On the possibility of an ever-increasing wealth concentration: Pasinetti, dual, and anti-dual equilibria in a Post-Keynesian framework

Ederer, Stefan; Rehm, Miriam Working Paper On the possibility of an ever-increasing wealth concentration: Pasinetti, dual, and anti-dual equilibria in a Post-Keynesian framework ifso working paper, No. 42 Provided in Cooperation with: University of Duisburg-Essen, Institute for Socioeconomics (ifso) Suggested Citation: Ederer, Stefan; Rehm, Miriam (2024) : On the possibility of an ever-increasing wealth concentration: Pasinetti, dual, and anti-dual equilibria in a Post-Keynesian framework, ifso working paper, No. 42, University of Duisburg-Essen, Institute for Socio-Economics (ifso), Duisburg This Version is available at: https://hdl.handle.net/10419/308036 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ uni-due.de/soziooekonomie/wp ifso working paper Stefan Ederer Miriam Rehm On the possibility of an ever-increasing wealth concentration: Pasinetti, dual, and anti-dual equilibria in a Post-Keynesian framework 2024 no.42 2 / 20 On the possibility of an ever-increasing wealth concentration: Pasinetti, dual, and anti-dual equilibria in a Post-Keynesian framework Stefan Ederer and Miriam Rehm Keywords: wealth distribution, Post-Keynesian model, aggregate demand, Pasinetti, profit rate JEL Codes: B22, D31, D33, E12, E21, E64, Abstract This paper develops a stock-flow consistent Post-Keynesian model in the Kalecki-Steindl tradition with endogenous wealth accumulation and distribution, which captures the key aspects of the Cambridge debate on (anti-)dual and Pasinetti equilibria. We find that a stable interior solution – that is, a Pasinetti equilibrium – is the most likely outcome, while the corner solutions of dual and anti-dual equilibria – both the euthanasia and the triumph of the rentier – are special cases of a standard Bhaduri-Marglin model. Endogenizing the profit share yields a two-dimensional dynamic system of the wealth concentration and the profit share, which is stable for a wide range of parameter values, as long as the concentration of wealth is not unrealistically low. An interior Pasinetti equilibrium thus remains the most likely outcome. However, for certain parameter combinations, the system may move onto an explosive trajectory with an ever-rising concentration of wealth and income in the hands of capitalists. Numerically illustrating the results of the analytical model shows that endogenizing the profit share leads to a more unequal wealth distribution, and a negative feedback effect between high wealth inequality, a high profit share, and growth. 1. Introduction In the 1960s, Luigi Pasinetti was at the center of a debate between mainstream and heterodox economists on the theoretical foundations of the distribution of wealth and its determinants. In Pasinetti (1962), he argues that capitalists always own a certain share of the total wealth of an economy, and that this share is determined by their savings rate. Samuelson and Modigliani (1966) respond that if workers save a high enough proportion of their income, they will accumulate wealth faster than capitalists, who will eventually disappear as a social class. Pasinetti (1966, 1974), in return, dismisses this so-called “dual equilibrium” as empirically irrelevant. Finally, Darity (1981) shows the possibility of an “anti-dual” equilibrium, in which all wealth is concentrated in the hands of capitalists. This debate was revived in the 2010s by Thomas Piketty’s best-selling book “Capital in the 21st Century”, albeit with reversed roles. Utilizing a neoclassical production function, Piketty (2014) argues that a rising ratio of wealth to income leads to a higher share of income for capitalists, which dynamically feeds back into a rising concentration of wealth, leading essentially to an anti-dual equilibrium unless prevented by policy measures such as a wealth tax. Many Post-Keynesians, on the other hand, while appreciating Piketty's empirical work and sympathizing with his policy conclusions, found his theoretical framework lacking. Building on Pasinetti’s (1962) “Cambridge equation”, they argue that the distribution of wealth is stable in the long run (e.g., Galbraith 2014, Palley 2014, Taylor, 2014). Recently, an increasing number of contributions incorporate the dynamics of wealth distribution based on differential saving rates into economic models in the Kalecki-Steindl (Ederer and Rehm, 2020a), Structuralist (Taylor et al., 2015) or Classical-Marxist (Petach and Tavani, 2020) tradition. Some include further elements of the Cambridge debate relevant to the distribution of wealth, such as differential returns or capitalists receiving a certain share of labor income (Ederer and Rehm, 2020b), and, overall, support Pasinetti’s position on the possibility of a stable distribution of wealth in the long run. This paper contributes to these debates in two ways: First, inspired by Dutt (1990), we discuss the conditions for dual, anti-dual and Pasinetti equilibria in a Kalecki-Steindl model with an exogenous profit share, thereby summarizing the arguments in the literature in a consistent yet flexible model in the Bhaduri-Marglin (1990) tradition. Our model includes all the channels that determine the dynamics of wealth accumulation and distribution discussed in the literature, namely labor income going to capitalists, workers saving, differential returns on assets for workers and capitalists, and corporate savings, thereby placing the discussion within a broader analysis. Second, we discuss the dynamics between the distributions of functional income and wealth, and their consequences for the equilibrium wealth share if the assumption of an exogenous profit share is dropped. In doing so, we build on Taylor (2014), but place both the dynamics of the profit share and the wealth share in the medium run, and explicitly discuss their mutual feedback. We thus extend the Kalecki-Steindl literature on the distribution of wealth by allowing for a reinforcing dynamic between the distributions of income and wealth. Our results show that the stable “Pasinetti equilibrium” of the wealth distribution is the most likely outcome both with an exogenous and with an endogenous profit share. In the first case, although both dual and anti-dual equilibria are theoretically possible, they are very special cases that exist only under certain, rather implausible parameter choices. In the second case, while a stable equilibrium wealth share exists for plausible parameter values, the mutual feedback between the income and wealth distributions could potentially de-stabilize the dynamic system, leading to a trajectory of an ever-increasing inequality and thereby an anti-dual “Piketty equilibrium”. However, both the wealth share and the profit share tend to self-stabilize, thus making a stable equilibrium a likely outcome if these forces are strong enough. We conclude, therefore, that while Piketty (2014) has brought to light an important channel through which inequality rises, that was neglected by Post-Keynesians, Pasinetti's (1962) insights into the dynamics of wealth distribution still carry the day. The paper is structured as follows: Section 2 summarizes the relevant literature. Section 3 outlines the model. Section 4 discusses the conditions for dual, anti-dual and Pasinetti equilibria. Section 5 analyses the dynamic interaction between the profit share and wealth distribution. Section 6 concludes. 2. Literature The seminal contribution of Pasinetti (1962) initiated a debate on the stability of the wealth distribution. By extending Kaldor’s (1956) saving function to allow for both workers and capitalists to 3 / 20 accumulate wealth and consequently also to receive profit income,1 Pasinetti (1962) finds that a stable wealth distribution is possible in the long run, and that it depends on the saving rates of workers and capitalists, as well as the functional income distribution. Samuelson and Modigliani (1966) and Meade (1966) respond by arguing that if workers' saving rate is high enough, they accumulate wealth faster than capitalists, so that the latter eventually vanish as a social class. Pasinetti (1966, 1974) responds that this so-called ‘dual equilibrium’ is empirically irrelevant. Later, Darity (1981) uncovers a third possibility – an ’anti-dual’ equilibrium – in which all wealth is concentrated in the hands of capitalists. This debate fruitfully linked to two other strands of literature, which focused on differential returns on the one hand, and corporate savings on the other hand. The former discussed the dynamics resulting from the assumption that workers receive an interest rate on their wealth ownership which is lower than the profit rate of capitalists. While Pasinetti (1962) still assumes these two rates to be equal, his seminal work (Pasinetti 1974, 1983) contributed to the discussion on differential returns following their conceptualization by Kahn (1959). Other work in this strand of literature includes Laing (1969), Balestra and Baranzini (1971), Harcourt (1972), Maneschi (1974), Moore (1974), Gupta (1977), and Fazi and Salvadori (1981). The reasons for differential returns across the wealth distribution might lie, among others, in more professional wealth management at higher wealth levels, the ability to take higher risk, or a higher likelihood of insider knowledge. Recent work provides empirical evidence for differential rates of return (Fagereng et al. 2020, Bach et al. 2020, Ederer et al. 2021, Petach and Tavani 2021, Bundesbank 2022). Differential returns also play a role in Piketty's (2014) argument for an everincreasing wealth concentration, that is, an anti-dual equilibrium. A second strand of the Post-Keynesian literature focused on retained earnings and thus saving by firms. The seminal work in this debate is Kaldor (1966), which defends the Kaldorian saving function in a response to Samuelson and Modigliani (1966) by arguing that retained earnings of firms are the reason why the saving rate out of profits is higher than the saving rate for wage income, given that the saving rate of firms is by definition equal to one. In a "neo-Pasinetti model", Kaldor includes firms' savings and thus shifts the analysis to a distinction between households and firms instead of workers and capitalists. Furthermore, the paper introduces a "valuation ratio", that is, the relation of the market value of shares to the reproduction cost of capital employed by firms, which acts as a mechanism equilibrating savings and investment; this concept was popularized as Tobin’s (1969) ‘q’. Chiang (1973), Moore (1974), and Darity (1981) generalize the analysis of the long-run equilibrium by distinguishing between workers, capitalists, and firms, all of whom save different fractions of their income. Darity (1981) explicitly models how the wealth accumulated by firms is distributed among socioeconomic classes by assuming that the wealth of firms is also indirectly owned by capitalists. These two strands of literature analyse the economy in its long-run equilibrium, in which the capital stock increases at the natural growth rate by assumption. Therefore, they usually do not formulate a general macroeconomic model which represents the full dynamics of wealth accumulation and distribution. Recently, a small but growing literature (Dutt, 1990; Palley, 2012; Taylor et al., 2015; Ederer and Rehm, 2020a) includes wealth accumulation and distribution into Post-Keynesian models in the tradition of Kalecki (1971) and Steindl (1952). These models allow for mixed income sources for both workers and capitalists and yield endogenous stable wealth and income distributions. Some of these contributions show that wealth concentration is likely to rise empirically in Europe (Ederer and Rehm, 2020a) and the US (Kumar et al., 2018). Ederer and Rehm (2020b) show that a Post-Keynesian model predicts dynamics that resemble Piketty’s empirical observations of a rising wealth to income ratio and a rising wealth concentration in the medium-run adjustment phase to the long-term equilibrium. Some of these contributions include some of the above-mentioned elements of the 1 Kaldor proposed different saving propensities for wage and profit income but neglected that workers then accumulate wealth and thus receive profit income. A ‘logical slip’, in Pasinetti’s (1962) words. 4 / 20 Cambridge debate relevant to the distribution of wealth, such as differential returns (Ederer and Rehm, 2020a, 2020b; Petach and Tavani, 2021) and the saving of firms (Palley, 2012; Taylor et al., 2015). The most relevant contribution to this paper, which discusses conditions of Pasinetti, dual and antidual equilibria within a Kalecki-Steindl framework, is Dutt (1990). It includes firms' savings as a factor co-determining the distribution of wealth and finds that in a “neutral” case, in which the wealth of firms is distributed the same way as the wealth held by households in the form of savings or other assets, dual and Pasinetti equilibria are the only possible outcomes. However, despite its prescience, Dutt (1990) has two drawbacks: First, it does not take differential returns into account, and second, it uses a “stagnationist” Kalecki-Steindl model. While state of the art at the time, it has since been replaced in the Kaleckian literature by the seminal Bhaduri and Marglin (1990) model due to its advances in terms of generality. As opposed to the Kaleckian tradition, structuralist models usually complement the demand side with an endogenous determination of the profit share. However, papers in the structuralist tradition typically focus on the cycles of growth and distribution, so-called “Goodwin cycles” (Barbosa-Filho and Taylor, 2006). The dynamics of the income distribution are thus placed in the short run. Those papers that include an endogenous wealth dynamic, including Taylor et al. (2015), Kumar et al. (2018), Taylor et al. (2019) usually do not analyze the feedback between the wealth and the income distribution. Taylor (2014) is the only paper that discusses the possibility of a Pasinetti equilibrium in such a framework, but retains the short-run dynamics of the profit share. Furthermore, demand in this model is unambiguously “profit-led” and it does not include differential returns or retained earnings of firms. Taylor (2014) finds that a dual equilibrium is only possible if the saving rates of workers and capitalists are equal. If capitalists save more, then a Pasinetti outcome is most likely. However, if the distribution of wealth has a strong impact on the profit share, then an anti-dual equilibrium becomes possible. This paper thus adds to these literatures by, first, developing a model of wealth dynamics in the modern Kalecki-Steindl tradition, incorporating mixed income of workers and capitalists that incorporates differential rates of return, and by analyzing the conditions for dual, anti-dual and Pasinetti equilibria. Second, we endogenize the profit share to bridge the gap to the structuralist strand of the literature, while retaining the medium-term perspective of the Kalecki-Steindl approach. 3. Model The model is a standard two-class, Post-Keynesian formulation in the tradition of Kalecki (1971) and Steindl (1952). Growth is endogenous and determined by the investment decisions of firms. Since Kaldor (1962) and Pasinetti (1962), we permit saving rates to differ between workers and capitalists. Building on Dutt (1990), Palley (2012, 2017), and Taylor et al. (2015), and closely following Ederer and Rehm (2020a, 2020b), the model features an endogenous accumulation and distribution of wealth. Both workers and capitalists receive labor and capital income, with the latter governed by differential rates of return on their wealth. Moreover, the model allows for corporate savings, which indirectly translate into the wealth of workers and capitalists via capital gains. As usual in the Kalecki-Steindl literature, the profit share 𝜋 is exogenous2 and determined by the relative power of workers and capitalists in the labor and product markets. Total income (𝑌) is thus divided between total profits (𝑅) and the wage bill (𝑊): 𝑅=𝜋𝑌 (1) 2 The assumption of a fixed profit share will be dropped in section 5. 5 / 20 𝑊=(1−𝜋)𝑌 (2) Retained earnings of firms (which are denoted by subscript f) are defined by the saving rate of firms 𝜂 on total profits: 𝑅𝑓=𝜂𝑅 (3) Distributed profits are therefore (1−𝜂)𝑅. Through their savings out of retained earnings, firms accumulate wealth (𝑉𝑓) over time, which is their capital (𝐾) minus loans (𝐿) and equity (𝐸):3 𝑉𝑓=𝐾−𝐿−𝐸. (4) In the spirit of López-Bernardo et al. (2016), we include a valuation ratio 𝑞, which is defined as the difference between firms’ net capital (capital minus loans) over their equity: 𝑞=𝐾−𝐿 𝐸. We assume a positive saving rate of firms and thus 𝑞 to be larger than 1: firms cannot have negative valuation in the long run. Plugging 𝑞 into equation (4) yields 𝑉𝑓=(𝑞−1)𝐸, where the factor 𝑞−1 represents capital gains, that is, the change in the valuation of equity, which is ultimately owned by households. Households receive a share of profits which corresponds to their share in this profit-generating wealth –that is, equity – but with differential returns between workers and capitalists. These are based on the empirical observation that the composition of wealth differs between workers and capitalists (Ederer and Rehm, 2020a; Ederer et al., 2020; Petach and Tavani, 2020): workers (denoted by subscript 𝑤) hold a larger share of their wealth in low-yield asset classes such as bank deposits, whereas capitalists (subscript 𝑟) tend to own businesses, and thus benefit more from the compound interest. We therefore follow a simplified portfolio choice approach (Godley and Lavoie, 2007), and assume deposits to be non-interest bearing, and the shares of profit-generating assets for workers (𝛾𝑤) and capitalists (𝛾𝑟) to be constant. Taking retained earnings into account implies that capital gains matter for wealth allocation for both workers and capitalists. We formulate this as the share of assets held in wealth generating assets (𝛾𝑟, 𝛾𝑤) depending on the valuation ratio (𝑞): 𝛾𝑤𝑉𝑤=𝑞𝐸𝑤 (5) 𝛾𝑟𝑉𝑟=𝑞𝐸𝑟 (6) The disposable income of capitalists then amounts to their share (𝛼) in total work income (𝑊) plus their share in distributed profit income ((1−𝜂)𝑅), with the latter being determined by their share in total wealth (𝑧) and the part of their wealth held in profit-generating assets (𝛾𝑟), normalized by total wealth (that is, capitalists’ and workers’ wealth shares (𝑧,1−𝑧) times their wealth shares held in profit-generating assets (𝛾𝑟, 𝛾𝑤)). Workers’ disposable income is formulated analogously: 𝑌𝑟=𝛼𝑊+𝛾𝑟𝑧 𝛾𝑤(1−𝑧)+𝛾𝑟𝑧(1−𝜂)𝑅 (7) 𝑌𝑤=(1−𝛼)𝑊+ 𝛾𝑤(1−𝑧) 𝛾𝑤(1−𝑧)+𝛾𝑟𝑧(1−𝜂)𝑅 (8) Total consumption (𝐶) out of disposable income yields differential saving rates of workers (𝑠𝑤) and capitalists (𝑠𝑟), with the assumption that 𝑠𝑟>𝑠𝑤: 3 For the balance sheet matrix and the transaction flow matrix of the model see Appendix A1. 6 / 20 𝐶=(1−𝑠𝑤)𝑌𝑤+(1−𝑠𝑟)𝑌𝑟 (9) Investment, that is the growth of the capital stock, is determined by animal spirits, capacity utilization (𝑢), and the profit share (𝜋). In the Bhaduri and Marglin (1990) tradition, this formulation allows for both wage-led and profit-led demand growth regimes: 𝐼=(𝛽0+𝛽1𝑢+𝛽2𝜋)𝐾 (10) The aggregate goods market is always in equilibrium, output equals demand. Since we abstract from all other sectors, total demand consists of consumption of households and investment of firms, so 𝑌= 𝐶+𝐼. To de-trend income, profits, and investment, we normalize them to the capital stock. This yields stable solutions for capacity utilization 𝑢=Y 𝐾 ⁄, the profit rate 𝑟=𝑅 𝐾 ⁄=𝜋𝑢, and the growth rate of the capital stock g=𝐼 𝐾 ⁄.4 The only asset in the model is the wealth ultimately owned by households (𝑉), which is equal to the capital stock (𝐾): 𝑉=𝐾. (11) Over time, both capitalists and workers accumulate wealth until the wealth share adjusts to its longterm equilibrium. Taking the derivative of the wealth share 𝑧=𝑉𝑟𝑉 ⁄ with respect to time and rewriting the differential equation yields: 𝑧󰇗=𝑉𝑟󰇗 𝑉−𝑉󰇗 𝑉𝑧=(𝑉󰇗𝑟 𝑉𝑟−𝑉󰇗 𝑉)𝑧 (12) This expression is a dynamic version of the famous Cambridge equation, since in its equilibrium (𝑧󰇗= 0)it boils down to Pasinetti's (1962) original expression 𝑠𝑟𝑟=𝑔 when capitalists do not receive any wage income, there are no differential returns for workers and capitalists, and firms do not save.5 It implies that the wealth share is stable when capitalists save exactly the amount that corresponds to their share in the increase in total capital (Palley, 2012, 2017b; Taylor, 2014). Palley (2017b) thus terms the Cambridge equation an “ownership equilibrium condition” since capitalists must save just enough to maintain their ownership share.6 The distribution of wealth stabilizes itself at its equilibrium since owning a higher (or lower) share of wealth leads to a lower (or higher, respectively) percentage increase in capitalists’ wealth than in total wealth. A detailed expression for equation (12)7 yields 𝑧󰇗=𝐴𝑢 (13) with 𝐴 being a quadratic function of 𝑧 in an inverse u-shape, as shown in Figure 1. Since 𝑢 is always positive for the model to be economically meaningful, the sign of 𝑧󰇗 depends uniquely on 𝐴, and setting 𝐴=0 yields the long-run equilibria for 𝑧. The two corner solutions of the model are 𝑧=0 (a euthanasia of the rentier) and 𝑧=1 (a triumph of rentiers). As discussed in Section 2, the Cambridge debate dubbed these "dual equilibrium" (Samuelson and Modigliani, 1966) and "anti-dual equilibrium" 4 For the short-run solution of the model, see Appendix A2. 5 That is, 𝛼=0,𝛾𝑤=𝛾𝑟=1, and 𝜂=0. 6 This is equivalent to saying that the wealth share z is stable if the (percentage) increase in capitalists’ wealth is equal to the (percentage) increase in total wealth, see the right-hand side of equation (12). 7 See Appendix for the details. 7 / 20 (Darity, 1981), respectively. A third option is a stable interior solution, a so-called "Pasinetti equilibrium" (Taylor 2014). We investigate the conditions for these equilibria next. [Figure 1 about here] 4. Dual, anti-dual, and Pasinetti equilibria We first examine the possibility of a dual equilibrium, in which all wealth is concentrated in the hands of workers. If 𝑧=0, then 𝐴=𝑠𝑟𝛼(1−𝜋)>0 in equation (13), which implies an increasing share of wealth for capitalists. If the latter receive a certain share of wage income (𝛼>0), they can always save and acquire wealth. Moreover, when the capitalists’ wealth share is low, the percentage increase in their wealth is higher than the percentage increase in total wealth, which means that the wealth share of capitalists rises. A dual equilibrium is therefore not possible. However, if capitalists do not receive any wage income (𝛼=0), the wealth distribution does not change at 𝑧=0, so that there is the possibility of a dual equilibrium. This equilibrium is nevertheless only stable under certain parameter conditions. For a reasonably small saving rate of workers, the slope of 𝐴 is positive at a zero capitalists’ wealth share, so that even the smallest deviation from the initial position 𝑧=0 entails a rising wealth concentration and the dual equilibrium is therefore not stable. Only if the workers’ saving rate is high, the slope of 𝐴 becomes negative at 𝑧=0, so that the dual equilibrium is the only stable outcome. The exact condition is 𝑠𝑤>[𝑠𝑟𝛾𝑟(1−𝜂)+(𝛾𝑟−𝛾𝑤)𝜂]𝜋 𝛾𝑤(1−𝜂𝜋) (14) Figure 1: Note: This figure shows the iso wealth curves 𝐴=0 for 𝑠𝑤=0.07,𝑠𝑟=0.24,𝛾𝑤=0.49,𝛾𝑟=0.91,𝛼= 0.06,𝜂=0.1; with 𝜋=0.29 (blue line) and 𝜋=0.39 (orange line). Source: Own elaboration. 8 / 20 Moore, B. J. 1974. The Pasinetti Paradox Revisited, The Review of Economic Studies, vol. 41, no. 2, 297-299 Palley, T. 2012. Wealth and wealth distribution in the neo-Kaleckian growth model, Journal of Post Keynesian Economics vol. 34, no. 3, 453–474 Palley, T. 2017. Wagevs. profit-led growth: the role of the distribution of wages in determining regime character, Cambridge Journal of Economics vol. 41, no. 1, 49–61 Palley, T. 2017b. Inequality and growth in neo-Kaleckian and Cambridge growth theory, Review of Keynesian Economics vol.5, no. 2, 146–69 Pasinetti, L. 1962. Income Distribution and Rate of Profit in Relation to the Rate of Economic Growth, Review of Economic Studies vol. 29 no. 4, 267–279 Pasinetti, L. 1966. Paradoxes in Capital Theory: A Symposium: Changes in the Rate of Profit and Switches of Techniques, Quarterly Journal of Economics vol. 80, no. 4, 503-517. Pasinetti, L. 1974. Growth and Income Distribution - Essays in Economic Theory, Cambridge: Cambridge University Press Petach, L. and Tavani, D. 2018. Income Shares, Secular Stagnation, and the Long-Run Distribution of Wealth, http://dx.doi.org/10.2139/ssrn.3198311 Piketty, T. 2014. Capital in the twenty-first century, Cambridge, Mass.: Harvard University Press. Samuelson, P. A., and F. Modigliani. 1966. “The Pasinetti Paradox in Neoclassical and more General Models.” The Review of Economic Studies 33 (4): 269–301. Steindl, J. 1952. Maturity and Stagnation in American Capitalism. New York: Monthly Review Press. Taylor, L. 2014. The Triumph of the Rentier? Thomas Piketty vs. Luigi Pasinetti and John Maynard Keynes, International Journal of Political Economy, vol. 43, no. 3, 4-17 Taylor, L., Ömer, Ö., Rezai, A. 2015. Wealth Concentration, Income Distribution, and Alternatives for the USA, INET Working Paper No. 17, https://www.ineteconomics.org/uploads/papers/WP17-Lance-Taylor-Income-dist-wealthconcentration-0915.pdf Taylor, L., Foley, D.K., Rezai, A. 2019. Demand drives growth all the way: Goodwin, Kaldor, Pasinetti and the Steady State, Cambridge Journal of Economics, vol. 43, no. 5, 1333-1352. Zamparelli, L. 2017. Wealth Distribution, Elasticity of Substitution, and Piketty: An 'Anti-Dual' Pasinetti Economy, Metroeconomica 68 (4), 927-946 15 / 20 A. Appendix A.1 Balance-sheet matrix Table A1 shows the financial balances of the economy. The economy consists of households (workers and capitalists), firms, and a very basic banking sector. Households own firm equities and bank deposits, which banks loan to firms. Firms’ wealth is made up of their capital minus equities and loans. Total capital in the economy equals total wealth ownership, and the financial balances of households’, capitalists’, and firms’ wealth, as well as banks’ balance sheets, sum to zero. Table A1: Balance sheet matrix of the endogenous wealth distribution model with capital gains Households Workers Capitalists Firms Banks Total Capital +𝐾 +𝐾 Equity +𝐸𝑤 +𝐸𝑟 −𝐸 0 Deposits +𝐷𝑤 +𝐷𝑟 −𝐷 0 Loans −𝐿 +𝐿 0 Wealth −𝑉𝑤 −𝑉𝑟 −𝑉𝑓 −𝑉 Total 0 0 0 0 0 Source: Own presentation. Table A2 shows the transactions in the economy. Households (workers and capitalists) receive wages paid by firms (𝑊) as well as distributed profits (𝑅𝑤,𝑟). Firms retain a part of profits (𝑅𝑓). Households’ consumption expenditures (𝐶) flow to firms. Firms invest (𝐼), financed by retained earnings, equity inflows (∆𝐸) and new loans (∆𝐿). Households finance new equity in firms and deposit their savings in banks. Loans flow from banks to firms. Since the model is stock-flow consistent, all rows and columns sum to zero. Table A2: Stocks and flows of the endogenous wealth distribution model with capital gains Households Firms Banks Workers Capitalists Current Capital Capital Total Consumption −𝐶𝑤 −𝐶𝑟 +𝐶 0 Investment +𝐼 −𝐼 0 Wages +𝑊𝑤 +𝑊𝑟 −𝑊 0 Profits +𝑅𝑤 +𝑅𝑟 −𝑅 +𝑅𝑓 0 Equity −∆𝐸𝑤 −∆𝐸𝑟 +∆𝐸 0 Deposits −∆𝐷𝑤 −∆𝐷𝑟 +∆𝐷 0 Loans +∆𝐿 −∆𝐿 0 Total 0 0 0 0 0 0 Source: Own presentation. 16 / 20 A.2 Short-run dynamics The short-run solution of the model assumes the wealth shares to be constant since wealth only accumulates over a longer time horizon. Capacity utilization is assumed to adjust, so that the equilibrium in the aggregate goods market (that is, the IS-condition) is fulfilled for any wealth share of capitalists (𝑧). The short-run solution for capacity utilization thus depends on the distribution of wealth which co-determines the aggregate saving rate 𝑠: 𝑢∗=𝛽0+𝛽2𝜋 𝑠−𝛽1 (17) 𝑠=𝑠𝑤[(1−𝛼)(1−𝜋)+𝛾𝑤(1−𝑧) 𝛾𝑤+(𝛾𝑟−𝛾𝑤)𝑧(1−𝜂)𝜋] +𝑠𝑟[𝛼(1−𝜋)+𝛾𝑟𝑧 𝛾𝑤+(𝛾𝑟−𝛾𝑤)𝑧(1−𝜂)𝜋]+𝜂𝜋 (18) Capacity utilization 𝑢 is positive if the (extended) Keynesian stability condition 𝑠−𝛽1 holds, which is commonly assumed in Post-Keynesian models in the Kalecki-Steindl tradition (Dutt 1990). Since the saving rate rises when the wealth share increases, capacity utilization falls (Ederer and Rehm, 2020a). The reason is that a higher wealth share transfers profit income to capitalists, which depresses total consumption due to their higher saving rate and their higher returns on wealth. Furthermore, since the growth rate 𝑔 positively depends on capacity utilization, a rise in the wealth share unambiguously depresses growth. From the definitions of the profit rate and the growth rate (𝑟=𝜋𝑢 and 𝑔=𝑠𝑢, respectively), it follows that the former is more sensitive to the wealth share than the latter, because the saving rate increases when the wealth concentration rises. This is the reason why the profit rate decreases faster than the growth rate when the wealth share rises, and the two variables will eventually fulfil the Cambridge equation. A.3 Long-run dynamics 𝐴=𝑠𝑟𝛼(1−𝜋)−[𝑠𝑤(1−𝛼)+𝑠𝑟𝛼](1−𝜋)𝑧 +[(𝑠𝑟𝛾𝑟−𝑠𝑤𝛾𝑤)(1−𝜂)𝜋+(𝛾𝑟−𝛾𝑤)𝜂𝜋 𝛾𝑤(1−𝑧)+𝛾𝑟𝑧](1−𝑧)𝑧 (19) 17 / 20 A.4 Conflict-inflation model With defining the target profit share as π𝑇𝑤 =1−Ω𝑇𝑤 =𝛾0−𝛾1𝑢, and the profit share given by the identity 𝜋=1−Ω, wages grow by 𝑤=𝜇1(𝛺𝑇𝑤 −𝛺)+𝑝+𝑎=𝜇1[𝜋−𝛾0+𝛾1𝑢]+𝑝+𝑎 , (20) For simplicity, we assume a constant target profit share 𝜋𝑇𝑟 =𝛿0: 𝑝=𝜏(𝜋𝑇𝑟 −𝜋)=𝜏(𝛿0−𝜋). (21) Taking time derivatives of the profit share 𝜋=𝑤𝐿 𝑝𝑌, we get 𝜋󰇗 =−(𝑤−𝑝−𝑎)=[𝜇1(𝛾0−𝛾1𝑢−𝜋)]𝜋=𝜇1𝐵𝜋 (22) with 𝐵=(𝛾0−𝛾1𝑢−𝜋). Setting 𝐵=0 yields the long-term equilibrium for the profit share. A.5 Jacobian of the dynamic system The elements of the Jacobian of the differential equation system around the long-term equilibria for z and 𝜋 are:14 𝑐11 =𝜕𝐴 𝜕𝑧 =−[𝑠𝑤+(𝑠𝑟−𝑠𝑤)𝛼](1−𝜋) −[(𝑠𝑟𝛾𝑟−𝑠𝑤𝛾𝑤)(1−𝜂)+(𝛾𝑟−𝛾𝑤)𝜂][𝛾𝑤(2𝑧−1)+(𝛾𝑟−𝛾𝑤)𝑧2]𝜋 [𝛾𝑤+(𝛾𝑟−𝛾𝑤)𝑧]2 (23) 𝑐12 =𝜕𝐴 𝜕𝜋=𝑠𝑤(1−𝛼)𝑧−𝑠𝑟𝛼(1−𝑧) +[(𝑠𝑟𝛾𝑟−𝑠𝑤𝛾𝑤)(1−𝜂)+(𝛾𝑟−𝛾𝑤)𝜂]𝑧(1−𝑧) 𝛾𝑤+(𝛾𝑟−𝛾𝑤)𝑧 (24) 𝑐21 =𝜕𝐵 𝜕𝑧 =−𝛾1𝜕𝑢 𝜕𝑧 (25) 𝑐22 =𝜕𝐵 𝜕𝜋=−𝛾1𝜕𝑢 𝜕𝜋−1 (26) 14 Recall that we obtain the equilibria by setting 𝐴=0 and 𝐵=0. 18 / 20 A.6 Numerical discussion of the stability of the dynamic system The condition for a stable equilibrium is 𝑐11𝑐22−𝑐12𝑐21 >0, in which case the dynamic system has two negative real Eigenvalues (see section 5). Figure A1 shows this condition is fulfilled for sufficiently large values of the wealth share and the profit share. For low values of these two variables, however, the condition for stability is not fulfilled (that is, the red area in Figure A1) and the dynamic system is unstable, which results in an increase of both the profit share and the wealth share, so that eventually they move into a range where the system is stable (that is, the green area in Figure A1). Moreover, the extent of the red area does not change much when the parameters are varied, because the effect of these changes on the entries in matrix C often work in the opposite directions (see section 5). Figure A1: Note: The figure shows the stability condition 𝑐11𝑐22−𝑐12𝑐21 as a function of 𝜋 and z. Positive values (stability) are displayed in green, negative (instability) in red. Parameters: 𝑠𝑤=0.07,𝑠𝑟=0.24,𝛾𝑤= 0.49,𝛾𝑟=0.91,𝛼=0.06,𝜂=0.1,β0=0.01,β1=0.01,β2=0.01,γ0=0.5,γ1=0.5). 19 / 20 Institute for Socio-Economics University of Duisburg-Essen Lotharstr. 65 47057 Duisburg Germany uni-due.de/soziooekonomie [email protected] ifso working paper ifso working papers are preliminary scholarly papers emerging from research at and around the Institute for Socio-Economics at the University of Duisburg-Essen. All ifso working papers at uni-due.de/soziooekonomie/wp ISSN 2699-7207 This work is licensed under a Creative Commons Attribution 4.0 International License