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Cambridge Journal of Economics
, Volume Advance Article – Feb 2, 2018

22 pages

/lp/ou_press/demographic-growth-harrodian-in-stability-and-the-supermultiplier-HcrUT10erh

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of the Cambridge Political Economy Society. All rights reserved.
- ISSN
- 0309-166X
- eISSN
- 1464-3545
- D.O.I.
- 10.1093/cje/bex082
- Publisher site
- See Article on Publisher Site

Abstract A basic Kaleckian model is enriched by three simple, intuitive assumptions. First, there is a redistributive system of the wage bill between employed and unemployed workers, the latter receiving subsistence income. Second, only individuals with an income above the subsistence level build savings. The combination of these two assumptions gives rise to an autonomous consumption component whose rate of growth depends on population growth. Consequently, the rate of capital accumulation spontaneously converges towards the rate of population growth (the supermultiplier effect), a dynamic that offers a solution to one of the two Harrodian instability problems. The third assumption corresponds to entrepreneurs’ attempts to adjust investments to restore the normal rate of capacity utilisation. Although this assumption usually generates knife-edge instability, we show here that the stabilising properties of the supermultiplier, provided that the accelerator effect is not overly strong, help overcome this instability and realise the normal rate of capacity utilisation. Therefore, the model may offer a simple, simultaneous solution to the two Harrodian instability problems. 1. Introduction This article presents a Post Keynesian model of economic growth that provides a common solution to the two Harrodian instability problems. First, Harrod (1939) shows that the entrepreneurs’ simple, intuitive behaviour, which involves adjusting their investment to reach their normal (or desired) rate of capacity utilisation, results in knife-edge instability. Instead of solving the problem, these attempts to adjust the capital stock increase the gap between the actual rate of capacity utilisation and its normal value as well as the gap between the actual growth rate and the warranted (normal capacity) growth rate. Regarding this issue, this article draws on previous models that have shown that knife-edge instability can be solved (depending on the value of the models’ parameters) with a combination of the abovementioned destabilising Harrodian behaviour of firms and a stabilising supermultiplier effect.1 Second, Harrod highlights the problems arising from the discrepancy between the actual growth rate and the natural growth rate (which is the sum of the growth rates of the labour force and of labour productivity). Accordingly, the unemployment rate should be subject to considerable instability. However, empirical observations do not confirm such instability. This empirical observation is often used by mainstream economists (for instance, in the Solow model of growth) to support the view that growth must ultimately be determined by supply-side factors, with economic growth adjusting to labour supply growth. From a demand-side perspective, the same observation can be explained by Marxian or Sraffian economists who claim that labour supply is (to a significant degree) endogenous with respect to output growth (through participation rates, disguised unemployment, migration, changes in the informal sector, etc.).2 The present article highlights another novel explanation. Our model shows that an income redistribution from workers to unemployed dependents may provide an account of why demographic growth may play a role in explaining output growth in a demand-led model. The core intuition is that the distributive device results in autonomous consumption expenditures that increase with the level of population. This autonomous consumption will be shown to drive the economic rate of growth. The second Harrodian instability problem is thereby overcome: the rate of employment stabilises, even though the unemployment rate can vary considerably and remain at persistently high levels. The very abundant heterodox literature on the two Harrodian instability problems can be divided into three large families. In the first family, knife-edge instability plays a central role in generating trade cycles, as shown in Hicks (1950). This instability must be impeded by other forces for the path to reach a ceiling and a floor and for the conjuncture to reverse. In Skott (1989, 2010) and Flaschel and Skott (2006), for instance, the reversal results from the negative impact of employment on investment: if the rate of employment is high, entrepreneurs find hiring additional workers difficult, which creates an incentive to slow down capital accumulation. The reverse occurs if the rate of employment is low. In Fazzari et al. (2013), the ceiling is determined by full employment, whereas the floor depends on a supermultiplier mechanism: the rate of accumulation then converges towards the rate of growth of a non-capacity-generating autonomous component of aggregate demand, such as government spending. Note that these trade cycle approaches provide an immediate solution to the second Harrodian problem mentioned above because the decline in employment during a recession is followed by a reversal when the cycle hits the floor; conversely, the rise in employment during an expansion stops and reverses when the cycle hits the ceiling. In the second family of approaches, the entrepreneurs’ investment behaviour that causes knife-edge instability is removed. The trade cycle is also neglected in favour of a steady-state equilibrium approach. Several mechanisms that enable the convergence between the actual and warranted rates of growth have been proposed, and they rest on alternative parameters: the income distribution in Cambridge models (Kaldor, 1957; Robinson, 1956, 1962), the profit retention ratio (Shaikh, 2007), the central bank rate of interest (Duménil and Lévy, 1999), etc. On their side, Kaleckian models rest on an endogenous determination of the actual rate of capacity utilisation and, in turn, on a lasting gap between the actual and warranted (normal capacity) rates of growth (Rowthorn, 1981; Dutt, 1984; Lavoie, 2014).3 In this long-run equilibrium framework, two distinct theoretical strategies have been proposed to reconcile the actual or warranted rates of growth with the natural rate of growth. On the one hand, reconciliation results from the endogenous adjustment of the natural rate of growth. For instance, if technical progress follows Kaldor-Verdoorn’s law (Dutt, 2006), a high rate of capital accumulation results in strong productivity gains and thus a rise in the natural rate of growth. On the other hand, the warranted rate of growth can converge towards the natural rate, for instance, because of counter-cyclical policies: a rise in the unemployment rate induces expansionary monetary and fiscal policies, which increase the rate of investment (Dutt, 2006).4 The third family of approaches retains the long-run steady-state equilibrium rather than the trade cycle analysis. However, the authors believe that the investment behaviour that causes knife-edge instability is so natural (because firms control their investment and target a normal rate of capacity) that ignoring this assumption weakens the Keynesian argument. However, how can knife-edge instability be overcome? In a recent article, Allain (2015) proposes a solution that combines the destabilising effect of the Harrodian firms’ behaviour with the stabilising effect of the supermultiplier mechanism. The latter, whose properties have been closely studied by Serrano (1995A, 1995B),5 rests on the presence of an autonomous aggregate demand component that grows at an exogenous rate. The main property is that the rate of accumulation converges towards the exogenously given growth rate of the autonomous component. However, this mechanism does not ensure that the rate of capacity utilisation converges towards its normal value. Allain’s (2015) innovation involves reintroducing the Harrodian firms’ behaviour and showing that, rather than generating knife-edge instability (but depending on the value of the model parameters), this destabilising behaviour is a necessary condition to restore the normal rate of capacity utilisation. The stability properties of this combination between the supermultiplier and the firms’ behaviour are analysed in detail by Freitas and Serrano (2015), who also suggest some similarities between their Sraffian approach and the Kaleckian approach adopted by Allain (2015), Lavoie (2016), Pariboni (2016) and Nah and Lavoie (2017). Nonetheless, some differences are present in the investment functions adopted. In Freitas and Serrano (2015), investment depends on the level of output and on a marginal propensity to invest,6 while it rests on the rate of growth expected by entrepreneurs and the gap between the actual and normal rates of capacity utilisation in the Kaleckian approach.7 Note also that the autonomous aggregate demand component, whose exogenous rate of growth drives the system dynamic, may vary from one model to another. Whereas Allain (2015) retains public spending, Nah and Lavoie (2017) consider exports. Two other models assume an autonomous consumption component: capitalists’ consumption for Lavoie (2016) or ‘that part of aggregate consumption financed by credit and, therefore, unrelated to the current level of output’ for Freitas and Serrano (2015, p. 261). Pariboni (2016) eventually consider a combination of autonomous consumption, public spending and exports. In the present article, we go a step further by addressing the other problem of instability, which relates to the discrepancy between the actual and natural rates of growth. For the sake of simplicity, we assume a constant labour productivity over time so that the natural rate of growth is equal to the demographic rate of growth. As the economic rate of growth is given by the exogenous rate of growth of the autonomous aggregate demand component, a simple solution is to assume that the government sets the rate of growth in public expenditures in accordance with the demographic rate of growth. However, this ad hoc solution does not reflect reality. Moreover, the aggregate demand component that is most naturally connected with demographic growth is household consumption: the more people there are, the more mouths there are to feed and the more bodies there are to clothe and shelter. Therefore, we propose introducing an autonomous consumption component (in addition to the endogenous component) into the model that grows at the demographic rate of growth. This autonomous component, whose bases differ from those given by Freitas and Serrano (2015) and Lavoie (2016), rests on the combination of three assumptions. First, an exogenous individual subsistence level of consumption exists. Second, a redistributive device between employed and unemployed workers makes the spending of unemployed people who do not receive any direct income from economic activity ‘effective’. We assume here a formal mechanism such as an unemployment benefits system. However, we would have achieved the same results by assuming an informal, more general device, such as the distribution mechanism at stake between breadwinners and dependents. Third, people who earn at the subsistence level completely consume their income, whereas those who earn more split their income between consumption and savings. For this autonomous consumption component to be effective, the wage bill must be high enough to enable everyone to consume at the subsistence level or higher. Interestingly, we will see that the two Harrodian instability problems prevail if this condition is not met. However, if the condition is fulfilled and the accelerator is not overly strong, knife-edge instability may be avoided, and the rate of employment may stabilise. However, there is no reason for the economy to reach full employment. In summary, we present a model with Kaleckian properties in the short run, notably and with the endogenous determination of both the rate of capacity utilisation and the rate of accumulation. However, in the long run, the two Harrodian instability problems will be tamed: the investment decisions made by entrepreneurs to achieve the normal rate of capacity utilisation may succeed (depending on the value of the model parameters) rather than generating knife-edge instability; moreover, the rate of accumulation converges towards the rate of demographic growth because the autonomous consumption of the growing population becomes the main engine of the demand-led economy. In addition, the model may illustrate the stabilising properties of the welfare state because the distributive mechanism at stake in the supermultiplier effect consists of an unemployment benefits system. Although our aim is essentially theoretical, it is supported by several pieces of empirical evidence. On the one hand, empirical data seemingly do not confirm the usual Kaleckian prediction of an endogenous rate of capacity utilisation in the long run,8 hence the need to explain the convergence towards its normal level. On the other hand, Lavoie (2016) reviews different articles showing that investment may be less autonomous than other aggregate demand components (such as government expenditure, exports, residential construction and consumer credit). This article is organised as follows. Section 2 presents the model assumptions, particularly those addressing the consumption behaviour of workers and the wage distribution between the employed and the unemployed that results from the unemployment benefits system. The short-run equilibrium that assumes an exogenous population-to-capital ratio and results in the usual Kaleckian outcomes is presented in Section 3. The supermultiplier equilibrium, which entails the convergence of the rate of economic growth towards the rate of demographic growth and thus the stabilisation of the rate of employment, is analysed in Section 4. However, this is not the long-run equilibrium of our model because of the lasting gap between the actual and normal rates of capacity utilisation. Therefore, the supermultiplier equilibrium must be taken as a benchmark for comparison with the long-run equilibrium developed in Section 5. At this stage, the destabilising behaviour of firms is introduced into the model, and this behaviour is shown to enable, depending on the value of some parameters, the convergence of the rate of capacity utilisation towards its normal level. However, all the previous sections depend on the condition that the wage bill is high enough to allow everyone to consume at the subsistence level. In Section 6, we discuss the meaning and implications of this condition, as well as the properties of the alternative economic regime if this condition is not fulfilled. We conclude in Section 7. 2. The model assumptions We adopt a basic Kaleckian framework, assuming a demand-restricted closed economy without a government, whose gross aggregate demand (Y) is given as follows: Y=C+I+δK, (1) where C, I, K and δ denote consumption, net investment, capital stock and the rate of capital depreciation, respectively. Assuming a technology with fixed coefficients, the aggregate production function is given as follows: Ys=qL=uνK, (2) where Ys and L correspond to the levels of production and labour, respectively. The labour productivity coefficient, q, is supposed to be exogenously given and not subject to productivity gains. As Y¯s is the production level at full capacity, u=YsY¯s represents the endogenous actual rate of capacity utilisation, whereas ν=KY¯s denotes the exogenous capital coefficient. Firms are also supposed to set prices according to a given mark-up on unit labour costs. Therefore, the gross real wage (before taxes) is given as follows: w=(1−π)q, (3) where π denotes the profit share, which is assumed to be exogenous.9 The active population, N, is supposed to grow at an exogenously given rate, n. However, only a fraction of N is employed. Therefore, the rate of employment is e=LN, whereas 1−e corresponds to the rate of unemployment. The firms’ rate of net capital accumulation is assumed to be given as follows: gi=IK=γ+γu(u−un), (4) where un is the normal (or desired) rate of capacity utilisation from the entrepreneurs’ perspective, and γu corresponds to the sensitivity of accumulation to the discrepancy between the actual and normal utilisation rates. Because u=un results in gi=γ, the γ parameter can be understood as the average expectation for the secular rate of growth (subject to animal spirits), as perceived by the managers of firms. Starting from this Kaleckian framework, a few original assumptions are added concerning both the wage bill distribution between employed workers and the unemployed and the resulting consumption behaviours. An unemployment benefits system is assumed to divide the wage bill between employed workers and the unemployed. To the greatest extent possible, unemployment benefits are equal to an individual ‘subsistence’ level of consumption ( α0), which rests on biological and sociological grounds. However, people are assumed to be able to live on an income lower than α0. Hence, two cases must be distinguished depending on condition (C1): wL>α0N. (C1) Consider first that (C1) is not fulfilled: the wage bill is too low to ensure that everyone in the population reaches the subsistence level. In this case, the wage bill is assumed to be uniformly shared between people through the benefits system. Workers and the unemployed consume their entire income, which falls below the subsistence level α0. The subsistence level does not give rise to ‘effective’ autonomous consumption because individual incomes are too low to make this level of expenditure ‘effective’. In addition, if capitalists are assumed to fully save the gross profit, the aggregate consumption is then equal to the wage bill. Conversely, if (C1) is fulfilled, the wage bill is high enough for employed workers to both contribute to the benefits system and keep a net wage higher than α0. Accordingly, each unemployed individual receives and consumes at the subsistence level, i.e. cU=α0. Unemployment benefits amount to (1−e)α0, while total resources amount to eτw, with τ representing the rate of contribution that balances the system: τ=1−eeα0w, (5) where 1−ee corresponds to the dependency ratio. In this economy, employed workers are assumed to share their net wages ( wL=(1−τ)w) between savings and consumption ( cL). More precisely, they are supposed to consume at the subsistence level plus a fraction of their remaining net wage, i.e. cL=α0+α(wL−α0), (6) where α<1 represents the employed workers’ marginal propensity to consume. In this economy, as above, the gross profit is assumed to be fully saved.10 Aggregate consumption then corresponds to C=N[ecL+(1−e)cU]. (7) Substituting cU, cL and the wage bill, wL=(1−π)Y, and then rearranging leads to C=α(1−π)Y+(1−α)α0N, (8) where α(1−π)Y corresponds to endogenous, induced consumption, whereas (1−α)α0N corresponds to autonomous consumption, which is made ‘effective’ because individual incomes are now high enough to make this level of expenditure ‘effective’.11 3. The short-run equilibrium In this and the two following sections, condition (C1) is assumed to be fulfilled, i.e. the wage bill is high enough to allow the unemployed to consume at the subsistence level and employed workers to both reach a higher level of consumption and save a fraction of their wages. The analyses of what occurs if (C1) is not fulfilled and the dynamics between the two different regimes are postponed to Section 6. In the short run, the population-to-capital ratio ( η=NK)12 and the firms’ expectations about the future growth rate (γ) are both assumed to be exogenously given. Both assumptions will be relaxed in the subsequent sections. Substituting (4) and (8) into (1), the short-run equilibrium rate of capacity utilisation is given as follows: u*=ν[γ+δ−γuun+(1−α)α0η]1−α(1−π)−νγu, (9) where the Keynesian stability condition is supposed to hold, i.e.13 1−α(1−π)−γz>0. (C2) For u* to be positive, the numerator of (9) must also be positive. In addition, the equilibrium rate of accumulation and the rate of employment, which can be deduced from (2), respectively correspond to g*=γ+γu(u*−un), (10) e*=u*ηνq. (11) According to the comparative static outcomes, summarised in Table 1.a, the model remains consistent with the basic Kaleckian framework: the economy is both stagnationist and wage-led; in addition, the paradox of thrift occurs because a rise in the propensity to save (through a decline in either the workers’ propensity to consume, α, or the subsistence level, α0) results in a decline in economic activity. In addition, the rate of employment remains procyclical with the rate of capacity utilisation. Table 1. Comparative static assuming that (C1) is fulfilled (we>α0) a. Short-run equilibrium ( η=N/K and γ are exogenous) γ γu un η π α, α0, δ and ν q u* + + / – – + – + 0 g* + + / – – + – + 0 e* + + / – – – – + – b. Supermultiplier equilibrium ( η is endogenous but γ is exogenous) n γ γu un π α, α0, δ and ν q u** + – – + 0 0 0 g** + 0 0 0 0 0 0 η∗∗ + – – + + – 0 e** – + + – – + – c. Long-run equilibrium (η and γ are endogenous) n γu un π α, α0, δ and ν q u∗∗∗ 0 0 + 0 0 0 γ∗∗∗ + 0 0 0 0 0 g∗∗∗ + 0 0 0 0 0 η∗∗∗ – 0 + + – 0 e∗∗∗ + 0 – – + – a. Short-run equilibrium ( η=N/K and γ are exogenous) γ γu un η π α, α0, δ and ν q u* + + / – – + – + 0 g* + + / – – + – + 0 e* + + / – – – – + – b. Supermultiplier equilibrium ( η is endogenous but γ is exogenous) n γ γu un π α, α0, δ and ν q u** + – – + 0 0 0 g** + 0 0 0 0 0 0 η∗∗ + – – + + – 0 e** – + + – – + – c. Long-run equilibrium (η and γ are endogenous) n γu un π α, α0, δ and ν q u∗∗∗ 0 0 + 0 0 0 γ∗∗∗ + 0 0 0 0 0 g∗∗∗ + 0 0 0 0 0 η∗∗∗ – 0 + + – 0 e∗∗∗ + 0 – – + – View Large Table 1. Comparative static assuming that (C1) is fulfilled (we>α0) a. Short-run equilibrium ( η=N/K and γ are exogenous) γ γu un η π α, α0, δ and ν q u* + + / – – + – + 0 g* + + / – – + – + 0 e* + + / – – – – + – b. Supermultiplier equilibrium ( η is endogenous but γ is exogenous) n γ γu un π α, α0, δ and ν q u** + – – + 0 0 0 g** + 0 0 0 0 0 0 η∗∗ + – – + + – 0 e** – + + – – + – c. Long-run equilibrium (η and γ are endogenous) n γu un π α, α0, δ and ν q u∗∗∗ 0 0 + 0 0 0 γ∗∗∗ + 0 0 0 0 0 g∗∗∗ + 0 0 0 0 0 η∗∗∗ – 0 + + – 0 e∗∗∗ + 0 – – + – a. Short-run equilibrium ( η=N/K and γ are exogenous) γ γu un η π α, α0, δ and ν q u* + + / – – + – + 0 g* + + / – – + – + 0 e* + + / – – – – + – b. Supermultiplier equilibrium ( η is endogenous but γ is exogenous) n γ γu un π α, α0, δ and ν q u** + – – + 0 0 0 g** + 0 0 0 0 0 0 η∗∗ + – – + + – 0 e** – + + – – + – c. Long-run equilibrium (η and γ are endogenous) n γu un π α, α0, δ and ν q u∗∗∗ 0 0 + 0 0 0 γ∗∗∗ + 0 0 0 0 0 g∗∗∗ + 0 0 0 0 0 η∗∗∗ – 0 + + – 0 e∗∗∗ + 0 – – + – View Large 4. The supermultiplier equilibrium In the supermultiplier model, η becomes endogenous because the capital stock increases at the endogenous rate ( g*), while the population increases at an exogenous rate (n). The resulting ‘long-run’ equilibrium thus combines the goods market equilibrium with a stationary population-to-capital ratio that results from g** being equal to n.14 This equilibrium (according to the constraints examined in Appendix 8.1) is given as follows: u**=n−γγu+un, (12) g**=n, (13) η**=(n−γ+γuun)[1−α(1−π)]−νγu(n+δ)γuν(1−α)α0, (14) e**=u**η**νq. (15) The comparative static results summarised in Table 1.b explain the convergence mechanisms at stake in the model. In the remainder of this section, the starting point is a supermultiplier equilibrium in which the population-to-capital ratio is stationary ( η=η**) because g**=n. Income distribution, savings and consumption. Starting from the supermultiplier equilibrium, we assume an exogenous change in the profit share (π), in the subsistence level of consumption ( α0) or in the employed workers’ marginal propensity to consume (α). Each of these changes directly affects the average marginal propensity to consume. In the same way, a change in the rate of capital depreciation (δ) affects the capitalists’ net savings. The short-run implications are consistent with the Kaleckian basis of the model (see Table 1.a): any change that reduces consumption (for instance, an increase in π) negatively affects u*, g* and e*. At this stage, the model is both stagnationist and wage-led. However, the fall in the rate of capital accumulation implies an increase in the η ratio (because g*<n), which exacerbates the employment situation but implies a reversal in both u and g (see Table 1.a for the short-run impacts of a change in η). The unemployment benefits system organises the wage bill distribution between employed workers (who save a fraction of their income) and the unemployed (who fully consume their benefits), resulting in a rise in ‘effective’ autonomous consumption that plays a counter-cyclical, stabilising role, i.e. the supermultiplier effect: aggregate demand is now supported by the autonomous component, (1−α)α0N in equation (8), which is growing with population.15 The recovery of u and g continues as long as g*<n because the continuing decline of the employment rate implies increased distribution between workers and the unemployed. The system eventually stabilises as g**=n, which demonstrates that the supermultiplier effect provides a solution to the Harrodian instability problem stemming from the discrepancy between economic and demographic growth. This property will be preserved in the long-run analysis below. Notably, even if u and g are not affected in the ‘long run’, the initial shock has persistent effects on the equilibrium. Indeed, the temporary slowdown in capital accumulation results in both a higher population-to-capital ratio ( η**) and a lower rate of employment ( e**). In other words, the solution provided for the Harrodian instability problem (resulting from the growing gap between n and g) is fully consistent with an economy in which the unemployment rate can remain at a high and steady level. Animal spirits. In the short run, we find the usual Keynesian outcomes: an increase in entrepreneurs’ expectations about the future growth rate (γ) implies a rise in u*, g* and e*. However, in the ‘long run’, the rise in g* results in a decline in the ratio η. The convergence mechanism then engages, though in the opposite direction of that seen previously: the rise in e* impedes the redistribution of wages between employed workers and the unemployed and then reduces the propensity to consume. The ensuing diminution in the rate of capacity utilisation slows the rate of capital accumulation, which converges back towards the rate of demographic growth. The positive impact of entrepreneurs’ optimism on economic growth is thus only transitory because g** remains equal to n at the supermultiplier equilibrium. Nevertheless, the transitory effects imply permanent consequences. The capital stock is permanently higher than it would have been without a rise in γ, but its rate of utilisation ( u**) diminishes. The positive short-run effect on the rate of employment persists in the ‘long run’, with a rise in e**. Demographic growth. Starting from the supermultiplier equilibrium, we now assume a positive shock to the rate of demographic growth. This shock entails an increase in the population-to-capital ratio (η). The same convergence mechanism engages as before: the decline in the rate of employment implies increased distribution between workers and the unemployed and then a rise in the weight of the ‘effective’ autonomous consumption that increases both the rate of capacity utilisation ( u**) and the rate of accumulation ( g**). The system stabilises when g**=n. However, because the rise in the rate of capital accumulation is gradual, its convergence towards the new value of n takes time, hence resulting in an increase in the population-to-capital ratio η** that stabilises at a higher level than in the initial situation. In parallel, there is a drop in the rate of employment e**. Serrano (1995A, 1995B) provides a very in-depth analysis of the supermultiplier properties.16 Note that the supermultiplier effect arises because the average propensity to consume is endogenous, despite a given marginal propensity to consume. Here, according to equation (8), the marginal propensity to consume is dCdY=α(1−π), while the average propensity to consume is CY=dCdY+NY(1−α)α0. Clearly, a necessary assumption for the supermultiplier effect is that employed workers save part of their wages (i.e. α<1). In that case, the consumption by the unemployed is partially financed by a cut in employed workers’ savings, which has a positive impact on aggregate demand. Otherwise, if α=1, unemployment benefits have no effect on aggregate demand because consumption by the unemployed is offset by an equal curtailment of employed workers’ consumption; as a result, u* no longer depends on the subsistence level of consumption, α0. Incidentally, the model outcomes strengthen the merits of income redistribution. As Keynesian economists generally claim, distribution from high to low incomes positively impacts economic activity and employment. However, this redistribution clearly also contributes to stabilising the economy as long as it gives rise to an autonomous demand that grows at the same rate as the population. In the current model, redistribution takes place through a formal unemployment benefits system. However, other welfare state devices (such as pay-as-you-go pension schemes or national health insurance funds) and, with perhaps a greater impact, informal income distribution mechanisms between breadwinners and their dependents can play a similar stabilising role. 5. The long-run equilibrium: a combination of the supermultiplier effect and the Harrodian behaviour of firms In the supermultiplier analysis presented above, the model provides a solution to the Harrodian instability problem that stems from the discrepancy between economic and demographic growth: the combination of autonomous consumption with an unemployed benefits system results in the convergence of the capital accumulation rate and the demographic growth rate ( g**=n). However, there is no reason for the rate of capacity utilisation ( u**) to be equal to its normal (or desired) value ( un), resulting in a lasting gap between g** and the warranted (normal capacity) rates of growth. More precisely, such equality would require the entrepreneurs’ expectations for the secular growth rate (γ) to be equal to the demographic growth rate (n). However, if entrepreneurs focus on their own business, there is no reason to assume that their expectations coincide, individually or on average, with the rate of demographic growth. Hence, there is a gap between u** and un, which may cause entrepreneurs to react in different ways.17 As in Allain (2015), we assume here that the abovementioned supermultiplier equilibrium cannot be the long-run equilibrium because entrepreneurs adjust their expectations (γ) to partially or completely compensate for the gap between these expectations and the actual rate of accumulation ( g*). Their reaction function should be: γ˙=ψ(g*−γ), (16) where the dot denotes the rate of change ( γ˙=dγ/dt) and ψ∈[0,1] represents the speed of the adjustment coefficient ( ψ=1 if the firms instantaneously fill the gap between g* and γ). Substituting with (10), the adjustment function becomes γ˙=ψγu(u*−un). (17) In most models, such behaviour exacerbates the situation because ∂γ˙/∂γ>0. A fall in u* induces a decrease in γ, which induces another fall in u*, and so on. This is the well-known Harrodian knife-edge problem. However, Allain (2015) has shown that this adjustment behaviour does not necessarily lead to instability in supermultiplier models, provided that there is a rather small value for the ψ coefficient, which is also the case in the current model. The long-run equilibrium solution, which combines the supermultiplier equilibrium with the condition that u=un, is given as follows: u***=un, (18) g***=γ***=n, (19) η***=un[1−α(1−π)]−ν(n+δ)ν(1−α)α0, (20) e***=unη***νq. (21) At this stage, the equilibrium rate of growth is finally equal to the warranted (normal capacity) rates of growth. According to the dynamics analysis (see details in Appendix 8.3), the condition under which the equilibrium is locally stable is given as follows: ψ<[1−α(1−π)]unν−(n+δ). (C3) In the Appendix, the right member of the inequality is shown to correspond to the share of autonomous consumption relative to the capital stock. Therefore, the meaning of this condition is that the destabilising Harrodian effect (the left member of the inequality, ψ) must be lower than the stabilising supermultiplier effect that depends on autonomous consumption (the right member) for the system to converge towards the steady-state equilibrium. Otherwise, the dynamic is destabilising. Either a high value of ψ or a high rate of gross capital accumulation prevents the condition from being fulfilled: the accelerator effect is then too strong, which generates knife-edge instability. By contrast, the system converges towards its long-run equilibrium if the accelerator effect is moderate, i.e. the values of ψ and n+δ are sufficiently low. In that case, the entrepreneurs’ adjustment behaviour not only does not result in instability but also becomes a necessary condition for the rate of capacity utilisation to converge towards its normal level. The relatively low level of the ψ parameter can raise questions. However, according to Freitas and Serrano (2015, p. 270), a ‘drastic adjustment is highly unrealistic, both because of the durability of fixed capital (which means that producers want normal utilisation only on average over the life of equipment and not at every moment) and also because producers know that demand fluctuates a good deal and therefore do not interpret every fluctuation in demand as indicative of a lasting change in the trend of demand’.18 The comparative static results regarding the long-run equilibrium are summarised in Table 1.c. Convergence. To explain the convergence mechanism, we start from the long-run equilibrium ( γ***=n and u***=un), and we assume an amelioration of the state of confidence that raises firms’ expectations about the future growth rate. In the short run, we observe the usual changes: a rise in γ results in a rise in u*, g* and e*. In the long run, firms are supposed to adjust the expected secular rate of growth to fill the gap between u* and un (see equation (17)), which leads to a rise in γ. If this is the only effect at work, it generates upward knife-edge instability: the rise in γ results in a rise in u that results in a rise in γ, etc. However, this destabilising effect may be offset by the supermultiplier effect: labour market improvement reduces the wage bill distribution between workers and the unemployed, resulting in a cut in the weight of the autonomous component in consumption that decreases the rate of capacity utilisation u. This stabilising supermultiplier effect dominates as long as (C3) holds. In that case, the destabilising rise in u is more than offset by the decrease in u. The rate of capacity utilisation converges back towards its normal value un, while the rate of accumulation converges back towards the rate of demographic growth. The two Harrodian instability problems are then simultaneously solved. Notably, entrepreneurs do not have to follow or even know the natural rate of growth for the rate of capacity utilisation to converge towards un. The convergence results from the combination of the supermultiplier effect (i.e. the application of the principle of effective demand in a context in which aggregate demand includes an autonomous component growing at the exogenous rate n) with the Harrod investment function (i.e. a progressive adjustment of expectations for the future growth rate in response to the discrepancy between u and un). Comparative static analysis. In the long run, u*** and g*** are given by un and n, respectively. Conversely, neither u*** nor g*** are impacted by a change in the parameters relating to technology, income distribution, consumption or savings (i.e. δ, ν, π, α0 or α). However, the transitory effects on the rates of capacity utilisation and capital accumulation impact the other endogenous variables in the long run. For instance, because a decrease in the profit share (as well as an increase in both α0 and α) results in an increase in both u* and g*, the capital stock momentarily increases at a faster rate than demographic growth. Although this divergence vanishes in the medium and long run, the consequences for capital stock are permanent, with a diminution in the long-run population-to-capital ratio ( η***). In addition, an economy endowed with more capital will hire more workers and produce more goods, leading to an increase in the long-run rate of employment ( e***). Of course, as in the supermultiplier model, the expected growth rate adjusts to an increase in the demographic growth rate. However, the consequences for the population-to-capital ratio and for the rate of employment are reversed. This reversal results from repeated attempts to adjust the rate of capacity utilisation. Because the rise in n implies a rise in u** (which departs from un), firms accelerate their capital accumulation. If condition (C3) holds, these repeated attempts succeed, and the rate of capacity utilisation returns to un. However, the higher rate of capital accumulation leads to a population-to-capital ratio that is lower than that in the initial situation, accompanied by an improved rate of employment. 6. A model with two growth regimes Our model exhibits two distinct growth regimes depending on the value of different parameters. For the sake of simplicity, although the aim of the article does not consider development economics, let us call the regime corresponding to the long-run equilibrium and the dynamics presented above the ‘developed economy’. This terminology refers to condition (C1) being met, i.e. the wage bill is high enough to ensure that everyone has a subsistence level of consumption. Substituting (20) and (21) into (C1), this condition can be rewritten in terms of the rate of gross capital accumulation: πunν<n+δ, (C1LR) where LR stands for ‘long run’. This condition simply reminds us that our ‘developed economy’ is an economy in which workers participate in the financing of gross capital accumulation through their savings. For this condition to be fulfilled, the savings from profits in the long-run equilibrium, πunK/ν, must be lower than the gross investment, (n+δ)K. Therefore, the rate of gross capital accumulation, n+δ, must be positive and high enough for (C1LR) to be fulfilled: if this rate is too low, the weakness of the supermultiplier effect results in a wage bill that is too low to finance the redistributive system. Notably, if (C1LR) holds, then η***<η˜ with η˜=un(1−π)να0. (C1LR’) Therefore, a ‘developed economy’ is also an economy in which the population-to-capital ratio is relatively low, a situation that necessitates a relatively large accumulation of capital over an important period. However, the range of the rates of gross capital accumulation for which the model is valid is limited because of the condition (C3): πunν<n+δ<[1−α(1−π)]unν−ψ. (C1LR and C3) In other words, n+δ must be high enough to induce high levels of aggregate demand but not too high to prevent system instability (resulting from a strong accelerator effect). By contrast, let us call the regime in which (C1) or (C1LR) is not fulfilled the ‘developing economy’. This situation occurs when the rate of gross capital accumulation, n+δ, is relatively low. The resulting weakness of the multiplier effect generates a low wage bill that cannot ensure that everyone reaches the subsistence level. In this case, both the unemployment benefits system and the workers’ saving behaviour are inoperative. The consumption expenditure is fully endogenous and is equal to the wage bill.19 Therefore, there is no room for any autonomous consumption growing at an exogenous rate; hence, the supermultiplier effect disappears. This situation is also characterised by a relatively high population-to-capital ratio (see C1LR’). Therefore, a ‘developing economy’ is an economy that has not accumulated sufficient capital over time. The equilibrium and dynamics of the ‘developing economy’ are given by the usual results of the basic Kaleckian framework, whose outcomes are well known. The equilibrium rates of capacity utilisation and economic growth are respectively given as follows (see details in Appendix 8.4): u#=ν(γ+δ−γuun)π−νγu, (22) g#=γ+γu(u#−un). (23) The rate of capacity utilisation can permanently depart from its normal level. Greater confidence (a higher γ) increases both capacity utilisation and accumulation. In addition, an increase in the profit share results in less economic activity (a stagnationist regime) and a lower rate of capital accumulation (wage-led growth). Notably, because the population-to-capital ratio (η) has no effect on u# or on g#, no mechanism allows g# and n to converge. Therefore, the system is subject to the Harrodian problem regarding the rate of employment instability. In addition, the dynamic of η depends on the gap between g# and n. If g#<n, the ratio η gradually increases, resulting in a continuous, unending decline in e#. By contrast, if g#>n, the ratio η gradually decreases, resulting in a continuous increase in e#, a dynamic that ends when the decline in η satisfies condition (C1LR’). At this stage, the wage bill is high enough to finance the redistributive device, and the economy becomes a ‘developed economy’. The ‘developing economy’ is also subject to the problem of knife-edge instability if entrepreneurs react to the gap between the actual and normal rates of capacity utilisation by adjusting their expected secular growth rate (γ) to target their normal rate of capacity utilisation: a fall in γ in response to u#<un implies a decline in u that widens the gap between u# and un, which results in a further decline in both γ and g#, and so on. The opposite occurs if u#>un, resulting in a cumulative increase in both γ and g#, which makes reaching the status of ‘developed economy’ possible. Therefore, the dynamics of a ‘developing economy’ are somewhat complex because they depend on both the gap between un and u# and the gap between n and g#. We do not investigate this issue in greater depth because this article focuses on what occurs if condition (C1) holds. However, we just call attention to the fact that a ‘developing economy’ may evolve and become a ‘developed economy’, depending on the value of different parameters. Likewise, a ‘developed economy’ can lapse into a ‘developing economy’ if a shock on some parameter leads to the invalidation of the (C1) condition, which may be the case if the rate of gross capital accumulation decreases, for instance. 7. Conclusion The model presented in this article is built on a Kaleckian foundation that is enriched by three distinct hypotheses. First, we assume that a formal or informal system organises the distribution of the wage bill between employed workers and the unemployed, the latter receiving the subsistence income. Second, only individuals who earn an income above the subsistence level are assumed to build savings. The combination of these two hypotheses gives rise to autonomous consumption expenditures with a growth rate given by the demographic growth rate. This crucial ingredient of the supermultiplier effect (Section 4) results in the convergence of the actual rate of capital accumulation towards the demographic growth rate. Therefore, this dynamic provides a solution for one of the two Harrodian instability problems. The core rationale behind this solution is that any change that positively affects the rate of accumulation (so that g>n) is now offset by an opposite change in the weight of autonomous consumption because a decrease in unemployment reduces the benefits, with the opposite being equally true. This mechanism incidentally recalls the positive, stabilising role of the welfare state in a capitalist economy. The third hypothesis introduced in the model corresponds to the entrepreneurs’ attempts to adjust their investment to restore the normal (or desired) rate of capacity utilisation. As is well known, this assumption has been removed from Keynesian long-run equilibrium models because it generates knife-edge instability. However, because firms control their investment and target a normal rate of capacity, this theory can be weakened if this destabilising behaviour is simply ignored. Allain (2015) has shown that knife-edge instability can be eliminated through the stabilising properties of the supermultiplier. We have followed the same path here and confirm the outcome: the attempts to adjust investment, provided that the accelerator is not too strong (cf. condition C3), not only do not degenerate into instability but also are necessary to restore the normal rate of capacity utilisation. As a consequence, both the economic growth rate and the capacity utilisation rate remain endogenous in the short run; however, their variations are transient, and these two rates may return to their position of rest in the long run. However, this long-run property does not suppress Keynesian outcomes because the transient variations lead to permanent changes in the levels of capital stock, production and employment. In other words, an exogenous shock to aggregate demand (via profit share, workers’ propensity to consume, counter-cyclical policies, etc.) will continue to have a lasting impact on economic activity and the rate of (un)employment. Of course, an autonomous consumption resting on the level of population appears to be a good natural candidate as an autonomous aggregate demand component because demographic growth is commonly assumed to be exogenous over a significant period.20 However, the literature addressing the supermultiplier suggests other potential candidates: government expenditure, exports, consumer credit, etc. These components likely can only be considered autonomous for a relatively short period of time, hence the need of a reflection on their choice and articulation. Such reflections may lead to the elaboration of demand-side accounting to identify the causes (or stages) of growth in the medium or long run.21 In addition, the analysis can be improved in many ways, primarily by including technical progress in the theoretical model.22 8. Appendix: mathematical model resolution 8.1. The supermultiplier effect: dynamic and parameter restrictions The dynamic of the supermultiplier model is given as follows: η˙=ηη^=η(n−g*). (24) The supermultiplier equilibrium, which combines (9) with a position of rest ( η˙=0), is the solution of the system: {η˙=0u=u* (25) Putting aside the obvious but irrelevant solution corresponding to η=0 (if N=0 or K→∞), the solution is given by equations (12) to (15). This equilibrium is subject to some conditions on the value of parameters: πu**ν<n+δ<[1−α(1−π)]u**ν, where the first inequality relates to condition (C1), while the second corresponds to the condition that η**>0. As these conditions are almost the same as those in the long-run equilibrium (in which u** is replaced by un), their interpretation and implications are discussed below. The stability conditions of the supermultiplier equilibrium depend on the two following derivatives: dη˙dη=−νγu(1−α)α0η**1−α(1−π)−νγu, (26) d2η˙dη2=−νγu(1−α)α01−α(1−π)−νγu. (27) Assuming that (C2) holds, the second derivative is negative. The function η˙(η) is then an inverted U-shaped relationship with two roots: η=0 and η=η**. It can then be shown that, if η**>0, the former solution is an unstable equilibrium, whereas the latter corresponds to the stable supermultiplier equilibrium. 8.2. A comparison with Matthews’s (1954) analysis As highlighted by an anonymous referee, the analysis proposed by Matthews (1954) more than half a century ago and the present model share some similarities. Matthews’s analysis is even more general, as he includes exogenous productivity (and exogenous productivity growth) in the autonomous consumption component. However, I believe that he is wrong or that his assumption, formulated in literary terms, should be better justified to be more convincing. In the model presented here, the labour productivity coefficient (q) does not appear in the consumption function (8) because a productivity gain (a rise in q) enables the same production level with fewer workers (2) who receive higher real wages (4). Therefore, the wage bill remains unchanged: the same wage bill must finance the subsistence income of the same population (N). The increase in labour productivity thus has no impact on the consumption function, even though it corresponds with a decrease in the employment rate (see equations (15) and (21)). In addition, Matthews’s (1954, p. 90) investment function raises additional problems, as it rests on a very strong accelerator effect because the investment has to completely fill the gap between the actual and the desired capital stock. With our notations, and introducing an additional parameter (λ), Matthews’s accumulation function can be rewritten as follows: gi=IK=λ(uun−1)+δ. (28) Since Matthews assumes that λ=1, the derivative dgi/du=1/un is greater than unity. However, because dgs/du<1, investment is more sensitive than savings to a change in economic activity, so the Keynesian stability condition does not hold. A simple solution may be to assume that investments fill only a fraction of the gap between the actual and desired capital stock. The marginal propensity to save ( λ/un) should be low enough to satisfy dgi/du<dgs/du. Under this condition, Matthews’s model can be shown to produce results similar to ours. Two points deserve special attention. First, a gap remains between the actual and normal rates of capacity utilisation and the normal level—or, in Matthews’s words, between the desired capital-output ratio and its long-term value (see the difference between ν and ν' in Matthews, 1954, p. 90). However, Matthews does not address the knife-edge instability problem. He does not introduce the destabilising behaviour as we have done in Section 5 or as Freitas and Serrano (2015) have done through a slow endogenous adjustment of the marginal propensity to invest. Second, once the issue of the Keynesian stability condition is resolved, Matthews’s system can be shown to converge towards its long-run equilibrium (because dη˙/dη<0) in the same way as the rate of capacity utilisation converges towards u** in our model. Therefore, Matthews’s analysis can hardly be taken as a model of endogenous (self-sustaining) cycles. 8.3. The long-run equilibrium: dynamic analysis The long-run equilibrium is the solution of the system: {γ˙=0η˙=0u=u* (29) where γ˙ and η˙ are given by (17) and (24), respectively. The unique solution of this system corresponds to equations (18) to (21). Note that the condition for η*** to be positive is as follows: 0<[1−α(1−π)]unν−(n+δ), (C4) where the right member of the inequality corresponds to the ratio of autonomous consumption (defined in equation (8)) to capital stock. Actually, the long-run equilibrium distribution of production between the three aggregate demand components (gross investment, induced consumption and autonomous consumption) in terms of capital stock is given as follows: unν=(n+δ)+α(1−π)unν+(1−α)α0η***, (30) where the two consumption components can be inferred from (8) and (18). Therefore, the share of autonomous consumption in terms of capital is given as follows: (1−α)α0η***=[1−α(1−π)]unν−(n+δ). (31) The right member of the equality must be positive for this share, as well as for η***, to be positive. This condition can be fulfilled for relatively low values of n+δ. However, this condition cannot be met if n+δ is too high. In this case, the long-run equilibrium production level is not high enough to satisfy both investment and induced consumption, and a fortiori there is no room for a third aggregate demand component. Therefore, condition (C4) limits the level of the long-run rate of gross capital accumulation ( n+δ). The local stability conditions, depending on the dynamics of both γ and η, can be analysed by means of the Jacobian matrix, which (after linearization) is given as follows: J=(∂η˙∂η∂η˙∂γ∂γ˙∂η∂γ˙∂γ)=(−γu(Dun−νBD)−(Dun−νB)(D+γuν)νADψγuνADψγuνD), (32) where A=(1−α)α0; B=n+δ−γuun; and D=1−α(1−π)−νγu. For the equilibrium to be stable, the matrix determinant must be positive, whereas the trace must be negative. The determinant is DET(J)=ψγuνADη***, (33) which is positive if conditions (C2) and (C4) are met. The trace is given as follows: TR(J)=−γuD([1−α(1−π)]un−ν(n+δ)−ψν). (34) Therefore, the following can be deduced: Tr(J)<0 ⇔ ψ<[1−α(1−π)]unν−(n+δ), (C3) which is more restrictive than (C4). 8.4. The regime of a ‘developing economy’ If (C1) is not fulfilled (i.e. if wL<α0N), the wage bill is assumed to be uniformly shared between people, each person receiving ew. Noting wU, the level of unemployment benefits, and wL, the net wage, it follows that wU=wL=(1−τ)w=ew<α0 with τ=1−e. (35) Workers and the unemployed consume their entire income ( cL=cU=ew). The subsistence level does not give rise to ‘effective’ autonomous consumption. In addition, because profits are assumed to be fully saved, the aggregate consumption amounts to the wage bill, C=wL. Substituting C and I (see equation (4)) into the aggregate demand function (1) and solving results in the goods market short-run equilibrium rate of capacity utilisation: u#=ν(γ+δ−γuun)π−νγu. (22) The denominator must be positive for the Keynesian stability condition to hold. The numerator must also be positive for u# to be positive. Note that (22) is equal to (9), provided that α=1 in (9). In other words, there is no supermultiplier effect if workers fully spend the wage bill, regardless of any consideration about the subsistence level of consumption. To give rise to the supermultiplier effect, a fraction of the consumption expenditures must be autonomous, and workers must save a fraction of their wages. The rate of accumulation that corresponds to the equilibrium of the ‘developing economy’ is given as follows: g#=γ+γu(u#−un). (23) The equilibrium employment rate depends on (2) and (22), then e#=u#ηνq. (36) Assuming that condition (C1) does not hold and substituting (3) and (36) results in the following condition regarding the population-to-capital ratio: η>u#(1−π)να0. (not C1LR’) The author is very grateful to Nicolas Canry and the three anonymous referees for their helpful comments and suggestions Bibliography Allain , O . 2015 . 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The Sraffian supermultiplier , PhD thesis, Faculty of Economics and Politics, University of Cambridge Shaikh , A . 2007 . ‘ A Proposed Synthesis of Classical and Keynesian Growth ’, New School for Social Research, Working Paper no. 5/2007 Skott , P . 1989 . Conflict and Effective Demand in Economic Growth , Cambridge , Cambridge University Press Google Scholar CrossRef Search ADS Skott , P . 2010 . Growth, instability and cycles: Harrodian and Kaleckian models of accumulation and income distribution , in Setterfield , M . (ed.), Handbook of Alternative Theories of Economic Growth , Cheltenham , Edward Elgar Publishing Google Scholar CrossRef Search ADS Skott , P . 2012 . Theoretical and empirical shortcomings of the Kaleckian investment function , Metroeconomica , vol. 63 , no. 1 , 109 – 38 Google Scholar CrossRef Search ADS Stirati , A . 1999 . Wages theory, Sraffian , in O’Hara , P. A . (ed.), Encyclopedia of Political Economy , London , Routledge Footnotes 1 See Allain (2015), Freitas and Serrano (2015), Lavoie (2016) and Dejuàn (2017). 2 See Stirati (1999). 3 See Hein et al. (2011) or Lavoie (2014, pp. 377–410) for a complete and detailed survey. 4 See also the abovementioned articles of Duménil and Lévy (1999) or Skott (2010), whose analyses fit quite well with such an argument. 5 See also Bortis (1997), Cesaratto et al. (2003), Dejuàn (2005, 2013) and Cesaratto (2015). 6 See also Dejuàn (2017) in which a different specification is adopted. 7 This Kaleckian specification will be adopted here. 8 See Skott (2010, 2012) and Dejuàn (2013), who show that the rate of utilisation does not really reflect the difference between rapidly growing countries and slowly growing ones. 9 This assumption is taken both for sake of simplicity and because there is no unanimity among Post Keynesians about the impact of economic conjuncture on income distribution. For many Post Keynesians (among others, Cambridgians), the profit share tends to increase as the economy approaches full employment because firms find responding to increases in aggregate demand more difficult. By contrast, for others (such as Kaleckians), the profit share tends to decrease because the rise in employment goes ahead with the greater bargaining power of workers. 10 This assumption makes avoiding the Pasinetti criticism possible: the workers’ savings are rewarded by a fraction of the profit, but this income is supposed to be fully saved. 11 I am grateful to an anonymous referee who advised me that my argument is very similar to that of Matthews (1954). A brief comment on his model is proposed in Appendix 8.2. 12 Because of the mathematical properties of the model, the use of this ratio is easier than the use of the usual capital intensity ratio. Note also that η considers the population level rather than the labour level. 13 Matthews (1954, p. 90) assumed a very strong accelerator effect because the investment had to completely fill the gap between the actual and desired capital stocks. Appendix 8.2 shows that this assumption does not make fulfilling the Keynesian stability condition possible. 14 As will clearly be shown below, the supermultiplier effect alone does not imply convergence between u and un. Therefore, the supermultiplier equilibrium cannot be a position of rest if entrepreneurs attempt to adjust their investments to target their normal rate of capacity utilisation, which is why the expression ‘long-run’ is put in quotation marks. 15 The causal link between demographic growth and the rise in aggregate demand is more direct than in the existing literature, where the variation in the rate of employment generates an incentive either for entrepreneurs to adjust their investment (Skott, 1989, 2010) or for the central bank to engage in counter-cyclical policies (Dutt, 2006). In Fazzari et al. (2013), the logic is somewhat different because aggregate demand is not necessarily impacted by demographic growth. Indeed, the rate of population growth affects the ceiling of the trade cycles via the usual supply-side argument: producing more than full employment is impossible. The floor of the cycles depends on government expenditure via a supermultiplier effect. However, the exogenously given rate of growth in government expenditure may differ from the demographic rate of growth, a divergence that impacts the amplitude of the cycles but does not eliminate them. 16 See also the recent articles of Cesaratto (2015), Freitas and Serrano (2015) and Lavoie (2016), among others. 17 See Hein et al. (2011, 2012) and Allain (2015), among others, for a detailed discussion. 18 Assuming a different Harrodian investment adjustment, Lavoie (2016) obtains a less restrictive condition on the ψ parameter, i.e. ψ<1. However, Lavoie recognises that ‘if I [Marc Lavoie] had made use of Allain’s Harrodian equation, based on the change in the γ parameter, I would have found exactly the same condition to achieve local dynamic stability’ (Lavoie, 2016, fn. 7, p. 191). 19 Formally, the low level of wages in the ‘developing economy’ does not allow workers to save. Therefore, the marginal propensity to consume out of wages is α=1. Substituting in (8), the aggregate consumption function becomes C=(1−π)Y, which is fully endogenous. 20 However, there is no truly exogenous variable in the ‘very’ long run. 21 See Freitas and Dweck (2013) for an earlier analysis in this direction focused on Brazil between 1970 and 2005. 22 Cesaratto et al. (2003) provide stimulating food for thought on this issue. © The Author(s) 2018. Published by Oxford University Press on behalf of the Cambridge Political Economy Society. All rights reserved.

Cambridge Journal of Economics – Oxford University Press

**Published: ** Feb 2, 2018

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