Minsky cycles in Keynesian models of growth and distribution

Transcription

1 Minsky cycles in Keynesian models of growth and distribution Soon Ryoo 1 Abstract This paper provides an alternative formalization of Minsky s theory of financial instability and examines the conditions under which perpetual cycles emerge from endogenous changes in financial practices. The main features of our model are found in its emphasis on (i) the interaction between debt and portfolio dynamics, (ii) the importance of margins of safety in the evolution of firms indebtedness, and (iii) the decisive role of the dynamics of capital gains and expectations in asset markets. The general framework of financial instability is combined with two Keynesian models of growth and distribution (Kaleckian vs. Kaldorian). Keyword Minsky cycles, financial instability, Kaleckian model, Kaldorian model JEL classification E12, E44 1 Introduction In his favorable review on Minsky s Stabilizing an Unstable Economy, James Tobin praised Minsky as the most sophisticated, analytical, and persuasive of those contemporary economists who believe that leverage is the Achilles heel of capitalism. Tobin, however, expressed his concerns about Minsky s theory: first, Minsky s theory suffers from a lack of a rigorous formal model and without one readers cannot cannot judge whether an undamped endogenous cycle follows from the assumptions or not ; second, Minsky s theory of prices, wages and profits are questionable and are not convincingly linked to the central message of the book, the financial theory of business cycles. 2 Partly motivated by the old Keynesian s concerns, the purpose of this paper is twofold. First, we give an alternative formalization to Minsky s theory of financial instability 3 and examine the dynamic properties of the model. We show that Minsky cycles emerge from the interaction between debt accumulation and portfolio transformations under a few assumptions. These assumptions are stated in the form of plausible restrictions over agents financial behavior and the effect of key financial variables on profitability and accumulation. Second, the paper places the general mechanism of instability and cycles in two Keynesian models (Kaleckian vs. Kaldorian models) where specific behavioral/structural assumptions about the real sector are explicitly made. By doing this, this paper attempts to clarify how Minskian insights can be linked to contemporary Post Keynesian models of growth and distribution. The main features of our model lie in its emphases on (i) the interaction between debt and portfolio dynamics, (ii) the importance of margins of safety in the evolution of firms indebtedness, and (iii) the decisive role of the dynamics of capital gains and expectations in asset markets. To highlight the features of our model, let us take a brief look at the existing literature. 1 Department of Accounting, Finance and Economics, Adelphi University, 1 South Avenue, Garden City, NY 11530, U.S.A. sryoo@adelphi.edu. I would like to thank Peter Skott for his useful suggestions for an early version of this paper and Tom Palley for his encouragement to submit this article. The usual caveat applies. 2 All quotes in this paragraph come from Tobin (1989, p.106). 3 Minsky s theory has received a formal treatment in a number of studies (among others, Taylor and O Connell (1985), Foley (1986), Skott (1994), and Fazzari et al. (2008)) 1

2 Portfolio choices play a central role in producing instability in Taylor and O Connell (1985) but the seminal paper pays little attention to endogenous transformation of firms liability structure. The importance of firms indebtedness has received growing attention in the post Keynesian/structuralist literature including Dutt (1995), Lavoie (1995), Hein (2007), Lima and Meirelles (2007), and Fazzari et al. (2008). In the body of the literature, portfolio decisions and asset prices play no role in the determination of firms indebtedness and the debt ratio of firms is determined as a residual of firms budget equation with no reference to firms margins of safety. 4 In this paper, the liability structure of the firm sector evolves according to firms/banks borrowing and lending decisions that are based upon what Minsky called the fundamental margin of safety (i.e. firms profitability relative to their payment obligations). Some of the previous studies, including Taylor and O Connell (1985), Delli Gatti and Gallegati (1990) and Flaschel, et al. (1998, Ch.12), analyze portfolio decision as well. In these models, however, capital gains play no role in portfolio dynamics. In our model, the dynamics of capital gains is fundamental in producing instability. This feature is shared by Asada et al. (2010) but their specification of portfolio decisions is different from ours and, more importantly, their analysis leaves out the issues of firms debt finance. The structure of agents balance sheets and flow-of-funds accounts is the same as in Skott (1989), Lavoie and Godley (2001/2) and Skott and Ryoo (2008). The purpose and analytic properties of these models are very different, however. Skott provides a Kaldorian model of business cycles where a Harrodian investment assumption makes the goods market unstable. Cycles are generated through the interaction of effective demand and labor market dynamics. The Lavoie and Godley simulations in which the goods market is stable due to the Kaleckian assumption focus primarily on comparative static properties of stable steady states. Skott and Ryoo (2008) examine steady state effects of financialization under various specifications of consumption, accumulation behavior and the labor market. In the present paper, the analysis of stability is a primary focus and financial elements are powerful driving forces of instability and cycles. The comparison of the models that have the same accounting relations points to the importance of structural/behavioral assumptions. The rest of the paper is structured as follows. Section 2 presents a model of Minsky cycles and section 3 analyzes the properties of the model. Section 4 consider a special example where saving and investment are given by the Classical saving behavior and Tobin s q investment function. Section 5 examines two versions of the model of Minsky cycles: Kaleckian and Kaldorian models. The final section concludes the paper. 2 A model of Minsky cycles This section presents a model of Minsky cycles. In the model, firms finance their 4 Minsky argued that lending and borrowing decisions are based on margins of safety and called the excess of firms profits from operations over payment commitments on outstanding debts the fundamental margin of safety. Other important margins of safety include the net worth of agents and the state of markets for financial instruments where refinancing can take place (Minsky, 1982, p.74). Kregel has placed a special emphasis on the role of margins of safety in Minsky s theory (Kregel, 2008). Ryoo (2012) stresses the critical role of margins of safety in Minskian debt dynamics and evaluates the argument by Lavoie and Seccareccia (2001), Hein (2007) and others who claim Minsky s financial instability hypothesis is invalidated by the paradox of debt. 2

3 expenditure by retained earnings, borrowing from banks and issuing new equities, and households make portfolio choices over two assets, stocks and deposits. The model focuses on the endogenous evolution of firms liability structure and households portfolio composition which are captured by the following two variables: (1), and are the level of firms outstanding debts, the price of capital goods and the quantity of capital goods. is the debt-capital ratio in the firm sector, an indicator of firms indebtedness. and are the unit-price and the number of stocks. Under some simplifying assumptions on the banking sector, 5 firms outstanding debts ( ) create the same amount of household deposits ( ) and therefore we will use and interchangeably. represents the ratio of stocks to deposits held by households, a measure of household portfolio preference for stocks. 6 Changes in and determines the trajectory of Tobin s average : (2) (3) to: The firms debt capital ratio and the household portfolio composition evolve according ( ) (4) [ ] (5) (6) where is the gross profit rate of firms, is the rate of return on equity and are the expected rate of return on equity. is the real interest rate on deposits and loans 7, which is taken as a parameter in this model. and are positive constants. A dot over a variable refers to its time derivative. Equation (4) captures Minsky s idea that the level of profits relative to the amount of payment obligations on debts ( the fundamental margin of safety ) has a foremost importance in 5 In this model, banks do not hold reserves and their only assets are loans to the firm sector; people do not hold cash and the loan rate of interest is assumed to equal the deposit rate. Given this assumption, banks loans to firms will return to the banking sector in the form of households deposits. 6 is an inverse measure of liquidity preference. 7 One may notice that we include firms interest payments are evaluated at the real interest rate in (4). By using the real rate in agents decisions on real qualities, our model becomes inflation-neutral. Some accounts of Minsky s theory deliberately introduce inflation-nonneutrality by making a real quantity depend on the nominal interest rate. In Fazzari et al. (2008), for instance, real investment is increasing in cash flow which in turn depends negatively on the nominal interest rate. Under the assumption of a fixed real interest rate, the nominal interest rate moves along with the expected inflation rate. The interaction between investment and inflation dynamics creates cycles. 3

4 the determination of the firms liability structure. If firms profitability is strong relative to their payments committed by debt contracts, firms desire to take and banks willingness to extend loans increase, thereby raising firms indebtedness. If firms profitability is relatively low, the firm debt ratio falls. Equation (4) is in line with Minsky s behavioral hypothesis: If recent experience is that outstanding debts are easily serviced, then there will be a tendency to stretch debt ratios; if recent experience includes episodes in which debt-servicing has been a burden and representative units have not fulfilled debt contracts, then acceptable debt ratios will decrease (Minsky, 1986, 187). Portfolio transformations are important driving forces behind cyclical fluctuations in Minsky s theory. Equation (5) shows that households desired portfolio share is positively related to the expected rate of return on equity ( ): given, a higher expected rate of return on equity justifies a higher weight of stocks in households portfolios. The desired portfolios, however, may not be immediately attained. The adjustment of the actual portfolio share to the desired ratio takes time due to conventional elements. A simple adaptive specification in (6) stresses the role of extrapolation in the formation of expectations in line with Minsky s perspective. For instance, Minsky states In making portfolio choices, economic units do not accept any one thing as a proven guide to the future state of the economy. Unless there are strong reasons for doing otherwise, they often are guided by extrapolation of the current situation or trend, even though they may have doubts about its reliability. Because of this underlying lack of confidence, expectations and hence present values of future incomes are inherently unstable...(minsky, 1982, p.131) The stock price dynamics implied in our specification of portfolio decisions 8 bear a resemblance to the behavioral literature of asset price dynamics (Beja and Goldmann (1980), Chiarella (1992), Sethi (1996), Brock and Hommes (1998)). Using the terminology in the literature, our specification (5) and (6) can be seen as a reduced form of the interaction between fundamentalists and chartists with constant wealth distribution between the two types. 9 The rate of return on equity is defined as (7) where stands for capital gains (A hat over a variable stands for the growth rate of the variable). Stock returns are related to the firms budget equation. To see this, consider the budget equation of firms. 8 The portfolio dynamics based on (5) and (6) is used in Skott (2012b) to study inequality-driven-financial instability. 9 The detailed argument is available upon request. The assumption of constant wealth distribution between the fundamental and chartists can be relaxed without affecting our main argument for instability in the subsequent sections. Friedman s classic argument (Friedman, 1953) suggests that rational speculators who buy low and sell high (fundamentalists) outperform chartists (e.g. trend chasers) and therefore chartists will eventually vanish: asset markets are self-stabilizing. DeLong et al. (1990), Chiarella (1992), Sethi (1996), and Brock and Hommes (1998) show that Friedman s argument against instability is not generally valid. 4

5 (8) where,, and are real investment spending, the nominal interest rate, dividend payments and gross profits ( ), respectively. Adding capital gains on stocks to both sides and rearranging them, we have: (9) Total returns on stocks (including capital gains) equal the excess of the change in the market value of firms and gross profits over the sum of interest payment and investment expenditure. Using (7), (9) and the definition of and, we rewrite (7) as [ ] ( ) (10) where and are the growth rate of capital stocks, the rate of profits and the rate of capital depreciation. The rate of return on equity has two components: [ ] ( ) and. The first component is related to the growth rate of the market value of firms. By the definition of in (3), the growth rate of the value of firms (deflated by output price) is equal to. In terms of and, equals the expression in the bracket in (10). [ ] (11) In a steady state, both and are zero and therefore must be constant ( ) and therefore the market value of firms grows at the same rate as that of physical capital stock,. Outside the steady state, if changes in and are such that, i.e., then the market value of firms grows faster than, which helps maintaining the rate of return on equity at a high level. 10 The second component represents the ratio of firms profits net of interest paid and gross investment to the value of stocks, which can be seen as an indicator of the fundamental earnings power of firms. Strong profitability relative to investment spending and payment commitments is associated with a high rate of return on equity Note that the rate of return on equity is increasing in which is in turn increasing in and. 11 A special case arises if and is determined only by real forces. We then have: This relation represents a version of the Modigliani-Miller irrelevance case (Modigliani and Miller 1958). The weighted average cost of capital must equal the return on real assets ( ) which is independent of firms financial structure ( ). 5

6 3 General Analysis The system (4)-(6), with the definition of in (10), leaves and undetermined. Financial decisions ( and ) feed back to and. The structure of the real sector needs to be specified to see how and affect and. For the moment, we reserve the description of the real sector for later sections and want to keep the analysis as general as possible. Thus we write and as general functions of and : Using (10), (12) and (13), the rate of return on equity can be written as (12) (13) [ ] [ ] (14) (15) (12) and (14) close the system of (4), (5) and (6). Now we make a few assumptions on the elements of the model. We assume that,, and are all continuous and differentiable with respect to their respective arguments. More substantive assumptions follow: Assumption 1 The desired stock-deposit ratio is bounded from above by and from below by. Both and are positive constants. (16) Assumption 2 There exists a profit-interest ratio that keeps the debt-capital ratio constant, i.e., there exists such that. Assumption 3 The profit function satisfies the following conditions for all in where, and is the initial value of : (17) (18) Assumption 4 Holding constant, the profit-interest ratio is diminishing as the debt-capital ratio increases. Formally, 6

7 (19) Let us examine the implications of these assumptions. Assumption 1, the boundedness of the desired portfolio function states there exists a minimum proportion of each asset in household portfolios and therefore the weight of each asset in household portfolios will not vanish. Note that the assumption of the boundedness of the desired portfolio share, together with equation (5), ensures that the trajectory of the actual portfolio share is bounded. Using Assumption 1 and (5), we obtain (see Appendix) (20) Assumption 2 is a minimal requirement for the existence of a steady state, stating that there exists a level of the fundamental margin of safety,, that is required to maintain a constant debt ratio. The first part in Assumption 3,, means that bullish (bearish) stock markets increase (decrease) firm profitability. 12 The positive dependence of firm profitability upon the state of stock markets is critical for Minsky s story: a stock market boom enhances firms profitability which makes it easy for firms to service their debts, encouraging firms to stretch their debt ratios and bankers to accept increasing leverage. The second part, condition (19), is also intuitively straightforward. If the current level of the debt ratio is negligible ( ), the profit-interest ratio is sufficiently high to induce firms to use debt finance. If the current level of the debt ratio is too high ( ), the profit-interest ratio becomes too low to keep the debt ratio constant. Assumption 4 is about how the profit-interest relation changes as firms indebtedness changes. Minsky assumes that increases in firms indebtedness gradually erode the fundamental margin of safety. For instance, Minsky maintains: the way in which investment spending is financed in a prosperous capitalist economy leads to an accumulation of indebtedness relative to the cash flows that enable units to fulfill their payment commitments. (Minsky, 1995) Assumption 4, along with Assumption 2 and 3, implies that, for a given value of, the debt dynamics (4) is stable and has a unique stationary debt-ratio. Let us denote this solution as. By definition, satisfies ( We call ) (21) the desired debt ratio (or the acceptable liability structure using 12 In all models in section 4 and 5, this positive effect of on works through the effect of stock market wealth on aggregate demand components. 7

8 Minsky s terminology). represents the liability structure that firms/bankers are content with, keeping the growth of firms outstanding debts in line with that of the value of capital stocks. The desired debt ratio depends positively on. Using (17) and (19), we have 13 (22) (22) has a straightforward interpretation: since a shift in household portfolios toward stocks stimulates firm profitability ( ), improved profitability supported by the stock market boom raises the debt-capital ratio that firms and bankers are willing to accept. The endogeneity of the acceptable liability structure has an important implication for the debt dynamics: if is allowed to vary, the desired debt ratio becomes a moving target that the actual debt ratio chases. If endogenous interactions make move ahead the actual debt ratio (either in the upward or downward direction), then the actual debt ratio may not settle down to a rest point. Perpetual cycles are a possible outcome. Lastly, the following assumption is added to the list of our assumptions. Assumption 5 Consider a hypothetical case where and. Then the portfolio composition, based on (5), converges to a unique stationary equilibrium. The condition required for this property is given by 14 [ ] [ ] (23) Assumption 5 means that if the debt-capital ratio remained at the desired ratio and households (stock market investors) made portfolio decisions with reference to only firms fundamental earnings power, financial markets would establish a unique and stable portfolio equilibrium. We will show that the stability of such a hypothetical portfolio dynamics can be turned into instability as endogenous changes in financial practices drive a wedge between the actual and the desired debt ratios, on the one hand, and between the expected rate of return on equity and the fundamental earnings power of firms, on the other hand. 3.1 Existence of a stationary solution A stationary equilibrium of the system (4)-(6) must satisfy: ( ) (24) 13 is continuous and differentiable since is continuous and differentiable. 14 If and, (5) is reduced to [ ] Condition (23) implies that the right-hand side of this equation is decreasing in, which, along with Assumption 1, ensures the uniqueness and stability of the stationary solution. Also see the next footnote for the uniqueness. 8 (25)

9 A steady state requires (i) the actual debt ratio must be equal to the desired debt ratio in a steady state path (eq.(24)); (ii) the actual portfolio composition must equal the desired composition (eq.(25)); (iii) household expectations on the rate of return on stocks must be met (eq.(26)). To examine the existence of a steady state, recall that, from Assumptions 3 and 4, (24) is equivalent to. Substituting this in (26) and then (26) in (25), we obtain: (26) (27) It is straightforward to check that assumptions 1 through 5 ensure the existence of a unique value of that satisfies (27) Stability Let us consider local stability properties of the unique stationary point. The local stability of the system depends critically on the magnitude of, and. If either or is sufficiently low, the stationary point is locally stable, whereas if, and are sufficiently high, then the stationary point loses stability, giving rise to a limit cycle via a Hopf bifurcation. 16 This condition for instability can be readily understood. High implies that a rise in the expected rate of return strongly increases the desire to hold stocks. With a high value of, such an increase in the desired portfolio will lead to a large shift in the actual portfolio toward stocks (equation 5). Therefore stock prices rise substantially and a large amount of capital gains are generated, 17 which helps keep the rate of return on equity above the expected rate, i.e.,. As a result, initially optimistic expectations in stock markets will be reinforced (equation (6)). With a high, such a further increase in will be substantial, which induces households to keep shifting their portfolio toward stocks as the desired portfolio ratio moves ahead of the actual portfolio ratio for a while (i.e. ). Destabilizing forces in stock markets interact with the debt dynamics. A boom in stock markets an increase in stimulates firms profitability due to Assumption 3 ( ). High profitability justifies high debt ratios during an expansion. Thus the stock market boom is accompanied with growing indebtedness in the firm sector. Increasing positive in turn will help maintain the rate of return on equity at a high level since the effect of on is positive in (14). 15 Defining, we find that and. Since is continuous in, there exists at least one stationary solution of in. If Assumption 5 is met, is strictly increasing in and therefore the stationary solution is unique. 16 The proof is available upon request. Given our five assumptions, the Hopf bifurcation, it can be also shown, is the only route through which the fixed point loses its local stability. 17 Recall that a high value of is associated with a high in (14). 9

10 The mechanism of this upward instability has a obvious limit because the desired portfolio approaches its upper bound. As a result, the portfolio adjustment toward stocks will slow down, which yields only a limited amount of capital gains. Stock markets will reach a turning point and will start to decline. This has a negative implication for firm profitability. As firm profitability declines, the level of firms indebtedness will turn out to be unwarranted. A deleveraging process will start at some point. Simulations show that stable limit cycles emerge from such an interaction between debt and portfolio dynamics (see section 4 and 5). In addition to the results from the local stability analysis, our system has an important global property: the model always generates bounded trajectories (see Appendix). A dynamical system with bounded trajectories and a unique stationary point must exhibit cyclical fluctuations if the stationary point is locally unstable. 18 Our local and global analysis thus are complementary, pointing out that the system can produce boom-bust cycles along the lines of Minsky s financial instability hypothesis. 4 A simple benchmark Our general analysis of Minsky cycles in section 3 was conducted without imposing particular structural/behavioral assumptions on the real side of the economy. This section considers a simple saving assumption to which Minsky s own analysis gives considerable weight, namely the case of profits equal investment. This case is derived from the Classical saving assumption that wage earners do not save and profit earners do not consume. Under this Classical saving assumption, we have: This special case of the investment-saving relation can be combined with Minsky s two-price theory of investment. As a shortcut, suppose that investment is a function of Tobin s, i.e., with. The profit and the accumulation rates are increasing in and : (29) and (30) provide a special case of (12) and (13). Assumptions 1 and 2 are independent of specific assumptions on savings and investment. (17) in Assumption 3 is satisfied here because and (18) can be rewritten as (28) (29) (30) (31) 18 In a planar system, the boundedness of trajectories and the existence of a unique unstable fixed point are sufficient to establish the existence of a stable limit cycle (the Poincare-Bendixson theorem) but in a higher dimensional system such as our three dimensional one, quasi-periodic or chaotic solutions cannot be ruled out. 10

11 Assumption 4 is related to the magnitude of Tobin s q effect. which means the elasticity of gross investment with respect to is less than Assumption 5 is met if (32) and [ ] ( ) (33) Assumptions 4 and 5 thus represents restrictions on the Tobin q effect. As pointed out in the previous section, assumptions 1 through 5 ensure that the system has a unique stationary point, the trajectories of the state variables are bounded and, as and increase, the system loses its local stability at the unique stationary point via a bifurcation, leading to perpetual cycles. Figure 1 shows some simulation results. 20 m a. ratio of debt to capital b. ratio of stocks to deposits time time c. rates of return on equity, e 0.20 e :dashed d. rates of profits and accumulation g:dashed r f, g time time Figure 1: Minsky cycles in a model with the Classical saving behavior 19 This assumption is plausible. If, and, then (32) boils down to. Empirically estimated q effect is far from being greater than Simulations for Figure 1 are based on: where [ ], [ ],, 0.2,,, 11

12 The story underlying Figure 1 is as follows. Increasing, combined with rising, produces capital gains which sustains high rates of return in stock markets. 21 The stock market boom raises and stimulates capital accumulation. The increase in the accumulation rate raises the profit rate. Strong profitability helps firms service their debts easily, thereby inducing firms to take more debt finance. Therefore increasing and reinforce each other and supports a long expansion. However, the desired share of stocks in household portfolios has an upper limit and increasing rates of return on equity during the boom will have only a limited impact on portfolio adjustments as the desired portfolio share approaches the upper limit. The resulting capital gains then become too small to keep the actual rate of return above the expected rate of return. As the relation between the actual and expected rates of return is reversed at some point, will start to decline. Falling stock markets have negative implications for aggregate demand and firm profitability. As falls passing through a critical level, the profit-interest ratio will fall below the level required to keep the debt-capital ratio rising. Deleveraging will take effect and accelerate the stock market downturn. 5 Minsky cycles in Kaleckian and Kaldorian models The benchmark case in section 4 is useful to illustrate the mechanism of cycles but the benefit comes at a cost. It is too simple to capture important behavioral and structural relations. The Classical saving assumption is heroic ; there is no explicit assumption about pricing/production; capacity utilization, one of the important elements in contemporary post Keynesian models, is not even mentioned. One may wonder how the analysis is related to contemporary structuralist/post-keynesian macroeconomic models. Therefore this section rebuilds our model of Minsky cycles upon explicit structural/ behavioral assumptions about the real sector. Technology The production process is given by a Leontief fixed coefficient production function. We assume no labor hoarding and normalizes the unit of labor to make labor productivity equal unity. Given that the technical output-capital ratio is exogenously given, the actual output-capital ratio serves as a measure of capacity utilization. Consumption/Saving Consumption depends positively on income and wealth. (34) (35) 21 The positive effect of on is important in the mechanism of cycles. Numerical experiments show that if this effect is absent, the system which is unstable otherwise can be stabilized. 12

13 where and are household income (the sum of wages, dividends and interest on deposits) and wealth (stocks plus deposits). Both are normalized by the value of capital stock. Using relevant definitions, is written as (36) (37) (38) Household income and consumption are increasing in and decreasing in as typical in Post Keynesian models. Along the lines of Kaldor (1966), the negative dependence of consumption on the profit share comes from the existence of corporate savings, rather than from differential saving propensities attached to members of different social classes. 22 It is readily seen from (36) and (37) that both and affect consumption positively: an increase in raises wealth and an increase in raises both income 23 and wealth. The expansionary effect of the financial variables on demand is a key channel through which financial cycles influence the real sector. Theories diverge when it comes to the specification of firms decisions on accumulation and pricing/production and the assumption regarding the labor market. We consider two cases: Kaleckian and and Kaldorian. 24 Our purpose here is to show that Minskian financial dynamics can be combined with alternative approaches to modeling the real sector and to examine main features of real-financial interactions in those models. 5.1 A Kaleckian model The baseline Kaleckian model assumes a constant markup (profit share) with perfectly elastic adjustment of output, perfectly elastic labor supply, and the weak response of accumulation to variations to the utilization rate. In this Kaleckian framework, the utilization rate adjusts to clear the goods market for a given profit share ( ), and a shift in aggregate demand produces a permanent change in the utilization rate. To highlight this feature, we consider a simple specification of investment behavior: An increase in the profit share reduces wages and increases profits but since a fraction of the increased profits is retained, there will be a net decrease in household income. 23 A rise in reduces dividends but raises interest income on deposits, which will more than offset the reduction in dividends. 24 For a recent debate regarding the specification of accumulation behavior (and the related literature), see Skott (2012a), Hein et al. (2012) and Skott and Zipperer (2012). The majority of Minsky-inspired models use the Kaleckian assumption on accumulation behavior but Foley (1986), Skott (1994) and Ryoo (2010) are exceptions. 25 Retaining the -effect on investment adds little to the analysis. Thus we drop from the investment function. Kaleckians are also interested in the effect of the profit share (or the markup) on accumulation but as the profit share is exogenous in the current setting and we do not focus on comparative statics, the profit share has been suppressed in (39). 13

14 (39) The product market equilibrium requires where and. In short-run Keynesian models, investment is less sensitive than savings in response to changes in utilization. The baseline Kaleckian model assumes that such a low sensitivity of investment applies to the long run as well. In the current framework, this condition can be rewritten as Given this assumption, we can express the equilibrium utilization rate as a function of financial variables. Utilization is increasing in and. Algebraically, (40) (41) (42) (43) (44) (45) An increase in raises income and wealth, thereby stimulating consumption and creating excess demand in the goods market. A higher utilization rate is needed to restore equilibrium in the goods market. A shift in household portfolios toward stocks (a rise in ) increases household wealth and consumption, which requires the utilization rate to rise to absorb the increase in demand. Using (43), we obtain the expressions for the accumulation rate and the profit rate as a function of and. Both and are increasing in and. ( ) (46) (47) The positive effect of on comes from the fact that an increase in raises 14

15 consumption which in turn increases utilization. Note that is positive, i.e. a rise in the debt-capital ratio increases utilization and profitability. This positive effect of on, however, should not be too large in order to satisfy the assumption of declining margin of safety (Assumption 4). This requirement can be written in terms of a constraint on the magnitude of the investment coefficient. Using (43) and (47), Assumption 4 condition (19) is rephrased as: This restriction on the sensitivity of accumulation to utilization ( ) is stricter than the stability condition (42). 26 Assumption 5, it can be shown, imposes a further constraint on the size of. 27 As in the model in section 4, the profit rate and the growth rate are increasing in and. The expression for the rate of return on equity will be more complicated but the analytic results must be qualitatively the same as in section 4. In this Kaleckian economy, the product market equilibrium is achieved through the variation of capacity utilization for a given profit share. The expansionary effects of the financial variables on the profit rate and the accumulation rate come from the induced changes in the utilization rate in this framework, not from Tobin s q effect on investment. Fluctuations in the financial variables cause aggregate demand to vary, which leads to variations in the utilization rate, the profit rate, the rate of accumulation and the rate of return on equity. Changes in these variables feed back into firms/banks and households financial decisions. Figure 2 shows some of the simulation results. 28 (48) 26 The condition (48) will be met if both consumption and investment functions are linear with a positive autonomous investment. The assumption of the linearity of consumption function can be relaxed to that of the homogeneity of of degree one, without affecting main analytic results. 27 Straightforward but tedious algebra shows that Assumption 5 can be written as [ ][ ] In a steady state, Tobin s q is likely close to unity. If, the condition is simplified to: [ ] 28 Simulations for Figure 2 assume: where, [ ],, 0.2,,,,,, 15

16 m a. ratio of debt to capital 1.4 b. ratio of stocks to deposits time time u 0.70 c. utilization rate g 0.05 d. accumulation rate time time Figure 2: Minsky cycles in a Kaleckian model 5.2 A Kaldorian model In the previous subsection, we combined the baseline Kaleckian growth model with our model of Minsky cycles. Without the financial dynamics, real variables in the Kaleckian model converge to their steady state values. This subsection integrates a Kaldorian model of business cycles (Skott (1989)) into our model of Minsky cycles. Our Kaldorian version is more complicated than the previous examples. A reason for such a complication is because the model has two mechanisms of cycles. One comes from Harrodian accumulation behavior in the real sector and the other from Minskian instability rooted in the financial side. The two mechanisms of cycles may not be synchronous, however. Harrodian instability has been seen as a mechanism of typical (short-run) business cycles but Minskian instability is best seen as a mechanism of long waves. Under profound uncertainty, agents financial practices are shaped by norms and conventions which slowly evolves over a long period of time and therefore are not greatly disturbed by the ups and downs of short-run business fluctuations. Endogenous changes in financial practices thus provide powerful forces of long waves of an economic system. This long-wave-interpretation of the financial instability hypothesis is found in Minsky s own writings. For instance, he argues that The more severe depressions of history occur after a period of good economic performance, with only minor cycles disturbing a generally expanding economy (Minsky, 1995, p. 85), and the mechanism which has generated the long swings centers around the cumulative changes in financial variables that take place over the long-swing 16

17 expansions and contractions (Minsky, 1964). 29 In our Kaldorian framework, the long wave interpretation paves the way for an attractive formulation of Minsky s financial instability hypothesis: the integrated system produces Minskian long waves around which Kaldorian short cycles oscillate. Following Ryoo (2010), two distinct cycles long and short are obtained by assuming that financial decisions over the longer-run are made with reference to the trend rates of profits and accumulation rather than their short-run variations. In other words, and in our model in section 2 refer to long-run trend rates rather than actual rates. More specifically, we define the trend rate of profits as (49) and take the trend rate of accumulation as constant (50) In equation (49), is the long-run average rate of utilization and, to simplify, we assume that is constant. The constancy of the long-run average rate of utilization is justified by the Harrodian argument that the rate of capacity utilization, over the longer-run, should not persistently deviate from a structurally determined desired rate. Thus the long-run average rate of utilization will be close to the desired rate of utilization. It should be noted that the simplifying assumption of strict exogeneity of can be relaxed without affecting the main results. For instance, we can take a moving average process to characterize an endogenous formation, i.e., where is the actual rate of utilization and is a positive constant. As long as is small, the average rates will exhibit sufficiently smooth variations compared to the actual rates. Thus the long-run trend can be conceptually distinguished from short-run fluctuations. Given the mild variations of long-run capacity utilization, the long-run average growth rate of capital will not be very different from that of output. In a labor constrained economy, output fluctuates around the natural rate of growth during a course of (short) business cycles. Thus (50) is a plausible approximation of the behavior of an economy over the longer-run. With fixed, the trend rate of profits varies as the profit share changes. Along the lines of Keynes (1930) and Kaldor (1955/56), the profit share adjusts to clear the goods market and therefore in (49) is the profit share that satisfies the condition for product market equilibrium: (51) (49) and (50) represent special cases of (12) and (13) in the present setting. Note that the 29 Ryoo (2010) proposes a Kaldorian model of Minskian long waves which is similar to the present model but a different specification of portfolio decisions. Palley (2011) recently called for understanding Minsky s theory through the lens of long swings. 17

18 profit share is increasing in and and so is. Financial variables influence aggregate demand which is reflected in the profit share. Assumptions 3, 4 and 5 imply some restrictions over structural and behavioral parameters of the model. 30 With these restrictions, the main analytic results in section 3 carry over to the present setting. The model produces cycles that emerge from endogenous changes in financial practices. A complete characterization of the model requires a system of short cycles. A simple modification of the Kaldorian model in Skott (1989) gives us a mechanism of short business cycles. The features of the Kaldorian approach are very different from the Kaleckian model. It assumes that the equality of savings and investment is achieved through the flexible adjustment of the profit share (markup); output adjustment is sluggish compared to markup adjustment; accumulation is more sensitive than savings in response to variations to the utilization rate, making the good market unstable. In addition, we consider a mature economy in which the state of the labor market (measured by the employment rate) constrains output expansion, thereby affecting accumulation. These assumptions are represented by (52) (53) The growth function (52) tells us that the adjustment of output is a costly process and both profit signals from the goods market and the state of the labor market are important in firms production decisions. 31 High profitability stimulates firms desire to expand their outputs and a tight labor market discourages firms from expanding production. (54) is a Harrodian assumption that states investment is more sensitive to changes in utilization than savings. Investment and saving behavior determines the equilibrium profit share ( ) in the goods market. Note that is distinguished from its long-run trend. and in (51) are replaced by and in the following equation: which gives us the solution for the profit share: (54) (55) (56) 30 The profit rate is increasing in and therefore the first part of Assumption 3 is always met. Assumption 4 and 5 can translate into constraints over the magnitude of the consumption wealth effect. Assumption 4 requires Assumption 5 will be met if [ ] [ ] 18 [ ] 31 Skott (1989) provides a behavioral foundation of equation (52) in detail.

19 The Harrodian assumption (54) makes the profit share increasing in : [ ] (57) From the definition of and, we have and. Using (52), (53) and (56), the system of short cycles can be written as [ ] (58) [ ] (59) If and are fixed, the dynamic properties of this system are essentially the same as those of Skott (1989). Under plausible assumptions, (58) and (59) yields a unique steady growth equilibrium and the steady state, it can be shown, is locally asymptotically unstable unless the negative effect of employment on output expansion is implausibly large. 32 If the boundedness of the trajectories is ensured, the system (40) and (41) will generate a limit cycle (See Skott 1989, Appendix 6C for the proof). The mechanism of cycles is the interaction between destabilizing goods market dynamics and stabilizing labor market dynamics. An increase in utilization raises aggregate demand and profitability, leading to strong output growth and a further increase in utilization. Such a positive feedback between demand and production tends to destabilize the system. Output growth, however, will be eventually limited because it increases the employment rate which negatively affects the condition of output expansion. 33 As our system of Minsky cycles produces long waves of the financial variables ( and ), the Kaldorian cycles fluctuate with reference to the trend set by Minskian long waves. Figure 3 and 4 presents some simulation results The trace and the determinant of the Jacobian matrix of (58) and (59) are given by The determinant is always positive. If the sign of Tr(J) is negative (positive), the equilibrium is locally asymptotically stable (unstable). The sign of the trace will be positive unless the negative employment effect is implausibly large (Skott, 1989). 33 The system is akin to the Goodwin cycle (Goodwin, 1967), which emphasizes the interaction between accumulation and conflict in the labor market. Unlike the Goodwin cycle where accumulation is passively determined by savings, effective demand plays a central role in the Kaldorian model. 34 The movement of m,, and are similar to that in Figure 1 and we do not reproduce them here. We use for this simulation: where, [ ],, 0.2,,,,,,,, [ ]. 19

20 0.45 s Figure 3: Profit share and employment in a Kaldorian model Figure 3 shows the trajectories of the employment rate (horizontal) and the profit share (vertical). In the Skott s Kaldorian model (Skott (1989)), the two variables exhibit clockwise cyclical behavior with a constant amplitude. In the present model, Minskian long waves constantly reset the values of financial variables ( and ) which provide the reference to the Kaldorian short cycles. As a result, during a long expansion (downturn) when and rise (fall) most of the time, short cycles spirals up (down) to the northeast (southwest). e, s a. profit share s :dashed e b. employment rate time time u c. utilization rate d. output growth and accumulation dashed Y, g time Figure 4: Long waves and short cycles in a Kaldorian model time 20

21 Figure 4 illustrates how some real variables evolve over time. The movement of the profit share driven by the financial factor plays a critical role in generating the pattern of long waves and short cycles. The interaction between effective demand and class conflict the mechanism of short cycles is intertwined with financial dynamics through its link to the profit share. The system of long waves provides a moving trend around which short cycles fluctuate. The system of short cycles that generate the fluctuations of utilization and accumulation around the desired and the natural rates justifies agents long-term expectations based on which they make financial decisions: the long-run average values of utilization and accumulation rates are close to the constant desired and the natural rates, respectively. The Kaldorian approach presented in this subsection offers a promising way of integrating Minsky s financial instability hypothesis and Harrod s instability principle. Both point to the impossibility of stable growth in the capitalist economy, providing mechanisms of cycles. In our Kaldorian framework, the Minskian instability creates long waves and the Harrodian instability produces short cycles. 6 Conclusion We have formalized Minsky s theory of financial instability in a general framework (section 2 and 3). Perpetual cycles emerge from the interaction between debt accumulation and portfolio transformations under plausible assumptions. These assumptions are not tied to specific structural/behavioral assumptions about real-sector relations. Therefore a variety of closures are possible. The simple case with the classical saving assumption and -investment function (section 4) provides a transparent mechanism of Minsky cycles but is not clear about the assumptions regarding firms pricing and production decisions and labor market conditions. In both Kaleckian and Kaldorian versions (section 5), financial elements are key determinants of aggregate demand and critical in producing instability and cycles. Due to the differences in the assumptions regarding pricing/production/accumulation decisions, however, the mechanism through which aggregate demand affects the economy comes in a different form. Most importantly, the assumption about accumulation behavior makes the goods market stable in the Kaleckian model but unstable in the Kaldorian model. In the Kaleckian version, fluctuations of financial factors drive those of real variables which otherwise converge to a stable steady growth path. In the Kaldorian version, the real sector itself has an inherent tendency toward cyclical behavior ( short cycles ), along with cyclical forces generated by endogenous changes in financial practices ( long waves ). The central message of this paper a key mechanism of Minsky s theory of instability and cycles can be formalized under plausible assumptions and can be combined with alternative Keynesian models is good news but the mere existence of plural approaches does not necessarily indicate a sound state of Keynesian theories. A careful evaluation of alternative macro models is necessary and left for future research. 21

22 Our formulation is a kind of what Minsky called a skeletal model of an capitalist economy, leaving aside a number of important aspects. The banking sector, although playing a fundamental role in making loans and therefore money in our model, is too simple and should be enriched. The model focuses exclusively on business debt in line with Minsky s own perspective but the recent history has shown that household debt can be a source of instability. 35 Open economy complications are completely left out of the analysis. Lastly, the actual trajectory of an economic system, as Minsky highly emphasized, is strongly shaped by a number of specific institutional factors and policy responses, and our simple formal analysis has obvious limitations in dealing with those issues. References Asada, T., Chiarella, C., Flaschel, P., Mouakil, T., Proaño, C. R., and Semmler, W. (2010), Stabilizing an unstable economy: On the choice of proper policy measures, Economics. The Open Access, Open Assessment E-Journal. No Beja, A., Goldman, M. B. (1980), On The dynamic behavior of prices in disequilibrium, The Journal of Finance, 25, Brock, W. A., Hommes, C. H. (1998), Heterogeneous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamics and Control, 22, Charpe, M., Flaschel, P., Proano, C., Semmler, W. (2009), Overconsumption, credit rationing and bailout monetary policy: A Minskyan perspective, Intervention. European Journal of Economics and Economic Policies. 6(2). Chiarella, C., (1992), The dynamics of speculative behavior, Annals of Operations Research, 37, Delli Gati, F., Gallegati, M. (1990) Financial instability, income distribution and the stock market, Journal of Post Keynesian Economics, DeLong, J. B., Shleifer, A., Summers, L. H., Waldmann, R. J. (1990), Positive feedback investment strategies and destabilizing rational speculation, The Journal of Finance, 45 (2), Dutt, A. K., (1995), Internal finance and monopoly power in capitalist economies: A reformulation of Steindl s growth model, Metroeconomica, 46(1), The issues of household debt have been examined in Kaleckian models by Palley (1994), Dutt (2006), Charpe et al. (2009), Palley (2010) and Kim and Isaac (2011). These models do not introduce the implications of asset prices for debt dynamics. Ryoo (2011) analyzes boom-bust cycles driven by the interaction between asset bubbles and household debt in a Kaldorian framework. 22