Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

The Importance of the Retention Ratio in a Kaleckian Modelwith Debt Accumulation

Hiroaki Sasaki and Shinya Fujita

Discussion Paper No. E-10-008

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

November 2010

The Importance of the Retention Ratio in a Kaleckian Modelwith Debt Accumulation
Hiroaki SASAKIƗ and Shinya FUJITA‡
Abstract
Hein (2007) investigates the effects of debt of firms and interest rates on output andgrowth, and obtains the following three results. First, the long-run equilibrium is stable
only if the short-run equilibrium is debt-led growth. Second, to obtain a positive
long-run equilibrium value of the debt-capital ratio under the debt-led growth regime,
extremely high interest rates are necessary. Third, the long-run equilibrium value of therate of capital accumulation is increasing in interest rates and is independent of income
distribution. However, these conclusions depend crucially on the assumption that theretention ratio of firms is equal to unity, and hence, that there is no dividend toshareholders. By relaxing the above assumption, we show in this paper that even thelong-run equilibrium under the debt-burdened growth regime can be stable and that thelong-run equilibrium value of the debt-capital ratio will be positive with plausible
interest rates irrespective of whether the long-run equilibrium is debt-led growth or
debt-burdened growth. Moreover, the effects of interest rates and income distribution
on capital accumulation differ from regime to regime.
1 Introduction
While many post-Keynesian researchers debate the relationship between income
distribution and economic growth, some Kaleckian researchers have investigated thelink between debt accumulation and economic growth. The economy is termed
debt-led if increases in the debt-capital ratio and the interest rate raise the rate ofcapacity utilization and/or the rate of capital accumulation. The economy iscorrespondingly termed debt-burdened if increases in the debt-capital ratio and theƗ Graduate School of Economics, Kyoto University. E-mail: sasaki@econ.kyoto-u.ac.jp‡

Graduate School of Economics, Nagoya University. E-mail: fujita@soec.nagoya-u.ac.jp
1interest rate lower the rate of capacity utilization and/or the rate of capital accumulation.
Models that consider the effect of debt-accumulation on economic growth include
Taylor (2004, ch. 8), Lima and Meirelles (2007), and Charles (2008a, 2008b).1 Taylor(2004, ch. 8) introduces the effect of interest payments into the investment function andshows the possibility of debt-led and debt-burdened regimes. In addition, he shows thatthe effect of fiscal expenditure depends on the regime. Both Lima and Meirelles (2007)
and Charles (2008a, 2008b) introduce a Minskyan taxonomy with regard to thefinancial fragility of the economy into Kaleckian models that consider
debt-accumulation. However, Lima and Meirelles’s model assumes that the savings
rate of capitalists is equal to the retention ratio of firms. Accordingly, their model
economy is neither debt-led nor debt-burdened. Charles’s model assumes that firms’
investment depends only on their retained profits, whereas in the usual Kaleckian
models, investment depends on the rate of capacity utilization and on interest payments.
Consequently, capital accumulation in Charles’s model necessarily leads to
debt-burdened.
Lavoie (1995) and Hein (2007) can be characterized as benchmark models of the“monetary theory of production” in that they clearly investigate the effects of monetary
variables such as interest rates and the debt-capital ratio on real variables such as
capacity utilization and capital accumulation. While Lavoie (1995) assumes thatinvestment is exogenously given, Hein (2007) considers a Kaleckian investment
function with a negative effect of interest payments, which is also adopted in Taylor(2004). In the short run of these models, disequilibrium of the goods market is adjusted
through changes in capacity utilization with the debt-capital ratio remaining constant.
1

The following are Kaleckian models that consider financial factors. Asada (2006)
presents a kind of Kaleckian model that considers debt-accumulation and analyticallyshows the existence of cyclical fluctuations. Charles (2010) presents a Kaleckian modelin which firms’ propensity to invest changes endogenously and shows the occurrenceof chaotic dynamics in a relatively simple framework. However, his result depends
crucially on the assumption that time is discrete. Hein (2010) presents a model in which
the outside finance-capital ratio is an endogenous variable; the fundamental structure of
the model is the same as that of Hein (2007). For Hein (2010), see also the empirical
study of Hein and van Treeck (2010). Setterfield (2009) introduces a monetary policy
termed the Pasinetti rule and investigates the effect of the policy.
2

Lavoie and Hein analyze how increases in the debt-capital ratio and interest rates affectthe rates of capacity utilization and capital accumulation. In the long run, the goods
market always clears and the debt-capital ratio becomes an endogenous variable. Hein
(2007) finds that, if the short-run equilibrium rate of capital accumulation is debt-led,the long-run dynamics of the debt-capital ratio are always stable. In contrast, if theshort-run equilibrium rate of capital accumulation is debt-burdened, the long-run
dynamics are always unstable.
However, Hein’s result that the long-run dynamics of the debt-capital ratio arestable only if the short-run equilibrium is debt-led growth is questionable from anempirical standpoint. For example, Hein and Schoder (2009) empirically study theeffect of the debt-capital ratio on the rate of capacity utilization in the US and Germany,and show that both countries are debt-burdened. It is implausible that the debt-capitalratio of these debt-burdened economies diverges to infinity.
In addition, in Hein (2007), extremely high interest rates are needed to obtain apositive long-run equilibrium value of the debt-capital ratio under the stable debt-led
growth regime. Hein and Schoder (2009) estimate parameters of the investment
function, from which we can deduce that unless interest rates are kept at 8% in the US
and 20% in Germany, the equilibrium debt-capital ratio of Hein (2007) will be
negative.
Moreover, in Hein (2007), the long-run equilibrium rate of capital accumulation
leads to the product of the savings rate of financial capitalists and the rate of interest.
From this, it follows that the paradox of thrift does not hold and that an increase in therate of interest leads to a rise in the rate of capital accumulation, which contradicts theusual Keynesian intuition.
In this paper, we show that Hein’s results depend crucially on the assumption thatthe retention ratio of firms is equal to unity, and hence, that there are no dividend
payments to shareholders. Indeed, if this assumption is relaxed—that is, if the retention
ratio of firms is less than unity—the long-run equilibrium will be stable even if theshort-run equilibrium is debt-burdened. In addition, under this relaxed assumption, thelong-run equilibrium value of the debt-capital ratio will be positive with plausible
3interest rates irrespective of whether the economy is debt-led or debt-burdened growth.
Moreover, the effects of interest rates and income distribution and interest rates oncapital accumulation differ from regime to regime.
The remainder of the paper is organized as follows. Section 2 briefly explains the
structure of Hein’s (2007) model and shows its drawbacks. Section 3 presents ourmodified model, in which the retention ratio of firms is strictly less than unity andshows some analytical results. Section 4 presents some numerical examples to ascertain
the results of the previous section and to clarify those aspects that cannot be known
analytically. Section 5 concludes the paper.
2 The basic structure of Hein’s (2007) modelThis section surveys the basic structure of Hein’s (2007) model and shows some special
results that arise from the assumption that the retention ratio is equal to unity.
We assume a closed economy without government. There exists a single good,which can be used for both production and consumption. Technological progress is not
explicitly considered: both the potential output-capital ratio and the output-labor ratio
are assumed to be constant. There are three agents in our economy: firms with excess
capacity, workers, and financial capitalists. Firms invest without issuing new shares,
using a part of profits and external funds that are financed by capitalists via banks,
which in turn set the nominal loan rate (interest rate).
The profit rate ( r ) is defined as the nominal profit ( Π ) divided by the price level( p ) and the existing capital stock ( K ).
r≡Π
Π Y Y h== u , h ∈ (0,1) ,pK pY Y K v(1)

where Y denotes real output and Y denotes potential output. The rate of profit iscomposed of three factors: the profit share ( h ), the rate of capacity utilization( u = Y / Y ), and the potential output-capital ratio ( v = Y / K ). 2 In the following
2

Hein (2007) considers the case where the mark-up rate in the price equation
elastically changes according to the interest rate. However, we abstract from this casebecause such an equation does not change our results very much.
4discussion, we assume v = 1 for the sake of simplicity.
According to the post-Keynesian “horizontalist” view (Moore (1988), Rochon(1999), and Fontana (2009)), we assume that firms can make use of loans, which arefinanced by capitalists, under a given nominal interest rate ( i ). Because firms hold thenominal sock of debt ( L ), they have to pay interest ( iL ) with a dividend to capitalists
in each period. Therefore, firms’ nominal profits are split into their retained profits( Π f ) and the capitalists’ income ( Π c ), which is composed of dividend and interest
receipts.
Πf

 Π
L 
 = s f (hu − iλ ) , s f ∈ (0,1) , i > 0 ,= s f 
−i
pK
pK 
 pK

(2)

Πc
= (1 − s f )(hu − iλ ) + iλ ,pK

(3)where s f denotes the retention ratio, which is assumed to be constant; andλ (≡ L / pK ) denotes the debt-capital ratio. The first term of the RHS in Equation (3)represents dividends, and the second term represents interest income.
We assume that capitalists save a constant fraction ( s f ) of their income, while
workers do not save. Total saving ( S ) is composed of retained profits and savings from
capitalists’ incomes. Using Equations (2) and (3), we obtain the aggregate saving
function.
S= s f (hu − iλ ) + s c (1 − s f )(hu − iλ ) + iλ , s c ∈ (0,1) .
K

[

]

(4)

Firms make an investment plan ( I ), which is given by the following desired
investment function:g≡I= α + βu + τh − θiλ , α > 0 , β > 0 , τ > 0 , θ > 0 .
K(5)Equation (5) shows that the rate of capital accumulation ( g ) responds positively to the
profit share and the rate of capacity utilization (Bhaduri and Marglin (1990)) andnegatively to interest payments per existing capital, iλ (Taylor (2004), and Taylor

5and Arnim (2008)).3
2.1 Short-run equilibrium
In the short run, disequilibrium between investment and saving is adjusted throughchanges in capacity utilization under given capital and debt stocks.
Hein (2007) assumes that s f = 1 for the sake of simplicity: firms do not pay a
dividend to capitalists. To see what happens in this case, this section accepts theassumption that s f = 1 . The equilibrium rate of capacity utilization is obtained from
Equations (4) and (5).
u=α + τh + (1 − s c − θ )iλ
.
h−β

(6)

This equilibrium will be stable if saving responds more elastically to variations incapacity utilization than investment, that is, if h − β > 0 , which we assume in thefollowing discussion.
If the reaction coefficient of investment to interest payments and capitalists’propensity to save are sufficiently small, that is, if 1 − sc − θ > 0 , increases in thedebt-capital ratio and the interest rate raise the rate of capacity utilization. This case iscalled the debt-led capacity utilization (DLCU, hereafter) regime in the recent
Kaleckian literature. If 1 − sc − θ < 0 , then increases in the debt-capital ratio and theinterest rate reduce the rate of capital accumulation; this case is called thedebt-burdened capacity utilization (DBCU, hereafter) regime.
Substituting Equation (6) in Equation (5), we get the rate of capital accumulationin the short-run equilibrium.
g=h(α + τh ) + [β (1 − sc ) − θh]iλ
.
h−β

3

(7)

Post-Keynesian models feature many variants of the desired investment function.
Hein (2006) uses a normal Kaleckian-type investment function that positively relates
investment per capital to the profit rate and the rate of capacity utilization. Charles(2008a, 2008b, 2008c) assumes that investment is an increasing function of retained
profits. Under this assumption, capital accumulation is debt-burdened in the short-run
equilibrium.
6

Whether capital accumulation is debt-led or debt-burdened depends on the profit shareand the reaction coefficient of investment to the rate of capacity utilization, in addition
to the coefficient of interest payment and to the capitalists’ savings propensity. If

β (1 − s c ) − θh > 0 , then the short-run equilibrium is debt-led growth (DLG, hereafter).
If, on the other hand, β (1 − s c ) − θh < 0 , then the short-run equilibrium isdebt-burdened growth (DBG, hereafter).
As we have seen, the condition for DLCU (DBCU) differs from the condition for
DLG (DBG). Note that DBCU is not compatible with DLG.4 Therefore, in Hein
(2007), there exist the following three cases for the debt regimes.
Case (a): the short-run equilibrium is both DLCU and DLG.
Case (b): the short-run equilibrium is both DBCU and DBG.
Case (c): the short-run equilibrium is both DLCU and DBG.
2.2 Long-run equilibrium
In the long run, the debt-capital ratio becomes an endogenous variable, while the goods
market always clears. The growth rate of the debt-capital ratio is given by


 L
L
− gλ ,− g λ =pK
L

λ = 

(8)

where inflation is assumed to be zero. Firms’ loans in each period are equal to thedifference between total investment and their internal profits. Since the goods market
always clears, loans are finally identical with the savings from capitalists’ incomes.5
4The condition for DBCU is given byx ≡ 1 − sc − θ < 0 .

The condition for DLG is given byy ≡ β (1 − s c ) − θh > 0 .
Using these two equations, we obtainy ≡ β ( x + θ ) − θh = βx + θ ( β − h) .

Here, x < 0 and β − h < 0 imply y < 0 . Therefore, DBCU necessarily leads to
DBG.
5
If the agents who save out of income are not capitalists but workers, firms’ loans are7LI= − (hu − iλ ) = s c iλ .
pK K

(9)

Substituting Equation (9) in Equation (8) yields

λ = (s c i − g )λ .

(10)

The long-run equilibrium debt-capital ratio ( λ* ) is derived from λ = 0 and from
Equations (7) and (10).

λ* =s c i (h − β ) − h(α + τh ).
i[β (1 − s c ) − θh](11)Here, we abstract a trivial solution, λ* = 0 , because it is economically meaningless. In
addition, empirical studies by Taylor and Arnim (2008) and Hein and Schoder (2009)
show that the debt-capital ratio is positive in advanced capitalistic countries; hence, we
assume that λ* > 0 .
The long-run equilibrium will be locally stable if the following condition issatisfied.

[β (1 − sc ) − θh]i λ* < 0 .
∂λ
∂g *
λ =−
=−
∂λ
∂λ
h−β

(12)

Because h − β > 0 and λ* > 0 , the necessary and sufficient condition for local
stability is given by β (1 − s c ) − θh > 0 . Note that this condition coincides with thecondition for DLG. Therefore, we obtain the following result.
Result 1. (Hein, 2007) Suppose that the long-run equilibrium debt-capital ratio is
positive. When s f = 1 , the long-run equilibrium is stable (unstable) if the short-run
equilibrium is DLG (DBG).
Now, consider the case where the long-run equilibrium is locally stable, in otherwords, where the short-run equilibrium is DLG (Case (a)). For the positive equilibrium
debt-capital ratio, the numerator of the RHS in Equation (11) must be positive (that is,
s c i(h − β ) − h(α + τh ) > 0 ) because its denominator is also positive. This yields

equivalent to workers’ savings (Dutt and Amadeo (1993)).
8

i > h(α + τh) /[ s c (h − β )] . Therefore, we obtain the following result.
Result 2. (Hein, 2007) To obtain a positive debt-capital ratio in the long-run
equilibrium, extremely high interest rates are needed.
For example, suppose that α = 0 , s c = 1 , and h = 0.3 , and substitute them in
the inequality. Then, the condition of λ∗ > 0 leads to i > 0.09τ / (0.3 − β ) . On thebasis of the empirical study of Hein and Schoder (2009) that finds that β = 0.14 andτ = 0.14 in the US and that β = 0.15 and τ = 0.33 in Germany, a positive λ∗requires i > 0.07875 for the US and i > 0.198 for Germany. Furthermore, as sc
gets smaller or α gets larger, a positive λ∗ requires higher interest rates.
Again, consider the case for the stable equilibrium. Provided thatβ (1 − s c ) − θh > 0 , the effect of the interest rate on the long-run equilibrium
debt-capital ratio is given by∂λ*h(α + τh )
= 2
> 0.
∂i
i [β (1 − s c ) − θh](13)Equation (13) contradicts the statement of Hein (2007, pp. 323-324) that an increase in
the interest rate can either raise or reduce the debt-capital ratio.
By substituting Equation (11) in Equation (6), we obtain the long-run equilibrium
rate of the capacity utilization ( u * ). Differentiating it with respect to i yieldss (1 − s c − θ )
∂u *
.
= c
[β (1 − sc ) − θh]
∂i

(14)

The denominator of the RHS of Equation (14) is positive because of the local stability
condition. Moreover, based on Footnote 4, its denominator is negative when its
numerator is negative, which contradicts the stability condition. Therefore, both thenumerator and denominator must be positive,6 which leads to the result that anincrease in the interest rate raises the rate of capacity utilization in the long run.
6

As mentioned above, the condition for the positive numerator of the RHS of
Equation (14), that is, 1 − s c − θ > 0 , implies that the short-run equilibrium is DLCU.
As a result, the stable long-run equilibrium can be compatible only with Case (a): theshort-run equilibrium is both DLCU and DLG.
9

Finally, we obtain the long-run equilibrium rate of capital accumulation g * = s c i .
From this, we obtain the following result.
Result 3. (Hein, 2007) A rise in the interest rate increases the rate of capital
accumulation irrespective of the debt regime.
This result seems strange from the Keynesian perspective that holds that lower
interest rates produce higher growth.7 Furthermore, capital accumulation in the longrun is independent of the profit share, which is contrary to the typical Kaleckian
proposition concerning the relation between growth and distribution.
3 Modified model: the case in which dividends are distributed
In this section, we explain how some analytical results of Hein (2007) are modified
when we take into account the case where firms distribute dividends to capitalists.
Assuming s f < 1 instead of s f = 1 and using Equations (4) and (5), we obtain the
short-run equilibrium rate of capacity utilization.
u=

α + τh + [ s f (1 − s c ) − θ ]iλ, σ ≡ s f + s c (1 − s f ) > 0 .
σh − β(15)Hereafter, we consider only the case where the short-run equilibrium is stable, which isequivalent to supposing that σh − β > 0 . We immediately find that the short-run
equilibrium is DLCU if s f (1 − s c ) − θ > 0 , while it is DBCU if s f (1 − s c ) − θ < 0 .
The short-run capital accumulation is given by
g = A + Bλ , A ≡[ βs f (1 − s c ) − σθh]iσh(α + τh).
> 0, B ≡σh − βσh − β(16)

The short-run equilibrium is DLG if βs f (1 − s c ) − σθh > 0 ( B > 0 ) and DBG if

βs f (1 − s c ) − σθh < 0 ( B < 0 ).
We also find, using the same method as in Footnote 4, that the condition for
DBCU contradicts the condition for DLG. Therefore, as in Hein (2007), there exist
7

In addition, g * = s c i denies the paradox of thrift that implies that an increase in thesavings propensity of capitalists leads to a decrease in the rate of capital accumulation.
10

three cases of debt regimes.
Case 1: the short-run equilibrium is both DLCU and DLG.8
Case 2: the short-run equilibrium is both DBCU and DBG.
Case 3: the short-run equilibrium is both DLCU and DBG.
Firms’ loans are given by the following equation.

LI= − s f (hu − iλ ) = s c (1 − s f )(hu − iλ ) + iλ .
pK K

[

]

(17)

Substituting Equations (16) and (17) in Equation (8) and rearranging, we obtain the
dynamical equation of the debt-capital ratio for the case where the retention ratio is
below unity.
sc s f− λ  g +
iλ
σ
σ


.
s c (1 − s f ) A  s c (1 − s f ) B
sc s f i 
2
=
+
− A+
 λ − Bλ
σ
σ
σ




λ ≡ G (λ ; i ) = 1 −

sf

(18)

Needless to say, substituting s f = 1 in Equation (18) yields Equation (10): ourmodified model boils down to Hein’s (2007) model.
3.1 Stability analysis in the long run: The DLG caseWe here assume that B > 0 . This corresponds to Case 1: the short-run equilibrium isboth DLCU and DLG. Then, the RHS of Equation (18) shows a parabola with its( )vertex oriented upwards in the λ, λ plane. The inflexion axis of the curve is givenbyλ =s c (1 − s f ) B − σA + s c s f i
2σB

.

(19)

If we assume that s f = 1 , the parabola passes through the origin of the plane, which is
identified with Hein (2007). Otherwise, an intercept of the parabola is inevitably
positive.
Figure 1 shows the case where 0 < s f < 1 and λ > 0 ; this case corresponds to

8

Case 1 is divided into Case 1-1 and Case 1-2 in the following analysis.
11Case 1-1 of Table 1 in the next section. Figure 2 shows the case where 0 < s f < 1 andλ < 0 ; this case implies Case 1-2 of Table 1. Figure 3 shows the special case wheres f = 1 and λ > 0 .9 From Figures 1 and 2, we can immediately understand thatwhen the retention rate is lower than unity, Equation (18) always intersects thehorizontal axis once within λ > 0 regardless of the sign of the inflexion axis. This
result is important because the equilibrium debt-capital ratio is always positive under
the DLG regime without any particular condition for the interest rate, which we impose
in Section 2.2. Once again, the inflexion axis must be positive in the case of s f = 1
for the positive equilibrium value (Figure 3). In other words, the interest rate must be
high enough to satisfy s c i (h − β ) − h(α + τh ) > 0 .
Proposition 1. The long-run equilibrium is stable if the short-run equilibrium is DLG.
Moreover, to obtain a positive debt-capital ratio in the long-run equilibrium, extremely
high interest rates are not needed.
We represent the long-run equilibrium debt-capital ratio ( λ* ) in Figure 1. Even if
an external shock moves the debt-capital ratio to the point λ0 on the horizontal axis,
this ratio continues to decrease in the range of λ < 0 until it converges to theequilibrium. In contrast, even if the debt-capital ratio happens to be below its
equilibrium, this ratio continues to increase because of λ > 0 and finally converges tothe equilibrium. This reasoning explains why a unique positive equilibrium under the
debt regime is always locally stable.
(Figures 1, 2, and 3 to be inserted here)
3.2 Stability analysis in the long run: the DBG case
Next, we consider the case where B < 0 , which means that the short-run equilibrium
9

We exclude the case where s f = 1 and λ < 0 because this case does not have apositive equilibrium value. For the same reason, we also abstract this case under the
DBG regime.
12

is DBG (either Case 2 or 3). In this case, the RHS of Equation (18) depicts a parabola( )with its vertex oriented downwards in the λ, λ plane. The necessary and sufficientcondition for the existence of the positive equilibrium value, which is equivalent to thecondition that the curve crosses the horizontal axis, is that the discriminant ofG (λ ; i ) = 0 is positive. However, it is difficult to specify this condition because our
0 < s f < 1 assumption requires very complicated calculations. Here, we assume thatthe discriminant is positive.10
Figure 4 shows the case where 0 < s f < 1 and λ > 0 ; Figure 4 corresponds toeither Case 2 or Case 3 in Table 1. Figure 5 shows the case where 0 < s f < 1 andλ < 0 , while Figure 6 shows the special case where s f = 1 and λ > 0 . There exists
at least one positive equilibrium debt-capital ratio if

λ > 0 , that is,s c (1 − s f ) B − σA + s c s f i < 0 is satisfied (Figures 4 and 6). However, in contrast to
Hein (2007), fulfilling this inequality does not require a higher interest rate. Thisinequality will be satisfied if parameters α , τ , and h are sufficiently large.
Proposition 2. The long-run equilibrium can be stable even if the short-run
equilibrium is DBG. Moreover, to obtain a positive debt-capital ratio in the long-run
equilibrium, extremely high interest rates are not needed.
Figure 4 shows that the case where 0 < s f < 1 and λ > 0 has multiple
equilibria ( λ1* ，λ*2 ). The smaller equilibrium, λ1* , is locally stable, while the larger one,

λ*2 , is locally unstable. Once more, there exists a unique positive but unstableequilibrium in the case where s f = 1 , as we mentioned in Section 2. 2.
(Figures 4, 5, and 6 to be inserted here)
3.3 The effect of the interest rate on capital accumulation in the long run
Although our modified model is somewhat complicated, it is possible to investigate the10

Numerical examples in the next section show the existence of the positive
equilibrium under plausible parameter settings.
13effect of the interest rate on the long-run equilibrium. In the following discussion, we
consider only the stable and positive equilibrium. The equilibrium in the long run is( )defined by λ = G λ* ; i = 0 , and its stability condition is given by

sc s f i  s c (1 − s f ) B∂G (λ ; i )
*
=
− A+
 − 2 Bλ < 0 .
∂λ λ =λ* 
σ
σ 

(20)

Note that the stable equilibrium always satisfies Equation (20). Totally differentiating( )G λ* ; i = 0 yields

( )

( )

 ∂G λ* ; i   ∂G λ* ; i dλ* / 
 ,
= −
*
di
 ∂i   ∂λ


(

)

(21)

where the denominator is negative and ∂G d * ; i / ∂i is given by

[

][

]

*
∂G λ* ; i λ*  s c (1 − s f ) − σλ βs f (1 − s c ) − σθh + s c s f (σh − β ) 
= 
.
σ 
σh − β
∂i


( )

(22)

Using g * and λ* , we can rewrite Equation (18) with λ* = 0 as follows:*g =s c s f iλ*σλ* − s c (1 − s f )

.

(23)

Equation (23) implies that the positive equilibrium values for the debt-capital ratio andfor the rate of capital accumulation impose the condition that σλ* − sc (1 − s f ) > 0 . We
suppose a situation where this inequality is always satisfied. Then, if the short-run( )equilibrium is DLG (Cases 1-1 and 1-2), the sign of ∂G λ* ; i / ∂i is ambiguous
because βs f (1 − s c ) − σθh > 0 , which leads to the ambiguous sign of dλ* / di .
However, if we consider the situation where the short-run equilibrium is DBG (Cases 2( )and 3), we obtain ∂G λ* ; i / ∂i > 0 because βs f (1 − s c ) − σθh < 0 , which leads to
dλ* / di > 0 .

Furthermore, differentiating the long-run equilibrium rate of capital accumulationg * = A + Bλ* with respect to the interest rate yields∂d * [ βs f (1 − s c ) − σθh]  *dλ*  λ + i.
=σh − β
∂i
di 


(24)

Proposition 3. A rise in the interest rate decreases the long-run equilibrium rate ofcapital accumulation if the short-run equilibrium is DBG.
14

In the case of βs f (1 − s c ) − σθh < 0 , we obtain dg * / di < 0 because dλ* / di > 0 .
This result is contrary to the result of Section 2.2. However, in the case ofβs f (1 − s c ) − σθh > 0 , this effect is not clear because the sign of dλ* / di is

ambiguous.
4 Numerical examples
This section presents numerical examples to show that the economically meaningful
long-run equilibrium actually exists under plausible parameter settings and that theresults of comparative statics analysis differ depending on the circumstances. In thissection, we investigate how increases in the interest rate and the profit share affect thelong-run equilibrium values of the debt-capital ratio, the rate of capital accumulation,and the rate of profit. Recall that in Hein (2007), the long-run equilibrium value ofcapital accumulation is given by g ∗ = sc i . Therefore, an increase in the interest ratenecessarily increases the rate of capital accumulation. In addition, an increase in the
profit share never affects the rate of capital accumulation. However, in our model, as
will be shown below, an increase in the profit share raises the rate of capital
accumulation in some cases and lowers it in others.
From the above analysis, we know that there exist situations where the short-run
equilibrium is DLCU or DBCU and DLG or DBG. Note that the combination of
DBCU and DLG is impossible. Therefore, four cases—Cases 1-1, 1-2, 2, and 3—arefeasible; these are listed in Table 1.
(Table 1 to be inserted here)
Recall that Cases 1-1 and 1-2 correspond to Figures 1 and 2, respectively, and that
Cases 2 and 3 correspond to Figure 4. Note that Figure 4 alone cannot distinguishwhether the short-run equilibrium is DBCU or DLCU.
We conduct the numerical analysis as follows. First, we find a benchmark for eachcase. Second, starting from each benchmark, we increase the interest rate (or profit
15

share) slightly and then compute the new equilibrium values. When conducting thenumerical analysis, we must consider that the short-run equilibrium belongs to thecorresponding case even after the interest rate (or profit share) changes. The parameters
for each benchmark are listed in Table 2. Naturally, the long-run equilibrium is stable inevery case.
(Table 2 to be inserted here)
4.1 Case 1-1
Table 3 shows an example that corresponds to Case 1-1. Here, we compute thelong-run equilibrium values by increasing the interest rate from 0.1 to 0.1001 and the
profit share from 0.3 to 0.301.
When the interest rate increases, all the long-run equilibrium values increase. This
result is similar to the result in Hein (2007). In Hein (2007), for the long-run
equilibrium to be stable, the short-run equilibrium must be DLG, which necessarily
makes the short-run equilibrium DLCU. In this case, an increase in the interest rateincreases all long-run equilibrium values.
When the profit share increases, the debt-capital ratio also increases. In addition,
the rate of capital accumulation decreases, and accordingly, the long-run equilibrium is
wage-led growth. Moreover, the rate of capacity utilization decreases, and accordingly,
the long-run equilibrium is stagnationist. These two results are typical of the Kaleckian
model.11 However, the rate of profit increases, that is, we never obtain the paradox ofcost in which an increase in the profit share lowers the rate of profit. In this respect, ourmodel is different from the typical Kaleckian model.
(Table 3 to be inserted here)
The economy is termed wage-led (profit-led) growth if an increase in the profit sharelowers (raises) the rate of capital accumulation. In addition, the economy is called
stagnationist (exhilarationist) if an increase in the profit share lowers (raises) the rate ofcapacity utilization. For more on this issue, see Bhaduri and Marglin (1990) andBlecker (2002).
11

16

4.2 Case 1-2
Table 4 shows an example that corresponds to Case 1-2. Here, we compute thelong-run equilibrium values by increasing the interest rate from 0.02 to 0.0201 and the
profit share from 0.4 to 0.401. Note that in this case, the rate of interest is small
compared to the rate in Case 1-1.
The effects of a rise in the interest rate on the long-run equilibrium values are thesame as those in Case 1-1; that is, all equilibrium values increase. The result that when
the interest rate increases, the rates of capital accumulation and capacity utilization
increase seems to be paradoxical. In the usual post-Keynesian theory, an increase in theinterest rate is expected to have a negative effect on output and output growth.
The effects of a rise in the profit share are the same as in Case 1-1, except for theeffect on the rate of profit. When the profit share increases, the rate of profit decreases,
implying the paradox of cost.
(Table 4 to be inserted here)
4.3 Case 2
Table 5 shows an example that corresponds to Case 2. Here, we compute the long-run
equilibrium values by increasing the interest rate from 0.02 to 0.0202 and the profit
share from 0.4 to 0.401.12
When the interest rate increases, the debt-capital ratio also increases, but the rateof capital accumulation, the rate of capacity utilization, and the rate of profit decrease.
The case in which a rise in the interest rate leads to a decline in the rates of capital
accumulation and capacity utilization is considered to be a normal case.
When the profit share increases, the debt-capital ratio decreases. In Cases 1-1 and
1-2 where the short-run equilibrium is DLG, the debt-capital ratio increases as a resultof a rise in the profit share. Therefore, in Case 2, the result we obtain is opposite to the12

To clearly discern the changes in the equilibrium values, we use 0.0202 instead of
0.0201.
17

results in Cases 1-1 and 1-2. In Case 2, the rate of capital accumulation increases,
which means that the long-run equilibrium is profit-led growth. Considering that wehave wage-led growth in Cases 1-1 and 1-2 where the short-run equilibrium is DLG,
we might say that there is a relationship between DLG and wage-led growth and that
there is a relationship between DBG and profit-led growth. However, note that thisinference is based on numerical examples and not on rigorous analytical solutions. In
Case 2, the rate of capacity utilization decreases, implying that the long-run equilibrium
is stagnationist, which is the same as in Cases 1-1 and 1-2. A rise in the profit share
increases the rate of profit. Consequently, we never obtain the paradox of cost.
(Table 5 to be inserted here)
4.4 Case 3
Table 6 shows an example that corresponds to Case 3. Here, we compute the long-run
equilibrium values by increasing the interest rate from 0.02 to 0.0201 and the profit
share from 0.4 to 0.401.
When the interest rate increases, the debt-capital ratio, the rate of capacity
utilization, and the rate of profit increase, whereas the rate of capital accumulation
decreases.
The effects of a rise in the profit share are the same as in Case 2.
(Table 6 to be inserted here)
5 Conclusions
Recent Kaleckian studies, taking into account firms’ debt accumulation, analyze how
the stability condition and the long-run effect of the interest rate on the real economy
differ according to debt regime (debt-led or debt-burdened). Hein (2007), one of therepresentatives among those research works, abstracts the distribution of dividends tocapitalists and assumes that the retention ratio is equal to unity. However, thisassumption leads to the following extreme results. First, the long-run equilibrium
18

debt-capital ratio is stable only if the short-run equilibrium is debt-led growth. Second,
the positive equilibrium debt-capital ratio under a debt-led growth regime requires anextremely high interest rate. Third, an increase in the interest rate always raises capital
accumulation in the long run, while the income distribution never affects capital
accumulation.
This paper investigates the stability condition and the long-run effect of theinterest rate when firms distribute dividends to financial capitalists, that is, when theretention rate is lower than unity. Our results are summarized as follows.
The long-run positive equilibrium debt-capital ratio can be stable not only under
the debt-led growth regime but also under the debt-burdened growth regime.
Regardless of the debt regime, this equilibrium does not impose any special conditionon the size of the interest rate. In addition, if the short-run equilibrium rate of capital
accumulation is debt-burdened, an increase in the interest rate reduces the equilibrium
rate of capital accumulation in the long run. Although the long-run effect of the interest
rate on capital accumulation under the debt-led regime is ambiguous, our numericalexamples using plausible parameters show that this effect seems to be positive.
Furthermore, the numerical examples show that a rise in the profit share decreases
long-run capital accumulation under the debt-led growth regime, while it increases
long-run capital accumulation under the debt-burdened regime. The paradoxical effectof income distribution on economic growth (that is, wage-led growth à la Kalecki)
seems to correspond to the puzzling relation between debt accumulation and economic
growth, that is, debt-led growth. These conclusions teach us that setting the retention
rate to unity for the sake of simplicity, unlike the “zero saving propensity hypothesis of
workers,” produces biased analytical results. Therefore, we must pay due attention tothis assumption when we construct a theoretical model.

19

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22

λ

λ* λ0
λ
Figure 1: 0 < s f < 1 and λ > 0 under the debt-led case

λ

λ*
λ
Figure 2: 0 < s f < 1 and λ < 0 under the debt-led case

λ

λ*
λ
Figure 3: s f = 1 and λ > 0 under the debt-led case

23

λ

λ*2

λ1*

λ

Figure 4: 0 < s f < 1 and λ > 0 under the debt-burdened caseλ

λFigure 5: 0 < s f < 1 and λ < 0 under the debt-burdened case

λ

λ*

λ

Figure 6: s f = 1 and λ > 0 under the debt-burdened case24

Table 1: Four cases in which the long-run equilibrium is meaningful and stable
Utilization

Accumulation

Case 1-1

DL

DL

Case 1-2

DL

DL

Case 2DB

DB

Case 3DL

DB

Shape of parabola
Convex upward,
positive axis
Convex upward,
negative axis
Convex downward,
positive axis
Convex downward,
positive axis

Table 2: Benchmark parameters for four cases
sc

sf

h

i

α

β

τ

θ

Case 1-1

0.7

0.98

0.3

0.1

0.05

0.05

0.02

0.025

Case 1-2

0.6

0.8

0.4

0.02

0.05

0.2

0.1

0.1

Case 2

0.6

0.7

0.4

0.02

0.05

0.1

0.1

0.3

Case 3

0.6

0.8

0.4

0.02

0.05

0.1

0.1

0.2

Table 3: Results for Case 1-1
Benchmark

Rise in interest
rate

Rise in profit share

Debt-capital ratio

0.949598

0.966359

0.959987

Accumulation rate

0.0700531

0.0701049

0.0700417

Utilization rate

0.328542

0.330464

0.328433

Profit rate

0.0985627

0.0991391

0.0988585

25

Table 4: Results for Case 1-2
Benchmark

Rise in interest
rate

Rise in profit share

Debt-capital ratio

0.137707

0.137745

0.137722

Accumulation rate

0.197589

0.197591

0.19722

Utilization rate

0.539321

0.53934

0.536976

Profit rate

0.215728

0.215736

0.215327

Table 5: Results for Case 2
Benchmark

Rise in interest
rate

Rise in profit share

Debt-capital ratio

0.221552

0.221739

0.22155

Accumulation rate

0.12435

0.124335

0.124365

Utilization rate

0.356791

0.356787

0.355945

Profit rate

0.142716

0.142715

0.142734

Table 6: Results for Case 3
Benchmark

Rise in interest
rate

Rise in profit share

Debt-capital ratio

0.142511

0.142577

0.142509

Accumulation rate

0.123140.123137

0.123162

Utilization rate

0.337097

0.337104

0.336316

Profit rate

0.134839

0.134842

0.134863

26