Consumer and Corporate Debt in a 3D Macroeconomic Model

Abstract

We build on the literature of consumer debt–income inequality nexus, by developing a post-Keynesian model of growth and income distribution that incorporates both consumer and corporate borrowing. Specifically, we examine a non-linear dynamic system of three differential equations of workers’ debt-to-capital ratio, corporate debt-to-capital ratio, and the accumulation rate. We conduct simulations to solve the system for the long-run equilibrium points and examine local stability using the Routh–Hurwitz conditions. Additionally, we conduct a comparative statics analysis and investigate the stability of the model using a measure of the maximum distance among the three equilibrium points. Our key findings suggest that although our model shares several quantitative and qualitative aspects with the 2 × 2 models, the inclusion of corporate debt alters the impact of parameter changes on macroeconomic stability. This incorporation increases the number of parameters that have differing effects on stability across various scenarios. The ratio of external-to-internal borrowing, along with the interest rate, is among the few parameters that consistently undermines macroeconomic stability across all scenarios.

1. Introduction

From the early 1980s until the outbreak of the global financial crisis in 2007/08, three key trends emerged across almost all developed economies: an increase in the share of consumption in gross domestic product (henceforth, GDP), a decrease in the personal savings rate, and a rise in personal income inequality (see [1] for Europe and [2] for the United States). Initially, the rise in the consumption share of GDP was attributed to the spending patterns of households at the top of the income distribution, primarily driven by higher returns from financial assets, while households at the lower end of the income spectrum experienced on average little or no consumption growth [3,4]. However, both theoretical and empirical evidence suggest that profit earners have a lower propensity to consume compared to wage earners. Consequently, an increase in personal income inequality would be expected to lead to a decline in the consumption share of GDP, rather than an increase. The missing link in this apparent “paradox” was largely uncovered after the global financial crisis of 2007/08, when economists recognized the relationship between income inequality and household debt.

Although empirical research supports the relationship between rising household debt and income inequality—particularly among middle- and lower-income classes [4,5,6]—the reasons why these households choose to borrow remain unclear. The current extensive literature of formal models attempting to interpret this relationship can be categorized into three main approaches. The first perspective draws on the work of Minsky, arguing that working-class households increase borrowing due to optimistic expectations about their future financial situation. The second approach finds its basis in the work of [7], which offers an interpretation more consistent with recent stylized facts. According to this view, working-class households emulate the consumption patterns of upper-class households or engage in conspicuous consumption competition with their peers—a phenomenon commonly referred to as “keeping up with the Joneses” [8,9,10,11,12,13,14,15,16]. The third category builds on the work of [17], who argue that historical consumption patterns are not easily reversed. From this perspective, consumer borrowing—and the resulting debt—during periods of pronounced income inequality and wage stagnation is seen as a consequence of working households’ efforts to maintain their established consumption trajectories. For further discussion, see [18], and for formal models, refer to [19,20,21].

Some recent contributions in the literature of the Sraffian Supermultiplier (SSM) model treat household borrowing as a factor influencing autonomous consumption. To date, this approach has not been linked to income inequality, while the borrowing function of firms (when incorporated) shares some similarities with ours. Ref. [22] integrates household debt into a supermultiplier growth model and finds that when consumption is the sole autonomous variable, higher consumption financed by credit does not lead to higher long-term indebtedness. Similarly, ref. [23], which develop a supermultiplier stock–flow consistent model of economic growth, reach the same conclusion. In their effort to categorize the existing literature, ref. [24] compare stock–flow consistent versions of extended canonical neo-Kaleckian and supermultiplier models that address either household or firm debt accumulation. Regarding the neo-Kaleckian models, ref. [24] find that firms’ leverage ratios can be either pro- or anti-cyclical, while in the household sector, the absence of the paradox of debt appears to be the most likely scenario.

This paper unites the theoretical literature that examines the interaction between consumer debt and capital accumulation with the one that examines the interaction between corporate debt and capital accumulation. It builds on the models of [19,20,21,25] for the consumer debt–rate of accumulation nexus and introduces corporate debt in a neo-Kaleckian framework. Our motivation is to explore the interaction between corporate debt, consumer debt, and the rate of accumulation both on the short and the long run, the equilibrium points, the growth regime, and the macroeconomic performance and stability. Corporate debt has been a central focus in research exploring the impact of the financial system on the real economy—initially through the effects of interest rates and more recently, through the influence of debt and debt servicing (e.g., [26]; Hein [27,28]; Charles [29]). However, its interaction with consumer debt has rarely been studied. To our knowledge, the only notable exception is the work of [30].

Our work diverges from [30] in several key respects. From a theoretical standpoint, while both are neo-Kaleckian models of growth and income distribution, we draw on the Steindlian approach outlined by [25], whereas the model of [30] is situated within the stock–flow consistent (SFC) framework. Another difference between the two models lies in the treatment of household borrowing behaviour. In our model, following [21], we link personal income inequality to household debt, along with other possible behaviours. In contrast, ref. [30] assume that working households set a credit target and borrow whenever their debt falls below this target. Another key difference lies in our treatment of investment. Following [25], we assume that investment is predetermined in the short run, with firms adjusting their actual investment rate to their desired rate in the long run. This approach leads to a three-dimensional non-linear dynamic system (comprising accumulation rate, consumer debt, and corporate debt dynamics), whereas [30] allows the investment rate to change in the short run and hence their dynamic model takes the simpler two-dimensional structure. Furthermore, in our model, corporate borrowing is assumed to increase with the profits of enterprise, that is, it depends on the gross profit share, the rate of interest, and the stock of corporate debt. This relationship reflects [31] the “principle of increasing risk”, which posits that a reduction in internal means of finance for real investment projects diminishes access to external financing in imperfectly competitive capital markets. In contrast, ref. [30] determine corporate borrowing based on the difference between investment and internal financing.

The remainder of the paper proceeds as follows: Section 2 examines the structure of the model. Section 3 examines the behaviour of the model in the short run and performs a comparative statics analysis. Section 4 conducts simulations to solve the dynamic model, examines the stability properties, performs comparative statics, and discusses issues of growth regimes and macroeconomic stability. Section 5 concludes.

2. Model Structure

2.1. Capitalists’ Consumption, Investment, and Borrowing

We assume a private closed economy. National income Y (Y) equals consumption (C), which consists of workers’ consumption $\left(C_w\right)$ and industrial capitalists’ consumption $\left(C_c\right)$, and investment (I).

$$Y=C+I=C_w+C_c+I$$

(1)

National income is allocated between wages, $wL$ ($wL$ stands for wages defined as the average wage, w, times the dependent employment, L; the definitions of all variables and parameters of the model are provided in

Table A1. Notation of variables and parameters.

Notation	Variable/Parameter
Y	National income
C	Aggregate consumption
$C_w$	Working households’ consumption
$C_I$	Industrial capitalists’ consumption
I	Investment
L	Homogeneous labour
K	Homogeneous capital stock
$wL$	Wages: average wage, w, times labour, L.
$w^t$	Wage target
$\Pi$	Gross profits
$\pi$	Gross profits share, $\pi =\Pi /Y$
i	Interest rate
$D_w$	Workers’ debt
${\delta }_w$	Working households’ debt-to-capital stock ${\delta }_w=D_w/K$
$D_I$	Industrial capitalists’ debt
$d_I$	Industrial capitalists’ debt-to-national income $d_I=D_I/Y$
${\delta }_I$	Industrial capitalists’ debt-to-capital stock ${\delta }_I=D_I/K$
$IndPr$	Industrial capitalists’ profit of enterprise
r	Gross profit rate, $r=\Pi /K$
$B_w$	Workers’ new borrowing
${\beta }_1$	Borrowing coefficient (wage gap)
${\beta }_2$	Borrowing coefficient (interest payments gap)
$\lambda$	Workers’ maximum debt service-to-income ratio
${\alpha }_0$	Conventional wage equation, constant term
${\alpha }_1$	Conventional wage equation, sensitivity of conventional wage to capacity utilization
s	Retention rate
g	Rate of accumulation ($g=I/K$)
$g^t$	Desired rate of accumulation
$B_I$	Industrial capitalists’ new borrowing
b	Proportion of external to internal finance
${\gamma }_0$	Investment coefficient, animal spirits
${\gamma }_r$	Investment coefficient, sensitivity of investment to the profits of enterprise

in the Appendix A), and gross profits, $\Pi$. Wages are split between interest payments on debt ($i{D}_w$) and net wages (${\left(wL\right)}_n$). While the functional distribution of income is treated as exogenous, it is, in a broader sense, endogenized through the endogenous variable of interest payments on debt.

$$Y=wL+\Pi ={\left(wL\right)}_n+i{D}_w+\Pi$$

(2)

The economy produces a single good utilizing two factors of production—homogeneous labour (L) and capital (K)—and employs a fixed-coefficient production function. There is no overhead labour in the production process. Labour supply is endogenous, with its growth rate equated to the rate of capital accumulation. Following the framework outlined by [32], we assume that the gross profit share ($\pi =\Pi /Y$) is determined by the mark-up that firms apply to unit labour costs in setting prices within oligopolistic markets. Furthermore, firms operate with excess capacity. All variables are expressed in real terms.

Industrial capitalists earn gross profits that are allocated between profits of enterprise ($IndPr$) and interest payments ($i{D}_I$). A portion of the profits of enterprise are consumed by industrial capitalists ($C_c$), while the remainder is saved as retained profits (R). The saving rate from profits net of interest payments, denoted as s, is assumed to be constant. These retained profits are subsequently saved and used to finance future investment endeavours.

$$\Pi =IndPr+i{D}_I=\underset{R}{\underbrace{sIndPr}}+\underset{C_c}{\underbrace{(1-s)IndPr}}+i{D}_I$$

(3)

From (3), it easily follows that industrial capitalists’ consumption function $C_c$ is given by:

$$C_c=(1-s)(\pi -i{d}_I)Y$$

(4)

where $i{d}_I=i{D}_I/Y$ are the industrial capitalists’ interest payments as a share of national income.

The gross profit rate is denoted by $r=\Pi /K$. Dividing both terms of the ratio by Y, we obtain the gross profit rate as the product of the profit share and the capacity utilization rate. In this formulation, we follow Taylor [33,34], Lima and Meirelles [35], Dutt [25], and Charles [36] and assume a constant technical capital–potential output ratio, and hence the capacity utilization rate is defined by the output–capital ratio.

$$r=\Pi /K=\pi u$$

(5)

Following [25,37,38], it is assumed that investment demand is predetermined in the short run. We denote the investment rate, that is the growth rate of K, as:

$$g=I/K$$

(6)

where I is gross investment, and for simplicity, there is no depreciation. In the long run, we assume that industrial capitalists adjust their actual investment rate to their desired rate of investment:

$$dg/dt=\Lambda (g^t-g),0<\Lambda \le 1$$

(7)

where $g^t$ denotes the desired rate of capital accumulation, and $\Lambda$ represents the speed of adjustment. This approach encapsulates the assumption put forth by Kalecki and Steindl that current variables influence investment with a lag of several periods. It yields a pure system of differential equations, enabling the dynamics of the economy to be portrayed in phase diagrams, rather than through a complex mixed system of difference–differential equations [25,37]. (This methodology, as introduced by Gandolfo [39], is widely employed within the post-Keynesian literature, see for instance [38,40,41,42,43,44,45,46,47]).

Desired investment depends on capitalists’ internal and external funds, namely the savings out of profits of enterprise, $sIndPr$, and corporate borrowing, $B_I$.

$$I^t={\gamma }_0+{\gamma }_{r}s(\pi -i{d}_I)Y+B_I,0<{\gamma }_0,{\gamma }_r<1$$

(8)

where ${\gamma }_0$ stands for the animal spirits of the private sector or the state of business confidence [27], ${\gamma }_r$ captures the sensitivity of the desired investment to the profits of enterprise, and $d_I=D_I/Y$ is the industrial capitalists’ debt as a share of national income.

We assume that industrial capitalists’ debt, $D_I$, adjusts to new borrowing, $B_I$, according to:

$$B_I={\displaystyle \frac{d{D}_I}{dt}}$$

(9)

The level of industrial capitalists’ borrowing is given by the following:

$$B_I=b(\pi -i{d}_I)Y,b>0$$

(10)

Industrial capitalists’ borrowing is a fraction of the profits of enterprise. For simplicity, we omit any financing costs other than interest. b is the proportion of external-to-internal finance and can be taken as a proxy of the leverage ratio. This ratio is determined either by bank lending practices or by industrial capitalists’ decisions. Equation (10) states that firms’ borrowing increases with gross profits and falls with interest payments. This relation captures [31] the “principle of increasing risk”, which posits that reduced internal financing for investments limits access to external financing in imperfectly competitive capital markets.

Inserting (10) into (8) and then normalizing by K, we obtain the desired rate of accumulation ($g^t$) with corporate borrowing:

$$g^t={\gamma }_0+({\gamma }_{r}s+b)(\pi u-i{\delta }_I),0<{\gamma }_0,{\gamma }_r<1$$

(11)

where $i{\delta }_I=i{D}_I/K$ is the corporate interest payments. This type of investment function is extensively used in the neo-Kaleckian literature (see Jarsulic [42], Charles [29,36], Isaac and Kim [30], among others) and differs from the canonical Kaleckian investment function in that capacity utilization is not considered in isolation but rather in conjunction with the profit share. This formulation has the advantage of facilitating the emergence of both profit-led and wage-led demand and growth regimes.

The economy is characterized as a mature monetary economy, where credit is generated endogenously. The interest rate is determined by the Central Bank, an assumption commonly adopted by post-Keynesian scholars [26].

2.2. Working Households Consumption and Borrowing

In this section, we present the consumption and borrowing behaviour of working households. The consumption of workers is financed by the sum of net wages (where the workers’ wage is derived by $Y=wL+\Pi =>wL=Y-\Pi =Y(1-\Pi /Y)=Y(1-\pi$), and the workers’ net wage is $wL-i{D}_w=Y(1-\pi )-i{D}_w$) and new borrowing ($B_w$):

$$C_w=Y(1-\pi )-i{D}_w+B_w$$

(12)

The workers’ debt, $D_w$, is assumed to adjust to new borrowing as follows:

$$B_w=d{D}_w/dt$$

(13)

We follow [21] and assume that the working households’ borrowing behaviour is determined by the subsequent double-adjustment mechanism equation:

$$B_w={\beta }_1[w^{t}L-Y(1-\pi )]+{\beta }_2[\lambda Y(1-\pi )-i{D}_w]$$

(14)

where $w^t$ denotes the wage target, or “conventional wage”, and serves as a benchmark for the standard of living that working households have historically established, $Y(1-\pi )$ represents the actual wage bill, while $\lambda (1-\pi )Y$ signifies the maximum affordable interest payment. (Equation (14) reflects empirical evidence that credit demand responds differentially to variations in income, interest rates, and interest payments. Specifically, credit demand responds more strongly to interest payments than borrowing costs, particularly for low- and middle-income borrowers [48,49]). $\lambda$ is the workers’ maximum debt service-to-income ratio, which is assumed constant, and $0<\lambda <1$ and reflects lending practices. (A $\lambda >30\%$ may indicate solvency risks for households with inadequate safety margins. Financialization, relaxed credit rationing, and predatory lending have fuelled unsustainable borrowing [50]. Gross and Souleles [51] found that consumers increased credit card debt following credit limit expansions.) The adjustment coefficients $0<{\beta }_1,{\beta }_2<1$ encapsulate the borrowing norms and motivations of households. The relative magnitudes of these coefficients assign proportional weights to the respective components influencing household decision-making.

The conventional wage ($w^{t}L$) is assumed to be a function of output. Therefore, by normalizing all variables by K and assuming that $L/K$ remains constant, changes in the conventional wage are contingent upon variations in the output–capital ratio u:

$$w^t={\alpha }_0+{\alpha }_{1}u$$

(15)

where ${\alpha }_i>0$ for $i=0,1$ are fixed parameters, and u is the output–capital ratio, $Y/K$, which is a measure of the rate of capacity utilization.

As previously stated, the conventional wage reflects the cost of the socially and historically determined standard of living achieved by working households, encompassing both privately purchased goods and services and those provided by the public sector. In this model, the conventional wage is influenced not only by capacity utilization but also by income inequality, particularly excessive inequality. Since the early 1980s, the rise in income inequality—driven by welfare state retrenchment, reductions in social spending, neoliberal privatization, and wage moderation—has compressed the standard of living for working households. This compression manifests in stagnant wages that fail to keep pace with economic growth, and an increasing reliance on the private sector for goods and services previously provided by the public sector. Consequently, the cost of maintaining the standard of living, $w^t$, has risen, rendering the actual wage insufficient to cover these costs. In Equation (15), this impact is captured by the parameter ${\alpha }_1$; higher economic activity combined with intense income inequality (high ${\alpha }_1$) leads to a greater increase in the conventional wage compared to periods of lower inequality. (Note that ${\alpha }_1$ is exogenous, with its value (low or high) assumed to define each scenario.) This mechanism links income inequality and household’s borrowing. In contrast, during periods of low income inequality, such as the Golden Age of Capitalism (1950–1969), wage growth tends to align with economic growth, with the cost of maintaining the standard of living closely matching actual wages, implying a lower value for ${\alpha }_1$.

Substituting Equation (15) into Equation (14) and normalizing by K, we derive the borrowing function for workers:

$$B_w/K={\beta }_1{\alpha }_0\ell +[{\beta }_1{\alpha }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )]u-{\beta }_{2}i{\delta }_w$$

(16)

where $\ell =L/K$ denotes the fixed labour intensity, ${\delta }_w=D_w/K$ represents the worker’s debt-to-capital ratio, and ${\beta }_1{\alpha }_0\ell$ indicates workers’ autonomous borrowing. The term $({\beta }_1-\lambda {\beta }_2)$ captures the sensitivity of workers’ borrowing to changes in the actual wage bill, with the condition $-1<{\beta }_1-\lambda {\beta }_2<1$, given that $0<{\beta }_1$, ${\beta }_2$, $\lambda <1$. Additionally, ${\beta }_1{\alpha }_1\ell$ reflects the responsiveness of workers’ borrowing to changes in the conventional wage, while $i{\beta }_2$ reflects the sensitivity of workers’ borrowing to changes in ${\delta }_w$.

The term ${\beta }_1{\alpha }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )$ in Equation (16) represents the responsiveness of workers’ borrowing to fluctuations in capacity utilization. It encapsulates the aggregate effect of variations in the conventional wage, the actual wage, and the maximum affordable interest payment. This expression serves as the foundational component for constructing various scenarios under different parameter configurations, as detailed in

Table 1. Scenario set-up.

Scenario	Inequality	Weight	Overall Impact	$\frac{\mathbf{\partial }\frac{{\mathbf{B}}_{\mathbf{w}}}{\mathbf{K}}}{\mathbf{\partial }\mathbf{u}}$
Scen. 1	${\beta }_1{\alpha }_1\ell$ high	${\beta }_1-\lambda {\beta }_2>0$	${\beta }_1{\alpha }_1\ell >({\beta }_1-\lambda {\beta }_2)(1-\pi )$	$>0$
Scen. 2	${\beta }_1{\alpha }_1\ell$ low	${\beta }_1-\lambda {\beta }_2>0$	${\beta }_1{\alpha }_1\ell <({\beta }_1-\lambda {\beta }_2)(1-\pi )$	$<0$
Scen. 3	${\beta }_1{\alpha }_1\ell$ low	${\beta }_1-\lambda {\beta }_2<0$	${\beta }_1{\alpha }_1\ell >({\beta }_1-\lambda {\beta }_2)(1-\pi )$	$>0$

.

In the first scenario, income inequality is high (${\beta }_1{\alpha }_1\ell$ is high) and workers’ borrowing decisions are more strongly influenced by changes in the wage gap than by fluctuations in the interest payment gap (i.e., ${\beta }_1>{\beta }_2\Rightarrow {\beta }_1-\lambda {\beta }_2>0$). An increase in economic activity will positively affect workers’ borrowing, as ${\beta }_1{\alpha }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )>0$. This outcome results from the substantial effect of capacity utilization on conventional wage and is driven by workers’ inclination to sustain their standard of living.

In the second scenario, income inequality is low (${\beta }_1{\alpha }_1\ell$ is low) and workers’ borrowing decisions are more influenced by changes in the wage gap than the interest payment gap (${\beta }_1>{\beta }_2\Rightarrow {\beta }_1-\lambda {\beta }_2>0$). As a result, an increase in economic activity (u) negatively impacts workers’ borrowing (${\beta }_1{\alpha }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )<0$), with working households borrowing primarily to smooth consumption.

In the third scenario, income inequality is low (${\beta }_1{\alpha }_1\ell$ is low) and workers’ borrowing decisions are more influenced by changes in the interest payment gap than the wage gap (${\beta }_1<{\beta }_2\Rightarrow {\beta }_1-\lambda {\beta }_2<0$). As a result, an increase in economic activity positively affects workers’ borrowing (${\beta }_1{\alpha }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )>0$), resembling a modest Minskyan process. (A fourth scenario arises with high income inequality, where working households prioritize the interest payment gap over the wage gap. However, this scenario offers limited theoretical insight, as it essentially represents an intensified Minskyan-type outcome (${\beta }_1{\alpha }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )>0$) driven by high income inequality (a higher ${\alpha }_1$). In this case, the wage gap’s influence on borrowing decisions is minimal (${\beta }_1<{\beta }_2$), reinforcing the core dynamics without adding new theoretical contributions.)

We obtain aggregate consumption by adding capitalists’ and working households’ consumption (Equation (4) and Equation (12), respectively), then insert Equations (14) and (15) into Equation (12) and normalize by K:

$$C/K={\beta }_1{\alpha }_0\ell +[{\beta }_1{\alpha }_1\ell +(1-{\beta }_1+\lambda {\beta }_2)(1-\pi )+(1-s)(\pi -i{d}_I)]u-(1+{\beta }_2)i{\delta }_w$$

(17)

3. Short-Run Equilibrium

In short-run equilibrium, the goods market clears. As firms function with excess capacity the level of output adjusts in response to aggregate demand. Investment is fixed at a point in time following [25,37,45], and it is assumed that the level of consumer and corporate debt, and the physical stock of capital are given. We insert Equations (17) and (6) into Equation (1) normalized by K and then solve for the short-run equilibrium value of capacity utilization, $u^{*}$, and the profit rate, $r^{*}$, according to Equation (5):

$$u^{*}={\displaystyle \frac{g+{\alpha }_0{\beta }_1\ell -(1+{\beta }_2)i{\delta }_w-(1-s)i{\delta }_I}{\pi s-[{\alpha }_1{\beta }_1\ell -(1-\pi )({\beta }_1-\lambda {\beta }_2)]}}$$

(18)

$$r^{*}=\pi u^{*}={\displaystyle \frac{\pi (g+{\alpha }_0{\beta }_1\ell -(1+{\beta }_2)i{\delta }_w-(1-s)i{\delta }_I}{\pi s-[{\alpha }_1{\beta }_1\ell -(1-\pi )({\beta }_1-\lambda {\beta }_2)]}}$$

(19)

The stability condition requires that the denominator in Equations (18) and (19), which is the standard multiplier, be positive. The denominator shows the impact of an increase in capacity utilization on saving: the first term represents the additional saving by industrial capitalists and the other two terms represent the additional consumption due to workers’ borrowing. This stability condition implies that saving increases with total income and particularly that the increase in saving by profits is greater than the increase in consumption (i.e., fall in savings) due to workers’ borrowing. (The denominator of Equation (18) is the sensitivity of saving to changes in capacity utilization, i.e., differentiating the saving function ${\displaystyle \frac{S}{K}}=s(\pi u-i{\delta }_I)-{\alpha }_0{\beta }_1\ell -[{\alpha }_1{\beta }_1\ell -(1-{\beta }_1+\lambda {\beta }_2)(1-\pi )]u+{\beta }_{2}i{\delta }_w$ with respect to u, we obtain ${\displaystyle \frac{𝜕\frac{S}{K}}{𝜕u}}=\pi s+(1-\pi )({\beta }_1-\lambda {\beta }_2)-{\alpha }_1{\beta }_1\ell$.) To ensure that we always have a positive $u^{*}$ and $r^{*}$, we require that the numerator in (18) and (19) be also positive.

Comparative Statics Analysis

Table 2. Short-run comparative statics.

	$\mathit{\pi }$	i	${\mathit{\delta }}_{\mathit{w}}$	${\mathit{\delta }}_{\mathit{I}}$	s	${\mathit{\alpha }}_0$	${\mathit{\alpha }}_1$	$\mathit{\lambda }$
u	?	−	−	−	−	+	+	+
r	?	−	−	−	−	+	+	+

reports the comparative static results for $u^{*}$ and $r^{*}$ derived from the temporary equilibria (18) and (19) with respect to $\pi$, i, ${\delta }_w$, ${\delta }_I$, s, ${\alpha }_0$, ${\alpha }_1$, and $\lambda$.

Given that the effects of the parameters on both the equilibrium values of capacity utilization and the profit rate exhibit the same sign, the subsequent analysis primarily concentrates on capacity utilization. It is observed that the parameters associated with the borrowing of working households (Equations (14) and (15)), i.e., $\lambda$, ${\alpha }_0$, and ${\alpha }_1$ contribute to an increase in borrowing. This, in turn, stimulates consumption, aggregate demand, and consequently, capacity utilization. Specifically, ${\alpha }_1$, which denotes the responsiveness of the conventional wage to fluctuations in economic activity, implies that a higher value of ${\alpha }_1$ amplifies the wage gap, thereby increasing borrowing by working households, as described by Equation (14). Additionally, the autonomous term ${\alpha }_0$ exerts a direct effect on the expansion of borrowing. Working households augment their borrowing in response to an increase in the permissible level of interest payments, represented by a higher value of $\lambda$. This, in turn, results in a rise in consumption, aggregate demand, and consequently, capacity utilization.

On the other hand, as workers increase their borrowing, they accumulate debt that necessitates servicing in each period. This represents a strain on aggregate demand, as it results in the redistribution of income from working households to capitalist households, which typically exhibit a lower marginal propensity to consume. As anticipated, both the interest rate and the debt-to-capital ratio exert a negative influence on capacity utilization. A higher industrial capitalist’s debt-to-capital ratio, ${\delta }_I$, affects negatively $u^{*}$ (and $r^{*}$) because it reduces the profits of enterprise and hence capitalists’ consumption. Note that there is no impact on investment because it is assumed to be given in the short run. The propensity to save out of profits in the short run exerts a negative effect on capacity utilization, commonly referred to as the “paradox of thrift”. This occurs because an increase in saving by capitalists reduces their current consumption, as they accumulate internal funds intended to finance future investment projects. Since investment is fixed in the short run, the immediate consequence is a reduction in aggregate demand and consequently, capacity utilization.

Equation (20) shows that the impact of the profit share on capacity utilization is ambiguous, and the demand regime may be wage- or profit-led:

$${\displaystyle \frac{𝜕u^{*}}{𝜕\pi }}=\left\{\begin{array}{cc}-{\displaystyle \frac{(s-{\beta }_1+\lambda {\beta }_2)u}{s\pi -[{\alpha }_1{\beta }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )]}}>0,\hfill & ifs<{\beta }_1-\lambda {\beta }_2(profit-led)\hfill \\ -{\displaystyle \frac{(s-{\beta }_1+\lambda {\beta }_2)u}{s\pi -[{\alpha }_1{\beta }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )]}}<0,\hfill & ifs>{\beta }_1-\lambda {\beta }_2(wage-led)\hfill \end{array}\right.$$

(20)

The sign of ${\displaystyle \frac{𝜕u^{*}}{𝜕\pi }}$ depends on the sign of the numerator, which, by rearranging, yields $(1-s)-[1-({\beta }_1-\lambda {\beta }_2)]$, namely, capitalists’ propensity to consume out of profits ($1-s$) and workers’ propensity to consume out of wages and borrowing ($1-({\beta }_1-\lambda {\beta }_2)$). If the propensity to consume (save) out of wages and borrowing is higher (lower) than out of profits (this assumption has been shown to be valid on theoretical, see [31,52], and empirical grounds, see [53,54], among many others), then the economy is wage-led ($s>{\beta }_1-\lambda {\beta }_2$, i.e., ${\displaystyle \frac{𝜕u^{*}}{𝜕\pi }}<0$). Otherwise the economy is profit-led.

The sign of ${\displaystyle \frac{𝜕r^{*}}{𝜕\pi }}$ depends on the sign of the numerator, which is the difference of the sensitivity of borrowing with respect to changes in conventional and actual wage. A positive numerator occurs in the first and third scenarios where income inequality is high and in that case, the paradox of costs applies. On the contrary, in the second scenario, where income inequality is low, the paradox of the costs does not apply.

$${\displaystyle \frac{𝜕r^{*}}{𝜕\pi }}=\left\{\begin{array}{c}-{\displaystyle \frac{({\alpha }_1{\beta }_1\ell -{\beta }_1+\lambda {\beta }_2)u}{s\pi -[{\alpha }_1{\beta }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )]}}<0,if{\alpha }_1{\beta }_1\ell >{\beta }_1-\lambda {\beta }_2\hfill \\ -{\displaystyle \frac{({\alpha }_1{\beta }_1\ell -{\beta }_1+\lambda {\beta }_2)u}{s\pi -[{\alpha }_1{\beta }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )]}}>0,if{\alpha }_1{\beta }_1\ell <{\beta }_1-\lambda {\beta }_2\hfill \end{array}\right.$$

(21)

4. Long-Run Equilibria: A 3D Model

In this section, we examine the three-dimensional (3D) dynamic system, where the dynamics of consumer and corporate debt ratios together with the rate of accumulation interact. Because of the model’s solutions’ complexity, we conducted simulations to perform comparative statics and derive the stability properties. Simulations were performed for all three parameter scenarios.

4.1. Working Households Debt Dynamics

The differential equation describing the path of ${\delta }_w$ is the following (over-hats denote growth rates):

$${\widehat{\delta }}_w={\widehat{D}}_w-\widehat{K}$$

(22)

and by inserting Equations (6), (16) and (18) into Equation (22), we obtain the growth rate of ${\delta }_w$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d{\delta }_w}{dt}}={\alpha }_0{\beta }_1\ell +[{\alpha }_1{\beta }_1\ell -(1-\pi )({\beta }_1-\lambda {\beta }_2)]\ast \\ \hfill \ast \left[{\displaystyle \frac{g+{\alpha }_0{\beta }_1\ell -(1+{\beta }_2)i{\delta }_w-(1-s)i{\delta }_I}{\pi s+(1-\pi )({\beta }_1-\lambda {\beta }_2)-{\alpha }_1{\beta }_1\ell }}\right]-{\beta }_{2}i{\delta }_w-g{\delta }_w\end{array}$$

(23)

The term $g{\delta }_w$ in Equation (23) indicates that the relation is non-linear and was investigated using a phase portrait analysis. For simplicity, henceforth, we use the following notation for the denominator: $\pi s+(1-\pi )({\beta }_1-\lambda {\beta }_2)-{\alpha }_1{\beta }_1\ell =\Gamma$.

4.2. Corporate Debt Dynamics

The dynamics of corporate debt are given by:

$${\widehat{\delta }}_I={\widehat{D}}_I-\widehat{K}$$

(24)

Inserting Equations (6), (10) and (18) into Equation (24), we obtain:

$${\displaystyle \frac{𝜕{\delta }_I}{𝜕t}}=b\left[\pi {\displaystyle \frac{g+{\alpha }_0{\beta }_1\ell -(1+{\beta }_2)i{\delta }_w-(1-s)i{\delta }_I}{\Gamma }}-i{\delta }_I\right]-g{\delta }_I$$

(25)

The growth rate of ${\delta }_I$ also exhibits a non-linear relationship due to the term $g{\delta }_I$; hence, we perform a phase portrait analysis.

4.3. Accumulation Rate Dynamics

The dynamic path of the accumulation rate, when both working household and corporate debt are endogenous, is specified by Equation (7), with Equations (11) and (18) substituted into it.

$$\begin{array}{c}\hfill {\displaystyle \frac{dg}{dt}}=\Lambda \{{\gamma }_0+(s{\gamma }_r+b)\left[\pi {\displaystyle \frac{g+{\alpha }_0{\beta }_1\ell -(1+{\beta }_2)i{\delta }_w-(1-s)i{\delta }_I}{\Gamma }}-i{\delta }_I\right]-g\}\end{array}$$

(26)

4.4. Deriving the Isoclines of the Differential Equations

To derive the isoclines of the differential equations presented in Equations (23)–(26), we begin by setting each of the equations equal to zero and solving for the accumulation rate g.

$$g={\displaystyle \frac{{\alpha }_0{\beta }_1\ell s\pi -i{\delta }_I(1-s)(s\pi -\Gamma )-i{\delta }_w[(1+{\beta }_2)s\pi -\Gamma ]}{\Gamma -s\pi +{\delta }_w\Gamma }}$$

(27)

$$g={\displaystyle \frac{b[{\alpha }_0{\beta }_1\ell \pi -i{\delta }_w\pi (1+{\beta }_2)-i{\delta }_I[\Gamma +\pi (1-s)]]}{{\delta }_I\Gamma -b\pi }}$$

(28)

$$g={\displaystyle \frac{{\gamma }_0\Gamma +{\alpha }_0{\beta }_1\ell (s{\gamma }_r+b)\pi -i{\delta }_I[\Gamma +(1-s)\pi ](s{\gamma }_r+b)}{\Gamma -(s{\gamma }_r+b)\pi }}-{\displaystyle \frac{(s{\gamma }_r+b)\pi (1+{\beta }_2)i{\delta }_w}{\Gamma -(s{\gamma }_r+b)\pi }}$$

(29)

Isocline (27) reflects a hyperbola with vertical and horizontal asymptotes equal to ${\delta }_w^{+}>(s\pi -\Gamma )/\Gamma$ and $g^{+}>-i[(1+{\beta }_2)s\pi -\Gamma ]/\Gamma$, respectively. Our focus lies on the first quadrant (for values ${\delta }_w>{\delta }_w^{+}$ and $g>g^{+})$, where the hyperbola is a convex line, specifically, ${\delta }_w\to \infty$, $g^{\prime }\left({\delta }_w\right)\to 0$.

The aforementioned properties suggest an inverse relationship between the variables ${\delta }_w$ and g. As borrowing intensifies, debt accumulation rises, and a larger fraction of both income and the incremental borrowing is allocated to servicing the existing debt obligations. This, in turn, results in a contraction of aggregate demand and subsequently dampens economic growth, given the debt-servicing burden. Note, however, that in the short run, borrowing exhibits an expansionary effect as it increases overall demand before the debt servicing impact becomes more pronounced. Furthermore, the responsiveness of growth g to changes in ${\delta }_w$ is not linear. Initially, an increase in ${\delta }_w$ generates a more substantial negative impact on growth, reflecting the higher marginal effect of rising debt on the real economy. However, as ${\delta }_w$ reaches higher levels, the marginal effect of further increases diminishes, leading to a progressively smaller impact on growth. This happens because at elevated levels of borrowing, the consumption component of aggregate demand becomes increasingly constrained by debt service obligations, such that further increases in borrowing contribute less to changes in overall demand. As ${\delta }_w$ approaches infinity, the growth rate g stabilizes, approaching an asymptote, thereby indicating a saturation point where additional debt no longer significantly influences growth.

Isocline (28), derived from the differential equation governing corporate debt dynamics, describes a hyperbolic relationship with asymptotic behaviour. Specifically, the vertical asymptote is given by ${\delta }_I^{+}=b\pi /\Gamma$, while the horizontal asymptote is represented by $g^{+}=-bi[\Gamma +\pi (1-s)]/\Gamma$. The isocline features two distinct branches. The first branch corresponds to combinations of positive corporate debt-to-capital ratios and positive rates of capital accumulation, which are of primary economic significance. The second branch reflects combinations of positive corporate debt-to-capital ratios coupled with negative rates of capital accumulation, but this branch is less relevant for the economic analysis under typical circumstances, as it implies an always contracting economy. The hyperbolic shape of the isocline is convex since when ${\delta }_I\to \infty$, $g^{\prime }\left({\delta }_I\right)\to 0$.

Isocline (29) is a downward-sloping line with a positive intercept and denominator. The denominator in isocline (29) represents the standard Keynesian macroeconomic stability condition, which requires the savings rate to respond more to changes in capacity utilization than the investment rate does.

According to [55], once income distribution is incorporated into the investment and savings functions, the Keynesian stability condition no longer guarantees self-correcting adjustments. Therefore, the “Robinsonian” macroeconomic long-run stability condition is required; that is, investment must be less responsive to the profit share than saving, implying that ${\displaystyle \frac{𝜕g_s}{𝜕\pi }}>{\displaystyle \frac{𝜕g_I}{𝜕\pi }}$ and consequently, $s-({\beta }_1-\lambda {\beta }_2)>s{\gamma }_r+b$.

4.5. Solving the Long-Run Model and Stability Properties

By solving the dynamic system governing the economic model, we identified three equilibrium points. However, these equilibrium points were characterized by mathematical complexity that rendered analytical solutions impractical for further interpretation. As a result, we conducted simulations to explore the precise nature and stability of these equilibria.

• Equilibrium Point A

$$\begin{array}{c}\hfill {\delta }_w\to {\displaystyle \frac{{\alpha }_0{\beta }_1\ell \pi }{i[{\beta }_1({\alpha }_1\ell +\pi -1)+{\beta }_2[\lambda (1-\pi )+\lambda +\pi ]]}},\\ \hfill g\to 0,\\ \hfill {\delta }_I\to -{\displaystyle \frac{{\alpha }_0{\beta }_1\ell \pi }{i[{\beta }_1({\alpha }_1\ell +\pi -1)+{\beta }_2[\lambda (1-\pi )+\lambda +\pi ]]}}\end{array}$$

(30)

Equilibrium Point B

$$\begin{array}{c}\hfill {\delta }_w\to {\displaystyle \frac{1}{2i\pi (1+{\beta }_2)(b+s{\gamma }_r)}}(-bi\pi -i{\beta }_1-bi{\beta }_1+i\pi {\beta }_1+bi\pi {\beta }_1\\ \hfill +b\ell \pi {\alpha }_0{\beta }_1+i\ell {\alpha }_1{\beta }_1+bi\ell {\alpha }_1{\beta }_1+i\lambda {\beta }_2+bi\lambda {\beta }_2-bi\pi {\beta }_2\\ \hfill +is\pi {\beta }_2-i\lambda \pi {\beta }_2-bi\lambda \pi {\beta }_2+\ell \pi {\alpha }_0{\beta }_1{\gamma }_r-i\pi {\beta }_2{\gamma }_r+\sqrt{\Delta }),\end{array}$$

(31)

$$\begin{array}{c}\hfill g\to {\displaystyle \frac{1}{2(b\pi -s\pi -{\beta }_1+\pi {\beta }_1+\ell {\alpha }_1{\beta }_1+\lambda {\beta }_2-\lambda \pi {\beta }_2+\pi s{\gamma }_r)}}(bi\pi -i{\beta }_1+bi{\beta }_1\\ \hfill +i\pi {\beta }_1-bi\pi {\beta }_1-b\ell \pi {\alpha }_0{\beta }_1+i\ell {\alpha }_1{\beta }_1-bi\ell {\alpha }_1{\beta }_1+i\lambda {\beta }_2-bi\lambda {\beta }_2\\ \hfill -bi\pi {\beta }_2+is\pi {\beta }_2-i\lambda \pi {\beta }_2+bi\lambda \pi {\beta }_2-\ell \pi {\alpha }_0{\beta }_1{\gamma }_r-i\pi {\beta }_2{\gamma }_r+\sqrt{\Delta }),\end{array}$$

(32)

$${\delta }_I\to b/(b+s{\gamma }_r)$$

(33)

Equilibrium Point C

$$\begin{array}{c}\hfill {\delta }_w\to {\displaystyle \frac{1}{2i\pi (1+{\beta }_2)(b+s{\gamma }_r)}}(-bi\pi -i{\beta }_1-bi{\beta }_1+i\pi {\beta }_1+bi\pi {\beta }_1\\ \hfill +b\ell \pi {\alpha }_0{\beta }_1+i\ell {\alpha }_1{\beta }_1+bi\ell {\alpha }_1{\beta }_1+i\lambda {\beta }_2+bi\lambda {\beta }_2-bi\pi {\beta }_2+is\pi {\beta }_2-i\lambda \pi {\beta }_2-bi\lambda \pi {\beta }_2\\ \hfill +\ell \pi {\alpha }_0{\beta }_1{\gamma }_r-i\pi {\beta }_2{\gamma }_r-\sqrt{\Delta },\end{array}$$

(34)

$$\begin{array}{c}\hfill g\to -{\displaystyle \frac{1}{2((-1+\pi +\ell {\alpha }_1){\beta }_1+(\lambda -\lambda \pi ){\beta }_2+\pi (b-s+s{\gamma }_r)))}}(-bi\pi +i{\beta }_1-bi{\beta }_1\\ \hfill -i\pi {\beta }_1+bi\pi {\beta }_1+b\ell \pi {\alpha }_0{\beta }_1-i\ell {\alpha }_1{\beta }_1+bi\ell {\alpha }_1{\beta }_1-i\lambda {\beta }_2+bi\lambda {\beta }_2+bi\pi {\beta }_2\\ \hfill -is\pi {\beta }_2+i\lambda \pi {\beta }_2-bi\lambda \pi {\beta }_2+\ell \pi {\alpha }_0{\beta }_1{\gamma }_r+i\pi {\beta }_2{\gamma }_r+\sqrt{\Delta },\end{array}$$

(35)

$${\delta }_I\to b/(b+s{\gamma }_r)$$

(36)

In Figure 1, we present the 3D diagram of the three isoclines and the three equilibrium points: A, B, and C.

In what follows, we deduce the stability properties of the dynamic system, consisting of isoclines (27)–(29). The stability analysis of a 3D dynamic system requires the examination of the Routh–Hurwitz necessary and sufficient conditions for local stability of the linearised system. We present the mathematical relations of these conditions and then the simulations performed.

• $Det\left[J\right]<0$;
• $Tr\left[J\right]<0$;
• $Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]>0$;
• $-Tr\left[J\right][Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]]+Det\left[J\right]>0$.

The Jacobian matrix is as follows:

$$J({\delta }_w,{\delta }_I,g)=\left|\begin{array}{ccc}-g-i{\beta }_2+(s\pi -\Gamma )u_{{\delta }_w}& {\displaystyle \frac{s\pi -\Gamma -{\delta }_w\Gamma }{\Gamma }}& {\displaystyle \frac{-i(1-s)(s\pi -\Gamma )}{\Gamma }}\\ {\displaystyle \frac{-i\Lambda (b+s{\gamma }_r)\pi (1+{\beta }_2)}{\Gamma }}& {\displaystyle \frac{-\Lambda [\Gamma -(b+s{\gamma }_r)\pi ]}{\Gamma }}& {\displaystyle \frac{-(b+s{\gamma }_r)i\Lambda [\Gamma +(1-s)\pi ]}{\Gamma }}\\ {\displaystyle \frac{-bi\pi (1+{\beta }_2)}{\Gamma }}& {\displaystyle \frac{-{\delta }_I\Gamma +b\pi }{\Gamma }}& -g-{\displaystyle \frac{bi[\Gamma +(1-s\left)\pi \right]}{\Gamma }}\end{array}\right|$$

(37)

The determinant and the trace of the Jacobian matrix are given by the following expressions:

$$\begin{array}{c}\hfill Det\left[J\right]={\displaystyle \frac{1}{\Gamma }}\Lambda (-(g-i)(g-bi(-1+{\delta }_I))+\\ \hfill +(-gis+b(g^2+i^{2}s(-1+{\delta }_I)+gi(-1+{\delta }_w+s+(1-s){\delta }_I)))+\\ \hfill +((g-i)i{\delta }_I\Gamma +(g^2+i^{2}s{\delta }_I+gi({\delta }_w+(1-s){\delta }_I))\pi )s{\gamma }_r+\\ \hfill +i\pi {\beta }_2(-gs+b((1+{\delta }_2)g+i(-1+{\delta }_I))+(g+{\delta }_{w}g+i{\delta }_I)s{\gamma }_r))\end{array}$$

(38)

and

$$\begin{array}{c}\hfill Tr\left[J\right]=-2g-ib{\displaystyle \frac{\Gamma +(1-s)\pi }{\Gamma }}-\Lambda {\displaystyle \frac{\Gamma -(b+s{\gamma }_r)\pi }{\Gamma }}-\\ \hfill -i{\beta }_2-{\displaystyle \frac{i(1+{\beta }_2)(s\pi -\Gamma )}{\Gamma }}\end{array}$$

(39)

4.6. Simulations

As previously noted, the solutions of the model are characterized by significant mathematical complexity, rendering the derivation of analytical solutions impractical for further interpretation. Consequently, the execution of simulations is required to examine the precise nature and stability of these equilibria.

To conduct a simple simulation in Mathematica software for solving a system of differential equations using specific parameter values, we followed four steps. First, we defined the system of differential equations. Second, we selected the initial parameter values for calibration. Third, we solved the system using the fourth-order Runge–Kutta method through the NDSolve function with the chosen parameter values. Finally, we adjusted the parameters and resolved the system to compare the results, thereby performing a comparative statics analysis.

To calibrate the model for parameter scenarios 1, 2, and 3, we utilized empirically grounded and plausible parameter values, which are provided in

Table 3. Calibration of the 3D model: scenarios 1, 2, and 3.

Parameter	Scen. 1	Scen. 2	Scen. 3	Source
${\alpha }_0$	$0.60$	$1.00$	$1.00$	Author’s calculations *
${\alpha }_1$	$1.00$	$0.80$	$1.00$	Author’s calculations *
${\beta }_1$	$0.30$	$0.41$	$0.07$	Author’s calculations *
${\beta }_2$	$0.06$	$0.10$	$0.40$	Author’s calculations *
ℓ	$0.55$	$0.31$	$0.30$	Author’s calculations *
i	$0.05$	$0.05$	$0.05$	(av. 1995–2010) OECD
s	$0.75$	$0.75$	$0.75$	[56]
$\lambda$	$0.30$	$0.30$	$0.30$	According to bank practices
$\pi$	$0.43$	$0.43$	$0.43$	Author’s calculations ** (AMECO)
$\Lambda$	$1.00$	$1.00$	$1.00$	Set for simplicity
${\gamma }_r$	$0.35$	$0.35$	$0.35$	[57]
${\gamma }_0$	$0.08$	$0.08$	$0.08$	[57]
b	$0.10$	$0.10$	$0.10$	Author’s calculations *** (BIS)
${\beta }_1{\alpha }_1\ell$	$0.165$	$0.102$	$0.021$
${\beta }_1-\lambda {\beta }_2$	$0.282$	$0.380$	$-0.05$
${\beta }_1{\alpha }_1\ell -({\beta }_1-\lambda {\beta }_2)(1-\pi )$	$0.004$	$-0.115$	$0.050$
$MaxDist$ ****	$26.368$	$27.966$	$2.536$

Notes: * Set in accordance with other parameters to satisfy the assumption of each scenario and the Keynesian stability condition. ** Gross operating surplus as % GDP, average 1995–2010, (annual macro-economic database of the European Commission’s Directorate General for Economic and Financial Affairs, AMECO). *** Credit to private non-financial sector (Bank of International Settlements, BIS) to gross operating surplus (AMECO), av. 2000–2017. **** We used the expression $Max\left[Norm\right(a-b),Norm(a-c),Norm(b-c\left)\right]$ to obtain the maximum pairwise distance among the three vectors a, b, and c.

. It is important to emphasize that for the first five behavioural parameters, located above the dashed line in the table, we selected values that not only aligned with empirical observations but also met the specific requirements of each scenario’s parameter configuration. These values ensured that the model accurately reflected the conditions necessary for each scenario’s distinct dynamics. In the last four lines, we present the computed values of the defining relationships for each scenario (see, Table 1), as well as the value of the formula that shows the maximum pairwise distance among the three equilibrium points (abbreviated as MaxDist).

Utilizing the previously outlined calibration, we then proceeded by solving the system of differential equations for the three parameter scenarios, and determining the equilibrium values of our endogenous variables ${\delta }_w^{*}$, ${\delta }_I^{*}$, and $g^{*}$. To solve the model in Mathematica, we employed the fourth-order Runge–Kutta method using the NDSolve function. After obtaining the equilibrium points, we plugged them into the Routh–Hurwitz conditions and found that in all scenarios, equilibrium point A satisfied the Routh–Hurwitz conditions for local stability. In particular, A was asymptotically stable, and B and C were unstable (saddle) points. Figure 2 provides a graphical overview of the 3D phase diagram, where the horizontal (yellow) hyperbolic surface reflects the ${\delta }_w^{*}$ isocline, the linear (light blue) surface the $g^{*}$ isocline, and the vertical (pink) hyperbolic surface the ${\delta }_I^{*}$ isocline. Obviously, A is an attractor, while B and C are repellors.

Table 4. Simulation results: Routh–Hurwitz necessary and sufficient conditions.

	Scenario 1
A(${\delta }_w^{*}=0.45$, ${\delta }_I^{*}=0.17$, $g^{*}=0.22$)	B(${\delta }_w^{*}=5.27$, ${\delta }_I^{*}=-1.17$, $g^{*}=0.02$)	C(${\delta }_w^{*}=18.94$, ${\delta }_I^{*}=-15.55$, $g^{*}=0.00$)
$Det\left[J\right]=-0.03$	$Det\left[J\right]=0.0009$	$Det\left[J\right]=-0.001$
$Tr\left[J\right]=-0.95$	$Tr\left[J\right]=-0.41$	$Tr\left[J\right]=-0.39$
$Det\left[J_1\right]=0.11$	$Det\left[J_1\right]=0.06$	$Det\left[J_1\right]=0.3$
$Det\left[J_2\right]=0.08$	$Det\left[J_2\right]=0.0003$	$Det\left[J_2\right]=0.00006$
$Det\left[J_3\right]=0.10$	$Det\left[J_3\right]=-0.16$	$Det\left[J_3\right]=-0.41$
$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=0.29$	$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=-0.1$	$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=-0.11$
$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=0.25$	$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=-0.04$	$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=-0.04$
	Scenario 2
A(${\delta }_w^{*}=0.26$, ${\delta }_I^{*}=0.15$, $g^{*}=0.18$)	B(${\delta }_w^{*}=6.35$, ${\delta }_I^{*}=-0.67$, $g^{*}=0.02$)	C(${\delta }_w^{*}=-14.34$, ${\delta }_I^{*}=17.95$, $g^{*}=0.00$)
$Det\left[J\right]=-0.03$	$Det\left[J\right]=0.003$	$Det\left[J\right]=-0.003$
$Tr\left[J\right]=-1.02$	$Tr\left[J\right]=-0.59$	$Tr\left[J\right]=-0.54$
$Det\left[J_1\right]=0.13$	$Det\left[J_1\right]=0.03$	$Det\left[J_1\right]=-0.49$
$Det\left[J_2\right]=0.06$	$Det\left[J_2\right]=0.0005$	$Det\left[J_2\right]=-0.00003$
$Det\left[J_3\right]=0.12$	$Det\left[J_3\right]=-0.14$	$Det\left[J_3\right]=0.35$
$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=0.3$	$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=-0.1$	$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=-0.15$
$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=0.22$	$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=-0.06$	$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=-0.08$
	Scenario 3
A(${\delta }_w^{*}=0.27$, ${\delta }_I^{*}=0.15$, $g^{*}=0.18$)	B(${\delta }_w^{*}=0.73$, ${\delta }_I^{*}=2.63$, $g^{*}=0.00$)	C(${\delta }_w^{*}=1.74$, ${\delta }_I^{*}=1.26$, $g^{*}=-0.02$)
$Det\left[J\right]=-0.02$	$Det\left[J\right]=0.0004$	$Det\left[J\right]=-0.0004$
$Tr\left[J\right]=-0.85$	$Tr\left[J\right]=-0.31$	$Tr\left[J\right]=-0.3$
$Det\left[J_1\right]=0.07$	$Det\left[J_1\right]=-0.05$	$Det\left[J_1\right]=-0.04$
$Det\left[J_2\right]=0.08$	$Det\left[J_2\right]=-0.0001$	$Det\left[J_2\right]=-0.002$
$Det\left[J_3\right]=0.08$	$Det\left[J_3\right]=-0.03$	$Det\left[J_3\right]=-0.05$
$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=0.24$	$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=-0.08$	$Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right]=-0.09$
$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=0.18$	$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=-0.02$	$-Tr\left[J\right](Det\left[J_1\right]+Det\left[J_2\right]+Det\left[J_3\right])=-0.03$

reports the Routh–Hurwitz necessary and sufficient conditions for local stability of the linearised system for each scenario.

In this framework, if the initial state of the economy is proximate to equilibrium point A, the system tends to converge towards this locally stable equilibrium. However, should the initial state lie near point B, the economy is prone to exhibiting excessive values of ${\delta }_w$) and low levels of growth. This outcome suggests that structural instability is an inherent feature of the model, even under the assumption of macroeconomic stability within the real economy, as prescribed by standard macroeconomic stability conditions. In particular, financial variables have the potential to exacerbate this instability, especially when long-term economic growth is sluggish. This insight is corroborated by the works of [37,38] and Charles [36], who also identified the destabilizing effects of financial factors under weak growth conditions. It is important to note that the stability status of the system is not affected by variations in borrowing behaviour or modifications to the parameter set; instead, such changes alter the intensity of the stability status, either strengthening or weakening it in response to perturbations [38]. Specifically, if the equilibrium points come closer, the risk of instability heightens, as a sudden external shock could trigger oscillatory movements, displacing the system from a stable trajectory to an unstable one. Conversely, when these equilibrium points diverge, the system’s stability improves, reducing the likelihood of such disruptive dynamics. This analysis of stability follows the mathematical stability methodology, as delineated by [38].

To conduct a simulation of the aforementioned stability analysis, ref. [21] employed the Euclidean distance in the context of a 2 × 2 model with two equilibrium points. An increase (or decrease) in the Euclidean distance between equilibrium points A’ and B’, relative to the initial distance between A and B, was interpreted as an indication of greater (or lesser) system stability. In this paper, which incorporates a 3D model, we utilized the expression $Max\left[Norm\right(a-b),Norm(a-c),Norm(b-c\left)\right]$ to obtain the maximum pairwise distance among the three equilibrium points, as a comparative measure of stability. An initial indication is provided by Table 3 (final row), where, based on the values derived from the MaxDist index, the second scenario (27.966) is identified as the most macro-economically stable, followed by the first (26.368), with the third (2.536) being the least stable.

In

Table 5. Simulation results: equilibrium points, wage gap, and debt sustainability.

			A
Scen.	${\delta }_w^{*}$	${\delta }_I^{*}$	$g^{*}$	$\mathit{w}.\mathit{g}.$	$\mathit{w}.\mathit{d}.\mathit{s}.$	$\mathit{c}.\mathit{d}.\mathit{s}.$
1	$0.45$	$0.17$	$0.22$	$0.31$	$0.07$	$0.03$
2	$0.26$	$0.15$	$0.18$	$0.09$	$0.03$	$0.02$
3	$0.27$	$0.15$	$0.18$	$0.12$	$0.03$	$0.02$
			B
Scen.	${\delta }_w^{*}$	${\delta }_I^{*}$	$g^{*}$	$\mathit{w}.\mathit{g}.$	$\mathit{w}.\mathit{d}.\mathit{s}.$	$\mathit{c}.\mathit{d}.\mathit{s}.$
1	$5.27$	$-1.17$	$0.02$	$0.34$	$-0.21$	$0.05$
2	$6.35$	$-0.67$	$0.02$	$0.45$	$-0.17$	$0.02$
3	$0.73$	$2.63$	$0.00$	$0.37$	$-0.04$	$-0.15$
			C
Scen.	${\delta }_w^{*}$	${\delta }_I^{*}$	$g^{*}$	$\mathit{w}.\mathit{g}.$	$\mathit{w}.\mathit{d}.\mathit{s}.$	$\mathit{c}.\mathit{d}.\mathit{s}.$
1	$18.94$	$-15.55$	$0.00$	$0.37$	$-1.01$	$0.83$
2	$-14.34$	$17.95$	$0.00$	$-0.19$	$0.73$	$-0.92$
3	$1.74$	$1.26$	$-0.02$	$0.43$	$-0.13$	$-0.09$

Note: w.g. stands for the wage gap, defined as $w^t\ell -u(1-\pi )$, w.d.s. stands for working households’ debt sustainability, defined as $B_w>i{\delta }_w^{*}$, and c.d.s. stands for capitalists’ debt sustainability, defined as $B_I>i{\delta }_I^{*}$. A positive w.d.s. or c.d.s. means working households or corporations’ debt is sustainable.

, we report the equilibrium values of our endogenous variables ${\delta }_w^{*}$, ${\delta }_I^{*}$, and $g^{*}$, the values of the wage gap (w.g.), the working households’ debt sustainability (w.d.s.), defined as $B_w>i{\delta }_w^{*}$, and the capitalists, debt sustainability (c.d.s.), defined as $B_I>i{\delta }_I^{*}$.

A first general observation is that the underlying motivations for workers’ borrowing, while maintaining the economic structure of the model, induce distinct quantitative and qualitative effects on the macroeconomic dynamics.

Second, we observe that across all three scenarios the equilibrium points A, B, and C exhibit a consistent pattern (see Block A, Block B and Block C irrespective of the type of scenario). Specifically, at equilibrium point A, both working households and capitalists are debtors and their debts are sustainable (new borrowing is higher than their interest payments). In addition, long-run growth is high. On the contrary, at equilibrium points B and C, working households and capitalists might be debtors or savers (negative debt implies they are saving) and when debtors, their debt is not sustainable (new borrowing is lower than their interest payments i.e., new borrowing is not enough to finance even past obligations in a process akin to Minsky’s Ponzi state of finance). At points B and C, working households (or capitalists) are heavily indebted, and growth is close to zero.

Third, from a mathematical perspective, there is a consistency between the mathematical notion of stability and the economic one. In all scenarios the asymptotically stable equilibrium point A corresponds to high levels of long-run growth and low levels of debt–capital ratios, while the unstable (saddle) points B and C correspond to extremely low levels of growth and high levels of debt–capital ratios. More importantly as we mentioned above, at equilibrium point A, both working households’ and corporate debt levels are sustainable, while at equilibrium points B and C, both households and corporations exhibit excessive debt accumulation, akin to a Ponzi scheme.

In the context of a cross-scenario analysis at equilibrium point A, we find that the first scenario outperforms the second and third scenarios in terms of long-run growth and debt sustainability for both households and corporations (where new borrowing exceeds interest payments), despite exhibiting higher levels of indebtedness. This superior performance arises from the more pronounced income inequality in the first scenario (note that at point A in the first scenario, the wage gap is significantly higher than in the other two scenarios), which leads to a more substantial increase in borrowing by working households. This heightened borrowing boosts long-term economic growth. The implication of this finding is that, in the presence of income inequality, elevated levels of borrowing—and consequently higher debt—can be sustained only under conditions of high accumulation rates. In contrast, the second and third scenarios yield lower levels of growth and debt-ratios, and working households and corporations are less debt-sustainable relative to the first scenario. (It is noteworthy that the Minskyan dynamics are constrained by the maximum allowable interest payment threshold ($\lambda =0.30$), a limit commonly imposed in standard banking practices. In the absence of this constraint, we would anticipate a higher performance in terms of economic outcomes. However, such an improvement would come at the cost of macroeconomic stability, as it would lead to an unsustainable accumulation of debt and increased financial instability).

4.7. Comparative Statics Analysis and Sources of Instability

In this section, we present the findings from the comparative statics analysis and examine the variations in stability status, as shown in

Table 6. Summary of comparative statics across all models. Stability analysis. Equilibrium points.

		Scen. 1				Scen. 2				Scen. 3
	${\mathit{\delta }}_{\mathit{w}}^{*}$	${\mathit{g}}^{*}$	${\mathit{\delta }}_{\mathit{I}}^{*}$	Stab.	${\mathit{\delta }}_{\mathit{w}}^{*}$	${\mathit{g}}^{*}$	${\mathit{\delta }}_{\mathit{I}}^{*}$	Stab.	${\mathit{\delta }}_{\mathit{w}}^{*}$	${\mathit{g}}^{*}$	${\mathit{\delta }}_{\mathit{I}}^{*}$	Stab.
${\alpha }_0$	+	+	+	+	+	+	+	+	+	+	+	-
${\alpha }_1$	+	+	+	-	+	+	+	+	+	+	+	+
${\beta }_1$	+	+	+	+	+	+	+	-	+	+	+	-
${\beta }_2$	+	+	+	-	+	+	+	+	+	+	+	+
$\lambda$	+	+	+	-	+	+	+	+	+	+	+	+
i	+	-	-	-	+	-	-	-	-	-	-	-
s	+	-	-	+	+	-	-	-	-	-	-	-
$\pi$	+	+	+	-	+	+	+	+	-	-	-	-
${\gamma }_0$	-	+	-	-	-	+	-	+	-	+	+	+
${\gamma }_r$	-	+	-	+	-	+	-	-	-	+	+	-
b	-	+	+	-	-	+	+	-	-	+	+	-

. Specifically, we analysed the effects of parameter changes on the equilibrium points and stability characteristics, providing insights into the underlying dynamics of the system. These results are essential for understanding the sensitivity of the equilibrium to shifts in exogenous variables and offer a comprehensive assessment of the system’s response to perturbations. The parameters under consideration were the animal spirits, ${\gamma }_0$, the sensitivity of investment to internal funds, ${\gamma }_r$, the interest rate, i, the adjustment borrowing parameters, ${\beta }_1$ and ${\beta }_2$, the retention rate, s, the profit share, $\pi$, the maximum debt-service-income ratio, $\lambda$, the sensitivity of the wage target to capacity utilization, ${\alpha }_1$, the constant term, ${\alpha }_0$, and the firms’ external to internal finance ratio, b.

Regarding the stability analysis, the MaxDist between the initial equilibrium points (ABC) and the new equilibrium points (A’B’C’) was compared to determine whether the distance between the new and old equilibrium points increased (indicating increased stability) or decreased (indicating reduced stability). Since it is more meaningful to conduct these analyses near the stable domain of the system, only the results related to the stable equilibrium point A are reported.

From the comparative statics analysis, several critical insights emerge. We observe that variations in parameters related to consumer borrowing (i.e., ${\beta }_1$, ${\beta }_2$, $\lambda$, ${\alpha }_0$, and ${\alpha }_1$) had a positive effect on the equilibrium values of the endogenous variables ${\delta }_w^{*}$, ${\delta }_I^{*}$, and $g^{*}$ across all model scenarios. This can be understood by recalling Equations (14) and (15), which showed that an increase in any of these parameters—assuming all other parameters remained constant (ceteris paribus)—positively impacted workers’ borrowing, $B_w$, both directly (via ${\beta }_1$, ${\beta }_2$, and $\lambda$) and indirectly through $w^t$ (via ${\alpha }_0$ and ${\alpha }_1$). In consequence, increasing borrowing enhances consumption, aggregate demand, and hence long-run growth. However, the effect on macroeconomic stability is contingent on the specific scenario being analysed, displaying varied directional shifts.

Regarding $\lambda$, in contrast, ref. [21], which focused solely on the interaction between consumer debt and the rate of capital accumulation, found that stability was enhanced in all scenarios. Similarly, in a model incorporating both consumer and corporate debt, ref. [30] also found that an increase in the consumer credit target (a parameter analogous to the function of $\lambda$) contributed to greater macroeconomic stability.

A higher ratio of external-to-internal corporate borrowing, b, had a positive effect on long-run growth and the corporate debt-to-capital ratio, while simultaneously leading to a reduction in the consumer debt-to-capital ratio. As for the first two variables, recalling Equations (10) and (11), we observe that a higher b, assuming all other parameters remained constant, positively impacted corporate borrowing. This, in turn, affected the desired accumulation rate, leading to higher long-run growth, as we know from Equation (7), where industrial capitalists adjust their actual investment rate to align with their desired rate of investment. On the other hand, a higher rate of accumulation not driven by finance-led consumption resulted in a decrease in the consumer debt-to-capital ratio.

Regarding macroeconomic stability, b, along with the interest rate, was one of the only parameters that consistently reduced stability across all scenarios. These observations highlight a trade-off between economic growth and macroeconomic stability.

An increase in animal spirits, ${\gamma }_0$, raised the desired rate of accumulation, thereby shifting the $dg/dt=0$ isocline upward without affecting the ${\displaystyle \frac{𝜕{\delta }_I}{𝜕t}}$ and ${\displaystyle \frac{𝜕{\delta }_w}{𝜕t}}$ isoclines. As a result, the initial increase in the desired investment function, for given debt levels (both workers and capitalists), tended to reduce the ${\delta }_w^{*}$ and ${\delta }_I^{*}$ ratios. As capacity utilization rose due to higher g (recall Equation (18)), its positive indirect impact on workers’ borrowing was weak, whereas, on the other hand, capitalists’ borrowing—especially in the third scenario—might increase as well. Macroeconomic stability decreased in the first scenario (so there is a trade-off between growth and macroeconomic stability) but improved in the second and third scenarios.

Regarding the impact of “animal spirits”, ref. [21] found a positive effect on g and a negative effect on ${\delta }_w$, but the stable region expanded in all scenarios. Similarly, ref. [30] found an increase in $g^{*}$, a decrease in both ${\delta }_w^{*}$ and ${\delta }_I^{*}$, and an expansion of the stable region.

A higher sensitivity of investment to the profit rate, ${\gamma }_r$, had a positive effect on long-run growth and a negative effect on the consumer debt-to-capital ratio in all scenarios (shifting the $dg/dt=0$ isocline clockwise while leaving the other isoclines unaffected). Corporate debt-to-capital ratios declined in the first and second scenarios, while an increase was observed in the third scenario. Regarding macroeconomic stability, its impact varied across the different scenarios. Furthermore, the analysis identified a trade-off between long-term growth and macroeconomic stability in the second and third scenarios.

An increase in the retention rate was contractionary in both the short and long run, in all scenarios, due to its negative impact on capacity utilization (see Table 2). This is consistent with the “paradox of thrift”. The impact on the corporate debt-to-capital ratio was also negative in all scenarios. A higher retention rate influenced workers’ borrowing only indirectly through its effect on u, and the specific sensitivity ${\displaystyle \frac{𝜕\frac{B_w}{K}}{𝜕u}}$ determined its impact (positive in the first and second scenarios, negative in the third). In the first scenario, however, the impact might have been expected to be negative; yet the interaction with other parameters resulted in a negative effect. The status of macroeconomic stability was scenario-dependent. Generally, a trade-off between long-run growth and macroeconomic stability was more likely to emerge within the framework of the income inequality scenario (the first scenario).

Another significant finding concerned the distributional parameter $\pi$. The results suggested that an increase in the profit share may have either a positive (as observed in scenarios 1 and 2) or a negative (as observed in scenario 3) impact on long-run growth. This implies that the nature of growth—whether profit- or wage-led—can be influenced by consumer borrowing behaviour. This differentiation arises from the ambiguity in the sign of ${\displaystyle \frac{𝜕g^t}{𝜕\pi }}$ (and ${\displaystyle \frac{𝜕u}{𝜕\pi }}$ in the short-run, see Table 2). The sign may be either positive or negative, implying that both the growth and the demand regime may be either wage- or profit-led. (In particular, by differentiating the desired investment function with respect to the profit share, we obtain ${\displaystyle \frac{𝜕g^t}{𝜕\pi }}=(s{\gamma }_r+b)(u+\pi {\displaystyle \frac{𝜕u}{𝜕\pi }})=(s{\gamma }_r+b)[-{\displaystyle \frac{u}{\Gamma }}[{\alpha }_1{\beta }_1\ell -({\beta }_1-\lambda {\beta }_2)]]$. Recall the relationship presented in Equation (20)).

In the context of the first scenario (increasing income inequality), an increase in the profit share exerted both direct and indirect effects on the borrowing behaviour of working households. The direct effect of an increase in the profit share was a corresponding reduction in the wage share, which led to an expansion of the gap between the wage target and the actual wage. As a result, household borrowing increased, thereby raising the debt-to-capital ratio. Conversely, the indirect effect arose from the reduction in capacity utilization resulting from the increase in the profit share. Given the positive relationship between capacity utilization and household borrowing (${\displaystyle \frac{𝜕\frac{B_w}{K}}{𝜕u}}>0$), this reduction in capacity utilization tended to mitigate borrowing. Simulation results indicated that the direct effect dominated, leading to an increase in the household debt-to-capital ratio.

Regarding the corporate debt-to-capital ratio, the effect was positive. As we can observe from Table 6, in all scenarios, corporate borrowing moved in the same direction as the change in $\pi$ (see Equation (10)).

In the case of the second scenario, household borrowing was subject to both direct and indirect effects resulting from an increase in the profit share. Both effects in that case were positive: the direct effect arose from the widening of the gap between the target wage and the actual wage, and the indirect effect was due to the negative relationship between household borrowing and capacity utilization (${\displaystyle \frac{𝜕\frac{B_w}{K}}{𝜕u}}<0$). Essentially, consumers mitigated the downward pressure on consumption, aggregate demand, and capacity utilization induced by an increasing profit share.

In the case of the third scenario, there were two direct effects and one indirect effect. Two opposing trends emerged in the direct effects. On the one hand, an increase in the share of profits led to a corresponding reduction in the wage share, resulting in the widening of the gap between the wage target and the actual wage. This triggered an increase in borrowing by working households. On the other hand, the gap between the maximum affordable interest payment and the actual payment decreased, leading households to reduce their borrowing. Given that in the third scenario, households placed more weight on the option based on the interest payment gap, the negative impact on borrowing dominated (recall Equation (14)). The indirect effect arose from the decline in capacity utilization due to the increase in the profit share. This decline in u, due to the positive relationship between capacity utilization and working households’ borrowing (${\displaystyle \frac{𝜕\frac{B_w}{K}}{𝜕u}}>0$), led to a reduction in borrowing. With respect to macroeconomic stability, it increased in scenario 2, while it declined in scenarios 1 and 3. It is important to note that the paradox of costs was only applicable in the third scenario.

Finally, regardless of the scenario, an increase in the interest rate exerted contractionary effects both in the short and long run and undermined macroeconomic stability. Specifically, it led to a reduction in long-run growth because the profits of enterprises (gross profits minus interest payments) decreased, resulting in a compression of the desired rate of accumulation (see Equation (11)). For the same reason, corporate borrowing declined, and consequently, the debt-to-capital ratio decreased, as the profits of enterprises diminished due to the increased cost of debt servicing (see Equation (10)). This causal chain aligns with the “principle of increasing risk”, where diminished internal means of financing investments reduce access to external financing in imperfectly competitive capital markets.

Regarding household borrowing, the following points are noted. According to the household borrowing function (Equation (16)), an increase in the interest rate affects household borrowing directly through the term ${\beta }_{2}i\delta w$, and indirectly through the negative effect the interest rate has on capacity utilization (see Table 2). In the first and second scenarios (where both have ${\displaystyle \frac{𝜕\frac{B_w}{K}}{𝜕u}}>0$) both impacts were negative. This negative impact was stronger in the case of the third scenario, as households placed more weight on the option based on the interest payment gap. In the first scenario, however, although the net effect of a higher interest rate was negative, the much stronger reduction in the desired rate of accumulation resulted in a higher consumer debt-to-capital ratio. In the second scenario, where working households smoothed the negative impact on consumption and hence the fall in the utilization rate (${\displaystyle \frac{𝜕\frac{B_w}{K}}{𝜕u}}<0$), households increased borrowing and hence debt accumulation.

The impact on long-run growth was reported as positive in [21,25], whereas [27] found a negative effect, and [30] identified no impact, although they observed a reduction in the stability region.

5. Conclusions

This paper integrated the literature that explores the interaction between consumer debt and capital accumulation with that which investigates the interaction between corporate debt and capital accumulation. In particular, it extended the models developed by [19,20,21,25], which captured the dynamics between consumer debt and the rate of capital accumulation, by incorporating the dimension of corporate debt. We conducted simulations using Mathematica software to solve the non-linear dynamic system of three differential equations: the workers’ debt-to-capital ratio, the corporate debt-to-capital ratio, and the accumulation rate. We then examined the local stability of the equilibrium points using the Routh–Hurwitz necessary and sufficient conditions for local stability. Finally, we performed a comparative statics analysis followed by a stability analysis. For the latter, we adhered to the theoretical framework proposed by [38] and employed a measure of the maximum distance among the three equilibrium points as a comparative measure of stability.

From the perspective of consumer borrowing, we assumed and analysed three distinct household borrowing behaviours: (a) sustaining their standard of living, rather than “getting ahead” beyond their class consumption patterns or “keeping up with the Joneses”; (b) smoothing their consumption in response to a decline in wage income; and (c) exhibiting a Minskyan-type borrowing behaviour, characterized by the accumulation of debt in a speculative manner. On the corporate borrowing side, we assumed firms’ borrowing decisions were contingent on the profits of enterprise. This assumption captured the Kaleckian “principle of increasing risk” that a reduction in internal financing sources for real investment projects diminishes access to external finance within imperfectly competitive capital markets.

The results of our analysis were consistent with previous works demonstrating that the various motivations underlying borrowing behaviour among working households exert both quantitative and qualitative effects on economic outcomes in the short and long run. Specifically, the particular motivation for borrowing, due to its distinct impact on capacity utilization, influenced the characteristics of the growth regime, debt dynamics, and macroeconomic stability. We also found that although the comparative statics analysis shared several similarities with the results of the 2 × 2 models, the inclusion of corporate debt predominantly altered the impact of parameter changes on the status of macroeconomic stability. Specifically, while a change in the parameters ${\alpha }_0$, ${\alpha }_1$, ${\beta }_1$, ${\beta }_2$, $\lambda$, ${\gamma }_0$, and ${\gamma }_r$ had the same effect across all scenarios in the 2 × 2 model, incorporating corporate debt led to differentiated effects depending on the scenario. The sole exceptions were the interest rate and the new parameter representing the ratio between external and internal borrowing, b, which exhibited the same negative impact in all scenarios. Furthermore, the inclusion of corporate debt not only changed the direction of the impact on macroeconomic stability but also increased the number of parameters for which the impact varied across scenarios. While in the 2 × 2 model, seven out of ten parameters had the same effect across all scenarios, in the 3 × 3 model, only the interest rate and the new parameter b had the same effect in all scenarios.

Our findings also highlight that under specific conditions, borrowing can mitigate the effects of contractionary economic policies. However, this benefit comes at the cost of increased household debt, which, when coupled with borrowing-induced consumption patterns, creates a trade-off between growth and macroeconomic stability. This outcome is particularly evident when examining shifts in income distribution within the context of income inequality exacerbated by market-oriented economic policies. These results underscore the need for policy-makers to carefully consider the implications of relying on borrowing as a tool for stimulating economic activity in contexts characterized by high income inequality. While consumer debt can be sustainable under conditions of strong growth, working households tend to become more indebted, and their standard of living increasingly diverges from their wage income compared to other scenarios. In the current economic era, achieving sustainable growth—defined as the combination of higher growth and macroeconomic stability—requires not only a redistribution of income in favour of working households but also a transformation of the market-oriented policies that have adversely affected workers’ living standards. Such a transformation should ensure that workers’ well-being is no longer dependent on excessive borrowing.

There are several limitations and shortcomings in our model. One limitation is that the financial sector is represented solely through debt-to-income ratios, without fully capturing its role as an active economic institution. Incorporating a more active role for the financial sector within the model framework would constitute a valuable enhancement. Moreover, as with any theoretical contribution, empirical econometric research is essential to assess the robustness and generalizability of the model’s results, thereby providing a solid foundation for validating its theoretical assumptions and conclusions. Lastly, the model could be further improved by incorporating behavioural corporate lending scenarios, drawing on insights from empirical and experimental studies on corporate behaviour.