Fiscal policy and social infrastructure provision under alternative growth and distribution regimes

Abstract

We built a Kaleckian dynamic model that can comprehensively analyse the growth, distribution, and employment rate of the government's social infrastructure and debt accumulation. The model allows for not only wage-led growth and profit-led growth regimes but also labour-market-led and goods-market-led distribution regimes. Particular attention is paid to the demand effects of fiscal policy and the productivity growth effect of social infrastructure investment. Our model derives the following results. A combination of alternative growth and distribution regimes is important for stability. When government debt also changes in the long-run, the Domar condition is required for stability. Despite the principally Kaleckian nature, the long-run economic growth rate depends not on demand or fiscal parameters but on supply-side parameters. We conclude that the government can still play an important role in stabilising the economy, improving the quality of social infrastructure, and achieving a resilient economy.

1 Introduction

The global economy has been facing a complex and structural challenge. Global financial crises and the COVID-19 pandemic have revealed the economy’s fragility, and climate change signals a fundamental ecological crisis. These crises show that the market economy does not function autonomously and requires resilient social, natural, and institutional foundations and that government policies can enhance them. Achieving economic growth with secured employment and fair income distribution is a major concern for policymakers. Additionally, a government’s fiscal sustainability remains an important but controversial issue (Herndon et al. 2014; Reinhart and Rogoff 2009). Crises urge economic theory to identify the government's roles to simultaneously establish a resilient economy through sustainable and proactive fiscal policy.

The provision of social common capital in normal times demonstrates resilience in emergencies. Uzawa (2005) highlights three social common capital categories as foundations for the market economy: natural environment, social infrastructure, and institutional capital, encompassing health care and education. The author emphasises that social common capital provides members of society with a cultural and human life while ensuring the sustainability and stability of an economy. Seguino (2012) underlines three key roles of public investment: stimulating demand and employment, creating productive capacity, and improving human development. Government spending boosts economic activities and equitable income distribution while also inducing private investment demand and significant productivity gains (Obst et al. 2020; Onaran et al. 2022; Oyvat and Onaran 2022).

The roles of government regarding economic growth have attracted theoretical studies in post-Keynesian literature. Commendatore et al. (2011) build a Kaleckian wage-led growth model with multiple equilibria in which the government has productive and unproductive expenditures. They show that when government expenditure raises wages more than its impact on labour productivity, it has an expansionary effect on capacity utilisation and growth rates. Dutt (2013) presents a Keynesian demand-led growth model in which government spending stimulates productivity growth and crowding-in effects for firms' capital accumulation. Ko (2019) and Parui (2021) address the absence of income distribution in Dutt's (2013) model and compare the impacts of a rise in government investment on growth and employment. These models commonly indicate that government expenditures positively impact economic growth. Hein (2018) and Hein and Woodgate (2021) identify the stability conditions in Kaleckian models when the government can create autonomous expenditure and debt accumulation. However, they do not consider the subsequent impacts of growth on income distribution or employment.

Tavani and Zamparelli's works (2016, 2017, 2020) are also relevant to our study. Tavani and Zamparelli (2017) employ demand-led models with public capital and show that the maximal growth rate is realised at a certain government spending ratio to social capital. In contrast, Tavani and Zamparelli (2016, 2020) use supply-led growth models in the public sector; they demonstrate (2016) that when the wage share is exogenously given by the conventional share, there is a tax rate that maximises the economic growth rate on a balanced growth path. Their 2020 model reveals that induced technical changes may mitigate the Goodwin cycles and distributional conflicts. However, as their models assume that savings create investment, they do not show the autonomous roles of private investment or demand-led growth mechanisms. Furthermore, the long-run consequences of fiscal sustainability are not clear.

Neo-classical economics has also extensively studied the government's roles in public service, capital, finance, and economic growth. Broadly, starting with Arrow and Kurz (1969), which considers the optimal public policy of resource allocation to maximise the discounted utility, the government plays an important role in the endogenous growth theory. Barro (1990), a seminal paper in this field, highlights that tax-financed government expenditure catalyses private production, leading to endogenous growth through its productivity-enhancing effects. Barro and Sala-i-Martin (2004) propose models where government services in production can counter diminishing returns of private capital, fostering endogenous growth. They also discuss how a decentralised market economy may face suboptimal growth rates due to congestion issues. Government finance matters as well. Futagami et al. (2008) develop an endogenous growth model with productive government services and a target government debt-GDP ratio, demonstrating how different finance methods impact public and private capital ratios and associated growth rates. Greiner and Semmler (2000) integrate government finance and public capital stock in economic growth analysis, finding that public deficits primarily used for public investment may not necessarily lead to lower growth rates. Neo-classical models generally explore the conditions for tax rates to achieve maximum growth rates. Empirical studies have shown a positive effect of public capital on economic growth. Aschauer (1989) suggests that core infrastructure, like highways and airports, significantly impacts productivity. Bom and Ligthart (2014) conducted a survey and meta-regression analysis, concluding that public capital is insufficient in OECD economies based on the average output elasticity of public capital. However, these models rely on optimisation by rational economic agents and are primarily supply-side growths. They ignore the uncertainty of the future or the bounded rationality of economic agents. Moreover, the models exclude the role of effective demand and involuntary unemployment.

We build a Kaleckian dynamic growth, distribution, and employment model with the government's social infrastructure provision and debt accumulation. Compared with the existing literature, our study has the following analytical novelties. First, our model is more comprehensive for growth and income distribution, allowing for both wage-led growth (WLG) and profit-led growth (PLG) regimes. Additionally, growth induces a change in employment, which subsequently positively or negatively changes the profit share. The positive and negative correlations between employment and profit share are goods-market-led (GML) and labour-market-led (LML) distribution regimes. Proaño et al. (2011) and Nishi and Stockhammer (2020a, b) specify the stability conditions for these alternative demand and distribution regimes; however, they do not incorporate the government economic policy or economic growth.

Second, we consider the effects of a proactive fiscal stance, signified by a tax cut and a rise in government propensities to consume and invest under alternative regimes. Both constitute an important part of effective demand in the short run, thus supporting the investment demand of firms. Particular attention has been paid to the productivity growth effects of social infrastructure investment. Our model differs from those of Tavani and Zamparelli (2016, 2017, 2020) because we explain the labour productivity effect of social infrastructure not only by its availability per private capital but also per labour input. In contrast, they explain it only by the former. Finally, along with these dynamics, government expenditure leads to long-term debt accumulation in the long-run. We sequentially elucidate how the stability of demand and distribution regimes is achieved and associated with government debt over different time horizons.

The remainder of this paper is organised as follows. Section 2 builds a dynamic Kaleckian growth, distribution, and employment model with the government's social infrastructure investment. It also outlines its short-run dynamics, in which the capacity utilisation rate is instantaneously determined. Section 3 proceeds to the analysis of long-run dynamics. Using numerical studies, we visually consider the nature of transitional dynamics and provide economic interpretations. Finally, Sect. 4 concludes the paper. The Appendices provide mathematical proofs of the main propositions.

2 Short-run analysis

The following variables are employed in our model: ${X}_{t}$: output; ${K}_{t}$: private capital stock; ${ S}_{t}$: social infrastructure stock; ${ L}_{t}$: labour demand; ${N}_{t}$: labour supply; ${C}_{t}$: private consumption; ${I}_{t}$: firms' investment; ${G}_{Ct}$: government consumption; ${G}_{It}$: government investment in social infrastructure; ${T}_{t}$: tax revenue; ${ \chi }_{t}$: social infrastructure-private capital ratio (capital composition); ${u}_{t}$: output-capital ratio (capacity utilisation rate); ${q}_{Lt}$: labour productivity; ${w}_{t}$: nominal wage; ${p}_{t}$: price; ${e}_{t}$: employment rate; ${m}_{t}$: profit share (${1-m}_{t}$: wage share); ${g}_{t}$: capital accumulation rate; and ${\delta }_{t}$: government debt-capital ratio (debt ratio). The subscript $t$ represents time in a continuous model. A dot over variable ${x}_{t}$ means its time derivative (i.e. ${{\dot{x}}_{t}=dx}_{t}/dt$), and a hat over variable indicates its rate of change (i.e.${\widehat{x}}_{t}={\dot{x}}_{t}/{x}_{t}$). These variables are introduced with subscript $t$ to show that they vary with time, but we omit them below for simplicity.

2.1 Production, income distribution, and effective demand

We consider a closed economy with household workers, private firms owned by capitalists, and the government sector. We assume that the final goods can be used not only for both consumption and private capital but also for the social infrastructure accumulated through government investment. Workers supply labour to private firms and earn wages in return. Firms employ the labour force provided by households and pay wages. Capitalists profit by managing firms and receive interest revenue by holding government bonds. Firms produce final goods using labour and capital stock in the presence of social infrastructure. We formalise the input–output relationship with the following Leontief-type fixed coefficient production function:

$$\begin{array}{*{20}c} {X = \min \left( {uK, q_{L} L} \right),} \\ \end{array}$$

(1)

where private capital stock and labour inputs are perfect complements. We assume no labour supply constraints, and the operating capital stock determines output under the effective demand constraint.

The total value added is distributed to workers and capitalists as wage and profit income, respectively.

$$\begin{array}{*{20}c} {pX = wL + rpK,} \\ \end{array}$$

(2)

where $wL$ is the wage bill, and $rpK$ is the profit income. Based on this, we denote profit share as

$$\begin{array}{*{20}c} {m = 1 - \frac{w}{{pq_{L} }},} \\ \end{array}$$

(3)

and accordingly, the wage share is expressed by $1 - m$.

Firms are oligopolistic in the goods market, and they set a mark-up over a unit labour cost to sell their goods. Then, the pricing is given by:

$$\begin{array}{*{20}c} {p = \left( {1 + \eta } \right)\frac{w}{{q_{L} }},} \\ \end{array}$$

(4)

where $\eta > 0$ is the mark-up rate. In the present setting, there is a one-to-one relationship of $m = \frac{\eta }{1 + \eta }$ between the profit share and mark-up rate, and the mark-up pricing equation can be denoted as:

$$\begin{array}{*{20}c} {p = \left( {\frac{1}{1 - m}} \right)\frac{w}{{q_{L} }}.} \\ \end{array}$$

(5)

The government levies taxes on wage and profit incomes at the same rate, $\tau \in \left( {0,1} \right)$, which comprises its revenue. Accordingly, tax revenue is

$$\begin{array}{*{20}c} {T = \tau \left( {wL + rpK} \right) = \tau puK.} \\ \end{array}$$

(6)

Government expenditure consists of government consumption, $G_{C}$, and investment in social infrastructure, $G_{I}$. We consider private economic activity as supported by government expenditure in two ways. First, government expenditure is not only a part of effective demand in the short run but generally helps firms expand investment and the associated capital accumulation through the crowding-in effect. Second, the government invests in social infrastructure, which generates its accumulation, that is, $\dot{S} = G_{I}$. The social infrastructure eventually supports the efficient production of goods in an economy by enhancing labour productivity growth in the long-run. Additionally, our model allows for fiscal deficit and debt finance when the government's total expenditure exceeds its tax income. The government then pays interest per unit of issued debt to the capitalists who purchase the bonds.

The government's propensity to consume and invest is denoted by $\theta_{C}$ and $\theta_{I}$, respectively. Then, we have

$$\begin{array}{*{20}c} {p(G_{C} + G_{I} ) = (\theta_{C} + \theta_{I} )\tau puK,} \\ \end{array}$$

(7)

where $G_{C}$ = $\theta_{C} \tau uK$ and $G_{I}$ = $\theta_{I} \tau uK$, and the size of $\tau$ determines the income redistribution from the private sector to the public sector. We call the tax rate and government expenditure propensities $\left( {\tau , \theta_{C} , \theta_{I} } \right)$ its fiscal stance. Because the tax income is $T = \tau puK,$ the degree of the government's primary deficit is measured as $\theta_{C} + \theta_{I} - 1 > 0$.

We assume that workers spend all their disposable income on final goods for private consumption demand. Capitalists spend a constant fraction of disposable profit income and receive interest income from government bonds, which for simplicity, we assume they save.¹ Then, private consumption demand is determined as:

$$\begin{array}{*{20}c} {\frac{C}{K} = \left( {1 - \tau } \right)\left( {1 - m} \right)u + \left( {1 - s} \right)\left( {1 - \tau } \right)mu.} \\ \end{array}$$

(8)

The firms' investment demand is given by the following equation:

$$\begin{array}{*{20}c} {g \equiv \frac{I}{K} = \alpha + \beta \left( {1 - \tau } \right)m + \gamma \left( {\frac{{G_{C} + G_{I} }}{K}} \right),} \\ \end{array}$$

(9)

and private capital accumulation proceeds per the realised investment, $\dot{K} = I$. $\alpha > 0$ is a constant term driven by the firm's animal spirits, and $\beta > 0$ represents the sensitivity of investment demand to a change in the after-tax profit share $\left( {1 - \tau } \right)m$. Importantly, private investment demand is complementary to government expenditure, for which sensitivity is measured by $\gamma > 0$.² This shows that increased government expenditure induces firms' investment and associated capital accumulation in the long-run. Thus, we define the size of $\gamma$ as the demand or crowding-in effect of government expenditure. By substituting Eq. (7) into Eq. (9), the firms’ investment demand is:

$$\begin{array}{*{20}c} {\frac{I}{K} = \alpha + \beta \left( {1 - \tau } \right)m + \gamma \left( {\theta_{C} + \theta_{I} } \right)\tau u.} \\ \end{array}$$

(10)

showing that fiscal stance $\left( {\tau , \theta_{C} , \theta_{I} } \right)$ is important for investment determination.³

2.2 Existence and stability of short-run steady state

The short-run is when the effects of firms' capital accumulation rate and the government's social infrastructure and debt accumulation have not yet been realised. Then, the demand and supply for goods are exclusively adjusted by the change in capacity utilisation rate $u$. Therefore, the short-run equilibrium in the goods market can be represented by

$$\begin{array}{*{20}c} {\frac{X}{K} = \frac{{C + I + G_{C} + G_{I} }}{K}.} \\ \end{array}$$

(11)

By substituting Eqs. (7), (8), and (10), we obtain the short-run steady-state condition in the rate of capacity utilisation term:

$$\begin{array}{*{20}c} {u = \frac{{\alpha + \beta \left( {1 - \tau } \right)m}}{{sm\left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}}.} \\ \end{array}$$

(12)

Substituting this value into Eq. (10), the short-run capital accumulation rate is obtained as:

$$\begin{array}{*{20}c} {g = \frac{{\left( {\alpha + \beta \left( {1 - \tau } \right)m} \right)\left( {sm\left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}{{sm\left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}},} \\ \end{array}$$

(13)

and the associated Keynesian stability condition is

$$\begin{array}{*{20}c} {m > \frac{\tau }{{s\left( {1 - \tau } \right)}}\left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right).} \\ \end{array}$$

(14)

It ensures a positive and economically meaningful value for the capacity utilisation and capital accumulation rates.

In the short run, the steady-state capacity utilisation rate rises following a fall in saving rate $s$ and profit share $m$. Thus, the paradoxes of thrift and cost hold true. An increase in the propensities of government consumption and investment, $\theta_{C} , \theta_{I}$, equally stimulates capacity utilisation rates, raising the capital accumulation rate in the short run. A rise in the tax rate $\tau$ increases the capacity utilisation rate, whereas its impact on capital accumulation is mixed.

Identifying growth regimes is important for elucidating the nature of long-run dynamics. A growth regime refers to the relationship between changes in income distribution and the economic growth rate. By differentiating $g$ with respect to $m$, we observe two growth regimes according to the following criterion:

$$\begin{array}{*{20}c} {\frac{{{\text{d}}g}}{{{\text{d}}m}} = \frac{{\left( {1 - \tau } \right)}}{{\left( {sm\left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)^{2} }}F\left( m \right),} \\ \end{array}$$

(15)

where

$$\begin{array}{*{20}c} {F\left( m \right) \equiv am^{2} + bm + c\frac{ > }{ < }0,} \\ \end{array}$$

(16)

and

$$a \equiv \left( {s - s\tau } \right)^{2} \beta,$$

$$b \equiv - 2s\left( {1 - \tau } \right)\tau \beta \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right),$$

$$c \equiv \tau \left( {\tau \beta \left( {\theta_{C} + \theta_{I} - 1} \right)\left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right) - s\alpha \gamma \left( {\theta_{C} + \theta_{I} } \right)} \right).$$

Thus, WLG and PLG regimes were established for $F\left( m \right) < 0,$ and $F\left( m \right) > 0$, respectively. The absolute value of $F\left( m \right)$ determines the degree of being wage-led and profit-led, which play important roles in stability analysis. Appendix 1 shows that there is a unique profit share $\tilde{m}$ that switches the growth regime between WLG and PLG in a domain. In this case, the economy may have multiple steady-state conditions. Note that the fiscal stance parameters ($\tau$, $\theta_{C}$, and $\theta_{I} )$ are included in $F\left( m \right)$. Thus, the government's fiscal stance shapes the type of growth regime, which we will consider in Sect. 3.3.

3 Long-run analysis

3.1 Long-run dynamics with social infrastructure accumulation

The long-run is when the accumulation of firms' capital, the government's social infrastructure, and its associated effects begin to fully realise. During this period, tax revenue varies according to the change in income distribution, affecting the size of government expenditure. Government expenditure stimulates effective demand, whereas the accumulation of social infrastructure enhances labour productivity growth. Moreover, these dynamics induce changes in income distribution and employment rates. Finally, the government accumulates debt based on the gap between government expenditure and tax income. Thus, the capacity utilisation rate, income distribution, and employment rate are endogenously determined as fast variables, whereas the government's debt ratio evolves as a slow variable following the fast variables. Fiscal sustainability is based on whether the government's debt-to-GDP ratio converges to a certain level.

The government spends tax revenue on investment in social infrastructure $G_{I}$ by $\theta_{I}$, generating the accumulation of social infrastructure, $\dot{S}$. Then, the social infrastructure grows at the following rate:

$$\begin{array}{*{20}c} {g_{S} = \frac{{\dot{S}}}{S} = \frac{{\theta_{I} \tau u}}{\chi } ,} \\ \end{array}$$

(17)

where $\chi = \frac{S}{K}$ is the capital composition (i.e. social infrastructure–private capital ratio), and $u$ is instantaneously determined by Eq. (12). For simplicity, we do not consider the depreciation rates of the capital stock or social infrastructure. Then, the dynamics of the capital composition $\chi$ are given by:

$$\begin{array}{*{20}c} {\frac{{\dot{\chi }}}{\chi } = \frac{{\dot{S}}}{S} - \frac{{\dot{K}}}{K}.} \\ \end{array}$$

(18)

By substituting Eqs. (12), (13), and (17) into (18), we obtain

$$\begin{array}{*{20}c} {\dot{\chi } = \left[ {\frac{{\left( {\alpha + \beta \left( {1 - \tau } \right)m} \right)\left( {\tau \theta_{I} - \left( {sm\left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)\chi } \right)}}{{\left( {sm\left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}} \right].} \\ \end{array}$$

(19)

We assume that labour productivity growth improves as per social infrastructure per private physical capital and per employed labour. Thus, labour productivity is endogenously determined as:

$$\begin{array}{*{20}c} {q_{L} = A\left( \frac{S}{K} \right)^{{\varepsilon_{1} }} \left( \frac{S}{L} \right)^{{\varepsilon_{2} }} ,} \\ \end{array}$$

(20)

where $\varepsilon_{1} \in \left( {0,1} \right)$ is the constant elasticity of labour productivity to social infrastructure per unit of private capital, and $\varepsilon_{2} \in \left( {0,1} \right)$ is that per unit of employed labour. $A$ represents the exogenous determinants of labour productivity, which grow at a constant rate of $\hat{A} = \varepsilon_{0}$. For example, social infrastructure is embodied in public capital as roads and public transportation, backing efficient logistics. It also includes broad institutional capital such as school education, healthcare, and skill training support. The availability of this kind of capital per person supports more productive work (Seguino 2012; Onaran et al. 2022; Oyvat and Onaran 2022); thus, as these effects are complementary to each other, labour productivity is defined so. Equation (20) expresses the productivity effects of social infrastructure, each of which is measured by $\varepsilon_{1}$ and $\varepsilon_{2}$. These are exogenous parameters but naturally differ according to the quality and attractiveness of government infrastructure provision. The better the quality or, the more innovative the project, the higher these values, and vice versa.

The structural form for labour productivity growth rate is

$$\begin{array}{*{20}c} {\hat{q}_{L} = \varepsilon_{0} + \varepsilon_{1} \left( {g_{S} - g} \right) + \varepsilon_{2} \left( {g_{S} - g_{L} } \right).} \\ \end{array}$$

(21)

Owing to the short-run steady state and production function, we have

$$\begin{array}{*{20}c} {u^{*} K = X = q_{L} L,} \\ \end{array}$$

(22)

and the labour demand rate is $g_{L} = g - \hat{q}_{L}$. By solving for the labour demand growth rate, $g_{L}$, we obtain:

$$\begin{array}{*{20}c} {g_{L} = \frac{{\left( {1 + \varepsilon_{1} } \right)g - (\varepsilon_{1} + \varepsilon_{2} )g_{S} - \varepsilon_{0} }}{{1 - \varepsilon_{2} }},} \\ \end{array}$$

(23)

and get the reduced form for labour productivity growth rate as:

$$\begin{array}{*{20}c} {\hat{q}_{L} = \left( {\frac{{\varepsilon_{1} + \varepsilon_{2} }}{{1 - \varepsilon_{2} }}} \right)\left( {g_{S} - g} \right) + \frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}.} \\ \end{array}$$

(24)

Meanwhile, the dynamics of the income distribution are derived using the mark-up pricing Eq. (5) as:

$$\begin{array}{*{20}c} {\dot{m} = \left( {1 - m} \right)\left( {\hat{p} - \hat{w} + \hat{q}_{L} } \right).} \\ \end{array}$$

(25)

We dynamically determine the income distribution based on the two types of Phillips curves (Asada et al. 2006), where wage and price inflation spiral each other. This framework details how income distribution is determined through cumulative changes in wages, prices, and labour productivity growth. According to the standard Phillips curve, the rate of change in wages and prices reacts to the deviation of the (un)employment rate from its natural rate, which does not accelerate inflation. We consider that the change in wage is indexed to price inflation and productivity growth, and an increase in the unit labour cost is passed through to price inflation by oligopolistic firms. Thus, wage and price inflation rates are partially determined by the tightness of the labour market, as measured by the employment gap. However, wage inflation is partly compensated by the indexation mechanism to ensure an adequate wage rate when productivity growth or price inflation occurs, whereas the dynamic cost pass-through mechanism partly determines price inflation.⁴ The structural forms for the wage and price inflation rates are:

$$\begin{array}{*{20}c} {\hat{w} = \mu_{1} \left( {e - e_{n} } \right) + \nu_{1} \left( {\hat{p} + \hat{q}_{L} } \right)} \\ \end{array}$$

(26)

$$\begin{array}{*{20}c} {\hat{p} = \mu_{2} \left( {e - e_{n} } \right) + \nu_{2} \left( {\hat{w} - \hat{q}_{L} } \right),} \\ \end{array}$$

(27)

where $e_{n}$ is the natural employment rate, which is assumed to be constant. Wage and price inflation rates react to the employment gap $e - e_{n}$ by $\mu_{1} > 0$ and $\mu_{2} > 0$, respectively. $\nu_{1} \in \left( {0,1} \right)$ represents the degree of wage indexation to price inflation and labour productivity growth. Similarly, $\nu_{2} \in \left( {0,1} \right)$ represents the degree of a firm's dynamic cost pass-through.

Solving these equations for price and wage inflation rates, we get

$$\begin{array}{*{20}c} {\hat{p} = \frac{{\mu_{2} + \nu_{2} \mu_{1} }}{{1 - \nu_{1} \nu_{2} }}\left( {e - e_{n} } \right) - \frac{{\nu_{2} \left( {1 - \nu_{1} } \right)}}{{1 - \nu_{1} \nu_{2} }}\hat{q}_{L} } \\ \end{array}$$

(28)

$$\begin{array}{*{20}c} {\hat{w} = \frac{{\mu_{1} + \nu_{1} \mu_{2} }}{{1 - \nu_{1} \nu_{2} }}\left( {e - e_{n} } \right) + \frac{{\nu_{1} \left( {1 - \nu_{2} } \right)}}{{1 - \nu_{1} \nu_{2} }}\hat{q}_{L} .} \\ \end{array}$$

(29)

By substituting these equations into Eq. (25), we obtain the dynamics of income distribution as:

$$\begin{array}{*{20}c} {\dot{m} = \left( {1 - m} \right)\left( {\left( {\rho_{p} - \rho_{w} } \right)\left( {e - e_{n} } \right) + \rho_{q} \left[ {\left( {\frac{{\varepsilon_{1} + \varepsilon_{2} }}{{1 - \varepsilon_{2} }}} \right)\left( {g_{S} - g} \right) + \frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right]} \right),} \\ \end{array}$$

(30)

where

$$\rho_{P} \equiv \frac{{\mu_{2} \left( {1 - \nu_{1} } \right)}}{{1 - \nu_{1} \nu_{2} }} >, 0$$

$$\rho_{W} \equiv \frac{{\mu_{1} \left( {1 - \nu_{2} } \right)}}{{1 - \nu_{1} \nu_{2} }} >, 0$$

$$\rho_{P} - \rho_{W} = \frac{{\mu_{2} \left( {1 - \nu_{1} } \right) - \mu_{1} \left( {1 - \nu_{2} } \right)}}{{1 - \nu_{1} \nu_{2} }}\frac{ > }{ < }0,$$

$$\rho_{q} \equiv \frac{{\left( {1 - \nu_{1} } \right)\left( {1 - \nu_{2} } \right)}}{{1 - \nu_{1} \nu_{2} }} > 0.$$

Equation (30) embodies the income distribution regime linking the change in employment rate and profit share. Profit share reacts both positively and negatively to a change in the employment gap. It increases, but the wage share decreases according to the positive gap in employment rates when $\rho_{P} - \rho_{W} > 0$, and vice versa when $\rho_{P} - \rho_{W} < 0$. Regarding the income distribution regime, we call $\rho_{P} - \rho_{W} > 0$ goods-market-led (GML) distribution regime and $\rho_{P} - \rho_{W} < 0$ labour-market-led (LML) distribution regime. The GML regime is shaped by the effect of large $\mu_{2}$ and $\nu_{2}$ and small $\mu_{1}$ and $\nu_{1}$, meaning that the rate of change in price inflation is larger than that in nominal wages when there is an employment gap. The driving force is pricing in the goods market, where firms have strong pass-through power of unit labour cost, and accordingly, price inflation is more sensitive to the employment gap than wage inflation. Conversely, the LML regime is shaped by the effect of large $\mu_{1}$ and $\nu_{1}$ and small $\mu_{2}$ and $\nu_{2}$, meaning that the rate of change in nominal wages is larger than in price inflation when there is an employment gap. The driving force is wage determination in the labour market, where workers have strong bargaining power to index price inflation and productivity gain; accordingly, wage inflation is more sensitive to the employment gap than price inflation.

Subsequently, with Eq. (22), the actual employment rate is determined by

$$\begin{array}{*{20}c} {e = \frac{{u^{*} K}}{{q_{L} N}},} \\ \end{array}$$

(31)

and the labour demand rate changes according to $g_{L} = g - \hat{q}_{L}$. Using Eqs. (24) and (31), the change in the employment rate is given by

$$\begin{array}{*{20}c} {\dot{e} = e\left( {g - \left( {\frac{{\varepsilon_{1} + \varepsilon_{2} }}{{1 - \varepsilon_{2} }}} \right)\left( {g_{S} - g} \right) - \frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }} - n} \right),} \\ \end{array}$$

(32)

which is the third state variable in our dynamic system. Equation (32) shows that the change in employment is led by private capital accumulation but is reduced by the productivity effect of social infrastructure accumulation during transitional dynamics.

Finally, we define the dynamics of the government debt ratio. In our model, the government sector depends on debt finance, as its expenditure always exceeds its revenue. The government also makes interest payments to capitalists based on debt stock. Then, the government budget constraint or the dynamics of debt is defined as:

$$\begin{array}{*{20}c} \begin{aligned} \dot{D} & = \left( {\theta_{C} + \theta_{I} } \right)\tau puK + i\delta pK - \tau puK \\ & = \left( {\theta_{C} + \theta_{I} - 1} \right)\tau puK + i\delta pK, \\ \end{aligned} \\ \end{array}$$

(33)

where $i$ is the nominal interest rate on the existing debt, and $\theta_{C}$ and $\theta_{I}$ are discretionary moves over unity according to the government's fiscal stance. Although we do not explicitly formalise monetary policy, the interest rate is exogenously controlled by the monetary authority.

Thus, the government debt ratio in real terms, $\delta = \frac{D}{pK},$ is an endogenous variable in our dynamic model,⁵ namely,

$$\begin{array}{*{20}c} {\frac{{\dot{\delta }}}{\delta } = \frac{{\dot{D}}}{D} - \hat{p} - g.} \\ \end{array}$$

(34)

Hence, with Eq. (33), we have

$$\begin{array}{*{20}c} {\dot{\delta } = \left( {\theta_{C} + \theta_{I} - 1} \right)\tau u + i\delta - \left( {\hat{p} + g} \right)\delta .} \\ \end{array}$$

(35)

We suppose the capacity utilisation, inflation, and output growth rates will follow the long-run steady-state values in our model in determining the dynamics of the debt ratio (35). By doing so, we elucidate how the stability of demand and distribution regimes may also guarantee long-term fiscal sustainability while maintaining mathematical traceability.

3.2 Existence of long-run steady state

Our differential equation system consists of the fast variables of capital composition $\chi$ (Eq. 19), profit share, $m$ (Eq. 25), and the employment rate $e$ (Eq. 32), where quantity adjustment in the goods market is instantaneous (Eq. 12). Based on these steady-state values, the slow variable of the government debt ratio eventually changes (see Eq. 35).

The long-run steady state for fast variables is a solution of the simultaneous equations for $\dot{\chi }=0,$ $\dot{m}=0,$ and $\dot{e}=0$. It follows from Eqs. (19), (25), and (32) that the nontrivial steady-state values for capital composition, profit share, and employment rate must satisfy the following relationship:

$$\begin{array}{*{20}c} {\chi^{*} = \frac{{\tau \theta_{I} }}{{sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)}}} \\ \end{array}$$

(36)

$$\begin{array}{*{20}c} {e^{*} = e_{n} - \left( {\frac{{\rho_{q} }}{{\rho_{P} - \rho_{W} }}} \right)\left( {\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right)} \\ \end{array}$$

(37)

$$\begin{array}{*{20}c} {\frac{{\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}{{sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}} = n + \frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }},} \\ \end{array}$$

(38)

and the capacity utilisation rate in the long-run is

$$\begin{array}{*{20}c} {u^{*} = \frac{{\alpha + \beta \left( {1 - \tau } \right)m^{*} }}{{sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}},} \\ \end{array}$$

(39)

which is also constant because profit share $m$ reaches a steady state. The conditions for the existence of long-run steady states are presented in Appendix 1. If these conditions are satisfied, Eqs. (36) and (38) simultaneously guarantee economically meaningful values for capital composition and profit share. In addition, the employment rate was independently determined using Eq. (37).

Finally, the dynamics of the government's debt ratio follow Eq. (35). When the steady state of the fast variables is stable, the economy has the following unique steady-state value for the government debt ratio:

$$\begin{array}{*{20}c} {\delta^{*} = \frac{{\left( {\theta_{C} + \theta_{I} - 1} \right)\tau u^{*} }}{{g^{*} - \left( {i - \hat{p}^{*} } \right)}}.} \\ \end{array}$$

(40)

The stability condition is provided in the next section and appendices, but when guaranteed, the long-run steady state can be characterised as follows. First, the model may generate multiple steady states, in which one is a WLG regime, and the other is a PLG regime depending on the initial value of the profit share.

Second, the long-run growth rate is equal to the natural growth rate, determined by the supply-side parameters. Using Eq. (38), the economic growth rate and the accumulation rate of social infrastructure are equally given by

$$\begin{array}{*{20}c} {g^{*} = n + \frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }},} \\ \end{array}$$

(41)

where supply-side parameters, such as autonomous productivity growth $\varepsilon_{0}$, productivity effect of social infrastructure $\varepsilon_{2}$ per employed worker and labour force, $n$ play a dominant role. Surprisingly, from a Kaleckian perspective, the demand parameters do not impact the economic growth rate, and the paradox of thrifts and costs does not arise in the long-run.

Third, the fiscal stance does not impact the long-run economic growth or employment rates. Different fiscal stances may temporarily affect the growth rates of social infrastructure and output during transitional dynamics, as Eq. (19) embodies. However, in the long-run steady state, the impacts of fiscal stance parameters are accommodated by changes in capital composition and income distribution, as Eqs. (36) and (38) indicate.

Fourth, productivity and wage-price parameters ultimately determine the actual employment rate and are independent of demand parameters. It persistently deviates from the exogenous natural rate as $\frac{{\rho_{q} }}{{\rho_{P} - \rho_{W} }}\hat{q}_{L}^{*}$. Additionally, the change in the steady-state value of labour productivity growth has different impacts on the profit share as a sign of $\rho_{P} - \rho_{W}$, namely, the type of income distribution regime.

Finally, the price and wage inflation rates remain constant and are given as:

$$\begin{array}{*{20}c} {\hat{p}^{*} = \frac{{\mu_{2} \left( {1 - \nu_{1} } \right)}}{{\mu_{1} \left( {1 - \nu_{2} } \right) - \mu_{2} \left( {1 - \nu_{1} } \right)}}\left( {\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right)} \\ \end{array}$$

(42)

$$\begin{array}{*{20}c} {\hat{w}^{*} = \frac{{\mu_{1} \left( {1 - \nu_{2} } \right)}}{{\mu_{1} \left( {1 - \nu_{2} } \right) - \mu_{2} \left( {1 - \nu_{1} } \right)}}\left( {\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right),} \\ \end{array}$$

(43)

and the real wage grows at the steady-state rate of labour productivity growth $\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}.$ If an economy has a GML regime $\rho_{P} - \rho_{W} > 0$, the steady-state inflation rates tend to be deflationary to a positive labour productivity growth rate, but if an economy has an LML regime $\rho_{P} - \rho_{W} < 0$, the steady-state inflation rates tend to be inflationary.

3.3 Stability of the long-run steady state

This section introduces the main propositions and discusses their economic implications; Appendix 2 provides detailed proofs.

Proposition 1

The steady state is a saddle path unstable in an economy with WLG/LML and PLG/GML regimes.

Proposition 2

The steady state of an economy under PLG/LML regimes is locally and asymptotically stable.

Proposition 3

If the degree of being wage-led is sufficiently weak in an economy with WLG/GML regimes, the economy's steady state is locally and asymptotically unstable. In contrast, if the degree of being wage-led is sufficiently strong, the local and asymptotic stability of the steady state depends on the elasticity of labour productivity to social infrastructure per physical capital, ${\varepsilon }_{1}$. Precisely,

• (1) When the elasticity of labour productivity to social infrastructure per physical capital ${\varepsilon }_{1}$ is sufficiently small (i.e. ${\varepsilon }_{1}<{\varepsilon }_{1N})$, the local and asymptotic stability of the steady state is guaranteed.

• (2) When the elasticity of labour productivity to social infrastructure per physical capital, ${\varepsilon }_{1}$, is moderate (i.e. ${\varepsilon }_{1N}<{\varepsilon }_{1})$ in this combination, the local and asymptotic stability of the steady state is guaranteed for ${{\varepsilon }_{1}\in (\varepsilon }_{1}, {\varepsilon }_{1}^{*})$, but not for ${\varepsilon }_{1}\in ({\varepsilon }_{1}^{*}, {\varepsilon }_{1D})$. Moreover, a limit cycle occurs by Hopf bifurcation for ${\varepsilon }_{1}$ sufficiently close to ${\varepsilon }_{1}^{*}$

Once the fast variables reach a steady state, the dynamics of the government debt ratio follow. When the stability conditions for fast variables are satisfied under the PLG/LML or WLG/GML regimes, we can also obtain Proposition 4.

Proposition 4

The long-run steady state of the government's debt ratio is locally and asymptotically stable as long as the nominal economic growth rate is higher than the nominal interest rate.

These propositions have economic implications. First, combining growth and distribution regimes is important for achieving economic stability. Proaño et al. (2011) and Nishi and Stockhammer (2020a, 2020b) identify the potential instability of an economy with WLG/LML regimes and an economy with PLG/GML regimes (Proposition 1). In this vein, our model elucidates that the government may avoid such instability by shifting the growth regime, causing a mismatch with the current distribution regime. For instance, a more proactive fiscal stance (i.e. rise in ${\theta }_{C}$, and ${\theta }_{I}$, or fall in $\tau$) ceteris paribus makes the nature of the growth regime more WLG style.⁶ Hence, the fiscal stance potentially matters in stabilising unstable combinations.

Second, the steady state of an economy with PLG/LML regimes is always stable (Proposition 2); however, the stability of WLG/GML regimes is conditional, where the productivity effect of social infrastructure per physical capital measured by ${\varepsilon }_{1}$ plays a crucial role (Proposition 3). It does not affect steady-state values because private capital eventually grows at the same rate as social infrastructure. However, the size of ${\varepsilon }_{1}$ has a lasting effect on labour productivity growth during transitional dynamics. If it is too strong, it may destabilise the steady state of WLG/GML regimes.

Furthermore, Proposition 4 indicates the Domar condition (Domar 1944). A higher economic growth rate helps to satisfy this condition, but the distribution regime matters because it affects the nominal economic growth rate through the steady-state inflation rate. This inflation rate tends to be deflationary in the GML regime, which makes it difficult to satisfy the stability condition. By contrast, as it tends to be inflationary in the LML regime, the stability condition is easier to guarantee. If the Domar condition is satisfied, the debt ratio converges to a certain level. Then, as the debt ratio and capital composition remain constant, social infrastructure, $S$, output, $X$, and government debt, $\frac{D}{p}$, grow at the same rate.

3.4 Numerical study

We have solved the analytical model for the main variables sequentially. However, as these variables change simultaneously in reality, it is reasonable to consider their long-run behaviours concurrently. Using a numerical study, this section considers simultaneous changes in endogenous variables and compares the initial and new steady states to determine their impact on the steady state.

Table 1 summarises the baseline values for the parameters that establish the PLG/LML and WLG/GML regimes. First, we provide these values for Eqs. (19), (25), (32), and (35). These parameters are chosen to ensure the stability of the fast variables and the Domar condition for the debt ratio for each combination. Second, using these parameters and some initial values, we derive tentative steady-state values for $({\chi }^{*}, {m}^{*}, {e}^{*}, {\delta }^{*} )$, which we subsequently use as the initial steady state for differential equation systems to be observed. The baseline parameters are selected to generate different stable regimes, and our purpose is to observe the transitional dynamics visually. Thus, we remark that even if some of them are chosen in light of economically reasonable values, they are not the ones obtained by calibration with real data.⁷ Finally, by pushing a 1% positive shock to the relevant parameters at time $t=10$, we visually follow the subsequent transitional dynamics. We show the transitional dynamics of PLG/LML and WLG/GML for $t=1000$ and $t=300$, respectively. These periods are chosen only for visually clear illustration, and the path eventually converges to a new steady state in both cases with time. We report the impacts of changes in fiscal stance parameters to save space and figures, and the effects of the other parameters are summarised in Table 2.

Table 1 Baseline parameters values for PLG/LML and WLG/GML regimes

Table 2 Results for comparative statics

Figure 1 illustrates the transitional path to a new steady state in the PLG/LML regimes after a 1% positive shock to ${\theta }_{C}$, ${\theta }_{I}$, and $\tau$ on the (a) economic growth rate, (b) capital composition, (c) profit share, (d) employment rate, and (e) government debt ratio. As Eqs. (37) and (41) indicate, increasing government expenditure propensities does not generally increase the long-run economic growth or employment rate. However, it positively impacts the capital composition $\chi$ while decreasing the profit share $m$. We confirm that capital composition and income distribution absorb changes in the fiscal stance parameters. The transitional path shows cyclical behaviours and a change in government investment propensity, ${\theta }_{I}$, generates the most fluctuating path, followed by consumption propensity, ${\theta }_{C}$, and tax rate $\tau .$ Graph (f) reports the combined plot for wage share ($1-m$) and employment rate ($e$) on the x- and y-axes, respectively. It shows clockwise cycles known as Goodwin cycles (Goodwin 1967). As Barbosa-Filho and Taylor (2006) and von Arnim and Barrales (2015) also show, a PLG regime with a Marxian profit squeeze mechanism (similar to the LML regime) can generate these cycles. Our analysis shows that such Goodwin cycles may also arise, even when a government sector exists in an economy with PLG/LML regimes. Finally, the government's debt ratio expands cyclically and reaches a higher ratio.

Fig. 1

Cyclical convergence to steady state in PLG/LML regimes

Figure 2 illustrates the transitional path to a new steady state in the WLG/GML regimes after the same shocks as those in Fig. 1.⁸ Long-run economic growth and employment rates are independent of increased fiscal stance parameters in these regimes. They merely generate cyclical behaviours of endogenous variables, of which the magnitude is the highest for ${\theta }_{C}$, followed by the consumption propensity, ${\theta }_{I}$, and tax rate $\tau .$ In contrast to the PLG/LML regimes, paradoxically, a rise in fiscal stance parameters negatively impacts the capital composition, $\chi$, while increasing the profit share, $m$. Additionally, Graph (f) plots the wage share and employment rate similarly to the above. It shows that the WLG/GML regimes generate anticlockwise cycles, in contrast to the PLG/LML regime cycles. Thus, this numerical study elucidates that an economy's growth and distribution regimes show different business cycles. Finally, the government's debt ratio cyclically increases due to a rise in expenditure propensities but decreases due to a higher tax rate.

Fig. 2

Cyclical convergence to steady state in WLG/GML regimes

3.5 Economic interpretation

Our analyses show that the effects of different shocks on cyclical behaviours in transitional dynamics and steady states depend on the combination of growth and distribution regimes. Fiscal policy has little impact on the long-run economic growth and employment rates. There are two theoretical reasons. First, when the employment rate is steady, the economic growth rate equals the sum of labour supply and labour productivity growth rates. Second, these parameters are determined independently of the fiscal stance. Therefore, supply-side parameters are crucial in determining long-run economic growth and employment rates.

What, then, is the government's role? Our model implies that the government is important for the following three reasons. First, a certain growth regime is required to stabilise an economy with a distributional regime. Thereupon, the fiscal stance is related to growth regime determination. For instance, a rise in tax rate weakens the profit effect, reducing the impact of profit share on the capacity utilisation rate in the capital accumulation process. These two effects make the nature of the growth regime more WLG-style. A rise in ${\theta }_{C}$ and ${\theta }_{I}$ also contributes to the WLG formation by enhancing the stagnationist nature and the crowding-in effect. The policies are particularly important when an economy currently has a GML distribution regime.

Second, the government can improve social infrastructure quality to enhance its accumulation's productivity effects. In particular, the size of ${\varepsilon }_{2}$ defines the growth and employment rates, which in reality, varies according to the efficiency of government investment in infrastructure. Even if the size of the social infrastructure per worker or capital is quantitatively identical, qualitatively more effective investment projects are associated with higher productivity. Our results show that the government should especially increase the quality of human-related social investments such as education, healthcare, and job training for long-term growth.

Finally, social infrastructure provision per se is important for building a resilient and sustainable market economy. For example, the institutional capital of healthcare enhances resilience against a pandemic, whereas environmental capital mitigates climate change. The model does not explicitly stipulate these effects, but they are important for sustainable economic growth. Depending on the combination of alternative regimes, fiscal stance differently affects capital composition that represents the relative size of social infrastructure. As Table 2 shows, a proactive fiscal stance matters: A rise in government expenditure increases the capital composition in the PLG/LML regimes, whereas a tax cut effectively increases it in the WLG/GML regimes. Although fiscal policy does not affect the long-run growth rate in both cases, it can provide a more affluent social infrastructure. This greatly benefits a market economy, as it works stably based on social and natural foundations.

We may name our result that social infrastructure provision by fiscal policies plays an important role in strengthening these foundations and building a resilient economy the Uzawa-Boyer proposition. Uzawa (2005) regards the social common capital, which includes social infrastructure, as crucial in maintaining human and cultural life, ensuring the right to live for citizens and restraining market instability. Boyer (2021) similarly advocates the anthropogenetic development mode, wherein an investment into healthcare, education, culture, or environment enhances economic resilience in times of crisis. Thus, both consider providing human-related social infrastructure and improving its quality not simply costs for the economy but enhances resilience and human development.⁹

4 Conclusion

We built a Kaleckian dynamic model of growth, distribution, and employment, in which government expenditure generates a crowding-in effect, social infrastructure provision, and debt accumulation. Our model analytically and numerically derives the following results. A combination of alternative growth and distribution regimes is important for stability. The WLG/GML and PLG/LML regimes may establish a stable steady-state, whereas the WLG/LML and PLG/GML regimes are unstable. Fiscal stance can partially shape the growth regime so that the government may avoid such unstable combinations. In addition, the cyclical behaviours of the WLG/GML and PLG/LML regimes are highly contrasting. When government debt changes, the Domar condition is required for stability. Moreover, the long-run economic growth rate depends not on demand or fiscal parameters but on supply-side parameters determining the natural growth rate.

Nevertheless, we conclude that the government still plays an important role in stabilising the economy, improving the quality of the social infrastructure, and achieving a resilient economy. First, the government may ensure stability by controlling the fiscal stance so that an appropriate growth regime is realised based on the distributional regime. Second, it can still improve the quality of the human-related social infrastructure to enhance its productivity effects and the associated rise in growth and employment rates. Finally, through changes in fiscal stance, social infrastructure provision contributes to enhancing the economy's social and natural foundations and, in turn, to resilient and sustainable growth. Hence, its provision matters both quantitatively and qualitatively to realise these purposes. Finally, we named these ideas the Uzawa–Boyer proposition.

This study has some remaining issues, both theoretically and empirically. Theoretically, alternative formalisation for investment function and productivity growth would be worth considering. Our investment function is based on the flow effect of government expenditure, but the stock effect of social infrastructure also induces the firms' investment. The more social infrastructure is provided, the more efficiently firms implement investment. It affects effective demand and labour and capital productivity growth as it increases the foundation for production.¹⁰ Such an alternative formulation may explain a proactive fiscal stance with a productivity growth effect and demand-led growth, which our model could not present. Our analysis shows that improving the quality of the human-related social infrastructure matters to establish economic growth, which we call the Uzawa-Boyer proposition. It is a theory-based proposition; hence, it should be subject to an empirical test to confirm if it is also true. This is worth pursuing since the social common capital and the anthropogenic development mode attract much attention in institutional and evolutionary economics.

Appendices

Appendix 1. Conditions for the existence of long-run steady states

The long-run steady state must simultaneously satisfy Eqs. (36), (37), and (38). As shown in Eq. (38), an economy may have two potential growth regimes: the shape of the economic growth rate is a convex quadrant in the domain of

$$\begin{array}{*{20}c} {m \in \left( {\frac{\tau }{{s\left( {1 - t} \right)}}\left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right),1} \right).} \\ \end{array}$$

(44)

Therefore, a unique profit share, $\tilde{m}$, switches the growth regime from WLG to PLG in this domain, and the actual growth rate (i.e. the LHS of Eq. (38)) takes the minimum value. The minimum growth rate is represented by:

$$\begin{array}{*{20}c} {g_{{\tilde{m}}} \equiv \frac{{\left( {\alpha + \beta \left( {1 - \tau } \right)\tilde{m}} \right)\left( {s\tilde{m}\left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}{{s\tilde{m}\left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}}.} \\ \end{array}$$

(45)

Hence, if

$$\begin{array}{*{20}c} {g_{{\tilde{m}}} < n + \frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \\ \end{array}$$

(46)

is satisfied in the above domain, the economy has two steady-state values for profit share and the associated growth regimes (i.e. WLG for smaller profit share and PLG for larger profit share). The value of a larger profit share must be less than unity to be economically meaningful. The capital composition is then determined by Eq. (36) according to the steady-state profit share. Independently, the employment rate is principally given by the parameters in Phillips curves.

Appendix 2. Proof of propositions 1 to 4

The dynamic system consists of Eqs. (19), (25), and (32), for which the Jacobian matrix, $J^{*}$, evaluated at the long-run steady state is given as follows:

$$j_{11} = \frac{{\partial \dot{\chi }}}{\partial \chi } = - \frac{{\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}{{sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}}$$

$$j_{12} = \frac{{\partial \dot{\chi }}}{\partial m} = - \frac{{s\left( {1 - \tau } \right)\tau \left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\theta_{I} }}{{\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{13} = \frac{{\partial \dot{\chi }}}{\partial e} = 0$$

$$j_{21} = \frac{{\partial \dot{m}}}{\partial \chi } = - \frac{{\left( {1 - m^{*} } \right)\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)^{2} \rho_{q} }}{{\tau \left( {1 - \varepsilon_{2} } \right)\theta_{I} \left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{22} = \frac{{\partial \dot{m}}}{\partial m} = - \frac{{\left( {1 - m^{*} } \right)s\left( {1 - \tau } \right)\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{23} = \frac{{\partial \dot{m}}}{\partial e} = \left( {1 - m^{*} } \right)\left( {\rho_{P} - \rho_{W} } \right)$$

$$j_{31} = \frac{{\partial \dot{e}}}{\partial \chi } = \frac{{e^{*} \left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)^{2} }}{{\tau \left( {1 - \varepsilon_{2} } \right)\theta_{I} \left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{32} = \frac{{\partial \dot{e}}}{\partial m} = e^{*} \left( {\left( {\frac{{1 + \varepsilon_{1} }}{{1 - \varepsilon_{2} }}} \right)\frac{\partial g}{{\partial m}} - \left( {\frac{{\varepsilon_{1} + \varepsilon_{2} }}{{1 - \varepsilon_{2} }}} \right)\frac{{\partial g_{s} }}{\partial m} } \right)$$

$$j_{33} = \frac{{\partial \dot{e}}}{\partial e} = 0$$

Note that, for simplicity, the steady-state value of $\chi^{*}$ is substituted into the Jacobian matrix elements to obtain these values.

The characteristic equation associated with the Jacobian matrix can be defined by

$$\begin{array}{*{20}c} {\lambda^{3} + a_{1} \lambda^{2} + a_{2} \lambda + a_{3} = 0,} \\ \end{array}$$

(47)

where $\lambda$ is the characteristic root. Coefficients $a_{1}$, $a_{2}$, and $a_{3}$ are given as:

$$\begin{array}{*{20}c} {a_{1} = g^{*} \left( {1 + \frac{{s\left( {1 - m^{*} } \right)\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}} \right)} \\ \end{array}$$

(48)

$$\begin{array}{*{20}c} {a_{2} = - \frac{{e^{*} \left( {1 - m^{*} } \right)\left( {1 - \tau } \right)\left( {\rho_{P} - \rho_{W} } \right)}}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)^{2} }}\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right),} \\ \end{array}$$

(49)

and

$$\begin{array}{*{20}c} {a_{3} = - \frac{{e^{*} \left( {1 - m^{*} } \right)g^{*} \left( {1 - \tau } \right)}}{{\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)^{2} }}\left( {\rho_{P} - \rho_{W} } \right)F\left( {m^{*} } \right),} \\ \end{array}$$

(50)

where

$$F\left( {m^{*} } \right) \equiv am^{*2} + bm^{*} + c\frac{ > }{ < }0$$

$$\omega_{1} \equiv s\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right) > 0$$

$$\omega_{2} \equiv \left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)\left( {s\alpha + \tau \beta \left( {\left( {1 + \gamma } \right)(\theta_{C} + \theta_{I} } \right) - 1} \right)) > 0$$

According to the Routh–Hurwitz criterion, the necessary and sufficient condition for the local asymptotic stability of the long-run steady state is:

$$\begin{array}{*{20}c} {a_{1} > 0, a_{2} > 0, a_{3} > 0,\; {\text{and}}\; a_{1} a_{2} - a_{3} > 0,} \\ \end{array}$$

(51)

where $a_{1} > 0$ is obvious under our assumptions, whereas the rest of the conditions are not a priori clear. Based on these preliminaries, we proceed to the proof of the propositions in order.

Proof of proposition 1. An economy with a WLG regime has $F\left({m}^{*}\right)<0$, whereas that with a PLG regime has $F\left({m}^{*}\right)>0$. In addition, ${\rho }_{P}-{\rho }_{W}<0$ in the LML regime, whereas ${\rho }_{P}-{\rho }_{W}>0$ in the GML regime. Based on the combination of these regimes, an economy with the WLG/LML and, or PLG/GML regimes does not satisfy ${a}_{3}>0$.

Q.E.D.

Proof of proposition 2. In an economy with PLG/LML regimes, we have $F\left( {m^{*} } \right) > 0$ and $\rho_{P} - \rho_{W} < 0$. Then, as the signs of both $\omega_{1}$ and $\omega_{2}$ are positive, $a_{2} > 0$ and $a_{3} > 0$ are ensured. Regarding $a_{1} a_{2} - a_{3}$, we have

$$\begin{array}{*{20}c} {a_{1} a_{2} - a_{3} = \frac{{e^{*} g^{*} \left( {1 - m^{*} } \right)\left( {1 - \tau } \right)\left( {\rho_{P} - \rho_{W} } \right)}}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)^{2} }} \cdot \Theta ,} \\ \end{array}$$

(52)

where

$\Theta \equiv F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right) - \left( {1 + \frac{{\left( {1 - m^{*} } \right)s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}} \right)\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right)$.

As $\rho_{P} - \rho_{W} < 0$ in the LML regime, the sign for $\Theta$ must be negative. Suppose $\Theta$ is negative; then, we have

$$\begin{array}{*{20}c} {F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right) < \left( {1 + \frac{{\left( {1 - m^{*} } \right)s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}} \right)\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right).} \\ \end{array}$$

(53)

It follows from this inequality that the following condition must be satisfied

$$\begin{array}{*{20}c} {\frac{{(1 - m^{*} )s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}} > - \left( {\frac{{F\left( {m^{*} } \right)\varepsilon_{2} + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}{{F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}} \right).} \\ \end{array}$$

(54)

Because $\omega_{1} > 0$, $\omega_{2} > 0$, and $F\left( {m^{*} } \right) > 0$ in the PLG regime, the sign of the RHS is always negative. In contrast, the value of the LHS is always positive, satisfying the above inequality. Hence, $a_{1} a_{2} - a_{3} > 0$ was also ensured under the PLG/LML regimes.

Q.E.D.

Proof of proposition 3. The steady-state values were independent of $\varepsilon_{1}$. In an economy with WLG/GML regimes, we have $F\left( {m^{*} } \right) < 0$ and $\rho_{P} - \rho_{W} > 0$. It follows from Eq. (49) that we need

$$\begin{array}{*{20}c} {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} < 0,} \\ \end{array}$$

(55)

to ensure $a_{2} > 0$. Because $\omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} > 0$, the absolute value of $F\left( {m^{*} } \right)$, which we call the degree of being wage-led, must be sufficiently large. In other words, if the degree of wage-led is sufficiently weak and $\omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} > \left| {F\left( {m^{*} } \right)} \right|$, we have $a_{2} < 0,$ and one of the stability conditions is violated.

Conversely, if the degree of being wage-led is sufficiently strong and $\omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} < \left| {F\left( {m^{*} } \right)} \right|$, we have $a_{2} > 0$ in the following conditions:

$$\begin{array}{*{20}c} {0 < \varepsilon_{1} < \varepsilon_{1D} \equiv - \left( {\frac{{F\left( {m^{*} } \right) + \omega_{2} \varepsilon_{2} }}{{\omega_{1} }}} \right),} \\ \end{array}$$

(56)

where the sign of $\varepsilon_{1D}$ is positive by a sufficiently strong degree of being wage-led.

Meanwhile, because $\rho_{P} - \rho_{W} > 0$ in an economy with a GML regime, we need $\Theta > 0$ to satisfy $a_{1} a_{2} - a_{3} > 0$. Thus, the following conditions must be guaranteed:

$$\begin{array}{*{20}c} {\frac{{(1 - m^{*} )s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}} > - \left( {\frac{{F\left( {m^{*} } \right)\varepsilon_{2} + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}{{F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}} \right),} \\ \end{array}$$

(57)

where the denominator of the RHS is negative for $a_{2} > 0$.

Let us consider both sides of inequality (57) as a function of $\varepsilon_{1}$, which does not affect the steady-state values of our system, and illustrate them on a plane coordinate in Fig. 3 below. Taking the value of $\varepsilon_{1}$ on the x-axis, the plot of the LHS has a positive slope and intercept. For the RHS of inequality (57), by differentiating it with respect to $\varepsilon_{1}$ in a row, we obtain

$$\frac{{\partial {\text{RHS}}}}{{\partial \varepsilon_{1} }} = - \frac{{F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right)\omega_{1} }}{{\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right)^{2} }} > 0,$$

and

$$\frac{{\partial^{2} {\text{RHS}}}}{{\partial \varepsilon_{1}^{2} }} = \frac{{ - 2F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right)\omega_{1}^{2} }}{{\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right)^{3} }} > 0.$$

Therefore, the plot of the RHS of Inequality (57) shows an increasing curve, asymptotically approaching the value of $\varepsilon_{1D}$.

Meanwhile, the sign of the numerator in (57) is not obvious, and we may have:

$$\begin{array}{*{20}c} {F\left( {m^{*} } \right)\varepsilon_{2} + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} \frac{ > }{ < }0,} \\ \end{array}$$

(58)

that is,

$$\begin{array}{*{20}c} {\varepsilon_{1} \frac{ > }{ < }\varepsilon_{1N} \equiv - \left( {\frac{{F\left( {m^{*} } \right) + \omega_{2} }}{{\omega_{1} }}} \right)\varepsilon_{2} = - \varepsilon_{2} ,} \\ \end{array}$$

(59)

where it is $\frac{{F\left( {m^{*} } \right) + \omega_{2} }}{{\omega_{1} }} = 1$. The following relationship is, therefore, confirmed:

$$\begin{array}{*{20}c} {\varepsilon_{1D} - \varepsilon_{1N} = - \frac{{F\left( {m^{*} } \right)}}{{\omega_{1} }}\left( {1 - \varepsilon_{2} } \right) > 0,} \\ \end{array}$$

(60)

and we always have $\varepsilon_{1D} > \varepsilon_{1N}$ in the WLG regime. The values of both sides in Eq. (57) are zero when $\varepsilon_{1} = \varepsilon_{1N} = - \varepsilon_{2}$ holds. Observing them together, we can find a unique value of $\varepsilon_{1}^{*} > 0$ between $0$, and $\varepsilon_{1D}$ guarantees that the value of LHS is equal to that of RHS, realising $a_{1} a_{2} - a_{3} = 0.$ These arguments are illustrated graphically in Fig. 3.

Fig. 3

Parameter configuration for stability condition $a_{1} a_{2} - a_{3}$

If $0 < \varepsilon_{1} < \varepsilon_{1}^{*}$, then both $a_{2} > 0$ and $a_{1} a_{2} - a_{3} > 0$ are guaranteed. If $\varepsilon_{1}^{*} < \varepsilon_{1}$, then both $a_{1} a_{2} - a_{3} > 0$ are violated. Moreover, if $\varepsilon_{1D} < \varepsilon_{1}$, then $a_{2} > 0$ is violated.

Thus, the following properties emerge according to the value of $\varepsilon_{1}$: for $0 < \varepsilon_{1} < \varepsilon_{1}^{*}$, the value of LHS is larger than that of RHS in Inequality (A14), and we have $a_{1} a_{2} - a_{3} > 0$. However, if $\varepsilon_{1}^{*} < \varepsilon_{1} < \varepsilon_{1D}$, we have $a_{1} a_{2} - a_{3} < 0$. Accordingly, there also exists a unique value of $0 < \varepsilon_{1}^{*} < \varepsilon_{1D}$, on which $a_{1} a_{2} - a_{3} = 0$ is established. Thus, a limit cycle occurs by Hopf bifurcation for $\varepsilon_{1}$ sufficiently close to $\varepsilon_{1}^{*}$ for the combination of WLG/GML regimes.

Q.E.D.

Proof of proposition 4

The dynamics of the debt ratio is

$$\begin{array}{*{20}c} \dot{\delta } = \left( {\theta_{C} + \theta_{I} - 1} \right)\tau u^{*} + i\delta - \left( {\hat{p}^{*} + g^{*} } \right)\delta , \end{array}$$

(61)

where the rates of capacity utilisation rate, inflation, and output growth as the fast variables will all follow the long-run steady-state values given by Eqs. (36), (37), and (38), respectively. By substituting them into Eq. (35), we obtain

$$\begin{array}{*{20}c} {\dot{\delta } = \frac{{\left( {\theta_{C} + \theta_{I} - 1} \right)\tau \left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)}}{{sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}} + i\delta - \left( {n + \frac{{\mu_{1} \left( {1 - \nu_{2} } \right)}}{{\mu_{1} \left( {1 - \nu_{2} } \right) - \mu_{2} \left( {1 - \nu_{1} } \right)}}\left( {\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right)} \right)\delta .} \\ \end{array}$$

(62)

It has the following unique steady-state value, $\delta^{*}$, shown by Eq. (40). The steady state of the government's debt ratio is locally and asymptotically stable if

$$\begin{array}{*{20}c} {\frac{{{\text{d}}\dot{\delta }}}{{{\text{d}}\delta }} < 0,} \\ \end{array}$$

(63)

is ensured at a steady state. Hence, we have

$$\begin{array}{*{20}c} {i < n + \frac{{\mu_{1} \left( {1 - \nu_{2} } \right)}}{{\mu_{1} \left( {1 - \nu_{2} } \right) - \mu_{2} \left( {1 - \nu_{1} } \right)}}\left( {\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right),} \\ \end{array}$$

(64)

which is equivalent to

$$\begin{array}{*{20}c} {i < \hat{p}^{*} + g^{*} .} \\ \end{array}$$

(65)

This shows that the nominal economic growth rate is higher than the nominal interest rate or the economic growth rate is higher than the real interest rate.

Q.E.D.