Stability of zero-growth economics analysed with a Minskyan model
aa r X i v : . [ q -f i n . E C ] N ov Stability of zero-growth economics analysed with a Minskyan model
Adam B. Barrett* Sackler Centre for Consciousness Science and
Department of Informatics , University of Sussex, Brighton BN1 9QJ, UK*[email protected] (correspondence)
Abstract
As humanity is becoming increasingly confronted by Earth’s finite biophysical limits, there isincreasing interest in questions about the stability and equitability of a zero-growth capitalisteconomy, most notably: if one maintains a positive interest rate for loans, can a zero-growtheconomy be stable? This question has been explored on a few different macroeconomic models,and both ‘yes’ and ‘no’ answers have been obtained. However, economies can become unstablewhether or not there is ongoing underlying growth in productivity with which to sustain growthin output. Here we attempt, for the first time, to assess via a model the relative stability ofgrowth versus no-growth scenarios. The model employed draws from Keen’s model of the Minskyfinancial instability hypothesis. The analysis focuses on dynamics as opposed to equilibrium, andscenarios of growth and no-growth of output (GDP) are obtained by tweaking a productivitygrowth input parameter. We confirm that, with or without growth, there can be both stable andunstable scenarios. To maintain stability, firms must not change their debt levels or target debtlevels too quickly. Further, according to the model, the wages share is higher for zero-growthscenarios, although there are more frequent substantial drops in employment.
As humanity is becoming increasingly confronted by Earth’s finite biophysical limits, there is anincreasing interest in questions about the stability and equitability of a zero-growth economy (Rezaiand Stagl, 2016; Hardt and O’Neill, 2017; Richters and Siemoneit, 2017a). In particular, there hasbeen a focus on the sustainability of a zero-growth economy that maintains a positive interest ratefor loans. There are now a variety of models on which this question has been posed explicitly, andboth ‘yes’ (Berg et al., 2015; Jackson and Victor 2015; Rosenbaum, 2015; Cahen-Fourot and Lavoie,2016) and ‘no’ (Binswanger, 2009) answers have been obtained as to whether a stable zero-growthstate is theoretically possible. Typically, the question is settled by the existence, or not, of a singleattractive fixed point (i.e. an equilibrium that is robust at least to small shocks) with economicallydesirable characteristics, namely positive profit and wage rates, and low unemployment (Richters andSiemoneit, 2017a). That is, the focus has been on demonstrating that there is some local stabilitywithin the system. However, real economies do not sit in equilibrium at a locally stable fixed point.They exhibit fluctuations, business cycles and, occasionally, severe crises, whether or not there isongoing underlying growth in productivity with which to sustain growth in output (Minsky, 1986;1een, 2011). This paper analyses zero-growth scenarios by focussing on global stability. Thus,a scenario is considered stable if its dynamics are characterised by fluctuations that do not growin severity; unstable scenarios will be characterised by run-away explosive behaviour (which wouldcorrespond to a crisis). The model employed is a non-linear dynamical system that incorporateselements of Minsky’s financial instability hypothesis (FIH) (Minsky, 1986; Minsky, 1992). Theanalysis involves the tweaking of a productivity growth parameter, set to either two percent or zeroto respectively produce growth and no-growth scenarios. In so doing, this paper is the first to attemptto compare the relative stability of a zero-growth economy with that of a growing economy.Key to the FIH is that serious macroeconomic instability arises as a result of firms desiringto vary their debt burden in response to changes in the profit share, and expectations about thefuture profit share. This idea was first put into a mathematical model by Keen (1995), and thereis now a substantial literature on Minskyan models that capture various dynamics related to theFIH; see Nikolaidi and Stockhammer (2017) for a recent survey. The original Keen (1995) modelconsisted of three coupled differential equations for the key variables: wage rate, employment rateand firm debt. It is derived from a few simple intuitive assumptions, and is capable of producing bothstable and unstable scenarios, depending on firms’ behaviour in relation to debt. It thus providesa useful starting point from which to build a simple model to compare the stability of growth andno-growth scenarios. Further, the presence of labour dynamics ( a la
Goodwin) enables comparisonof employment and wage rates between growth and no-growth scenarios. However, in the originalmodel, investment is a direct function only of the profit share of output, i.e. investment decisionsare based purely on recent profit. Since investment must depend on growth, it is necessary forthe present study to extend the model. Further, it is realistic for investment decisions to have anadditional explicit direct dependence on debt, (i.e. beyond the indirect dependence due merely toprofit itself depending on debt). Thus, rather than employing the Keen (1995) model in its originalform, the investment dynamics here have terms added from a recent model of Dafermos (2017)to include an explicit direct dependence on growth and debt. With output determined by theinvestment dynamics, consumption will be the accommodating, or residual, variable in the model.The outline of the paper is as follows. Section 2 presents the details of the model. The dynamicalvariables in the model are the wage rate, the employment rate, firm debt and target firm debt.Further equations express GDP, growth rate and profit share in terms of these variables. In Section2.1 the analysis pipeline is presented. Section 2.2 demonstrates that the modelled dynamics can formpart of a stock-flow consistent framework. In Section 2.3 the parameters used in the simulations arewritten down and explained. Section 3 presents the simulation results. Scenarios of constant positiveproductivity growth and constant zero productivity growth are shown, demonstrating stable andunstable runs for both cases. Then, more realistic scenarios of fluctuating productivity growth areexplored, with comparisons between scenarios in which mean growth is positive and in which meangrowth is zero. Further, transitions from a positive to zero productivity growth era are considered.The paper concludes with Discussion and Concluding Remarks sections. Running the original Keen (1995) model with parameters that produce a stable scenario with two percent produc-tivity growth led to run-away behaviour when productivity growth was switched to zero (simulation not shown). SeeAppendix C for an explanation of this, and further discussion of the original Keen model. The Model
This section describes the model and its assumptions in detail. As mentioned in the Introduction,most of the pieces of the model are taken from that of Keen (1995), but the debt dynamics areinspired by the recent model of Dafermos (2017). The notation and presentation are drawn fromGrasselli and Costa Lima (2012). Further, the model is an extension of the Goodwin (1967) growthcycle model, which consisted of just two equations for the wage and employment rates, and containedno debt, only reinvestment of profit.It is assumed that there is full capital utilisation and a constant rate of return ν − on capital K : Y = K/ν = aL, (1)where Y is the yearly output, a is productivity and L is labour employed. The yearly wage bill isdenoted W , firm debt is denoted D , and the interest rate by r . The yearly profit Π is defined asoutput minus the yearly wage bill minus the yearly interest payments, that is Π =: Y − W − rD .Concerning investment, it is assumed that all profits are either reinvested or used to pay down debts.Thus the rate of investment I is given by I = ˙ D + Π . (2)This is admittedly a simple model of finance, however the concern in this paper is to constructjust one possible economic model with interest-bearing debt and no growth imperative; for furtherdiscussion of finance see Section 2.2 and the Discussion. Given the rate of depreciation of capital δ we have ˙ K = I − δK . (3)From (1), (2) and (3) we have ˙ Y = 1 ν ( ˙ D + Π − δK ) , (4)an expression we will use further down to derive the growth rate in terms of profit and debt. Pro-ductivity growth is denoted by α , and a constant population size N is assumed. Thus˙ a = αa . (5)Using (1) and (5) it can be derived that the employment rate λ =: L/N satisfies˙ λ = λ ( g − α ) , (6)where g =: ˙ Y /Y is growth (of output). The rate of change of wages w per unit of labour is anincreasing function of the employment rate λ ,˙ w = Φ( λ ) w , (7) The dot here denotes derivative with respect to time. Note the continuous time formulation implies that profitand investment here are both rates. The term yearly profit is used in place of profit rate to avoid confusion, as profitrate commonly refers to a rate of return on capital. Φ explicitly in Section 2.3 below. Note that in additionto being an increasing function, the Phillips curve should satisfy Φ(0) < λ approaches 1 from below, as the employment rate cannot rise higher than 1 (given thatit starts positive, Eq. (6) ensures that it can’t drop below zero). In practice, in the simulations, anexceptional line was included in the code to implement that if λ exceeds 0.99, and Eq. (6) indicatesthat λ should rise further, then that equation is overridden, and ˙ λ is set to zero for the givenintegration step. This is just a simple way of imposing that there is a limited labour pool.The equation for the wages share of output ω =: wL/Y is derived from (1), (5) and (7) as˙ ω = ω [Φ( λ ) − α ] . (8)The equations (6) and (8) for the employment rate and wages share are the same as those of theGoodwin (1967) model, except growth g itself satisfies different dynamics in the present model, aswill be described below.Considering now the debt dynamics, following Dafermos (2017), the rate of change of debt is takento be proportional to the difference between the target debt and the current debt. The equation forthis, expressed in terms of normalised debt d =: D/Y is˙ d = θ ( d T − d ) (9)(henceforth, when the term debt is used, normalised debt is implied). The parameter θ heredetermines the rate at which debt moves towards the target level; θ − is the length of time it takesfor the difference between debt and target debt to drop by a factor of e , all other variables remainingconstant. Note that, in practice, target debt may never become close to being realised, as all thevariables of the system remain in continuous flux. The target debt has a tendency to move towardsa benchmark that depends on the current growth rate and profit share π =: Π /Y :˙ d T = θ ( d + η g + η π − d T ) . (10)The parameter θ determines the timescale on which target debt moves towards the benchmark d + η g + η π . The parameter d is a constant, and η and η respectively determine how stronglythe benchmark debt is affected by changes in growth rate and profit share. As mentioned above, inthe original Keen (1995) model, the investment rate was taken as a function only of profit, with thesimplifying assumption that firms pay attention only to profits and not to debt at all. The targetdebt equation (10) differs from that in Dafermos (2017) by depending additionally on the profit shareas well as the growth rate. Note that these dynamics are designed to model ‘normal times’, and theonset of a crisis, but not the behaviour of the economy after crisis onset. A crisis is assumed to haveoccurred if at any point in a simulation, investment becomes less than zero as a result of the changeof debt becoming sufficiently negative. Throughout the paper, the term ‘Phillips curve’ refers to that linking the rate of employment with wage growth,rather than that linking wage growth and inflation. This equation implies that non-normalised debt D satisfies ˙ D = θ ( d T − d ) Y + dg . Thus, it is assumed that therate of increase of debt depends not just on how far away the current stock of debt is from the current target, but alsoon the current growth rate of the economy, so as to achieve the desired move of the debt-to-output ratio toward thetarget. ω , employment rate λ , debt d and target debt d T . The profitshare π , growth rate g and output Y (GDP) can be written in terms of these variables. The profitshare is given by π = 1 − ω − rd , (11)where r is the interest rate on loans. Using (1), (4), (9), and some basic calculus, the growth ratecan be expressed as g = π + θ ( d T − d ) − δνν − d . (12)The output derives, by definition and basic calculus, from the integral of the growth rate Y ( t ) = Y exp (cid:18)Z tt g d t (cid:19) , (13)where Y is output at some initial time t . Finally, the four coupled differential equations that specifythe dynamics of the system are equations (10), (9), (6) and (8):˙ d T = θ ( d + η g + η π − d T ) (14)˙ d = θ ( d T − d ) (15)˙ λ = λ ( g − α ) (16)˙ ω = ω [Φ( λ ) − α ] . (17) As mentioned in the Introduction, recent analyses of zero-growth economics have focused on thefixed points (equilibria) of the model systems (Richters and Siemoneit, 2017a). Here however, thefocus is on non-equilibrium dynamics. The system will be classed as stable if it is not prone to acrisis (characterised by explosive run-away behaviours), even if it does not converge to a fixed point.The justification for this is that real economies are constantly fluctuating and exhibit oscillations; itis unrealistic to expect convergence to a fixed point.In practice the system tends to oscillate around the theoretical fixed point, and thus it remainsinformative to write down the equations for it. There exists one economically desirable fixed point,i.e. one with a positive employment rate λ > ω >
0. Setting the left-handside of each of the equations (14)–(17) to zero, assuming λ > ω > λ = Φ − ( α ) , (18)¯ d T = ¯ d = 11 + η α [ d + η α + η ν ( δ + α )] , (19)¯ ω = 1 − ( α + δ ) ν − ( r − α ) ¯ d , (20)¯ π = δν + α ( ν − ¯ d ) , (21)¯ g = α . (22) Specifically the product rule ˙ D = ˙ dY + ˙ Y d . Households Firms Banks Foreign ΣNet financial assets S − D D − S − F F S - D F S + D + F Deposits S - - F S + F Loans - - D - D Financial liabilities -
D S + F - D + S + F Deposits - - S + F - S + F Loans - D - - D The nature of the fixed point, i.e., whether it is attractive or repulsive, can be formally assessed bythe signs of the eigenvalues of the Jacobian matrix, see Appendix B for details. There is no simpleset of conditions on the parameters for the fixed point to be attractive. In the scenarios carried outbelow (Section 3), there are cases for which the fixed point is attractive and cases for which it isrepulsive.In the simulations, the system is started close to, but not at, the fixed point. When all parametersare held constant, three possible behaviours are exhibited (i) convergence to the fixed point; (ii)oscillations around the fixed point, with an amplitude that eventually stabilises; (iii) oscillationsthat grow in amplitude, until a crisis is reached and the simulation is stopped (investment becomesnegative and the model breaks down). Scenarios exhibiting either of the first two behaviours areconsidered stable scenarios, whilst only the third scenario is considered unstable. Further scenariosare considered in which productivity growth fluctuates randomly around a fixed mean. In thesescenarios, the nature of the fixed point becomes irrelevant, as the fluctuations in productivity growthtrigger ongoing oscillations in all the variables whether the fixed point is attractive or repulsive. Thisanalysis pipeline differs from that employed by Richters and Siemoneit (2017a), which classified amodel scenario as stable if and only if there exists an attractive fixed point with desirable economiccharacteristics.
In this section it is demonstrated that the model can fit into a stock-flow consistent framework. Table1 provides a financial balance sheet for the model, and Table 2 provides a transaction flow matrixconsistent with the model. Note that not all the flows in the transaction flow matrix are specifiedexplicitly in the equations of the model, and that Tables 1 and 2 do not provide the unique stock-flowconsistent framework that is compatible with the model. It rather provides a useful simple exampleframework for conceptualising the model, and demonstrating its consistency. It is assumed that theflows that are not specified explicitly do not affect the long run stability of the system. The financialassets are household savings S , firm debt D , and net debt F to the rest of the world (RoW). Theflows that have not been defined in the previous section are domestic household consumption C ,interest on savings i S , interest to the RoW i F , imports M , sub-divided into those for consumption M C and those for investment M I , and exports X . Note that domestic consumption, imports andexports must satisfy the accounting identity Y = C + I + X − M . (23)6able 2: A transaction flow matrix for the model
Households Firms Banks Foreign ΣCurrent Capital Current CapitalWages ωY − ωY − C C − M C M C I − M I -I M I X − X − rD rD i S − i S − i F i F rD − i S − i F − ( rD − i S − i F ) 0Firm profits − Π Π 0Net new loans ˙ D - ˙ D − ˙ S ˙ S F − ˙ F
0Σ 0 0 0 0 0 0 0
The lack of a role for consumption in the stability of the model constitutes a departure fromseveral recent analyses of zero-growth economics (Richters and Siemoneit, 2017a). However, thoseanalyses assumed constant rates of consumption out of wealth and income. This is reasonable fortheir fixed point analyses, which explore the system only in the immediate neighbourhood of thefixed point. However, for the dynamical analyses in the present study, including an explicit rolefor consumption would involve adding further parameters to the model to specify how consumptionrates depend on all of the dynamical variables, particularly on the current growth rate and wagesshare. Further assumptions would have to be made about the availability of credit to households forconsumption beyond income. It is beyond the scope of this study to consider diverse debt behaviourfor households; it is rather assumed that if households are not overly indebted, then consumptiondoes not play a role in stability of zero growth macroeconomics. Thus, for the present study the roleof consumption is ignored. The addition of a foreign sector, capable in theory of smoothly consumingoutput that is not consumed domestically makes this assumption more reasonable than if the foreignsector were excluded. It is left for future work to introduce roles for domestic consumption, importsand exports in this modelling framework.Note that the equations of the model impose that all investment is financed by firm profit and(domestic) bank lending, rather than through households or banks taking up firm equity, or house-holds lending to firms. It is further assumed here that banks distribute all their profits to households.The model also neglects to include financial speculation. Incorporating explicit details of realisticmodern-day finance into the model is left for future work, see Discussion.
For the constants in the model, typical values are chosen, taken from Jackson and Victor (2015).The interest rate on loans is r = 0 .
05, i.e. 5%. The depreciation rate is δ = 0 .
07, since typical valuesin advanced economies are around 6-8%. The capital to income ratio is ν = 3; the current value forthis in Canada is a little under 3, while in the UK the value for this is around 5. The Phillips curve7 is drawn from Keen (2013) and is given byΦ( λ ) = 0 . λ − . − . , (24)so that Φ(0 .
95) = 0 and Φ(0) ≈ − .
01. Note that at the fixed point (18)–(22) only the employmentrate depends on the Phillips curve. In particular, neither the profit or wages share at the fixed pointdepend on the Phillips curve.
This section presents the results of the simulations. Scenarios with constant positive and zero pro-ductivity growth are explored, as well as scenarios in which productivity growth fluctuates, and inwhich there is a transition from positive to zero productivity growth. Further, the dependence ofstability on the debt behaviour parameters θ , θ , η , η and d is investigated. Stability is assessedbased on whether or not a crisis occurs, where a crisis is defined as occurring if a moment is reachedat which investment turns negative as a result of rapid debt pay-off. When a crisis occurs, the modelis assumed to have broken down, and the simulation is halted.Fig. 1 shows two percent and zero constant productivity growth scenarios for several choices ofthe debt behaviour parameters. In the top row, the strength of dependence of benchmark debt oncurrent growth and profit share are respectively η = 5, η = 2, while the rates of convergence ofdebt to target debt and target debt to benchmark debt are given by θ = θ = 0 .
25, corresponding toa timescale of 4 years (for an e -fold convergence). The constant d = 0 .
5. This leads to actual debtratios in the various scenarios presented in Fig. 1 lying in the same range as those currently typicalin advanced economies. Simulations are initialised with all variables assigned the values they takeat the fixed point except for the employment rate λ , which is initialised at its fixed point value minus0.01, so as to avoid a constant equilibrium scenario. It can be seen that for these parameter choicesthe system is stable for both positive and zero productivity growth, although the zero growth caseexhibits higher fluctuations in employment. GDP growth fluctuates close to productivity growth,as one would expect, given that a constant population size is assumed. Note that 250 years wassufficient to display the behaviour of these and all other subsequent parameter choices. Continuingthe simulation for longer merely resulted in repetitive oscillatory behaviour. In the second row ofFig. 1, the debt change parameters θ and θ are both increased to 0.5, corresponding to a timescaleof 2 years for ( e -fold) convergence of debt to target debt and target debt to benchmark debt. Thisled to a crisis occurring during the two percent productivity growth run, while the zero growth runremained stable, albeit with oscillations. In the third row, θ and θ are increased further to 0.75,and a crisis occurs for both positive and zero growth cases. Finally, in the bottom row of Fig. 1, θ and θ are maintained at 0.75, while η and η are reduced, respectively to 3 and 1. This leads againto a stable outcome for both two percent and zero productivity growth. The unstable scenario inthe left panel of the second row, i.e. two percent productivity growth, θ = θ = 0 . η = 5, η = 2,could be rendered stable by decreasing any one of the debt behaviour parameters, e.g. by changing This was a typical value for advanced countries during the economically stable period 1981-2006; see OECD dataat https://data.oecd.org. As obtained from the OECD’s table entitled ‘Debt of non-financial corporations, as a percentage of GDP’. Availablefrom http://stats.oecd.org/index.aspx?queryid=34814 mploymentFirm debt to GDP ratioWages share of GDPProfit share of GDPGDP growth (productivity growth dashed) Time (years)
Figure 1: Example two percent (left) and zero (right) constant productivity growth runs for differentdebt behaviour parameters. In the top row θ = θ = 0 . η = 5, η = 2. In the second row, θ and θ are increased to 0.5, leading to instability for the α = 0 .
02 case. In the third row, θ and θ are increased to 0.75, leading to instability for both the positive and zero growth cases. The fourthrow shows stability of positive and zero growth cases for θ = θ = 0 . η = 3, η = 1. In eachpanel d = 0 .
5. All variables are started at the value they take at the fixed point, except for λ whichis initialised at ¯ λ − .
01. 9ither θ to 0.25, θ to 0.25, η to 3 or η to 1, or from reducing the constant d to 0.3. In general, thesystem has potential to move from being stable to unstable if any of the debt behaviour parameters θ , θ , η , η and d are increased from a given stable scenario. In summary, Fig. 1 demonstratesthat the model allows for both stable and unstable economic scenarios, and, in concordance withMinsky (1986, 1992), the greater the variability in debt, the more likely the scenario ends in crisis.The model can be stable for zero productivity growth as well as for positive productivity growth, andwe have even found a scenario in which the model is stable for zero but not two percent productivitygrowth.The model assumes a constant interest rate. Incorporating a dynamical interest rate, as wellas interactions between the interest rate and the other parameters and variables of the model isbeyond the scope of this paper. However, if everything else is held constant, and the interest rateis increased, the dynamics generally become more stable. Fig. 4 in Appendix A shows an exampleof this, namely by reproducing the scenarios in the second row of Fig. 1 but with a higher interestrate of r = 0 .
1. With this higher interest rate, the α = 0 .
02 case no longer exhibits a crisis, and thesize of the oscillations for the α = 0 case are smaller than for the lower interest rate of r = 0 . α changes at the beginning of each year. It is independently regenerated eachyear, from a normal distribution with constant mean (0.02 in the left panels and 0 in the rightpanels) and a standard deviation of 0.01. In the top row of Fig. 2, the scenarios from the toprow of Fig. 1 are reproduced with such fluctuating productivity growth. Both scenarios remainstable, although there are some sizeable drops in employment for the zero growth case, includingone drop down to almost 0.6 during the 250 simulated years. In the middle row of this figure,the scenarios from the bottom row of Fig. 1 are reproduced. Once again, both scenarios remainstable. In this case the fluctuations in employment are comparable for both two percent and zerogrowth. In the bottom row of Fig. 2, it is demonstrated that stochastic productivity growth leads tosubstantial fluctuations in employment and the profit and wages shares even if debt is held almostconstant by the debt behaviour parameters; the scenario θ = θ = 0 . η = 0, η = 0 is plotted.(These scenarios lead to only very small fluctuations in these variables if productivity growth is setconstant rather than fluctuating stochastically.) In this case, the fluctuations in employment andprofit and wages shares are bigger for the zero growth case. In Fig. 5 in Appendix A, Monte Carlosimulations are shown for each of the scenarios in Fig. 2, namely mean and standard deviation over1000 implementations are plotted. The Monte Carlo simulations demonstrate that the behaviourseen in the single implementations in Fig. 2 are typical. Such a distribution reflects real data from the UK from the period 1987-2006, during which mean an-nual productivity growth was 2 . .
22% (according to the OECD’s table athttps://data.oecd.org/lprdty/labour-productivity-and-utilisation.htm F = 0 . p = 0 .
81) and no significant correlation from one year to the next( r = 0 . p = 0 . mploymentFirm debt to GDP ratioWages share of GDPProfit share of GDPGDP growth (productivity growth dashed) Time (years)
Figure 2: Stochastic productivity growth runs. (Left) Two percent mean productivity growth.(Right) Zero mean productivity growth. In all panels d = 0 .
5. All variables are started at the valuethey take at the fixed point, except for λ which is initialised at ¯ λ − .
01. See main text for furtherdetails. 11 mploymentFirm debt to GDP ratioWages share of GDPProfit share of GDP
Time (years)
Time (years) (b)0 50 100 150 200 250
Time (years)
Time (years)
GDP growth (productivity growth dashed)
Figure 3: Transition from positive growth to zero growth. (a) Constant two percent productivitygrowth for t <
50 years, and zero productivity growth thereafter. (b) Constant two percent pro-ductivity growth for t <
50 years; productivity growth decreasing linearly from two percent to zerobetween t = 50 years and t = 60 years; zero productivity growth thereafter. (c, d) Stochastic pro-ductivity growth with mean rate of two percent for t <
50 years and mean rate of zero for t ≥ θ = θ = 0 . η = 3, η = 1, d = 0 .
5. In allpanels the dotted lines show the transition points in productivity growth behaviour.12ig. 3 shows scenarios for the transition from a positive growth economy to a zero-growth econ-omy, under the debt behaviour parameters θ = θ = 0 . η = 3, η = 1, d = 0 .
5. Fig. 3(a) showsconstant two percent productivity growth prior to 50 years, followed by zero productivity growththereafter. The system remains stable following the end of growth, although there is a temporarysubstantial drop in employment near the beginning of the zero-growth era, with a low of 0.863. InFig. 3(b), the change in productivity growth is instead implemented gradually, linearly decreasingfrom 0.02 to 0 over the course of a decade from 50 to 60 years. In this scenario the low in employmentis instead 0.881, thus there is not a huge apparent advantage of a gradual over a sudden curtailingof growth. In Fig. 3(c,d), two runs are shown in which productivity growth is stochastic as in Fig. 2,with mean 0.02 and standard deviation 0.01 before 50 years, and mean 0 and standard deviation 0.01after 50 years. In Fig. 3(c) there is no substantial drop in employment in the period immediatelyafter mean growth goes to zero, while in Fig. 3(d) a substantial drop in employment is observed inthis period. In the long run, however, in both of these fluctuating productivity growth runs, thereare occasional substantial drops in employment after growth has ended. More positively for workers,all of the scenarios in Fig. 3, and indeed in the other figures above, show a higher mean wages shareof output during zero-growth simulations compared with two percent productivity growth simula-tions. In summary, the model implies a stable transition to a post-growth economy, albeit with somefluctuations in the level of employment in the absence of an active government.
In summary, we have found that the model can produce stable and unstable runs, both for a positivegrowth scenario and a zero growth scenario. Further, the simulations suggest that there is no loss ofstability when the economy transitions from positive to zero growth. On the contrary, parameterswere found that produced a stable run only for the zero growth case and not for the two percentgrowth case. In general the system is less stable the greater the dependence of the target debt on profitshare and instantaneous growth, and the faster the rates of convergence of debt to target debt andtarget debt to benchmark debt. This is consistent with debt-deflation theories of economic crisis anddepression (Fisher, 1932; Fisher, 1933; Minsky, 1986; Minsky, 1992; Keen, 2000). The employmentrate was generally less stable for zero growth scenarios than for positive growth scenarios. However,the mean wages share of output was higher for zero growth runs than for positive growth runs withthe same parameters.
The question of whether a capitalist economy with interest-bearing debt has a growth imperative haspreviously received a range of answers from a variety of models pertaining to the stability and viabilityof states of various variables. The Binswanger (2009) model, which concluded that a desirable zero-growth state was not possible, made some restrictive assumptions, namely that of a constant growthin firm debt at all times, equal to the growth rate of the economy. Further, the wage bill was assumedto be a constant proportion of firm debt. Cahen-Fourot and Lavoie (2016) showed that the Kaleckiand Cambridge equations do allow for the possibility of a stationary zero-growth economy, although For further critique of this model see Richters and Siemoneit (2017a), in particular for the problematic feature thatbanks are constantly removing money, in the form of retained profits, from the system. θ = θ = 0 . η = 3, η = 1, theformer scenario is one of constant equilibrium with constant productivity, and the latter is one ofsubstantial fluctuations in all variables, as a result of simply allowing productivity growth to fluctuaterealistically. We have held other variables constant, notably the interest rate, and (implicitly) prices.Future research could explore fluctuations of these parameters, or incorporate dynamic prices as in,say Grasselli and Huu (2015).How is stability achieved in the model for the zero-growth case? A standard concern is thata positive interest rate leads to an exponentially growing stock of debt in the absence of growth(Douthwaite, 2000; Farley et al., 2013). However, in the many stable scenarios plotted, the profitshare (defined as output minus wages minus interest) is positive, and thus firms are (at the aggregatemacro level) able to keep up with interest payments to prevent an exponential growth of debt. Thedesired investment (profit plus new debt) fluctuates, but on average just covers deprecation of thecapital stock (average growth of the capital stock is zero). On average, there is precisely zero profit left14ver after the costs of replenishing the capital stock. Following Richters and Siemoneit (2017a), weconsider this a viable scenario; however, firm owners’ income must be considered as either negligibleor simply part of the wage bill.In the model, debt dynamics are assumed to be determined by a dynamical target debt thatdepends on growth and the current profit share. Investment is a ‘residual variable’, that is fixedgiven the assumptions about debt. The results however are likely to generalise to related models onwhich debt is the residual, and investment is determined explicitly as a function of profit, debt andgrowth. In Appendix C, we show for a general class of such models [based on the Keen (1995) model]that there exist, irrespective of whether productivity growth is positive or zero: (i) an economicallydesirable fixed point (positive wages and employment, and finite debt) that may or may not beattractive; (ii) an attractive fixed point with infinite debt. Behaviour close to the economicallydesirable fixed point will generally be complex and non-linear; however there is no mathematicalreason to expect a demarcation in behaviour between scenarios in which the productivity growthparameter is zero and in which it takes a small positive value e.g. two percent. We do however findthere to be no stable zero growth scenario for the original Keen (1995) model, on which investmentdecisions are based purely on profit (with no direct dependence on debt).It is notable that the model shows the wages share to increase when growth decreases to zero (in-dependent of Phillips curve parameters). This is because Piketty’s (2014) famous analysis posited theopposite, leading to concerns of there being an incompatibility between sustainability (low growth)and equality (high wages share). Piketty’s analysis was neoclassical in nature, and considered onlyan equilibrium scenario, assuming a constant rate of return on existing wealth, which leads to ever-increasing inequality if output (and the workers’ wages share of it) doesn’t grow at least as fastas this rate of return. Here, profits, wages and production are placed into a dynamical stock-flowconsistent model, and a different conclusion emerges. Our finding here also contrasts somewhat withthat of Jackson and Victor (2016). In that paper, the question whether slow growth leads to risinginequality was explored with a stock-flow consistent model with a constant elasticity of substitutionproduction function (a production function associated with neoclassical studies). It was found thatonly for relatively small values for the elasticity of substitution between labour and capital did in-equality not rise for low growth. Here we have utilised the production function Y = K/ν , which ismore common in the post-Keynesian literature (Fontana and Sawyer, 2016), and was the one used inthe original Keen (1995) Minsky model. Our production function does however make the simplify-ing assumption of full capacity utilisation. More detailed post-Keynesian models would incorporateincomplete capacity utilisation, see e.g. Fontana and Sawyer (2016).There are obviously many significant omissions to the simple model. As mentioned above, con-sumer demand dynamics are not modelled, and it is assumed (implicitly) that the supply-drivenoutput can be smoothly absorbed by international markets. The government sector is notably ab-sent. Minsky advocated a big government to stabilise unstable economies (Minsky, 1986; Minsky1992). Indeed, other studies on similar models have shown that countercyclical government spendingcan enhance stability (Dafermos, 2017; Costa Lima et al., 2014). The model does not incorporatea financial sector, nor households taking up firm equity, or corporate bonds. Future work will ex-plore the extent to which the modern financial system creates a growth imperative, and in whatways it could be tweaked to improve the viability of low- or no-growth economics. Further, only asingle country is considered. With the profit share decreased for the no-growth compared to growthscenario, in an open-border global economy, capital would flow out of the borders of a no-growth15ountry to a growth country, with potential to cause a crisis from lack of investment (Lawn, 2005,2011). Further work ought to analyse the extent to which restricting the international mobility ofcapital would be necessary during the transition to a zero-growth economy. For a recent ecologicalmacroeconomics study with much more detailed modelling see Dafermos et al. (2017).The model is macroeconomic in nature and does not address the existence of a growth imperativeat the single firm level. Gordon and Rosenthal (2003) analysed this, and concluded there wasa growth imperative based on the volatility of profits of typical large firms on the stock market.However, Richters and Siemoneit (2017b) pointed out that several key assumptions of this analysiswere unrealistic, for example, the constant investment rates and personal drawing rates. Further,a zero-growth macroeconomic era would likely see reduced volatility of profits, as debt/investmentbehaviour would likely become less volatile. Thus, there is scope for further combined micro- andmacroeconomic analysis of zero growth at the single firm level. Of course in a zero-growth economy,there will still be some businesses that grow alongside others that shrink, and the dynamics ofeconomic transformation and creative destruction will still occur (Jackson, 2009; Malmaeus andAlfredsson, 2017).We have not presented arguments for or against desiring zero growth in productivity and/oroutput, or for the feasibility of long-run zero productivity growth. We have rather attempted tomodel the consequences of this, should this occur. It is notable that Keynes (1936) envisaged aneventual end to growth. Further, some mainstream economists do now consider that, irrespective ofpolicy, the “new normal” growth rate is 1% or lower, possibly due to environmental factors startingto substantially counteract productivity advances from technological development (Malmaeus andAlfredsson, 2017). For recent discussion of prospects for growth and ideologies about growth, seee.g. Malmaeus and Alfredsson (2017) or Rezai and Stagl (2016). The conclusions of this paper remainvalid whether one is interested in the properties of zero-growth economics for reasons of ecologicalconcern (Meadows et al., 1972; Jackson, 2009) or of practical necessity.
We have analysed the relative stability of positive and zero growth scenarios on a dynamical macroe-conomic model with Minskyan features, namely of increasing instability for greater debt behaviourvolatility. We found that, all else being equal, zero productivity growth is, if anything, more likelyto lead to long-term stability than positive productivity growth, albeit with perhaps a somewhatgreater short-term volatility in the oscillatory cycle. Further, according to the model, the end ofgrowth would increase the wages share of output, and hence would not in itself exacerbate inequal-ity. The model contained a basic monetary circuit, and demonstrated the possibility of zero-growtheconomics with a positive interest rate for loans. Further work will analyse the extent to which otheraspects of finance in the modern economy create a growth imperative.
Acknowledgements
ABB is funded by EPSRC grant EP/L005131/1. The Sackler Centre for Consciousness Science issupported by the Dr. Mortimer and Theresa Sackler Foundation. I am grateful to Yannis Dafermos,Tim Jackson, Salvador Pueyo and Oliver Richters for feedback on a first draft of this paper. Two16 mploymentFirm debt to GDP ratioWages share of GDPProfit share of GDPGDP growth (productivity growth dashed)
Figure 4: Example scenarios with higher interest rate. Simulations identical to those in the secondrow of Fig. 1, but with the higher interest rate of r = 0 .
1. In both panels d = 0 .
5. All variables arestarted at the fixed point, except for λ which is initialised at ¯ λ − . AppendixA Further Simulations
Fig. 4 shows example scenarios with a higher interest rate of r = 0.1. Fig. 5 shows mean andstandard deviation over 1000 simulations of each of the stochastic productivity growth scenariosshown in Fig. 2.
B Analysis of fixed point
This section presents analysis of the fixed point (18)–(22). Defining the vector x = ( d T , d, λ, ω ) T ,the Jacobian J is given by J ij =: ∂ ˙ x i /∂x j , and can be computed at the fixed point as¯ J = η θ θ ν − ¯ d − θ η θ ( α − r − θ ) ν − ¯ d − η θ r − η θ ν − ¯ d − η θ θ − θ θ ¯ λν − ¯ d ¯ λ ( α − r − θ ) ν − ¯ d − ¯ λν − ¯ d ω Φ ′ (¯ λ ) 0 . (25)17
50 100 150 200 25000.20.40.60.811.20 50 100 150 200 250
Time (years)
EmploymentFirm debt to GDP ratioWages share of GDPProfit share of GDPGDP growth (productivity growth dashed)
Figure 5: Monte Carlo simulations of stochastic productivity growth scenarios. Mean over 1000implementations of the simulations shown in Fig. 2. (Left) Two percent mean productivity growth.(Right) Zero mean productivity growth. In all panels d = 0 .
5. All variables are started at the fixedpoint, except for λ which is initialised at ¯ λ − .
01. Dotted lines show mean plus/minus one standarddeviation across implementations. See main text for further details.18his can be re-expressed as¯ J = θ − θ − K K − K r − K − K θ − θ θ K − K K − K K , (26)where K = ¯ λ ( ν − ¯ d ) − , K = r + θ − α , K = η θ , K = ¯ ω Φ ′ (¯ λ ) , K = η θ ( ν − ¯ d ) − . (27)The characteristic polynomial is then χ ( x ) = x + p x + p x + p x + p , (28)where p = θ + θ (1 − K ) , (29) p = K K + θ θ − θ K + θ ( K K + K r ) (30) p = K K [ θ (1 + K ) + θ ] (31) p = K K θ [ K ( r + θ − K ) + θ ] . (32)The Routh-Hurwitz criterion for the fixed point to be attractive is p i > , for 0 ≤ i ≤ , p p > p , and p p p > p + p p . (33)These do not translate into any simple condition on the parameters for the fixed point to be attrac-tive. For the simulations carried out, there were sometimes eigenvalues with positive real parts andsometimes not, indicating that, in the range of parameter space explored, there are cases for whichthe fixed point is attractive and cases for which it is repulsive. In all simulations there was at leastone pair of complex eigenvalues, which explains the observed oscillatory behaviour. C Analysis of similar models with an explicit investment function,and debt as a residual variable
In this Appendix, we show firstly that the original Keen (1995) model becomes unstable whenproductivity growth α →
0. Then we show that this does not happen if the model is modifiedso that the investment function depends explicitly on debt. Further, for this latter scenario, wedemonstrate that the fixed point structure is not dependent on whether productivity growth is zeroor positive. These models are similar to the main model presented in this paper, however they havedebt as the residual variable in the capital account of firms, as opposed to investment.In the original Keen (1995) model, investment is purely a (increasing) function of the profit share, I = κ ( π ) ν K . (34)19his leads to the system of equations (Grasselli and Costa Lima, 2012):˙ ω = ω [Φ( λ ) − α ] , (35)˙ λ = λ ( g − α ) , (36)˙ d = κ ( π ) − π − gd , (37) g = κ ( π ) ν − δ . (38)Assuming κ is such that there exists a ¯ π ∈ (0 ,
1) for which κ (¯ π ) = ν ( α + δ ) , (39)there is a single economically desirable fixed point ( λ > ω > d finite), given by¯ λ = Φ − ( α ) , (40)¯ d = κ (¯ π ) − ¯ πα , (41)¯ ω = 1 − ¯ π − r ¯ d . (42)For stable scenarios, the system oscillates close to this fixed point (Grasselli and Costa Lima, 2012).On this model, stability can not be maintained as productivity growth α →
0, because (41) impliesthat the debt at the fixed point goes to infinity.We now consider modification to this model so that the investment function has an additionaldirect dependence on debt, i.e. we replace κ ( π ) above with κ ( π, d ). Then (41) becomes κ (¯ π, ¯ d ) = ¯ π + α ¯ d . (43)There is a broad space of functions κ ( π, d ) for which a solution to this equation exists for both α = 0 and small positive values of α . Thus, in general there will be an economically desirable fixedpoint for both α = 0 and, say α = 0 .
02. Such a system also has a fixed point with infinite debt,( ω, λ, d ) = (0 , , ∞ ), and a possible cause of run-away behaviour is this fixed point being attractive.The existence of this fixed point is easily verified by considering the transformed system with d replaced by u =: 1 /d [again following Grasselli and Lima (2012)],˙ ω = ω [Φ( λ ) − α ] , (44)˙ λ = λ ( g − α ) , (45)˙ u = [ u (1 − ω ) − r + g − uκ ( π, u − )] u . (46)The Jacobian at this fixed point ( ω, λ, u ) = (0 , ,
0) is given by J (0 , ,
0) =
Φ(0) − α κ ( −∞ , ∞ ) − ν ( α + δ ) ν
00 0 κ ( −∞ , ∞ ) − ν ( r + δ ) ν . (47) We don’t consider κ here to depend on recent (productivity or output) growth. It is straightforward to see thatthe results here generalise to such cases; the crucial factor is that κ has a direct dependence on debt, and that there isa solution to the equivalent of (43), with all variables taking economically desirable values. <
0, and given that α and r are greater or equal to zero, these components are indeed negative if κ ( −∞ , ∞ ) ν < δ , (48)that is if the rate of investment is less than the rate of depreciation of capital in the worst case limitscenario of infinite loss (negative profit share) and infinite debt. This would normally be assumedto be the case. Thus, the fixed point at infinite debt is typically stable irrespective of the valueof productivity growth α . We conclude that the fixed point structure is not dependent on whetherproductivity growth is zero or positive. Hence, from this analysis, there is no reason that a generalisedKeen model, with an investment function that depends explicitly on debt, should be any more orless stable when productivity growth is zero, as opposed to say two percent. D Supplementary data