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This paper examines the dynamics of Keynesian models that incorporate feedback e§ects from the labor market to income distribution, investment, aggregate demand and output. A baseline version of the model can generate endogenous growth cycles, but cumulative divergence and economic collapse also become possible for plausible parameter values. Extensions of the model that include monetary and Öscal policy show greater robustness: the local instability of the stationary point leads to limit cycles (rather than complete collapse), even when large, destabilizing changes are made to parameters describing the private sector. The robustness of the general approach is reinforced by the endogeneity of the Öscal and monetary policy rules.


Introduction
Unlike weather-related áuctuations in agricultural societies, business cycles and recurrent crises in capitalist economies cannot be explained by purely natural causes.In principle, they could arise as a result of shocks operating on an otherwise stable economic system.The technology shocks emphasized by real business cycle theory represents a prominent example, and this approach has been extended in the DSGE literature to include a slew of other shocks.The resulting shock-generated áuctuations are ëexogenousí: the economy would converge to a steady growth path in the absence of shocks.
Endogenous business cycles, by contrast, are created by the economic system, even in the absence of external shocks.The analysis of endogenous cycles in capitalist economies goes back to Marxís general law of capitalist accumulation, as described by Marx (1867, chapter 25) and formalized by Goodwin  (1967).A di §erent theory of endogenous cycles was developed more than 70 years later, when Keynesian multiplier-accelerator models became mainstream, following early contributions from Harrod (1936) and Samuelson (1939): the steady growth path is locally unstable, but nonlinearities in the relations that drive these models can prevent cumulative divergence (Kaldor 1940, Hicks 1950,  Goodwin 1951).
The work by Peter Flaschel and his coauthors stands out as a major contribution to the development and reÖnement of dynamic models in a Keynesian tradition, often augmented by Marx-Goodwin elements that endogenize the distribution of income. 1 Unlike Goodwinís formalization of Marxís general law, which uses a reduced-form real-wage Phillips curve to describe movements in real wages, Flaschel and his coauthors have typically speciÖed separate equations for wage and price ináation to determine the evolution of the wage share, with the explicit equations for ináation also leading naturally to an analysis of monetary policy.This paper adds elements that have been emphasized by Flaschel and his coauthors to the áex-output model in Skott (2015, 2023).Harrodian multiplier-accelerator mechanisms represent the main source of instability, but the model is extended to include destabilizing interactions between income distribution, consumption and aggregate demand (ëdestabilizing Rose e §ectsí, in Flaschelís terminology) as well as economic policy.
The paper makes three main points.First, using empirically based functional forms and parameter values, it is shown that the endogenous cycles generated by the model provide a good Öt with observed cyclical patterns for the US economy.Second, the properties of nonlinear dynamic systems can be very sensitive to the precise speciÖcation of the system, and it would appear to raise serious questions about the empirical relevance of the model if small changes in parameters can lead to unbounded divergence.These outcomes may be neutralized by strong stabilizing e §ects of automatic Öscal stabilizers and monetary policy.Economic policy, third, is endogenous, and this endogeneity strengthens the robustness argument: shifts in private sector behavior that threaten to produce cumulative divergence or signiÖcant increases in the amplitude of the áuctuations of key macroeconomic variables will almost certainly provoke changes in economic policy.
Section 2 describes and analyzes a stripped-down model of a pure capitalist economy along the lines of Skott (2015).The extended model in section 3 includes wage and price Phillips curves, the ináuence of expectations and lags on investment, and Öscal and monetary policy.The simulations of the model are presented and discussed in section 4. The concluding section 5 discusses the main results and their robustness.

A baseline model 2.1 Assumptions
Most Örms cannot adjust their employment levels instantaneously to meet variations in demand.The hiring process is costly as well as time consuming, and most jobs require some on-the-job training.Like the capital stock, the level of employment is therefore best treated as a state variable in dynamic models of the business cycle.
The net hiring rate reacts to movements in the demand for output, while adjustments in investment aim to keep capital utilization rates at levels that Örms deem desirable.If the trajectory of demand were perfectly foreseen, the process would be likely to guide the economy to a steady growth path. 2 In the absence of perfect foresight, however, the dynamics of employment and investment are driven mainly by the behavioral responses to evolving discrepancies between outcomes and expectations.In this setting, the steady growth path need not be stable.
If the productivity of labor and the level of output are predetermined, unanticipated movements in demand lead to discrepancies between output and demand.These discrepancies can show up as quantity rationing, price adjustments or unplanned changes in inventories.Rationing occurs but is generally insignificant in capitalist economies, and inventories are procyclical: at the frequencies that are relevant for business cycles, they tend to amplify the áuctuations, rather than act as bu §ers against unanticipated changes in demand.This leaves price adjustments and windfall proÖts as the most plausible manifestation of shortrun disequilibrium when output is predetermined.Emphasizing price stickiness, new Keynesians and most post Keynesians reject this approach. 3My reading of the evidence suggests that prices are, in fact, much more áexible than commonly believed; Abe and Tonogi (2010), for instance, Önd that the price of the average item changes roughly every 3 days in a Japanese data set with 3 billion daily observations. 4In previous work I have therefore analyzed models in which adjustments in prices and proÖt shares clear the goods market (Skott  1989a, 1989b, Skott and Zipperer 2012).
Building on Skott (2015), this paper takes a di §erent route.Production lags make output predetermined in many sectors of the economy, but there are also businesses without production lags and predetermined levels of output.Haircuts, for instance, are not produced prior to demand, and the consumption of haircuts cannot deviate from the production of haircuts.Similar reasoning applies to many other services.Hair dressers, department stores and restaurants need workers as well as Öxed capital.If they have excess capacity of both workers and capital, however, their levels of output may become perfectly elastic: the work intensity and labor productivity react to changes in demand.
The model describes a ëáex-output economyí along these lines.There is excess capacity of both labor and capital, and output adjusts instantaneously to the level of demand, within the limits imposed by labor and capital capacity.
Output and employment Using a Leontief production function, the capacity constraints are given by Y = yL = K  minfy max L;  max Kg where Y; L; K; y and  denote real output, employment, capital, labor productivity and the output capital ratio.For simplicity, there is no technical change, 5 and the maximum levels of the output-labor and output-capital ratios, y max and  max ; are constant.The Leontief production function implies that the values of y and  are proportional to (and can be used as indicators of) the utilization rates of labor and capital, respectively. 6or a number of reasons ñ including volatility of demand ñ Örms typically aim for some level of ënormalí or ëdesiredí degree of excess capacity of both labor and capital.The presence of excess capacity allows movements in output to absorb unanticipated demand shocks without any direct rationing, adjustments in prices or unplanned changes in inventories.Employment responds gradually to these movements, expanding when demand is high and the utilization rate of labor exceeds its normal level.The expansion gets blunted, however, if capital constraints become binding for a signiÖcant proportion of Örms.This happens when the employment capital ratio is high and the average utilization rate of capital approaches its upper limit (if yL=K approaches  max ).Low rates of unemployment, moreover, will raise the search and hiring costs and a §ect the general business climate, thereby putting downward pressure on new hiring.
Formally, the growth rate of employment is speciÖed as a function of signals from the output and input markets (the utilization rate of labor, the employment capital ratio and the employment rate): where l = L=K and e = L=N denote the employment capital ratio and the employment rate.ëDotsí and ëhatsí over a variable will be used throughout the paper to denote the rates of change and proportional growth rate of the variable; i.e., L = _ L=L = (dL=dt)=L: Investment Capital adjusts more sluggishly than labor, and the accumulation rate is unlikely to be ináuenced signiÖcantly by temporary shocks to demand.The fast-moving utilization rate of labor therefore will have no or only limited direct impacts on investment.But investment does react to more sustained changes in demand and, with employment responding to changes in labor utilization, any such changes will be reáected in the employment capital ratio.These considerations suggest a simple speciÖcation with the accumulation rate as an increasing function of the employment-capital ratio: where g K = K and  are the growth rate of the capital stock and the rate of depreciation.Although this speciÖcation of investment may seem unusual, it is closely related to standard investment functions that relate accumulation to the utilization rate of capital.By deÖnition the output capital rate  is equal to the product yl.Thus, equation ( 2) is consistent with the speciÖcation in models that deÖne accumulation as a function of  and assume y = 1: With labor productivity as a jump variable, however, a simple substitution of  = yl into a standard accumulation function (I=K = f () = f (yl)) would be unreasonable: Örmsí investment decisions do not to respond instantaneously to demand shocks that may turn out to be very short-lived.Movements in the labor capital ratio l; by contrast, capture longer term changes that can be expected to a §ect investment.
Private consumption Real consumption (C) depends on labor income and household wealth, The parameters c and  are the marginal propensities to consume out of wages and wealth.Household wealth, , is taken to be equal to the capital stock (K):7 Equation ( 3) is consistent with an old Keynesian consumption function based on the life cycle hypothesis as well as with classical and post-Keynesian theories that emphasize di §erential saving rates out of proÖts and wages (see Skott 2023, chapter 3 for a more detailed discussion).Equations ( 3)-( 4) imply that the ratio of saving to capital can be written Income distribution The real wage is taken as constant in the baseline version of the model. 8The distribution of income need not stay constant, however: cyclical movements in labor productivity will a §ect the proÖt share, which is deÖnitionally related to the real wage and the productivity of labor, Equilibrium conditions The equilibrium condition for the goods market requires that I = S (7)

Implications
Equations ( 2) and ( 5)-(7) determine the utilization rate of labor (y) as a function of the employment capital ratio (l): or The dynamics of the economic system now follow from equations ( 1)-( 2) and (9).If the labor force grows at a constant rate, n, the model produces a two-dimensional system of di §erential equations in the state variables e and and l: _ e = e[h(y(l); l; e)  n] The system (10)- (11) has at most one non-trivial stationary solution with l > 0; e > 0. To see this, note that if ê = 0; then h(y(l); l; e) = n: The stationarity of both e and l therefore requires that f (l) =  + n.The accumulation function f is monotonically increasing, and this equation can have at most one solution, l  .A meaningful solution exists if the net accumulation rate falls below the natural growth rate for a utilization rate of zero but rises above the natural rate for su¢ciently high utilization rates; formally, it is required that f (0) <  + n and that f (l) >  + n for l values above some value.A capitalist economy would not be viable if these behaviorally plausible conditions on accumulation failed to be satisÖed.The conditions are not su¢cient, however.
The function h(y(l); l; e) is monotonically decreasing in e and, having pinned down l  ; equation (11) can be used to determine e  : a solution with 0 < e  1 exists if h(y(l  ); l  ; 0) > n and h(y(l  ); l  ; 1)  n.The second of these conditions must hold, almost by deÖnition: employment cannot increase faster than the labor force if the economy is at full employment.For present purposes, I shall assume that the Örst condition is also satisÖed.If the condition fails to be met, the economy will experience secular stagnation: aggregate demand will be insu¢cient (in the absence of government intervention) to support a steady growth path with a growth rate that equals the natural growth rate n; Skott (2023, chapter 11) discusses secular stagnation and functional Önance.
Assuming the existence of a stationary solution, the local stability properties are determined by the trace and determinant of the Jacobian matrix (evaluated at the stationary point).The Jacobian is given by e  h 3


The trace and determinant are Local stability requires a positive determinant and a negative trace.The determinant is unambiguously positive, but the trace, which contains both positive and negative terms, cannot be signed without additional assumptions about relative magnitudes.A case can be made that the trace will be positive and the stationary solution locally unstable, but stability cannot be excluded; see Appendix A.

Simulations
The simulation in Ögure 1a illustrates a case with local instability.The natural growth rate, the depreciation rate and the real wage are set at n = 0:03;  = 0:07 and != 0:7: The consumption parameters are c = 1;  = 0:05, yielding a saving capital ratio of S K = yl  0:05 The investment function is taken to be linear; by deÖnition, however, gross investment cannot be negative.Thus, The employment expansion function (1), by contrast, is likely to be highly non-linear.Managerial resources "create a fundamental and inescapable limit to the amount of expansion a Örm can undertake at any time" (Penrose 1959, p.  48), and very rapid contractions of the labor force may also impose high costs as morale and productivity su §er. 9 Additional nonlinearities are associated with the employment rate and the employment capital ratio.It matters little for the business climate and the ability of Örms to Önd and hire workers with the required skills whether the employment rate is 50 percent or 51 percent; an increase from 95 to 96 percent, by contrast, could have a large impact.Analogously, small changes in the employment capital ratio have no signiÖcant e §ect on the prevalence of capital constraints if the initial value of l is low.Thus, the sensitivity of employment growth to changes in the aggregate employment rate and the employment capital ratio will be weak at low levels of e and l but strengthen with increases in the levels of employment or capacity utilization.These nonlinearities are captured by the following speciÖcation of the growth rate employment: The functional forms and parameter values of the investment and employment expansion functions draw on the speciÖcations and econometric estimates in Skott and Zipperer (2012).Their regression results imply that investment reacts strongly to sustained changes in demand and that the growth rate of output responds positively to the proÖt share (which is related to the utilization rate of labor) and negatively to the employment rate and capital utilization.The present model di §ers from the one in Skott and Zipperer (2012) with respect to price and output áexibility, but the parameters are in line with their estimates.

Figure 1a about here
With these functional forms and parameters, the model has a stationary solution at y  = 1; l  = 0:5; e  = 0:9;   = 0:3: Figure 1a depicts the outcome over 500 periods with the initial values set to l(0) = 0:52; e(0) = 0:91 The stationary solution is locally unstable, and the model produces convergence to a limit cycle with clockwise cycles in the (e; ); (e; ); (e; Ŷ ); (; Ŷ ) and (; Ŷ ) spaces.These movements are consistent with observed patterns in the US economy (Zipperer  and Skott 2011).
Inevitably, highly stylized models like ( 10)-( 11) also fail to capture many properties of real-world cycles; the model, for instance, produces a static, reducedform functional relation between the proÖt share and the output capital ratio, which precludes the clockwise cycles in the (; ) space seen in the data.More importantly, like most simulations of non-linear dynamic systems, the results are sensitive to changes in parameter values and functional forms.The functional forms and parameters are, I would argue, behaviorally and empirically plausible, but other plausible parameters and functional forms can produce very di §erent results.
The fragility of the results can be illustrated by changing the parameter in the investment function.If the coe¢cient on l is reduced slightly from 1 to 0:985, the limit cycle disappears and the stationary solution becomes locally stable; increasing the coe¢cient above 1:075, on the other hand, generates unbounded áuctuations and, eventually, a complete collapse with the employment rate converging to zero.
The window of cycles is not quite as narrow as these qualitative results suggest.For practical purposes it makes little di §erence whether the system produces slow oscillatory convergence to the stationary solution or convergence to a closed orbit.But the point remains: the simulation results are fragile, and I deliberately chose a sensitivity of investment to changes in utilization at the low end of the long-run estimates in Skott and Zipperer (2012) in order to illustrate the possibility of limit cycles.Figures 1b and 1c depict a stable case with an investment parameter of 0.97 and a case of cumulative divergence with an investment parameter of 1.08, respectively.

Figures 1b and 1c about here
The fragility also shows up if the real wage is endogenized.Using a linear real-wage function, let != 0:7 + a(e  0:9) The real wage now responds to deviations from the stationary value of e but, keeping the functional forms and parameter values of the original simulation, the stationary solution is unchanged.The sensitivity of the outcome to changes in the parameter a; can be illustrated by noting that (retaining all other parameters in the simulation) the economy heads for collapse with e !0 for any a value above 0.033; parameter values below -0.01, by contrast, make the stationary point stable.Following Flaschel and his coauthors, the real wage can also, more realistically, follow a dynamic ëreal-wage Phillips curveí, 10 In this case positive coe¢cients on (e  e  ) are highly destabilizing, while negative coe¢cients on (    ) are stabilizing.Figures 2a-2b illustrate the patterns for (e; ). Figure 2a, which uses the parameters  1 = 0:01 and  2 = 0; shows how even a tiny employment e §ect on real wage growth is su¢cient to produce divergence.In Ögure 2b with  1 = 0 and  2 = 0:1, the stabilizing negative feedback from capital utilization to real-wage growth produces convergence to the steady growth path. 11In both cases, the investment, consumption and employment expansion functions as well as the initial values are as in Ögure 1a.
Figures 2a and 2b about here 10 The cyclical pattern of real wages is not entirely clear, but many studies suggest that the level or growth rate of real wages depends positively on the employment rate.Abraham and Haltiwanger (1995) suggest that the evidence is inconclusive. 11For values of  1 below 0.1 there is convergence to a limit cycle with progressively smaller amplitude as  2 increases.

An extended model with monetary and Öscal policy
The complete absence in the baseline model of a public sector and economic policy misrepresents modern capitalist economies.Central banks routinely adjust interest rates in response to economic performance, automatic stabilizers are signiÖcant, even in capitalist economies with relatively small public sectors, and discretionary Öscal policy can be and has been used, especially when monetary policy and automatic stabilizers prove inadequate to prevent major recessions.Furthermore, the sensitivity of the baseline model to cyclical movements in the real wage make it questionable to freeze the real wage by assumption, and the speciÖcation of investment also has serious weaknesses.The extensions in this section address these problems at the cost of increasing the dimensionality of the dynamic system. 12Harrodian multiplier-accelerator mechanisms are still the main source of instability, with feedback e §ects from the labor market playing a key role in turning local instability into bounded áuctuations.But the Harrodian mechanism becomes more complex; endogenous movements in real wages and ináation contribute to the dynamics; important stabilizing feedback e §ects from the labor market are mediated by economic policy and automatic Öscal stabilizers.

Extensions
Fiscal and monetary policy Government spending on goods and services is relatively stable, a feature that can be captured in a stylized way by linking spending to potential output.There are two natural indicators of potential output: the level of output associated with a normal utilization of the capital stock (  K) and the level of output associated with having the employment rate and the productivity of labor at their steady growth values (y  e  N ).A simple linear speciÖcation along these lines assumes that13 where N is the labor force and e  ; y  and   denote the employment rate, labor productivity and output capital ratio along the steady growth path.
Leaving out discretionary Öscal responses, however, would underestimate the ináuence of Öscal stabilization, especially in states with high unemployment and interest rates at the zero lower bound.The US Öscal stimulus in 2009 and again, more dramatically, during the COVID pandemic provide recent examples.Thus, I shall add a third, cubic term: this term is relatively unimportant when the employment rate is close to its steady growth value; positive when e is below the steady growth value and increasing rapidly as e declines; negative when the economy is overheating and decreasing rapidly as employment rises signiÖcantly above the steady growth path. 14igh debt ratios, Önally, often generate downward pressures on government spending.The strength of this e §ect is not obvious, but the ësustainability of government debtí has been a constant theme in economic policy, and it would seem clear that there is some negative feedback from debt levels to government spending.
Formally, let where B is nominal government debt.The extended model includes an endogenous determination of ináation.Thus, the price level cannot be normalized to one, and nominal debt must deáated by the price p; unlike B; the variables G and K are in real terms.Tax revenues may be determined largely by current income, but many transfers depend on the employment rate; unemployment beneÖts represent a prime example.As a simple speciÖcation, assume that net taxes (in real terms) are given by 15 T The dynamics of nominal government debt follows the standard equation where r is the nominal rate of interest on government debt.The tax, transfer and spending parameters will be calibrated to ensure that a balanced government budget and zero debt become consistent with steady growth. 16DeÖcits (surpluses) will appear in recessions (booms), however, leading to áuctuations in public debt; because of asymmetries between expansions and downturns, the average level of debt over the cycle will not in general be equal to zero.The 14 Almost by deÖnition, discretionary policy will be lumpy and irregular; it will not follow a smooth function.The cubic speciÖcation should be seen as a simple, mechanical way of approximating these less regular responses and the increasing likelihood of discretionary intervention when the employment rate deviates signiÖcantly from its steady growth value. 15Interest payments on government debt are typically taxed, and tax revenues could be speciÖed as Simulation results with this alternative speciÖcation are very similar to those reported in section 4. 16 Skott (2023, chapter 11) considers Öscal policy and the role of public debt in the long run if weak demand threatens to produce secular stagnation (cf.above, p. 6).
trajectory of the debt capital ratio (b = B=(pK)) is determined by Taylor rules for monetary policy specify the target interest rate as an increasing function of ináation and the deviation of output from potential output.The evidence shows signiÖcant inertia, however, with the adjustments happening gradually (Clarida et al. 2000).Formally, let The central bank would like to implement a change _ r d that is proportional to the di §erence between the its target rate r T and the actual rate r, with the parameter  r indicating the speed of adjustment.The trajectory of interest rates is subject to a ZLB-constraint, however: the actual nominal rate is constrained to be positive, 17 r = maxf0; r d g Wages, prices and the proÖt share Using speciÖcations that follow the analysis and empirical results in Flaschel and Krolzig (2006) and Diallo et al.
(2011), it is assumed that the rates of wage and price ináation are determined by separate, expectations augmented Phillips curves.Wage ináation is a function of the employment rate, the utilization rate of capital, the proÖt share and expected price ináation, while price ináation depends on the utilization rate of capital, wage ináation, the proÖt share and the expected ináation rate: 18 where w; p and x are the nominal wage, price and expected price ináation.
Expected ináation follows an adaptive process ) 17 The policy interest rate has been negative in some economies, including the EU.It makes no substantial di §erence, however, whether the lower bound is at 0; 0:5% or 1%: Following standard practice, I shall assume that the zero lower bound is in fact zero. 18Diallo et al. (2011) also include endogenous variations in the utilization of labor.Unlike in the present model, however, they assume perfect áexibility with respect to the number of working hours.The productivity per hour is taken to be constant, with changes in hours a §ecting total wage income rather than the proÖt share.
Using equations ( 24) and ( 25), the ináation rate and the dynamics of the real wage can be written as where Phillips curves often leave out the utilization rate of capital.Both employment and utilization ináuence potential output, however, which supports speciÖcations like equation ( 27) that include both variables. 19Equation ( 28) implies that the real wage can be ëlabor market ledí or ëgoods market ledí, in the terminology of Diallo et al. ( 2011): depending on parameter values and on the reduced-form correlation between e and , the growth of real wages may be positively or negatively related to the employment rate.Equations ( 24)-( 27) overestimate the downward áexibility of nominal wages and prices, which even in the depth of the great depression did not fall at an annual rate below -10%. 20The imposition of a lower limit on the ináation rate approximates this stickiness in a simple way: Investment Investment plays a central role in all Keynesian theory ñ especially so in dynamic models with Harrodian instability ñ and the baseline investment function in equation ( 2) leaves out potentially important complications.First, investment requires Önancing, and an increase in the costs of Önance (in the real rate of interest) is likely to exert a dampening ináuence.Furthermore, an inability of some Örms to obtain external Önance (or, more generally, their inability to obtain it on reasonable terms) implies that the proÖt share may have a direct positive ináuence on investment (e.g.Fazzari et al. 1988), an ináuence that is reinforced by a more general e §ect of the proÖt share on animal spirits.Second, there are likely to be feedback e §ects from the labor market to accumulation: the size of the reserve army of the unemployed impacts the general business climate and the willingness of Örms to invest.The accumulation rate, third, will be a §ected by expected future growth rates of demand.If Örms were identical and the economy áuctuated around a steady growth path, the expected medium-term growth rate could be taken as constant: the expectations could become anchored to the long-run steady growth rate.But Örms di §er.Some Örms experience high growth rates while others stagnate or decline.Hence, even if aggregate output áuctuates around a steady growth path with a constant employment rate, the growth expectations of individual Örms will not be anchored to the steady growth rate.When observing an increase in demand, each Örm has to disentangle the role of temporary shifts (which can be Örm speciÖc or related to aggregate demand) from sustained shifts in demand for its output.Weighing these possibilities, Örms are likely to adjust their medium-term demand expectations partially in response to movements in aggregate demand. 21ourth, investment is subject to both decision and implementation lags.The speciÖcation of investment as a function of the employment capital ratio (rather than the output capital ratio) excludes an immediate impact of demand shocks.The ináuence comes gradually as employment responds to changes in the utilization rate of labor.This indirect introduction of a delay between changes in demand and investment goes some way towards capturing the decision lags.But some investment projects ñ the construction of major new plants, for instance ñ involve signiÖcant lags between the decision to invest and the start of actual investment as well as prolonged periods of ongoing investment.To approximate these lags between decision and implementation in a continuous-time setting, a distinction can be made between actual and target investment, with actual accumulation rates adjusting towards the target rate.This gradual adjustment of investment is analogous to the capital adjustment principle, but now applied to investment levels rather than capital stocks.
Formally, adding the real interest rate (r  x), the proÖt share, the employment rate and the expected, medium-term growth rate () to the target investment function and introducing an implementation lag, let where (I=K) T denotes the target rate of gross accumulation.As in section 2, gross investment is subject to a non-negativity constraint.
Private consumption and saving Consumption (C) now depends on disposable labor income, employment-dependent transfers (which are treated like wage income) and household wealth,

Short run equilibrium and dynamic system
The equilibrium condition for the goods market includes government spending and taxation Using equations ( 6), ( 17)-( 18) and ( 34), equation (34) determines the utilization rate of labor as an implicit function of l; g K and b: The extended model generates the following system of di §erential equations: There are eight state variables (l; g K ; ; e; x; !; b; r d ), with ; r; p; G=K; T =K and y determined statically as functions of these state variables (equations ( 6), ( 17)-( 18), ( 22), ( 29) and ( 35)).
Even basic properties of the system are indeterminate without additional assumptions.Depending on functional forms and parameters, for instance, the system may have no stationary solution or it could have multiple solutions.Fortunately, we have relatively good information about some of the relations.The investment function is the main exception, but even in this case, ballpark values of the parameters can be identiÖed.
The next section uses empirically based calibrations and simulations to illustrate important properties of the system.As in section 2, variations in the sensitivity of investment to changes in the employment capital ratio will be used to examine the robustness of the model.

Simulations
All simulations retain the employment growth function h(y; l; e) in equation ( 14) with respect to both functional form and parameter values; the argument for this speciÖcation, which was outlined in section 2, still applies.The investment function is also kept linear, as in section 2, but variables have been added and the speciÖcation now applies to target investment:

Simulations with given policy rules
The simulations in Ögure 3 use the following parameter values  ( x ).Instead, it is assumed that expected ináation is equal to an unweighted moving average over the previous 12 quarters (Flaschel and Krolzig) or a weighted 12 quarter moving average with linearly declining weights (Diallo et al.).These assumptions provide an average lag that is comparable to that obtained by the autoregressive formulation with  x = 0:3: This speed of adjustment of ináation expectations is also consistent with Blanchardís (2016) Öndings.
The government spending and tax parameters imply that the government budget is balanced along a steady growth path if there is no initial debt.The share of government consumption in GDP is 0.2, below the share in the US from 1951-1992 and above the average for 1992-2021.The parameter describing employment dependent transfers is set to 0.04.The average unemployment replacement ratio in the US is about 0.4, but only a fraction of the unemployed, ranging between 9 percent in Mississippi and 55 percent in Massachusetts, get unemployment beneÖts.Relatively low paid workers experience greater employment volatility, and a one percent increase in unemployment may only reduce aggregate wage income by about one third of a percent of GDP.Assuming that about 30 percent of the unemployed receive beneÖts at an average replacement ratio of 0.4, the direct Öscal e §ect of increased unemployment makes  2 equal to 0.04.This calculation leaves out other sources of increased transfers, including food stamps and medicaid; thus, the parameter value may be on the low side.The parameter  3 describing discretionary policy is calibrated based on the US stimulus package in 2009, when a fall in the employment rate of about 5 percentage points was met with a discretionary Öscal stimulus of about 5 percent of GDP spread over about two years. 22The value of  4 implies that a 10 percentage point increase in the debt to GDP ratio reduces the share of discretionary in GDP spending by half a percentage point.
The Taylor rule and its parameter values follow the Öndings in Clarida et al.  (2000) and Diallo et al. (2011).The natural growth rate (n); the consumption parameters (c; ) and the depreciation rate () match standard assumptions.
There is greater uncertainty with respect to the parameters of the investment function.My reading of the available evidence suggests large positive long-run e §ects of utilization, some positive e §ect of the proÖt share and some negative e §ects of the employment rate and real interest rates. 23Including the interest rate is standard, but the cost of Önance may have limited e §ects on investment in recessions.The relevant risk adjusted rate on corporate loans and bonds, moreover, does not always move with the policy rate; the spread between the policy rate and the risk adjusted interest rate on business Önance tends to increase in recessions.As an extreme example, the policy rate ñ the federal funds rate ñ was reduced by 5 percentage points between July 2007 and the end of 2008.Yet, during that same period the yield on corporate Baa bonds increased by more then 2 percentage points. 24The investment parameters reáect this evidence and represent, I would argue, reasonable ballpark estimates.
The functional forms and parameter values imply that the model has a stationary solution E 1 with l  = 0:5; e  = 0:9; y  = 1; g  K = 0:03; x  = 0:02;   = 0:03; r  = 0:05; ! = 0:625; b  = 0. Simulations with a range of di §erent initial values indicate that, given all other parameters, this stationary solution is locally stable if the investment parameter  1 is below 0:89 and locally unstable for values above 0:89.The convergence is oscillatory in the stable case with  1 below 0:89 and quite slow for  1 values above 0.5: In the unstable case, the trajectories remain bounded, converging to a limit cycle even for values of  1 that are far above any plausible range.The amplitude of the asymptotic cycle depends on  1 : it is zero for  1 = 0:89; increases gradually with  1 , and for  1 = 10 the convergence is to a limit cycle with employment áuctuating between 0.79 and 0.98. 25gure 3 about here Figure 3 shows bivariate patterns for (e; ); (e; ); (e; p); (e; r) and (e,!) from a simulation with the  1 parameter at the borderline value ( 1 = 0:89): The initial values of the employment rate, the employment capital ratio and the debt ratio are e 0 = 0:92 and l 0 = 0:51, with all other state variables equal to their values at the stationary solution E 1 ; the Ögure depicts patterns for the Örst 100 periods.The orientations of the cycles match the regular patterns of all bivariate áuctuations identiÖed by Zipperer and Skott (2011) as well as the ëlabor-market ledí movements in real wages emphasized by Diallo et al. (2011).26 The relative amplitudes of the key variables are roughly correct, but with the implied amplitude of the capital utilization rate on the low side, relative to the amplitudes of the proÖt share and the employment rate.With  1 at the threshold of instability, the stationary solution is stable or, equivalently, there is convergence to a limit cycle with zero amplitude.But the convergence is slow and oscillatory, with a cycle length of about 10 years.Figure 4 about here A variation of the simulation removes the feedback from debt to government spending (setting  4 = 0).If the initial value of the debt ratio is positive, the system now produces divergence, even if positive debt is the only deviation of the initial position from the stationary solution: the debt ratio rises without limit; 25 Using the stationary values associated with E 1 in the equations for government consumption, taxes, real wages and ináation, the system has a second non-trivial stationary solution (E 2 ).The interest and ináation rates are at their lower bounds at this solution, while the values of the other state variables depend on the parameters.When  1 = 0:89; this solution has e  = 0:865; l  = 0:494;   = 0:489; y  = 0:990;   = 0:404;   = 0:03; x  = 0:1; g  K = 0:03; r d = 0; ! = 0:590; b  = 0:387: The stationary solution at E 2 is unstable.In the case with  4 = , however, the value of the debt ratio b has no ináuence on the other state variables, and the trajectories of l; g K ; ; e; x; !; r d may converge to the stationary solution associated with E 2 : The basin of attraction for this outcome is small: it appears, for instance, that if  1 = 0:89, all trajectories converge to the ëgood solutioní E 1 as long as the initial value of expected ináation exceeds 0:03.E 2 , moreover, is a saddlepoint and only by a áuke will trajectories that start within this basin of attraction that also generate convergence of b to b  ; the debt ratio will almost certainly exhibit cumulative divergence.This divergence would generate unbounded ñ and thereby unsustainable ñ movements in C and G, with the C + G staying constant.see Ögure 4, which uses b 0 = 0:03. 27Intuitively, debt has an expansionary e §ect on consumption (which is not being o §set by lower government spending when  4 = 0), and monetary policy makers react to the expansion by increasing the rate of interest, which raises the growth rate of government debt.As the debt ratio rises, the expansionary e §ect on rising debt on private consumption comes to dominate the contractionary e §ects of rising interest rates on investment, generating an explosive path of rising e; :; y; !; r; x and b: 28 A negative initial debt ratio, conversely, can generate explosive growth in private sector debt and downward divergence with private consumption going to zero.This outcome requires that the nominal interest rate hits the ZLB, leading to progressively falling ináation and increasing real rates of interest. 29t may be noted that an increase in the minimum ináation rate tends to dampen the explosiveness of the movements in the debt ratio.This dampening e §ect is intuitively obvious in the case with negative public debt, where the destabilizing rise in real interest rates happens because interest rates are non-negative and ináation rates fall below zero.It also applies when debt is positive, however: the áoor under ináation tends to reduce the real interest rate in downturns, thereby dampening or eliminating the trend rate of growth in the debt ratio.In other words, nominal wage stickiness can be stabilizing if monetary policy is used for stabilization and nominal interest rates are subject to a zero lower bound, a position long advocated by Keynesians.

Simulations with induced changes in policy
The simulation in Ögure 5 raises the investment parameter  1 to 1.2, keeping all other parameters as in table 1.The initial values are e 0 = 0:92; l 0 = 0:51; with all other state variables at the stationary values associated with E 1 .The increase in  1 strengthens the destabilizing Harrodian forces, and the trajectories converge to a limit cycle.Asymptotically the interest rate becomes stuck at the ZLB, but the other other variables áuctuate; the employment rate, for instance moves between 0.83 and 0.95. Figure 5 shows bivariate patterns for (e; ); (e; p); (e; r) and (e; !): Figure 5 about here This outcome will almost certainly lead to changes in the policy rules.Central banks will react to a state in which their policy instrument ceases to be 27 Similar qualitative results hold as long as an increase in debt has expansionary e §ects.This happens if  4 is below the propensity to consume out of wealth.The two parameters are equal in the simulation depicted in Ögure 3a, which removes the feedback from the debt level to the other variables in the system. 28Exogenous limits on labor productivity productivity and the utilization rate of capital must curtail this process at some point.When that happens, the goods market can no longer clear through movements in output and productivity.Before it happens, however, there will almost certainly be changes in economic policy and private-sector behavior.The simulations have not imposed these upper limits. 29Interactions between Öscal and monetary policy are discussed by Bell-Kelton and Ballinger (2008), Ryoo and Skott (2017), Franke (2019), Mason and Jayadev (2018) and Franke (2019).e §ective, and recurrent states of deep recession and signiÖcant overheating put pressure on Öscal policy makers to respond more aggressively.A likely result is increases in both the transfer parameter t 2 and the discretionary policy parameter  3 : These changes are stabilizing.If t 2 is raised to 0.1 and  3 to 200, for instance, e will áuctuate between 0.86 and 0.93, rather than between 0.83 and 0.95. 30An alternative (or supplementary) change could be to let Öscal policy respond directly to ináation, especially when interest rates are at the zero lower bound and monetary policy has become ine §ective; this innovation ñ making government spending depend inversely on the ináation rate ñ reinforces the stabilizing e §ect.
The reduction in the amplitude of the áuctuations has an important implication: the nominal interest rate moves o § the zero lower bound for most of the cycle, and monetary policy regains traction.Thus, stability can be obtained by combining the Öscal changes with increases in the adjustment speed to  r = 3 and the sensitivity of the target interest rate to ináation to  1 = 4.These changes in the monetary policy parameters would have been ine §ective, by contrast, if the Öscal rules had not changed; the amplitude of the limit cycle would still be as in Ögure 5. Figure 6 illustrates the case with t 2 = 0:1;  3 = 200 and an unchanged monetary policy Figure 7 adds a more aggressive monetary policy ( r = 3;  1 = 4); the convergence to E 1 is slow, and only the Örst 100 periods are shown: The initial values in Ögures 6 and 7 are the same as in Ögure 5.
Figures 6 and 7 about here

Conclusion
The analysis in this paper has pursued themes that go back to Karl Marx and Roy Harrod: locally unstable steady growth paths ñ a likely outcome in capitalist economies ñ form the basis for a theory of endogenous growth cycles; that is, for the integration of business cycles and economic growth.Unlike in DSGE models and other theories that take the steady growth path as intrinsically stable, there is no need for exogenous shocks to generate deviations from steady growth and for imposing particular structures of autocorrelation and time-varying volatility of these shocks to match the cyclical patterns in the data. 31he irregularity of observed business cycles is sometimes cited as evidence against endogenous cycles. 32This argument is weak: if shocks are added to models of endogenous cycles, the cycles lose their regularity.The issue that separates the two approaches concerns the local stability of the steady growth path in the absence of shocks, a question that is orthogonal to the source and magnitude of exogenous shocks. 33he models in this paper assume that employment and the capital stock respond gradually to signals from the goods and labor markets.Labor productivity (the utilization rate of labor), however, is taken to be perfectly áexible, allowing output adjustments to clear the goods market in the short run. 34Using empirically based behavioral functions and policy rules, the simulations of the extended model generated cyclical patterns that are consistent with US evidence. 35 stripped-down version of the model without a public sector also matched many of the features of US data.Like many nonlinear dynamic models, however, the properties of this version are very sensitive to changes in parameter values: plausible parameter values can reproduce many empirical patterns, but other equally plausible parameter generate very di §erent outcomes, including cumulative divergence.This fragility of the results represents a serious weakness if the model is to be applied directly to real-world economies: there are no good reasons why private decision makers, acting to promote their own individual interests, would choose behaviors that produce the observed macroeconomic patterns.
The fragility of the qualitative results of the stripped-down model suggests that, except by a áuke, unregulated capitalism will be likely to descend into chaos and collapse.This implication may appear to be at odds with the evidence.In fact, however, there is no evidence to support or reject this implication: unregulated capitalism has never existed as a dominant mode of production.It has coexisted with non-capitalist sectors, mainly agricultural sectors in its early stages but with the public sector becoming increasingly important in later stages. 36y is perturbed by disturbances of various types and sizes at more or less random intervals, and that those disturbances then propagate through the economy.
Where the major macroeconomic schools of thought di §er is in their hypotheses concerning these shocks and propagation mechanisms. 33Econometric work based on linear speciÖcations may also seem to favor local stability.But if, in fact, the data have been generated by a nonlinear system with local instability and bounded áuctuations, the conclusions from linear regressions will be biased, tending to support parameter values that imply local stability.The stability conclusions are easily reversed when the estimation allows for nonlinearities (Beaudry et al. (2017, 2020)). 34The economy may contain both áex-price and áex-output sectors.Skott (2023, chapter 10) analyzes a áex-price model with a public sector and, perhaps surprisingly, the properties and patterns predicted by the two models are quite similar.Thus, the models may also provide a good starting point for analyzing an aggregate economy that contains both types of sectors. 35The cyclical patterns in smaller and more open economies tend to be less regular, a Önding that is not surprising: the movements of macroeconomic aggregates will not be governed by purely domestic interactions in economies that are strongly ináuenced by international trade and capital mobility.
The extended model includes empirically motivated modiÖcations of investment behavior and, following Flaschel and his coauthors, of wage and price setting.More importantly, it adds a public sector with stylized Öscal and monetary policy rules.This addition greatly enhances the robustness of the qualitative outcomes: minor changes in parameter values no longer lead to global divergence, even if the policy rules are kept constant.
Policy rules are not constant, however.Unlike the production, investment and consumption decisions by individual Örms and households, furthermore, they are chosen and implemented centrally.The political process behind economic policy is complex, and there can be no presumption of optimal rules, even if ëoptimalityí could be deÖned meaningfully in a world of uncertainty and conáicting interests.But it should be uncontroversial to suggest that deep depressions or explosive ináation lead to changes in economic policy.New policy tools may be devised ñ central banks reacted to the Önancial crisis and a binding zero lower bound by adding quantitative and qualitative easing to their toolbox ñ and traditional tools may be used more forcefully, as illustrated by the Öscal stimulus after the Önancial crisis and during the COVID pandemic as well as by the willingness of Japanese policy makers to allow large increases in public debt when a stagnating economy needed persistent stimulus. 37hese observations weaken critiques of nonlinear dynamic models that point to potentially large di §erences in outcomes following a change in private behavior, keeping constant all policy rules.The prevailing policy rules are not independent of private sector behavior.Had private sector behavior been di §erent in a way that signiÖcantly a §ected outcomes, policy rules would probably also have been di §erent.
The increase in the labor capital ratio also a §ects both investment and the short-run equilibrium level of employment (the latter associated with y = y  and Y = y  L): dI = f 0 Kdl dY eq = y  dL eq = mdI = mf 0 Kdl where m; Y eq and L eq are the investment multiplier and the short-run equilibrium levels of output and employment.If it takes T periods for the multiplier to work itself out, with T so short that the capital stock can be taken as constant, we have Now return to the model in equations ( 10)- (11) and observe that the trace of the Jacobian can be written38 Hence, with a multiplier well above one and a strong sensitivity of accumulation rates to changes in employment capital ratio in the neighborhood of the stationary point, rapid convergence to short-run equilibrium (a small value of T ) is destabilizing: the Örst term on the right hand side of equation (37) will be positive, and local stability requires a strong negative e §ect from employment to the growth rate of output. 39