Discussion Papers In Economics
And BusinessInverted u-shaped relationship between inequality
and economic performance
Yuichiro Matsumoto
Discussion Paper 14-19

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInverted u-shaped relationship between inequality
and economic performance
Yuichiro Matsumoto
Discussion Paper 14-19

April 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInverted u-shaped relationship between inequality
and economic performance∗
Yuichiro Matsumoto†

Abstract
This paper shows that in a borrowing-constrained economy, a medianlevel of inequality stimulates investment, whereas low and high levels of
inequality dampen investment. This nonlinearity is a result of two eﬀects.
There are more rich individuals in an equal economy than in an unequal
economy. Therefore, more individuals can invest. However, in an equal
economy, rich individuals have less wealth than they would in an unequal
economy, which can dampen investment. My paper shows that these two
eﬀects produce nonlinearity if investment is indivisible.

*JEL Classiﬁcations:O11,O16*Keywords: Income inequality; Economic development; Indivisible investment; Misallocation of capital.

1

Introduction

This paper shows that in a borrowing-constrained economy, there is an inverted
u-shaped relationship between income distribution and output (see ﬁgure 1). In
an extremely unequal economy, the ratio of agents who can invest is low andoutput also becomes because there are many more poor individuals than in an
equal economy. However, in an extremely equal economy, rich agents cannot
invest and the output level is low because rich individuals have less wealth thanthey would in an unequal economy. My paper shows that these two forces
produce a nonlinear relation between inequality and productivity.
In the empirical research, various studies examine the relationship between
inequality and economic growth. On the one hand, Alesina and Rodrik (1994)
and Persson and Tabellini (1994) state that inequality has a negative eﬀect on
growth. On the other hand, Li and Zou (1998) and Forbes (2000) state that
inequality has a positive eﬀect on growth. Banerjee and Duﬂo (2003) ﬁnd an
inverted u-shaped relationship between inequality and growth.1 In addition,∗ This work was supported in part by Grants for Excellent Graduate Schools, MEXT,
Japan.
† Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan. E-mail address: u847103e@ecs.osaka-u.ac.jp
1 Chen (2003) also reports a similar relationship. Barro (2000) identiﬁes another type of
nonlinearity.

1Figure 1: Inequality and productivity

2

Banerjee and Duﬂo (2003) propose that the diﬀerent results found in the literature are a result of nonlinearity. My paper aims to provide an explanation forthis inverted u-shaped relationship.
My model can be regarded as a formalized version of the idea of Banerjee
and Duﬂo (2003). They propose that nonconcave production functions produce
nonlinearity. In an economy with a concave production function, the marginal
productivity of the capital of rich agents is lower than that of poor agents
because of decreasing returns to capital. Redistribution from rich agents withlow marginal capital productivity to poor agents with high marginal capital
productivity improves productivity. However, this relation does not necessarily
hold if the production function is not concave. The authors only provide a brief
explanation for this result because their main focus is on empirical issues. My
model is similar to their explanation in that I assume investment indivisibility.2
Indivisibility produces nondecreasing returns. Hence, redistribution from therich to the poor can dampen eﬃciency.3 4 Also, it is important that only withthis frequently used concept, I can produce inverted u-shaped relation.
My model is also related to the literature on the misallocation of capital.5
Banerjee and Moll (2010) deﬁne two types of capital misallocation. First, intensive margin capital misallocation takes place if the marginal product of capital isunequal among agents. Second, extensive margin capital misallocation occurs if
the number of investors is smaller than the optimum. Extensive margin capital
misallocation results from ﬁxed entry costs or indivisibility.
The model of Benabou (1996) can be viewed as a model of intensive margin
capital misallocation. He constructs a neoclassical growth model with an imperfect ﬁnancial market and states that the marginal product of capital is notequal among agents due to ﬁnancial market imperfection. In addition, unequalincome distribution produces unequal marginal products of capital. That is, hismodel states that inequality produces intensive margin capital misallocation.
There is no extensive margin misallocation because all agents invest.
In contrast to his model, productivity is determined by the number of agents
who participate in an investment project in my model. Hence, my model analyzes how the distribution of income aﬀects eﬃciency in an economy in which
extensive margin capital misallocation exists.
The remainder of the paper is organized as follows. Section 2 introduces
the basic structure of the model. Section 3 analyzes a case in which factor
prices are exogenously given, as in Galor and Zeira (1993). Section 4 analyzes a
2 There is one shortcoming in this paper. Matsuyama (2011) claims that it is extreme toassume that there is only one indivisible project in an economy. If we allow many indivisible
projects, the eﬀect of redistribution changes. Despite this shortcoming, it is important toinvestigate models with indivisible investment because many models use this assumption.
3 Many papers address income distribution and indivisible investment. However, thesemodels do not ﬁnd an inverted u-shaped relationship.
4 Alesina and Rodrik (1994), Persson and Tabellini (1994), and Li and Zou (1998) use
a political economics model to explain the relationship between economic performance andinequality.
5 Buera and Moll (2012), Caballero et al. (2008), Hsieh and Klenow (2009),Midrigan andXu (2010), Moll (2010), and Song et al. (2011) also address the misallocation of capital.

3

Figure 2: The relationship between β and the endowment of a rich agent
case in which wages are endogenously determined, as in Banerjee and Newman(1993) and Ghatak and Jiang (2002). Section 5 analyzes a case in which interest
rates are endogenously determined, as in Aghion and Bolton (1997). Section
6 analyzes a case in which the prices of these two factors are endogenously
determined. Section 7 concludes.

2

The model

There is a continuum of agents. Each agent has an indivisible investment projectthat converts one unit of goods at the beginning of the period into y units ofgoods at the end of the period. There are two types of agent: rich and poor.
Each rich agent has er units of goods, which can be used to capital goods toinvest. Each poor agent has ep units of goods at the beginning of the period.
The initial endowment of the poor ep and the mean level of endowment ē areexogenously determined and ep < ē. However, the endowment of rich agents err
is endogenously determined and satisﬁes βer + (1 − β)ep = ē and dedβ < 0 (seeﬁgure 2). The ratio of rich agents is β. A high value of β indicates an equalincome distribution.6
Each agent can choose between becoming a worker or an employer and has
L units of labor. By becoming a worker, the agent earns the competitive wage
rate v. In addition, she lends her endowment and earns interest at a rate of r̄ perunit of endowment. r̄ is the exogenously determined world interest rate. Thus,
6 Similar class structures are used in Acemoglu and Robinson (2005) and Foellmi and
Zweimüller (2006).

4

by becoming a worker, an agent with e units of endowment at the beginning of
the period will have r̄e + vL at the end of the period.
To become an employer, an agent needs to use one unit of capital to employ
one unit of labor as well as all of her labor force L. She needs to borrow 1−e unitsof money to invest. The interest rate paid to lenders is r. This rate equals r̄ insections 3 and 4. However, in section 5, I introduce an aggregate borrowing limit
and r can therefore diﬀer from r̄. Details related to the aggregate borrowing
limit are discussed in section 5. Thus, the following inequality determines theinvestment threshold.
r̄e + Lv ≤ y − r(1 − e) − v(1)The left-hand side of (1) represents the opportunity cost of investing andthe right-hand side represents the return on the investment. If r̄e + Lv <y − r(1 − e) − v is satisﬁed, the return on the investment is higher than its
opportunity cost. Agents decide to invest if the borrowing constraint is notbinding. The borrowing constraint is determined as follows.
r(1 − e) ≤ λ(y − v)

(2)That is, borrowers can only pay a fraction of their revenues λ(y − v). The
assumption that (0 ≤ λ ≤ 1) in (2) has many justiﬁcations. For example,
borrowers can steal a fraction of his proﬁt equal to (1 − λ)(y − v). If lenders
lend more than λ(y − v) of capital, borrowers have an incentive to steal. Hence,
lenders do not lend more than this amount of capital.7 8 If this constraint issatisﬁed, agents can borrow to invest. Agents invest only if the investment is
proﬁtable and the borrowing constraint is satisﬁed, that is, if (1) and (2) aresatisﬁed.
I impose two assumptions throughout this paper. First, I assume that investment is productive, that is, that y > r̄. Second, I assume that an agent cannot
invest when e = ē, v = 0, and r = r̄, or, put diﬀerently, when λy ≤ r̄(1 − ē).
This assumption implies that poor agents cannot invest. Also, rich agents caninvest if and only if r̄(1 − er ) ≤ λy holds when v = 0 and r = r̄. I deﬁne athreshold ê that satisﬁes r̄(1 − ê) = λy. This threshold implies that only richagents whose endowments are higher than ê can invest when v = 0 and r = r̄. I
also deﬁne a threshold β̂ that satisﬁes β̂ê + (1 − β̂)ep = ē. This threshold implies
that only if β is lower than β̂ do rich agents have an endowment greater than ê.

3

Exogenous factor prices

In this section, I assume that the interest rate and wage are constant at r = r̄and v = 0, respectively. Proﬁtability condition (1) is satisﬁed automatically.
7 See also Aghion et al. (1999), Holmstrom and Tirole (1998), and Kiyotaki and Moore(1997) for other micro foundations.
8 Inequality (2) means that only wealthy agents can invest. See Blanchﬂower and Shadforth(2007) and Evans and Jovanovic (1989) for the empirical validity of this argument.

5Hence, I only focus on borrowing constraint (2). Borrowing constraint (2) for
rich agents can be rewritten as follows.
r̄(1 − er ) ≤ λyRich agents invest if and only if (3) is satisﬁed (this is equivalent to β < β̂).
If β < β̂, aggregate investment is the ratio of rich since all rich agents invest andpoor agents cannot invest. Otherwise, aggregate investment is zero because allagents cannot invest. We deﬁne Y as an economy’s total output. These relation
satisﬁed.
{
βy if β ≤ β̂Y =0
if β̂ < βThis nonmonotonicity induces an inverted u-shaped curve (see ﬁgure 3).
In this section, investment returns are exogenously determined because factor
prices are constant. One may think that if factor prices were endogenously
determined, then the inverted u-shaped relationship would not hold because
whether borrowing constraint (2) is satisﬁed is determined not only by theendowment but also by factor prices. This means that the borrowing limit isalso endogenously determined. In the following sections, we endogenize factor
prices. Will the inverted u-shaped relationship remain intact?

4

Endogenous wages

In this section, I assume that the interest rate is constant at r = r̄. Unlike theprevious section, however, I now include labor market and wage determination
mechanisms. Poor agents become workers because they cannot invest if v ≥ 0.
They are indiﬀerent to whether they become workers or are unemployed if v = 0.
Thus, the labor supply of poor agents lsp can be written as follows.
{(1 − β)Lv>0lsp =[0, (1 − β)L]
v=0
y−r
r)
Rich agents demand labor if v ≤ v̂ ≡ min{ L+1, y − r(1−e}. That is,
λproﬁtability constraint (1) and borrowing constraint (2) are satisﬁed. If v > v̂,either constraint (1) or (2) is violated. Hence, all rich agents cannot invest andsupply labor. The labor supply is lsr = 0. Otherwise, if v̂ > v, constraints (1)and (2) are satisﬁed. Hence, all rich agents choose to invest. The labor supply
is lsr = β.
y−rr)
If v = v̂, there can be two outcomes. First, if v = L+1≤ y − r(1−e, proﬁts
λfrom investment are zero. Rich agents are indiﬀerent to whether they invest
or lend money. The labor supply lsr and labor demand ldr of rich agents arerespectively determined as follows.

6Figure 3: β and productivity

7
if v > v̂βL
lsr = [0, βL] if v = v̂0if v < v̂
if v > v̂
0ldr = [0, β] if v = v̂βif v < v̂
y−r
r)
< L+1
, there are positive proﬁts from investment.
Second, if v = y − r(1−eλBorrowing constraints hamper the price mechanism. At ﬁrst, I consider a casewithout borrowing constraint (2). If there is an excess supply of labor at thiswage rate, the price mechanism reduces the wage rate. If there is an excess
demand for labor, the rich agents who do not meet labor can oﬀer wage v > v̂.
However, the price mechanism does not work if borrowing constraint (2) isbinding.
Rich agents cannot oﬀer wage v > v̂ even if it is proﬁtable because theycannot borrow when v > v̂. To balance the labor market, some agents must be
credit-rationed, meaning that they cannot borrow beyond a certain limit. The
labor supply lsr L and labor demand of rich agents are determined by equations(6) and (7), respectively.
Although the labor supply and demand are determined by equations (6) and(7) in these two cases, there is one diﬀerence. If (1) is binding, rich agents gain
equal utility since they are indiﬀerent to whether they invest or lend money. If(2) is binding, this ex post equality among rich agents no longer holds. Credit
rationing hampers utility level equality among rich agents since constraint (1)is not binding.
Figure 4 shows that if β < (1 − β)L and β < β̂ hold, then the equilibrium
wage is zero because more labor is supplied by poor agents than is demanded byrich agents. The equilibrium level of employment and investment is β because
all rich agents invest. Figure 5 shows that if (1 − β)L < β and β < β̂ hold,then the equilibrium wage is v = v̂. If the equilibrium wage is zero, equilibrium
investment is β. If the equilibrium wage is higher than zero, resource constraints
determine the equilibrium level of investment. The following three equations
deﬁne the market equilibrium.

lsp + lsr = ldr(3)

lsr = L(β − ldr )(4)

lsp = (1 − β)L(5)Equation (3) represents labor market equilibrium conditions. It means thatthe labor supply must be equal to the labor demand. Equations (4) and (5)represent resource constraints. Equation (4) states that if lsr units of rich agents8

Figure 4: Labor market equilibrium if β < (1 − β)L holds
invest, then the remaining β − lsr units of rich agents supply L units of labor.
Because the wage rate is positive, agents who do not invest supply labor. The
equilibrium level of investment is equal to the labor demand ldr because thelabor demand is equal to the number of agents who invest.
If β < β̂, then all agents cannot invest since v̂ < 0.
In summary, an inverted u-shaped curve is identiﬁed.
L
if β ≤ L+1
βyL
L
Y = L+1(6)
y if L+1< β ≤ β̂0if β̂ < βL
If β̂ ≤ L+1, then v = 0 irrespective of β since the labor demand is less thanthe labor supplied by poor agents. This case is the same as in section 3 and we
can think of section 3 as special case of this section.
The trapezoid presented in ﬁgure 6 indicates that there is an upper bound of
productivity. At the upper bound, full employment occurs. In section 3, there
is no full factor employment level. Hence, ﬁgure 3 does not present a trapezoid
but rather a triangle.

9Figure 5: Labor market equilibrium if (1 − β)L ≤ β holds

Figure 6: Inequality and output

10

5

Endogenous interest rate

I analyze a model with an endogenous interest rate and exogenous wages. In
contrast to section 4, v is constant at zero. In addition, capital inﬂows are
restricted. That is, domestic borrowers cannot borrow more than S units ofcapital from abroad. To simplify the analysis, I introduce the concept of ﬁnancial intermediaries and agents can now borrow only through these lenders.
Why would agents borrow money through ﬁnancial intermediaries? Financial
intermediaries are superior with regard to lending because they have more information about borrowers than other lenders. Intermediaries can also borrowat the world interest rate r̄. However, there is a limit to how much can be borrowed from abroad.9 Why are ﬁnancial intermediaries constrained to borrow
only in foreign borrowing? For example, foreign lender valued only a fractionof collateral of ﬁnancial intermediary. Hence, only domestic lenders are willingto lend to intermediaries above the limit S.
I assume that the ﬁnancial intermediary sector is competitive. Financial
intermediaries lend money at interest rate r and borrow and lend as follows. Ifr̄ ≤ r < r̂, intermediaries decide to lend to rich agents and rich agents decideto borrow money. If r̂ < r, rich agents cannot borrow because either constraint(1) or (2) is violated. If r̂ = r, credit rationing occurs, as in section 4.
Poor agents cannot invest if r ≥ r̄. The capital supply of poor agents is(1 − β)ep .
As in the previous sections, rich agents can borrow money to invest if profitability constraint (1) and borrowing constraint (2) are satisﬁed. I deﬁne
λyr̂ ≡ min{y, 1−e}. r̂ indicates that if r is higher than r̂, either constraintr(1) or (2) is violated.
Rich agents supply capital if r > r̂ and demand capital if r < r̂. If r = r̂,there are two possibilities. First, the proﬁtability constraint is binding andrich agents are indiﬀerent to whether they invest or lend money. Second, theborrowing constraint is binding. As in section 4, credit rationing occurs. Somerich agents cannot borrow money even if (1) and (2) are satisﬁed. Rich agentscannot post interest rate r > r̂ even if it is proﬁtable since they cannot borrowat r > r̂. To balance the capital market, some agents must be credit-rationed,
meaning that they cannot borrow beyond a certain limit. I denote the capital
supply of rich agents to ﬁnancial intermediaries by ksr and the capital demandof rich agents by kdr . The following equations are satisﬁed.

if r > r̂βerksr = [0, βer ] if r = r̂
(7)


0
if r < r̂
if r > r̂0kdr = [0, β(1 − er )] if r = r̂(8)β(1 − er )if r < r̂
9 Similar to the borrowing limit used incabkris.

11

Figure 7: Credit market equilibrium if β ≤ ē + S) holds
Figure 7 states that if β ≤ ē + S), then the equilibrium interest rate is r̄ andthe equilibrium level of investment is β. Figure 8 states that if ē + S < β < β̂,
then the equilibrium interest rate is r̂ and the equilibrium level of investment isē + S. In this case, resource constraints determine equilibrium, as in section 4.
I deﬁne I as the investment level. The equilibrium condition can be written as
follows.
ksr + ksp + S = kdr(9)

I − kdr = βer − ksr(10)ksp = (1 − β)ep

(11)

Equation (10) represents capital market equilibrium conditions. If r > r̄,
ﬁnancial intermediaries borrow money as possible and lend money to rich agentsas possible. Equations (11) and (12) represent resource constraints. The lefthand side of (11) means that capital used in rich investor. The right-hand sideof (12) represents the endowment of rich agents minus how much they lend.
Equation (12) represents how much poor agents lend. Figure 9 shows that ifē + S ≤ β̂, the relationship between inequality and output is similar to the12Figure 8: Credit market equilibrium if ē + S < β holds

13Figure 9: Inequality and output
case in section 4 (see equation (6)). If there is no full employment of capital(ē + S > β̂), the relationship is similar to the case in section 3.

if (β ≤ ē + S)
βy(12)
Y = (ē + S)y if (ē + S < β ≤ β̂)0
if (β̂ < β)6Endogenous wages and interest rate

I analyze a case in which wages and the interest rate are endogenously determined. This is slightly diﬃcult because, in contrast to the previous sections,factor prices move simultaneously. The equilibrium investment level I is equal
to the labor demand ldr . The equilibrium conditions can be written as follows.
lsp + lsr = ldr{≤ L(β − ldr ) if v = 0lsr= L(β − ldr ) if v ≥ 0{≤ (1 − β)L if v = 0lsp= (1 − β)L if v > 0{≥ kdr if r = r̄
ksr + ksp + S= kdr if r > r̄
14

(13)

(14)

(15)

(16)

I − kdr = βer − ksr(17)

ksp = (1 − β)ep

(18)

L
If L+1< ē + S, then the level of investment cannot take a value higher thanLLbecauseif I = kdr = ldr > L+1, labor market equilibrium conditions (13),L+1(14), and (15) are not satisﬁed. Since kdr < s̄ + S, ksp + S < kdr is satisﬁed.

Hence, r is equal to r̄ irrespective of β from (16). This case can be analyzed asin section 4.
L
if β ≤ L+1
βyL
L
Y = L+1(19)
y if L+1< β ≤ β̂0if β̂ < βL
If ē + S < L+1, then the level of investment cannot take a value higher thanē + S because if this condition is satisﬁed, capital market equilibrium conditions(16), (17), and (18) are not satisﬁed. Since ldr < lsr + lsp , v is zero irrespectiveof β from (14). This case can be analyzed as in section 5.10
if β ≤ ē + S
βy(20)
Y = (ē + S) if ē + S < β ≤ β̂0if β̂ < β7Conclusion

In this model, inequality aﬀects the ratio and wealth of rich agents. In section
3, changes in inequality have a nonlinear eﬀect on output. From sections 4 to
6, there is a full employment level β and at this full employment level, smallchanges in inequality do not change output in these regions. However, the modelwith factor price movement produces an inverted u-shaped curve like the modelin section 3. Even if factor prices are endogenously determined, indivisible
investments produce an inverted u-shaped curve.

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