as technical details, time difference, etc., but not because of any wrong-doing

on the firms’part. In other words, we abstract from any incentive problem on the firms’

side. Banks have full knowledge of the arrival time of repayments, and make decisions

accordingly.

Given(A, id, Dn, E, B−n

), the bank chooses

(Bn, Zn, Kn

)to maximize the present

value of the sum of current and future profits:

Vn (gn1, gn2, ZnD, Kn) = max(Bn,Zn,Kn)

{Πbn + βVn

(gn1, gn2, ZnD, Kn

)}

s.t. ρKn + Zn ≥ γ

[∫a∈A

gn1 (a) da+Bnφb(Bn, B−n, E

)]+ κBn

+ (1 + id) ρZnD + κf (12)

ρKn + Zn +

∫a∈A

agn2 (a) da ≥ Dn + γ

[∫a∈A

gn1 (a) da+Bnφb(Bn, B−n, E

)]+ κBn

+ (1 + id) ρZnD + κf + Kn (13)

6There is no dispersion of contract terms in a symmetric banking equilibrium where all banks and all fund-seeking firms are identical.

14

Kn ≥ 0 (14)

ZnD = Zn −Dn (15)

gn1 (a) =

(1− λf

) [gn1 (a) +Bnφ

b(Bn, B−n, E

)],

if a = A(1− λf

)gn1 (a) , if a 6= A

(16)

gn2 (a) =

(1− s) gn2 (a) + λf

[gn1 (a) +Bnφ

b(Bn, B−n, E

)],

if a = A

(1− s) gn2 (a) + λfgn1 (a) , if a 6= A.

(17)

Condition (12) is the liquidity constraint faced by bank n by the end of subperiod 1

of t. We refer to it as the lending liquidity constraint (lending constraint hereafter). The

right-hand side of (12) consists of the costs and payments that must be covered by the

bank within subperiod 1. They are lending costs associated with γ and κ, the payment

of principal and interests on previous demand deposits, (1 + id) ρZnD, and the fixed cost.

The left-hand side of (12) consists of the available funds for covering the above costs

and payments. There are two sources of such funds: bank savings at the beginning of

subperiod 1, ρKn and newly issued demand deposits, Zn. Because of the delay in arrival

of loan repayments, demand deposits can be particularly helpful for relieving the liquidity

condition of banks, since bank savings can be costly due to inflation.

Condition (13) is another liquidity constraint faced by bank n by the end of subperiod

3 of t, when workers withdraw their demand deposits to purchase DM goods. We refer to

this constraint as the withdrawal liquidity constraint (withdrawal constraint hereafter).

Comparing with condition (12), the additional funds that the bank receives is the loan

repayment∫a∈A agn2 (a) da that arrives in subperiod 3. On the expense side, there are

two additional items faced by bank n: withdrawal demand Dn and bank’s savings Kn.

Condition (14) regulates that the bank’s savings must be nonnegative. Condition

(15) states that the total balance of demand deposits at the end of subperiod 3 of t is

equal to the amount of deposits issued in subperiod 1 of t, Zn, minus the total amount

of withdrawal in subperiod 3 of t, Dn. Conditions (16)-(17) are the laws of motion

for the distributions of contracts owned by a bank. The distributions are over a, the

repayment requirement of the contracts. In general, the stocks of contracts owned by

15

a bank may have different loan requirements, depending on the economic conditions at

the time of creating contracts. Recall that the bargained contracts have the term a = A

in the current CM. Therefore, at the beginning of period t + 1, the type-1 contracts

with particular requirement a = A consist of the previous and the newly-added type-

1 contracts of a = A that are not matched with a worker in period t. Similarly, the

evolution of type-2 contracts with a = A must include those existing type-2 contracts

that survive the separation shock, and those newly added to the pool of type-2 contracts.

Finally, measures of the contracts with any requirement a 6= A evolve in their due courses

given LM shocks, without adding to existing stocks.

Let ε and ξ be the Lagrangian multipliers for the bank’s liquidity constraints (12) and

(13), respectively. Using the envelop conditions and first-order conditions, we derive the

optimal conditions for interior solutions of(Bn, Zn, Kn

)as follows

[φb(Bn, B−n, E

)+Bn

∂φb

∂Bn

] βλf (1+ξ)A1−β(1−s) − γ (1 + ε+ ξ)

1− β(1− λf )= κ (1 + ε+ ξ) (18)

βρ (1 + id) = 1− κd1 + ε+ ξ

(19)

βρ (1 + ε+ ξ)− (1 + ξ) ≤ 0, Kn ≥ 0, (20)

where the last condition holds with complementary slackness.

3.4 Bargaining Solutions

Let ηb be the loan offi cer’s bargaining power in CM. Before sending them to CM, bank

n instructs its loan offi cers the marginal value of a type-1 contract with a given term a,

i.e., Vn1 (a) = ∂Vn/∂gn1 (a). In CM, loan offi cers and firms engage in Nash bargaining.

If they agree on a deal, the loan offi cer brings the value Vn1 (a) back to the bank, and the

firm receives U f1 (a). If no deal, then the bank receives nothing and the firm retains the

value βIf0 .7 Therefore, the bargaining solution of a splits the surplus between the two

parties according toVn1 (a)

U f1 (a)− βIf0

=ηb

1− ηb. (21)

7Clearly, the number of banks affects firms’outside option in the bargaining process. Specifically, a highernumber of banks leads to a higher matching probability for a firm in the CM. This improves a firm’s value ofoutside options during bargaining. Rejecting an offi cer’s offer and waiting to participate the CM in the nextperiod becomes less costly for a firm thanks to a higher matching rate.

16

The solution to this bargaining problem yields the following optimal condition:

βλfa(1 + ξ)− γ(1 + ε+ ξ)[1− β(1− s)]βλf{y + αf [ρd− c(q)]− w − a}

=ηb

1− ηb. (22)

In LM, firms and workers engage in Nash bargaining, taking the loan contract that

the firm has with a bank as given. Effectively given a, the wage rate w (a) solves

V f2 (a)− βU f

1 (a)

V h1 (z)− V h

0 (z)=

ηf1− ηf

, (23)

where ηf is the firm’s bargaining power. If both parties agree, the firm receives V f2 (a)

and the worker receives V h1 (z). Otherwise, the firm and worker continue with values

βU f1 (a) and V h

0 (z), respectively. Due to the linearity of W he (z), the optimal condition is

given by

(1− β)[1− β(1− s− λh)][1− β(1− s)][1− (1− λf )β]

y + αf [ρd− c(q)]− w − aw − (b+ `)

=ηf

1− ηf.

In DM, firms and workers bargain over the terms of trade (q, d) according to the

Kalai protocol. Let µ be the bargaining power of a worker. The surplus for a worker is

u(q)−β(1 + id)ρd. Similarly, the firm’s surplus is given by β[ρd− c(q)]. Kalai bargaining

solves

max(q,d)

u(q)− β(1 + id)ρd st.u(q)− β(1 + id)ρd

β[ρd− c(q)] =µ

1− µ and d ≤ z. (24)

The resulting terms of trade depend on z if and only if the constraint d ≤ z binds.

Lemma 1 The optimal bargaining solution in DM is given by

d (z) =

z∗, if z ≥ z∗

z, if z < z∗

q (z) =

q∗, if z ≥ z∗

g−1 (βρz) , if z < z∗

where g (·) is defined in (25) and q∗ is implicitly defined by u′(q∗)/c′(q∗) = β(1 + id) and

βρz∗ = g(q∗),

βρd = g(q) ≡ (1− µ)u(q) + µβc(q)

(1− µ)(1 + id) + µ. (25)

The proof of the above lemma, as well as all other proofs, are provided in Appendix A.17

The results in this lemma are standard for variations of the Lagos-Wright environment.

As will be shown later, in any equilibrium with money and banking, a worker will spend

all of her real balances in a DM trade. That is, d = z and q = g−1 (βρz) because there is

no precautionary motive of saving through holding money, due to quasi-linear preferences.

4 Equilibrium

4.1 Definitions

Let the distributions of type-i contracts aggregated across all N banks be denoted by

gi (a) =∑N

n=1 gni (a) for i = 1, 2. We have the following equilibrium definition:

Definition 2 A symmetric banking equilibrium(Zn, Bn > 0

)consists of values{

(Vn)Nn=1,(Uhe , V

he ,W

he

)e=0,1

,(If0 , U

f1 , V

f2 ,W

f2

)},

decision rules{

(X, z) ,(Bn, Zn, Kn, Dn

)Nn=1

}, measures {B,E, u, v}, prices and terms of

trade {w (a) , q (a) , d (a) , id}, and distributions {gi (a)}i=1,2 such that given policy (b, T, π),

the following are satisfied:

1. All decisions are optimal;

2. All bargaining results are optimal in (21), (23) and (24);

3. Free entry of firms and banks is such that

If0 = 0. (26)

Πbn = 0, ∀ n. (27)

4. Matching in the credit and labor markets satisfies:

L (B,E) = N (u, v) . (28)

N (u, v) = (1− u)s. (29)

5. All competitive markets clear. In particular, clearing of money and demand-deposit

18

markets requires:

M

P=

N∑n=1

{∫a∈A

agn2 (a) da+ ρKn + Zn − γ[∫

a∈Agn1 (a) da+Bnφ

b(Bn, B−n, E

)]−κBn − (1 + id) ρZnD − κf}. (30)

z =N∑n=1

Zn. (31)

6. Consistency: The aggregate distributions satisfy:

gi (a) =

N∑n=1

gni (a) , i = 1, 2 and ∀ a,

where gni (a) obeys laws of motion (16)-(17);

7. Symmetry:(Bn, Zn, Kn

)are respectively the same for all n in any time period.

Thus, B = NBn, and αhz = NDn.

8. Government balances budget:

bu = T + πM

P. (32)

Conditions 1-3 and 6-7 are self-explanatory. Condition 4 ensures that the inflows and

outflows to the stock of type-1 firms (those with a bank contract but without a worker),

and those to the stock of jobs are balanced in equilibrium. The money market clearing

condition (30) is particularly worth mentioning. The right-hand side of (30) is the money

demand at the end of subperiod 1 of t. In particular, loan repayments by productive

firms must be in the form of fiat money for two reasons: first, it is the only way for

the repayments to arrive at the banks in subperiod 3 since real goods do not last across

subperiods. Second, banks need money to accommodate withdrawal demand in DM in

subperiod 3. For the same reasons, any balance of a bank at the end of subperiod 1,

ρKn+ Zn−γ[∫a∈A gn1 (a) da+Bnφ

b(Bn, B−n, E

)]−κBn− (1 + id) ρZnD−κf , must also

be carried forward as money.8 Given that both money and deposit markets clear, as in

(30) and (31), the GM must also clear by Walras’s law. Condition 8 stipulates that the

UI program is supported by taxes imposed on workers and seigniorage.8Thus the definition of money demand in our economy is different from the standard definition of money

demand in Lagos-Wright type of monetary search models, which is the total amount of fiat money held byworkers and is given by g(q)/{(1− u){αf [g(q)− c(q)] + y}}.

19

Definition 3 A stationary and symmetric banking equilibrium is one in which all real

values and distributions are time-invariant. In particular, the contract term remains the

same over time at a = A, and

g1 (A) =

(1

λf− 1

)L (B,E) (33)

g2 (A) =L (B,E)

s(34)

gi (a) = 0 ∀ a 6= A ∈ A.

Equations (33) and (34) are derived from (16) and (17) given gni (A) = gni (A) for all

n and i in the steady state. All wages and terms of trade in DM are constant over time

because of the constant contract term. That is, there is no dispersion of contract terms,

wages or goods prices in the steady state. Let w = w (A) represent the steady-state wage

from this point on.

4.2 Characterizing the Stationary Banking Equilibrium

We focus on the case of 1 +π > β, although we do consider the limit as 1 +π → β, which

is the Friedman rule. We study the stationary equilibrium and therefore drop all time

subscripts hereafter. First, we characterize workers’optimal choice of z as follows.

Proposition 1 In the steady state, worker’s optimal decisions are such that:

(i) If id satisfies βρ (1 + id) = 1, then d < z .

(ii) If id satisfies βρ (1 + id) < 1, then d = z = g(q)/βρ and q satisfies

αhu′(q)

g′(q)+ (1− αh)(1 + id) =

1

βρ. (35)

The intuition of Proposition 1 is straightforward. When βρ (1 + id) = 1, bank deposits

generate real returns that compensate exactly for discounting. Thus, workers are indif-

ferent between consuming the GM goods and making deposits. When the deposit rate

less than compensates for discounting, workers will only deposit the necessary amount

(for purchasing the DM goods) z into the bank.

Next, firm’s entry condition (7) implies that in the steady state

βφfλfy + αf [ρd− c(q)]− w − A

1− β (1− s) = k[1− β(1− λf )]. (36)

20

Lastly, if we allow free entry of banks, then it implies thatΠbn = 0, which endogenously

determines the equilibrium number of banks N . Thus, N must satisfy

(ρ−1)Kn+[(1− ρ (1 + id)) (1−αh)−κd]Zn+Agn2 (A)−γ[gn1 (A)+Bnφb (B,E)]−κBn−κf = 0.

(37)

We are now ready to characterize the stationary equilibrium. First, we assume that

k ≤β

1−β(1−s) [y + g(q∗)β− c(q∗)− (b+ l)]− γ

(1− β)(1− ηf )1−β(1−s)ηf

+ 11−ηb

, (38)

which is a necessary condition for firms’entry to CM (see the Appendix for the derivation

of this condition).

Recall that ε and ξ are the Lagrangian multipliers for the bank’s liquidity constraints

(12) and (13), respectively. For the rest of the paper, we conduct our analyses in two

steps. We first hold N as exogenous, to better understand the influence of the number

of banks on the nature of the equilibrium. Afterwards, we will endogenize N by allowing

for bank entry. The following proposition establishes possible types of equilibria when

the number of banks is taken exogenously.

Proposition 2 If exists, a stationary banking equilibrium must have d = z and the

equilibrium deposit rate id satisfy (19). Moreover, six possible types of stationary equilibria

can occur depending on the number of banks N and other parameter values.

(I) ε = ξ = Kn = 0. Neither of the two liquidity constraints binds for banks. Banks do

not retain earnings. This type of steady state can be determined by solving (B,E, u, v, A)

from (18), (22), (28), (29), and (36);

(II) ε > 0 and ξ = Kn = 0. Lending constraint (12) binds but withdrawal constraint

(13) does not bind. Banks choose not to retain earnings. This type of steady state can

be determined by solving (B,E, u, v, A, ε) from (12) with equality, (18), (22), (28), (29),

and (36);

(III) ε > 0, ξ = 0 and Kn > 0. Lending constraint (12) binds but withdrawal

constraint (13) does not bind. Banks carry a strictly positive balance of Kn. This type

of steady state can be determined by solving (B,E, u, v, A, ε,Kn) from (12) with equality,

(18), (22), (20) (28), (29), and (36);21

(IV) ε > 0, ξ > 0 and Kn > 0. Both constraints (12) and (13) bind for banks.

Banks carry a strictly positive balance of Kn. This type of steady state can be determined

by solving (B,E, u, v, A, ε, ξ,Kn) from (12) with equality, (13) with equality, (18), (20),

(22), (28), (29), and (36);

(V) ε = 0, ξ > 0 and Kn = 0. Lending constraint (12) does not bind but withdrawal

constraint (13) binds. Banks choose Kn = 0. This type of steady state can be determined

by solving (B,E, u, v, A, ξ) from (13) with equality, (18), (22), (28), (29), and (36);

(VI) ε > 0, ξ > 0 and Kn = 0. Both constraints (12) and (13) bind for banks. Banks

choose Kn = 0. This type of steady state can be determined by solving (B,E, u, v, A, ε, ξ)

from (12) with equality, (13) with equality, (18), (22), (28), (29), and (36).

The presence of the constraints naturally generates different types of stationary bank-

ing equilibria. In the next section, our quantitative analysis suggests that the (exogenous)

number of banks has an important influence on which type of equilibrium arises under a

given set of parameters. Figure 3 summarizes how equilibrium switches types as N in-

creases. In type I equilibrium, neither constraint binds. Banks have suffi cient funding to

cover loan issuance in the first subperiod and deposit withdrawal in the third subperiod.

This is more likely to occur when the number of banks is low so that each bank has a

relatively large share of the market and is not liquidity constrained. The proof of Propo-

sition 2 shows that a non-binding lending constraint (12) implies Kn = 0. Intuitively, a

non-binding lending constraint implies that banks have enough liquidity and so there is

no need to rely on costly internal savings.

As the number of banks rises, each bank receives a smaller share of deposits, which

tightens banks’lending constraint and withdrawal constraint. Either a type II equilibrium

in which banks’lending constraint (12) binds or a type V equilibrium in which banks’

withdrawal constraint (13) binds will arise. In either case, banks do not yet resort to

their own savings as a source of funding (Kn = 0). The intuition is the following: With

more banks, it is true that increased demand for deposits from banks will stimulate the

aggregate deposits supplied by all workers. But banks will be more likely to face binding

constraints with a higher number of banks in competition, as long as the drop in the

deposit market share of each bank dominates the induced increase in deposits supply.

22

(13) binds VI 𝑲𝑲𝒏𝒏 > 𝟎𝟎 (12) binds II 𝑲𝑲𝒏𝒏 > 𝟎𝟎 III (13) binds (13) binds V VI 𝑲𝑲𝒏𝒏 > 𝟎𝟎 (12) binds Number of Banks

Type I equilibrium,

(12)&(13) not bind,

𝑲𝑲𝒏𝒏 = 𝟎𝟎

Type IV equilibrium,

(12)&(13) bind,

𝑲𝑲𝒏𝒏 > 𝟎𝟎

Figure 3: Types of Stationary Equilibria

When the lending constraint (12) binds first, the economy is in a type II equilibrium.

In this case, as the number of banks increases, the tension of liquidity needs associated

with competing with more banks eventually pushes banks to either have a binding with-

drawal constraint (13) or to save on their own. With two binding constraints and no bank

savings, the economy will be in a type VI equilibrium. In this case, banks are constrained

but do not find it optimal to save on their own. However, it is also possible that banks

begin to save (Kn > 0) before the withdrawal constraint (13) binds. In this case, the

economy switches from a type II to a type III equilibrium.

When the withdrawal constraint (13) binds first, the economy is in a type V equilib-

rium. In this case, having more banks in the economy can push banks to have a binding

lending constraint. In other words, the economy switches from a type V to a type VI

equilibrium. Note that banks do not save as long as the lending constraint (12) does not

bind.

Eventually, when the number of banks is suffi ciently high, competition is so severe

that both constraints bind and banks must save to fund future lending (Kn > 0). As a

result, the economy switches into a type IV equilibrium from either a type VI or a type

III equilibrium. We will conduct quantitative analysis in the next section to explore the

nature of these different types of banking equilibria. Our numerical results suggest that

23

types V and VI equilibria do not exist under reasonable parameter values.

The characterization of banking equilibria with an endogenous N is straightforward:

each banking equilibrium is solved as described in Proposition 2 with the addition of

(37) to solve for N . With an endogenous N , there are three possible types of banking

equilibria corresponding to types I — III above. Types IV—VI equilibria cannot occur

because the zero-profit condition implies that (13) never binds, i.e., ξ = 0.

5 Quantitative Analysis

We now use quantitative analysis to further explore the properties of banking equilibria

and examine the effects of inflation. To conduct this analysis, we calibrate the model

using the U.S. data over the time period 1998-2008. As mentioned previously, first we

explore the equilibrium taking N as exogenous. This will help us better understand the

nature of each type of equilibria as outlined in Proposition 2. Then we allow free entry

of banks to determine the number of banks endogenously. With this, we study how the

changes in inflation affects the nature of equilibrium and welfare.

5.1 Parameterization

The model period is set to be one quarter. Following the macro-labor literature, the

analysis in this section uses the following specifications: In LM, the matching function is

assumed to be N (u, v) = uv/(uφ + vφ)1φ . In DM, the utility function has the functional

form of u (q) = Aq1−α/ (1− α), and the cost function is c (q) = qθ, where θ ≥ 1. The

matching function in DM is given byM (1, 1− u) = (1− u) / (2− u). Thus, the matching

probabilities of buyers (workers) and sellers (firms) are given by αh = (1− u) / (2− u) ,

and αf = 1/ (2− u). Moreover, the matching function in CM is assumed to be L(B,E) =

BE/(Bψ + Eψ)1ψ .

Given these specifications, the parameters to be determined include the preference

parameters (β,A, α), the matching function parameters (φ, ψ), the technology parameters(y, s, ηf , ηb, µ, θ

), cost parameters (γ, κd, κf , κ, k), the policy parameters (b, `, ρ), and the

number of banks operating in the economy N .

Preference parameters. The discount factor β is set to 0.992 to match the annual real

24

interest rate in the U.S., which is 3 percent.

Matching function parameters. We set the LM matching function elasticity parameter

φ to 1.25, which is consistent with Petrosky-Nadeau et al. (2018). Following Petrosky-

Nadeau and Wasmer (2015), the CM matching function elasticity ψ is set to 1.35.

Technology parameters. The output y is normalized to 1. The job separation rate is

s = 0.05, which is used in Andolfatto (1996) and is within the range of the estimates

reported by Davis et al. (2006). Next, the firm’s bargaining power in LM ηf is set to

0.9, targeting the mid-range of 0.85 to 0.95, according to Petrosky-Nadeau and Wasmer

(2015) and Hagedorn and Manovskii (2008), respectively. Bethune et al. (2020) and

Petrosky-Nadeau and Wasmer (2015) set the loan offi cer’s bargaining power in CM to

0.52 and 0.62, respectively. We compromise with the midpoint and set ηb = 0.56. Finally,

the curvature parameter θ in the cost function c(q) is set to 1.35.

Cost parameters. According to Silva and Toledo (2009), the average labor cost of

hiring one worker is 3 to 4.5 percent of quarterly wages of a new hire. Hence, we set

γ = 0.03 × w. We will conduct a robustness check by varying its value to 0.045 × w.

Bank’s variable cost of handling demand deposits κd is obtained from the Call Reports

data. Using quarterly information on banks’income statement in the U.S. in 2003, we

find that κd = 0.001658.9

Policy parameters. The inflation rate π = 1/ρ− 1 is the quarterly CPI inflation rate

reported by the Federal Reserve Bank. The calibrations of the two remaining policy

parameters, i.e., the value of leisure ` and UI benefits b, are more controversial in the

search literature. Shimer (2005) sets ` = 0 and finds that the unemployment rate is not

responsive enough to changes in productivity. Alternatively, Hagedorn and Manovskii

(2008) argue that if this parameter is calibrated to match hiring costs, b + ` should be

set to 0.95 of wages, resulting in a response of unemployment to productivity that is

more consistent with the data. Here, we follow Hagedorn and Manovskii (2008) and set

(b+ `) = 0.95× w.

The number of banks N plays a crucial role in our model. To determine the value of

N , we follow the standard approach in the literature and use the Herfindahl-Hirschman9We choose year 2003 because some key statistics on the banking sector used in calibration is only available

from a 2003 financial survey. We refer the reader to the Appendix for a detailed description of the data andcalibration method.

25

Index (HHI) for the U.S. banking sector as an empirical counterpart to N . The HHI

is a commonly accepted measure of market concentration, which is calculated by taking

the sum of squared market share of each firm competing in a specific market. The

inverse of the HHI yields the number of “effective competitors,”or the number of equal-

sized firms that would produce an equivalent HHI score.10 Black and Kowalik (2017)

find that the HHI of U.S. banks for the deposit market is 0.12 in 2003, which suggests

that N = 1/0.12 ' 8. This is considered as a moderately concentrated market by the

U.S. Department of Justice (DOJ).11 To check the validity of N , we compute banks’

profit margin and compare it with its empirical counterpart from 2003 Call Reports data.

With N = 8, our model generates a profit margin of 3.19%, which is fairly close to the

observed profit margin of 3.18% for commercial banks with demand deposit greater than

USD 100, 000.12 Thus, we consider N = 8 as a valid account of concentration of the U.S.

banking sector.

Table 1 summarizes the parameter values used in calibration. The remaining parame-

ters (A,α, µ, k, κf , κ) are calibrated using the following strategy: (i) the two preference

parameters (A,α) are chosen to match money demand M/ (pY ) between 1998 and 2008.

The money demand curve is stable during this period, and (ii) worker’s share of trade

surplus in DM (µ) is set for a target retail markup of 20% —the upper bound of the

reported range, which is between 10% and 20%. We will conduct a robustness check by

varying its value to 0.15. In the model,

M

pY=

[1− ρ (1 + id) (1− αh)]z + ρNKn − γ[g1 (A) +Bφb (B,E)]− κB −Nκf + Ag2(A)

(1− u)y + αh[ρd− c(q)] , and

10For the banking sector, the HHI can be calculated by the sum of squared market share by each bank ina given market. For example, suppose there are ten banks in a market. The five largest banks each has 15percent market share, and the remaining five banks each has 5 percent market share. The resulting HHI wouldbe 5(0.15)2 + 5(0.05)2 = 0.125. Under this definition, an extremely competitive market would have an HHIapproaching zero, while a pure monopoly would have an HHI of one. One complication arising in the caseof banking sector is related to the definition of market share. In the literature, the market share has beenapproximated by a bank’s asset share, deposit share, or loan share. It would be ideal to use the HHI computedbased on loan share in our calibration. However, we use the HHI based on deposit share as proxy due to dataavailability.11The DOJ divides the spectrum of market concentration into three roughly delineated categories that can be

broadly characterized as unconcentrated (HHI below 0.1), moderately concentrated (HHI 0.1− 0.18), and highlyconcentrated (HHI above 0.18).12To facilitate comparison, we define bank’s profit margin in our model as the ratio of bank’s cash profit to its

total cost, which is given by (Πbn + κz/N)/{(1 + id) ρ(1− αh)z/N + γ[gn1 (A) +Bnφb (B,E)] + κBn + κf}.

26

Markup =ρd

c′(q)q− 1.

Table 1

Parameter Values in the Calibration

Notation Parameter Value

β Discount factor 0.992

φ Elasticity of the LM matching function 1.25

ψ Elasticity of the CM matching function 1.35

y Productivity 1

s Separation rate 0.05

ηf Firm’s bargaining power in LM 0.9

ηb Loan offi cer’s bargaining power in CM 0.56

θ Curvature parameter in c (q) 1.35

γ Vacancies posting costs 0.03w

κd Bank’s variable cost of handling deposits 0.001658

b UI benefits 0.5w

` Value of leisure 0.45w

N Number of banks 8

The three cost parameters (k, κf , κ) are calibrated to match the following targets:

(i) the average unemployment rate is 0.06 in 2003; (ii) the 2003 National Survey of

Small Business Finances reports that 12% of surveyed firms have no relationships with

commercial banks. Let F be the total measure of firms with a bank loan. In our model, the

measure of firms seeking a loan at a particular point in time is E. Thus the total measure

of firms is E + F . The aforementioned statistic suggests F/(F + E) = 1 − 0.12 = 0.88.

In a steady state, the inflow and outflow to F must be balanced, which is φfE = sF .

This implies F/(F + E) = φf/(s + φf ) = 0.88. Given s = 0.05, we solve for φf =

0.88 × 0.05/0.12 = 0.37; and (iii) the financial sector’s value added as a share of U.S.

GDP in 2003 is 2.5%. In our model, this share is

Σ =N × Πbn

(1− u)y + αh[ρd− c(q)] .

Table 2 reports the calibrated values of parameters. The first column corresponds

to the baseline calibration described above. For robustness checks, we also present two

alternative calibrations in the other columns. In the first alternative, labeled Markup, µ is

determined to match the mark-up rate of 0.15 rather than 0.2. In the second alternative,

27

labelled Cost, we set firms’vacancies posting costs to 0.045×w. We use these alternative

calibrations to illustrate how the results depend on the respective parameters.

Table 2

Calibrated Parameter Values

Notations Description Baseline Markup Cost

A Level parameter in u (q) 1.033 1.075 1.033

α Curvature parameter in u (q) 0.319 0.259 0.319

µ Worker’s share of trade surplus in the DM 0.382 0.344 0.382

k Firm’s cost of entering CM 2.440 2.429 2.458

κf Bank’s fixed cost (∗10−3) 6.511 6.422 6.509

κ Cost of sending loan offi cers to CM 6.562 6.458 6.537

Figure 4 compares the model-predicted money demand curve with the data between

1998 and 2008 in the baseline case. The relationship generated by the model are similar

to the basic pattern observed in the data.

annual inflation (%)-2 0 2 4 6 8 10

mone

y dem

and

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

Figure 4: Money Demand

5.2 Results

Using the calibrated parameters, we compute the steady states. We summarize our

numerical results into two categories: One set of results are obtained taking the number

of banks N as given. For the second set of results, we endogenize N by activating the

bank’s free-entry condition.

28

5.2.1 Equilibrium with an Exogenous N

We first analyze the equilibrium in which the number of banks is taken as given. Figure

5 illustrates how a set of equilibrium variables vary with the number of banks in the

baseline model. As is indicated in Proposition 2, in theory, six types of equilibria may

exist. However, using calibrated parameter values, we find that types V and VI equilibria

do not occur. In fact, as the value of N increases, our model features four non-overlapping

regions in which types I to IV equilibria exist.

In Figure 5, the four different types of banking equilibria are represented by four

different colors. The blue color represents type I equilibrium (1 ≤ N ≤ 9.075) in which

neither the lending constraint nor the withdrawal constraint binds and banks do not save.

As N increases, the lending constraint tightens and the banking equilibrium switches

to type II. The red color represents type II equilibrium (9.075 < N ≤ 9.285) in which

only the lending constraint binds and banks do not save. Any further increase in N

induces the banking equilibrium switches to type III. The black color represents type III

equilibrium (9.285 < N ≤ 12.560) in which only the lending constraint binds and banks

start to save. Lastly, the banking equilibrium switches to type IV as the withdrawal

constraint binds. The green color represents type IV equilibrium (12.560 < N ≤ 16) in

which both constraints bind and banks save.

Figure 6 depicts how welfare varies with the number of banks in the economy for the

baseline and two alternative specifications. Welfare in our model is defined as

W =1

1− β {(1− u)y + αh[u(q)− c(q)] + u`− vγ − Ek − κB −Nκf − ρzκd}, (39)

which includes payoff for all workers and excludes all costs. Figure 6 displays a hump-

shape relationship between the number of banks and welfare. Welfare first increases and

then decreases as N rises. This result is robust across all specifications considered. In

the baseline case, the welfare-maximizing number of banks is given by the integer that

offers a welfare level closest to the peak, which is N = 3.13

To understand these results, let us analyze each type of equilibrium corresponding to

the rise in N . In a type-I equilibrium (colored blue), none of the banks faces any binding13For simplicity, we treat the number of banks as a continuous variable when we compute these results. We

add dots in Figure 6 to pin down the points corresponding to the number of banks being integers.

29

number of banks2 4 6 8 10 12 14 16

Bn

0.01

0.03

0.05Number of Loan Officers Sent by Each Bank

number of banks2 4 6 8 10 12 14 16

N*B

n

0.056

0.058

Total Measure of Loan Officers

number of banks2 4 6 8 10 12 14 16

A

0.5

0.55

Loan Repayment

number of banks2 4 6 8 10 12 14 16

phi b

0.8

0.82

Loan Officers Matching Probability in CM

2 4 6 8 10 12 14 16

phi f

0.32

0.34

0.36

Firms Matching Probability in CM

number of banks2 4 6 8 10 12 14 16

lam

bda

f

0.4

0.5

Firms Matching Probability in LM

2 4 6 8 10 12 14 16

u

0.06

0.07

Unemployment

2 4 6 8 10 12 14 16

e0.1

0.12

number of banks Vacancy

number of banks2 4 6 8 10 12 14 16

w

0.160.2

0.240.28

Wage

number of banks2 4 6 8 10 12 14 16

q

0.658

0.6582

0.6584

number of banks Consumption in DM

2 4 6 8 10 12 14 16

z

0.92540.92560.9258

number of banksDeposit Held by Each Worker

2 4 6 8 10 12 14 16

id

1.0064

1.0065

1.0065Real Deposit Rate

number of banks2 4 6 8 10 12 14 16

Kn

×10-3

0

1

2

Saving by Each Bank

number of banks2 4 6 8 10 12 14 16

prof

it

00.05

0.10.15

number of banks Profit of Each Bank

number of banks2 4 6 8 10 12 14 16ag

greg

ate

prof

it

1.051.1

1.15

number of banksProfit Margin of Each Bank

Figure 5: Equilibrium Variables

liquidity constraint. In such an equilibrium, aggregate lending increases as N goes up,

which is reflected by the rising total measure of loan offi cers, B = N×Bn. Meanwhile, the

loan rate declines as indicated by the lowered loan repayment. This outcome is consistent

with a Cournot context that the presence of more oligopolistic firms (banks) intensifies

competition, which stimulates quantities (loans) supplied and reduces prices (interest

rates) at the same time. Such expanded lending brings an immediate macroeconomic

impact that more firms obtain funding to create vacancies, more workers find jobs, more

output is produced, and thus both consumption and savings (deposit per worker) rise.

All of these factors contribute to the rise in welfare for N < 3.

However, welfare peaks at N = 3. This obviously steps outside the realm of Cournot

competition, which would predict a sustained rise in welfare with N . Indeed, the decline

in welfare starting at N = 3 (while the equilibrium remaining type-I) is caused by an

exacerbated negative search externality as more banks compete for market share. The

intuition is the following: As triggered by the rise inB, the matching probability for a loan

offi cer in CM, φb = L (1, E/B), decreases along the blue segment. This effectively raises

the variable cost of sending offi cers to CM because each offi cer being sent is less likely

to successfully issue a new loan. Therefore, from N = 3 onwards, the negative welfare30

2 4 6 8

number of banks10 12 14 16

welfa

re

50

52

54

56

58

60

62benchmark

markup = 0.15

gamma = 0.045w

Figure 6: Welfare

effect of congestion overpowers the positive effect of competition in CM. Although these

large banks have market power and thus do take into account their own influence on

the matching probabilities (i.e., B = Bn + B−n), the nature of an oligopoly is such that

no single bank can fully internalize the negative search externality of it sending offi cers

to crowd CM. This also indicates that such negative effect on welfare can only worsen

with more banks in the economy, as they lose market power and move toward completely

ignoring the externality they impose on other banks. An important message here is that

perfect competition among banks can be welfare-costly, for a sector such as banking,

where firms “shop”around banks for loans.

The segment in red is of type-II within a tiny range of 9.075 < N ≤ 9.285 (and thus no

integer ofN supports such an equilibrium), in which case only the lending constraint binds

and banks still do not save. A key observation here is that the competitive deposit rate,

id, starts to rise. This indicates that as more banks compete in lending, the aggregate

bank demand for funds/deposits has risen significantly to the extent that the lending

liquidity constraint starts to bind for the banks. A higher deposit rate implies a higher

cost of lending for banks. As a result, aggregate lending declines and banks negotiate for

a higher loan repayment, in order to compensate for the higher cost of funds. As lending

becomes more costly, vacancies and wages both decrease. Nevertheless, aggregate deposit

increases sharply as induced by the sharp in rise id, which leads to higher consumption

in DM due to a wealth effect.31

As N ascends further, the economy moves into the black-colored type-III equilibrium,

in which only the lending constraint binds and banks retain earnings. Banks continue

to be constrained in obtaining funds to lend. Yet the deposit rate id is rather flat along

this segment. Instead of solely relying on deposits, banks in this type of equilibrium

find it optimal to retain earnings to help alleviate the demand for costly deposits. On

the lending side, this type of equilibrium behaves in a similar way to that of type-I

equilibrium in that aggregate lending (mildly) expands, and the loan repayment (mildly)

decreases and the profit margin of banks (mildly) declines, as more banks compete in

the market. Accordingly, vacancies and wages both (slightly) increase. Nevertheless, the

overall welfare in this equilibrium decreases further away from the peak (N = 3) due

to costly lending resulted from a worsening probability of making loans per offi cer and

costly deposits.

Finally, the banking equilibrium switches to type IV in the region of (12.560 < N ≤ 16),

which is marked as green. In this segment, both liquidity constraints bind while banks

retain earnings. At this point the number of competing banks is so high that banks sim-

ply do not have suffi cient amount of retained earnings to escape from a situation of fierce

competition for deposits. This situation is similar to the type-II equilibrium but much

more pervasive. As N further increases, the deposit rate, aggregate deposit, consumption

in DM, and bank savings all rise accordingly. However, banks are seriously constrained

on the lending side, which is clearly indicated by the decreasing number of offi cers in

CM, the higher loan repayment, and the rise in the matching probability of offi cers. As a

result, vacancies and wages are harmed. Even though consumption is boosted by higher

returns from deposits, welfare still declines due to the severe liquidity problems that limit

bank lending.

To summarize, our framework demonstrates a novel result that competition in the

banking sector improves welfare only to a certain extent. In general, banks must deal with

issues of two categories, one stemming from the oligopolistic structure of the sector, and

the other from a macroeconomic stand point. First, there is a negative search externality

associated with oligopolistic banks competing to lend. This corresponds to how banks, in

reality, choose to open more branches and hire more bankers, in an effort to have better

access to clients and maintain market share. The problem with such an endeavor is that

32

the market can become congested in that the average amount of loans created by each

branch/offi cer decreases, which effectively raises the cost of lending.

The macroeconomic issue faced by banks is that how much funds a bank can “gener-

ate”for (the producer side of) the economy is limited by how much funds it can gather

from (the consumer side of) the same economy. Having more banks means more intense

competition for deposits, which drives up the deposit rate and exacerbates the liquidity

problems for banks. This also raises the cost of lending. Overall, it is socially optimal

to have a small, yet greater than one, number of banks in order to have some healthy

competition among banks but at the same time, avoid market congestion and bidding

wars for deposits.

It is a well-known fact that the banking industry is a heavily-regulated industry due

to the important roles played by banks in the economy. In the U.S. banking system,

banks must be chartered by either the state or federal government. Our results imply

that bank charters can be used as a policy instrument to improve welfare by regulating

the number of banks.14

We also investigate the effects of inflation on the economy in the baseline model. Hold-

ing the number of banks constant at N = 8, Figures C1 and C2 in the Appendix report

the effects of inflation on equilibrium variables and welfare. A remarkable observation

is that inflation changes the nature of banking equilibrium, as all curves change colors

from blue (type I) to red (type II) and finally black (type III) with the rise of annual

inflation from the lower bound of −2.53% up to 50%.15 All else equal, inflation makes

workers value money less and thus deposit less, which makes banks’liquidity constraints

more likely to bind. This causes the equilibrium to switch types as inflation increases.

Moreover, Figure C2 demonstrates that inflation reduces welfare given a fixed N . The

key to this result is because aggregate lending declines continuously with inflation, which

results in higher unemployment, less vacancies and lower output.14The model in Huang (2019) has a similar policy implication, however, the mechanisms behind our results are

different.15As mentioned earlier, we focus on equilibrium with id ≥ 0. According to condition (19), at the Friedman

rule (i.e., 1/ρ = 1 + π = β = 0.992, which yields a zero nominal interest rate), no banking equilibrium withid ≥ 0 exists. Thus, the lower bound on 1 + π that allows a banking equilium with id ≥ 0 to exist is givenby 1 + π = β/(1 − κd) = 0.992/0.998. In other words, annual inflation must be greater than or equal to(1− 6.4× 10−3)4 − 1 = −2. 53%.

33

5.2.2 Equilibrium with an Endogenous N

While the banking industry is certainly not characterized by free entry in practice, both

banking profitability and regulations are important factors that influence the creation of

new bank charters. In this subsection, we consider how banking profitability affects bank

entry. In this way, we endogenize the number of banks to study how inflation affects

the economy. Note that the free-entry condition by banks indicates that the equilibrium

number of banks N is determined by Πbn (N) = 0.

8 8.5 9 9.5 10 10.5

number of banks11 11.5 12 12.5 13

prof

it

×10-3

0

0.5

1

1.5

2

2.5

3

Figure 7: Inflation, Bank Profit and the Number of Banks

Using the calibrated parameter values for the baseline model, Figure 7 illustrates how

Πbn (N) varies with N for two different levels of inflation, 2% (benchmark, solid line) and

−2.53% (lower bound, dashed line), respectively. The intersection of the profit function

and the zero-profit condition determines N in equilibrium. A few observations are worth

noting here: First, inflation worsens the liquidity constraint. In particular, the solid line

turns red and then black respectively at much lower levels of N than the dashed line

does. Second, inflation improves bank profits for both type-II and type-III equilibria.

This is seen as the red and the black segments together on the solid line stay above

their counterparts on the dashed line. All else equal, inflation has a positive effect on

bank profit as banks reduce lending by sending fewer offi cers to CM. A higher matching

probability in the CM implies that bank profit continues to rise despite the rise in the

real deposit rate. Third, inflation encourages bank entry. A unique banking equilibrium

34

exists at N = 11.731 when inflation is at the lower bound, but at N = 12.340 given a 2%

annual inflation.

Figures C3 in the Appendix depicts how welfare and endogenous variables respond to

inflation ranging from −2.53% to 50%.16 It is striking to see that with endogenous entry,

all equilibria consistently rest on type III (as indicated by the black color of all curves

in this figure) for such a wide range of inflation considered. As explained before, in a

type-III equilibrium the lending constraint binds and banks choose to retain earnings to

ease their liquidity needs. Therefore, our findings suggest that entry of banks supports

an equilibrium that the liquidity constraint does bind, but only to the extent that the

retained earnings are suffi cient to avoid a fierce competition among banks for deposits.

Note that welfare decreases monotonically with inflation in Figures C3. Indeed, inflation

decreases welfare as it leads to more bank entry and thus a more congested CM and

worsening liquidity constraints.17

5.3 Discussions

5.3.1 Market power of banks

To further investigate the role of market power of banks, we study a variant environment

where we force the competitive behavior on banks. In particular, we assume that all

banks take the matching probability φb in CM as given. Given this assumption, all banks

ignore their individual impacts on the aggregate supply of loans. Nevertheless, it is worth

pointing out as a caveat that this is not exactly a case of perfect competition because

there are only a small number of banks and each of them does a have significant impact

on the equilibrium lending supply. Put it another way, our assumption makes the banks16We also compute lending rate in the model, which is given by Aλf/γ− 1. With baseline paremeters in 2003,

we obtain a lending rate of 18.80 in the exogenous N case and 18.92 in the endogenous N case. These large valuesare mainly due to the special feature of our model’s lending mechanism: the amount of loan repayment (A) isdetermined by bargaining between oligopolistic banks and firms. First, oligopolistic banks have more bargainingpower (0.56) than firms. Second, banks lend the amount of γ to firms every period until firms are matched inthe LM. Finally, firms are subject to default risk s. When negotiabting A with firms, banks take these factorsinto account. Thus, the loan repayment A should compensate banks for the large amount of loan and risk.Numerical results suggest that: (i) Lending rate decreases with N , which is a standard result of competition;

(ii) Lending rate increases with inflation. This is because inflation makes workers deposit less. Banks must offerhigher deposit rates to attract deposits, which requires higher lending rates to sustain.17Note that depending on parameter values, multiple equilibria could arise if the zero-profit line and profit

function intersect multiple times. In this case, the type II equilibrium is unstable and inflation can switch theeconomy to a unique banking equilibrium (either type I or type III). This, however, does not occur given ourcalibration to the U.S. economy.

35

not aware of the true market power that they have. The purpose of this exercise is to

illustrate the outcome of intensified competition among oligopolistic banks.

In this variant model, there are still six possible types of banking equilibrium, depend-

ing on the number of banks and other parameters. Using the baseline parameters, Figure

C4 reports equilibria of the variant model (blue dashed line) and compares them with

those of the baseline model (black line). There are two key observations here, one on the

lending side and the other on the deposit side. In terms of lending, the total measure

of loan offi cers (B) for N < 12 is clearly larger with the variant. Unlike the baseline,

here no bank internalizes the negative search externality it imposes on other banks when

choosing the amount of offi cers to send to CM. As has been discussed previously, a more

congested CM makes lending become more costly for banks and thus can reduce welfare.

The second key observation is that the deposit rate for N > 11 is higher in the variant

model. This indicates that the liquidity problem becomes much more prominent for the

variant, as a result of intense competition for deposits. This exacerbates the negative

effect on welfare.

Figure C5 compares welfare across these two scenarios. Welfare decreases monotoni-

cally with N in the variant model. Note that welfare is higher in the variant than in the

baseline for N < 12. This is because the amount of aggregate lending (reflected by B)

is much higher in the former. The positive effect of having higher supply of more afford-

able loans (lower loan repayment requirement) dominates the two negative welfare effects

mentioned above. That being said, such dominance eventually disappears for N ≥ 12.

This is because the banks are truly losing market power as N rises, and so is the bene-

fit of having the oligopolistic banks behaving more competitively. Once N goes beyond

12, both economies can be considered as approaching the case of “perfect competition”

because the number of loan offi cers sent by each bank barely changes with N any more.

5.3.2 Money creation

In our model, banks accept deposits and then issue loans in the form of fiat money to

firms. We do not allow the banks to issue inside money (i.e., their own banknotes). In

practice, when commercial banks issue loans, they expand the amount of bank deposits.

The banking system can expand the money supply of an economy beyond the amount

36

created or targeted by the central bank, creating most of the broad money in a process

called money creation. It would be interesting and important to know howmoney creation

can potentially affect our results.

To address this issue, we consider an alternative environment where commercial banks

create money, in the form of bank deposits, by making new loans. When a bank makes

a loan with amount of γ to a firm, it credits firm’s bank account with a bank deposit

of the size of the loan. As a result, banks do not face any liquidity constraint. Instead,

an exogenous reserve requirement ratio χ set by the government generates the only con-

straint that a bank faces, which is given by γ[∫a∈A gn1 (a) da+Bnφ

b(Bn, B−n, E

)]≤

(1−χ){γ[∫a∈A gn1 (a) da+Bnφ

b(Bn, B−n, E

)]+ Zn

}. Everything else remains the same

as in the baseline model.

The analytical solution to this alternative optimization problem shows that banks do

not save (Kn = 0) in any banking equilibrium. The intuition is simple: given that banks

are not liquidity constrained in the sense that they can create the same size of deposits as

loans, banks optimally choose not to save because saving is subject to the inflation tax.

In this case, two types of banking equilibria exist. When the number of banks is low so

that each bank has a relative large share of deposits, the reserve requirement constraint

is not binding. Banks optimally choose the amount of loan to make. This equilibrium

corresponds to the type I equilibrium in the baseline model. As the number of banks rises,

each bank receives a smaller share of deposits, which tightens banks’reserve constraint,

an equilibrium in which banks’reserve constraint binds arises.

We then use quantitative analysis to further explore the equilibria in this alternative

environment with money creation. Using baseline parameters and setting the reserve

requirement ratio χ to 0.1 (regulatory reserve ratio in the U.S.), we find that banks’

reserve requirement never binds in equilibrium. This is due to the significant multiplier

effect, which is measured by 1/χ. Clearly, this numerical result depends on the value of

χ. In the extreme case where χ = 0.999, the second type of equilibrium arises at N = 3.

Moreover, welfare still responds to the number of banks non-monotonically in a hump

shape, same as in the baseline model displayed in Figure 6.

To summarize, with money creation and reserve requirement, the span of type-I equi-

librium in the baseline model increases significantly. With a reasonable value of reserve

37

ratio, only type-I equilibrium exists. In other words, banks are never constrained when

they are allowed to create money in the form of deposits. It is helpful to clarify that the

results we obtained here are built upon the assumption that inside money from various

banks circulates in a non-distinguishable way. That is, inside money from all banks ap-

pears to be the same. Sun (2007) considers an alternative scenario where inside money

is distinguishable by bank. This complicates the decision to issue inside money because

now banks have a way of competing for market share by affecting the purchasing power

of its own inside money. That argument would still apply to our model. It is reasonable

to expect that if distinguishable bank money is allowed, the equilibrium deposit rate will

be much higher than what we find in the above because banks choose to promise a higher

deposit rate to enhance the purchasing power of their own notes. Obviously, the liquidity

constraints will exacerbate in this situation and thus, the span of type-I equilibrium is

likely to shrink (likely in a significant manner) relative to the case we consider in the

above. Overall, it is important to realize that the welfare level we obtained here through

non-distinguishable inside money is at best an upper bound for an extension to allow for

money creation, although the intuition for the hump-shaped welfare effect in Figure 6

should still apply to a case with distinguishable-money as studied by Sun (2007).

5.3.3 Liquidity mismatch

As mentioned earlier, liquidity mismatch problem is present and severe in the banking

sector. To examine the implication of liquidity mismatch on our model, we remove

mismatch of cashflows and assume loan repayment always arrives on time during the first

subperiod. As a result, the lending constraint (12) does not exist any more and only the

withdrawal constraint (13) exists. Everything else remains same. This is a special case

of the baseline model in which constraint (12) never binds, i.e., ε = 0.

Without liquidity mismatch, we can show that banks do not save (Kn = 0) in a

banking equilibrium. Once the liquidity mismatch problem is removed, optimizing banks

can perfectly plan the other two sources of funding (loan repayment and newly accepted

demand deposits) in advance such that the withdrawal constraint is satisfied. Therefore,

banks do no need to rely on their own savings as an additional source for funding because

saving is subject to the inflation tax.

38

Specifically, when N is exogenous, two types of equilibria exist in this special case: (I)

constraint (13) not binding. This corresponds to the type I equilibrium in the baseline

model. (II) constraint (13) binds. This corresponds to the type V equilibrium in the base-

line model. The first scenario arises when the number of banks is low so that each bank

has a relative large share of the market and is not liquidity constrained. As N increases,

each bank receives a smaller share of deposits, which tighten banks’liquidity constraint.

As a result, scenario II arises. For the endogenous N case, only type I equilibrium exists

because banks’zero-profit condition implies that the withdrawal constraint never binds.

6 Conclusion

We have constructed a tractable macroeconomic model of money and banking to study the

behavior and economic impact of oligopolistic banks. Our model has two key features: (i)

banks as oligopolists in a frictional credit market, and (ii) liquidity constraints for banks

that arise from timing mismatch of cashflows. We find that it is optimal to have a small,

yet greater than one, number of banks. That is, an oligopolistic banking sector is welfare-

maximizing. When the number of banks is low and banks are not liquidity constrained,

competition improves welfare. However, there are two negative implications associated

with more banks crowding the credit market: First, it becomes more diffi cult for each

loan offi cer to make loans as the rate at which they meet with clients declines. This

drop in lending opportunities and the resulted rise in lending costs eventually dominates

and makes welfare decrease with the number of banks. Second, at some point all banks

become liquidity constrained as the presence of more banks renders them competing for

deposits and driving up the deposit rate. This further exacerbates the negative impact

on welfare.

Overall, as the exogenous number of banks rises, we find the equilibrium switches

types from (i) banks not being liquidity constrained, to (ii) banks being constrained

on lending but need not retain earnings to improve liquidity conditions, to (iii) banks

being constrained on lending and resorting to retained earnings, and finally, to (iv) banks

being constrained on both lending and meeting withdraw demands while having to retain

earnings. When calibrated to the U.S. economy, inflation causes the equilibrium to switch

39

types consecutively in the order from the first to the third type described above, as the

number of banks held fixed. Nevertheless, when bank entry is allowed, the equilibrium

consistently rests on the third type. Our model also predicts that inflation reduces welfare

monotonically.

40

References

[1] Ahnert, T. and D. Martinez-Miera (2020). “Bank Runs, Bank Competition and

Opacity.”Manuscript.

[2] Andolfatto, D. (1996). “Business Cycles and Labor-Market Search.”American Eco-

nomic Review 86, 112-132.

[3] Bai, J., A. Krishnamurthy and C.H. Weymuller (2018). "Measuring Liquidity Mis-

match in the Banking Sector". The Journal of Finance 73, 51-93.

[4] Berentsen, A., Menzio, G. and Wright, R. (2011). “Inflation and Unemployment in

the Long Run.”American Economic Review 101, 371-98.

[5] Bethune Z., G. Rocheteau G., T. Wong and C. Zhang (2020). “Lending Relationships

and Optimal Monetary Policy.”Federal Reserve Bank of Richmond Working Paper

20-13.

[6] Black, L. and M. Kowalik (2017). “The Changing Role of Small Banks in Small

Business Lending.”Manuscript.

[7] Brunnermeier, M.K., G. Gorton and A. Krishnamurthy (2012). "Risk Topography."

NBER Macroeconomics Annual 26,149-176.

[8] Brunnermeier, M.K., G. Gorton and A. Krishnamurthy (2013). "Liquidity Mismatch

Measurement." Risk Topography: Systemic Risk and Macro Modeling. University of

Chicago Press, 99-112.

[9] Burdett, K. and K. Judd (1983). “Equilibrium Price Dispersion.” Econometrica

51(4), 955-969.

[10] Cavalcanti, R. and N. Wallace (1999). “Inside and Outside Money as Alternative

Media of Exchange.”Journal of Money, Credit, and Banking 31, 443-457.

[11] Chiu, J, M. Davoodalhosseini, J. Jiang and Y. Zhu (2020). “Bank Market Power and

Central Bank Digital Currency: Theory and Quantitative Assessment.”Manuscript.

41

[12] Corbae, D. and P. D’Erasmo (2013). “A Quantitative Model of Banking Industry

Dynamics.”Manuscript.

[13] Davis, S. J., R. J. Faberman and J. Haltiwanger (2006). “The Flow Approach to

Labor Markets: New Data Sources and Micro-macro Links.”Journal of Economic

Perspectives 20, 3—26.

[14] Diamond D. and P. Dybvig (1983). “Bank runs, Deposit Insurance, and Liquidity.”

Journal of Political Economy 91, 401—419.

[15] Dreschler I., A. Savov and P. Schnabl (2017). "The Deposit Channel of Monetary

Policy." Quarterly Journal of Economics 132, 1819-1976.

[16] English, W.B., S.J. Van den Heuvel and E. Zakrajšek (2018). "Interest Rate Risk

and Bank Equity Valuations." Journal of Monetary Economics 98, 80-97.

[17] Freixas, X. and J.C. Rochet (2008). “Microeconomics of Banking.”Cambridge: MIT

Press.

[18] Gertler, M. and N. Kiyotaki (2010). “Financial Intermediation and Credit Policy in

Business Cycle Analysis.”Handbook of Monetary Economics, Volume 3A, Elsevier,

547-599.

[19] Gertler, M. and N. Kiyotaki (2015). “Banking, Liquidity and Bank Runs in an Infinite

Horizon Economy.”American Economic Review 105, 2011-2043.

[20] Gertler, M., N. Kiyotaki and A. Prestipino (2016). “Wholesale Banking and Bank

Runs in Macroeconomic Modelling of Financial Crises.” Handbook of Macroeco-

nomics Volume 2, Elsevier, 1345-1425.

[21] Gorton G. and A. Winton (2003). “Financial Intermediation.” Handbook of the

Economics of Finance, Volume 1, Elsevier, 431-552.

[22] Haddad, V. and D. A. Sraer (2019). "The Banking View of Bond Risk Premia".

NBER Working Paper No. 26369.

[23] Hagedorn, M. and I. Manovskii (2008). “The Cyclical Behavior of Equilibrium Un-

employment and Vacancies Revisited.”American Economic Review 98, 1692-1706.42

[24] Head, A., T. Kam and L. Ng (2020). “Equilibrium Imperfect Competition and Bank-

ing: Interest Pass Through and Optimal Policy.”Manuscript.

[25] Huang K. (2019). “On the Number and Size of Banks: Effi ciency and Equilibrium.”

Manuscript.

[26] Kalai, E. (1977). “Proportional Solutions to Bargaining Situations: Interpersonal

Utility Comparisons.”Econometrica 45, 1623—1630.

[27] Lagos, R. and R. Wright (2005). “A Unified Framework for Monetary Theory and

Policy Analysis.”Journal of Political Economy 113, 463-484.

[28] Petrosky-Nadeau, N. and E. Wasmer (2015). “Macroeconomic Dynamics in a Model

of Goods, Labor, and Credit Market Frictions.”Journal of Monetary Economics 72,

97-113.

[29] Petrosky-Nadeau, N., L. Zhang, and L. Kuehn (2018). “Endogenous Disasters.”

American Economic Review 108, 2212-45.

[30] Shimer, R. (2005). “The Cyclical Behavior of Equilibrium Unemployment and Va-

cancies.”American Economic Review 95, 25-49.

[31] Silva, J. and M. Toledo (2009). “Labor Turnover Costs and the Cyclical Behavior of

Vacancies and Unemployment.”Macroeconomic Dynamics 13, 76-96.

[32] Sun, H. (2007). “Aggregate Uncertainty, Money and Banking.”Journal of Monetary

Economics 54, 1929-1948.

[33] Sun, H. (2011). “Money, Markets and Dynamic Credit.”Macroeconomic Dynamics

15, 42-61.

[34] Wasmer, E. and P. Weil (2004). “The Macroeconomics of Credit and Labor Market

Imperfections.”American Economic Review 94, 944-963.

[35] Williamson, S. (2004). “Limited participation, Private money, and Credit in a Spatial

Model of Money.”Economic Theory 24, 857-875.

[36] Williamson, S. and R. Wright (2010a). “NewMonetarist Economics: Models.”Hand-

book of Macroeconomics, Volume 3, Amsterdam: Elsevier, 25-96.43

[37] Williamson, S. and R. Wright (2010b). “New Monetarist Economics: Methods.”

Federal Reserve Bank of St. Louis Review 92, 265-302.

44

A Proofs

A.1 Proof of Lemma 1

The Kalai bargaining problem in the DM is outlined in (24). If d < z, we can rearrange thesurplus sharing rule to get (25). Then we can eliminate ρd in the objective function. Thebargaining problem is reduced to max

qu(q)− β (1 + id) c(q). Therefore, the optimal choice of q

solves u′(q∗)/c′(q∗) = β(1 + id). Accordingly, the optimal choice of payment d = z∗, where z∗ isgiven by βρz∗ = g (q∗). If d = z, q can be directly solved from (25). That is, q (z) = g−1 (βρz) .

A.2 Proof of Proposition 1

Consider condition (6), which defines optimal choice of z. Notice that u′(q)∂q/∂z − βρ(1 +id)∂d/∂z is zero for all z ≥ z∗ by Lemma 1. Hence, βρ(1 + id) > 1 implies that the problem ofchoosing z has no solution, because the objective function max{

z

αh[u(q)− β(1 + id)ρd]− z[1−

βρ(1 + id)]} is strictly increasing for all z ≥ z∗. This result means that any equilibrium mustsatisfy βρ(1 + id) ≤ 1. Given βρ(1 + id) ≤ 1, the objective function is non-increasing in z forz ≥ z∗.

The slope of the objective function as z → z∗ from below is proportional to −[1−βρ(1+id)]+αh[u′(q)βρ/g′(q)−βρ(1+id)], where u′(q)βρ/g′(q)−βρ(1+id) is the worker’s marginal gain frombringing an additional unit of deposit into the DM. For any µ ∈ (0, 1), unless βρ(1 + id) = 1,the slope of the objective function as z → z∗ is strictly negative, and therefore any solutionmust satisfy z < z∗. This completes the proof of the claim that d = z as long as we are not inthe extreme case of βρ(1 + id) = 1.

A.3 Necessary Condition for Firm Entry

Note that firms’free-entry condition (36) can be rewritten as:

k = φfβ

1−β(1−s){y + αf [ρd− c(q)]− (b+ l)} − γ(1 + ε)[1λf− ( 1

λf− 1)β

][(1− β)(1− ηf )1−β(1−s−λ

h)ηf

+ 11−ηb

]. (40)

Given that ρd − c(q) = g(q)/β − c(q) = (1 − µ)[u(q) − (1 + id)βc(q)]/[(1 − µ)(1 + id) + µ] isstrictly increasing with q for q < q∗, the term ρd− c(q) is maximized at q = q∗. Moreover, theright-hand side of (40) is strictly increasing in firm matching rates and strictly decreasing in thehousehold matching rate λh and the bank’s multiplier ε. Thus the right-hand side is maximizedgiven φf = αf = λf = 1 and λh = ε = 0. Thus, a necessary condition for firm entry is given by(38).

A.4 Proof of Proposition 2

Now consider the decisions of individual bank n. The first-order condition for Zn > 0 is givenby

1− κd + ε+ ξ + βV3 = 0. (41)

The first-order condition for Bn > 0 is[φb(Bn, B−n, E

)+Bn

∂φb

∂Bn

] [(1− λf

)βV1 + λfβV2 − γ (1 + ε+ ξ)

]= κ (1 + ε+ ξ) . (42)

45

The first-order condition for Kn is

βV4 − ξ − 1 ≤ 0, Kn ≥ 0. (43)

Moreover, the Envelope Theorem yields

V1 = −γ (1 + ε+ ξ) + β[(

1− λf)V1 + λf V2

](44)

V2 = A(1 + ξ) + β (1− s) V2 (45)

V3 = −ρ (1 + id) (1 + ε+ ξ) (46)

V4 = ρ (1 + ε+ ξ) (47)

Equations (41) and (46) imply optimal condition for Zn is 1+ε+ξ−κd−βρ (1 + id) (1 + ε+ ξ) =0. Thus,

βρ (1 + id) = 1− κd1 + ε+ ξ

< 1, (48)

given κd > 0. Therefore, the money constraint must be binding for all workers, i.e., d = z.Equations (42), (44) and (45) imply the optimal condition for Bn is (18). Furthermore, (43)

and (47) imply the optimal condition for Kn is

βρ (1 + ε+ ξ)− (1 + ξ) ≤ 0, Kn ≥ 0. (49)

We now describe possible equilibria. First, consider ε = 0. Then condition (49) yieldsβρ (1 + ξ)− (1 + ξ) < 0, given that βρ < 1. Thus, ε = 0 implies Kn = 0. That is, a non-bindinglending constraint (12) must mean that the bank does not carry a positive balance of Kn. Thereare two possible cases. If the withdrawal constraint does not bind, neither constraint binds andKn = 0. The equilibrium is type I in which (B,E, u, v,A) are solved from (18), (22), (28), (29),and (36). If the withdrawal constraint binds, ξ > 0 and the equilibrium is type V. The valuesof (B,E, u, v,A, ξ) are determined by (13) with equality, (18), (22), (28), (29), and (36).

Second, consider ε > 0 and Kn = 0. The lending constraint binds and banks do not saveyet. The binding lending constraint (12) implies,

[1− ρ (1 + id) (1− αh)]z + ρNKn − γ[g1 (A) +Bφb (B,E)]− κB −Nκf = 0. (50)

There are two cases to consider depending on whether the withdrawal constraint binds. If thewithdrawal constraint does not bind, the equilibrium is type II. The values of (B,E, u, v,A, ξ)are solved from (18), (22), (28), (29), (36) and (50). If the withdrawal constraint binds, thebinding lending constraint (12) implies that (13) can be simplified into

agn2 (a)−Dn = Kn. (51)

The equilibrium is type VI. The values of (B,E, u, v,A, ε, ξ) are solved from (18), (22), (28),(29), (36), , (50) and (51).

Lastly, consider ε > 0 and Kn > 0. The lending constraint binds and banks save. Then(49) yields (20). Combined with (48), we have (1 + ε + ξ − κd)/(1 + id) = 1 + ξ. Thus,ε = (1 + ξ)id + κd > 0. That is, lending constraint (12) must be binding as long as the bankholds a positive balance of Kn. There are two possible cases here, depending on whetherconstraint (13) binds or not. When the constraint (13) does not bind, ξ = 0. The equilibrium is

type III. The values of(B,E, u, v,A, ε, Kn

)are determined by (18), (22), (28), (29), (36), (50)

and (20). When the constraint (13) binds, both constraints bind and Kn > 0. The equilibrium

46

is type IV. The values of(B,E, u, v,A, ε, ξ, Kn

)are determined by (18), (20), (22), (28), (29),

(36) (50) and (51).

A.5 Welfare

Welfare in our model is defined as

W = (1− u)[αhW he (0) + (1− αh)W h

e (ρz)] + u[αhW hu (0) + (1− αh)W h

u (ρz)]− Ek

1− β .

Note that ∂W he (ρz)/∂(ρz) = 1 + id. Hence, W h

e (ρz) = (1 + id)ρz + W he (0) and W h

u (ρz) =(1 + id)ρz+W h

u (0). From (5) we know that W he (0)−W h

u (0) = [w− (b+ `)]/[1− β(1− s− λh)].Therefore,W = W h

u (0)+(1−u)[w−(b+`)]/[1−β(1−s−λh)]+(1−αh)(1+id)ρz−(Ek)/(1−β).We know that

W hu (0) =

b+ `+ βλhw1−β(1−s) + [1 + βλh

1−β(1−s) ]{Π + ρ�b,−1 − T + αhu(q)− z[1− βρ(1 + id)(1− αh)]}

1− β2λhs1−β(1−s) − β(1− λh)

.

Together with the facts that Π = αf [ρd−c(q)]−w−A, �b,−1 = nΠbn = nρKn+(1−αh)z[1−(1 + id) ρ]− zκd − γ[g1 (A) +Bφb (B,E)]− κB − nκf +Ag2 (A)− nKn, T = bu− (1− ρ)M/p,and the money-market clearing condition M/p = Ag2 (A) + nρKn + z[1− (1− αh) (1 + id) ρ]−γ[g1 (A) + Bφb (B,E)] − κB − nκf , the welfare function can be simplified into (1 − β)W =(1− u)y + αh[u(q)− c(q)] + u`− vγ − Ek − κB −Nκf − ρzκd.

B Data

Data source is Bank Regulatory Call Reports unless otherwise stated. The Call Reportsdata are constructed with bank data from the Consolidated Reports of Condition and Incomeavailable from the Federal Reserve Bank of Chicago.1 The data set contains quarterly incomestatement and balance sheet data that all federally insured banks are required to submit tothe Federal Deposit Insurance Corporation (FDIC). We download the data from WRDS usingsample SAS code by Drechsler et al. (2017). We focus on year 2003 because one of the keystatistics on banking sector used in calibration (φf ) is only available from the 2003 NationalSurvey of Small Business Finances.

Bank’s variable cost of handling demand deposits (κd): unfortunately data on costof handling demand deposit is not available. However, we observe service charges on domesticdeposits (RIAD4080). Here we assume the proportional handling costs are identical acrossdifferent types of deposits (demand and time). We then divide service charges on domesticdeposits by total domestic deposits (RCON2200) to compute κd for each bank in each quarterof 2003. Lastly, we take the average across all banks.

Financial sector’s value added as a share of GDP: we calculate this share using theindustry value added table provided by the Bureau of Economic Analysis (BEA). In 2003, thisrepresents approximately 7.4 percent of GDP. To this fraction, we subtract the share in GDPof household financial services and insurance available from National Income and ProductsAccount Table 1.5.5, which is 4.9 percent of GDP in 2003. Therefore, the financial sector’sshare of aggregate value added we match in our numerical exercise is 2.5 percent.

Profit margin of banks: from the Call Reports data, we first subtract interest expense ontime deposits (RIAD517 + RIAD518) from interest expense on total deposits (RIAD4170) toobtain interest expense on demand deposits. We then divide the sum of interest income on com-

47

mercial and industrial loans (RIAD4012) and interest income on agricultural loans (RIAD4024)by bank’s cost, which is given by the sum of interest expense on demand deposits and expenseon premises (RIAD4217). We subtract 1 from this ratio to obtain the net profit margin for eachbank in each quarter of 2003. Then we take the average across all banks with demand depositsgreater than 100, 000 and aggregate to the annual level. This leads to an empirical counterpartof profit margin that is comparable to our model.

Money demand: to compute money demand, we divide quarterly M1 (seasonally adjusted,series MANMM101USQ189S) by GDP (seasonally adjusted). Both time series are downloadedfrom FRED.

C Additional Figures

annual inflation (%)0 10 20 30 40 50

Bn

×10-3

7.1

7.2

7.3

Number of Loan Officers Sent by Each Bank

0 10 40 50

N*B

n

0.057

0.0575

0.058

0.0585

Total Measure of Loan Officers

annual inflation (%)0 10 20 30 40 50

A

0.51

0.52

0.53

Loan Repayment

0 10 20 30 40 50

u

0.06

0.065

Unemployment

annual inflation (%)0 10 20 30 40 50

e

0.1

0.12

20 30 annual inflation (%)

Vacancy

annual inflation (%)0 10 20 30 40 50

w

0.2

0.25

Wage

annual inflation (%)0 10 20 30 40 50

q

0.6

0.65

annual inflation (%) Consumption in DM

0 10 20 30 40 50

z

0.86

0.88

0.9

0.92

Deposit Held by Each Worker

annual inflation (%)0 10 20 30 40 50

real

id

1.0064

1.0065

Real Deposit Rate

annual inflation (%)0 10 20 30 40 50

Kn

×10-3

0

1

2

Saving by Each Bank

annual inflation (%)0 10 20 30 40 50

prof

it

×10-3

4

6

8

annual inflation (%) Profit of Each Bank

annual inflation (%)0 10 20 30 40 50

prof

it m

argi

n

1.04

1.06

1.08

Profit Margin of Each Bank

Figure C1: Inflation and Equilibrium Variables (N = 8)

annual inflation (%)0 5 10 15 20 25 30 35 40 45 50

welfar

e

56.5

57

57.5

58

58.5

59

Figure C2: Inflation and Welfare (N = 8)

48

annual inflation (%)0 10 20 30 40 50

Bn

×10-3

4

5Number of Loan Officers Sent by Each Bank

0 10 40 50

N*B

n

0.0575

0.058

0.0585

Total Measure of Loan Officers

annual inflation (%)0 10 20 30 40 50

A

0.5

0.52

Loan Repayment

0 10 20 30 40 50

u

0.06

0.065

Unemployment

annual inflation (%)0 10 20 30 40 50

e 0.12

0.14

20 30 annual inflation (%)

Vacancy

annual inflation (%)0 10 20 30 40 50

w

0.22

0.24

0.26

0.28

Wage

annual inflation (%)0 10 20 30 40 50

q

0.6

0.65

annual inflation (%) Consumption in DM

annual inflation (%)0 10 20 30 40 50

z

0.86

0.88

0.9

0.92

Deposit Held by Each Worker

annual inflation (%)0 10 20 30 40 50

real

id

1.0064

1.0065

Real Deposit Rate

annual inflation (%)0 10 20 30 40 50

kn

×10-3

2

3

4

Saving by Each Bank

annual inflation (%)0 10 20 30 40 50

N

12

14

16

Number of Banks

annual inflation (%)0 10 20 30 40 50

wel

fare

50

55

Welfare

Figure C3: Inflation and Equilibrium Variables (Endogenous N )

number of banks2 4 6 8 10 12 14 16

Bn

0.01

0.03

0.05

Number of Loan Officers Sent by Each Bank

number of banks2 4 6 8 10 12 14 16

N*B

n

0.056

0.058

Total Measure of Loan Officers

number of banks2 4 6 8 10 12 14 16

A

0.5

0.55

Loan Repayment

number of banks2 4 6 8 10 12 14 16

phi b

0.8

0.82

Loan Officers Matching Probability in CM

2 4 6 8 10 12 14 16

phi f

0.320.340.36

Firms Matching Probability in CM

number of banks2 4 6 8 10 12 14 16

lam

bda

f

0.4

0.5

Firms Matching Probability in LM

2 4 6 8 10 12 14 16

u

0.060.0650.07

0.075

Unemployment

2 4 6 8 10 12 14 16

e

0.10.120.14

number of banks Vacancy

number of banks2 4 6 8 10 12 14 16

w

0.20.25

0.3Wage

number of banks2 4 6 8 10 12 14 16

q

0.658

0.6585

number of banks Consumption in DM

2 4 6 8 10 12 14 16

z

0.92540.92560.9258

number of banksDeposit Held by Each Worker

2 4 6 8 10 12 14 16

real

id

1.0064

1.0065

1.0065

Real Deposit Rate

number of banks2 4 6 8 10 12 14 16

Kn

×10-3

0

1

2

Saving by Each Bank

number of banks2 4 6 8 10 12 14 16

prof

it

00.05

0.10.15

number of banks Profit of Each Bank

number of banks2 4 6 8 10 12 14 16ag

greg

ate

prof

it

1.051.1

1.15

number of banksProfit Margin of Each Bank

Figure C4: Equilibrium Variables in the Variant Model

49

number of banks2 4 6 8 10 12 14 16

wel

fare

52

54

56

58

60

62

64benchmarkvariant

Figure C5: Welfare and Market Power

50