a welfare analysis of financial development
TRANSCRIPT
A Welfare Analysis of Financial Development
Jurgen von Hagen∗ and Haiping Zhang†
September 2008
Abstract
This paper analyzes the long-run and short-run welfare implications of finan-
cial development to different individuals in an overlapping generations model of the
closed economy with financial frictions. Financial development enables the produc-
tive individuals to borrow and invest more. As a result, the improvement in resource
allocation makes aggregate production more efficient.
The productive individuals may benefit or lose from financial development in
the long run as well as in the short run, depending on the initial degree of financial
development and the magnitude of its change. The unproductive individuals benefit
in the long run from the improvement in the lending opportunity. However, they
may lose in the short run, if the initial degree of financial development is very low.
The short-run and long-run gains and losses from financial development in
the intra- and intergenerational dimensions play an important role in determining
whether and how public policies may achieve Pareto improvement in the process of
financial development.
JEL Classification: E44, F41
Keywords: Financial frictions, Financial Development, Welfare Analysis
∗University of Bonn, Indiana University and CEPR. Lennestrasse. 37, D-53113 Bonn, Germany.
E-mail: [email protected]†Corresponding author. School of Economics, Singapore Management University. 90 Stamford Road,
Singapore 178903. E-mail: [email protected]
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1 Introduction
Various forms of financial contracts, intermediaries, and markets emerge and evolve over
time to improve resource allocation by acquiring information, enforcing commitments,
and facilitating transactions (Levine, 2006). Different types and combinations of infor-
mation, enforcement and transaction costs have motivated distinct financial contracts,
intermediaries, and markets across countries with different legal, regulatory, and market
structures (Porta, de Silanes, Shleifer, and Vishny, 1997, 1998).
Financial development facilitates saving and investment, which fosters economic growth
unambiguously. However, theory provides opposite predictions on the welfare implications
of financial development to different individuals. Some claim that by ameliorating infor-
mation and transaction costs and allowing more credit-constrained individuals (who are
normally the poor) to obtaining external finance, financial development will have a dis-
proportionately beneficial impact on the poor (Aghion and Bolton, 1997; Banerjee and
Newman, 1993; Galor and Zeira, 1993), while others argue that it is primarily the rich and
politically connected who benefit from financial development, especially at early stages of
economic development (Haber, 2005; Lamoreaux, 1994). Greenwood and Jovanovic (1990)
show how the interaction between financial and economic development can give rise to an
inverted U-shaped curve of income inequality and financial intermediary development.
We develop an overlapping generations model with financial frictions and analyze the
long-run and short-run welfare implications of financial development to different individ-
uals in the intra- and inter-generational dimensions.
Due to financial frictions, individuals with productive projects can borrow only a frac-
tion of their future output. The constrained credit demand depresses the loan rate and
some resources are inefficiently allocated to unproductive projects. In other words, in-
vestment in the unproductive projects can be considered as the potential credit supply
which is larger for a smaller degree of financial development. Financial development ame-
liorates information asymmetry on the credit market and enables productive individuals
to borrow against a larger fraction of their future output.
The improvement in resource allocation enhances production efficiency. On the one
hand, the increase in the credit demand of those productive individuals tends to push
up the loan rate. On the other hand, the increase in aggregate output reduces the price
of output and makes the projects of unproductive individuals less profitable. Thus, un-
productive individuals prefer to supply more credit instead of investing into their own
unproductive projects. The increase in the credit supply tends to reduce the loan rate.
Given a small initial degree of financial development, potential credit supply is abundant.
For a marginal improvement in the financial sector, the increase in the credit supply may
dominate that in the credit demand and the loan rate may decline. While, given a large
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initial degree of financial development, potential credit supply is scarce. For a marginal
improvement in the financial sector, the increase in the credit demand may dominate
that in the credit supply and the loan rate rises. In this sense, the loan rate may have a
U-shaped pattern with respect to financial development.
The equity rate is defined as the rate of return on the own funds of productive individ-
uals invested in their projects. By borrowing at the loan rate lower than the equity rate,
productive individuals benefit from the leveraged investment. The equity rate is affected
positively by the debt-equity ratio and the relative price of output, negatively by the loan
rate. Financial development monotonically raises the debt-equity ratio, reduces the price
of output and has the U-shaped effect on the loan rate, as mentioned above. Its net effect
on the equity rate depends on the interactions of the three factors. Overall, the equity
rate has a hump-shaped pattern with respect to financial development.
Financial development may have opposite welfare implications on different individuals
in the intra- and intergenerational dimensions, depending on the initial degree of financial
development and the magnitude of its change. Unproductive individuals acting as lenders
in our model benefit in the long run from financial development due to the increase in
lending opportunity and the wage income. However, the welfare gains of the unproduc-
tive individuals in the later generations may come at the cost of the welfare losses of
the unproductive individuals in the earlier generations. The long-run welfare pattern of
productive individuals with respect to financial development is driven mainly by that of
the equity rate. That is, if the financial sector is initially underdeveloped, productive
individuals benefit in the long run from financial development due to the increase in the
equity rate and the wage income; if the financial sector is moderately developed, they lose
from financial development because the decrease in the equity rate dominates the rise in
the wage income. Such opposite welfare implications in the intra- and intergenerational
dimensions may motive us to consider relevant public policies for Pareto improvement in
the process of financial development.
The rest of this paper is organized as follows. Section 2 describes the model economy.
Section 3 analyzes the long-run efficiency and welfare effects of financial development.
Section 4 discusses the short-run dynamics with respect to an marginal improvement in
the financial sector under different initial degrees of financial development. Section 5
summarizes the main findings. Appendix collects some technical issues.
2 Model
The basic framework is the overlapping generations model with two-period lives a la
Diamond (1965) and Bernanke and Gertler (1989). There are two goods in the economy:
the final good and the intermediate good. The final good can be either consumed or used
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as capital to produce intermediate goods. The final good is chosen as the numeraire and
vt denotes the price of intermediate good in period t. There is no aggregate uncertainty
in the model economy. There is no population growth and the population size of each
generation is normalized to one. Each generation consists of two types of individuals, i.e.,
entrepreneurs and households, each of mass η and 1− η, respectively.
Individuals born in period t have the identical additive logarithm preference as follows,
U jt = ρ ln cj1,t + β ln cj2,t+1 + φ ln bjt+1, (1)
where the superscript j ∈ h, e denotes household or entrepreneur, respectively; cj1,t and
cj2,t+1 denote the consumption of individual j when young and when old, respectively; bjt+1
denotes the bequest given to their respective offsprings in period t + 1. In other words,
parents derive utility directly from the size of the bequest.1
When young, individuals are endowed with a unit of labor and earn the wage income
as well as receive the bequest from their parents contemporaneously. At the end of the
first period of life, individuals invest final goods as capital in their respective projects. At
the beginning of the next period, their projects produce intermediate goods and capital
depreciates at a rate of δ after production. Then, intermediate goods are used together
with the labor of the young generation to produce final goods contemporaneously.
By assumption, the project of entrepreneurs is more productive than that of house-
holds. Thus, entrepreneurs prefer to borrow from households for the project investment.
Due to limited commitment problem, they can only borrow against a fraction of their
future project outcome. The efficiency of the legal and financial institutions ultimately
determines the strictness of the borrowing constraints, i.e., entrepreneurs can borrow
against a larger fraction of their future project outcome in the country with better pro-
tection of creditors, more efficient legal system, and more liquid asset market, etc.
2.1 Households
A representative household born in period t receives the bequest bht , earns the wage income
wt, consumes ch1,t, invests kht in his project, and lends dt = bht +wt − kht − ch1,t at the gross
loan rate of rt. His project has the decreasing return to scale and produces G(kht ) units
of intermediate goods in period t+ 1, where G′(kht ) > 0, G′′(kht ) < 0, and G′(0) = R. As
shown later, the decreasing-return-to-scale project helps generate an upward-sloping credit
supply curve in equilibrium. Besides the project revenue vt+1G(kht ) and the depreciated
capital (1 − δ)kht , the household also receives the loan repayment rtdt in period t + 1.
Then, the household consumes ch2,t+1 and leaves bequest bht+1 to his offspring.
1Barro (1974) assumes that parents care about the lifetime utility of their children instead of the
size of the bequest. See Andreoni (1989) for more detailed descriptions of the bequest motive and the
warm-glow utility function.
4
In period t, the household chooses kht , ch1,t, c
h2,t+1, b
ht+1 to maximize their lifetime utility
(1) subject to the life-time budget constraints,
ch1,t +ch2,t+1
rt+bht+1
rt= wt + bht +
vt+1G(kht ) + (1− δ)khtrt
− kht . (2)
The first order conditions are,
rt = vt+1G′(kht ) + (1− δ), ch1,t =
ρ
ρ+ β + φ(wt + bht + Ψt), (3)
bht+1 =rtφ
ρ+ β + φ(wt + bht + Ψt), ch2,t+1 =
rtβ
ρ+ β + φ(wt + bht + Ψt), (4)
where Ψt ≡ vt+1G(kht )+(1−δ)khtrt
− kht denotes the discounted profit of the household project.
2.2 Entrepreneurs
A representative entrepreneur born in period t finances the project investment ket using
own funds nt = wt + bet − ce1,t and loans zt = ket −nt. In period t+ 1, the project produces
Rket units of intermediate goods; after repaying the debt of rtzt, the entrepreneur consumes
ce2,t+1 and leaves bequest bet+1 to his offspring.
Given the assumption of G′(0) = R and G′′(kht ) < 0, the entrepreneur’s project has
a larger marginal product than that of the household, vt+1R > vt+1G′(kht ), Thus, the
entrepreneur prefers to finance the project investment using loans. Due to limited com-
mitment problem, he can only borrow against a fraction of the future project outcome,
rtzt ≤ θt(Rvt+1 + 1− δ)ket . (5)
Following Matsuyama (2004, 2007, 2008), we use θt ∈ [0, 1] to measures the degree of
financial development in the economy. θt is higher in the country with more sophisticated
financial and legal system, better creditor protection, and etc.2 Financial development is
modeled here as an increase in θt. The equity rate in period t is defined as the rate of
return to the own funds of the entrepreneur invested in period t,
Γt ≡(Rvt+1 + 1− δ)ket − rtzt
ket − zt= rt + [(Rvt+1 + 1− δ)− rt]
ketnt≥ rt, (6)
which should be no less than the loan rate; otherwise, the young entrepreneur would rather
lend than borrow. It can be considered as the entrepreneur’s participation constraint. The
2The pledgeability, θ, can be argued in various forms of agency costs story, e.g., the inalienability
of human capital of entrepreneurs by Hart and Moore (1994) or costly state verification by Townsend
(1979), or unobservable project (effort) choices by Holmstrom and Tirole (1997). See Tirole (2006) for a
comprehensive overview of different models of financial contracting. This paper analyzes the implications
of financial development on the borrowing constraints of different individuals. Thus, we choose the
simplest form of borrowing constraints.
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participation constraint is equivalent to rt ≤ Rvt+1 + 1 − δ. Intuitively, only if the loan
rate is lower than the marginal outcome of the entrepreneur’s project, the entrepreneur
would like to finance his project using loans and the equity rate is higher than the loan
rate; if the loan rate is equal to the marginal outcome of the entrepreneur’s project, the
equity rate is equal to the loan rate and the entrepreneur may not borrow to the limit.
In period t, the entrepreneur chooses ket , zet , c
e1,t, c
e2,t+1, b
et+1 to maximize his life-time
utility (1) subject to the period budget constraints (7) and (8), the borrowing constraints
(5) and the participation constraints (9):
ce1,t + ket = wt + bet + zt, (7)
ce2,t+1 + bet+1 + rtzt = (Rvt+1 + 1− δ)ket , (8)
rt ≤ Rvt+1 + 1− δ. (9)
Note that only one of the two constraints (5) and (9) is strictly binding in equilibrium.
The equilibrium conditions are,
Γt =
1−θt1
Rvt+1+1−δ−θtrt
, if rt < Rvt+1 + 1− δ,
rt if rt = Rvt+1 + 1− δ.
, ce1,t =ρ
ρ+ β + φ(wt + bet), (10)
bet+1 =Γtφ
ρ+ β + φ(wt + bet ), ce2,t+1 =
Γtβ
ρ+ β + φ(wt + bet). (11)
Intuitively, if rt < Rvt+1 + 1− δ, the entrepreneur prefer to borrow to the limit. For each
unit of capital invested in period t, the marginal revenue in period t+1 amounts to Rvt+1
and the post-production value of capital is 1−δ. The entrepreneur finances his investment
using θt(Rvt+1+1−δ)rit
units of loan and 1− θt(Rvt+1+1−δ)rit
units of own funds in period t. After
repaying the debt in period t + 1, the entrepreneur gets (1 − θit)(Rvt+1 + 1 − δ) units as
the net return. Thus, the equity return is defined as the ratio of the net return over the
net worth, Γt = (1−θt)(Rvt+1+1−δ)1− θt(Rvt+1+1−δ)
rt
= 1−θt1
Rvt+1+1−δ−θtrt
. If rt = Rvt+1 + 1 − δ, the entrepreneur
may not borrow to the limit and according to equation (6), the equity rate is equal to the
loan rate, Γt = rt = Rvt+1 + 1− δ.
2.3 Aggregate Production and Market Equilibrium
In period t, final goods are produced from intermediate goods Mt and the labor input of
young generation L = 1. The wage rate and the price of intermediate good are equal to
their respective marginal revenue,
Yt = Mαt L
1−α, where Mt = ηRket−1 + (1− η)G(kht−1), and L = 1, (12)
vtMt = αYt, and wtL = (1− α)Yt. (13)
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Aggregate capital stock is defined as follows,
Kt = ηket + (1− η)kht . (14)
The credit market and goods market clear in period t,
ηzt = (1− η)dt or η[ket + ce1,t − (wt + bet )] = (1− η)[wt + bht − (kht + ch1,t)], (15)
Yt = η(ce2,t + ce1,t) + (1− η)(ch2,t + ch1,t) +Kt − (1− δ)Kt−1. (16)
Definition 1. Given the degree of financial development θt, market equilibrium is a set
of allocations of households, kht , ch1,t, ch2,t, bht , entrepreneurs, ket , zt, ce1,t, ce2,t, bet, aggre-
gate variables, Yt, Kt,Mt, wt, vt, together with the loan rate and the equity rate rt,Γt,satisfying equations (3)-(5), (9)-(15).
3 Long-Run Analysis of Financial Development
This section analyzes how financial development affects production efficiency and welfare
of households and entrepreneurs in the long-run. The mechanism driving our results
is discussed first in the baseline model where capital is fully depreciated, δ = 1, and
individuals only care about consumption when old, ρ = φ = 0, β = 1. Afterwards, it
is shown that adding other elements into the baseline model does not change the core
results qualitatively.
3.1 The Baseline Model
Proposition 1. Let θU ≡ 1− η denote the threshold value of financial development. For
any θt ∈ [θU , 1], economic allocation is independent of θ and efficient in the long run in
the sense that entrepreneurs are not credit constrained, intermediate goods are produced
only by entrepreneurs in the steady state, kht = 0, and the loan rate is equal to the equity
rate at r = Γ = Rv.
Proof. Let θU denote the threshold value where intermediate goods are only produced by
entrepreneurs, kht = 0, and the entrepreneur’s borrowing constraint (5) is binding. In this
case, the loan rate is equal to the equity rate at the threshold, rt = vt+1G′(0) = vt+1R = Γt.
The credit market clearing implies D = (1 − η)w = Z = ηz. Aggregate investment is
only undertaken by the young entrepreneurs, I = w = ηke. Given per capita investment
and borrowing of young entrepreneurs, ke = wη
and z = (1−η)wη
, the binding borrowing
constraint rz = θURvke implies (1−η)wη
= θUwη
, or θU = (1− η).
In the baseline model, the wage income wt is the crucial variable that determines the
model dynamics. Consider the frictionless case where financial development is above the
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threshold value, θ ≥ 1− η, and aggregate production is efficient. In aggregate, the wage
income of the young generation, wt, is all invested into the entrepreneurs’ project in period
t and Rwt units of intermediate goods are produced in period t+1. Then, final goods are
produced from intermediate goods and labor in period t + 1, Yt+1 = (Rwt)α. The wage
income dynamics is determined by, wt+1 = (1−α)Yt+1 = (1−α)(Rwt)α. Given α ∈ (0, 1),
the phase diagram of the wage income is concave and it has a unique cross-point with
the 45 degree line for w > 0, i.e., w = [(1− α)Rα]1
1−α . The slope of the phase diagram at
the steady state is α < 1. Therefore, the model economy converges globally to the steady
state which is unique.
Proposition 2. For θt ∈ [0, θU), aggregate production is inefficient in the long run in the
sense that entrepreneurs are credit constrained and households have the positive project
investment, kht > 0. Financial development results in the monotonic decline in the house-
hold project investment,∂kht∂θt
< 0, and the monotonic increase in the wage income, ∂wt∂θt
> 0,
in the long run.
Proof. Given young individuals invest all their wage income into the project in period t,
wt = ηket + (1 − η)kht , and the equilibrium loan rate is rt = vt+1G′(kht ), the borrowing
constraints of entrepreneurs can be reformulated into
θt
[η
(1− η)
wt(wt − kht )
+ 1
]=G′(kht )
R. (17)
Consider the left- and right-hand sides of equation (17) as two functions of kht , given the
wage income and the degree of financial development. Graphically, the right-hand side of
equation (17) is a monotonically downward-sloping curve with an intercept of one on the
vertical axis, while the left-hand side of equation (17) is a monotonically upward-sloping
curve with an intercept of θt1−η . Given θ ∈ [0, 1 − η), the two curves must intersect once
and only once at kht > 0 and the equilibrium loan rate is lower than the marginal revenue
of the entrepreneurs’ project and the equity rate, rt = vt+1G′(kht ) < vt+1R < Γt. It verifies
that the borrowing constraints of entrepreneurs are binding.
Aggregate output of intermediate goods is Mt+1 = Rwt−Λt, where Λt ≡ (1−η)[Rkht −G(kht )] defines the direct efficiency loss due to the inefficient project investment of house-
holds. Since dΛtdkht
= (1 − η)[R − G′(kht )] > 0, the direct efficiency loss is larger for an
increase in kht . The wage income dynamics is determined by wt+1 = (1− α)(Rwt − Λt)α.
Comparing with the frictionless case of θ ∈ [1 − η, 1], the phase diagram of the wage
income is lower in the case of financial underdevelopment which further reduces aggregate
investment and output. It can be considered as the indirect efficiency loss.
We use proof by contradiction to show∂kht∂θt
< 0. Suppose an increase in θ results in
the increase in kht . The direct efficiency loss rises due to dΛtdkht
> 0, which then leads to a
lower wt in the steady state. The rise in kht leads to a decline in the right-hand side of
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equation (17). The increase in θt and the increase inkhtwt
jointly raise the left-hand side of
equation (17). The borrowing constraint does not hold. Thus, the presumption is false
and∂kht∂θt
< 0 must be true. Since θ has the monotonic negative effect on kht and hence on
Λt, it has monotonic positive effect on the wage income, ∂wt∂θt
> 0, in the long run.
The slope of the phase diagram of the wage income is
dwt+1
dwt= (1− α)α [Rwt − (1− η)Λt]
α−1R
[1− (1− η)
(1− G′(kht )
R
)dkhtdwt
](18)
For wt → 0, kht → 0. Thus,G′(kht )
R→ 1 and Λt → 0. The second square bracket on
the right hand side converges to one; given α ∈ (0, 1), the first square bracket on the
right-hand side is converging to infinity. Thus, the slope of the phase diagram of the
wage income converges to infinity for wt → 0. In order to prove the existence and the
uniqueness of the steady state, it is sufficient to show that wt+1 is a concave function of
wt. Since wt+1 = (1− α)Mαt+1 is a concave function of Mt+1, it is sufficient to show that
Mt+1 = Rwt − Λt is a concave function of wt or, −Λt is a concave function of wt. Since
−Λt is a concave function of kht , it is sufficient to show that kht is a concave function of
wt in order to prove that wt+1 is a concave function of wt.
Take θt as given, equation (17) shows that kht is a function of wt. We use proof by
contradiction to showdkhtdwt
> 0. Suppose an increase in wt would result in the decrease
in kht . The right-hand-side value of equation (17) would rise due to the concavity of the
household project, while the left-hand-side value would decline. The borrowing constraint
would not hold, which is false. Thus,dkhtdwt
> 0.
Since the borrowing constraint is always binding in the constrained equilibrium, the
equality should also hold if we take the first derivative of both sides with respect to wt,
θη
(1− η)
wt(wt − kht )
2
(dkhtdwt
− khtwt
)=G′′(kht )
R
dkhtdwt
. (19)
Given G′′(kht ) < 0 anddkhtdwt
> 0, we getdkhtdwt
<khtwt
, which implies that kht is a concave
function of wt. Thus, the steady state of the constrained equilibrium is unique and stable.
3.1.1 A Numerical Example
Our paper intends to provide a conceptual framework to think about the efficiency and
welfare implications of capital account liberalization. Thus, we focus here more on its
qualitative results instead of its quantitative relevance. As an analytical solution is not
obtainable, we use a numerical example to show the intuition explicitly.
We set α = 0.6 so that the wage income of the young generation accounts for 40% of
aggregate income. Note that capital in our model not only includes physical capital but
9
also human capital. Intuitively, When young, individuals are unskilled and the wage in-
come of unskilled labor accounts for 40% of aggregate income; while when old, individuals
become skilled labor and they are paid mainly for their human capital which accounts
for 60% of aggregate income. In other words, young individuals in the aggregate invest
their wage income for the revenue of intermediate goods at the average gross return ofα
1−α = 1.5 in the steady state. Our qualitative results are independent of the parameter
value of α. The parameter values of R and η do not matter for our qualitative results.
We normalize R = 1 implying that entrepreneurs produce intermediate goods one-to-one
from final goods. We set η = 0.2 implying that entrepreneurs account for 20% of the
population.
The household project takes the following functional form, G(kht ) = Rkht −(kht )1+ψ
1+ψ,
where ψ > 0. Figure 1 compares the respective marginal product of the projects of
entrepreneurs and households, given R = 1 and ψ ∈ 0.25, 1. The horizontal axis denotes
the project investment. Given ψ, due to the decreasing return to scale, the difference in
the marginal product of entrepreneurs and households increases in the project investment
of households, d[R−G′(ih)]
dih> 0. Thus, for a larger initial project investment of households, a
marginal transfer of resource from households to entrepreneurs can generate larger output
gains due to the larger difference in the marginal products. Second, take the project
investment at 0.5 as an example. For ψ = 1, the difference in the marginal products of
entrepreneurs and households is measured by AB; for ψ = 0.25, the difference is measured
by AD, which is larger than AB. Thus, for a smaller ψ, a marginal transfer of resource
from households to entrepreneurs can generate larger output gains.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1A
B
D
E
G(i)ψ=1G(i)ψ=0.25
RiH
Figure 1: The Difference in the Marginal Products: ψ ∈ 0.25, 1
Figure 2 shows the phase diagram of the wage income under various degrees of financial
10
development, representing the case of financial autarky θ = 0 where no borrowing and
lending takes place, the case of financial underdevelopment θ = 0.4 where entrepreneurs
are subject to borrowing constraint, the frictionless case θ ≥ 0.8 where entrepreneurs
are not subject to borrowing constraint, respectively. The horizontal axis denotes the
wage income in period t and the vertical axis in period t+ 1. For more explicit graphical
illustration, we set ψ = 0.25.
0 0.02 0.04 0.06 0.08 0.1 0.120
0.02
0.04
0.06
0.08
0.1
0.12
θ=0
θ≥0.8θ=0.4
A
B
D
wt
wt+1
Figure 2: The Phase Diagram of the Wage Income: θ ∈ 0, 0.4, 0.8
According to Proposition 1, for θ ∈ [1 − η, 1], aggregate investment is undertaken by
entrepreneurs only. The phase diagram of the wage income has a concave form and the
model economy converges globally to the steady state denoted by point A. According to
Proposition 2, for θ ∈ [0, 1 − η), the direct and indirect efficiency losses shift down the
phase diagram for every initial value of the wage income and the steady state value of
the wage income rises in θ. Given the Cobb-Douglas aggregate production function, the
model economy converges globally to the steady state.
The parameter value of ψ does not affect our main results qualitatively but does affect
the loan-rate pattern with respect to financial development. See appendix A for detailed
discussion. If not specified, we set ψ = 1 as the default value.
We drop the time subscript of the relevant variables for investigating the long-run
steady state of the model economy. Figure 3 shows the steady-state values of endoge-
nous variables under θ ∈ [0, 1] which is denoted by the horizontal axis. Let ∆X ≡[X(θ∈[0,1])X(θ=0)
− 1]100 denote the percentage difference of variable X under the case of θ ∈
[0, 1] and under the case of autarky θ = 0.
In the case of financial autarky, θ = 0, entrepreneurs cannot borrow against their
future project outcome and have to finance their project investment using their wage
11
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
Individual Investment
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
Output and Social Welfare
0 0.2 0.4 0.6 0.8 11.4
1.45
1.5
1.55
1.6
1.65
Rates of Return
0 0.2 0.4 0.6 0.8 1
−1.95
−1.9
−1.85
−1.8
Individual Welfare
ie
ih
Γ
r
Ue
Uh
ΔUs
ΔY
Figure 3: Long-Run Allocation: θ ∈ [0, 1]
income, ke = w. The equity rate is simply the marginal product of their project, Γ = Rv.
Despite of the inactive credit market, the (underlying) loan rate is equal to the marginal
revenue of the households project, r = vG′(kh) = v(R−kh) = v(R−w) < vR = Γ, which
is smaller than the equity rate.
Financial development is modeled as an increase in θ which enables entrepreneurs to
borrow against a larger fraction of their future project revenue and expend their current
project investment. As long as r < vR, entrepreneurs always borrow up to the limit.
The improvement in resource allocation increases aggregate output of intermediate good
and final good. Given the constant labor input in the final good production, the rise in
aggregate input of intermediate goods increases the wage rate and reduces the price of
intermediate good.
According to Proposition 1, if financial development is above the threshold value,
θ ∈ [1 − η, 1], households do not invest in their own project but lend all their wage
income to entrepreneurs. According to Proposition 2, if financial development is below
the threshold value, θ ∈ [0, 1 − η), due to constrained credit demand, households have
positive project investment which is inefficient and can be regarded as potential credit
supply. Potential credit supply is larger if the degree of financial development is lower.
Financial development monotonically increases the loan rate in the long run, as shown
in the third panel of figure 3. Financial development affects both the credit demand and
the credit supply. On the one hand, the rise in θ enables entrepreneurs to borrow against
a larger fraction of their future project revenue and the rise in the credit demand tends
to push up the loan rate. On the other hand, the decrease in the price of intermediate
12
good makes the project investment less attractive for households and they lend more to
entrepreneurs rather than invest in own projects. The rise in the credit supply tends to
reduce the loan rate. Given the parameter value ψ = 1, the rise in the credit demand
always dominates the rise in the credit supply, and the loan rate rises monotonically in θ.
Appendix A shows that the parameter value of ψ may affect the loan rate pattern. How-
ever, it can be shown that the shape of the loan rate pattern does not affect qualitatively
the long-run welfare and efficiency implication of financial development.
Financial development has the non-monotonic impact on the equity rate in the long
run, as shown in the third panel of figure 3. We can decompose the equity rate by
substituting ket = nt + zt into equation (6) and rewriting the equity rate as follows,
Γt = Rvt+1 + (Rvt+1 − rt)ztnt. (20)
Intuitively, for each unit of net worth invested in the project, the entrepreneur can obtain
Rvt+1 as the marginal revenue of his own funds. Additionally, he can get ztnt
units of
loan. After repaying the debt at the loan rate rt, the entrepreneur can obtain the extra
return of (Rvt+1 − rt)ztnt
. Thus, the equity rate is affected by three factors: it rises in the
debt-equity ratio and the price of intermediate good but decreases in the loan rate.
As θ rises from 0 to θU , the debt-equity ratio and the loan rate increase and the price
of intermediate good declines monotonically. The net effect of financial development on
the equity rate depends on the interactions of the three factors. For a small initial value
of θ, the increase in debt-equity ratio dominates the rise in the loan rate and the decline
in the price of intermediate good so that the equity rate rises in θ. For a relatively large
initial value of θ, the rise in the loan rate and the decrease in the price of intermediate
good dominate the rise in the debt-equity ratio so that the equity rate decreases in θ. As
θ rises further, the equity rate and the loan rate tend to converge for θ → θU . See the
third panel of figure 3 for the hump-shaped equity-rate pattern.3
As shown in Proposition 1, when financial development is at its threshold value, θU ,
intermediate goods are produced only by entrepreneurs, kh = 0, and the loan rate is
equal to the equity rate, r = vG′(0) = vR = Γ. Any further increase in θ does not affect
economic allocation. Despite of the same form, Γ = vR, in the case of financial autarky
and in the frictionless case, the equity rate is lower in the latter case, due to the lower
price of intermediate good.
Financial development has a non-monotonic effect on the welfare of entrepreneurs,
as shown in the fourth panel of figure 3. In the case of financial autarky and in the
frictionless case, the equity rate has the same form, Γ = vR, and so has the period-2
consumption of entrepreneurs, ce = wΓ = wvR = (1− α)αY 2− 1α . Since aggregate output
3As shown in figure 9 in appendix A, the choice of the parameter value of ψ does not affect the
hump-shaped equity rate pattern qualitatively.
13
rises in θ, the period-2 consumption of entrepreneurs is higher in the frictionless case
than in the case of financial autarky, given α = 0.6. For θ ∈ [0, 1 − η], the wage income
has a monotonically increasing pattern while the equity rate has a hump-shaped pattern.
Overall, the welfare of entrepreneurs also has a hump-shaped pattern. In other words,
whether entrepreneurs benefit from financial development in the long run depends on the
initial degree of financial development and the magnitude of its change.
Financial development improves the welfare of households monotonically, as shown
in the fourth panel of figure 3. Intuitively, in the case of financial autarky, households
have the same wage income as entrepreneurs but the less productive projects. Their
project revenue is smaller than that of entrepreneurs, vt+1G(wt) < vt+1Rwt, and so is
their second-period consumption. In the frictionless case, households lend their wage
income to entrepreneurs at the loan rate equal to the equity rate and they have the same
second-period consumption as entrepreneurs. As mentioned above, entrepreneurs have
the higher second-period consumption in the frictionless case than in the case of financial
autarky and so do households. For θ ∈ [0, 1−η], whether households benefit monotonically
from financial development depends on four factors, i.e., the loan rate, the wage income,
the price of intermediate good, and Ωt given the households’ second-period consumption
cht+1 = rt(wt − kht ) + vt+1G(kht ) = rtwt + vt+1Ωt, where Ωt ≡ G(kht ) − G′(kht )kht > 0 and
dΩtdkht
= −G′′(kht ) > 0. As mentioned above, the increase in θ raises the wage income
and the loan rate but reduces the price of intermediate good and the household project
investment. Overall, the increases in the first two factors dominate the declines in the
last two factors and households benefit monotonically from financial development.
Social welfare defined as the weighted sum of households and entrepreneurs lifetime
utility, U s = ηU e + (1− η)Uh, rises monotonically in θ. See the second panel of figure 3.
3.2 Alternative Model Specifications
As shown in subsection 3.1, the interactions of the three factors give rise to the hump-
shaped pattern of the equity rate. The long-run welfare implications of financial de-
velopment to entrepreneurs are mainly driven by the hump-shaped equity-rate pattern.
In order to check the robustness of the hump-shaped equity-rate pattern in the baseline
model, we extend the baseline model by including partial depreciation of capital, including
consumption when young, and including the bequest motive, respectively.
3.2.1 Partial Depreciation of Capital
In the baseline model, capital depreciates fully δ = 1 after the production of intermediate
goods. In order to analyze the effect of partial depreciation of capital, we set δ = 0.5 and
keep other parameter values same as in the baseline model. Figure 4 shows the long-run
14
patterns of endogenous variables under θ ∈ [0, 1] which is denoted by the horizontal axis.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
Individual Investment
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
Output and Social Welfare
0 0.2 0.4 0.6 0.8 11.9
1.95
2
2.05
2.1
2.15
Rates of Return
0 0.2 0.4 0.6 0.8 1
−1.66
−1.64
−1.62
−1.6
−1.58
−1.56
−1.54
−1.52
Individual Welfare
ie
ih
Uh
Ue
ΔUs
ΔY
Figure 4: The Effect of Partial Capital Depreciation: δ = 0.5
Introducing partial depreciation of capital only affects the levels of endogenous vari-
ables but not their non-monotonic patterns. Intuitively, the wage income of the young
generation is the only intergenerational linkage in our model economy, which is affected by
aggregate output of intermediate goods produced by the old generation. Since financial
development improves capital allocation across individuals, aggregate output of interme-
diate goods rises and so does the wage income. Thus, the young generation benefits from
the positive income effect.
Partial depreciation of capital does not add new channels to the intergenerational
linkage. Despite that, each generation still benefits directly from the positive wealth
effects of residual value of capital. According to the equilibrium condition, r = vR+1−δ,the smaller depreciation rate results in the higher loan rate. However, the hump-shaped
pattern of the equity rate and the rising pattern of the loan rate are unaffected. As
a result, the long-run welfare implications of financial development to households and
entrepreneurs are qualitatively similar as in the baseline model.
3.2.2 Consumption in Two Periods
In the baseline model, individuals only consume when old, β = 1 and ρ = 0. In order
to analyze the effects of allowing consumption when young, we set ρ = 1 and keep other
parameter values same as in the baseline model. Figure 5 shows the long-run patterns of
some endogenous variables under θ ∈ [0, 1] which is denoted by the horizontal axis.
Letting individuals consume when young reduces the amount of capital invested for
15
0 0.2 0.4 0.6 0.8 1
0
0.02
0.04
0.06
0.08
Individual Investment
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
Output and Social Welfare
0 0.2 0.4 0.6 0.8 1
2.98
3
3.02
3.04
3.06
Rates of Return
0 0.2 0.4 0.6 0.8 1
−6.97
−6.96
−6.95
−6.94
Individual Welfare
ie
ih
Γ
r
Ue
Uh
ΔUs
ΔY
Figure 5: The Effect of Two-Period Consumption: ρ = 1
producing intermediate goods and hence output but does not change the basic mechanism.
As a result, it only affects the levels of endogenous variables but not their respective
patterns. Thus, the long-run welfare implications of financial development to households
and entrepreneurs are qualitatively similar as in the baseline model.
3.2.3 The Bequest Motive
In the baseline model, agents only care about their own consumption when old without
leaving bequest to their offsprings, φ = 0. In order to analyze the effect of the bequest
motive, we set φ = 0.25 and keep other parameter values same as in the baseline model.
Figure 6 shows the long-run patterns of some endogenous variables with respect to θ ∈[0, 1] which is denoted by the horizontal axis.
Besides the wage income, bequest becomes another intergenerational linkage. Intu-
itively, if individuals receive bequest when young, they can have more for investment than
in the baseline model. Additionally, the larger project investment leads to larger aggre-
gate output of intermediate goods and thus the higher wage rate as well as the lower price
of intermediate good. The positive income (wage) effect indirectly increases the project
investment of the next generation and reduces the price of intermediate good. The long-
run patterns of the loan rate and the equity rate are similar as in the baseline model,
except that their levels are lower than in the baseline model. Consider the frictionless
case θ ∈ [1 − η, 1] where the loan rate is equal to the equity rate, r = Γ = vR. Ceteris
paribus, the lower price of intermediate good in the model with the bequest motive results
in the lower interest rates than in the baseline model.
16
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Individual Investment
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
Output and Social Welfare
0 0.2 0.4 0.6 0.8 1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
Rates of Return
0 0.2 0.4 0.6 0.8 1
−2.6
−2.5
−2.4
−2.3
−2.2
Individual Welfare
ie
ih
Γ
r
Ue
Uh
Us
Y
Figure 6: The Effect of Bequest: φ = 0.25
In sum, the mechanism of the baseline model holds even if we consider partial de-
preciation of capital, consumption when young, and the bequest motive. For simplicity,
we stick to the parameter values in the baseline model for the long-run analysis in this
section and for the dynamic analysis in the next section.
4 Dynamic Analysis of Financial Development
The model economy has a structure of overlapping generations. Individuals in early and
later generations may be affected in a different way during the transitional process of
financial development. We analyze the model dynamics with respect to a 10-percentage-
point increase in θ in period 0 in two cases, i.e., given the model economy at the steady
state with θ constant at 10%, 60% before period 0. Figure 7 and 8 show the impulse
responses of endogenous variables in the two cases, respectively.
Consider the case where the initial degree of financial development is θ = 10%. See
Figure 7. An increase in θ from 10% to 20% in period 0 expands the entrepreneurs’
borrowing capacity and entrepreneurs increase their project investment by 14%. The
excessive credit demand of entrepreneurs pushes up the loan rate in period 0. As the
wage income is unchanged in period 0, the rise in the loan rate induces young households
to lend more to entrepreneurs by reducing their own project investment by 4%. The
improvement in resource allocation in period 0 increases aggregate output of intermediate
good and final good in period 1. Since the price of intermediate good is negatively
related to aggregate output of final good, v1 = αY1
M1= αY
1− 1α
1 , the price of intermediate
17
0 2 4 6−5
0
5
10
15
Individual Investment
0 2 4 6
0
0.1
0.2
0.3
0.4
Output and Social Welfare
0 2 4 6
0
0.2
0.4
0.6
0.8
1
1.2
Rates of Return
0 2 4 6
0
0.2
0.4
0.6
0.8
Individual Welfare
ie
ih
Γ
r
Ue
Uh
Us
Y
Figure 7: Dynamic Analysis: An Increase in θ from 0.1 to 0.2
good declines and the project revenue vG(kh0 ) declines. The period-1 consumption of
households born in period 0 is determined by the sum of project revenue and the loan
repayment. Given the degree of financial development is quite low, θ = 0.10, the size of
the loan repayment in period 1 is relatively small in comparison with the size of project
revenue. As a result, the decrease in the project revenue dominates the rise in the loan
repayment and hence, households born in period 0 is worse off than those born before
period 0. Due to the positive wage income effect, households born after period 1 are better
off than their predecessors. In this sense, the welfare gains of households in the latter
generations come at the cost of the welfare losses of households in the early generation.
An increase in θ from 10% to 20% enables entrepreneurs to borrow more and increase
the project investment by nearly 14% in period 0. The equity rate rises by 1.2%. Ac-
cordingly, after the debt repayment in period 1, the consumption of entrepreneurs born
in period 1 is higher than the previous level, ce1 = w0Γ0, given the predetermined wage
income in period 0. Therefore, entrepreneurs born in period 0 benefit from financial devel-
opment. It takes a few periods for capital accumulation and the economy to approach to
the long-run steady state. Due to the rise in the wage income over time, later generations
of entrepreneurs benefit from financial development even more than early generations.
Social welfare measured by the weighted sum of welfare of each generation converges
gradually to its long-run level which is higher than before the change.
Consider the case where the initial degree of financial development is θ = 60%. See
figure 8. The rise in θ from 60% to 70% in period 0 enables entrepreneurs to borrow and
investment more. Their excessive credit demand pushes the loan rate which jointly with
18
0 2 4 6
−40
−20
0
20
40
Individual Investment
0 2 4 6
0
0.2
0.4
0.6
0.8
1
Output and Social Welfare
0 2 4 6
−1.5
−1
−0.5
0
0.5
1
1.5
2
Rates of Return
0 2 4 6−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Individual Welfare
ie
ih
Ue
Uh
Us
Y
r
Γ
Figure 8: Dynamic Analysis: An Increase in θ from 0.6 to 0.7
the decline in the price of intermediate good induce households to reduce investment and
increase lending to entrepreneurs in period 0. Since the households’ project investment is
quite small, their project revenue only accounts for a small fraction of their second-period
income. The increase in the loan repayment dominates the decline in the households’
project revenue and their period-1 consumption rises. Thus, households born in period 0
benefit from financial development. Due to capital accumulation and the rise in the wage
income, households born after period 0 are continuingly better off over time.
A marginal increase in θ affects the equity rate through three channels, i.e., the increase
in the loan rate, the decline in the price of intermediate good, and the increase in the
debt-equity ratio. According to figure 3, for a large initial value of θ, e.g., θ = 0.6, the
potential credit supply, i.e., the household project investment, is small. For an increase in
θ from 60% to 70% in period 0, the first and the second effects dominates the third effect
and the equity rate declines by 1% in period 0. Given the predetermined wage income, the
decline in the equity rate reduces the entrepreneurs’ income and hence their consumption
in period 1. This way, entrepreneurs born in period 0 lose from financial development.
The equity rate converge to the level which is 1.5% lower than before the change. The
decline in the equity rate dominates the rise in the wage income. Thus, entrepreneurs in
both earlier and later generations lose from financial development.
19
5 Conclusion
This paper develops an overlapping generations model with financial frictions and ana-
lyzes the short-run and long-run welfare implications of financial development to different
individuals in a closed economy. The productive individuals may benefit or lose from
financial development in the long run as well as in the short run, depending on the initial
degree of financial development and the magnitude of its change. The unproductive indi-
viduals benefit strictly in the long run from the increases in the lending possibility and the
wage income due to financial development. However, the long-run welfare gains of those
unproductive individuals in the later generations may come at the cost of the welfare
losses of those unproductive individuals in the early generations. The driving mechanism
is the non-monotonic patterns of the loan rate and the equity rate with respect to the
degree of financial development.
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A Robustness Test: ψ
Figure 9 shows the steady-state values of endogenous variables with respect to financial
development, given ψ = 0.25, and the horizontal axis denotes θ ∈ [0, 1]. Compare figure
3 and 9, the only qualitative difference is the loan rate pattern with respect to financial
development. For ψ = 1, figure 3 shows that the loan rate rises monotonically in θ, while
for ψ = 0.25, figure 9 shows that the loan rate has a U-shaped pattern.
As mentioned in subsection 3, for θ ∈ [0, θU), the project investment of households
is positive, which is inefficient and can be approximately regarded as potential credit
21
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
Individual Investment
0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
60
70
Output and Social Welfare
0 0.2 0.4 0.6 0.8 11
1.5
2
2.5
3
3.5
Rates of Return
0 0.2 0.4 0.6 0.8 1
−2.6
−2.4
−2.2
−2
−1.8
−1.6
−1.4
−1.2
Individual Welfare
ie
ih
Γ
r
Ue
Uh
ΔUs
ΔY
Figure 9: Long-Run Allocation: θ ∈ [0, 1]
supply. The lower the degree of financial development, the larger the household project
investment and the larger the potential credit supply are.
According to figure 1, a marginal increase in θ raises aggregate output of intermediate
goods to a larger extent under ψ = 0.25 and than under ψ = 1, given the initial value
of θ = 0. The decline in the price of intermediate good is also more significant under
ψ = 0.25 than under ψ = 1. Comparing with ψ = 1 and a small initial value of θ,
an improvement in financial development reduces the price of intermediate good more
significantly under ψ = 0.25 and it induces households to supply more credit on the
market. Given that households have abundant potential credit supply under a low initial
value of θ, the increase in the credit supply of households may dominate that in the credit
demand of entrepreneurs and the loan rate may decline in θ under ψ = 0.25. While, for
a relatively large initial value of θ, e.g., θ = 0.6, the project investment of households
is small and the potential credit supply is scarce. A further increase in θ still induce
the increase in the credit supply due to the decline in the price of intermediate good.
However, given the scarce potential credit supply, the increase in the credit demand of
entrepreneurs may dominate the increase in the credit supply of households and the loan
rate rises. For financial development above the threshold value, θU , households do not
invest in own projects but lend all their wage income to entrepreneurs. As a result, the
parameter value of ψ does not affect economic allocation in the frictionless case.
22
Although the parameter value of ψ matters for the loan rate pattern, our qualitative
results of welfare implications actually depend on the hump-shaped pattern of the equity
rate rather than on the loan rate pattern. Therefore, we set ψ = 1 for simplicity in the
analysis. It can be shown that our main results still hold for other values of ψ.
23