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

Income Distribution, Debt Accumulation, and Financial Fragility in a
Kaleckian Model with Labor Supply Constraints

Hiroaki Sasaki and Shinya Fujita

Discussion Paper No. E-12-007

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

September 2012

Income distribution, debt accumulation, and financial fragility
in a Kaleckian model with labor supply constraints
Hiroaki SASAKI † and Shinya FUJITA ‡Abstract
This paper investigates the effect of changes in the retention ratio, profit share,interest rate, and natural rate of growth on the rate of capital accumulation and thefinancial structure of firms by using a Kaleckian growth model with labor supply
constraints. We show that if the economy exhibits a debt-burdened regime,
depending on certain conditions, there could be cyclical fluctuations such that thefinancial structure of firms changes periodically from speculative finance to Ponzi
finance.
Keywords: Financial structure, labor supply constraints, Kaleckian model, cyclicalfluctuations
JEL Classification: E12; E21; E22; E32; E44
1. Introduction
We introduce labor supply constraints into a Kaleckian model with debt
accumulation and investigate the dynamics of the rate of capital accumulation,
debt-capital ratio, and capital-labor supply ratio. We further analyze how changes
in the labor supply growth, retention ratio of firms, profit share, and interest rateaffect the equilibrium values of endogenous variables and the Minskyan financial
structure of firms.
In the era of financialization, it is a stylized fact that we observe a tendency of
income distribution in favor of shareholders. For example, Stockhammer (2004)
points out the increased income share of rentiers in France, Germany, the UK, andthe US. Epstein and Power (2003) reveal that in the OECD countries, after the
1980s, the income share of rentiers has been rapidly increasing and the profit share†

Graduate School of Economics, Kyoto University‡

Graduate School of Economics, Nagoya University1

of non-financial corporations mildly increasing. Skott and Ryoo (2008) show thatin the US economy, the retention ratio of firms declined from 1970 to 2005.
This tendency also holds true in the Japanese economy. The corporate
governance of Japanese firms has turned to shareholder value orientation after thelate 1990s, and accordingly the propensity to distribute dividend has increased. In
other words, the retention ratio has decreased. 1 The main reason for this is thatthe old custom of mutual shareholding collapsed and the number of foreign
shareholders increased rapidly. As Figure 1 shows, the retention ratio of Japanesefirms has consistently exceeded 60% from the late 1960s through the late 1980s,
but has been consistently less than 30% from the late 1990s to the present day. 2
These changes in income distribution in Japan are more dramatic than in the US
economy, which is regarded as a typical example of financialization.
[Figure 1 around here]Manyempirical

andtheoreticalstudiesinvestigatetheeffectoffinancialization in terms of income distribution in favor of shareholders on themacro economy. 3 To begin with, we turn to empirical studies.
Stockhammer (2004), which is already cited above, shows that in France, theUK, and the US, the increased income share of rentiers depresses equipment
investment of firms. A related study is Orhangazi (2008). He empirically showsthat financialization has a negative impact on capital accumulation in the US.

1

If we wish to examine the structure of income distribution from the viewpoint ofshareholder value orientation, it is appropriate to use the retention ratio that regulates the
distribution between shareholders and firms rather than the sum of interest payments anddividends.
2

The retention ratio is calculated by dividing the retained earnings (which is defined as thenet profits of a term  dividend at the term end  remuneration of officials at the term end) by
profits (which is defined as the retained earnings  dividend at the term end).
3

For the microeconomic effects of financialization on firms’ behavior, refer to Dallery(2009), in which the effect of financialization on the finance frontier and the expansion
frontier are analyzed from various angles.
2Increases in financial investment and the opportunity of financial profitability
crowd out real investment by changing the incentives of managers. Moreover,increased payments to financial markets depress real investments by decreasing
the availability of internal funds. Van Treeck (2008) shows that increases in the
profit share and the income share of rentiers can be sources of increased
profitability by increasing consumption demand on the one hand and not leading tocapital accumulation on the other hand. Onaran, Stockhammer, and Grafl (2011)
analyze how financialization and the resulting changes in income distribution
affect the aggregate demand in the US economy. They conclude that changes inincome distribution in favor of capital have a positive effect on consumption
demand though the wealth effect, but has a negative effect on investment demand,and these two effects offset each other and, consequently, in total, have a neutral
effect on the aggregate demand.
However, the former two studies (i.e., Stockhammer, 2004; Orhangazi, 2008)
estimate

only

investment

functions

and

do

not

considertheeffectoffinancialization through consumption demand. Moreover, they are short-run
analysis in that they do not consider capital/debt accumulation; in other words,
they do not consider endogenous changes in the debt-capital ratio. The latter two
studies (i.e., van Treeck, 2008; Onaran, Stockhammer, and Grafl, 2011) consider
the effect of financialization on both consumption and investment, and hence are
more general than the former two studies. However, these are also short-run
analysis in that they too do not consider capital accumulation like the former
studies.
Next, we turn to theoretical studies.
Hein and van Treeck (2010a) analytically investigate the effect of increased
shareholder power on the macro variables by using a Kaleckian model. 4 They use

4

In this respect, we consider Hein and van Treeck (2010b) as a related study. They

theoretically analyze whether financialization has an expansive effect or contractive effecton the macro economy from some aspects. Using a Kaleckian model, Hein (2010)
analytically investigates how an increasing shareholder power can affect the macro economy.
He considers the medium term, where the outside finance-capital ratio changes, as well as
the short term.
3

two types of investment functions: the Kalecki-type investment function andBhaduri and Marglin’s (1990) investment function. They show that in accordancewith the size of parameters, various demand regimes are produced. Hein (2012)
builds a Kaleckian model that considers the effect of a rise in shareholder poweron productivity growth, and theoretically investigates its effect on the rate ofcapital accumulation and growth rate of labor productivity. He concludes that even
if the effect of the rising shareholder power is considered, the total effect offinancialization on the macro economy is likely to be negative.
These theoretical studies do not consider capital accumulation and hence canbe said to be short-run analysis, similar to the above-mentioned empirical studies.
A long-run theoretical analysis that considers capital accumulation is Skottand Ryoo (2008). This study first shows that the US economy experiences
financialization, by using some economic indicators. Then, using a Harrodianmodel and a Kaleckian model (without labor supply constraints), they theoreticallyinvestigate the effect of financialization on the macro variables. Limiting our
discussion to a Kaleckian model in this paper, we find that the effect offinancialization on the macro economy is expansionary.
As stated at the beginning, we investigate the effect of financialization on themacro variables by using a Kaleckian model. In order to carry out a long-run
analysis, we incorporate labor supply constraints into a Kaleckian model with debt
accumulation. If we intend to investigate the financialization of a capitalist
economy in which near-full employment is attained, we need to build a matureeconomic model in which labor supply constraints bind.
To our knowledge, Ryoo and Skott (2008) is the only study that examines
financialization by using a Kaleckian model with labor supply constraints. This
study is a continuation of Skott and Ryoo (2008), and theoretically shows that theeffect of financialization is expansionary even in a labor-constrained economy. In
the present paper, we also build a similar Kaleckian model. However, our method
and objective are largely different from those of Ryoo and Skott (2008).
First, Ryoo and Skott (2008) use numerical calculations to conduct theircomparative statics analysis. This is because their model is complicated and,hence, analytical solutions are hard to obtain. In this paper, in contrast, we
simplify their somehow complicated settings and obtain our main results of4

comparative statics analysis by using an analytical method.
Second, their analysis is limited to the steady-state equilibrium. Hence, they
do not consider the stability and transitional dynamics of the long-run equilibrium.
In this paper, in contrast, we explicitly investigate the stability and transitional
dynamics of the long-run equilibrium. According to our analysis, the long-run
equilibrium can be stable or unstable, and depending on conditions, there can beperpetual business cycles.
Moreover, we employ the Minskyan taxonomy of the financial structure offirms. That is, we investigate whether the financial structure of firms becomes
hedge finance, speculative finance, or Ponzi finance.
As existing studies that investigate the financial structure of firms using
Kaleckian models, we consider Foley (2003), Lima and Meirelles (2007), Charles(2008a), and Nishi (2012). In these models, the financial structure of firms isshown to endogenously change along with transition to the steady state, andcontinues to stay as speculative or Ponzi finance after the economy reaches thesteady-state equilibrium. 5
In contrast, we show that depending on conditions, there can occur cyclicalfluctuations such that the financial structure of firms changes periodically from
speculative finance and Ponzi finance. Moreover, by using numerical simulations,
we show that a decrease in the retention ratio, an increase in the profit share, adecrease in the interest rate, and an increase in the natural rate of growth candiminish cyclical fluctuations.
The rest of the paper is organized as follows. Section 2 derives the dynamical
equations of rate of capital accumulation, debt-capital ratio, and capital-effective
labor supply ratio. Section 3 obtains the steady-state equilibrium, investigates
which financial regime (hedge finance, speculative finance, or Ponzi finance)
firms belong to, and in addition examines the stability of the steady-state5

A Kaleckian model of Fujita and Sasaki (2011) analytically examines the effect offinancialization on macro-economic variables, and investigates how financialization affects
the financial structure of firms in the long-run equilibrium by using Minskyan taxonomy.
However, they analyze only the long-run equilibrium, but not the transitional dynamics tothe long-run equilibrium.
5equilibrium. Section 4 investigates the effects of changes in the retention ratio offirms, profit share, interest rate, and the natural rate of growth on the financial
structure of firms, rate of capacity utilization, capital-effective labor supply ratio,
and rate of employment. Section 5, by using numerical simulations, analyzes how
the financial structure of firms evolves along the transitional path to the steadystate. Section 6 concludes the paper.
2. ModelWe consider a one-sector, one-good, closed economy, with no government sector.
In addition, there are three types of agents—the firms, households, and banks. The
firms produce using the capital stock and labor services supplied by thehouseholds. According to the post-Keynesian Horizontalist approach (Moore,
1988; Rochon, 1999), we assume that firms borrow their investment funds fromhouseholds via the banking sector at a constant nominal lending interest rate. The
firms then engage in investments using the borrowings and their retained earnings.
In addition, they do not issue new equity for investment, but issue equity at thetime when they start a business. These equities are all held by the households.
Therefore, the households receive wage income, interest income, and dividend
income (Taylor, 2004; Skott and Ryoo, 2008; Ryoo and Skott, 2008). The
households spend a constant ratio of their total income on consumption and savethe rest. The banks lend the deposits they receive from the households to the firms
and, in addition, set a nominal lending interest rate. However, we assume that thedeposit interest rates and the lending interest rates are equal, and hence the banks
incur no cost and make no profit.
2.1 Dynamics of the rate of capital accumulation
Manufacturing firms produce with a fixed coefficient production function. In thispaper, we assume that the potential output-capital ratio is constant. If we denotethe actual output as Y , then we can denote the rate of capacity utilization asu  Y / K . 6 Firms obtain profits, which are obtained by subtracting the dividend

6Strictly speaking, the rate of capacity utilization in this paper is the output-capital ratio.

The rate of capacity utilization is decomposed into the output-capital ratio and the6and interest payments with regard to their real debt L , and so the profits of firms(i.e., the retained earnings) divided by K , that is,  f , are obtained byfL s f  u  i  , s f  0,1 ,   0,1 , i  0 ,
K
K


(1)

where s f denotes the retention ratio,  the profit share, and i the nominalinterest rate, all assumed to be constant.
The households in the economy receive wage income, interest income, anddividend income, and so their total income divided by K , that is,  h , is obtained
by
h
L
L

 1   u  i  1  s f  u  i  .
K
K
K


(2)

Let a constant s h be the households’ propensity to save. From equations (1)and (2), the actual rate of capital accumulation (i.e., the real saving divided bycapital stock) is obtained bySg   s f u  i   s h 1  s f  u  s f i , sh  0,1 ,
K





(3)

where   L / K denotes the debt-capital ratio.
From equation (3), we obtain the relationship between the rate of capacity
utilization and the rate of capital accumulation:g  s f 1  s h i.
u
s f  1  s h   s h(4)

We assume that the firms’ target rate of capital accumulation g d is an increasing
function of the profit rate and capacity utilization rate (Kalecki, 1954; Taylor,2004; Hein, 2006) and a decreasing function of the employment rate e ( Ryoo andSkott, 2008; Abe, 2012; Skott and Zipperer, 2012).
g d    s f u  i   u  e ,   0 ,   0 ,   0 ,   0 .

(5)

The negative effect of the employment rate on investment is justified by thefollowing two factors. First, as the employment rate increases and the economy
approaches full employment, the bargaining power of workers increases while thecapital-potential output ratio. If we assume that the capital-potential output ratio is constant,then the rate of capacity utilization and the output-capital ratio change in the same direction.
Therefore, we can regard the output-capital ratio as the rate of capacity utilization.
7

firms’ animal spirit decreases. Second, if the economy is near full employment, thefirms’ opportunity to employ more workers and expand production is lost, andtheir incentive to accumulate capital stock is also lost (Ryoo and Skott, 2008).
Next, if we denote employment as E , consider A  Y / E the level of laborproductivity, and in addition denote the labor supply as N , the employment rate
can be shown as follows:
e

~
E Y K
 
 uk
N K AN(6)~
where k  K /( AN ) denotes the capital-effective labor supply ratio. Substituting
equation (6) into equation (5), we obtain~g d    s f u  i     k u .

(5’)

We assume that the actual rate of capital accumulation is adjusted according to thedifference between the target rate of capital accumulation and the actual rate ofcapital accumulation (Skott and Zipperer, 2012):g   g d  g ,   0 ,(7)

where the dot over the variable denotes the time derivative of the variable, and 
the speed of adjustment. Substituting equations (4) and (5’) into equation (7), we
obtain







~
~
 s f  1  s h     s h    k  k 1  s h   s h~g  g g ,  , k   gs f i    . (8)s f  1  s h   s h
s f  1  s h   s h







2.2 Dynamics of the debt-capital ratio

Log-differentiating the debt-capital ratio, we obtain L K L       g .
K
L K

(9)

The increment of debt is equal to the actual investment less the retained earnings.
Therefore, with equation (4), the increment of debt can be given as follows:s h 1  s f  s f shL g  s f u  i  
g
i .
K
s f  1  s h   s h
s f  1  s h   s h(10)

Substituting equation (10) into equation (9), we obtain the dynamical equation of
the debt-capital ratio:8    g ,   



 g

s h 1  s f    s f  1  s h   s h 
s f  1  s h   s h

s f shs f  1  s h   s h

i .

(11)

2.3 Dynamics of the capital-effective labor supply ratio

We assume that the productivity and supply of labor grow at constant rates n1 andn 2 , respectively. Log-differentiating the capital-effective labor supply ratio, we~
obtain the dynamical equation of k ,~ ~~k  k g    g  n k , n  0(12)where n  n1  n 2 , which we call the “natural rate of growth.”
3. Properties of the steady-state equilibrium
3.1 Steady-state values

From the above analysis, the dynamical system of the rate of capital accumulation,
debt-capital ratio, and capital-effective labor supply ratio can be given as follows:~~ s f  1  s h     s h    k  k 1  s h   s h~gs f i    , (8)g  g g ,  , k   
s f  1  s h   s h
s f  1  s h   s h









    g ,   



 g

s h 1  s f    s f  1  s h   s h 
s f  1  s h   s h

s f shs f  1  s h   s h

i ,

(11)

~ ~
~
k  k  g    g  n k .

(12)

The steady-state equilibrium is a situation where~
steady-state values be denoted as g * , * , and k * .

~g    k  0 . Let the

To begin with, equation (12) indicates that the steady-state value of the rate ofcapital accumulation is equal to the natural rate of growth: 7g*  n(13)

Next, substituting equation (13) into equation (11), we obtain the steady-statevalue of the debt-capital ratio as follows:s h n1  s f  .
* 
s f  1  s h   s h n  s f s h i



(14)



If the interest rate is sufficiently small, thens  1  s   s n  s s i  0 is
f

h

h

f

h

satisfied, implying that the steady-state value of the debt-capital ratio is positive.
7~*

We here exclude trivial solution k 0.
9Moreover, equation (14) can be rewritten as follows:s h n1  s f  .
* 
s h n1  s f    s f n  s h i (14’)

Note here that *  1 holds under n  s h i  0 , *  1 holds under n  s h i  0 ,and *  1 holds under n  s h i  0 . The situation where the steady-state value ofthe debt-capital ratio is greater than or equal to unity seems to be unrealistic. Thus,
it is appropriate to assume that n  s h i  0 in the following analysis. Under thiss  1  s   s n  s s i  0 is also satisfied. Furthermore, equation

assumption,

f

h

h

f

h

(14) can be rewritten as follows:ns f s h i*s  1  s   s   s 1  s    0 .

(15)

*

f

h

h

h

f

The denominator of the right-hand side of equation (15) is positive. This isbecause substituting equation (14) intos f s i 1  s f    0 .

s  1  s   s   s 1  s    0 yields
*

f

h

h

h

f

2
h

Finally, substituting equations (13) and (14) into equation (8) yields thesteady-state value of the capital-effective labor supply ratio as follows:**~ * s f  1  s h   s h   s f  1  s h     s h   g  s f  1  s h    s h ik .
 g *  s f 1  s h i*



 






(16)

The denominator of the right-hand side of equation (16) is positive, whereas thenumerator is ambiguous. If the trend term in the target rate capital accumulation,

 , is sufficiently large, then the steady-state value of the capital-effective labor
supply ratio becomes positive.
From the above analysis, the following assumptions are required in order toobtain economically meaningful steady-state values.
Assumption 1. n  s h i  0 holds.
Assumption 2.

s  1  s   s   s  1  s     s   g  s  1  s   s i  0
*

f

h

h

f

h

h

*

f

h

h

holds.
3.2 Financial structure of firms

Minsky (1975, 1982) classified the financial structure of firms into three regimes
10

based on cash-flow accounting: hedge finance, speculative finance, and Ponzi
finance. Our next task is to find out the financial regime the steady-state financial
structure locates in. Earlier studies that specify the taxonomy of the financial
structure using Kaleckian models include Foley (2003, pp.158–160), Meirelles andLima (2006, pp.102–104), Lima and Meirelles (2007, pp.570–572), Charles (2008a,
pp.320–321), Fujita and Sasaki (2011, pp.8–10), and Nishi (2012, pp.12–14). 8[Table 1 around here]
Table 1 shows the condition for each financial regime, based on the present
notation. Hedge finance is a situation where the financial structure is the mostsound; it is defined as a situation where the profits of firms are larger than orequal to their total expenditure, that is, the sum of their investment, interest
payments, and dividends. Rearranging the condition for hedge finance obtains thefollowing relation:g  s f u  i   0 .

(17)Speculative finance is defined as a situation where the profits of firms are lessthan the sum of their investment, interest payments, and dividends but larger than
the sum of their interest payments and dividends. Rearranging the conditions for
speculative finance obtains the following relations:g  s f u  i   0 and u  i .

(18)

Ponzi finance is a situation where the finance structure is the most fragile; it isdefined as a situation where the profits of firms are less than the sum of theirinterest payments and dividends. Rearranging the condition for Ponzi financeobtains the following relation:
8

These earlier studies include only investment and interest payments in firms’ expenditure;they do not consider dividends. However, we should not abstract from dividends, because it
is one of the most important expenditure items under financialization. In addition, Minsky
himself explicitly explains that dividend is a component of the firms’ expenditure. “The cash
payments made by a unit over a relevant time period equal the spending on current labor andpurchased inputs, tax payments, the remittance due to debts that fall due, and dividends.”(Minsky, 1982, p.24)
11u  i .

(19)

In which financial regime is the steady-state equilibrium located? First, by
substituting equation (4) into equation (17), we obtain the following inequality:
1  s f  g  s f i  0 .
(20)
Equation (20) contradicts the fact that both the rate of capital accumulationand debt-capital ratio show positive signs, which implies that the steady-state
equilibrium is not hedge finance. In addition, this means that the former conditionof equation (18) necessarily holds.
Second, substituting equation (4) into the latter inequality of equation (18) andrearranging, we obtain the following relation:
g  s h i  0 .

(21)

If equation (21) holds, the financial structure of firms becomes a speculative
regime; otherwise, it becomes a Ponzi finance regime. Substituting equations (13)and (14) into equation (21) produces the following inequality:n  s h i  0 .

(22)Equation (22) identifies with Assumption 1. Thus, the financial position offirms in the steady-state equilibrium necessarily becomes speculative financewhen the steady-state value of the debt-capital ratio is smaller than unity.
Moreover, if n  s h i  0 holds (i.e., the financial structure is in the Ponzi financeregime), the steady-state value of the debt-capital ratio is larger than unity, and ifn  s h i  0 holds (i.e., the financial structure is in a boundary between speculativeand Ponzi finance regimes), the steady-state value is equal to unity.
3.3 Stability

We next consider the local stability condition of the steady-state equilibrium. The
elements of the Jacobian matrix J that correspond to the system of thedifferential equations are given as follows:~s f  1  s h     s h    k *
g
 
 0,
J 11 
gs f  1  s h   s h







~
s f   k * 1  s h   s h ig,J 12 
s f  1  s h   s h

12

(23)

(24)

g *  s f 1  s h i*g
 0,
J 13  ~  
s f  1  s h   s hk
J 21  s h 1  s f    s f  1  s h   s h 

 0,
g
s f  1  s h   s h



(25)

*



(26)

s f  1  s h   s h g *  s f s h i


 0,
J 22 
s f  1  s h   s h

(27)


J 23  ~  0 ,
k

(28)

~
k ~ *J 31  k  0,
g

(29)

~
k
J 32 
 0,


(30)

~
k
J 33  ~  0 .
k

(31)

Note that all the elements are evaluated at the steady-state values.
First, we suppose the situation where quantity adjustment in the goods market
is stable—that is, the Keynesian stability condition is always satisfied. In thissituation, we have J 11  0 .
Second, we introduce the following definition.

~Definition 1. We define   k * 1  s h    s h  0 as the economy where the rate of~capital accumulation is debt led, and   k * 1  s h   s h  0 as the economywhere the rate of capital accumulation is debt burdened. 9
The element J 12 shows the effect of an increase in the debt-capital ratio on
the rate of capital accumulation. If its sign is positive, the economy in the9

In our model, the regime of rate of capital accumulation depends on the level of the~endogenous variable, k ; this differs from earlier researches (Lavoie, 1995; Taylor, 2004;
Hein, 2007; Sasaki and Fujita, 2012). For example, if the steady-state value of thecapital-effective labor supply ratio is sufficiently large, the rate of capital accumulation in
the steady state exhibits a debt-burdened regime, and vice versa.
13steady-state equilibrium is debt led, and if the sign is negative, the economy isdebt burdened.
Finally, J 13  0 , J 21  0 , and J 22  0 are obtained under Assumption 1.
The characteristic equation that corresponds to the Jacobian matrix J is given as(32)
q 3  a1 q 2  a 2 q  a3  0 ,where q denotes a characteristic root. The coefficients of equation (32) are givenby
a1   trJ   J 11  J 22  0 ,(33)

a 2  J 11 J 22  J 12 J 21  J 13 J 31 ,(34)

a3  detJ  J 13 J 22 J 31  0 ,(35)

where trJ denotes the trace of J and detJ denotes the determinant of J .
The necessary and sufficient condition for local stability of the steady-state
equilibrium is given by a1  0 , a 2  0 , a3  0 , and a1 a 2  a3  0 . We clearly knowthat a1  0 and a3  0 , whereas we do not know the signs of a 2 and a1 a 2  a3 . In
what follows, we investigate whether or not the latter two coefficients are
positive.
If the rate of capital accumulation is debt led, we have J 12  0 , andaccordingly, a 2  0 . However, if the rate of capital accumulation is debt burdened,
J 12  0 is obtained, and hence the sign of a 2 is ambiguous. We therefore assume

the following.
Assumption 3. Even if the rate of capital accumulation exhibits a debt-burdened

regime, a 2  0 holds.
This assumption is satisfied in the situation where the trend term  of thetarget rate of capital accumulation is sufficiently large. 10
We next examine the sign of a1 a 2  a3 . The coefficients of the characteristic
equation can be rewritten as follows:a1   1   2 ,  1  

10

J 11



See the appendix.
14 0 ,  2   J 22  0 ,(33’)

a 2   3 ,  3 

J 11 J 22  J 12 J 21  J 13 J 31

a 3   4 ,  4 J 13 J 22 J 31



 0,

(34’)

0,

(35’)

a1 a 2  a3  f      1  3   2  3   4  .

(36)

Note here that  1 ,  2 ,  3 , and  4 are all positive and independent of the levelof  . From this, f   shows a parabola, with its vertex oriented downward in the , a1a2  a3  plane. Furthermore, this quadratic function passes the origin and 4   2  3  / 1 3 ,0 . Now, we rewrite  4   2  3 as follows:
J 22  J 11 J 22  J 12 J 21 
4  23 

.



(37)

If the rate of capital accumulation exhibits a debt-led regime,  4   2  3  0 is
obtained, because J 12  0 . Then, f    0 is always satisfied for the range of  0 . Thus, the steady-state equilibrium is locally stable in the debt-led case,because we have a1  0 , a 2  0 , a3  0 , and a1 a 2  a3  0 .
However, if the rate of capital accumulation exhibits a debt-burdened regime,
J 11 J 22  J 12 J 21 shows an ambiguous sign, because J 12  0 . 11 In the case of
J 11 J 22  J 12 J 21  0 , we have

 4   2  3  0 , and the steady-state equilibrium

becomes stable, as was in the debt-led case.
In contrast, in the case of J 11 J 22  J 12 J 21  0 , we have  4   2  3  0 . 12 Then,for  0,  4   2  3  / 1  3  ,weobtaina1  0 ,a2  0 ,a3  0 ,andf    a1 a 2  a3  0 , which implies that the steady-state equilibrium becomes
locally unstable. On the other hand, for the range of    4   2  3  /  1  3 , weobtain a1  0 , a 2  0 , a3  0 , andf    a1 a 2  a3  0 , which implies that thesteady-state equilibrium becomes locally stable. Finally, the Hopf bifurcation
occurs11at   4   2  3  /  1  3

becausea1  0 ,a2  0 ,a3  0 ,and

We cannot prove the determinant factors of the sign of J 11 J 22  J 12 J 21 analytically. The

numerical simulation in section 5.2, however, indicates that when the rate of capital
accumulation is debt burdened, a decrease in the retention ratio, an increase in the profit
share, a decline in the interest rate, and a rise in the natural growth rate raise the value ofJ 11 J 22  J 12 J 21 and contribute to the local stability of the steady-state equilibrium.
12Note that J 11 J 22  J 12 J 21  0 is not necessarily consistent with Assumption 3.
15

f    a1 a 2  a3  0 are satisfied. This means that there exists a continuous familyof non-constant, periodic solutions of the system around    4   2  3  /  1  3 .
Summarizing these results, we obtain the following propositions.
Proposition 1. Suppose that in the steady-state equilibrium the rate of capital

accumulation is debt led. Then, the steady-state equilibrium is locally stable.
Proposition 2. Suppose that in the steady-state equilibrium the rate of capital

accumulation is debt burdened. Then, in the case of J 11 J 22  J 12 J 21  0 , thesteady-state equilibrium is locally stable. In the case of J 11 J 22  J 12 J 21  0 , ahigher adjustment speed of the rate of capital accumulation makes the steady-state
equilibrium stable, and a lower adjustment speed of the rate of capital
accumulation makes the steady-state equilibrium unstable. Moreover, a limit cycle
occurs when the adjustment speed of the rate of capital accumulation approaches
near some value.
4 Comparative statics analysis
4.1 The debt-capital ratio

As section 3.2 has shown, the threshold between speculative finance and Ponzi
finance in the steady-state equilibrium is a situation where the debt-capital ratio
becomes unity. Accordingly, if the steady-state value of the debt-capital ratio
increases and approaches unity, then the financial structure of firms becomes morefragile. On the other hand, if the steady-state value of the debt-capital ratio
decreases, then the financial structure of firms becomes more healthy.
The effects of changes in the retention ratio, profit share, interest rate, andnatural rate of growth on the steady-state value of the debt-capital ratio are given
as follows:s h nn  s h i 
d*

0
ds f
 s f  1  sh   sh n  s f sh i2(38)s f s h nn  s f s h i 
d*

0
d
 s f  1  sh   sh n  s f sh i2(39)s f s h2 n1  s f  
d*
0

di
 s f  1  sh   sh n  s f sh i2

(40)













16

s f s h2 i 1  s f  
d*
0

dn
 s f  1  sh   sh n  s f sh i2





(41)

Note that n  s f s h i  0 holds under Assumption 1. Thus, increases in the retention
ratio and profit share decrease the debt-capital ratio, because increases in theretention ratio and profit share lower the ratio of borrowings to investment.
In addition, an increase in the interest rate raises the increasing speed of debt
accumulation, and hence increases the debt-capital ratio.
Furthermore, since an increase in the natural rate of growth is equivalent to anincrease in the rate of capital accumulation, it lowers the debt-capital ratio.
4.2 The rate of capacity utilization

Substituting equations (13) and (14) into equation (4), we obtain the steady-statevalue of the rate of capacity utilization:nn  s f s h i .
u* 
s f  1  s h   s h n  s f s h i





(42)

From Assumption 1, we have u *  0 .
The effects of changes in the retention ratio, profit share, interest rate, andnatural rate of growth on the steady-state value of the rate of capacity utilization
are given as follows:n 2 1  s h n  s h i 
du *

 0,
ds f
 s f  1  sh   sh n  s f sh i2(43)

s f n 2 1  s h n  s f s h i 
du *

 0,
d
 s f  1  sh   sh n  s f sh i2(44)s f s h n 2 1  s h 1  s f  
du *
 0,

di
 s f  1  sh   sh n  s f sh i2

(45)

















du * n s f  1  s h   s h n  s f s h i s f s h i n  s f s h i .
dn s f  1  sh   sh n  s f sh i2





(46)

Equation (43) shows that an increase in the retention ratio decreases the rate ofcapacity utilization, while equation (44) shows that an increase in the profit sharedecreases the rate of capacity utilization. In other words, in an economy with labor
supply constraints, the steady-state equilibrium exhibits the stagnationist regime(Blecker, 2002). The reasons for obtaining these results are as follows. An
17increase in the retention ratio depresses the income of households through adecrease in dividends on the one hand and a decrease in the debt-capital ratio andthe resultant decrease in the interest income on the other hand. An increase in the
profit share lowers the income of households through a decrease in wage income.
These decreases in the income of households depress consumption demand, which
results in a decrease in the rate of capacity utilization. 13
Equation (45) shows that a decrease in the interest rate lowers the rate ofcapacity utilization. In other words, the rate of capacity utilization exhibits a
debt-led regime in the steady state (Lavoie, 1995; Hein, 2007). As stated above,
this is also because a decrease in the interest rate lowers the income of households.
Note that the results of equations (43) to (45) are the same as those in Ryoo andSkott (2008), which are obtained by numerical calculations. 14
Finally, equation (46) shows the effect of an increase in the natural rate of
growth on the rate of capacity utilization, but its sign is ambiguous.
4.3 The capital-effective labor supply ratio

We now examine the effects of changes in the retention ratio, profit share, interest
rate, and natural rate of growth on the capital-effective labor supply ratio.

13Here, one might consider increases in the retention ratio and profit share offsetting

decreases in consumption demand by increasing the invest demand through an increase inprofits of firms. However, since labor supply constraints bind in the long run, the rate ofcapital accumulation is equal to the natural rate of growth given exogenously. Hence, theeffects of increases in the retention ratio and profit share are entirely reflected on a decreasein the rate of capacity utilization as a decrease in consumption demand. In contrast, in a dual
economy where labor supply constraints do not bind, and hence the rate of capital
accumulation is not equal to the natural rate of growth, it is possible that an increase in theretention ratio raises the rate of capacity utilization.
14

However, there are some differences between Ryoo and Skott (2008) and our model withregard to the specifications of the investment and consumption functions. In addition, while
new issues of equity are not considered in our model, it is considered in Ryoo and Skott(2008). These facts imply that the results in Ryoo and Skott (2008) can be obtained without
using numerical simulations if somehow the complicated settings are ignored.
18~j  A J  A1 J 22 
dk *,
 31 2 12
det J
ds f

(47)

~
j B J  B1 J 22 
dk *
 31 2 12
,
det J
d
~
j C J  C1 J 22 
dk *,
 31 2 12
det J
di
~
~
dk *  J 11 J 22  J 12 J 21 k *

,
dn
det J
where



(48)
(49)
(50)





   k * 1  s h   s h n  s h i* s h n  s h i* 
,

 0,
A1 
A
2
s f  1  sh   sh 2
s f  1  sh   sh 2

B1 



~





s f   k * 1  s h   s h n  s f 1  s h i* ~s  1  s   s 

2

f



h

~

, B2 

s f sh n  s f 1  sh i*s  1  s   s 

2

f

h

h

  0,hs f   k * 1  s h   s h *s f s h *
, C2  
 0.
C1  
s f  1  s h   s h
s f  1  s h   s hNote that we have det J  0 .
The effect of an increase in the retention ratio on the capital-effective labor
supply ratio can be explained as follows. Since equation (21) holds in the steadystate, we obtain n  s h i*  0 . Accordingly, we obtain A1  0 when the rate ofcapital accumulation is debt led, A1  0 when the rate of capital accumulation isdebt

burdened,andA2  0irrespectiveofwhether

the

rate

of

capitalaccumulation is debt led or debt burdened. In addition, we obtain J 12  0 when
the rate of capital accumulation is debt led, while we obtain J 12  0 when the rate~of capital accumulation is debt burdened. Accordingly, we obtain dk * / ds f  0
because we have A2 J 12  A1 J 22  0 in the debt-led case. In contrast, we obtain~dk * / ds f  0 because we have A2 J 12  A1 J 22  0 in the debt-burdened case.
Next, we obtain both

J 12  0and

B1  0when the rate of capital

accumulation is debt led, while we obtain both J 12  0 and B1  0 when the rate~of capital accumulation is debt burdened. Accordingly, we obtain dk * / d  0
because we have B2 J 12  B1 J 22  0 in the debt-led case. In contrast, we obtain~dk * / d  0 because we have B2 J 12  B1 J 22  0 in the debt-burdened case.
In addition, we obtain both J 12  0 and C1  0 when the rate of capital
19

accumulation is debt led, while we obtain both J 12  0 and C1  0 when the rate~of capital accumulation is debt burdened. Accordingly, we obtain dk * / di  0
because we have C 2 J 12  C1 J 22  0 in the debt-led case. In contrast, we obtain~dk * / di  0 because we have C 2 J 12  C1 J 22  0 in the debt-burdened case.
~Moreover, we obtain dk * / dn  0 because we have J 11 J 22  J 12 J 21  0 when~
the rate of capital accumulation is debt led. However, the sign of dk * / dn isambiguous when the rate of capital accumulation is debt burdened.
4.4 The employment rate~
The steady-state value of the rate of employment is given by e  u * k * .

Accordingly, the effect of an increase in the retention ratio on the rate of
employment is give by~de * du * ~ * dk * *
k 
u .

ds f
ds f
ds f

(51)

When the rate of capital accumulation is debt led, we obtain both du * / ds f  0 and~dk * / ds f  0 , thereby leading to de * / ds f  0 . However, when the rate of capital~
accumulation is debt burdened, we obtain dk * / ds f  0 , and hence the effect of anincrease in the retention ratio on the rate of employment is ambiguous.
From similar deductions, we can examine the effect of changes in otherparameters on the rate of employment. Suppose that the rate of capital
accumulation is debt led. An increase in the profit share decreases both the rate ofcapacity utilization and the capital-effective labor supply ratio, which decreases
the rate of employment. In addition, a decrease in the interest rate lowers both therate of capacity utilization and the capital-effective labor supply ratio, which
decreases the rate of employment.
However, in the debt-burdened case, the effects of an increase in the profit
share and a decrease in the interest rate on the rate of employment are ambiguous
because these lower the rate of employment and raise the capital-effective labor
supply ratio. In addition, the effect of an increase in the natural rate of growth on~
the rate of employment is ambiguous, because the sign of dk * / dn is ambiguous.
Summarizing the above results, we obtain the following propositions.
Proposition 3. A decrease in the retention ratio of firms worsens the financial
20

structure of firms. In contrast, an increase in the profit share, a decrease in theinterest rate, and an increase in the natural rate of growth improve the financial
structure of firms.

Proposition 4. A decrease in the retention ratio increases the rate of capacity
utilization. In contrast, an increase in the profit share and a decrease in theinterest rate decrease the rate of capacity utilization.

Proposition 5. Suppose the economy exhibits debt-led capital accumulation in thesteady-state equilibrium. Then, a decrease in the retention ratio of firms increases
both the capital-effective labor supply ratio and the rate of employment. Moreover,an increase in the profit share and a decrease in the interest rate decrease both thecapital-effective labor supply ratio and the rate of employment.

Proposition 6. Suppose that the economy exhibits debt-burdened capital
accumulation in the steady-state equilibrium. Then, a decrease in the retention
ratio of firms decreases the capital-effective labor supply ratio. However, an
increase in the profit share and a decrease in the interest rate increase thecapital-effective labor supply ratio. Moreover, their effects on the rate of
employment can be positive or negative.

5 Changes in financial structure along the transitional dynamics
5.1 Debt-led caseSection 3.2 has analytically shown that the financial structure of firms follows a

speculative finance regime in the steady-state equilibrium. However, there exists acase where the financial structure of firms goes away from the domain of
speculative finance and enters into the domain of Ponzi finance along thetransitional dynamics, which we will show in this section by using numericalsimulations.
Let us consider the case where the rate of capital accumulation is debt led. In
our numerical example, we set our parameters as follows: n  0.03 , s f  0.8 ,s h  0.2 ,   0.35 , i  0.05 ,   0.025 ,   0.03 ,   0.04 ,   0.05 , and   0.1 .

In this setting, we obtain economically meaningful steady-state values such as
21





~
~
g *  0.03 , *  0.915 , and k *  0.095 . Moreover, from   k * 1  s h   s h  0.022 ,we can easily verify that the rate of capital accumulation is debt led in thesteady-state equilibrium.
Setting additionally the initial conditions to g 0   0.035 ,  0   0.98 , and~k 0  0.3 , we obtain the transitional dynamics of the endogenous variables, which

are depicted as solid lines in Figures 2, 3, and 4. As these solid lines show, all of
the endogenous variables converge monotonically to their steady-state values.
Furthermore, Figure 5 shows the transition of u  i . As equation (20) indicates,
the sign of u  i becomes negative when the financial structure of firms
exhibits the Ponzi finance regime. Although u  i shows a sharp decline once,
it does not become negative under the present setting, which implies that thefinancial structure of firms never falls into the Ponzi finance regime.
Now, leaving the other parameters unchanged, we raise the initial value of the~~capital-effective labor supply ratio from k 0  0.3 to k 0  0.5 . Then, thetransitional dynamics of the endogenous values change to the dashed lines in
Figures 2, 3, 4, and 5. Figure 3 shows that when the initial value is given relatively
far away from the steady-state value, the debt-capital ratio overshoots unity.
Figure 5 shows that the sign of u  i becomes negative in a certain period andfinally converges to the steady state in the positive area. This phenomenon isinterpreted as follows. A relatively small shock in an economy keeps the financial
structure of firms in the speculative finance area. By contrast, a relatively large
shock in an economy temporally drives firms into the Ponzi finance regime. Firms
eventually converge to the steady state in the speculative area, but it is not certain
whether they can precisely predict the convergence. Unless firms can predict theconvergence toward the steady state, they will not produce; if the worst happens,
they would go bankrupt due to default.
[Figures 2, 3, 4, and 5 around here]
5.2 Debt-burdened case
Next, we consider the case where the rate of capital accumulation is debt burdenedin the steady state. Here, we set our parameters as follows: n  0.015 , s f  0.8 ,s h  0.1 ,   0.35 , i  0.05 ,   0.1 ,  0.03 ,   0.04 , and   0.05 . We
22

~
additionally set the initial conditions to g 0   0.02 ,  0   0.8 , and k 0  10 . In
this example, we obtain economically meaningful steady-state values such as~g *  0.015 , *  0.843 , and k *  13.998 . Moreover, we can verify that the rate of~capital accumulation is debt burdened because   k * 1  s h    s h  0.022 holds.

In this setting, if the adjustment speed of the rate of capital accumulation isassumed to be   0.000735 , a limit cycle occurs under the debt-burdened case,which we have described in section 3.3. The solid lines in Figures 6, 7, and 8 show
the transitional dynamics of the rate of capital accumulation, debt-capital ratio,
and capital-effective labor supply ratio. Moreover, the solid line in Figure 9
indicates the transitional dynamics of u  i , and its sign repeatedly becomes
positive and negative, implying that there exists cyclical fluctuations between
speculative and the Ponzi finance regime. 15
The mechanism of the financial structure of firms to circulate between
speculative and Ponzi finance is explained as follows. Suppose a shock to an
economy leads to a reduction in the capital-effective labor. This would decrease
the employment rate, and the firms’ animal spirits would start to rise. Increases in
the incentives for investment would raise the rate of capital accumulation anddecrease the debt-capital ratio. Then, the financial position becomes speculative
finance. However, a rise in the rate of capital accumulation has a positive impact
on both the capital-effective labor supply ratio and the employment rate. When theemployment rate approaches its peak, the rate of capital accumulation begins topass through the downturn phase. A decrease in the rate of capital accumulation
increases the debt-capital ratio, which in turn drives the financial position to Ponzi
finance. In the situation where the debt-capital ratio is sufficiently high, the rate

15

There is a large difference between the earlier studies that consider the dynamics of thefinancial structure of firms (Foley, 2003; Meirelles and Lima, 2006; Lima and Meirelles,
2007; Charles, 2008a, 2008b; and Nishi, 2012) and our model. The results of the former
studies show that the firms’ financial position keeps to stay in either the speculative or Ponzi
finance regime once the endogenous variables converge to the steady-state equilibrium. In
contrast, the key point of our model is that the financial position shows a cyclical behavior
between the two regimes, because a mature economy with debt accumulation produces
cyclical movements of both real and financial variables.
23

of capital accumulation starts to decrease due to the property of debt burden andthe economy returns to its initial position, in which the capital-effective labor
supply ratio is reduced.
[Figures 6, 7, 8, and 9 around here]
Furthermore, the range of fluctuations in the business cycle depends on theparameters of the model. If we decrease the retention ratio from s f  0.8 tos f  0.79 keeping all other parameters unchanged, the transitional dynamics ofeach variable would change from the solid lines to the dotted lines in Figures 6, 7,
8, and 9; a decline of the retention ratio would suppress the range of the cycle.
One reason for this is that the absolute value of J 11 J 22  J 12 J 21 , which is theinstability factor of the steady state that we have shown in section 3.3, is reduced
by a decrease in the retention ratio. We can use similar analysis for the otherparameters; a rise in the profit share, a decrease in the interest rate, and anincrease in the natural growth rate contributes to suppression of the range offluctuation, the stability of the economy. These results are summarized in Table
2. 16[Table 2 around here]
6. Conclusion
This paper has presented a Kaleckian model in which both finance and labor
supply constraints influence economic growth. Using this model, we haveinvestigated analytically the impacts of changes in the retention ratio, profit share,
rate of interest, and natural rate of growth on capital/debt accumulation and on
firms’ financial structure in a mature economy. Our results are summarized as

16

Note here that our numerical simulation results are obtained on the basis of plausible

parameter constellations. As we described above, the range of fluctuation depends on theabsolute value of J 11 J 22  J 12 J 21 . However, we cannot find the analytical results of the
impact of a change in a parameter on the absolute value of J 11 J 22  J 12 J 21 because ofcomplicated calculations.
24

follows.
Stability analysis . If the rate of capital accumulation exhibits a debt-led regime,

the steady-state equilibrium is locally stable. On the other hand, if the rate ofcapital accumulation exhibits a debt-burdened regime, there exist three cases: (1)the steady-state equilibrium is locally stable, (2) the steady-state equilibrium islocally unstable, and (3) a limit cycle occurs when the adjustment speed of the rateof accumulation approaches to some value.
Comparative statics analysis . A decrease in the retention ratio raises the rateof capacity utilization, whereas it makes the financial structure fragile. Moreover,it increases the capital-effective labor supply ratio and employment rate in thedebt-led

case.

However,adeclineintheretention

ratio

reducesthecapital-effective labor supply ratio in the debt-burdened case; consequently, theeffect of a decrease in the retention ratio on employment is ambiguous. Moreover,an increase in the profit share and a decrease in the interest rate reduce the rate ofcapacity utilization and improve the financial position of firms. Then, these leadto a decline in the capital-effective labor supply ratio and employment rate in thedebt-led case. In contrast, an increase in the profit share and a decrease in theinterest rate raise the capital-effective labor supply ratio, and accordingly thesehave an ambiguous effect on the employment rate in the debt-burdened case.
Finally, a rise in the natural rate of growth caused by increases in the laborproductivity growth and labor supply growth has a positive impact on the financial
structure of firms, although its impact on the rate of capacity utilization,
capital-effective labor supply ratio, and rate of employment is not clear, at leastfrom an analytical point of view.
Transitional dynamics of financial structure. In the steady-state equilibrium,

the financial structure of firms becomes speculative finance irrespective ofwhether the economy exhibits debt-led or debt-burdened growth. However, there
is a difference in the transitional dynamics of the financial position between thedebt-led case and debt-burdened case. Our numerical simulations have revealed
that in the debt-led case, a relatively large shock brings on an overshoot to thedebt-capital ratio, and accordingly the financial structure temporally goes into aPonzi finance regime; on the other hand, if there occurs a limit cycle in thedebt-burdened case, the financial position circulates between speculative finance25

and Ponzi finance regimes. Furthermore, we have clarified that decreases in theretention ratio and interest rate and increases in the profit share and natural rate of
growth suppress the range of cyclical fluctuations.
Finally, we sum up the total effects of pro-shareholder income redistribution
on the macroeconomic performance in the financialization era. As we have shown,a decrease in the retention ratio has both advantages and disadvantages; it worsens
the financial structure of firms, but stabilizes the economy. In contrast, a rise in
the profit share contributes to the financial position as well as the stability of theeconomy. Thus, we are inclined to conclude that an increase in the profit share ismore effective for the economic performance than a decline in the retention ratio.
However, raising the profit share is not so easy, because it produces inconsistentredistribution for households in that a rise in the profit share may lead to adecrease in the wage income and an increase in dividend income. If the number ofcapitalist-type households who hope for an increase in the dividends are larger
than worker-type households who prefer a rise in the wage income, increases in
the profit share is likely to be realized. Otherwise, it is not likely. Therefore, thetotal assessment of income distribution under financialization largely depends on
the orientation of firms and choice of households.
Appendix: A sufficient condition for fulfilling Assumption 3
In what follows, we show that when the trend term of the target rate of capital
accumulation  is sufficiently large, Assumption 3 (i.e., a 2  0 ) is fulfilled. Oneof the coefficients of the characteristic equation, a 2 , is given bya 2  J 11 J 22  J 12 J 21  J 13 J 31~
 s f  1  s h     s h    k *  s f  1  s h   s h g *  s f s h i2
s f  1  s h   s h~
s f i   k * 1  s h   s h s h 1  s f    s f  1  s h   s h *  .
2s f  1  s h   s h~
k * g *  s f 1  s h i*s f  1  s h   s h





























(A1)

~
From equation (16), we have dk * / d  0 . Thus, the larger  is, the larger

the first term of the right-hand side in equation (A1). The second and third terms
can be rewritten as follows:26 J 12 J 21  J 13 J 31  







s f i 1  s h    s h  s f  1  s h   s h *  s h 1  s f  s  1  s   s ~
k s  1  s   s g  s s i 1  s 1  s  s  1  s   s 
2

f

h

*

h

*

f

h

h

f

h

h

.

(A2)

f

2

f

h

h

The sign of the first line of the right-hand side in equation (A2) is ambiguous,~
although the sign of the second line is strictly positive. From dk * / d  0 , if  issufficiently large, the second line becomes large and dominates the sign of theright-hand side; in other words,  J12 J 21  J13 J 31 is likely to become positive. Thus,a sufficiently large  satisfies Assumption 3.

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29

Figures and Tables
Figure 1: The retention ratio in Japan and the U.S. (1960–2009)

Source: Financial Statements Statistics of Corporations by Industry, Policy ResearchInstitute, Ministry of Finance, Japan. National Income and Product Accounts, Bureau of
Economic Analysis, U.S. Department of Commerce.

Figure 2: Transition of rate of capital accumulation in the debt-led case
g
0.04

0.03

0.02

0.01

0.00

0

500

1000

1500

2000

2500

3000

t

Figure 3: Transition of debt-capital ratio in the debt-led case

30

l
1.0
0.8
0.6
0.4
0.2
0.0

0

500

1000

1500

2000

2500

3000

t

Figure 4: Transition of capital-effective labor supply ratio in the debt-led case
1.0

é
k

0.8

0.6

0.4

0.2

0.0

0

500

1000

1500

2000

2500

3000

t

Figure 5: Transition of u  i in the debt-led casepu- il

0.012
0.010
0.008
0.006
0.004
0.002
0.000

500

1000

1500

2000

2500

3000

t

-0.002

Figure 6: Transition of rate of capital accumulation in the debt-burdened case
31

g
0.020

0.015

0.010

0.005

0.000
15000

16000

17000

18000

19000

20000

t

Figure 7: Transition of debt-capital ratio in the debt-burdened case
l
1.5

1.0

0.5

0.0
15000

16000

17000

18000

19000

20000

t

Figure 8: Transition of capital-effective labor supply ratio in the debt-burdened
case
30

é
k

25
20
15
10
5
0
15000

16000

17000

18000

19000

20000

32

t

Figure 9: Transition of u  i in the debt-burdened regime
pu- il

0.015
0.010
0.005
0.000

16000

17000

18000

19000

20000

t

-0.005-0.010-0.015

Table 1: Taxonomy of the financial structures of firms
Finance regime

Definition of each finance regime

Steady-state equilibriumHedge

u  g  i  1  s f u  i NoSpeculativeu  g  i  1  s f u  i  and

Yes

Ponzi

u  i  1  s f u  i Nou  i  1  s f u  i 

Table 2: The effect of parametric changes on the range of fluctuation

Range of fluctuation

sf



i

n

+

-

+

-

33