I am grateful to the two anonymous referees and editors of Revue de la régulation for their comments on the original manuscript.
Any remaining errors are my own.

Introduction

1This study builds a growth regime model to analyse demand, income distribution, and employment rate with an endogenous determination of Non-Accelerating Inflation Rate of Unemployment (NAIRU) based on the growth regime approach. It also examines the causes and effects of these endogenous dynamics from the Régulationist and post-Keynesian perspectives.

2A growth regime (or accumulation regime) is a core concept in the Régulation theory. It is defined as “the ensemble of macroeconomic growth processes that absorb distortions and imperfections constantly arising from the process itself, ensuring a general and coherent capital accumulation process” (Boyer, 2015, p. 61). Modelling growth regimes analytically identify the dynamics and diversity of economic growth patterns. They emphasise that institutional structures determine the key macroeconomic parameters.

3An institution that coordinates the key parameters within an economic growth regime is called ‘mode de régulation (régulation mode)’. The key parameters reflecting the effect of institutions not only constrain and expand the interactions of different economic behaviours but also pilot the dynamics of the growth regime. Thus, growth regime analysis investigates how the régulation mode can coordinate inherent conflicts and contradictions and realise stable economic growth. The novelty of Régulationists is synthesising the dynamics of effective demand, productivity growth, and income distributional conflict (e.g. Boyer, 1988; Boyer & Petit, 1988, 1991). Régulationists also consider how these dynamics vary depending on institutional and structural characteristics and why some spur economic growth, while others lead to crises.

4Growth regime analysis has also been developed by post-Keynesians (e.g. Lavoie & Stockhammer, 2013; Setterfield, 2016; Blecker & Setterfield, 2019). The post-Keynesian growth regime analysis formalises a wage-led or a profit-led demand (WLD/PLD) growth regime. The prototype of the wage-led growth regime has been formalised by Rowthorn (1981), among others. Certainly, a rise in wage share increases wage earners’ consumption demand, while it compresses profit share and firms’ investment demand. A wage-led growth regime is established when the increase in consumption demand is higher than the decrease in investment demand, thus accelerating economic growth. Bhaduri and Marglin (1990) critically develop a growth regime analysis to explain the mechanism behind the profit-led growth regime. A lower wage share decreases wage earners’ consumption demand, whilst the profit share increases. As the profit share increase positively stimulates investment demand, a higher economic growth rate is realised.

5We develop a growth regime analysis based on the Régulation theory and post-Keynesian modelling to consider the causes of economic growth and its effects on income distribution and employment. We also expand the analytical perspective to investigate the endogenous dynamics of NAIRU, the rate of unemployment at which inflation does not accelerate. NAIRU initially attracted attention due to the persistently high unemployment rate in Europe in the 1980s and early 1990s (Blanchard & Summers, 1986; Stockhammer, 2004). Later, the determinants of NAIRU attracted controversy again due to a resurgence in the unemployment rates of most OECD countries caused by the financial crisis of 2007 and 2008 (Heimberger, 2019).

6The controversy regarding NAIRU is broadly divided into two contrasting economics research streams. According to Layard, Nickell and Jackman (2005), NAIRU has both mainstream (or classical) and Keynesian elements. Sawyer (2001, 2002) claims that the mainstream explanation for NAIRU’s general characters is determined by exogenous supply-side factors in the long period, independent of demand shock and history. In this sense, NAIRU is similar to the notion of Friedman’s “natural rate of unemployment”, which “[…] is the level that would be ground out by the Walrasian system of general equilibrium equations, provided there is imbedded in them the actual structural characteristics of the labor and commodity markets […]” (1968, p. 8). The supply-side view is also applied in the New Keynesian model. For example, Carlin and Soskice (2014) present a simple model wherein “the equilibrium unemployment rate is determined of the structural or supply-side features of the economy that lie behind the wage-setting and price-setting curves” (p. 63).

7By contrast, the Régulation theory emphasises that the labour market outcome depends not only on the market mechanism but also on the conflict and compromise between capital and labour. Accordingly, the dynamics of wage and unemployment are affected by how the conflict and compromise are coordinated. For example, Boyer (2015, p. 127) explains the contrast in the dynamics of wage and employment between the competitive and Fordist modes of régulation.

8Meanwhile, post-Keynesians have little faith in NAIRU (Lavoie, 2014, p. 542). Even when they suppose its existence and estimate it, they retain critical attitudes to the mainstream assumptions. For instance, by critically illustrating the model of Layard, Nickell and Jackman (2005), Sawyer (2001) concludes that exploring how NAIRU may shift over time with aggregated demand and income distribution, as well as supply-side policies, is important. This study retains Sawyers’ (2001) caveat supposing the existence of NAIRU. Then, following post-Keynesian arguments, we investigate the dynamics of NAIRU in terms of demand and income distribution sides, as well as on the supply side and its endogenous change.

9Thus, for post-Keynesians, NAIRU is not a unique attractor determined by supply-side factors. NAIRU may also shift once a demand shock impacts the actual employment rate. Further, it is endogenous to the actual employment rate through several channels. For example, Hein and Stockhammer (2011, p. 127) emphasise insider bargaining for wage determination, deskilling of unemployment, conventional wage norms, capital shortages, and changes in the interest rate as the main channels. Empirically, Heimberger, Kapeller and Schütz (2017) find that NAIRU for 14 European countries is driven by cyclical factors, such as capital accumulation and boom-bust patterns in the housing markets. Jump and Stockhammer (2019) demonstrate that a one-percentage-point demand shock to the unemployment rate results in a long-run increase of the expected unemployment rate (i.e. a proxy for NAIRU) for Germany, France, and the UK. Heimberger (2019) provides evidence from OECD countries that real investment plays an important role in the endogenous determination of NAIRU. Even when taking a post-Keynesian approach, it should be noted that market institutions may also affect the endogenous NAIRU dynamics (Arestis, Ferreiro & Gómez, 2020).

10We theoretically approach demand, distribution, and employment with endogenous NAIRU dynamics. For our approach, Lavoie (2006), Proaño et al. (2011), Sasaki (2013), and Nishi and Stockhammer (2020a, 2020b) are particularly relevant. Proaño et al. (2011) and Nishi and Stockhammer (2020a, 2020b) identify stable and unstable feedback mechanisms between the output level and income distribution dynamics. However, because the models of Proaño et al. (2011) and Nishi and Stockhammer (2020b) exclusively examine the output level dynamics, the nature of the output growth rate cannot be explored. In contrast to these models, Lavoie (2006) analytically explains demand-led hysteresis in the natural output growth rate. Nonetheless, the effect of income distribution is excluded in Lavoie (2006). Moreover, as all of these models do not incorporate the employment rate, the feedback from the labour market to the goods market cannot be explained.

11Our study contributes to the literature as follows. In contrast to Lavoie (2006), Proaño et al. (2011), and Nishi and Stockhammer (2020a, 2020b), we incorporate not only the economic growth and income distribution but also their feedback linkage to the (un)employment rate. Therefore, a multi-feedback mechanism among demand, distribution, and employment is analytically identified. In this sense, our study is similar to that of Sasaki (2013). However, unlike Sasaki’s model, our model allows for the endogenous determination of NAIRU. Moreover, in our model, an endogenous change in NAIRU affects the income distribution and effective demand through the institutional structures for wage and price determination.

12Our approach is largely inspired by Nishi and Stockhammer (2020a, 2020b) in defining the demand and distribution regimes, and in numerically analysing the nature of growth regimes. We also accept the endogeneity of natural variables provided in their studies. However, this study substantially extends their model in the following areas. Nishi and Stockhammer (2020a, 2020b) exclude relevant variables crucial for growth regime analysis à la Régulationist: capital accumulation, economic growth, or employment rate dynamics do not exist. Moreover, due to the zero labour productivity, the cause and effect of the productivity growth regime cannot be examined. Our study thoroughly addresses these remaining problems. Furthermore, it considers the institutional structures of wage and price dynamics inspired by the Régulation theory, which were absent in Nishi and Stockhammer (2020a, 2020b). Precisely, this study (1) asks how demand and distribution regimes are established; (2) identifies the stability conditions for the multi-feedback mechanism among demand, distribution, and employment under different regimes; and (3) reveals how different degrees of NAIRU endogeneity affect the dynamic behaviour of macroeconomic variables within a growth regime.

13In the following sections, our approach seeks to synthesise the ideas of Régulationists and post-Keynesians. To build an institutional macroeconomic model (section 2), we start by defining the general framework for income generation, which is an open system with some degrees of freedom. Then, we close the system by introducing the key parameters or behavioural equations that reflect the prevalent institutional structures of the economy. The demand and distribution regimes are established under these institutional structures. Section 3 defines the economic growth regime as the ensemble of the demand and distribution regimes supported by institutional structures. By doing so, this study finds that the WLD and PLD regimes require particular institutional structures that affect wage and price determinations for the stability of the steady-state. It clearly presents the Régulationist implication that an economy has particular business cycles and growth patterns specific to the institutional structures shaping its demand and distribution regimes. Our model generates perpetual business cycles under WLD/LML and PLD/GML regimes when combined with a certain degree of NAIRU endogeneity. We also compare the theoretical results with empirical evidence given by recent studies. Finally, section 4 concludes the study.

1. Growth model with endogenous NAIRU dynamics

This section builds a macroeconomic model of growth, distribution, and (un)employment rates, including NAIRU. Our analytical procedure can be summarised as follows:

1. We introduce a general framework (accounting identities) for income generation. The general framework is an open system with some degrees of freedom. Then, the system is closed by determining the critical parameters or introducing behavioural equations that reflect the prevalent institutional structures of an economy. Therefore, the closure is chosen depending on the institutional information.
2. Principle endogenous variables can be regarded as functions of institutions. The effects of institutional coordination (i.e. régulation mode) drive the macroeconomic regularity of the principal variables (e.g. output growth, income distribution, and employment rate). The changes in the key parameters generate the variety and stability of the economic growth regime.
3. On this basis, we explore how and why an economy displays a specific economic growth trajectory and the instability associated with the institutional structures. This is an important research agenda in the Régulation theory (Boyer & Saillard, 2002, Part. V).

14Our institutional structures model obtains an idea from structural macroeconomics (e.g. Taylor, 2004; Setterfield, 2007). It also aims to reveal the macroeconomic dynamics generated by the particular structures of an economy. Thus, structural macroeconomics employs the institutional understanding of an economy to identify how a macro model should be closed (Taylor, 2004, p. 182).

16Régulationists and post-Keynesians share a common issue regarding growth regime analysis, and our integration of the post-Keynesian modelling approach with the Régulation theory fills the gap of both theories. Particularly after the 2008 global financial crisis, advancing the synthesis of post-Keynesian and Régulation research is expected to formalise endogenous destabilisation processes in an institutional configuration. Boyer (2011) introduces post-Keynesian studies (e.g. Nishi, 2012) that explore the endogenous dynamics of finance and growth, and underline the importance of understanding such models in light of institutional foundations. Meanwhile, Lavoie (2007), Setterfield (2011), and Stockhammer (2011) find that Régulationists and post-Keynesians have many similarities regarding the dynamics of capitalism. For example, the analytical contents of Régulationists and post-Keynesians display broad similarities in their treatments of the income-generating process, crisis-prone nature of capitalism, and institutional contingency of capitalist growth and development (Setterfield, 2011). Post-Keynesians do not assume that economic growth and income distribution are uniquely determined but rather commonly consider that these dynamics can vary depending on the institutional and structural characteristics of an economy. Further, post-Keynesianism has long highlighted the role of institutions as stabilisers of inherently unstable capitalist economies and radically uncertain worlds (Arestis, 1996). Nonetheless, post-Keynesians have not combined the modelling approach with the implication of the Régulation theory. Therefore, we attempt to fill the gap between these two research streams.

1.1. General framework

17In this study, we use the following notations: Y: actual output level, L: labour demand, K: capital stock, S: saving, C: consumption, I: investment demand, N: labour supply, u: capacity utilisation rate, g: capital accumulation rate (or output growth rate), q: labour productivity, p: price, w: nominal wage, m: profit share, e: employment rate, t: time. The variables with dot marks represent change (e.g. ẋ_t = dx_t/dt), and those with hat marks represent the growth rate (e.g. x̂_t = ẋ_t/x_t).

where we assume that the ratio of the maximum output to capital is unity. Then, we can use the output-capital ratio u_t as the capacity utilisation rate. Following the Keynesian theory, the labour demand is determined by effective demand and L_t = Y_t/q_t. The actual employment rate is

19Accordingly, after the actual employment rate is determined, the actual unemployment rate 1 – e_t can be obtained. This relationship also holds for the natural employment rate e_nt; after the natural employment rate is determined, the natural unemployment rate 1 – e_nt can be derived. As e_nt is a natural employment rate wherein wage and price inflations are non-accelerating, the natural unemployment rate 1 – e_nt is a non-accelerating inflation rate of unemployment. Therefore, we consider the dynamics of NAIRU based on the actual and natural employment rates and show that their dynamic patterns are also specific to the institutional structures of an economy.

The total expenditure is the sum of consumption and investment. The goods market equilibrium is

We assume that the goods market is cleared by an instantaneous quantity adjustment. Then, the IS balance is S_t = I_t, or

Focusing on the dynamics of the income distribution, the total income p_tY_t is distributed to the wage bill W_tL_t and profit r_tp_tK_t Then, the profit share is

by which, we can also decompose the profit rate as follows:

20Thus, the general framework provides accounting identities for income generation. However, we still have unknown variables (u_t, m_t, r_t, g_t, e_t, e_nt). We explore how these endogenous variables are determined.

1.2. Modelling demand regimes

21The demand regime is defined as a dynamic causality from the income distribution to the output (economic) growth rate. This aspect is realised using Equation (4), and we characterise the saving and investment behaviours. We assume that saving (or consumption demand) follows the Cambridge expression of aggregate demand (Lavoie, 2014, p. 311): workers consume all their wage income, while capitalists consume only part of their profit income. Then, the total savings normalised by the capital stock are

Where s  ∈ (0.1] is the constant saving rate of capitalists.

22We employ the linearised Bhaduri and Marglin (1990) model to introduce the accumulation function as

where α reflects capitalists’ animal spirits for investment, β represents the sensitivity to the profit share, and γ approximates the accelerator effect on investment. The capital accumulation pattern (i.e. the relative sizes of α, β, and γ) plays a crucial role in dividing the demand regime into WLD or PLD regimes. Investment is not only a source of effective demand but also creates new capital stock. Therefore, K̇_t = I_t is established, and g_t = K̇_t/K_t measures the capital accumulation rate. The capacity utilisation rate accommodates the gap between investment and saving at each time point. Then, it is instantaneously determined using Equations (7) and (8) as

23where we assume that the Keynesian stability condition is satisfied and the denominator is positive. A simple calculation for Equation (9) shows that a decrease in the savings rate and wage share raises the capacity utilisation rate . The former is a demand-led expansion, whereas the latter is called the stagnationist demand formation (Lavoie, 2014).

By substituting the capacity utilisation rate into Equation (7), the capital accumulation rate is

where shows the property of the demand regime. If f(m_t) is positive, the economy has a PLD regime, and a WLD regime in the opposite case. Note that f(m_t) is a convex function regarding m_t. As the Keynesian stability condition is imposed, the demand regime can be defined for , under which f(m_t) has the following characteristics:

25From these inequalities, if then is obtained, and the economy has a WLD regime only. Conversely, if , there exists a profit share such that . Accordingly, is obtained for and the economy has a WLD regime. By contrast, is obtained for , and the economy has a PLD regime. Therefore, there exists a distribution value that switches the demand regime.

26Table 1 summarises the driving force for the demand regime formation based on inequality (13). A WLD regime is more likely to be established when (i) capitalists’ animal spirits to realise the desired capital accumulation is active (i.e. a large α); (ii) the accelerator effect γ is large, but capital accumulation is less sensitive to profit (i.e. β is small); (iii) capitalists’ consumption is large, and their saving rate s is low. In summary, an economy has a WLD regime when it is more sensitive to buoyant demand conditions. Conversely, a PLD regime is more likely to be established under the opposite conditions. That is, when (iv) the capitalists’ animal spirits are less active (i.e. a small α); (v) the accelerator effect γ is small, but capital accumulation is more sensitive to profit (i.e. β is large); and (vi) the savings rate of capitalists s is high. In summary, an economy has a PLD when it is more sensitive to profitability. When a PLD regime exists, a change in the profit share endogenously switches the demand regime of an economy between WLD and PLD at the threshold .

Table 1. Driving forces for demand regime formation

Demand regime	Driving forces for demand formation
WLD regime:	Buoyant animal spirits (large α)
	Strong accelerator effect (large γ) and restricted profit effect (small β)
	Capitalists’ active consumption (low s)
PLD regime:	Stagnant animal spirits (small α)
	Weak accelerator effect (mal γ) and strong profit effect (large β)
	Capitalists’ inactive consumption (high s)

Source: author

27Based on the above argument, we identify two types of demand regimes.

Definition 1. We define as a PLD regime and as a WLD regime.

Similarly, the profit rate changes proportionally to the capital accumulation rate because of the Cambridge equation, sr_t = g_t. Then, we have

28Under the PLD regime, the profit rate increases with a higher profit share, whereas it increases with a higher wage share under the WLD regime. As the profit share is , the demand regime is linked to productivity growth through income distribution.

1.3. Modelling income distribution regimes

29The income distribution regime is defined as the dynamic causality from employment rates to income distribution. Income distribution is also a result of how productivity gains are distributed to workers and capitalists and how institutional structures affect their distributional conflicts. We extend the conflictive wage-price spiral model of Nishi and Stockhammer (2020a) to formalise the income distribution regime by including the productivity growth dynamics.

The structural forms for wage and price inflation are as follows:

Workers and capitalists dynamically change wages and prices to close the aspiration gap (first term). Simultaneously, the workers aim to link price inflation and productivity growth to wage increases, whereas capitalists aim to pass through a rise in the unit labour cost to the price increase (the second term), resulting in wage and price changes. The magnitudes of 1 – θ and θ ∈ (0,1) represent the relative strength of the bargaining power of workers’ and capitalists’, respectively. As workers’ (capitalists’) bargaining power increases, they have a stronger influence on the wage (price) increase, given the aspiration gap for income distribution. The second term describes the wage and price determination based on abstracted institutional and market structures. κ_w ∈ (0,1) simultaneously measures the degree of wage indexation to labour productivity growth and price inflation. It implies that workers strive to ensure an adequate wage rate when there is productivity growth or price inflation. Using the Régulation theory terminology, this value can reflect the régulation mode for the wage-labour nexus, namely the direct and indirect determinants of the wage income through the bargaining compromise between the capital and labour relationship. κ_p ∈ (0,1) represents the pass-through rate from the change in the unit labour cost to the price. Using the Régulation terminology again, this value can reflect the régulation mode for the competition form, namely how the units of accumulation (i.e. firms) compete with each other. A high value represents a more monopolistic price determination, and capitalists can easily mark prices over the average production costs; conversely, a low value represents a more competitive régulation mode.

30Nishi and Stockhammer (2020a) formalise aspirations (i.e. the actual norms that workers and capitalists seek) for the profit shares of workers (m_w) and capitalists (m_p) in terms of the output gap. Rather than reformulating the output gap, we reformulate the aspirations for income share in terms of the employment gap:

32Finally, we develop Boyer’s (1988) and Boyer and Petit’s (1988, 1991) models to build a productivity growth regime. The models include the effects of economic growth and changes in employment as follows:

where ρ₀(е_t) has ρ₀'(е_t) > 0 representing the labour-saving technical change. This effect embodies capitalists adopting a new production technique or innovations that save labour inputs. It also prevents a reduction in the rate of profit because of an increase in the wage costs triggered by increased employment (Sasaki, 2013; Tavani & Zamparelli, 2017). Meanwhile, ρ₁ simultaneously measures the mixed impact of capital deepening and the effect of dynamic increasing returns to scale. The idea is that new capital equipment embodies new technologies, enhancing productivity growth. Additionally, as stated by Smith (1976 [1776]), Young (1928), and Kaldor (1966; 1970), the larger the market expansion (i.e. economic growth), the higher the productivity growth through the more profound division of labour and specialisation. Productivity growth is not only a cause for the change in income distribution that impacts the dynamics of the demand regime but also an effect of output growth.

33By taking the log differentiation for the profit share, the dynamics of the income distribution are given as

Rearranging Equations (15)–(20), we obtain the dynamics of the profit share:

where

34The dynamic relationship between income distribution and the employment gap in Equation (21) defines the income distribution regime. Although the signs of and are necessarily positive, that of can be either positive or negative and depends on the relative bargaining power and the size of the indexation and pass-through in the wage and price spirals. Since it is concerned with the different impacts of the tightness in labour and (approximated) goods markets on the changes in income distributional norms, institutions matter how income distribution is realised over a conflict. Based on this argument, we identify two types of income distribution regimes.

35Definition 2. A labour market-led (LML) distribution regime refers to the case in which an increase in the employment gap leads to a decrease in the profit share. That is, holds. By contrast, a goods market-led (GML) distribution regime refers to the case in which an increase in the gap leads to a rise in the profit share. That is, holds.

Table 2. Income distribution regimes and institutional foundations

Distribution regime	Institutional structures determining parameters
LML regime:	Competitive wage formation and workers’ aspiration is elastic: large μw
	Strong wage indexation to price and productivity growth: large κw
	Strong bargaining power of workers: a small θ
GML regime:	Competitive price formation and capitalists’ aspiration is elastic: large μp
	Large path-through of unit labour cost on price: large κp
	Strong bargaining power of capitalists: large θ

Source: author

36Our definition of LML and GML distribution regimes is based on the sign of the coefficient of employment gap, which is analogous to Nishi and Stockhammer (2020a, 2020b), who defined them by the output gap. Here, we highlight how the formation of income distribution regimes is associated with institutional and market structures affecting wage and price dynamics in Table 2.

37When workers’ bargaining positions are dominant (i.e. large1 – θ), and their wage share aspirations largely respond to an employment gap (i.e. large μ_w), the economy probably has an LML regime. It is called the profit-squeeze distributive regime (Barbosa-Filho & Taylor, 2006) because a rise in overall economic activity levels decreases the profit share. This property is enhanced when capitalists’ aspirations hardly change to an employment gap (i.e. small μ_p), and the wage indexation to productivity growth and price inflation is stronger than the pass-through power of capitalists (i.e. large κ_w and small κ_p). Under the opposite conditions, an economy is more likely to have a GML regime. That is, this happens when capitalists’ bargaining position is dominant (i.e. large θ), and their profit share aspirations broadly change with the employment gap (i.e. large μ_p). It is enhanced when workers’ wage share aspirations hardly change to an employment gap (i.e. small μ_p), and the pass-through power of capitalists’ is stronger than that of workers (i.e. large κ_p and small κ_w).

1.4. Employment and endogenous dynamics of NAIRU

38The second state variable is the employment rate е_t = Y_t/q_tN_t, wherein the labour force grows at an exogenously given growth rate, . Hence, the change in the employment rate follows:

As the actual employment rate changes proportionally to the output growth rate, the demand regime also determines the change in the employment rate in response to a change in the income distribution.

39Finally, we introduce an endogenous determination of the natural (un)employment rate (i.e. the proxy of NAIRU) into our model. When the actual employment rate is below the natural one, the natural employment rate declines. Conversely, when the actual employment rate is above the natural rate, the natural employment rate rises. Formalising these dynamics in a simple adaptive manner, the change in the natural employment rate is

Equation (23) shows that the change in the natural employment rate is demand-led (Sawyer, 2001; Jump & Stockhammer, 2019; Heimberger, 2019). After the output growth is realised through capital accumulation, it changes the actual employment rate in Equation (22). Subsequently, the actual employment rate affects the natural employment rate in Equation (23), which, in turn, provides feedback to the income distribution in Equation (21). The natural rate is also institutionally contingent, determined by demand and income distribution regimes. The positive Φ measures the degree of endogeneity for the natural rate. A small value implies that the evolution of the natural employment rate is sticky to the current rate, while a large value indicates that it changes quickly according to the historical gap.

2. Growth regime analysis

2.1. The steady state and nature of the demand and distribution regimes

40Our model had six unknown variables (i.e.u_t,m_t,r_t,g_t,e_t,e_nt) and six equations, i.e. (9), (10), (14), (21), (22), and (23) to close the model. The steady state is a situation under which is realised. Hence, we can respectively derive the steady state conditions for the actual and natural employment rates and profit share as follows:

where asterisks denote the steady state value. As the capacity utilisation rate is also constant in the steady state, the output growth rate is the same as the capital accumulation rate, . We call Equations (24)–(26) the economic growth regime, the ensemble of the demand and distribution regimes piloted by the institutions affecting wage and price determinations. Additionally, the wage and price inflation rates are

41As the wage and price inflation rates are also constant at , we can call and the non-accelerating actual and natural rate of employment, respectively. Hence, the associated unemployment rate is , which is NAIRU in our model. Note that the inflation rate is not always zero but stable at NAIRU. The stability conditions are presented in the following subsection. Table 3 shows that a shock on the relevant parameters has similar and different impacts on the endogenous variables for the stable cases. The details are provided in Appendix 1.

Table 3. The impacts of change in a parameter under different growth regimes

An increase in		WLD/LML	WLD/GML	PLD/LML	PLD/GML

Note: A rise in g_t is led by a lower saving rate and higher values for α, β, and γ.

Source: author

42First, a rise in the economic growth rate (g_t) increases both the rates of the actual and natural employment and the profit share. Second, a rise in the growth rate of the labour force (n) decreases both the rates of actual employment and the profit share. Finally, the change in prevalent norms has contrasting effects, depending on the demand regime.

43Our analysis may highlight a cause of secular stagnation from the perspectives of post-Keynesian and Régulationist. The former emphasises the negative effect of a decrease in wage share in the neoliberalism age as a cause of the secular stagnation (Petach & Tavani, 2020); the latter attributes it to the erosion of stable wage–labour nexus by the collapse of the Fordist capital-labour accord and the increasingly important role of the financial market with change in political class coalition (Boyer, 2010). Although the present model does not include the alliance of managers and financiers, a theoretically possible implication is as follows: If wage share is squeezed to ensure a high return to financiers under a WLD regime economy, the actual and natural employment rates, and the economic growth rate, persistently shift downward.

2.2. Stability, instability, and cycles

44We identify the conditions for local asymptotic stability, instability, and the existence of limit cycles by a Hopf bifurcation for the steady state derived for each growth regime. As the proof for the proposition is lengthy, we provide it in Appendix 2.

Proposition 1. In an economy with a PLD regime, if the causal impacts among profit share, output growth, and productivity growth are strong (i.e. sensitive to each other), the steady state is saddle-path unstable.

When this relationship is weak, we still have the following propositions, depending on the combination of demand and distribution regimes.

Proposition 2. In an economy with PLD/GML or WLD/LML regimes, if the causal impacts among profit share, output growth, and employment rate are weak, the steady state of the economy is locally and asymptotically stable for any positive value of Φ.

By contrast,

Proposition 3. If the causal impacts among profit share, output growth, and employment rate are strong under PLD/GML regimes, there exists a positive value such that the steady state is locally stable for , locally unstable for , and a limit cycle occurs by a Hopf bifurcation for a Φ sufficiently close to . Similarly, if the causal impacts are strong under WLD/LML regimes, there exists a positive value such that the steady state is locally stable for , locally unstable for , and a limit cycle occurs by a Hopf bifurcation for a Φ sufficiently close to .

Proposition 4. In an economy with PLD/LML or WLD/GML regimes, the steady state is locally asymptotically stable for any positive value of Φ.

45The propositions are arranged comparably with Nishi and Stockhammer (2020a), who demonstrate that instability in PLD/LML or WLD/GML regimes may arise from natural output endogeneity and speed of actual output adjustment. Meanwhile, the instability in our model of these regimes arises from the endogeneity in the natural employment rate. Naturally, this difference is from the structures of the models. Whilst Nishi and Stockhammer (2020a) allow for a dynamic adjustment of the goods market, the adjustment is instantaneous in our model. Nonetheless, the instability remains in our model because of the reasons we consider below. Additionally, the PLD/LML regimes in Nishi and Stockhammer (2020a) are unconditionally stable, whereas the PLD regime in our model has potential instability. This result is because our model allows for endogenous productivity growth, thus promoting profit-led growth and increased profit share, whereas their model does not particularly have productivity growth dynamics.

46It would be pertinent to consider the economic interpretation in more detail. Proposition 1 indicates that the PLD regime has the following potential instability. A positive shock to the profit share augments the output growth rate, thereby increasing the employment rate under this regime. Although the initial increase in the profit share decreases due to the negative self-feedback, the rise in output and employment growth rates boosts productivity growth. Subsequently, it pushes up the profit share. When this causal impact is dominant and strong, there is positive feedback among the profit share, labour productivity, and output growth as follows:

and WLD/LML regimes involve potential instability in that

47Proposition 2 shows that, if the relationships among profit share, output growth, and employment rate are weak (i.e. the first line of (PLD/GML) and (WLD/LML)), ceteris paribus, the potential positive feedback from profit share to output growth and employment rate to profit share is restrained. Then, the dynamics converge to a steady state with a negative self-feedback of the profit share.

48Conversely, Proposition 3 shows that, if these relationships (i.e. the first line of (PLD/GML) and (WLD/LML)) are strong for these regimes, the degree of endogeneity in the natural employment rate may restrain the instability. Assume that the natural employment rate largely changes endogenously to the actual rate under the PLD/GML regimes. An increase in the profit share raises the output growth rate, thus positively stimulating the employment rate. Then, because the natural employment rate is largely led by the actual rate, the subsequent fall in the profit share is proportionally large. Hence, when the endogenous change in the natural employment rate is strong, it stabilises the economy with PLD/GML regimes. Similar reasoning applies to the stabilisation process for WLD/LML regimes with .

49However, Proposition 3 also implies that, when the endogeneity degree of the natural employment rate is moderate, given the strong feedback among profit share, output growth, and employment rate (i.e. from the first lines), the economy may experience perpetual cycles. Intuitively, the cycles of the PLD/GML regimes can be explained as follows. A rise in the profit share augments the output growth rate, which, in turn, increases the employment rate. The rise in the employment rate has two impacts on the profit share: it directly increases the profit share and indirectly decreases the profit share through the endogenous change in the natural rate (i.e. the second line). If the former and the latter occur alternatively, and due to the moderate change in the natural rate, these impacts lead to perpetual cycles. The same token can explain the WLD/GML regime case.

50Finally, Proposition 4 states that an economy with PLD/LML or WLD/GML regimes realises a stable, steady state. It is mainly because of the negative feedback between profit share and output growth in these cases. For instance, when an economy has a PLD regime, the LML regime facilitates stable dynamics. A positive shock to the profit share increases the output growth and actual employment rate, under which the LML regime decreases the profit share. The decrease in the profit share, in turn, lowers the output growth and the employment rate. Then, it pushes the profit share again and, thus, the dynamics converge to the steady state. The stable dynamics of WLD/GML can be explained by the same logic.

51Propositions 2 and 3 show that the natural rate adjustment speed (Φ) and the elasticity of the distribution norms (μ_w and μ_p) are critical parameters of stability. Principally, these key parameters are exogenously and independently determined, and it is by chance that they are combined to achieve stable growth. Therefore, these propositions reveal that a certain régulation mode is required for the stability specific to each regime.

2.3. Numerical analysis

We visually confirm the configuration of the cycles for PLD/GML and WLD/LML regimes by numerical examples.

Table 4. Parameters for different growth regimes

(a)PLD/GML regime

Demand regime

Distribution regime

Others

b)WLD/LML regime

Demand regime

Distribution regime

Others

Source: author

52The procedure for the numerical analysis follows that of Nishi and Stockhammer (2020a). Tables 4 (a) and (b) summarise the parameters used to build an economy with PLD/GML and WLD/LML regimes, respectively. The parameters in (a) and (b) for the demand regimes generate PLD and WLD regimes, respectively. As the WLD regime can arise only when the steady state profit share is low, the target profit shares of workers, , and capitalists, , are set accordingly low.

53The parameters for the distribution regimes are similarly set, and the GML and LML regimes are generated exclusively by μ_w and μ_p to highlight their impacts. The steady state values generated by these parameters for the PLD/GML regimes are , whereas those generated for the WLD/LML regimes are .

54We set the degree of endogeneity for natural employment rate Φ to 0.003556 for the PLD/GML regimes, which is sufficiently close to the Hopf bifurcation value. Setting the initial values to (m_o,e_o,e_no) = (0.350,0.950,0.950), Figure 1 projects the solution paths in the three dimensions for all variables. Figure 2 shows the two dimensions of the actual employment rate and the wage share to allow the comparison with the literature. Similarly, we set the degree of endogeneity for natural employment rate Φ to 0.032851 for the WLD/LML regimes. Setting the initial values to (m_o,e_o,e_no) = (0.214,0.9272,0.9272), Figures 3 and 4 are similar to Figures 1 and 2, respectively.

55Each figure illustrates the solution paths from t = 100 to t = 10000, showing that each variable traces cyclical paths. A small and temporary shock hitting the system leads the economy to persistent cycles, wherein the natural (un)employment rate also follows endogenous fluctuations. Then, we can visually confirm that the wage share and employment rate move in a contrasting manner. The wage share and employment rates move in an anti-clockwise manner under the PLD/GML regimes (Figure 2), whereas their evolution is clockwise under the WLD/LML ones (Figure 4).

56Regarding Figures 1 and 2, Barbosa-Filho and Taylor (2006) and von Arnin and Barrales (2015) show that there can exist an anti-clockwise cycle between the wage share and output levels or capacity utilisation rate which they call Goodwin cycles. They explain anti-clockwise cycles by the downward sloping effective demand and upward sloping distribution curve, which correspond to the PLD/LML regimes as per our definition. However, Nishi and Stockhammer (2020a) show that such cycles can also be established by PLD/GML regimes for the output level. Unlike Nishi and Stockhammer (2020a), our model is more successful in comparing the cyclical nature with Goodwin’s (1967) original configuration as it explicitly incorporates the employment rate and income distribution.

57Figures 3 and 4 indicate that the WLD/LML regimes produce clockwise cycles. This result is also similar to that of Nishi and Stockhammer (2020a). However, our model shows that clockwise cycles are plausible between the wage share and employment rates in a growth process, which Nishi and Stockhammer (2020a) do not consider.

58Finally, the numerical study presents an important implication for the Régulation theory in that business cycles and growth patterns are associated with the institutional structures of the economy. Régulationists ask how and why the capitalist economy, which is fundamentally composed of conflicts and contradictions between different actors, can experience high and stable economic growth and then fall into a crisis. A general answer is that growth and crises depend on the institutions in an economy, as demonstrated in our model. Macroeconomic performance differs depending on the demand and distribution regimes. These regimes are generated by the institutional structures that determine key macroeconomic parameters. Then, these parameters shape the trajectory of the growth, distribution, and employment dynamics. Consequently, they generate highly contrasting dynamics, as we present both analytically and numerically.

2.4. Discussion

This section discusses the representative economies by comparing recent studies on both demand/growth and distribution regimes (see Table 5).

Table 5. Empirics of demand and distribution regimes

Authors	Country and periods	Demand/Growth and Distribution regimes
Barbosa-Filho and Taylor (2006)	US (1948—2002)	PLD and LML
Tavani et al. (2011)	US (1956—1998)	PLD and LML
Rezai (2013)	US (1953–2011)	PLD and LML
Barrales-Ruiz et al (2021)	US (1949—2020)	PLD and LML
Sonoda (2017)	Japan (1977—2007)	PLD and GML
Sonoda (2021)	Japan (1977—2007)	PLD and GML

Source: author

60Sonoda (2017, 2021) analyses the Japanese demand and distribution regimes closely to the present model. By estimating these regimes that comprise two types of the Phillips curve, where the natural employment rate is also included, Sonoda (2017, 2021) reveals that the Japanese economy exhibits the PLD/GML regimes for 1977–2007. Additionally, Sonoda (2021) find that an active discretionary fiscal policy was not sufficient to support the effective demand, and that the Japanese economy had evolved with potential instability.

61In terms of reality, our model serves to consider whether the existence of these regimes is theoretically plausible and to identify supporting institutions and (in)stability mechanisms involved in each regime. The historical succession of one regime to another occurs through the changes in the key parameters in the model. For example, if the growth regime consists of the PLD/LML regimes, such as the US case partially, it is because of the strong profit and weak accelerator effects on investment for the PLD regime and on account of elastic aspiration of workers, effective wage indexations, and strong bargaining power of the workers for the LML regime. Conversely, if the growth regime has a GML regime, such as the Japanese experience, it is because of the elastic aspiration and strong bargaining power of capitalists, and effective path-through of unit labour cost.

62An important lesson is that each parameter that supports a regime is not a priori combined to functionally realise high and stable growth. That is, a growth regime consisting of demand and distribution regimes results from an unintended combination of institutional effects. Specifically, the Régulation theory originally emphasises the socio-political compromises (Boyer, 2015), and the forms of institutions are dependent on the interest of different social groups (Amable, 2016). For instance, if capitalists and managers promote wage suppression, it inevitably leads to economic stagnation and a higher unemployment rate in an economy with the WLD regime. Conversely, if workers want to keep a strong wage indexation similar to the end of the wage-led Fordism, it forms the LML regime. Consequently, such an economy with WLD/LML regimes may experience unstable cycles or profit-squeeze stagnation. Thus, our model confirms the Régulationist implication that institutional structures shape specific crises.

Conclusions

63This study analytically investigated the demand, distribution, and employment rate with an endogenously determined NAIRU based on the growth regime approach. Our post-Keynesian model incorporates the Régulation theory: the accumulation and growth processes are institutional and historical contingencies of capitalism.

64The questions posed in the introduction can be answered. First, the WLD or PLD regime is established by the parameters determining saving and investment demand, which is common with post-Keynesian modelling. However, our model showed that under the GML and LML regimes, the distribution regime is highly shaped by the institutional structures that affect wages and prices. Therefore, we identified four types of growth regimes: PLD/GML, PLD/LML, WLD/GML, and WLD/LML. Second, the multi-feedback mechanism among demand, distribution, and employment under the different growth regimes requires particular causal impacts among the principal variables to realise stability. Further, the relationship between profit share, output growth, and productivity growth must be weak for an economy with a PLD regime.

65Additionally, if an economy with PLD/GML or WLD/LML regimes has a strong relationship between profit share, output growth, and employment rate, a certain endogeneity degree of NAIRU induces perpetual cycles. By contrast, an economy with PLD/LML or WLD/GML regimes realises a locally asymptotically stable steady state, regardless of the endogeneity of NAIRU. Third, the degree of NAIRU endogeneity differently affects the trajectory of macroeconomic variables across different growth regimes. It strongly influences the stability of PLD/GML and WLD/LML regimes with strong feedback profit shares, output growth, and employment rates, whereas it is independent of the stability of PLD/LML and WLD/GML regimes.

66Using the Régulationist terminology, we may infer that, if the régulation mode pilots demand and income distribution regimes consistently by chance, an economic growth regime is stably realised; in the opposite case, it is unstable. Not all regimes necessarily react similarly to the same exogenous shock. While the dynamics of an economy with PLD/LML or WLD/GML regimes converge to a steady growth path, those with PLD/GML or WLD/LML regimes may present persistent cycles. Moreover, the nature of the cycles differs: PLD/GML regimes generate anti-clockwise cycles in the wage share and employment rate plane, while the WLD/LML regimes generate clockwise cycles. Hence, our growth regime approach of the endogenous NAIRU dynamics presents the primary message of the Régulation theory: an economy has business cycles and growth patterns specific to its institutional structures (Labrousse, 1976; Boyer, 2015). The long-run paths of NAIRU also follow this implication.