Download 3. Debt Dynamics and Long

Debt Dynamics, Capital Accumulation and Macroeconomic Instability
Min-Chang Ko and Sangheon Lee
1. Introduction
Post Keynesian-Kaleckian(abb. PK) models of growth and distribution were developed
with the studies of Rothorn (1981), Dutt (1984), Taylor (1986), Amadeo (1986) and
others. They presented the stagnationist thesis that economic growth would be
obstructed by deteriorating distribution of income. Bhaduri and Marglin (1990)
extended this thesis by formulating a modified PK model where alternative regimes of
growth co-exist within a same theoretical space. They argue that, when income
distribution deteriorates, a growth rate is retarded in a stagnation regime but is
accelerated in an exhilaration regime.
The argument of Bhaduri and Marlin produced lots of discussions, and motivated
empirical studies of the relationship between income distribution and growth. The first
empirical study of PK growth model was given by Bowles and Boyer (1995). They
estimated savings, investment and net export functions of various countries, and argued
for alternative regimes of growth for different countries. According to their study, UK
and UK show wage-led growth but France, German and Japan show profit-led growth.
Their study was followed by others who adopt a similar method, e.g., Stockhammer and
Onaran (2004), Naastepad and Storm (2007), Hein and Vogen (2008), Stockhammer,
Onaran and Ederer (2009).
Theoretical studies, on the other hand, attempted to develop the arguments of Bhaduri
and Marglin by introducing monetary variables of interest rate and debt ratio. Monetary
1
PK models of growth and distribution were proposed by, among others, Taylor (1985),
Dutt (1992), Lavoie (1995a), and Hein (2003, 2006, 2007). Recent studies attempted to
combine Minsky’s financial instability hypothesis with PK arguments by introducing
debt dynamics into dynamic PK models, e.g., Charles (2008), Lima and Meirelles (2007)
and Lee and Ko (2011).
This paper considers what Lavoie (1995a) calls Minsky-Steindl model, a monetary PK
model. In this model Lavoie takes seriously the discussions of debts in Steindl (1952)
and Minsky (1975). He attempts to formalize their arguments, and examines how
growth and distribution are affected by monetary variables of interest rate and debt ratio
of firms. He first assumes that a debt ratio is constant in the short-run, and shows that a
higher interest rate would result in a higher rate as well as a lower rate of capital
accumulation. He calls the former situation a “puzzling case.” Let us call the latter
situation the conventional case. He also examines the effects of higher interest rate in
long-run situation of flexible debt ratio. He argues that, when a long-run equilibrium is
stable, the only possible situation is the puzzling case. In his model a conventional case
occurs only if a long-run equilibrium is unstable. Hein (2007) extends Lavoie’s model
to incorporate the effects of interest rate on income distribution, and show that Lavoie’s
main proposals are maintained. Minsky’s financial instability hypothesis suggests, on
the one hand, that a long-run equilibrium is unstable and, on the other hand, that an
interest rate, an accumulation rate, and a debt ratio rise simultaneously in a boom phase
of business cycles. In the arguments of Lavoie and Hein, a higher rate of interests brings
out a lower rate of accumulation and a higher debt ratio, i.e., Steindl’s paradox of debts,
but there is no Minskian instability.
The models of Lavoie and Hein do not formulate debt dynamics explicitly. In this paper
we will construct a dynamic system of monetary PK model of growth and distribution
which introduces the equation of debt dynamics. Our model is a developed version of
Lavoie (1995a) and Hein (2006, 2007), and is consistent with their arguments in the
2
short-run situation. But we will argue that, when a long-run equilibrium is stable, both
the conventional case as well as Lavoie’s puzzling case would possibly occur. We will
also connect these considerations with Minsky’s financial instability hypothesis, and
proposes some policy prescriptions to deal with a Minskian meltdown.
In the next section we present a monetary PK model of growth and distribution. It is
based on Lavoie (1995a) with some modifications. Section 3 presents a dynamic PK
model with an equation of debt dynamics. It is shown that a standard PK model would
explain the existence alternative accumulation regimes when debt services are
introduced. Section 4 discusses what our dynamic model suggests to Minsky’s
hypothesis and macro-economic instability. It also proposes some policy measures to
deal with a Minskian meltdown. The final section summarizes our arguments.
2. The Basic Model and a Short-run Equilibrium
Consider, for simplicity, a one-good economy with households and firms. Households
consist of workers and Kaldorian rentier-capitalists. Workers supply labor services to
firms, earn only wage incomes, spend all their incomes on consumption, and make no
savings. Firms employ workers, produce output, make investment, and accumulate
productive capacity. We follow Kaldor(1966, p. 311) to distinguish firms of production
units and rentier-capitalists of their financial sources. Firms demand funds to finance
investment expenditure. These funds are partly provided by internal reserves and partly
by external borrowings. Rentier-capitalists earn dividends as owners of firms'
productive capacity. They also earn interests as suppliers of firms' external finances.
Rentier-capitalists make savings out of these incomes, lend them to firms, and satisfy
firms' demand for external finances. Aggregate savings of this economy consist of firms'
reserves of current profits and savings of rentier-capitalists. A Keynesian equilibrium is
achieved through output adjustments at the point of investment equal to aggregate
3
savings. We further assume that a price level remains constant for given distributive
factors of a money wage rate and a profit share.
The aggregate real income (Y) is, by the accounting identity, a sum of real wage bills
(ωL) and real profits (Π).1 We assume that workers make no savings, and that rentiercapitalists adopt a same marginal propensity to savings (𝑠𝑟 ) for dividend and interest
earnings. Then real consumption expenditure (C) is given as the following equation (2).
Y = ωL + Π …………………………………………………………………………... (1)
C = ωL + (1 − 𝑠𝑟 )(1 − 𝜃)(Π − 𝑖D) + (1 − 𝑠𝑟 )𝑖𝐷 ………………………………….. (2)
θ is a retention ratio of firms out of net profits. Net profits are defined as gross profits
minus interest payments which, in turn, are determined by an interest rate (𝑖) and firms’
real debts (D) owed to rentier-capitalists. Therefore the second and the third term of
equation (2) depict real consumption expenditure of rentier-capitalists out of dividend
earnings and interest earnings respectively. We rearrange equation (1) and (2) and get
the following equation (3). Here 𝑙 denotes a labor coefficient L⁄Y, and ℎ′ a ratio of
net profits over aggregate income (Π − 𝑖D)⁄Y. We assume that a labor coefficient 𝑙 is
a technically given constant in a short-period.
C = {1 − 𝑠𝑟 (1 − 𝜔𝑙) − θh′ (1 − 𝑠𝑟 )}𝑌 ……………………………………………..... (3)
The Keynesian condition of good market equilibrium is given by the equation condition
Y = C + ΔK where ΔK is investment. By substituting equation (3) for consumption,
1
It would be more accurate to say that aggregate income is a sum of aggregate wage
bills and aggregate profits. As we are mostly concerned with aggregate variables in a onegood economy, so the term aggregate may be safely omitted.
4
we get an equilibrium level of aggregate income Y ∗ .
1
′ (1−𝑠 ) ΔK
(1−𝑤𝑙)+θh
𝑟
𝑟
Y∗ = 𝑠
…………………………………………………………… (4)
The investment multiplier of our model is different from the traditional formula due to
two factors–a retention ratio out of net profits and interest payments on debts. It is
worth noting that these two factors do not affect the multiplier when a retention ratio is
zero or when there are no net profits. In both cases rentier-capitalists earn all gross
profits. It is also worth noting that we do not yet consider disbursements of retained
profits to finance investment or repay debts. If these financial channels are operative,
they affect debt dynamics of firms and so ℎ′ would change over time.
Post Keynesian models of growth and distribution focus upon growth rates, rather than
levels, of aggregate income. We need more analytical tools to locate the above equation
(4) within a meaningful dynamic model. As is well-known, e.g., Lavoie(1995b), the
standard PK model consists of three equations–a pricing equation, a profits–cost curve
and an effective demand curve. The PK pricing equation is founded on Kalecki’s
argument. It says that firms determine prices by mark-ups over average variable costs
while mark-up ratios are set to reflect degrees of monopoly. Let 𝑚 denote a mark-up
ratio. We simply assume that m is constant. Equation (6) represents a pricing equation
where 𝑝 denotes a price level and 𝑤 a money wage rate. It leads to the equation (6),
the relation between profit shares and mark-up ratios. We follow Taylor (1985) to
define capacity utilization u as a ratio of aggregate income Y over capital stock K.
Then equation (7) expresses the relation between capacity utilization, a mark-up ratio,
and a profit rate 𝑟.
𝑝 = (1 + 𝑚)𝑤𝑙 ………………………………………………………………………. (5)
5
Π
𝑌
𝑚
= 1+𝑚
Y
…………………………………………………………………..…………. (6)
𝑌Π
u = K = Π𝐾 =
1+𝑚
𝑚
𝑟 ……………………………………………………...………….. (7)
Consider that a real wage rate is exogenously given. Equations (5) and (6) determine a
mark-up ratio and a profit share independently of capacity utilization. Equation (7) then
says that, for a given real wage rate, a higher level of capacity utilization corresponds to
a higher profit rate. We may put together equation (4) and (7) and determine an
equilibrium level of capacity utilization and an equilibrium profit rate for exogenously
given values of real wage rate ω and accumulation rate ΔK⁄𝐾 .2 Alternatively, we will
introduce an autonomous investment function and determine equilibrium by the
Keynesian condition of savings equal to investment.
S = θ(Π − 𝑖D) + 𝑠𝑟 (1 − 𝜃)(Π − 𝑖D) + 𝑠𝑟 𝑖𝐷 ………………………………………... (8)
𝑔 𝑆 = 𝛼𝑟 − 𝛽𝑖𝑑 ……………………………………………………………………….. (9)
𝑤ℎ𝑒𝑟𝑒 𝛼 = 𝑠𝑟 + 𝛽, 𝛽 = 𝜃(1 − 𝑠𝑟 ), 𝑎𝑛𝑑 𝛼, 𝛽 > 0.
The saving function of equation (8) corresponds to the consumption function of
equation (3). It says that aggregate savings consist of firms’ internal retention out of net
profits and the savings of rentier-capitalists out of dividend and interest earnings. θ
represents for a retention ratio and 𝑠𝑟 for a propensity to savings of rentier-capitalists.
We divide both sides of equation (8) by capital stock K and derive equation (9). Here
𝑔 𝑆 and 𝑑 denote 𝑆⁄𝐾 and 𝐷⁄𝐾 respectively.
The equation (9) is a simplified version of Lavoie’s saving function. Lavoie (1995a)
2
We will denote the rate of time changes of a variable X as ΔX, and its percentage rate
as 𝑔𝑋 . When there is no confusion, we will simply omit the subscript X of 𝑔𝑋 .
6
assumes that rentier-capitalists take different propensities to savings for dividends and
interest earnings. We follows Charles (2008) and assume that rentier-capitalists apply a
single propensity to savings to their total financial incomes.
As for an investment function, PK economists usually present that an accumulation rate
of capital stock is a linear function of capacity utilization and a profit factor. The
positive effects of capacity utilization on investment denote the familiar Harrodian
principle of accelerator. The positive effects of a profit factor on investment represent
some financial considerations. When a profit rate is taken as a determinant of
accumulation rate, firms are assumed to finance investment partly from external sources.
Lower profit rates suggest that borrowing firms have weak capacity to create cash
inflows. Then lenders evaluate lower the credit worthiness of firms, and constrain credit
supplies to finance investment. In this case there appears the so-called stagnation thesis
within PK models of growth and distribution, i.e., a positive relationship between real
wage rates and accumulation rates.3 Bhaduri and Marglin (1990) present an alternative
investment function where a profit share is taken as a determinant of accumulation rate.
Then they show that there may appear a positive relation between profit rates and
accumulation rates.4 We do not follow Bhaduri and Marglin for the simple reason that a
profit rate is a better indicator to depict firm’s financial condition.
Gross profits are divided into interest payments and net profits, and net profits are
disposed into dividends and internal reserves. Lavoie (1995a) and Hein (2006) take a
rate of net profits as a determinant of accumulation rate, and Lavoie proposes that gross
profit rates and interest rates would affect investment differently. We will incorporate a
3
See, for details, Rowthorn (1981), Dutt (1984), Amadeo (1986), Lavoie (1995b).
4
See Hein (2007) for a recent discussion.
7
modified version of Lavoie’s idea. An interest rate is combined with a debt ratio in our
investment function, and this composite factor affects an accumulation rate negatively.
This setting would be more appropriate to depict the financial effects of interest
payments on investment. It also allows to examine the effects of debts on investment.5
We will propose in the next section that internal reserves are used to finance investment
partially, and discuss debt dynamics. Our investment function, equation (10), works as a
main channel of interaction between accumulation rates and debt dynamics.
𝑔𝑑 = γ0 + 𝛾𝑢 𝑢 + 𝛾𝑟 𝑟 − 𝛾𝑑 𝑖𝑑, 𝛾0 , 𝛾𝑢 , 𝛾𝑑 > 0 ………………………………………. (10)
This investment function says that financial constraints of firms are alleviated and more
credits are provided to finance investment alongside with higher profit rates and lower
debt ratios. The equilibrium of our macro-model is determined by equation (11), the
condition of savings equal to investment. We assume à la Lavoie (1995a) and Hein
(2006, 2007), that a debt ratio of firms is constant in the short-run. Then the values of
short-run equilibrium (𝑟 ∗ , 𝑢∗ , 𝑔∗ ) are given by equations (12), (13), and (14).
𝑔 𝑠 = 𝑔𝑑 ……………………………………………………………………………... (11)
𝑚(𝛾 −𝛾𝑑 𝑖𝑑+𝛽𝑖𝑑)
𝑢 (1+𝑚)−𝛾𝑟 𝑚
𝑟 ∗ = 𝛼𝑚−𝛾0
𝑢∗ =
……………………………………………………………….. (12)
(1+𝑚)(𝛾0 −𝛾𝑑 𝑖𝑑+𝛽𝑖𝑑)
𝛼𝑚−𝛾𝑢 (1+𝑚)−𝛾𝑟 𝑚
……………………………………………………………...
(13)
𝑔∗ =
5
𝛾0 𝛼𝑚−𝛾𝑑 𝛼𝑚𝑖𝑑+𝛾𝑢 𝛽(1+𝑚)𝑖𝑑+𝛾𝑟 𝛽𝑚𝑖𝑑
𝛼𝑚−𝛾𝑢 (1+𝑚)−𝛾𝑟 𝑚
……………………………………………… (14)
Our investment function may be taken as a revised formula of Steindl’s arguments.
Steindl (1952) considers investment as a function of capacity utilization, profit rates, and
firms’ debt ratios.
8
The followings are comparative static analyses of short-run equilibrium with respect to
changes of interest rate. The comparative static effects of changes in a mark-up ratio
would be similarly examined.
∂r∗
𝜕𝑖
∂𝑢∗
𝜕𝑖
∂𝑔∗
𝜕𝑖
(β−𝛾𝑑 )𝑚𝑑
= 𝛼𝑚−𝛾
𝑢 (1+𝑚)−𝛾𝑟 𝑚
(β−𝛾𝑑 )(1+𝑚)𝑑
= 𝛼𝑚−𝛾
𝑢 (1+𝑚)−𝛾𝑟 𝑚
=
………………………………………………………………. (15)
………………………………………………………………. (16)
[γ𝑟 𝛽−𝛾𝑢 𝛽(1+𝑚)−𝛾𝑑 𝛼𝑚]𝑑
𝛼𝑚−𝛾𝑢 (1+𝑚)−𝛾𝑟 𝑚
………………………………………………………… (17)
The output adjustment process of Keynesian equilibrium is stable when the elasticity of
savings with respect to capacity utilization is greater than that of investment,
𝑑𝑔𝑑 6
.
𝑑𝑢
𝑑𝑔𝑠
𝑑𝑢
>
We derive this stability condition, equation (19), from equation (7), (9), and (10).
We assume that this condition holds. Then the signs of comparative static analyses are
given as equation (19)~(22).7
𝛼𝑚 > 𝛾𝑢 (1 + 𝑚) + 𝛾𝑟 𝑚 ……………………………………………………………. (18)
β − 𝛾𝑑 > 0 ⇒
6
∂𝑟 ∗
𝜕𝑖
> 0,
𝜕𝑢∗
𝜕𝑖
> 0. …………………………………………………. (19)
The term elasticity is not an exact term to express the concerned relationship. But it
conveniently describes significations of the context without any confusion.
7
In equation (19)~(22) sign conditions of profit rates are different from those of
accumulation rates. Hence it may happen that a higher interest rate brings out a higher rate
of profits and a higher rate of accumulation. It suggests that alternative regimes of
accumulation would arise not only from the investment function of Bhaduri and Marglin
(1990) but also from our saving function.
9
β − 𝛾𝑑 < 0 ⇒
∂𝑟 ∗
𝜕𝑖
< 0,
𝜕𝑢∗
𝜕𝑖
< 0. …………………………………………………. (20)
γ𝑟 𝛽 − 𝛾𝑢 𝛽(1 + 𝑚) − 𝛾𝑑 𝛼𝑚 > 0 ⟹
γ𝑟 𝛽 − 𝛾𝑢 𝛽(1 + 𝑚) − 𝛾𝑑 𝛼𝑚 < 0 ⟹
∂𝑔∗
𝜕𝑖
∂𝑔∗
𝜕𝑖
> 0 ……………………………………. (21)
< 0 ……………………………………. (22)
The above results are equivalent to the analyses of Lavoie (1995a) except those
deviations due to the simplifying assumption of a single propensity to savings out of
financial incomes and the specification of capacity utilization as a determinant of
investment. It is obvious that a higher interest rate does not uniformly affect either profit
rates, capacity utilization, or accumulation rates. Consider, for example, that the
elasticity of savings with respect to an interest rate, β, is greater(smaller) than that of
capacity utilization, 𝛾𝑑 . Then a higher interest rate increases(decreases) a rate of profits
and capacity utilization.
Let us think of how a higher rate of interests may increase a rate of accumulation. 8 The
consumption function of equation (2) says that a higher interest rate raises the earnings
of rentier-capitalists and so their consumption expenditure. It immediately increases
capacity utilization and, if a sign condition is proper, a rate of accumulation would rise.
3. Debt Dynamics and Long-run Equilibrium
8
Hein and Ochsen (2003) construct a PK model of growth and distribution and present
an empirical analysis of how interest rates affect rates of accumulation and of profits. They
conclude that “a positive relation between interest rate variations and changes in economic
activity as well as in capital stock growth seem to be empirically possible in some countries
and in some time periods,” Hein and Oshsen (2003, p. 427).
10
Comparative static analyses of the previous section consider a Keynesian equilibrium
with the short-run condition of constant debt ratio. We will examine in this section debt
dynamics of how a debt ratio changes over a time sequence. Debt dynamics is a familiar
theme in the PK literature. For example, Lavoie discusses debt dynamics with
simplifying assumptions to the effect that “the new loans contracted by firms
correspond to the savings of bond holders,” Lavoie (1995, p. 168). We will explicitly
formulate the financial constraint of firms, and present a more general argument of debt
dynamics.
Firms would finance investment either through internal reserves or through external
borrowings. We may safely introduce banks to our model by separating rentiercapitalists into rentiers-industrial capitalists and rentier-financial capitalists. But it does
not make much difference even if firms are assumed to borrow directly from rentiercapitalists. So we maintain our previous model.
∆D + Π = (1 − θ)(Π − iD) + iD + ΔK …………………………………………….. (23)
∆D + θ(Π − iD) = ∆K ……………………………………………………………… (24)
∆d + d𝑔 + θ(r − id) = 𝑔 …………………………………………………………... (25)
Rentier-capitalists own titles of firms’ capital stock, and provide to firms external
finances. So the financial constraint of firms is given as equation (23). ∆D and ∆K
denote changes of firms’ debt and of capital stock respectively. The left- and the righthand side of equation (23) represent sources and uses of firms’ finances. It says that
gross profits of firms are disposed into dividends, interest payments, and internal
finances of investment expenditure. When internal finances are insufficient, firms
require external finances and raise debts. Equation (24) is reduced to equation (23). It
says that investment is financed by external borrowings and internal reserves.
Depending on investment expenditure greater or less than internal reserves, firms obtain
11
new debts or pay back old debts. Equation (25) is derived from equation (24). Here d
represents for a debt ratio of firms, 𝐷⁄𝐾 , and ∆d for a rate of time changes of d.
1
𝛽
r = α 𝑔 + 𝛼 𝑖𝑑 ……………………………………………………………………….. (26)
∆d + (𝑔 −
𝑎𝑔
α−β
𝛼
𝜃
𝜃𝑖) 𝑑 = (1 − 𝛼) 𝑔 ………………………………………………….. (27)
𝑎𝑏𝜃𝑖
𝑑 = 𝑔−𝑏𝜃𝑖 = 𝑎 + 𝑔−𝑏𝜃𝑖 ……………………………………………………………… (28)
𝑠 (1−θ)
𝑟
where 𝑎 = 𝜃+𝑠
𝑟 (1−θ)
> 0 and b = 𝜃+𝑠
𝑠𝑟
𝑟 (1−θ)
> 0.
Our method of debt dynamics is to examine how a short-run equilibrium moves in the
(𝑔, d) space. We define a long-run equilibrium condition as ∆d = 0.9 So a long-run
equilibrium represents the position where a short-run equilibrium remains standstill. We
rearrange the savings function of equation (9), and present it as a Cambridge-type
equation (26). Then we derive equation (27) from equations (25) and (26). The long-run
equilibrium locus, equation (28), is derived by imposing the condition ∆d = 0 to
equation (27). The long-run equilibrium locus is given as a hyperbolic curve. An point
on this curve may become a long-run equilibrium. But only those points will be realized
that satisfy the short-run equilibrium condition.
1
d = 𝜙𝑖 [−𝛾0 𝛼𝑚 + {𝛼𝑚 − 𝛾𝑢 (1 + 𝑚) − 𝛾𝑟 𝑚}𝑔] …………………………………… (29)
where ϕ = 𝛾𝑢 𝛽(1 + 𝑚) + 𝛾𝑟 𝛽𝑚 − 𝛾𝑑 𝛼𝑚.
We rearrange equation (14), a short-run equilibrium rate of accumulation, and get
equation (29). It expresses a debt ratio, d, as a linear function of accumulation rate 𝑔.
9
Our definition of long-run equilibrium is equivalent to Hein (2006, 2007) and Lavoie
(1995a).
12
We will call it Keynesian equilibrium line. Due to the stability condition of short-run
Keynesian equilibrium, the sign condition of the slope of this line depends on the sign
of ϕ. When ϕ is positive, the slope of Keynesian equilibrium line is positive. So a
debt ratio and an accumulation rate move in the same direction. When ϕ is negative, a
debt ratio and an accumulation rate move in the opposite direction.
The dynamic system of our study consists of equations (28) and (29). The economy is
assumed to exist on the Keynesian equilibrium line, and a long-run equilibrium is
determined at the intersection of this line and the long-run equilibrium locus. We
examined stability properties of long-period equilibrium in terms of phase diagram
analysis, and report the results in the following figures.10
<Figure 1> & <Figure 2>
Here the notations “d = f(𝑔)” and “𝑑̇ = 0” represent for the Keynesian equilibrium line
and the long-run equilibrium locus respectively. As is clearly shown, there exist two
different regimes of accumulation in our model. Each regime may have a stable longrun equilibrium with positive values of debt ratio and accumulation rate.
Now let us consider the effects of a higher interest rate on a stable long-run equilibrium
in the positive quadrant of (𝑔, d) space. When ϕ is positive, a higher interest rate
shifts the Keynesian equilibrium line downward and the long-run equilibrium locus
upward. Hence a long-run equilibrium rate of accumulation rises always. But it is not
obvious whether a long-run debt ratio rises or falls. The involved mechanism is as
follows. When an interest rate rises, the consumption expenditure out of interest
10
See Lee and Ko (2011) for the details of phase diagram analysis.
13
earnings rises. It raises capacity utilization and profits, and so brings out positive effects
on investment. As these positive effects dominate the negative effect of higher interest
rate on investment, so an accumulation rate rises.11 But the effect on a debt ratio is not
clear, since profits rise alongside with a higher accumulation rate. The so-called
Lavoie’s puzzling case may happen such that a higher interest rate increases both of an
accumulation rate and a debt ratio.
<Figure 3>
Consider the case that ϕ is negative. As an interest rate rises, the long-run equilibrium
locus shifts upward and the Keynesian equilibrium line shifts downward with a flatter
slope. Therefore the so-called Steindl’s case may happen such that a debt ratio rises and
an accumulation rate falls. A higher interest rate raises interest payments which, in turn,
would reduce investment and raise a debt ratio.12
<Figure 4>
Lavoie (1995a) assets that, in the long-run of flexible debt ratios, a long-run equilibrium
11
Consider, for convenience, the case of 𝛾𝑢 equal to zero. Then the condition ϕ > 0 is
reduced to
𝛾𝑟
𝛼
>
𝛾𝑑
𝛽
. As the sign condition α > β > 0 holds, so the negative effect of interest
payments on investment, 𝛾𝑑 , is less than the positive effect of profits on investment, 𝛾𝑟 .
12
Bhaduri and Marglin (1990) argues for the existence of alternative growth regimes in
terms of dual effects on investment of higher real wage rate. Their arguments would be
incorporated into our model through the analysis of changes in a mark-up ratio m. Our
discussion, on the other hand, shows the existence of alternative regimes in terms of
changes in an interest rate and the subsequent changes of debt ratio. In this sense our
model may be read as a financial version of what Bhaduri and Marglin (1990) argues.
14
is stable only in the puzzling case. He also argues that the Steindl’s case happens only
when a long-run equilibrium is unstable. Our above discussion does not support
Lavoie’s assertion. It shows that a stable long-run equilibrium may exist in which a
higher interest rate decreases an accumulation rate. By suggestion, the perspective of
Lavoie (1995a) and Hein (2006, 2007) would not be maintained that makes a
connection between the existence of alternative accumulation regimes and the stability
of long-run equilibrium. When a short-run Keynesian equilibrium is stable, a higher
interest rate would increase or decrease an accumulation rate depending on parameter
conditions. And this phenomenon may still occur even if a long-run Keynesian
equilibrium is stable.
4. An Interpretation of Macroeconomic Instability
The basic idea of Minsky’s financial instability hypothesis is based on the proposition
that an interest rate, an accumulation rate and a debt ratio would possibly rise
concurrently.
13
It says that the macro-economy becomes unstable when this
concurrence occurs. Lavoie (1995a) and Hein (2006, 2007) argue that such a
concurrence happens only if a long-run equilibrium is stable. Accordingly, they think
13
Minsky proposes that, in the boom phase of business cycles, firms form optimistic
expectations of the future. Accordingly, they tend to accept risks more willingly and increase
borrowings. As their leverage ratios rise, their financial conditions are transformed from
hedge, through speculative, to Ponzi status. A higher interest rate deteriorates their financial
conditions furthermore, and ultimately a Minskian meltdown occurs. Lavoie and Seccareccia
(2001, p.87) argues in this context that Minsky’s financial instability hypothesis presupposes
“a positive correlation between the aggregate leverage ratio of the non-financial corporate
sector and real output growth.”
15
that Minsky’s hypothesis is not persuasive in the macroeconomic context. Our
arguments in the previous sections suggest that not only the puzzling case but also the
Steindl’s case when a long-run equilibrium is stable. In this section we will consider a
comparative dynamic analysis of how a higher interest rate would cause
macroeconomic instability. The main idea is to examine the shifts of long-run
equilibrium and show the emergence of structural instability.
Let us first consider how a long-run equilibrium shifts with a higher interest rate in the
Steindl’s case. It is shown in Figure 5 that a higher interest rate shifts a long-run
equilibrium from E to E′. So a debt ratio rises and an accumulation rate falls. However,
when an interest rises successively, there arises a situation where a long-run equilibrium
does not exist. It corresponds to the Keynesian equilibrium line of d′′ = f(𝑔) and the
long-run equilibrium locus of 𝑑̇ ′′ = 0 in Figure 5. A short-run equilibrium is located
on the Keynesian equilibrium line, and moves to the northwest direction. Therefore a
debt ratio rises and an accumulation rate falls continuously in the context of long-run
disequilibrium. We may call this situation a stagnation regime where a Minskian
meltdown would be a final solution.
<Figure 5>
A similar story holds in the Lavoie’s puzzling case. As is shown in Figure 6, a higher
interest rate shifts a long-run equilibrium from E to E′. So a debt ratio and an
accumulation rate rise together. As an interest rise further, however, a net profit margin
may shrink so severely that what Verselli calls a margin of safety would disappear.14
14
We reinterpret what Vercelli (2011) suggests as a margin of safety. It implies in our
discussion a gap between a realized leverage ratio and its maximum value beyond which
firms do not go for fear of insolvency.
16
Then firms would modify their investment strategies due to the overburdening of debts.
For example, they would decrease sensitivities of investment to capacity utilization and
profit rates, γ𝑢 and 𝛾𝑟 , and increase that of interest payments, 𝛾𝑑 . These revisions of
parameters of investment function may transform the sigh condition of ϕ from negative
to positive. The d′′ = f(𝑔) line of Figure 6 depicts the effect of this change. As a longrun equilibrium locus shifts to 𝑑̇ ′′ = 0, there exist no long-run equilibrium in this
situation. Then the story of stagnation regime follows. This discussion of how a
stagnation regime emerges out of changes in investment strategies corresponds to the
arguments of Vercelli (2000, 2001). He proposes that the essential property of Minsky’s
financial instability hypothesis is not dynamic instability but structural instability.15 In
our model structural instability comes from changes of parameters which reflect firms’
decisions in face of insolvency risks.
<Figure 6>
We interpreted in the above that a stagnation regime or Minskian meltdown corresponds
to a situation of long-run disequilibrium where a debt ratio rises and an accumulation
rate falls continuously. Our arguments suggest how to manage Minskian meltdown and
stabilize the macro-economy. A long-run equilibrium would be recovered, when a slope
of Keynesian equilibrium line becomes positive or the long-run equilibrium locus shifts
downward. We will consider some policy prescriptions which would induce changes of
parameters to bring out these effects.
15
Vercelli (2011) interprets Minsky’s financial instability as a phase of business cycles in
the Lotka-Volterra equation system. He suggests that, when a financial condition of firms
encroaches a margin of safety, firms’ behavior change fundamentally and a Minskian
moment begins. So the critical condition of Vercelli’s reading of Minsky’s hypothesis is the
positive correlation between accumulation rates and debt ratios.
17
Firstly, a monetary policy of lower interest rate serves to stabilize a macro-economy. A
lower interest rate reduces the burden of firms to pay interests, and it affects capital
accumulation positively. In our model a lower interest rate serves to recover a long-run
equilibrium, since it shifts the long-run equilibrium locus downward and the Keynesian
equilibrium upward.
Secondly, financial expenditure of government to clear firms’ debts would alleviate
interest payments of firms and so reduce insolvency risks of firms. Financial structures
of firms become more robust, and it serves to reduce lender’s risks of providing
investment finances. It would create more optimistic environments of investment, and
firms would transform their investment strategies correspondingly. In our model the
Keynesian equilibrium line shifts, and its slope may change from negative to positive.
On the other hand, as is explain in the below, government financial expenditure would
move the long-run equilibrium locus downward.
We will modify the financial constraint of firms to reflect government financial
expenditure, 𝐹𝐺 , and derive the long-run equilibrium locus correspondingly. For
simplicity, it is assumed that that government financial expenditure is proportional to
capital stock, 𝐹𝐺 = 𝑓𝐺 𝐾.
ΔD + Π = (1 + θ)(Π − 𝑖D) + 𝑖(D − F𝐺 ) + Δ𝐾 ……………………………………. (30)
d=𝑎+
𝑎𝑏𝜃𝑖−𝑓𝐺
𝑔−𝑏𝜃𝑖
……………………………………………………………………...
(31)
We will only consider the movement of long-run equilibrium locus. The Keynesian
equilibrium line is assumed as fixed. When government financial expenditure rises,
equation (31) says that the 𝑑̇ = 0 curve shifts downward. A stable long-run
18
equilibrium E may emerge, while there is no long-run equilibrium previously. That is,
government financial expenditure may change a regime of accumulation from the runaway process of stagnation to the process of convergence toward a stable long-run
equilibrium. In this case government financial expenditure would stabilize the economy
when the economy is already located in the stagnation regime of Minskian meltdown.
<Figure 7>
Thirdly, a policy measure may be taken to decrease a propensity to savings of rentiercapitalists. It would lead to higher capacity utilization which, in turn, induces a higher
rate of accumulation. Then firms earn more profits, and it may bring out a lower ratio of
debts. Therefore a policy measure of increasing consumption expenditure of rentiercapitalists would stabilize the economy. In our model a lower propensity to savings of
rentier-capitalists shifts the long-run equilibrium locus downward. The Keynesian
equilibrium line may shift such that its slope changes from negative to positive. A longrun equilibrium may emerge in this case, the economy is stabilized.
5. Conclusion
This study considers a simple model of debt dynamics within the framework of the Post
Keynesian theory of growth and distribution. The proposed model is a modified version
of what Lavoie presents as a Minsky-Steindl model. We simplify Lavoie’s model by
assuming a single propensity to savings out of the financial income of dividends and
interest earnings. But we generalize his investment function and introduce as
determinants capacity utilization as well as rates of profits and interests and a debt ratio.
Comparative static analyses of our model are not different from Lavoie’s in the shortrun condition of a constant debt ratio. It is shown that a higher rate of interests brings
out Lavoie’s puzzling case of higher rates of profits and accumulation as well as the
19
conventional case of lower rates of profits and accumulation
We also examine the long-run situation of flexible debt ratios. We introduce a financial
constraint of firms, formulate an equation of debt dynamics, and connect it with the
macro-model of growth and distribution. Our dynamic system consists of two equations.
One is the accumulation path of short-run Keynesian equilibrium. The other is the longrun equilibrium locus which reflects financial flows of firms to maintain a constant debt
ratio. A long-run equilibrium is determined at the intersection of these two curves. We
adopt phase analyses and determine movements of short-run Keynesian equilibrium on
the accumulation path. When an interest rises, there appear exist two distinct regimes of
accumulation in our model. A ratio of debts always rises in the stagnation regime, and
an accumulation rate always rises in the regime of Lavoie’s puzzling case. These two
regimes correspond to two cases of short-run situation. So our study does not support
the assertion of Lavoie (1995a) and Hein (2006, 2007) that Steindl’s paradox of debts
does not arise when a long-run equilibrium is stable.
According to Lavoie and Seccareccia (2001), the central proposition of Minsky’s
financial instability hypothesis is not effective. They assert that the positive correlation
between an accumulation rate and a leverage ratio does not hold. We show that this
positive correlation may happen in our model, and that a Minskian meltdown would
occur due to structural instability. A higher rate of interests may change the sign
condition of dynamic system such that the regime of Lavoie’s puzzling case is
transformed into the stagnation regime. When an interest rate rises successively, no
long-run equilibrium tends to exist in the stagnation regime. If a long-run equilibrium
does not exist in the conventional regime, the macro-economy is located on the runaway
process where a debt ratio rises and an accumulation rate falls continuously. A Minskian
meltdown is an ultimate solution of this process.
We consider some policy prescriptions to stabilize the macro-economy located on the
20
runaway process of stagnation regime. Firstly, an expansive monetary policy of lower
interest rate reduces interest payments of firms. It brings out higher capacity utilization
and a lower debt ratio both of which have positive effects on accumulation. Secondly,
government may intervene to clear firms’ debts. On the one hand, it motivates
investment by directly reducing a debt ratio of firms. On the other hand, it decreases
lender’s risks and creates an optimistic atmosphere where firms change their investment
strategies accordingly. Both of these effects contribute to the emergence of stable longrun equilibrium. Thirdly, government may take some measures to decrease a propensity
to savings of rentier-capitalists. It increases capacity utilization, decreases a debt ratio of
firms, and tends to stabilize the macro-economy.
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23
FIGURES
𝑑
𝑑
𝑑 = 𝑓(𝑔)
𝐸
𝑑̇ = 0
𝑎
𝑔
𝑏𝜃𝑖
𝐸 𝑑̇ = 0
𝑎
𝑏𝜃𝑖
𝑔
𝑑 = 𝑓(𝑔)
Figure 1. Long-run equilibrium in case of
𝛾𝑢 𝛽(1 + 𝑚) + 𝛾𝑟 𝛽𝑚 − 𝛾𝑑 𝛼𝑚 > 0
Figure 2. Long-run equilibrium in case of
𝛾𝑢 𝛽(1 + 𝑚) + 𝛾𝑟 𝛽𝑚 − 𝛾𝑑 𝛼𝑚 < 0
𝑑
𝑑 = 𝑓(𝑔)
𝑑′ = 𝑓(𝑔)
𝐸′
𝐸
𝑑̇ ′ = 0
𝑑̇ = 0
𝑎
𝑏𝜃𝑖
𝑔
𝑏𝜃𝑖′
Figure 3. Effects of the rising interest rate:
puzzling case
24
𝑑
𝐸′
𝐸
𝑎
𝑏𝜃𝑖
𝑑̇ ′ = 0
𝑑̇ = 0
𝑔
𝑏𝜃𝑖′
𝑑′ = 𝑓(𝑔)
𝑑 = 𝑓(𝑔)
Figure 4. Effects of the rising interest rate:
Steindlian case
𝑑
𝐸′
𝐸
𝑎
𝑏𝜃𝑖
𝑏𝜃𝑖′
𝑑̇ ′′ = 0
𝑑̇ ′ = 0
𝑑̇ = 0
𝑔
𝑏𝜃𝑖′′
𝑑′′ = 𝑓(𝑔)
𝑑′ = 𝑓(𝑔)
𝑑 = 𝑓(𝑔)
Figure 5. Stagnation regime:
dynamic instability
25
𝑑
𝑑 = 𝑓(𝑔)
𝑑′ = 𝑓(𝑔)
𝐸
𝐸′
𝑑̇ ′′ = 0
̇′ = 0
𝑑
𝑑̇ = 0
𝑎
𝑏𝜃𝑖
𝑏𝜃𝑖′
𝑔
𝑏𝜃𝑖′′
𝑑′′ = 𝑓(𝑔)
Figure 6. Stagnation regime:
parametric structural instability
𝑑
𝑎
𝐸
𝑏𝜃𝑖′
𝑑̇ = 0
𝑑̇ ′ = 0
𝑔
𝑑 = 𝑓(𝑔)
Figure 7. Stabilizing effects of government financial expenditure
26