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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. 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Vercelli, A. (2011): A perspective on Minsky moments: revisiting the core of the Financial Instability Hypothesis, Review of Political Economy, 23, pp. 49-67. 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