ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E14-13

Entrepreneurship, Financial Intermediation,and Inequality
byTakanori Adachi

December 2014

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Entrepreneurship, Financial Intermediation, andInequality
Takanori AdachiyNovember 19, 2014

Abstract
This paper provides a simple dynamic framework to study the long-run relationship between …nancial intermediation and wealth inequality. By considering two types of entrepreneurial …nancing (self-…nancing and intermediated…nancing), I show that wealth inequality is more severe in an economy where
all …nancing is intermediated than in an economy where some entrepreners
rely on self-…nancing. This result is consistent with the augmented Kuznets
hypothesis that a large scale operation of production and the …nancial intermediary development are associated with higher inequality.

Keywords: Entrepreneurship; Financial Intermediation; Inequality.
JEL classi…cation: E21, L26.

I am grateful to Munetomo Ando, Ryo Arawatari, and Makoto Nakajima for helpful discussions on an early version of the paper. I also acknowledge a …ancial grant from the the Yamada
Foundation and a Grant-in-Aid for Scienti…c Research (A) (23243049) from the Japanese Ministryof Education, Science, Sports, and Culture. Any remaining errors are my own.
ySchool of Economics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan.
E-mail: adachi.t@soec.nagoya-u.ac.jp

1

Introduction

What is the relationship between …nancial intermediation and inequality? Consider
the case where one needs to raise outside capital for production. Under capital
market imperfections, one’s borrowing opportunities would be associated with hisor her wealth level. On one hand, …nancial intermediation might ease inequality
by making less wealthy people capable to borrow. On the other hand, one may
also expect that …nancial intermediation would help richer people borrow more andmore, accelerating inequality. The same arguments also hold for the situation where
opportunities of …nancial access become broadened by …nancial liberalization suchas deregulation or capital account liberalization.
Although it is di¢ cult to tell whether inequality per se is bene…cial or harmful, it would be important to understand how the degree of inequality in a society
is determined (see, e.g., Piketty (2014)). One may argue that inequality mattersto the working of social infrastructure such as political stability (see, e.g., Alesina
and Perotti (1996)) and public safety (see, e.g., Fajnzylber, Lederman, and Loayza(1998)). If so, policymakers would be concerned about what determines the degree of inequality. This might be the case especially when they want to consider
the balance of economic organization and non-economic arrangement in their society. Traditionally, what they are expected to do to cure inequality is to engage
in redistribution, that is, ex-post public transfers. However, they may also wantto know ex-ante private factors which generate inequality, given that they have anappropriate incentive and instrument to a¤ect inequality.
Taking inequality per se seriously, this paper provides a simple dynamic framework to study the long-run relationship between …nancial intermediation and wealthinequality. More speci…cally, I consider a deterministic dynamic model à la Matsuyama (2000), who analyzes how inequality arises in an economy with an imperfectcapital market. In contrast to Matsuyama (2000), my model allows di¤erent typesof …nance (and can eliminate the steady state equilibrium with perfect equality). I
assume that some positive …xed amount of capital is necessary to do production (i.e.
non-convex technology). Then, in the presence of the capital market imperfection1

(due to imperfect enforcement), a household with little wealth cannot borrow anywhere even if he wants to start production, while he may borrow from a …nancial
intermediary when he has enough wealth. A borrower always has an option of simply defaulting, and as a result, repayment is enforceable only with some inevitablecost. The role of …nancial intermediaries (FIs) is thus to alleviate this enforcement problem by doing some costly monitoring activities. Speci…cally, the model
below assumes that when the borrower defaults (which never happens in equilibrium, though),1 FIs can get back some fraction of money (one can think of FIs in
the model as local banks, investment banks, securities agencies, and so on). Intermediated borrowing, however, might not be so bene…cial for entrepreneurs with alarger amount of wealth because it is costly in nature. This paper aims to capture
this feature in a formal model. While uncertainty nor asymmetric information is
not considered to maintain analytical tractability, I provide a full characterizationof the steady-state equilibria to obtain interesting insights about the relationshipbetween …nance and inequality.
Speci…cally, I provide a theoretical basis for the augmented Kuznets hypothesis(see the next section for the related literature); (i) inequality persists in a country
and varies across countries, and (ii) a large scale operation of production and the…nancial intermediary development are associated with higher inequality. I consider
two types of …nancing, self-…nancing and intermediated …nancing, and assume that
there is imperfect enforcement in the capital market, which makes room for …nancial intermediaries endowed with the monitoring technology to play a role in theeconomy. Other economic agents are households (potential entrepreneurs), who areheterogenous only in the level of their wealth. In the analysis below, the interest
rate is endogenously determined. I provide the steady-state characterization to see
what kind of …nancial pattern prevails in the economy as well as the characteristicsof the wealth distribution. Although there is no such heterogeneity as talent, perfectequality never arises in any steady-state equilibria (i.e., inequality persists) unlessthe initial wealth distribution is too skewed toward rich or poor. It is also found1

Note also that default is always voluntary; since there is no uncertainty in the present paper,
debt overhang is not an issue.

2

that wealth inequality is severe for a lower equilibrium interest rate, and that wealthinequality is more severe in an economy where all …nancing is intermediated than in
an economy where some entrepreners rely on self-…nancing. It is also shown that fora wide range of parameters (concerning the bene…t and the cost of monitoring) there
are two continua of the steady-state equilibria; one is where all entrepreneurs rely
on …nancial intermediation, and the other is where some of (richer) borrowing entrepreneurs self-…nance. The multiplicity comes from the mutually reinforcing e¤ects;
in the former type of equilibria, the equilibrium interest rate is low and the supplyof capital (the number of poorer agents) is large. These two e¤ects are mutuallydependent, and inequality is severe. In the second type of equilibria, however, theequilibrium interest rate is high and the supply of capital (the number of poorer
agents) is small. Inequality is less severe. These features are consistent with the…ndings of Clarke, Xu, and Zou (2013) (see the next section). My formal model
suggests that even if two economies that have similar values of parameters, they
may end up di¤erent types of economy in terms of …nancial structure.
The rest of the paper is organized as follows. The next section brie‡y reviews
the related literature. Then, I present a dynamic model in Section 3. Section 4
provides the steady-state analysis, followed by Section 5 where the e¤ects of capital
account liberalization are discussed. Section 6 concludes the paper.

2

Related LiteratureThis paper is related to the literature which investigates macroeconomic or development implications in the presence of capital market imperfections, which is started
with a seminal article by Galor and Zeira (1993).2 These papers are concernedwith the long-run e¤ects of the imperfect capital market on the wealth distribution.
They are basically silent on the di¤erences in …nance. An exception is a paperby Chakraborty and Ray (2006), who investigate the issue of bank-based versus
market-based lending in an Ak-type endogenous growth model. Speci…cally, they2

Followed by, e.g., Banerjee and Newman (1993), Aghion and Bolton (1997), Piketty (1997),
Matsuyama (2000) and many others.

3extend Holmström and Tirole’s (1997) incentive model (moral hazard with respect
to the project choice) of …nancial intermediation to a dynamic context. As in thepresent paper, the role of bank monitoring is to mitigate the agency problem, andeach entrepreneur chooses how she borrows the working capital. Chakraborty and
Ray (2006) focus on the balanced growth paths, and compare the growth rate of percapita GDP (and other macro variables) in the market-based system with the one in
the bank-based system. In their model, however, there is no room for a mixed structure of di¤erent …nance types to arise: in the long run, all entrepreneurs (except forthe ones who can borrow from nowhere) in the economy borrow either from banks
or from markets, depending on exogenous parameter values concerning monitoringe¤ects and costs. In other words, direct and intermediated lending cannot coexist inan economy in Chakraborty and Ray’s (2006) model. On the contrary, Chakraborty
and Ray (2007) allow three types of households to emerge in the steady state; (i)those who cannot borrow, (ii) those who borrow some amount from a bank, and (iii)
those who rely only on the credit market. They focus on two features of a …nancial
system; its depth and structure. By …nancial depth, they mean how large the proportion of unconstrained borrowers is. Financial structure captures the fraction ofborrowers who rely only on the market among them. Basically, the initial inequality
entirely determines to which …nancial system the economy converges; the more unequal in the initial stage, the less developed the economy’s …nancial system remains.
This is because Chakraborty and Ray (2006, 2007) consider a small open economy.
Namely, they assume that the interest rate is exogenously given. In the following
model, the interest rate is endogenously determined as a component of equilibrium.
As such, the initial distribution is just one factor to determine the characteristics ofthe steady state.
The formal model presented below is also motivated by empirical …ndings oninequality.3 Based on a time series data of England, Germany and the US available
3

Note that the target of these empirical studies is on income inequality rather than wealthinequality. This is because it is not easy to …nd an appropriate index or a good proxy for wealthinequality (the GINI index for land ownership is typically used). Although it is well held that
wealth inequality is more severe than income inequality, we do not go into details on this issue inthe present paper. The bene…t of considering a theoretical model is that one can systematically4at that time, Kuznets (1955) attempted to o¤er a broad hypothetical view on therelationship between economic development and income inequality, which has been
known as the Kuznets inverted U-shaped hypothesis. It states that as an economy
develops income inequality rises, but in the later stage of development it mitigates.
Kuznets (1955) attributed this change to the migration shift from the traditional
agricultural sector to the modern industrial sector, where the wage dispersion islarge. The congestion in the modern sector eventually makes the traditional sector
again attractive, which eases inequality.
Di¤erent from Kuznets’ (1955) original reasoning, Greenwood and Jovanovic(1990) o¤er a theoretical model to endogenously derive the inverted U-shaped curve,
focusing on the role of …nancial intermediation. In their model, individuals caninvest in a risky but pro…table project only when they pay a …xed membershipfee to join a …nancial intermediary coalition. This …xed cost …rst has a role toprevent poorer individuals from accumulating wealth, which exacerbates inequality,
However, the more rich individuals join these coalitions, the lower the entry fee
becomes (because the average cost of the coalition declines as the number of members
increases), which eventually vanishes inequality.
However, a number of empirical studies (such as Deininger and Squire (1998),
Li, Squire, and Zou (1998), and Clarke, Xu, and Zou (2013)) …nd little support
for the Kuznets inverted U-shaped relationship between income inequality and thelevel of income per capita. Rather, as Li, Squire, and Zou (1998) show, the degreeof inequality seems to have a nature to persist within an economy, but it varies
across economies. Based on their cross-country empirical study, Clarke, Xu, andZou (2013) also cast doubt on the role of …nancial intermediation in Greenwood
and Jovanovic’s (1990) story. Considering these points, I model inequality as aperpetuating phenomenon even in the long run. In particular, equality never arises
in any steady state equilibria in my model. This is in sharp contrast to the models
which derive wealth distribution but allow perfect equality to arise as one of theequilibria.4 I also interpret institutional di¤erences in …nance as main causes ofdeal with both inequalities.
4
Mookherjee and Ray (2003) is an exception.

5

generating these international varieties in equality, if other possibly related factors
are controlled.
Even though Kuznets’(1955) original inverted U-shaped hypothesis has gained
little empirical support, Kuznets’ (1955) analysis could be still insightful. If themodern technology which entrepreneurs adopt needs high leverage, …nancial intermediation would help already rich people borrow more, keeping poorer households
from starting the project, who remain suppliers of capital. In this way, inequality
might be associated with the prevalence of the modern technology via …nancial intermediation. Indeed, Clarke, Xu, and Zou (2013) …nd this e¤ect in their empirical
study, and call it an augmented Kuznets hypothesis: as Kuznets (1955) suggested,
sectorial structure matters to inequality. In particular, a large scale operation of
production (measured by added value of non-agricultural sectors divided by GDP)and the …nancial intermediary development (measured by the amount of bank assets
or of private credit divided by GDP) are associated with higher inequality (measured
by the GINI index of income). To the best of my knowledge, there is no theoreticalmodel that formalizes this e¤ect of …nancial intermediation on inequality. Although
my simple model cannot replicate all of their empirical results, one may …nd it usefulin terms of investigating the role of …nancial intermediation from various points ofview.
It is also interesting to see the e¤ects of capital account liberalization on inequality. Based on the data of eleven emerging markets which ex equity market
liberalization during the period of 1986 to 1995, Das and Mohapatra (2003) …nd
that average middle class income share (not the absolute value) decreased while theaverage income share of the highest class increased (and the lowest class experienced
little change in their share). After analyzing the steady-state, Section 6 incorporates capital account liberalization into the dynamics of my model, and considers
its e¤ects on the wealth distribution.

6

3

The Model

In this section, I describe a formal dynamic model of household behavior and …nancial intermediation. In particular, I incorporate …nancial intermediation into a
dynamic model à la Matsuyama (2000). I …rst explain the production technology,
imperfect enforceability, and the role of …nancial intermediation. Then, I determine
the equilibrium interest rate in each period, and illustrate equilibrium dynamics of
the wealth distribution and the interest rate.

3.1

Economic Environment

The economy is closed with an in…nite, discrete time horizon t = 0,1,2,:::. The word“closed”implies that the interest rate is endogenously determined in the model. In
Section 5, I consider the e¤ects of capital account liberalization on the dynamics.
In this economy, there is a continuum of dynastic families which live forever. The
total mass is normalized to be one, and there is no population growth. Each agent
in a dynastic family is risk-neutral and lives for one period only (reproducing oneson).5 Also, there is a competitive …nancial sector which consists of a large number…nancial intermediaries (FIs). I assume that intermediaries and households aredi¤erent agents.
In this economy, there is only one type of good, which can be consumed or bemade for bequest (to be explained shortly). In each period t, an identical household
has the following deterministic production technology,6 which is non-convex (becauseof discontinuity):f (kt ) =where kt

8
< kt if kt:

0 if 0q,
kt < q,0 is his investment level and the unit revenue is normalized one,7 which

5

The male pronouns are used throughout to represent a generic agent.
Under the …xed cost of starting production and borrowing constraints (like the ones introduced
later), households might consider joint borrowing and joint production. I simply assume away thispossibility in the present paper.
7
In the following analysis, this normalization will make the equilibrium per-unit interest ratealways less than one, which seems odd at …rst (because repayment is less than the amount of
6

7

accrues when his investment is over the normalized …xed level, which is not too large
but not too small, either (kt = q 2 [1; 2)). This …xed cost can be literally taken

as physical capital, or can be thought of entrepreneurial human capital. Note thatthe output is linearly increasing after the …xed threshold level of capital. I assume
that each household has access to an alternative “backyard” storage technology,
whose per-unit return is

0 for any input level (i.e. no …xed cost is necessary to

generate a return). This can be also considered as a traditional technology such as
small-scale agriculture. Restrictions on

will be added later on. We also assume

that each household earns an exogenous nonrandom revenue, which is normalized
to one and is common to all households and non pecuniary (so households cannot
borrow or lend a part of this income). This is just a technical assumption to yieldsteady-state results in this non-growth model.8
Let at0 denote the wealth of a household in generation t (which is inheritedfrom his parent at t

1; to be explained in Subsection 3.6). The wealth level is the

only source of household heterogeneity. The distribution of wealth across households
is denoted by the measure Gt (a) de…ned on Borel subsets of [0; 1), and the initial
wealth distribution G0 (a) is given.

Given his inherited wealth at (and under the constraints explained later), a
household maximizes his income. He can consume it by himself or can make itbequest to his son at t + 1 (to be explained in Subsection 3.6). We call a household
an entrepreneur when he earns revenue from using the production technology. For
simplicity, we assume that capital for production fully depreciates in one period.
This assumption would particularly be …tted when the capital is interpreted as
human capital. In this economy there is another revenue-generating opportunity
for households (other than the backyard storage technology), namely a one-period
competitive capital market (to be explained in the next paragraph), where a per-unit
gross interest rate rt

0 accrues when a household has saved some of his wealth inborrowing!). However, for analytical purposes, it is more important to note that it still earns apositive amount, and this normlization does not invalidate the main thrust of the results.
8This exogenous income should not be large; otherwise, all the households become rich enoughand they can easily overcome the investment threshold caused by the non-convexity of the technology. It can be veri…ed that normalizing it to one satis…es this requirement.

8

an FI. Thus, the opportunity cost of using capital kt for production is maxf , rt gkt .
The timing of decision making in period t is as follows. At the beginning a newagent in a household is given the inherited wealth from his parent. He divides it
into savings on the backyard storage (st ) and the rest (atst ). Then, he decides

whether he becomes an entrepreneur or not, and he divides (atst ) into the partfor production (only when he becomes an entrepreneur) and the part for savings in
the savings market.

3.2

Financial Arrangement

Although as a deposit saver a household can save any amount of money in an FI,
he cannot borrow any amount from an FI. In addition, he cannot borrow directlyfrom other households; he must borrow from an FI, otherwise, he must self-…nance.
These features come from the enforcement problem and the role of FIs, which is
explained below in this subsection. First, we look at the role of FIs adopted in thepresent paper.
Following Holmström and Tirole (1997), I assume that the role of FIs is tomonitor an entrepreneur to mitigate his opportunistic behavior. I also assume that
only FIs are endowed with monitoring technology, or one may assume that households have a prohibitively costly monitoring activity because of, say, the lack of
specialization. Monitoring is a broad concept, and one can think of various typesof monitoring. In my model, monitoring takes place when lending is made, andit determines what the lender can do in the case of default. As such, it may also
include the cost of writing a contract regarding what will be legally done in the caseof default, and/or the cost of FIs’service (via renegotiation) in the case of default.
It may include sending experts to the company boards. FIs o¤er a …xed interest
rate rt on savings deposits, and earn revenue using the deposits to make loans toentrepreneurs who borrow from them. The rate charged for a loan made by an FI
is the (gross) lending rate it . The lending-deposit rate spread is the return to the
FI for providing the …nancial service. An FI will choose whether to monitor or not;there is no choice for intensity of monitoring. For simplicity, I assume that there9

is no …xed cost for monitoring and that the marginal cost of monitoring (per theamount of lending) is> 0, which is exogenous9 and constant over time (that is,

the monitoring technology has constant returns to scale). This can be understood asFI’s disutility of labor for monitoring, or their human capital value (both of which
are assumed not to be tradeable). I also assume that the monitoring cost is sunkwhen lending is made (in this sense monitoring here is ex-ante one so that free riding
is not an issue). In addition, I assume that there is a large number of FIs. Perfectcompetition in the …nancial sector implies that deposits each FI receives are equal
to loans issued by that FI and
it = r t +(1)

so that FIs make zero pro…t in equilibrium.10 Note that the possibility of FI’s
incentive problem is assumed away. One may think that an FI can make a credible
commitment, caring about its reputation.
If there were no enforcement problem, however, no household wants to borrowfrom an FI because it should be more expensive than when he can borrow directlyfrom other households.11 In order to validate the existence of FIs, I assume thatenforcement in the capital market is imperfect; an entrepreneur always has an option of simply defaulting.12 Speci…cally, suppose that he borrows bt . When the
borrower does not honor his repayment it bt , however, the lender cannot seize all of
the entrepreneur’s revenue due to the imperfect enforcement. In other words, anentrepreneur can pledge only up to some fraction of his revenue. This amount iscalled his pledgeable revenue, and the following modeling assumption is made.
Assumption 1. Ex-ante monitoring by an FI is necessary for the borrower’s
9

Ando and Yanagawa (2004) construct a model where monitoring technology is endogenous.
We do not assume monotoring has increasing returns to scale, which is incompatible withperfect competition. See Allen and Gale (2000, Ch.8) and Allen and Gale (2004) for an analysis
of competition in the banking sector.
11
And, if the interest rate is below one, he wants to borrow an in…nite amount of capital becauseof the linearity of the production technology. As is seen below, the equilibrium interest rate under
the perfect enforcement should be one.
12
In the present paper, I simply assume away the role of intertemporal incentives such as reputational concerns. Since my interests lie in macroeconomic issues, and I do not pose microeconomic
problems in, say, a small community, this would not cause a serious ‡aw.
10

10

pledgeable revenue to be positive.
Speci…cally, I assume that when an entrepreneur tries to circumvent intermediated borrowing, his pledgeable revenue is zero.13 It becomes kt , however, when herelies on intermediated borrowing. Then, Assumption 1 is expressed by 0
Note that< 1.14is less than one, meaning that FIs can improve but cannot perfectly

correct the enforcement problem. The parametercan be understood as capturing

the e¤ectiveness of monitoring. It may be strengthened by the e¢ ciency of the economy’s legal system for protecting investors (but the …nancial sector is necessary to
work it).15 This can be interpreted as the situation where upon default the fraction
1of the production revenue perishes due to the costly renegotiation process.

One may also interpret this as a reminiscent of the costly state veri…cation (CSV)
problem; veri…ability of the project is not without cost.16
From the borrower’s side, these pledgeable revenues are also the default cost,which is seized by the lender upon default.17 Thus, if an entrepreneur circumvents
intermediated borrowing, he cannot borrow kt and save st in the backyard storage
technology over his wealth level at . That is, he can borrow money within his wealthlevel which he has saved. In this sense, we call this situation self-…nancing. This is13This might seem a strong assumption. However, if one allows non-zero pledgeable income whenan entrepreneur escapes intermediated borrowing, the steady state analysis becomes complicated,and yields less interesting results.
14
Note that = 1 means the perfect enforcement: as is shown below, any household with wt 0
does not su¤er from the borrowing constraint (to be formalized shortly).
15
Seminal papers on the international di¤erences of the legal system in protecting investors are
La Porta, Lopez-de-Silanes, Shleifer, and Vishny (1997, 1998).
16
For more on the formulation of this problem, see, e.g., Freixas and Rochet (2008).
17
Note that Kiyotaki and Moore’s (1997) justi…cation for the borrowing constraint can also
be applied in the present context. They propose the following story. Production technology
is speci…c to the borrower, and if the lender succeeds the borrower’s production he cannot do
production as much as the lender can do. Knowing this, the lender can suggest renegotiating theinitial contract to reduce his repayment, and the borrower will accept this as long as the suggesed
repayment is not below the level of he can earn by succeeding the borrower’s production. If one
assumes that it is the lender who must succeed the borrower’s production, and that (unmodeled)“shadow” middilemen, who have an inferior technology, are actually between lending households
and borrowing entrepreneurs, then Kiyotaki and Moore’s (1997) story applies. If the relationshipbetween lending households are borrowing entrepreneurs is literallhy direct, then it does not hold.
However, this seems less natural.

11expressed byrt (ktat + s t )

(2)

0.

We assume that households are allowed either to self-…nance or to borrow all
his capital from an FI only. Then, the borrowing constraint under intermediatedborrowing becomes(rt + )(kt

at + s t )

kt .

(3)

Note that in equilibrium default never happens since this inequality must hold.18
I also assume limited liability; borrower’s payment cannot exceed his total income.

3.3

Optimal Investment Decisions

Now we turn to the revenue structure of households. Consider a household withat and suppose that he saves st

at in the backyard storage technology (this does

not happen in equilibrium in the analysis below). If he invests kt , and when heself-…nances (that is, kt

atst ), his total income becomes 1 + st + F (kt ) + rt (atmaxf , rt gkt . So, it can be written as
8)st + minf1; 1 rt gkt + rt at< 1 (rt

st )

:

1

(rt

)st + rt at

maxf ; rt gkt

if kt

q,

if 0

kt < q.

On the other hand, when he borrows from an FI, it becomes 1 + st + F (kt ) +rt (atasst )maxf , it gkt (with kt satisfying constraint (3)). Thus, it can be written
8
< 1
:

1

it gkt + rt at(rt)st + minf1

;1

(rt

)st + rt at

maxf ; it gkt

if kt

q,

if 0

kt < q.

Here, it is implicitly assumed that a household saves atst on deposits in an

FI and after that he draws for self-…nancing or borrows from an FI. It is seen thatgiven his investment level kt only households whose wealth satisfy at18st

kt canA similar borrowing constraint is adopted by Holmström and Tirole (1997), where production
is stochastic. This makes dynamic analysis less tractable. Indeed, Chakraborty and Ray (2006)
incorporate Holmström and Tirole’s (1997) incentive problem into the dynamic model. However,they assume that the most e¢ cient technology out of three is deterministic.

12

self-…nance his investment to become an entrepreneur. Similarly, a household whose
wealth after the saving in the storage technology at

st is greater than or equal to=(rt + )]kt can borrow capital from an FI. Since the minimum investment[1level to become an entrepreneur is kt = q, only a household with at

st

q can

become an entrepreneur by self-…nancing. Similarly, a household whose net wealthst is greater than or equal to [1at=(rt + )]q can borrow capital from an FI.

Now, in order to analyze the e¤ects of the enforcement problem, I make thefollowing assumption.
Assumption 2.

>+ .

This assumption states that the monitoring cost is not so high, and hence monitoring is socially desirable given the enforcement problem. It can be also statedthat the monitoring e¤ect is high enough, and that the backyard storage technology
is not so productive. Note that this assumption implies< 1.

Notice at this moment that if there were no enforcement problem then no households would be free from borrowing constraint (and there would be no …nancial sector). As a result, the equilibrium interest rate would be rt = 1, which is the only
possible case. This is because if rt > 1, all households, irrespective of his wealth at ,would want to become a lender, and if rt < 1, all households would want to become
an entrepreneur, both of which imply the capital market would not clear. Notethat no households want to use the backyard storage technology because< 1. In

this case, all households are indi¤erent between borrowing and lending, obtaining
1 + at . So, how the wealth in the economy is divided to lending and borrowing is
indeterminate. The GDP in this economy is one (the identical revenue times the
population) for any period.19 These arguments can be summarized in the following
proposition.
Proposition 1. There is no income inequality across households if there are no19

In the following analysis, I ignore the value of the exogenous revenue when we calcurate theaggregate income measures such as the GDP and the GINI index. This eases computation, andenables more direct intermpretation of the results.

13

enforcement problems (the GINI index for earnings inequality is zero). That is,irrespective of the wealth level, all households earn a net in‡ow one in any period.
Note that there can be income inequality in period t: given the wealth level at ,a household obtains 1 + at . However, as we will see in the next section, this income
inequality and wealth inequality in the perfect world disappears in the long run.
This is also a consequence of the assumption that there is no talent or endowment
heterogeneity.
On the contrary, if there were no …nancial intermediary sector in our imperfectworld, income inequality would arise. It is easy to see that a household has no way
to use the backyard storage technology if his wealth level at is less than q, earning
per-unit income
at(his total revenue is 1 + at ), while it will earn one if he hasq (his total revenue is 1 + at ). The aggregate earnings in period t in this caseis Gt (q) + (1Gt (q)) = 1(1)Gt (q). The GINI index for income inequality is

calculated as[Gt (q)]2
1 (1)Gt (q)1(1)Gt (q)(1 Gt (q))=,
1 (1)Gt (q)IncomeGIN ItN F I = 1

Gt (q)+ 1 [1(1)Gt (q)Gt (q)]

from which we have @IncomeGIN ItN F I =@ < 0; the more e¢ cient the storage technology is the less severe income inequality is. The sign of @IncomeGIN ItN F I =@Gt (q)is ambiguous. This is because an increase in Gt (q) can imply either upward or downward shift. In the next section, wealth inequality in this case will be considered.
Now, I come back to the world with the imperfect enforcement and …nancial
intermediation to investigate the e¤ects of …nancial intermediation on equality. First,
I obtain the following lemma about the lower bound of the equilibrium interest rate.
Lemma 1. In equilibrium, it must be the case that rt >Proof. Suppose that rt.

happens. Then, the borrowing constraint (3) is nolonger binding for any household so that every household with at
14

st < [1=rt ]qdemands in…nite capital, meaning there is excess demand of capital in thiseconomy. QED
Using this lemma, I obtain the following lemma that states that savings in the
backyard storage technology do not happen in equilibrium.
Lemma 2. In equilibrium, no households use the storage technology. That is, for
any household, st = 0 for any t.
Proof. Suppose st > 0 for some t. Then, by reducing some amount of st and byplacing that on deposits in an FI, a household obtains, by Lemma 1, a per-unit
gain rt >, which is greater than the backyard storage technology’s returnby Assumption 2. QED
Since a household does not choose st = 0, it is seen that given his investment
level kt only a household with atq can become an entrepreneur by self-…nancing(as is explained below, he may optimally borrow from an FI). Similarly, a household
whose wealth at is greater than or equal to [1=(rt + )]q (a(rt )) can borrow

capital from an FI (as is also explained below, he may optimally self-…nances when
his wealth is high enough). Notice that a(rt ) < q for any rt >.

Since rt is endogenously determined, the threshold a(rt ) is also endogenously
determined as a function of rt , while q is not. Simple algebra shows that:2 q
q@ 2 a(rt )@a(rt )
=
>
0,
=
< 0,
2
2
@rt
(rt + )@rt
(rt + )3
which means that a( ) is strictly increasing and concave. This fact will be used in thenext section. The …rst inequality shows that the higher the interest rate the tighterborrowing constraint (3) is. The latter relationship implies that a(rt ) is concavewith respect to rt . Then, the following lemma is obtained.
Lemma 3. For all rt >, a(rt ) > 0. That is, if Gt (a(rt )) > 0, there exist

households (with at < a(rt )) who cannot obtain any external …nance so that they
become a net lender.
15

Proof. It is immediate from limrt #a(rt ) = 0 and @a(rt )=@rt > 0. QED

It is also veri…ed that:@a(rt )@a(rt )q@a(rt )
q
=1
> 0,
=
< 0,
=
> 0,
@q
rt +
@
rt +
@
(rt + )
which means that the more the necessary amount of the …xed capital, the less e¤ective the monitoring, and the more costly the monitoring, the severer the thresholda(rt ) is.
If the interest rate is exogenous from the initial period, then a(rt ) is always aconstant. Thus, the initial wealth distribution G0 completely determines who canbe a borrower and who remains a lender, and there is no social mobility acrossentrepreneurs and non entrepreneurs because there is no such jumping process as
uncertainty in the present model. If the interest rate is endogenous, however, theinitial wealth distribution is not a sole determinant. Also, if capital account liberalization occurs in some period t (which means that all agents in this economy
take the interest rate as exogenously given), the threshold becomes a constant fromthat period on. Conversely, suppose that the government stops regulation on theinterest rate in some period t. Before that period the interest rate is exogenous on
the interest rate, but now it becomes endogenous.
The following lemma determines the upper bound of the equilibrium interest
rate.
Lemma 4. In equilibrium, it must be the case that rt
Proof. Suppose that rt > 1

1.

holds. Then, a household with at < q does not

become an entrepreneur. This is because he can earn 1 + rt at by choosing kt = 0,
while his income is 1 + (1rt )kt + rt at , which is less than 1 + rt at , when hebecomes an entrepreneur. Thus, intermediated borrowing is relatively costly
compared to the bene…t from production, so there are no households who borrowfrom an FI, meaning there is excess supply of capital in this economy. QED
I then obtain the following lemma about the optimal amount of capital, given
the …nancial decision (self-…nance or intermediated …nance).
16

Lemma 5. In equilibrium, an entrepreneurs chooses kt = at when he self-…nances,and kt = k(at ; rt )at =(1

Proof. If rt < 1

is the case, then his income is strictly increasing in kt=[rt + ]) > at when he borrows from an FI.

irrespective of whether he self-…nances or borrows from an FI. Thus, he wants toborrow up to the level where the borrowing constraint (2) or (3) binds. If, then households with at 2 [a(1rt = 1); q) are indi¤erent between

becoming an entrepreneur and choosing kt = 0. QED
Note here that if the optimal investment level is allowed to be less than his
wealth, he may not want to save his money if he is quali…ed to borrow only from abank. My formulation on the production technology excludes this situation.

3.4

Optimal Financial Decisions and Income Inequality

First, it is immediate to see that when an entrepreneur has no access to capital his
income is strictly smaller than the one he can borrow from an FI. This is because
the total income when he self-…nances is 1 + at , while the one when he borrows froman FI is1+
and for rt < 1

1

rt =(rt + )1=(rt + ), we have rt < [1at ,rt =(rt + )]=[1=(rt + )]. Note herethat an entrepreneur who can self-…nance earns the same amount of income he could
earn in the perfect world.
Now, it can be veri…ed that [1rt =(rt + )]=[1=(rt + )] > rt , rt < 1,which is assured by Assumption 2. So, both incomes are strictly increasing functionsof at . By considering the two slopes 1 and f[1g=[1rt =(rt + )]one obtains the following lemma.
Lemma 6. For any
re 2 (;1

2 [0; 1) and any

2 (0;rt =(rt + )]=[1) such that for rt 2 (
[1

; re]

17

=(rt + )],) there exists a unique=(rt + )]1

holds, where the equality holds if and only if rt = re. For rt 2 (er; 1[1rt =(rt + )]=[1), it is that=(rt + )] < 1.

Proof. The slope of the entrepreneur’s income when he borrows from an FI is
continuous, di¤erentiable and strictly decreasing in rt 2 (
@
@rt
=

1

;1

) because

rt =(rt + )1=(rt + )
)2

(rt +

1

rt +

(rt +

)2

1

rt
rt +

< 0.

Now, it is veri…ed that
1>1

= lim

1

rt "1

rt =(rt + )1=(rt + )

and
lim

1

rt #

rt =(rt + )1=(rt + )= 1 > 1.

Thus, the statement in the lemma follows from the Intermediate Value Theorem.
QED
Indeed, an explicit solution for re is obtained: re = re( ; ) = (

2

)=( +

). This lemma shows that for a su¢ ciently low interest rate, there are no self…nancing entrepreneurs in the economy. This result might at …rst seem odd, but
the reason is clear: under intermediated lending, an entrepreneur can borrow morethan under direct lending, and this e¤ect is greater as the interest rate becomes
lower. Summarizing the argument so far, I obtain the following proposition, whichstates that a household’s optimal …nancial decisions, given the interest rate rt andthe inherited level of wealth at .
Proposition 2. Suppose that (a)< rtre. Then, a household does not

become an entrepreneur if his wealth is at 2 [0; a(rt )), but he borrows from an FI to

become an entrepreneur if his wealth is atre < rt1a(rt ). Next, suppose that (b). Then, a household does not become an entrepreneur if his wealth is

at 2 [0; a(rt )), but he borrows from an FI if his wealth is at 2 [a(rt ); q) or

self-…nances if atq.

18

Based on this proposition, one can derive what kind of income inequality arisesin this economy, which is summarized in the following corollary.
Corollary 1. In equilibrium, a household with at < a(rt ) earns the interest proceedrt from his savings, and one with ata(rt ) earns the return one, which is greaterthan rt , from the project.
Note that the total revenue of an entrepreneur depends on whether he has self…nanced or borrowed from an FI, and his inherited wealth level at . As in the case of
no …nancial intermediation, the GINI index for income inequality can be calculated:
IncomeGIN It (rt ) =(1rt )Gt (a(rt ))[1 Gt (a(rt ))].
1 (1 rt )Gt (a(rt ))If Gt (q) > Gt (a(rt )) and Gt (q)(1 Gt (q)) > Gt (a(rt ))[1 Gt (a(rt ))], then IncomeGIN ItN F I >IncomeGIN It (rt ).

3.5

Existence and Uniqueness of the Equilibrium Interest
Rate for a Fixed Wealth Distribution

Now, I verify the existence of the equilibrium interest rate rt in period t, given
the wealth distribution Gt . First, I look at the demand side of capital (i.e. theeconomy’s total investment). In the case of re < rt < 1, the aggregate demandfor intermediated capital isI

D (rt ) =1

Z q
1
adGt (a),=(rt + ) a(rt )while the aggregate amount of self-…nancing is
Z 1
S
D (rt ) =adGt (a)qso that the aggregate demand for capital is D(rt ) = DI (rt )+DS (rt ) for rt 2 (e
r; 1

).

Note that for a …xed Gt , DS (rt ) is indeed a constant. For this range, D(rt ) is
continuous, and decreasing in rt . It is not strictly decreasing, though, since there19can be a positive mass on a(rt ) or on q. These properties also hold for the case of< rt < re, whereD(rt ) =1

Z 1
1
adGt (a)=(rt + ) a(rt )since there is no self-…nancing.
Now consider the case of rt = re. In this case, households with atr)a(eare indi¤erent between borrowing self-…nancing and borrowing from an FI. So, thedemand for intermediated capital is between zero andZ 1
1
adGt (a)1=(er + ) a(er)so that the aggregate aggregate demand for capital D(rt ) is continuous at rt = re.
Last, consider the case of rt = 1. As explained above, households withat 2 [a(1); q) are indi¤erent between becoming an entrepreneur (by borrowing

at 2 [a(1); q) can borrow up to kt = at . Thus, the demand for intermediatedfrom an FI) and choosing kt = 0. Due to the borrowing constraint (3), household
capital when rt = 1

is a correspondence:
D(1) = 0;1
1

Z q

adGt (a) .

1

It can be veri…ed that D(rt ) is continuous at rt = 1. Overall, the aggregate

demand is a continuous function of rt in the relevant range.
Next, I turn my attention to the supply side of capital. For any rt 2 (;1],

the aggregate supply of capital (i.e. the economy’s total savings) is
Z 1
adGt (a),
Kt =
where

R

0

denotes Lebesgue integral. Note that it does not depend on rt . It is,however, endogenously determined since it depends on Gt . The equilibrium interest
rate in period t is determined by the usual market clearing condition: Kt = D(rt ).
It is immediate to see the following proposition holds.
Proposition 3. The equilibrium interest rate rt 2 (unique.

20;1] exists and is3.6

Equilibrium Dynamics of the Wealth Distribution andthe Interest Rate

Given the initial wealth distribution G0 , the market clearing condition also has arole to recursively determine the dynamics of the equilibrium interest rate frt g1
t=0

together with the wealth dynamics fGt g1
t=1 (which is caused by dynastic motivation
explained below). Here I assume that expectations are fully rational (or, players

have perfect foresight since there is no uncertainty). First, take any period t, andsuppose that re < rt < 1. Then, the wealth dynamics in this case is described by
8(1 + at )for at q,
>
>
>
>
>
>
<
1
rt =(rt + )at+1 = at+1 (at ; rt ) =1+
atfor at 2 [a(rt ); q),>1=(rt + )
>
>
>
>
>
:
(1 + rt at )for at 2 [0; a(rt )),where

2 (0; 1) is the parameter which describes how a household cares about his

next generation; a household consumes fraction (1fraction) of his income, and leaves

to his son.20 One may interpret this as an exogenous parameter of savingrate as in the Solow growth model.
Similarly, for rt 2 (at+1 = at+1 (at ; rt ) =

8
>
>
<
>
>
:

; re] the wealth dynamics is described by1+

1

rt =(rt + )1=(rt + )atfor ata(rt ),

for at 2 [0; a(rt )).

(1 + rt at )Thus, for any equilibrium interest rate rt , the wealth transition at to at+1 isobtained. The wealth distribution dynamics fGt g1
t=1 is determined by the following
20

This dynamics can be derived from generation t’s entrepreneur’s utility maximization problem
if we assume the following “warm-glow” utility function:u = (1) ln c +ln b,where c is the amount of his own consumption and b is that of his bequest to his son. The indirect
utility as a function of the realized income I becomes)1u(I) = (1I,

which is linear, so this formulation is consistent with the assumption that households are riskneutral.

21

law of motion:
Gt =Z

at (at 1 ; rt 1 )dGt 1 .

Instead of investigating the dynamics per se, I focus on the steady-state to study
which …nancial pattern arises as well as the wealth distribution in the economy. Notethat the limit wealth distribution G1 (a) will have positive mass on …xed points ofmapping at+1 = at+1 (at ; r1 ), where r1 is a limit interest rate. Thus, my goal is
to analyze the properties of these …xed points. As is seen in the next section, thelimit wealth distribution G1 includes a …nite number of mass points. While itdoes not resemble any conceivable wealth distribution in the real world, one canobtain interesting insights about the relationship between …nance and inequality.
For example, the steady-state GINI index of wealth can be analytically computed.

4

Steady-State Analysis

In this section, I study the relationship between inequality and …nancial intermediation in the steady state. First of all, it is immediate to see that if there
were no enforcement problem, one would have r1 = 1 from the beginning of theeconomy, and in the steady state all households would have the same wealth level
a1 = =(1), provided thatis 1 + =(1) = 1=(1< 1. In each period, the income of all households). This is because for any period t the wealth dynamics

is described by at+1 = (1 + at ), and the steady-state wealth level is the …xed pointof this mapping. Since there is no heterogeneity among households with respect
to production ability or consumption preferences, the di¤erence in the wealth level
eventually vanishes. There would be no room for FIs from the beginning, and there
would be no income or wealth inequality in the long run. These arguments are
summarized in the following proposition.
Proposition 4. There is no income or wealth inequality across households in thelong run if there are no enforcement problems (the GINI indices for income andwealth inequalities are zero in the long run). That is, the wealth level of any

22household converges to=(1), and the income level of any household to

1=(1) in the long run.

4.1

Steady-State without FIs

Before going to analyze the roles of …nancial intermediation, I check the steadystate without FIs. Recall that in each t households with at < q has no way to use
the backyard storage technology, while those with atThus, the wealth dynamics is described by8< (1 + at )at+1 =:(1 + at )for atq can invest in the project.

q,

for at 2 [0; q).

I assume that the …xed cost for production is not large and that the returnof the storage technology is not large either. Speci…cally, I make the following
assumption to ensure the existence of the steady state where both rich households(entrepreneurs) and poor households (lenders) coexist in the long run. Then thewealth dynamics is described as in Figure 1.
Assumption 3. 2

q > (1 + ).

Note that because q < 2 (see Subsection 3.1), the …rst part of the assumption
imposes a restriction on the range of the warm-glow parameter:
also that since q

1 it must be the case that

Assumption 2), I have 0< minf2 [q=2; 1). Note

1=2. Together with

; 1= ; q=

1g, and indeed sup ;q

It is seen that there are two …xed points of the mapping, =(1households, and =(1< 1 (from=.

) for richer) for poorer households. Notice that the initial wealthdistribution G0 completely determines the destiny of a household: the wealth of
those with a0 < q (the fraction of which is G0 (q)) converges tothat of those with a0q (the fraction of which is 1=(1), andG0 (q)) to =(1). Unless

G0 (q) = 0 or G0 (q) = 1, inequality necessarily arises.
Let the aggregate national wealth and the GINI index for wealth inequality
NF I
NF I
in the steady state by denoted by N W1
and by W ealthGIN I1, respectively.

23

Figure 1: Wealth dynamics when there are no FIs.

Then, it is veri…ed that (note that
NF I
N W1
=

(1

< 1 from Assumption 2):)G0 (q) + (1
(1
)(1

)[1
)

G0 (q)]and(1)G0 (q)[1 G0 (q)](1
)G0 (q) + (1)[1 G0 (q)]2
(1
)G0 (q)[1 G0 (q)]
=
.
NF I
(1
)(1
)N W1

NF I
W ealthGIN I1
=

NF I
NF I
As is expected, it is seen that @N W1=@ > 0. I also have @N W1=@G0 (q) <NF I
0, though the sign of @W ealthGIN I1=@G0 (q) is indeterminate. If G0 (q) > 1=2,

NF I
NF I
then @W ealthGIN I1=@G0 (q) > 0, and for @W ealthGIN I1=@G0 (q) < 0 to

hold, it is necessary that G0 (q) < 1=2. This means that if there are many agents
whose initial wealth is smaller than q (so that G0 (q) > 1=2), then an increase in
G0 (q) worsens the inequality. In order to see the operation of this benchmark case,consider the following numerical example.
Example 1. Suppose that
NF I
N W1
(

= 3=4) = 3= 3=4,= 4=15, and q = 1. Then, we have

NF I
33G0 (q)=16 and W ealthGIN I1( = 3=4) = 33G0 (q)[1

24

NF I
( = 3=4). Now, suppose that
G0 (q)]=16N W1

NF I
this case, N W1
( = 7=8) = 7

539G0 (q)[1= 7=8 (and= 4=15, q = 1). In

NF I
77G0 (q)=120 and W ealthGIN I1( = 7=8) =NF I( = 7=8). We can verify that 7
G0 (q)]=92N W1

77G0 (q)=120 >

NF I
NF I
33G0 (q)=16 and W ealthGIN I1( = 3=4) > W ealthGIN I1( = 7=8) for3

any G0 (q) 2 (0; 1). Thus, in this case, the higheris, the higher the national wealthand the less severe the inequality is.

4.2

Financial Intermediation and Inequality

Now, I consider the role …nancial intermediation in the wealth dynamics. Recall
that there are …ve parameters; (i)2 [0; 1); the e¤ect of monitoring, (ii)> 0; theunit cost of monitoring, (iii) ; the “warm-glow”parameter, (iv) q; the …xed cost for
production, and (v) ; the per-unit return of the backyard storage technology. As
is seen below,

does not appear in the case of active FIs because of Assumption 2.

It is veri…ed that the original Kuznets hypothesis does not hold; inequality persists
in any steady state equilibria for any initial distribution G0 . This is an importantfeature because many existing models cannot exclude perfect equality out of theequilibria.21 For comparison with the case of no FIs, I maintain the …rst part ofAssumption 3 (i.e., 2q) as well as Assumption 2 (i.e.,

>

+ ).

For notational convenience, I de…ne the following function:
r1
H(r1 ) = 1,
r1 +which is equal to (1r1 ) + r1 a(r1 )=q so that H(r1 ) > 0 because 1> r1 .

There are three possible …xed points of the mapping at+1 = at+1 (at ; r1 ), which
are

21

8
>
>
,
a (r1 ) =
>
>
>
1
r1
>
>
>
>
>
<
H(r1 )a (r1 ) = 1
q>a(r1 )
>
>
>
>
>
>
>
>
>
,
: a1 =
1

An exception is Mookherjee and Ray (2003).

25

1

, and

where all the households who cannot borrow in the steady-state have wealth a (r1 ),those who borrow from an FI have a (r1 ), and those who self-…nance have a1 .
Now consider the relationship among a (r1 ), a (r1 ), and a1 . Note thata1 > a (r1 ) for any r1 2 (;1], a (r1 ) R a (r1 ) , r1 R 1, and

a1 R a (r1 ) , r1 R re( ; ). Note also that the domain of these functions is
extended to (; 1] for the purpose of drawing the lines in the …gure, though r1

is indeed no greater than 1. For all r1 2 (;18
2@a (r1 )
>
>
=
>0
>
>
< @r1
(1
r1 )2

], it is veri…ed that>>
@ 2 a (r1 )
2 3
>
>
:
=
> 0,
2
@r1
(1
r1 )3

so that a (r1 ) is strictly increasing and convex. Note that a (r1 ) is bounded above
zero because limr1 #a (r1 ) = =[1()] and< 1. It is also veri…edthat@a (r1 )
=
@r1

2

q

qH(r1 ) + a(r1 )<0(r1 + )2 [a(r1 )qH(r1 )]2

so that a (r1 ) is strictly decreasing. One can see that limr1 #a (r1 ) = 1,

which comes from the fact that for any at > 0, k(at ; rt ) ! 1 as r1 #;entrepreneurs want to borrow in…nitely if there is no borrowing limit because of thelinearity of the production technology. This also implies that the limit interest rateis never equal tobecause that is inconsistent with the de…nition of steady state.

Since we have

< rt

1

in any period t, we know that

< r1

1

in any steady state. Figure 2 is a graphical summary of the arguments so far. This
…gure is useful to investigate the existence of steady-state equilibria in the following
analysis. From this …gure, it is immediate to obtain the following proposition.
Proposition 5. Perfect equality never arises and …nancial intermediation neverdisappears in any steady state unless G0 (q) = 1 or G0 (q) = 0.
This result stands in contrast to the existing literature which allows perfectequality to arise as one of the equilibria. To see why this result holds in our model,
26

Figure 2: Relationship among a (r1 ), a (r1 ) and a1 .
note that perfect equality requires r1 = 1. In this case, the wealth dynamics
should be at+1 = (1 + at ) for all households. This should mean that provided that=(1)

wealth, =(1q, all households become an entrepreneur and have the same amount of). In this case, however, all entrepreneurs should rely on self-…nanceso that no FIs survive, which means that the capital market does not clear.
Next, observe a higher interest rate bene…ts net lenders. This is because giventhat they cannot borrow from anywhere the only source of poor households’income
is lending. Note also that
2 2
@a (r1 )
q (1
r1 )
=
> 0,
2
@
[a(r1 )
qH(r1 )] (r1 + )which means that the better the loan enforcement is the greater the entrepreneur’s
wealth is, while a change indoes not a¤ect the lender’s income. Recall thatgiven the wealth distribution a change inborrowing constraint since @a(r1 )=@ =alleviates inequality by mitigating the
1=(r1 + ) < 0.

In what follows, I study a steady-state equilibrium, and consider the implication
for inequality in the economy.
27

4.2.1Equilibria where All Financing is IntermediatedFirst, I consider the case of

< r1

needs
q

re. First, for the steady-state to exist, one

H(r1 )< 1,a(r1 )

which can be written as
r1 >

(1
1

)+
+ ( + )

rb( ; ; ).

Otherwise, there is no steady-state because the wealth of entrepreneurs does notconverge to a certain level. If this is not the case, then there are two …xed points,
a1 and a1 , with there being no capital market if the following is satis…ed:a (r1 ) < a(r1 )a (r1 ).

Here, it can be veri…ed that rb( ; ; ) < re( ; ) as long as

to ensure the existence of steady-state equilibria with r1 2 (some r1 2 (; re] that satis…es a (r1 ) < a(r1 )< 1. In order; re] (that is, fora (r1 ) to exist), the following

condition is necessary:a(br( ; ; )) < a (br( ; ; )):This condition actually always holds as long as
noting that this condition is equivalent to a(br)[1+ ( + )]=(1 +> 0; it can be veri…ed by

qH(b
r) <

and that =(br+ )=) holds

We have two cases for r1 2 (br; re] to consist of a steady state equilibrium: (i)a (er) and rb < rL+

when a(er( ; )) > a (er( ; )) holds, and (ii) when a(er)

re,where rL+ is the larger solution of a (r1 ) = a(r1 ) if any. Note here that a (r1 ) =a(r1 ) is equivalent tog(r1 )

2
r1[1=q + ()]r1 +(1=q) = 0,which has at most two solutions. Note that a (r1 ) > a(r1 ) , g(r1 ) > 0.
28

+In case (i), the equilibrium limit interest rate is r1 2 (max[br; rL ]; min[rH; re]],+
where rL is the smaller solution of g(r1 ) = 0 and rH
is a solution of a (r1 ) = a(r1 ),which is unique because a ( ) is strictly decreasing and a( ) is strictly increasing.
; re) exists by the Interme-

Note that if a(er) > a (er), then such an rL 2 (diate Value Theorem because and limrt #a (r1 ), and thea(rt ) = 0, limr1 #monotonicities of a( ) and of a ( ). It is veri…ed that rL+ 2 (er; 1= ) exists becauser) > a (er)
a( ) is bounded below q and limr1 "1= a (r1 ) = 1. Now, it is seen that a(eis equivalent tof( )) 2 +[(1

2+(1+ =q) +(1+1=q)

1]

[1

=q +

Here, it is veri…ed that f (1) = (1 + )2 =q > 0, f 0 (1) = (121) + =q > 0, and f ( ) = [letting+and+ (1 + 2=q)

2(1

1=q)

(1 1=q)] < 0.
)+ (+ =q +

2(1)]. Thus,

be de…ned by the larger and the smaller solution of f ( ) = 0,

respectively,22 one can verify that f ( ) < 0 if and only if max[

; ]<

<

+

.

If case (ii) happens, then the equilibrium limit interest rate is r1 2 (max[br; rL ]; rL+ ].

For rL+re to exist, the determinant of g(r1 ) = 0 should be positive, which impliesh(z)where z=q so that 1

1

for z

1

)2

2 [

2

z 2 + 2 ( + )z +

(

)2

4

> 0,

=2. Note that h0 (z) = 2z + 2 ( + ) > 0

z<1. Thus, it is deduced that h(z) > 0 , h(1) = (1)2 +

2(+ ( + )] > 0.

Summarizing the argument above, I obtain the following proposition. Figure 3
illustrates one case of this equilibrium.
Proposition 6. If max[
(1
22

)2 +

+

; ]<

<

2

)2

(

or if
2 [

+ ( + )] > 0

There always exist two solutions for f ( ) = 0 since the determinant is[2+ (1 + =q) + (1 + 1=q)

which is positive because 1=q > 0 and q

1]2 + 4 (1
1.

29

)[(1

=q) +

(1

1=q)],

Figure 3: One case of the economy where all …nancing is intermediated.

and rb( ; ; ) < rL+ ( ; ; ; q), then there exists a steady-state equilibrium with+
r1 2 (max[rL ; rb], min[rL+ ; rH
; re]]. In this equilibrium, the wealth of entrepreneurs is

a (r1 ), and that of lending households is a (r1 ).

Bold lines on the wealth levels and on the interest rate in Figure 3 depict acontinuum of the steady-state equilibria where all entrepreneurs rely on …nancial
intermediation and r1 2 (rL ; re].

Now, consider the capital market clearing condition. Letting XL (r1 ) be thefraction of net lenders in the economy (“L” connotes the interest rate is low), I
obtainXL (r1 )a (r1 ) + [1

XL (r1 )]a (r1 ) = [1XL (r1 )]which implies
XL (r1 ) =

[1

(1
(1
30

r1 )
.
)](r1 + )1

a (r1 ),=(r1 + )It can be shown that @XL (r1 )=@r1 =

(1 +

)=[1

(1

)](r1 + )2 < 0,which means that the greater the limit interest rate is, the less households are net
lenders, which seems a natural consequence.
Let the aggregate wealth in the economy when r1 2 (N WL (r1 ). Then, it is shown that

N WL (r1 ) =+ r1 +:(1)](r1 + )[1

; re] be denoted by

Similarly, the GINI index for wealth inequality in the steady state is given by
XL (r1 )a (r1 )N WL (r1 )XL (r1 )a (r1 )+ 1 [1 XL (r1 )]]N WL (r1 )XL (r1 )[1 XL (r1 )][a (r1 ) a (r1 )].
=N WL (r1 )W ealthGIN IL (r1 ) = 1

4.2.2

XL (r1 )Equilibria where Not All Financing is IntermediatedNow, I consider the case of re < r1

. There are three possible …xed points,1

a (r1 ), a (r1 ) and a1 if the following is satis…ed:a (r1 ) < a(r1 )a (r1 ) < q

a1 .

Thus, I need to ask whether there exists r1 2 (er; 1] that satis…es the above

relationship. First, it must be the case that=(1which is rewritten as)q,q=(1 + q). Recall now the restriction of the …xed cost(1

q < 2) and the …rst part of Assumption.3 (

q=2). It is veri…ed thatq=2

q=(1 + q). Thus, this condition actually always holds under the assumptions.
Since a (r1 ) is strictly increasing, there are two cases for r1 2 (er; 1consist of a steady state equilibrium: (i) when a(1holds, and (ii) when a(1)

solution of g(r1 ) = 0 if any.

a (1) and re < rL+
31) > a (11

) = a (1] to), where rL+ is the larger

In case (i), it is veri…ed that a(1=fq[1)]g, or(1< (1) is equivalent to) > a (1)q=[1 + (1< 1)q]. The equilibrium limit)(1+interest rate is r1 2 (max[rH ; rL ], rH], where rH is de…ned bya (rH ) = q.
+++
Recall that rH
is de…ned by a (rH) = a(rH). Since a ( ) is strictly decreasing,it is shown that a (r1 ) < q for r1 > rH . It is also shown that a (r1 )r1+
rH. Indeed, one can derive explicit forms of solutions:8(1=q) + (1
=q)
(1
)
>
>
r =
, and
>
>
< H
1 (1 + 1=q) + ( + )

>
>
>
+
>
=
: rH

1

a(r1 ) for(1 + 1=q)+,(1 + 1=q) + ( + )+where rH < rH
always holds.

If case (ii) happens, then the equilibrium limit interest rate is r1 2 (max[rH ; rL ]; rL+ ].

As in the last subsection, for rL+
2

)2

(

the following proposition.
Proposition 8. If q=2
(1

)2 +

(

)2 +

+ ( + )] > 0. Summarizing the arguments so far, I obtain2 [2

re to exist, it must be the case that (1

)2

< (1
2 [

)q=[1 + (1)(1)q] or if+ ( + )] > 0, then there exists a steady-state+
equilibrium with r1 2 (max[rL ; rH ], min(rL+ ; rH
;1)]. In this equilibrium, thewealth of entrepreneurs who self-…nance is a1 , that of entrepreneurs who borrowfrom an FI is a (r1 ), and that of lending households is a (r1 ).
Figure 4 illustrates one case of this equilibrium. Bold lines on the wealth levels
and on the interest rate depict a continuum of the steady-state equilibria with r1 2+
(rH ; rH].

Now, consider the capital market clearing condition. Letting XH (r1 ) andYH (r1 ) be the fractions of net lenders and of entrepreneurs who rely on intermediated borrowing, respectively (“H ”connotes the interest rate is high), I obtain
XH (r1 )a (r1 ) + YH (r1 )a (r1 ) + [132

XH (r1 )YH (r1 )]a1

Figure 4: One case of the economy where not all …nancing is intermediated.

= YH (r1 )1a (r1 )
+ [1=(r1 + )XH (r1 )YH (r1 )]a1 ,which implies1

r1

XH (r1 ) =[1+ ( + )]r1 + [1(1

YH (r1 ).

z)]

Together with XH (r1 ) + YH (r1 ) = G0 (q), it is veri…ed that
XH (r1 ) =G0 (q)(1
r1 )
.
[1
(1
)](r1 + )It can be shown that @XH (r1 )=@r1 =G0 (q)(1+

)=[1

(1

)](r1 + )2 <0, which means that the greater the limit interest rate is, the less households are
net lenders, which seems, again, a natural consequence.
Let the aggregate wealth in the economy when r1 2 (er; 1

N WH (r1 ). Then, I obtain
N WH (r1 ) =1

G0 (q)(1
r1 )[1(1)](r1 + )33

] be denoted by[1+(r1 + )+ ( + )]r1 + [1
(1

1

(1

)]

G0 (q)).

It is not easy, however, to determine whether N WH (r1 ) is larger than N WL (r1 ).
Meanwhile, one can compare the fractions of lending households. It is veri…ed that
XL > XH because it is equivalent to
[1

(1
(1

r1 )
G0 (q)(1
r1 )>,)](r1 + )[1
(1)](r1 + ); re], and the one in the right hand side

where r1 in the left hand side is r1 2 (is r1 2 (er; 1]. Thus, more poor lending households are there in the equilibrium; re] than in the equilibrium with r1 2 (er; 1with r1 2 (]. The GINI index

for wealth inequality is given by
W ealthGIN IH (r1 ) =XH (r1 )YH (r1 )[a (r1 ) a (r1 )]N WH (r1 )+[1 XH (r1 ) YH (r1 )]a (r1 )) + YH (r1 )(a1
XH (r1 )(a1
N WH (r1 )a (r1 ))

Finally, notice that from Propositions 6 and 8, we know that if max[
<

+

and

> max[q=2; (1

.

; ]<

)] (and Assumptions 2 and 3), there)=[ + (1can exist two continua; one is with r1 2 (max[br; rL ], min[er; rL+ ]], and the other+with r1 2 (rH , min(rH;1)). Since> 1=2, it is veri…ed that (21= ) lies

between 0 and 1. This situation is depicted in Figure 5. Recall for a high monitoringe¤ect (+), there does not exist a steady-state equilibrium with r1 2 (; re]because it contradicts with the de…nition of the steady-state. Thus in this region,r; rL+ ]] is permitted as a steady-state equilibrium. If
only r1 2 (max[br; rL ], min[ethe cost of monitoring is so low that+
r1 2 (rH , min(rH;1<21= , then the equilibrium with)) does not arise; all entrepreneurs want to borrow from an

FI because the borrowing rate is low enough.

34

Figure 5: One case of two continua of multiple steady-state equilibria.

5

The E¤ects of Capital Account Liberalizationon the Wealth Distribution

In this section, I consider the e¤ects of capital account liberalization on the wealthdistribution. By capital account liberalization (CAL), I mean the situation where
agents in this economy get access to the international capital market populated withrisk neutral foreign investors whose opportunity cost is rW > 0. As such, agents inthis economy takes this interest rate as given. I assume that CAL does not change
the parameters of the model, and consider the steady-state of the economy after
CAL is introduced.
First, suppose that the economy is in the steady-state with r1 2 (max[br; rL ],

min[rL+ ; rL+ ; re]], and that the world interest rate is rW 2 (br; r1 ) (note that if rW < rbthen after CAL the economy does not go to a steady-state; borrowing entrepreneurs
get in…nitely wealthier). From Figure 3, it is easy to see that borrowing entrepre-

neurs support this regime change, while lending households are against it, as longas it does not make the threshold a(r1 ) greater than a (r1 ). Note here that it is35

implicitly assumed that the FI sector still behaves competitively, so that they haveto reduce the deposit rate to rW as well as the lending rate to rW + . If majority
voting is necessary for the regime change and XL (r1 ) > 1=2, then this economy does
not implement CAL. However, if entrepreneurs can engage in lobbying, then CAL
could be implemented. If the world interest rate is rW > r1 , then entrepreneurs
want to oppose to this regime change, while lending households welcome it.
Next, suppose that the economy is in the steady-state with r1 2 (max[rL ; rH ],+min[rL+ ; rH
;1]], and that the world interest rate is rW 2 (rH ; r1 ). From Figure 4,

it is clear that entrepreneurs borrowing from an FI welcome CAL, lending households
are against it, and self-…nancing entrepreneurs are neutral. However, if the worldinterest rate is rW 2 (max[br; rL ], min[rL+ ; rL+ ; re]], then all self-…nancing entrepreneur
now support CAL because their successors (i.e., their sons) will eventually becomericher. Whether CAL is implemented or not depends on the political decisionmaking system as well as the steady-state fractions of lending households and ofentrepreneurs (self-…nance or borrowing). Also, in this model, …nancial sectors,
whether domestic or foreign, have a passive role because of the assumed perfectcompetition. If they are gaining rent due to the regulation or to the increasing
return to scale of monitoring, then they will play a substantial role in the politicaldecisions for CAL.

6

Concluding Remarks

In this paper, I have constructed a simple dynamic model to investigate the long-run
relationship between types of …nancing (self-…nance or intermediated …nance) andthe wealth distribution in an economy. Speci…cally, I have provided the steady-statecharacterization to see which …nance pattern prevails in the economy as well as thecharacteristics of the wealth distribution. For any steady-state, the wealth inequality is higher and the interest rate is lower under the economy where intermediatedcapital is dominant. It is shown that for a wide range of parameters (concerning
the bene…t and the cost of monitoring) there are two continua of the steady-state
equilibria; one is where all entrepreneurs rely on …nancial intermediation, and the36other is where some borrowing entrepreneurs self-…nance. The source of multiplicity is based on self-ful…lling prophecy; because the interest rate is low few richer
borrowing entrepreneurs can borrow more, which makes the low interest rate is selfful…lling due to the existence of many poor lending households and the opposite
situation is also self-ful…lling. The e¤ects of capital account liberalization on thewealth distribution was also discussed.
There still remains important questions unanswered in the present paper. In
particular, how do …nancial crises a¤ect the change in …nancial systems as a whole?
How important are political processes for the evolution and changes in …nancial
systems? Looking back at historical experiences (see, e.g., Allen and Gale (2000,
Ch.2), Allen and Gale (2004a) and Bolton (2003)), we observe that it is often the casethat characteristic features of …nancial systems change only if …nancial crises occur,
regardless of whether they are due to domestic factors or to international pressure.
It thus might suggest that …nancial crises have a positive e¤ect on the working ofan economy, of course with an inevitable turmoil, though. Analytical framework isnecessary to investigate the issue of how political elements matter to the evolution
and changes in …nancial systems (however, see, e.g., Bolton and Rosenthal (2002)).
Another interesting issue would be on the e¤ect of international technologicaldi¤usion on the evolution and changes in domestic …nancial systems. What is therelationship between international technological di¤usion and domestic evolution of…nancial systems? Based on their (1999) analysis, Allen and Gale (2000, Ch.13) suggest that the less innovative industry an economy has the bank-oriented system that
economy adopts. However, domestic industrial progress and …nancial characteristics in one economy might be a result of international technological di¤usion rather
than of its original endowment. Naturally, this issue should be investigated witha full-‡edged dynamic model. There has been an extensive literature investigating
the issue of technological di¤usion and economic growth. The literature, however,is silent on the di¤erences in …nancial systems. These and other issues concerning
the evolution and changes in …nancial systems are left for future research.

37

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