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

Formalizing Debt-led and Debt-burdened Growth Regimes
with Endogenous Macrodynamics of Minskian Financial
Structure: A Long-run Analysis

Hiroshi Nishi

Discussion Paper No. E-10-016

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

March 2011

Formalizing Debt-led and Debt-burdened Growth Regimes with
Endogenous Macrodynamics of the Minskian Financial
Structure:
A Long-Run Analysis
Hiroshi NISHI
Faculty of Economics, Hannan University,
5-4-33, Amami Higashi, Matsubara-shi, Osaka, 580-8502, Japan
E-mail: nishi@hannan-u.ac.jp
JEL classification: E12, E22, E44, O42
Keywords: Minskian financial structure, Economic growth, Post-Keynesian model
Abstract
In this paper, we formally derive a version of the Minskian taxonomy of
the firms’ financial structure (hedge, speculative, and Ponzi types), under the
economic growth context in the long run. As for the economic growth, we formalize the mechanism of debt-led (debt-burdened) growth where the economy
expands as the debt variables increase (decrease). By explicitly introducing therelationship between the finance growth regime and Minskian taxonomy in a
dynamic model, the model in this paper enables us to evaluate whether or not
the economic growth regime is sounded in terms of the firms’ financial positions.

1 Introduction
The purpose of this paper is to examine the dynamic characteristics of growth andfinance in the long run. It also aims at generalizing the finance growth regimes andconsidering their relationships with the Minskian financial structure. In this paper,
the finance growth regime indicates debt-led or debt-burdened growth, whereas the
Minskian financial structure refers to hedge, speculative, and Ponzi finance of the
Minskian taxonomy.
Many post-Keynesian studies have focused on the relationship between income
distribution and aggregate demand or growth regimes, which demonstrate either
wage-led or profit-led growth regimes (Bhaduri [2007]; Lavoie [2010]). The profitled growth regime indicates a phase wherein the output expands as the profit variables
increase. On the contrary, the wage-led growth regime indicates a phase wherein the
output expands as the wage variables increase. These studies are inspired from thediscussions on growth and distribution by Robinson, Kaldor, and Kalecki.
Minsky (Minsky [1975]; Minsky [1982]; Minsky [1986]) is another intellectual
source of post-Keynesian economics. Minsky explained the fluctuation in investment in a capitalist economy by focusing on not only the income distribution but
also the link between investment and finance. According to him, capital accumulation is aﬀected by the long-run expectation of an entrepreneur with regard to cash
flow. When economic booms continue in an economy and the entrepreneurs’ expectations become optimistic, we have active capital accumulation. However, withcapital accumulation, the firms are more dependent on external finance. As for the
result, the firms’ leverage ratio gradually increases, and their financial position thus
becomes fragile.
The Minskian and Kaleckian models with debt accumulation try to link economic
growth with the Minskian financial structure. Foley [2003] is a seminal paper thatexamined the Minskian taxonomy in the context of post-Keynesian growth theory.
He extended the Taylor and O’connell [1985] type of short-run growth model, andexplained that the economy tends to be closer to the Ponzi regime because of a low
growth rate and profit rate. On the basis of the model in Foley [2003], Lima andMeirelles [2007] also investigate the stability of debt and the interest rates dynamics
under the Minskian taxonomy. They showed given an interest rate that is lower than
the growth rate, the long-run equilibrium solution under the hedge finance would
be stable through the procyclical banking markup policy. In contrast, they showedthat the Ponzi regime would be unstable regardless of the types of interest rate policy. Moreover, Charles [2008b] presents a dynamic model with a Minskian financial
structure that tends to be unstable. He explained the unstable mechanism by intro1

ducing a parameter that describes the state of the financial structure à la Minsky.
Other noteworthy works in this field are on the formalization of economic growthin terms of debt-led or debt-burdened regimes. If a rise in the debt ratio (in a numberof cases, the debt-capital ratio) stimulates capital accumulation, then this capital accumulation pattern is referred to as a debt-led pattern. On the contrary, if the rise in
debt ratio restrains capital accumulation, the capital accumulation pattern is referred
to as a debt-burdened pattern.
Although the above studies on the Minskian taxonomy are examined in termsof economic growth, they do not consider the type of finance growth regimes in a
suﬃcient manner. For example, Lima and Meirelles [2007] assume that the saving
rates of productive and financial capitalists are equal in analyzing the macrodynamics
of the debt regime. As for the results, their model cannot capture the mechanism ofdebt-led and debt-burdened growth. On the other hand, the model of Foley [2003]
covers only debt-burdened growth, due to the depressing eﬀects of an increase in
the interest rate and capital inflow. Moreover, the models of Charles [2008a] andCharles [2008b] are viable only under debt-burdened growth, which is due to thelarge negative impact of the interest payments on investment. These studies can
explain only a phase of the finance growth regimes, and thus, are shallow.
In contrast, Hein [2006] and Hein [2007] allow both debt-led and debt-burdened
growth regimes. While the former is always stable, the latter is unstable in the longrun when the debt-capital ratio also evolves. However, the results obtained by Hein
[2007] are in question, since Hein and Shoder [2009] empirically shows that debtburdened growth actually exists in the U.S. and Germany. Thus, from an empiricalpoint of view, it is hard to consider that the debt ratio does not shrink or expand
infinitely. Sasaki and Fujita [2010] constructed a model that allows both stable debtled and stable debt-burdened growth regimes. They relaxed the assumption in themodels of Hein [2006] and Hein [2007] that the retention ration of the firms is unity,and proved that the long-run equilibrium value of the debt-capital ratio is positive
with plausible interest rates irrespective of whether the long-run equilibrium is debtled growth or debt-burdened growth. While this is a general model that allows both
stable debt-led and stable debt-burdened growth regimes, it still does not consider
the Minskian financial structure. The same is true for the recently popular stock-flow
consistent models (Lavoie and Godley [2001]; Dos Santos and Zezza [2008]; Treeck[2007]).
To summarize, the existing literature cannot generally consider the relationshipamong the debt-led and debt-burdened growth regimes and the Minskian financial
structure. Some works have revealed the mechanism of finance growth regimes2

(Hein [2006]; Hein [2007]; Sasaki and Fujita [2010]), without considering the Minskian taxonomy. Other works have examined the stability of the Minskian taxonomy(Foley [2003]; Lima and Meirelles [2007]; Charles [2008b]), without considering
the possibility of debt-led and debt-burdened growth regimes. Therefore, the relationship among the debt-led and debt-burdened growth regimes and the Minskian
financial structure is yet to be clarified.
The novelty of this paper is that we explicitly explain the relationship between thefinance growth regime and the Minskian taxonomy in a dynamic model, whereas related literature discusses these mechanisms separately. Thus, we formalize not only
the mechanisms of the finance growth regime more generally (i.e., our model formalizes both the debt-led and debt-burdened growth regimes) but also considers how
their association with the Minskian taxonomy (i.e., hedge, speculative, and Ponzi
regimes). Indeed, if these relationships are captured by a macrodynamic model, we
can understand whether or not the economic growth regime is sounded in terms of
the firms’ financial positions.
Our attempt also contributes to Minsky’s explanations in his financial instability
hypothesis. His argument with regard to this issue emphasizes that the debt ratio
increases with the boom in capital accumulation. In terms of the financial growth
regime, this implies that the economy is a debt-led growth regime. However, as explained above, the growth regime can be both debt-led and debt-burdened. Therefore,by considering them together, we can have a more detailed understanding of the possibility of the financial instability hypothesis in the case where capital accumulation
may expand as the debt ratio decreases.
The remainder of this paper is organized as follows. Section 2 presents the basic structure of our model, where we classify the finance growth regimes and the
Minskian financial structure. Section 3 examines the dynamic characteristics of thedebt-led and debt-burdened growth regimes, while focusing on their relationships
with the Minskian financial structure. Section 4 concludes the paper.

2 ModelThe following is a list of the main notations used in this paper. X: output (total
income), X ∗ : potential output, K: capital stock, E: eﬀective employment level, 1 − π:wage share, π: profit share, X ∗ /K = ν: potential output-capital ratio (constant andset to unity for simplicity), u = X/K: output-capital ratio (eﬀective demand), r =πu: profit rate, S : total savings, I: investment demand, g: actual rate of capital
accumulation in the short-run, gL : actual rate of capital accumulation in the long3

run, W: nominal wage, R: profit (net operating revenue), i: nominal interest rate, λ:debt-capital ratio, t: time.
The economy is closed and has no government fiscal expenditure. A single goodthat is used for both investment and consumption is produced with labour and capital, which are combined through a fixed coeﬃcient technology. For simplicity, we
assume that there is no technological change in production. Three classes—firms,
workers, and capitalists (who own the firms)—are supposed to exist in the economy.
The workers provide labour and receive wage income, with the wage bill being WE.
The firms receive net operating revenue R, which is the surplus over wages. Hence,
the functional distribution of income is given by
PX = WE + R,

(1)

where PX is the total income.
From equation (1), the relationship among profit rate, capacity utilization, andprofit share can be expressed asr = πu,(2)

where we assume the income distribution share to be constant. The profit rate r isthen defined as the net operating revenue R divided by the value of capital stock.
Following Lima and Meirelles [2007] and many post-Keynesian analyses, we
assume that the firms can make use of loans, which are financed by capitalists undera given interest rate, i. The firms must therefore pay debt service iD on their debt
stock, along with a dividend to the capitalists in each period. We assume the dividendrate to be (1− sF ), and the retention rate to be sF . Hence, the firms’ net profit is R−iD,
and the capitalists’ total income is (1 − sF )(R − iD) + iD.
We assume that the three classes diﬀerent saving behaviours. The firms retain aconstant fraction sF of their net profits, capitalists save a constant fraction sC of theirincome, while workers spend all their wage income. At the macroeconomic level,
the total savings S is composed of the profits retained by the firms and the savings
from the capitalists’ income. Therefore, the aggregate savings as a proposition of thecapital stock is given byS= sC [(1 − sF )(r − iλ) + iλ] + sF (r − iλ).
K

(3)

The firms make an investment plan I, which is given by the following desired
investment function:g=I= α + βr − γiλ,
K
4

(4)

where α, β and γ are positive parameters, and g denotes investment in the short run
as a ratio of the existing physical capital stock. α stands for the motivation to accumulate that might be aﬀected by the long-run economic trends. The profit rate r isdefined as the net operating revenue R divided by the value of capital stock. We also
assume that the investment is negatively aﬀected by iλ due to the interest payments.
In this setting, we basically follow Lima and Meirelles [2007]. The only diﬀerence
lies in the forms of investment function. In our model, the interest payments negatively aﬀect the capital accumulation rate, while in the model of Lima and Meirelles[2007], the interest rate restrains the capital accumulation rate.

2.1 Behaviour of the Model in the Short run
The short run is defined as a time span along which the capital stock K and the debt
stock D are taken as given. The disequilibrium between investment and savings isadjusted through the changes in capacity utilization.
The short-run equilibrium value of capacity utilization rate is obtained from eqs.
(3) and (4) as follows:u=

α + {sF (1 − sC ) − γ}iλ,π(∆ − β)(5)where ∆ = sF (1 − sC ) + sC . The short-run equilibrium is stable if the denominatorof this eq. (5) is positive. Since π ∈ (0, 1), the necessary and suﬃcient condition forstability is ∆ − β > 0. This is well known as the Keynesian stability condition, which
we assume to be the case. We also assume that the numerator of eq. (5) is positive,
which will ensure a positive equilibrium value of u.
Using the short-run equilibrium value of the capacity utilization rate, we canobtain the short-run profit rate and accumulation rate as follows:r=

α + {sF (1 − sC ) − γ}iλ,(∆ − β)g = A + Biλ,

(6)
(7)

α∆
sF (1 − sC )(β − γ) − γsC
and B =.
∆−β(∆ − β)Some characteristics should be mentioned regarding the short-run capacity utilization, profit rate, and capital accumulation rate1 . In our model, a rise in the profit
share will lead to a decrease in the capacity utilization. This is because the redistribution of income from the firms who save to the workers who do not, raises consumption demand, which increases capacity utilization. This case is referred to as

where A =1

The results below are explained by almost the same mechanism as in Lima and Meirelles [2007].

5

the stagnationist regime in the post-Keynesian literature. However, a change in theincome distribution share does not have any impact on the profit rate and the rate ofcapital accumulation. This is because an increase in the profit share will lower thecapacity utilization to the same extent.
In turn, the impact of a change in the debt-capital ratio on the short-run equilibrium values of capacity utilization and capital accumulation rate is ambiguous:[]∂usF (1 − sC ) − γ
=
i,
(8)
∂λ
π(∆ − β)[]∂gsF (1 − sC )(β − γ) − γsC= Bi =
i.
(9)
∂λ
∆−β
A change in the debt-capital ratio also aﬀects the level of eﬀective demand. If thereaction coeﬃcient of investment to interest payments γ is larger than the productof the firms’ retention ratio and the capitalists’ propensity to consume sF (1 − sC ),an increase in the debt-capital ratio decreases the capacity utilization rate. In the
opposite case, an increase in the debt-capital ratio raises the capacity utilization rate.
The short-run finance growth regimes depend on the profit eﬀect on investment, in
addition to the interest payment and the savings parameters. If sF (1 − sC )(β − γ) −γsC > 0, then the short-run finance growth regime is debt-led growth. In this case, anincrease in the debt-capital ratio raises the capital accumulation rate. In the opposite
case ((sF (1 − sC )(β − γ) − γsC < 0)), the short-run finance growth regime is debtburdened growth. Here, an increase in the debt-capital ratio decreases the capital
accumulation rate.

2.2 Behavior of the Model in the Long Run
In the short run, the capacity utilization adjusts the imbalance in IS . As a result,the short-run capital accumulation rate is determined as g = A + Biλ. However, we
suppose that in the long run, with capacity utilization and profit rate as determined in
the short run, the capital stock K and the debt stock D also evolve2 . As both the debt
stock and the capital stock change, the long-run capital accumulation rate and debt
ratio, too, change.
We consider the evolution of the long-run rate of capital accumulation gL withthe dynamics of the debt ratio λ. Following Foley [2003], the long-run rate of capital
accumulation is assumed to be determined according to the dynamic form of the2

The post-Keynesian macrodynamic models with such periodic analysis include the household
debt models in Dutt [2005] and Dutt [2006b] and the model in Charles [2008b].

6

short-run accumulation rate3 . That is, in the long run, the dynamics of the capital
accumulation rate follow the change in the short-run accumulation rate due to thefinancial activity over time. This is obtained by diﬀerentiating eq. (7) with respect totime:g˙L = Biλ̇.

(10)In addition, we suppose that the firms’ new loans in each period Ḋ = dD/dt are
equal to the diﬀerence between the long-run capital accumulation and the firms’ net
profit, as we will derive later. That is, Ḋ = gL K − (rK − iD). Since λ = D/K, we haveλ̇ = gL − r + iλ − gL λ.

(11)Eqs. (10) and (11) define the long-run economic dynamics. These two equations
comprise the so-called zero root system wherein one of the eigen values of the Jacobian matrix is zero. In a two-dimensional model, the other eigen value is the trace of
the Jacobian matrix4 . In the long-run steady state wherein g˙L = λ̇ = 0, we havegL =r − iλ.
1−λ(12)Since the profit rate r is determined in the short-run eq. (6), in the long-run steadystate, we get
gL =

[
]
1
α − (sC + γ − β)iλ ,(1 − λ)(∆ − β)(13)where we assume sC + γ − β > 0, so that we can obtain economically meaningful
solutions. Using this equation, we will depict below the steady state locus of thiseconomy in the (λ, gL ) plane. By diﬀerentiating eq. (13) with respect to λ, we get
the slope of the steady state locus:
[
]
∂gL
1
=
α
−
(s
+
γ
−
β)i
,
C
∂λ
(1 − λ)2 (∆ − β)(14)All points on this locus constitute the long-run steady state. However, we have toconsider another condition that determines the transitional dynamics. From eq. (10),3In Foley [2003], the short-run capital accumulation is determined by a given interest rate and thestate of confidence. The long-run accumulation rate is determined by the exogenous target equilibrium
rate of accumulation, while the dynamics are derived from the short-run capital accumulation rate.
4
Giavazzi and Wyplosz [1985] explain the solution of the zero root system. Bhaduri [2007], Dutt[2006a], and Lavoie [2010] are the dynamic analyses using the zero root system in the post-Keynesian
economics. However, these studies examine growth and distribution, and not financial issues.

7

we have g˙L = Biλ̇. Therefore, the equation for the transitional dynamics is obtainedby integrating this equation with respect to time:gL (t) = Biλ(t) + gL (0) − Biλ(0),(15)

where the sign of B determines whether the growth regime is debt-led or debtburdened. In this system, it is easy to understand that the dynamics for the long-run
steady state depend on the initial values of gL and λ in eq. (15). Hence, the long-run
steady state of the growth regime has path dependency. If the initial condition varies,
the long-run position of the steady state also varies accordingly.
The local stability of the dynamical system eqs. (10) and (11) can be examinedthrough a routine calculation of the Jacobian matrix. However, we consider a larger
neighbourhood domain of the steady state, focusing on global stability. λ ∈ (0, 1) isassumed as an upper and lower limit for the debt ratio. We state the following scalarfunction in order to examine global stability:
1
V(t) = (gL − g∗L )2 .
2

(16)This scalar function satisfies the condition for using the second method of Liapunov (Gandolfo [1997])5 . Therefore, stability within the domain is guaranteed bythis method.
Substituting eq. (16) in eqs. (15) and (13), and diﬀerentiating eq. (16) withrespect to time, we obtain
(
)
[
]
1
α − (sC + γ − β)i2˙V(t) =(gL − r + iλ − gL λ) Bi −
.
(17)
1−λ
(1 − λ)2 (∆ − β)α − (sC + γ − β)i< 0.
(1 − λ)2 (∆ − β)While B determines whether the growth regime is debt-led or debt-burdened, thesecond term in the large parentheses in equation (14) represents the slope of long-run
steady state locus. In other words, the combination of the type of growth regime andthe slope of the long-run steady state locus plays an important role in global stability.
As we have seen, the sign of B can be either positive or negative depending on thefinance growth regime. As for α − (sC + γ − β)i, it can be either positive or negative
depending on the Minskian financial structure, as we will explain later. From thediscussion above, we get the following proposition.
Therefore, for λ ∈ (0, 1), global stability requires Bi −In fact, it satisfies Theorem 23.1 in Gandolfo [1997]. For λ ∈ (0, 1), all partial derivatives are
assumed to exist with respect to gL − g∗L , and are deemed to be continuous. In addition, V(t) is positive
definite.
58Proposition 1. The global stability of a dynamic system is assured, if the absoluteα − (sC + γ − β)ivalue of Bi is smaller than that of.
(1 − λ)2 (∆ − β)In this section, we showed that the long-run steady state of the growth regime
has path dependency. We also presented a condition for the global stability of the
dynamics of this economy. In the next section, we will show that the long-run steadystate of the locus is also aﬀected by the Minskian financial structure, which is related
to global stability.

2.3 Minskian Financial Structure
Foley [2003] and Lima and Meirelles [2007] formalized what Minsky descriptively
explained as the financial instability hypothesis. They classified the conditions forhedge, speculative, and Ponzi regimes, and examined the dynamic stability of eachregime.
Following their studies6 , we set the cash flow identity such that it equates thefirm’s source of funds from net operating revenue R and new borrowing that is thechange in debt Ḋ, to its usage of funds for long-run investment gL K and interest
payments iD. Therefore, we haveḊ = gL K − R + iD.

(18)

Using this relationship, we define the Minskian financial structure as follows:
Hedge financial structure The hedge financing firms can fulfil their contractual
payment obligations through their net operating revenue obtained as the result of
economic activity in the short run. By considering current borrowing and debt service
as contractual payment obligations, we formalize the hedge finance by the following

6Here, we employ the accounting categories of Lima and Meirelles [2007]. By abstracting fromthe payment of dividends for simplicity, the cash flow identity equates the firms’ source of funds fromnet operating revenue R and new borrowing Ḋ, to its usage of funds for long-run capital accumulation
gL K and debt service iD. We then have R + Ḋ ≡ gL K + iD. Hence, the change in debt is derived byḊ = gL K − R + iD. By normalizing through the positive capital stock K, we can express the change
in debt as in eq. (18).

9equation7 :R ≥ Ḋ + iD.

(19)

We will examine this condition on per capita basis. By dividing both sides of eq.
(19) with positive K, we get:R/K ≥ Ḋ/K + iD/K.

(20)

Using eqs. (6) and (18), we investigate the following equation:
R/K − (Ḋ/K + iD/K) = r − (gL − r + iλ + iλ)(α − (sC + γ − β)iλ)=(1 − 2λ).
(∆ − β)(1 − λ)(21)Therefore, with α − (sC + γ − β)iλ) > 0, the financial structure is hedge finance
if 0 < λ ≤ 1/2, and is not hedge finance, if λ > 1/2. This result is suggestive, since
it implies that the firms must ensure that their debt ratio is less than half, in order toretain hedge finance.
Speculative financial structure According to Minsky’s characterization, the speculative firms can meet their debt service even as they are unable to repay the principle
from their net operating revenue. Our model formalizes speculative finance as a casewhere a firm’s net operating revenue can cover its debt service, while the debt ratio
is more than half (i.e., R/K < Ḋ/K + iD/K). The mathematical formalization in the
speculative financial structure is this as follows:R ≥ iD.

(22)

As in the above exercise, by dividing both sides of eq. (22) with positive K, we
consider the condition for speculative finance:
R/K ≥ iD/K.

7

(23)

Minskian studies varies in their mathematical formalization of the Minskian taxonomy: ((Minsky[1986]), Foley [2003], Lima and Meirelles [2007] and Charles [2008b]). In particular, our definitionof hedge finance is diﬀerent from that employed by Foley [2003] and Lima and Meirelles [2007].
While firms do not borrow in the hedge financial structure in their models (i.e., Ḋ < 0), the negativeinterest payment eﬀect remains. In contrast, our model allows the firms to borrow even in the hedge
financial structure. This modification will be persuasive when we employ an investment function with
the negative interest payment eﬀect.

10

Using eq. (6), we can obtain the boundary for speculative finance:
R/K − iD/K = r − iλ=1
(α − (sC + γ − β)iλ).
(∆ − β)(1 − λ)(24)Therefore, given λ > 1/2, if condition α − (sC + γ − β)iλ ≥ 0 is satisfied, the financial
structure is of the speculative type.
Ponzi financial structure In Minsky’s original argument, the Ponzi firms are unable to pay not only the principle but also the debt service from their net operating
revenue. In other words, the Ponzi firms are borrowing to pay part of their debt
service. Such a condition for Ponzi finance can be derived directly from the above
results. This means that a firm’s net operating revenue cannot cover its debt service.
The mathematical formalization in the Ponzi financial structure is this as follows:
R < iD.

(25)

By referring to eq. (24), we can obtain the boundary for Ponzi finance:
R/K − iD/K = r − iλ=1
(α − (sC + γ − β)iλ).
(∆ − β)(1 − λ)(26)

Therefore, given λ > 1/2, if condition α − (sC + γ − β)iλ < 0 is satisfied, the financial
structure is of the Ponzi type.
It is shown that the financial structure can be distinguished according to the debt
position and the interest rate. We now summarize the basic condition. If 0 < λ ≤ 1/2,
the financial structure is hedge finance. If 1/2 < λ ≤ α/(sC + γ − β)i, the financial
structure is speculative finance. If 1/2 < α/(sC + γ − β)i < λ, the financial structure
is Ponzi finance.

2.4 Growth Trajectory and Financial Structure
Our model involves the endogenous transformation of the Minskian financial strucαture. In particular, it shows two cases with regard to the regime change. As(sC + γ − β)i
describes the boundary between the speculative and Ponzi financial structures, we denote it by λS P . In addition, the relationship between λS P and 1 plays an importantrole in the endogenous transformation of the Minskian financial structure.
Given a constant positive parameter α, we can depict the relationship among theslope of the steady state locus, the Minskian financial structure, and the economic
11

growth rate as in Figure 1. In this figure, we define a function R j (λ) = α−(sC +γ−β)iλ,which concerns the steady state path eq. (13), and the financial structure of (21), (24),and (25). Using this function, we can integrally consider the relationships between
the growth trajectory and the change in financial structure. The slope of the steadystate locus, g˙L = λ̇ = 0, can be distinguished by evaluating the sign of R j (λ) at λ = 1.
If R j (1) > 0, the slope is upwards, but if R j (1) < 0, the slope is downwards.
R j = α − (sC + γ − β)iλg∗L > 0
0

1/2

λS P

1

λS P

g∗L < 0λ
R2 (λ)R1 (λ)
Figure 1: Discriminant for the Slope of the Steady State Locus, and the Economic
Growth Rate
Note: R j (λ) : i = 1, 2 is defined on the basis of α − (sC + β − γ)iλ; λS P is given by α/(sC + γ − β)i,
which distinguishes between the speculative and Ponzi financial structures. For 0 < λ ≤ 1/2,
the financial structure is of the hedge type; for 1/2 < λ ≤ λS P , it is of the speculative type; for
λS P < λ, it is of the Ponzi type. As this figure indicates, if λS P > 1, the Ponzi regime does notarise and the slope of the steady state locus g˙L = λ̇ = 0 is always upwards. On the other hand,
λS P < 1 results in an endogenous change in the Minskian financial structure and the slope of thesteady state locus g˙L = λ̇ = 0 is always downwards.

Using eq. R j (λ), we distinguish some cases. If α/(sC +γ−β)i > 1, α−(sC +γ−β)i >0 and the slope of the steady state locus in eq. (14) is always positive. As R2 (λ)
in Figure 1 indicates, this parametrical configuration does not result in the Ponzi
financial structure given the possible debt ratio λ ∈ (0, 1). The financial structure isof the hedge type for 0 < λ ≤ 1/2, and of the speculative type for 1/2 < λ < 1. Wecollectively refer to these cases as the non-Ponzi regime.
On the other hand, if α/(sC + γ − β)i < 1, then α − (sC + γ − β)i < 0 and the slopeof the steady state locus in eq. (14) is always negative regardless of the financial
structure. As R1 (λ) in Figure 1 indicates, this parametrical configuration allows three
types of Minskian financial structure. For 0 < λ ≤ 1/2, the financial structure isof the hedge type; for 1/2 < λ ≤ λS P , the financial structure is of the speculative
type. For these two financial structures, the growth rate at the steady state (13) is still
12positive. However, for λS P < λ, the financial structure is of the Ponzi type, and thegrowth rate at the steady state is negative. We refer to this case as the Ponzi regime.
λ:(endogenous)
1
PWP

S NP
S WP

1/2
HNP

α
λS P = (sC +γ−β)iHWP0i:(exogenous)
iNW

imax

Figure 2: Condition for the Minskian Financial Structure
2αα
and imax = (sC +γ−β); H refers to the hedge financial structure; S , the speculaNote: iNW = (sC +γ−β)tive financial structure; P, the Ponzi financial structure. These are the conditions at the long-run
steady state. The subscript NP implies that the regime is of the non-Ponzi type, the subscript
WP implies that the financial structure is of the Ponzi type. The curve derived by function λS P
distinguishes the area between the speculative finance and Ponzi financial structure.

Interestingly enough, the above finance conditions aﬀect the growth rate in thelong-run steady state. Under a non-Ponzi regime, the condition for the hedge andspeculative financial structure always ensures that α − (sC + γ − β)iλ > 0, whichindicates that the long-run equilibrium capital accumulation rate in eq. (13) is always
positive. On the contrary, under the Ponzi regime, α − (sC + γ − β)iλ < 0, whichindicates that the long-run equilibrium capital accumulation rate in eq. (13) may be
negative. In this sense, the long-run growth rate is not independent of the financial
structure.
The change in the financial structure can also be explained in terms of the interest
rate and debt ratio, as in Figure 2. If the interest rate is smaller than iNW , the economy
does not include the Ponzi financial structure, regardless of the value of debt ratio.
In this case, for λ ∈ (0, 1), λ is always smaller than λS P that distinguishes the area
for the Ponzi regime. In this case, given an interest rate, the financial structure cantransform only between the hedge and speculative financial structure according to thedebt level. On the other hand, when the interest rate lies between iNW and imax , theeconomy involves three types of financial structure including the Ponzi finance. Ifi ∈ (iNW , imax ), the financial structure changes according to the value of λ. While for13λ ≤ 1/2, the financial structure is always of the hedge type, for λ > 1/2, it changes
depending on the relationship with λS P . If 1/2 < λ ≤ λS P , the financial structure is
still speculative, but in the case of 1/2 < λS P < λ, it results in the Ponzi finance.
Thus, if the interest rate is relatively high (i ∈ (iNW , imax )), the financial structure cantransform among the hedge, speculative, and Ponzi finance types according to thedebt ratio. Thus, a rise in the debt ratio leads the economy only to the speculative
finance when the interest rate is relatively low. However, if the interest rate is high,an increase in the debt ratio easily leads to the endogenous transformation of the
Minskian financial structure from the hedge to Ponzi type.
In the next section, we will investigate the dynamic stability of the debt-led anddebt-burdened growth regimes under each Minskian financial structure.

3 Growth Regimes and Endogenous Macrodynamics
of the Minskian Financial Structure
3.1 Dynamics of the Finance Growth Regimes: Non-Ponzi Caseαα
> 1 and> i (in(sC + β − γ)i(sC + β − γ)
terms of the interest rate). In this case, the slope of g˙L = λ̇ = 0 is always positive,
from eq. (14). Under these conditions, the finance growth regime can either be debtled if B > 0 (for example, due to a large sF ), or be debt-burdened if B < 0 (forexample, due to a small sF ). Let us consider the global stability of each case. Figure
3 illustrates the case of debt-led growth under a non-Ponzi financial structure.
The conditions for the non-Ponzi case are

Proposition 2. Debt-led growth under a non-Ponzi regime is conditionally stable.
Proof. The sign of Bi is positive under debt-led growth, whereas the slope of theg˙L = λ̇ = 0 locus is also positive. Therefore, if and only if the absolute value of Bi issmaller than that of g˙L = λ̇ = 0, global stability is assured.
Suppose first that global stability is assured. As the arrows in this figure indicate,if one starts from initial position A, the transitional dynamics trace the solid line
depicted by eq. (15). When the global stability condition is assured, a possible caseis that it leads to point E1 where the locus g˙L = λ̇ = 0 intersects the transitional
dynamics line. On the contrary, if one starts from initial position B, the transitional
dynamics trace the solid line from B, which too is depicted by eq. (15). When the
global stability condition is assured, it leads to point E2 , where the locus g˙L = λ̇ = 0
intersects the transitional dynamics line. The steady state in the long run diﬀers
according to the initial position, and thus, the dynamics have hysteresis.
14g˙L = λ̇ = 0

gL

E1
Speculative

A

B
E2
Hedge
0

1/2

1

λ

Figure 3: Debt-led Growth under a Non-Ponzi Regime
Note: The hedge finance corresponds to the steady state locus defined in λ ∈ (0, 1/2), whereas
the speculative finance corresponds to the steady state locus defined in λ ∈ (1/2, 1). The solidlines depict the stable path to the steady state, and the broken lines depict the unstable path.

As proposition 2 stipulates, however, convergence may not occur. If the slopeof the transitional dynamics is much larger than that of the locus g˙L = λ̇ = 0, we
have a state where both the economic growth and debt ratio are low. For example,
when the economy starts from initial position B with a high Bi, it may ride on theunstable path depicted by the broken line. The path from B with a high value of Bi
does not intersect the locus g˙L = λ̇ = 0. Thus, the cumulative contraction of thesetwo variables continues without some policy stimulation for economic growth.
On the other hand, even if the economy starts from initial position A with a highBi, it will lead to a steady state. This is because the slope of g˙L = λ̇ = 0 also
increases, according to the rise in the debt ratio. As a result, the transitional pathintersects the locus g˙L = λ̇ = 0 when the debt ratio and the capital accumulation
rate are high (λ  1). This is mathematically explained as follows. The slope of thelocus g˙L = λ̇ = 0 is given by eq. (14), and we get lim ∂gL /∂λ = ∞. On the contrary,
λ→1
the value of Bi is always finite in the neighbourhood of λ = 1. Therefore, we get
˙ < 0 in eq. (17), and the stability condition is assured. Hence, when high debtV(t)led economic growth depends on the debt, the hedge financial structure can be neverrecovered.
We now investigate global stability in the case of debt-burdened growth under anon-Ponzi financial structure.
Proposition 3. Debt-burdened growth under a non-Ponzi regime is globally stable.

15

g˙L = λ̇ = 0
gL

E2

SpeculativeA
E3

E4
HedgeE1

B
0

1/2

1

λ

Figure 4: Debt-burdened Growth under a Non-Ponzi Regime
Note: The hedge finance corresponds to the steady state locus defined in λ ∈ (0, 1/2); the speculative finance corresponds to the steady state locus defined in λ ∈ (1/2, 1). The solid lines depict
the stable path to the steady state.

Proof. The sign of Bi is negative under debt-burdened growth, whereas the slope of
˙ <0
the locus g˙L = λ̇ = 0 is positive. Therefore, the global stability condition V(t)being assured in equation (17) is trivial.
Given proposition 3, convergence occurs in this regime. Some possible cases are
depicted in the phase diagram (Figure 4). The direction depends on the initial valueof the long-run growth rate and debt ratio, and the impact of debt on the economic
growth. Let us consider four cases. First, if the initial position is near the area
corresponding to the hedge financial structure, then the economy remains in this area
as long as the negative impact of debt burden on the economic growth is strong,i.e., the absolute value of Bi is large. This case is depicted by the path from A to
E1 . Second, even if the initial position is near the area corresponding to the hedge
financial structure, the speculative finance will arise because the negative impact
of debt burden on the economic growth is weak, i.e., the absolute value of Bi issmall. This is depicted by the path from A to E3 , wherein the dynamics involve an
endogenous change in financial structure between the hedge and speculative types.
Third, if initially, the debt ratio is high and the growth rate is low, the speculative
financial structure will arise due to the large negative impact of debt burden on theeconomic growth, i.e., the large absolute value of Bi. This case is depicted by the16path from B to E2 . Lastly, even from the same initial position as in the third case, thehedge finance will arise because the negative impact of debt burden on the economic
growth is weak, i.e., the absolute value of Bi is small. This case is depicted by thepath from B to E4 . This is also a case wherein the dynamics involve an endogenouschange in financial structure between the hedge and speculative types.

3.2 Dynamics of the Finance Growth Regimes: Ponzi CaseAs we have formalized above, if the interest rate is set at a high level, it may result inan endogenous change in the firms’ financial position from hedge to Ponzi. The conααditions for the Ponzi case are< 1 and< i (in term of the(sC + β − γ)i(sC + β − γ)interest rate). In this case, the slope of g˙L = λ̇ = 0 is always negative, from eq. (14).
Here, both debt-led (B > 0) and debt-burdened (B < 0) growth regimes are possible.
As in the previous section, let us consider the global stability of each regime. Figure
5 illustrates a phase diagram of debt-led growth under the Ponzi financial structure.
The stability of the debt-led growth regime under the Ponzi financial structure can beobtained as follows.
Proposition 4. Debt-led growth under the Ponzi regime is globally unstable.
Proof. The sign of Bi is positive under debt-led growth. On the other hand, the slopeof the locus g˙L = λ̇ = 0 is always negative. Therefore, Bi is always larger thang˙L = λ̇ = 0. As a result, this economy is globally unstable.

Using Figure 5, we consider some possibilities concerning the dynamic properties of this economy. In the macrodynamics of an economy with the Ponzi regime,
we can explain that the economy has hysteresis and that the steady state in the longrun diﬀers according to the initial position. However, as proposition 4 stipulates,
convergence never occurs in this case. If one starts from initial position A, the transitional dynamics trace the broken line depicted by eq. (15). Since this economy isglobally unstable, the transitional dynamic path from A never intersects the steadystate locus. As a result, it necessarily leads to point E1 , where the value of λ is close
to unity. In this example, we initially start from the hedge financial structure, which
will go through a high growth phase. However, if the economy originally is close to
the Ponzi financial structure (e.g., at point B), it will experience negative economic
growth, as indicated by the broken line to point E2 . In both cases, the firms’ financial
positions are never sounded, since the firms’ debt ratio is high and their equity ratio
is almost squeezed in these cases.

17

gL

E1

A
Hedge
C
E3

Speculative
D
1/2

0

λS P

1

λ

E2

E4

B
Ponzi
g˙L = λ̇ = 0
Figure 5: Debt-led Growth under the Ponzi Regime
Note: The hedge financial structure corresponds to the steady state locus defined in λ ∈ (0, 1/2);
the speculative financial structure corresponds to the steady state locus defined in λ ∈ (1/2, λS P );the Ponzi financial structure is on the steady state locus defined in λ ∈ (λS P , 1). The solid lines
depict the stable path to the steady state, and the broken lines depict the unstable path.

On the contrary, if one starts from initial position C, the transitional dynamics
trace the broken line that leads to point E3 . In addition, the transitional dynamic pathnear the area corresponding to the speculative financial structure (area near D) leads
to point E4 . As we have shown above, the direction of each path depends on thevalue of Bi. Therefore, if the impact of Bi is relatively strong—that is, the slope ofthe transitional dynamics is steep—we may easily have negative growth.
Finally, we consider the dynamic property of the debt-burdened growth regime
under the Ponzi regime.
Proposition 5. The global stability of the debt-burdened growth regime under thePonzi regime is conditionally stable.
Proof. The sign of Bi is negative under debt-burdened growth, whereas the slope ofthe locus g˙L = λ̇ = 0 is also negative under the Ponzi regime. Therefore, if andonly if the absolute value of Bi is larger than that of g˙L = λ̇ = 0, global stability is
assured.

18

gL

A
E4

Hedge

E1
E2
Speculative

E3
0

1/2

λS P

1

λ

B
Ponzi
E3
C g˙L = λ̇ = 0
Figure 6: Debt-burdened Growth under the Ponzi Regime
Note: The hedge financial structure corresponds to the steady state locus defined in λ ∈ (0, 1/2);
the speculative financial structure corresponds to the steady state locus defined in λ ∈ (1/2, λS P );the Ponzi financial structure is on the steady state locus defined in λ ∈ (λS P , 1). The solid lines
depict the stable path to the steady state, and the broken lines depict the unstable path.

As in the discussion above, dynamic stability can be distinguished according to
the absolute value of Bi and the slope of g˙L = λ̇ = 0. The transitional dynamics
from initial position A results in E1 (for example, in the stable case due to the large
absolute value of Bi). In contrast, the transitional dynamics from A never intersect
the locus g˙L = λ̇ = 0, and leads to E2 (for example, in the unstable case due to thesmall absolute value of Bi).
Interestingly, if the economy is near the area corresponding to the Ponzi financial
structure, its financial structure may remain of the Ponzi type. For example, whenthe economy starts from initial position B, it rides on the stable path depicted by
the solid line. Since the debt ratio is close to unity in this situation, the absolutevalue of the slope of the locus g˙L = λ̇ = 0 as calculated by eq. (14) is close toinfinity. However, the absolute value of Bi remains finite even in the neighbourhood
˙ < 0 in eq. (17), and the stability condition isof λ = 1. Therefore, we get V(t)assured. Thus, the transitional path intersects the locus g˙L = λ̇ = 0 when the debt
ratio is high and growth is negative (position E3 ). Hence, if the economy depends(by nature) on a high debt ratio, it is possible its financial structure remains of the
Ponzi type. The steady state point E3 is not desirable for the firms’ financial position
19and macroeconomic condition.
However, since the economy has hysteresis, other cases may also be possible.
Even if the economy initially depends on a high debt ratio, it may trace a financiallysounded stable dynamics path. An example is depicted by the solid line from position
C in Figure 6. In this case, even if the debt ratio is high at the beginning, the economy
may expand while reducing the debt ratio. As a result, the economy will lead to thesteady state E4 , where the financial structure is of the hedge type.

4 Conclusion
Thus far, the Minskian and Kaleckian models with debt accumulation have examined the mechanism of debt-led and debt-burdened growth regimes, and the Minskian financial structure separately. However, the interrelationships between the finance growth regimes and the Minskian financial structure have been ambiguous. In
contrast, this paper explicitly examines the relationship between the finance growth
regime and the Minskian taxonomy in a dynamic model. Our macrodynamic modelsheds light on whether or not the economic growth regime is sounded while focusing
on the firms’ financial positions.
Table 1 summarizes our results on the interrelationships between the finance
growth regimes and the Minskian financial structure. The main results are summarized as follows. (i) Given the parameters in the IS balance, the interest rate
determines the type of Minskian financial structure under both Ponzi and non-Ponzi
financial structures. In this sense, the monetary policy via the interest rate plays animportant role in preventing the deterioration of the firms’ financial structure. If theinterest rate is set as relatively low, only the hedge and speculative financial structures will arise. If it is set as relatively high, not only the hedge and speculative
financial structures, but also the Ponzi type will appear. (ii) The debt ratio and thegrowth rate of the economy change in the long-run dynamics, in which the finance
growth regimes concern the direction of these variables. According to the debt ratio
determined at the steady state, the financial structure of the economy is determined
finally. The higher the debt ratio, the more unlikely is the hedge financial structure
to arise. (iii) Our results show that whereas debt-led growth is conditionally stable
at the most, debt-burdened growth is conditionally stable at the least. Therefore, ourmodel can explain that a Minskian phase, where the debt ratio increases with capital
accumulation, may involve unstable dynamics. In addition, under the Ponzi regime,
nonconditional stability is not assured in both finance growth regimes.

20Non Ponzi Regime
Interest rate
0 < i < iNW
Hedge
0 < λ ≤ 1/2
1/2 < λ < 1
Debt ratio
Speculative
Ponzi−
Stability
Debt-led growth
conditionally stableDebt-burdened growthstablePonzi Regime
iNW < i < imax
0 < λ ≤ 1/2
1/2 < λ ≤ λS P
λS P < λ < 1unstableconditionally stable

Table 1: Minskian Taxonomy and Stablity of Economic Growthαα2α, iNW = (sC +γ−β)and imax = (sC +γ−β).
Note: λS P = (sC +γ−β)i

Introducing the Minskian taxonomy in a dynamic model plays an important role
for the post-Keynesian economic growth theory, because it enables us to evaluate
whether or not the firms’ finance position is robust in the process of economic growth.
Even if an economy attains high growth, it is not financially desirable as long as thefinancial structure has the momentum to be fragile.
We hope the results in this paper will provide useful foundations for further research, as there exist only a few works on the post-Keynesian economic growth andthe financial structure. For this purpose, some extensions are required. For instance,
our model can evaluate the Minskian financial structure only in the steady state.
Therefore, an extended model that can examine the Minskian financial structure in
the transitional dynamics should be presented. In addition, in our model, the Minskian financial structure changes from the hedge to speculative type at λ = 1/2. Thisboundary may be deterministic, even though it is economically understandable. This
result may come from the definition of the hedge financial structure in eq. (21). In
fact, the firms’ financial regime may be of the non-hedge type depending on the profitability conditions, even if their debt ratio is smaller than half. An extended modelthat captures the changing financial structure at a nondeterministic debt ratio will beinteresting. Lastly, our analysis focused on external finance via borrowing. However,other methods such as equity finance are possible in general. We did not included
these methods into our model for simplicity. If, however, we introduce these methods
into the model, then the growth trajectory will be also aﬀected by them, because thefirms will not necessarily depend on debt finance. These issues will be left for future
research.

21

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