Public debt and income inequality in an endogenous growth model with elastic labor supply

Abstract

In this study, we examine the relationships among long-run public debt policy, economic growth, and income inequality by extending a representative consumer theory of distribution and using an endogenous growth model with a sustainable fiscal rule, elastic labor supply, and initial endowment of heterogeneous wealth. We show that fiscal policy with strict (loose) budgetary discipline decreases (increases) the economic growth rate in the short run. In contrast, it increases (decreases) the economic growth rate in the long run. Then, we show that fiscal policy with strict (loose) budgetary discipline decreases (increases) income inequality in both the short and long run. First, we show that a balanced growth path is a locally stable saddle point. Then, we consider the relationships among public debt, the economic growth rate, and income inequality. In this model, the average leisure and the public-debt-to-private-capital ratio determine the rate of return and wage rate, affecting both the economic growth rate and income distribution across agents. We conclude that there is a negative relation between the economic growth rate and the measure of income inequality in the long run by conducting numerical simulations using US data. The policy implication of this paper is that the government has a public debt policy that achieves both faster economic growth and the improvement of income inequality.

Introduction

The aftermath of the coronavirus disease 2019 (COVID-19) crisis has raised questions about the relationships among long-run fiscal policy stance, economic growth, and income inequality. The COVID-19 pandemic has led the US economy to increase the public-debt-to-gross-domestic-product (GDP) ratio sharply. However, income inequality has been on an upward trend. The gross-public-debt-to-GDP ratio in the United States was 108.5% in 2019, but in 2020, this ratio increased to 133.9% (IMF [20]). In addition, in 1975, the top 1% income share of the United States accounted for 12.1%, but in 2021, this share increased to 21.9% (WID. World [37]).

Many studies have examined the relationship between public debt and economic growth. For example, Greiner [18] compares three fiscal rules that satisfy fiscal sustainability: the balanced budget rule, a budgetary rule where the growth rate of public debt is lower than the equilibrium growth rate, and a rule where the rate of public debt grows at the same rate as the equilibrium growth rate. Bokan et al. [9] derive a growth-maximizing public debt ratio. The above studies suggest that the government should conduct fiscal policy from a long-run perspective.¹ However, except for Borissov and Kalk [10] and Maebayashi and Konishi [23], few theoretical studies have examined the effects of public debt on income inequality. In recent years, empirical studies have reported a positive correlation between income inequality and public debt (e.g., Azzimonti et al. [2]; Bartak et al. [5]; Luo [22]). Azzimonti et al. [2] suggest that an uninsurable idiosyncratic risk that yields income inequality has enhanced the demand for government debt through political influence. In other words, their research shows a causal relationship in which income inequality accumulates public debt. However, Bartak et al. [5] emphasize that this relationship is induced by demand stabilization rather than political action. Luo [22] insists that labor income inequality leads to higher debt levels.

Although it is crucial to stabilize the economy through fiscal and monetary policy during a recession, the government should conduct sustainable fiscal policy from a long-run standpoint. Some economists have argued the importance of preserving fiscal and monetary policy spaces for a downturn. Romer and Romer [29] show that in the past 24 economic crises, the drops in output were less than 1% when there were policy spaces, whereas if there were no policy spaces, these drops were close to 10%. Barro’s [3] tax-smoothing theory is a pioneering approach to these countercyclical policies from fiscal policy, showing that the government acts to minimize tax distortion and that, as a result, the budget deficit expands during the recession and shrinks in times of peace. The tax-smoothing theory of Barro [3] provides a theoretical basis for considering long-run public finance and is one of the leading frameworks for subsequent theoretical and empirical studies.² Another famous study, Bohn [8], has applied the tax-smoothing theory of Barro [3] and examined the conditions in which public finance is sustainable, showing that public finance is sustainable if the primary-surplus-to-GDP ratio has a positive relationship with the public-debt-to-GDP ratio. By developing this research, Mendoza and Ostry [25] and Aldama and Creel [1] verify whether or not US public finance is sustainable. As a result, they conclude that US public finance is eventually sustainable. The above literature has led almost all economists and policymakers to focus on long-run fiscal sustainability rather than short-run budget deficits.

Since growth rate and income inequality are endogenous variables, if public debt affects the economic growth rate, then it can determine the size of income inequality. This relation leads to the following essential question: how does the stabilization of public debt impact economic growth and income inequality? This work considers the relationships among stabilizing public debt, economic growth, and income inequality. We extend García-Peñalosa and Turnovsky’s [15] endogenous growth model, which has endogenized labor supply and introduces consumer initial wealth heterogeneity using Bohn’s [8] fiscal rule.

Of course, many sources generate heterogeneity among agents. For example, Becker [6] examines the heterogeneous time preference rate, while Krusell and Smith [21] consider idiosyncratic shocks. However, we employ only the heterogeneity of the initial wealth endowment, which is one of the essential sources of inequality. For example, Piketty [28] shows that heterogenous wealth is the most crucial source of income inequality. This assumption, established by Caselli and Ventura’s [11] representative consumer theory of distribution (RCTD), facilitates the analysis of the relationship between aggregate and distributional dynamics (see also Turnovsky [33]; Turnovsky [35]).

An important mechanism yielding income inequality is that aggregate variables are determined independently from distribution dynamics, and distribution variables are determined after aggregate variables are determined. Public debt affects income inequality in this case, but the opposite effect does not exist. Therefore, it is possible to consider only the influence of public debt on income inequality. This mechanism is consistent with stylized facts. Chancel [12] suggests that since the 1980s, the private sector’s wealth, including public bonds, has increased, while that of the government sector has declined. Saez and Zuckman [31] note that the government has held a balanced budget and insist that this policy is a reason for the improvement in income inequality after the Second World War.

We have clarified that the government’s policy with a strong (weak) reaction of the public-debt-to-GDP ratio to the primary-surplus-to-GDP ratio decreases (increases) the economic growth rate in the short run. In contrast, it increases (decreases) the economic growth rate in the long run. Then, we have shown that this fiscal policy with strict (loose) budgetary discipline decreases (increases) income inequality in both the short and long run. The critical factors that generate these results are the public-debt-to-private-capital ratio, heterogeneity of wealth, and time allocation to either leisure or to labor supply. Specifically, this aspect relates to decreasing (increasing) crowding out, increasing (decreasing) labor supply, and difference in consumer behavior regarding wealth heterogeneity. Many empirical studies show a negative correlation between wealth and labor supply. For example, Holtz-Eakin et al. [19] show that in the United States, individuals with a strong heritage decrease labor participation.

The long-run relationship between strict (loose) fiscal discipline and income inequality can be summarized as follows. The strict (loose) fiscal discipline improves (hampers) private capital accumulation and lowers (raises) interest rates. As a result, the marginal productivity of labor increases (decreases); hence, agents decrease (increase) their leisure time. A rise (fall) in the marginal productivity of labor, namely, an increase (a decrease) in wage rates, increases (decreases) the income of poorer agents in terms of wealth; therefore, this effect decreases (increases) income inequality.

The remainder of this paper is organized as follows. The next section describes the decentralized economy. In the subsequent section, we derive the dynamic system of the economy and explain the steady-state characteristics followed by which the income distribution is examined. In the penultimate section, numerical simulations are performed. Finally, the conclusions and remaining problems are described.

Model

In this section, we describe the structure of the economy. By incorporating public debt, we extend García-Peñalosa and Turnovsky’s [15] endogenous growth model. We assume a closed economy, which is populated by symmetric firms that produce output using labor input and private capital, by infinitely lived agents who are similar in all respects except for their initial endowments of wealth, and by a government that receives income taxation and obtains revenue from issuing government bonds.

Description of the decentralized economy

Firms

The productive sector is populated by a large number of symmetric firms, each indexed by $j$. We normalize the mass of firms as 1. Each firm has a homogeneous production technology and produces output in perfectly competitive markets. As in Romer [30], representative firm $j$’s production technology is represented by

$$ \begin{array}{*{20}c} {Y_{j} = F\left( {L_{j} K,K_{j} } \right),} \\ \end{array} $$

(1a)

where $Y_{j}$, $L_{j}$, and $K_{j}$ denote firm $j$’s output level, labor input, and private capital stock, respectively. Unindexed $K$ is an economy-wide capital stock corresponding to knowledge and playing a role similar to that of public goods. We assume that Eq. (1a) has neoclassical properties of constant returns to scale in both the labor $L_{j}$ and private capital stock $K_{j}$ and has positive but diminishing marginal productivities of the production factor. Each firm faces identical production conditions. Hence, we assume that $K = \mathop \smallint \limits_{0}^{1} K_{j} {\text{d}}j$ and $L = \mathop \smallint \limits_{0}^{1} L_{j} {\text{d}}j$, where $L$ is the average economy-wide labor supply. We can rewrite the economy-wide production function as

$$ \begin{array}{*{20}c} {Y = F\left( {LK,K} \right) = f\left( L \right)K, f^{\prime}\left( L \right) > 0, f^{\prime\prime}\left( L \right) < 0.} \\ \end{array} $$

(1b)

Equation (1b) implies that aggregate private capital yields a positive externality.

Factor prices are determined by their respective marginal productivity. From differentiating the production function with respect to each production factor and assuming symmetric firms, we obtain the following:

$$ \begin{array}{*{20}c} {\left( {\frac{\partial F}{{\partial L_{j} }}} \right)_{{K_{j} = K, L_{j} = L}} = f^{\prime}\left( L \right)K \equiv w\left( L \right)K} \\ \end{array} , $$

(2a)

$$ \begin{array}{*{20}c} {\left( {\frac{\partial F}{{\partial K_{j} }}} \right)_{{K_{j} = K, L_{j} = L}} = f\left( L \right) - Lf^{\prime}\left( L \right) \equiv r\left( L \right)} \\ \end{array} . $$

(2b)

Equations (2a) and (2b) imply that the wage rate, $w\left(L\right)K$, depends on labor input and private capital stock, while the interest rate depends only on labor input, which means that in a growing economy, the average economy-wide labor supply is relatively scarce, and the wage rate rises with capital accumulation.³

Consumers

The household sector is populated by a large number of consumers, each indexed by $i$. We normalize the mass of consumers as 1. These consumers are identical in all respects except for their initial endowments of wealth, ${W}_{i,0}$.

Since the economy is growing, we focus on the share of consumer $i$’s wealth to total economy-wide wealth; namely, ${\omega }_{i}\equiv {W}_{i}/W$, where ${W}_{i}$ and $W$ denote consumer $i$’s wealth and average economy-wide wealth, respectively. We normalize the mean of relative wealth as 1 and assume that the distribution function $D({\omega }_{i})$ distributes relative wealth with variance ${s}_{\omega }^{2}$. ${s}_{\omega }^{2}$ denotes the measure of wealth inequality. Therefore, wealthier (poorer) consumers’ relative wealth is more (less) than 1. Wealth is divided into private capital and public bonds, which satisfies no arbitrage condition ($W=K+B,\left(1-{\tau }_{r}\right){r}_{K}={r}_{B}=\left(1-{\tau }_{r}\right)r).$

Each consumer endowed with a unit of time can allocate it either to leisure ${l}_{i}$ or to labor supply ${L}_{i}\equiv 1-{l}_{i}$. A representative consumer maximizes the integral of the isoelastic lifetime utility, which depends on consumption ${C}_{i}$ and leisure ${l}_{i}$:

$$ \begin{array}{*{20}c} {\max \mathop \smallint \limits_{0}^{\infty } \frac{1}{\gamma }\left( {C_{i} \left( t \right)l_{i}^{\eta } } \right)^{\gamma } e^{ - \rho t} {\text{d}}t,\quad {\text{with}} - \infty \left\langle {\gamma \left\langle {1,\eta } \right\rangle 0, 1} \right\rangle \gamma \left( {1 + \eta } \right),} \\ \end{array} $$

(3)

subject to the wealth accumulation equation:

$$ \begin{array}{*{20}c} {\dot{W}_{i} \left( t \right) = \left( {1 - \tau_{r} } \right)r\left( L \right)W_{i} + \left( {1 - \tau_{w} } \right)\left( {1 - l_{i} } \right)w\left( L \right)K - C_{i} ,} \\ \end{array} $$

(4)

where $\rho > 0$, $\eta$, $\tau_{r} \in \left( {0,1} \right)$, and $\tau_{w} \in \left( {0,1} \right)$ represent the time preference rate, utility elasticity of leisure, capital-income tax rate, and labor-income tax rate, respectively. $\varepsilon \equiv 1/\left( {1 - \gamma } \right)$ is the intertemporal elasticity of substitution.⁴ The preponderance of empirical evidence suggests that this value is below unity. That is, we assume that $\gamma <0$. The dot over the variables indicates the time derivative.⁵

The optimization problem of this representative consumer is to choose consumption, leisure, and capital accumulation rates to maximize Eq. (3) subject to Eq. (4). The first-order conditions of utility maximization are as follows:

$$ \begin{array}{*{20}c} {C_{i}^{\gamma - 1} l_{i}^{\eta \gamma } = \lambda_{i} ,} \\ \end{array} $$

(5a)

$$ \begin{array}{*{20}c} {\eta C_{i}^{\gamma } l_{i}^{\eta \gamma - 1} = \left( {1 - \tau_{w} } \right)w\left( L \right)K\lambda_{i} ,} \\ \end{array} $$

(5b)

$$ \begin{array}{*{20}c} {\left( {1 - \tau_{r} } \right)r\left( L \right) = \rho - \frac{{\dot{\lambda }_{i} }}{{\lambda_{i} }},} \\ \end{array} $$

(5c)

where ${\lambda }_{i}$ is consumer $i$’s shadow value of wealth. Eq. $\left(5\mathrm{a}\right)$ implies that consumer $i$’s marginal utility of consumption equals consumer $i$’s shadow value of wealth. Eq. $\left(5\mathrm{b}\right)$ indicates that consumer $i$’s marginal utility of leisure is equal to the after-tax wage rate. Eq. (5c) implies wealth return is equal to the returns of consumption. We also hold the transversality condition as

$$ \begin{array}{*{20}c} {\mathop {\lim }\limits_{t \to \infty } \lambda_{i} W_{i} e^{ - \rho t} = 0.} \\ \end{array} $$

(5d)

Government

The public sector, namely, the government, receives income taxation and obtains revenue from issuing government bonds. Here, we focus only on the effects of public debt and assume that government consumption is unproductive and has no impact on utility.

The government budget constraint is described by

$$ \begin{array}{*{20}c} {\dot{B} = r\left( L \right)B\left( {1 - \tau_{r} } \right) - \left[ {\left( {1 - \tau_{r} } \right)r\left( L \right)K + \left( {1 - \tau_{w} } \right)w\left( L \right)KL - C_{p} } \right] = r\left( L \right)B\left( {1 - \tau_{r} } \right) - S,} \\ \end{array} $$

(6)

where ${C}_{p}$ is the government consumption and $S>0$ is the primary surplus. The government follows a sustainable fiscal policy, as examined by Bohn [8], and we assume that the primary-surplus-to-GDP ratio is a positive linear function of the debt-to-GDP ratio as follows:

$$ \begin{array}{*{20}c} {\frac{S}{Y} = \phi + \beta \frac{B}{Y},} \\ \end{array} $$

(7)

where $\phi \in {\mathbb{R}}$ is a parameter that determines the level of primary surplus when GDP varies, and $\beta > 0$ is a parameter that determines how strongly the primary surplus reacts to changes in public debt. For example, $\phi$ implies how much the primary surplus reacts to fiscal policy changes in a recession, and $\beta$ implies how fiscal discipline is strict in the long run. By inserting Eq. (7) into Eq. (6), we obtain⁶

$$ \begin{array}{*{20}c} {\dot{B}\left( t \right) = \left( {r\left( L \right)\left( {1 - \tau_{r} } \right) - \beta } \right)B - \phi f\left( L \right)K.} \\ \end{array} $$

(6′)

If public finance is sustainable and intertemporal budget constraint is satisfied, then we hold no-Ponzi game (NPG) condition as follows:

$$ \begin{array}{*{20}c} {\mathop {\lim }\limits_{t \to \infty } e^{{ - \mathop \smallint \limits_{0}^{t} \left( {1 - \tau_{r} } \right)r\left( \xi \right){\text{d}}\xi }} B\left( t \right) = 0.} \\ \end{array} $$

(8)

Market clearing condition

By aggregating $l_{i}$, we describe the market clearing condition in terms of labor by

$$ \begin{array}{*{20}c} {\mathop \int \limits_{0}^{1} L_{j} {\text{d}}j = \mathop \int \limits_{0}^{1} \left( {1 - l_{i} } \right){\text{d}}i.} \\ \end{array} $$

(9)

The left-hand side denotes aggregate labor demand, while the right-hand side denotes aggregate labor supply. We can express Eqs. (2a) and (2b) as functions of $l$ using Eq. (9), namely

$$ \begin{array}{*{20}c} {w\left( L \right) = f^{\prime}\left( L \right) = f^{\prime}\left( {1 - l} \right) \equiv w\left( l \right),} \\ \end{array} $$

(10a)

$$ \begin{array}{*{20}c} {r\left( L \right) = f\left( L \right) - Lf^{\prime}\left( L \right) = f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right) \equiv r\left( l \right).} \\ \end{array} $$

(10b)

Dynamics

In this section, we derive the dynamic system that constructs the economy. In the endogenous growth model of García-Peñalosa and Turnovsky [15], the economy is always in a steady state, and the initial wealth distribution is equal to the steady-state wealth distribution. However, in this model, there are transitional dynamics. Similar to other economic variables, wealth distribution changes along the time path and converges to a steady state. In other words, the initial wealth distribution is usually different from that of the steady state. From this viewpoint, the analytical method of this paper relies on Turnovsky and García-Peñalosa [36] and Turnovsky [34], who focus on the analysis of transitional dynamics.

Dynamics of average leisure and the public-debt-to-private-capital ratio

Since national wealth is composed of private capital and public bonds, as discussed in Appendix 1, we describe the economy as a differential equation with three variables, $B$, $l$, and $K$. The average economy-wide private capital growth rate and the average economy-wide leisure growth rate can be represented as

$$ \begin{array}{*{20}c} {\frac{{\dot{K}}}{K} = \left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)w\left( l \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta \frac{B}{K} + \phi f\left( {1 - l} \right),} \\ \end{array} $$

(11)

$$ \frac{{\dot{l}}}{l} = \frac{{\rho - \left( {1 - \tau_{r} } \right)r\left( l \right) - \left( {\gamma - 1} \right)\left[ {\left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)w\left( l \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta \frac{B}{K} + \phi f\left( {1 - l} \right)} \right]}}{{\left( {\gamma - 1} \right)\left( {\frac{{w^{\prime}\left( l \right)l}}{w\left( l \right)} + 1} \right) + \eta \gamma }}. $$

(12)

The system, consisting of Eqs. (6′), (11), and (12), implies a three-dimensional simultaneous differential equation with economy-wide private capital, public debt, and leisure. Furthermore, by setting $b = B/K$, we can rewrite this system as

$$ \begin{array}{*{20}c} {\dot{l} = \frac{{G\left( {l,b} \right)}}{H\left( l \right)},} \\ \end{array} $$

(13)

$$ \begin{array}{*{20}c} {\dot{b} = - \left( {1 - \tau_{w} } \right)w\left( l \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right)b - \beta \left( {1 + b} \right)b - \left( {1 + b} \right)\phi f\left( {1 - l} \right),} \\ \end{array} $$

(14)

where

$$ \begin{array}{*{20}c} {G\left( {l,b} \right) = \frac{{\left( {1 - \tau_{r} } \right)r\left( l \right) - \rho }}{1 - \gamma } - \left[ {\left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)w\left( l \right)\left( {\left( {1 - l} \right) - \frac{l}{\eta }} \right) + \beta b + \phi f\left( {1 - l} \right)} \right],} \\ \end{array} $$

(15a)

$$ \begin{array}{*{20}c} {H\left( l \right) = \frac{{w^{\prime}\left( l \right)}}{w\left( l \right)} + \frac{{1 - \gamma \left( {1 + \eta } \right)}}{{\left( {1 - \gamma } \right)l}} > 0.} \\ \end{array} $$

(15b)

Equations (13), (14), (15a), and (15b) imply that the dynamics of the aggregate variable do not depend on distributional variables such as $\omega_{i}$. Equations (13) and (14) play an essential role in our model.

Balanced growth path

Here, we consider the property of a balanced growth path (BGP) to clarify the fiscal rule consistent with the government intertemporal budget constraint and NPG condition. Let us define the market equilibrium and the BGP as follows:

Definition 1

Market equilibrium is a sequence of variables $\left\{ {l\left( t \right), K\left( t \right), B\left( t \right)} \right\}_{t = 0}^{\infty }$ and factor prices $\left\{ {w\left( t \right), r\left( t \right)} \right\}_{t = 0}^{\infty }$ satisfying firm’s profit maximization, consumer’s utility maximization, market clearing condition, and government’s intertemporal budget constraint.

Definition 2

BGP is a growth path of the market equilibrium in $\left\{ {w\left( \infty \right),r\left( \infty \right)} \right\}$.

First, leisure is bounded, and the growth rate of leisure is zero in the steady state, namely, $\dot{l}/l = 0.$ Therefore, the time derivative of Eq. (1a) leads to

$$ \begin{array}{*{20}c} {\frac{{\dot{Y}}}{Y} = - \frac{{f^{\prime}\left( 1-l \right)l}}{f\left( 1-l \right)}\frac{{\dot{l}}}{l} + \frac{{\dot{K}}}{K} = \frac{{\dot{K}}}{K}.} \\ \end{array} $$

(16)

We obtain the following Condition 1 using Eq. (16).

Condition 1: In the BGP, economy-wide consumption, private capital, and output have the same constant growth rate:

$$ \begin{array}{*{20}c} {\psi \equiv \frac{{\dot{C}}}{C} = \frac{{\dot{K}}}{K} = \frac{{\dot{Y}}}{Y}.} \\ \end{array} $$

(17)

Next, we consider the fiscal rule consistent with the government intertemporal budget constraint and the NPG condition. The NPG condition in the BGP, using Eq. (8), is represented as

$$ \begin{array}{*{20}c} {\mathop {\lim }\limits_{t \to \infty } e^{{ - \mathop \smallint \limits_{0}^{t} \left( {1 - \tau_{r} } \right)r\left( \xi \right){\text{d}}\xi }} B\left( t \right) = \mathop {\lim }\limits_{t \to \infty } B_{0} e^{{\left( {g_{B} - \left( {1 - \tau_{r} } \right)r^{*} } \right)t}} = 0,} \\ \end{array} $$

(8′)

where $r^{*}$ denotes the long-run interest rate, $B_{0}$ denotes initial public debt, and $g_{B}$ denotes the growth rate of public debt. Therefore, if $g_{B} < \left( {1 - \tau_{r} } \right)r^{*}$, then the NPG condition is satisfied.

All agents face the same transversality condition, and the aggregate transversality condition is described as

$$ \begin{array}{*{20}c} {\mathop {\lim }\limits_{t \to \infty } \lambda We^{ - \rho t} = 0.} \\ \end{array} $$

(5d′)

Transversality condition (5d′) denotes the relationship between the equilibrium growth rate (17) and the after-tax interest rate as

$$ \begin{array}{*{20}c} {\left( {1 - \tau_{r} } \right)r\left( {l^{*} } \right) > \psi .} \\ \end{array} $$

(18)

This inequality implies that the after-tax interest rate exceeds the equilibrium growth rate. We obtain the fiscal rule satisfying the government intertemporal budget constraint and NPG condition as $g_{B} \le \psi$. In addition, the transversality condition is found to be

$$ \begin{array}{*{20}c} {\left( {1 - \tau_{w} } \right)\left( {1 - l^{*} } \right)w\left( {l^{*} } \right)K < C,} \\ \end{array} $$

(19a)

$$ \begin{array}{*{20}c} {l^{*} > \frac{\eta }{1 + \eta }.} \\ \end{array} $$

(19b)

Equation (19a) indicates that the economy-wide after-tax labor income is lower than the economy-wide consumption, while Eq. (19b) shows the restriction in economy-wide leisure.

Case $g_{B} = 0$ is the balanced budget rule. The rules $g_{B} = 0$ and $0 < g_{B} < \psi$ obtain the same result. In this paper, to analyze public finance with fiscal space during the recession, we focus only on the rule where the rate of public debt grows at the same rate as the equilibrium growth rate, $g_{B} = \psi$⁷.

Existence, uniqueness, and stability of the BGP

Now, we examine the property of the BGP. We obtain Propositions 1 and 2, which refer to the existence, uniqueness, and stability of the BGP.

Proposition 1

We assume a CES (constant elasticity of substitution) production function. If

$$ \beta > \frac{{\left[ {\rho - \left\{ {\left( {1 - \tau_{r} } \right)\left( {1 - \alpha } \right)\gamma - \left( {\left( {1 - \tau_{w} } \right)\alpha + \phi } \right)\left( {1 - \gamma } \right)} \right\}f\left( 1 \right)} \right]\left\{ {\rho - \left( {1 - \tau_{r} } \right)\left( {1 - \alpha } \right)f\left( 1 \right)\gamma } \right\}}}{{\left[ {\rho - \left\{ {\left( {1 - \tau_{r} } \right)\left( {1 - \alpha } \right)\gamma - \left( {1 - \tau_{w} } \right)\alpha } \right\}f\left( 1 \right)} \right]\left( {1 - \gamma } \right)}} > 0 $$

and

$$ \left( {1 - \tau_{w} } \right)\left[ {\frac{1 - \alpha }{\sigma }\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) - \left( {\frac{1 + \eta }{\eta }} \right)} \right] > \phi > \left( {1 - \tau_{r} } \right)\frac{\gamma }{1 - \gamma }\frac{1 - \alpha }{\sigma } + \left( {1 - \tau_{w} } \right)\left[ {\frac{1 - \alpha }{\sigma }\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) - \left( {\frac{1 + \eta }{\eta }} \right)} \right], $$

then, a BGP exists in $0 < l < 1$ and $b \ge 0$, where $\alpha$ is share of labor income and $\sigma$ is elasticity of substitution.

Proof: See Appendix A2.

Proposition 2

If the production function is CES , BGP is a locally stable saddle point.

Proof: See Appendix A3.

The adjustment paths of $l$ and $b$ are, respectively,

$$ \begin{array}{*{20}c} {b\left( t \right) = b^{*} + \left( {b\left( 0 \right) - b^{*} } \right)e^{\mu t} ,} \\ \end{array} $$

(20a)

$$ \begin{array}{*{20}c} {l\left( t \right) = l^{*} + \frac{{a_{12} }}{{\mu - a_{11} }}\left( {b\left( t \right) - b^{*} } \right) = l^{*} + \frac{{\mu - a_{22} }}{{a_{21} }}\left( {b\left( t \right) - b^{*} } \right),} \\ \end{array} $$

(20b)

where $\mu$ denotes a stable eigenvalue, $b$ is pre-determined (state) variable, and $l$ is jump valuable. Therefore, if there are policy shocks, $l$ is change instantaneously, and then $b$ and $l$ converge to a new steady state. As discussed in Appendix 3, we show that, $\mu - a_{11} < 0$ and $a_{12} < 0$. Intuitively, Eqs. (20a) and (20b) denote that public debt accumulation (retrenchment) rises (falls) the public-debt-to-private-capital ratio, which increases (decreases) the interest rate and decreases (increases) the wage rate. Therefore, agents decrease (increase) labor supply and increase (decrease) leisure time during transition.

Dynamics of relative wealth

Next, we consider the dynamics of relative wealth. Using the definition of $\omega_{i} \left( t \right) \equiv W_{i} /W$, we obtain the differential equation of relative wealth as follows:

$$ \dot{\omega }_{i} \left( t \right) = \left( {\frac{{\dot{W}_{i} }}{{W_{i} }} - \frac{{\dot{W}}}{W}} \right)\omega_{i} = \left( {1 - \tau_{w} } \right)\frac{{f^{\prime}\left( {1 - l} \right)}}{1 + b}\left[ {\left( {1 - l_{i} - \frac{{l_{i} }}{\eta }} \right) - \left( {1 - l - \frac{l}{\eta }} \right)\omega_{i} } \right]. $$

(21)

Equation (21) describes the evolution of relative wealth. Using equation $\dot{l}/l = 0$, in the steady state, the dynamic equation of agent $i$’s leisure time is found to satisfy the following condition:

$$ \begin{array}{*{20}c} {\frac{{\dot{l}_{i} }}{{l_{i} }} = \frac{{\dot{l}}}{l} = 0.} \\ \end{array} $$

(22)

Equation (22) shows that there exists a steady-state value, $l_{i}^{*}$. Proposition 1 and this relation give

$$ \begin{array}{*{20}c} {l_{i}^{*} = z_{i} l^{*} .} \\ \end{array} $$

(22′)

From Eq. (22′), relative leisure is found to be a constant rate, $z_{i}$, over time. Noting Eq. (19b), the coefficient of $\omega_{i}$ is positive, which implies that the only solution consistent with long-run stability and the transversality condition is that the value of the parentheses in the right-hand side of Eq. (21) is zero. That is, relative leisure satisfies the following relative labor supply function:

$$ \begin{array}{*{20}c} {l_{i}^{*} - l^{*} = \left( {l^{*} - \frac{\eta }{1 + \eta }} \right)\left( {\omega_{i}^{*} - 1} \right).} \\ \end{array} $$

(23)

Equation (23) yields a positive relationship between agent $i$’s leisure, $l_{i}$ and relative wealth, $\omega_{i}$, which means that wealthier agents’ marginal utility of wealth (future consumption) is smaller than that of poorer agents.⁸

Then, we derive agent $i$’s relative wealth and leisure dynamics. By noting Proposition 1 and the existence of $l_{i}^{*}$ and $\omega_{i}^{*}$, linearizing Eq. (21) around the steady state $l^{*} , b^{*}$, and $\omega_{i}^{*}$, with calculations as shown in Appendix A4, yields the stable adjustment path as follows:

$$ \begin{array}{*{20}c} {\omega_{i} \left( t \right) - 1 = \delta \left( t \right)\left( {\omega_{i}^{*} - 1} \right),} \\ \end{array} $$

(24a)

where

$$ \begin{array}{*{20}c} {\delta \left( t \right) \equiv \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l^{*} } \right)}}{{\left( {1 - l^{*} } \right)\left( {1 + b^{*} } \right)\left( {a_{33} - \mu } \right)}}\left( {1 - \frac{l\left( t \right)}{{l^{*} }}} \right) + 1.} \\ \end{array} $$

(25a)

Setting $t = 0$ in Eqs. (24a) and (25a), we obtain

$$ \begin{array}{*{20}c} {\omega_{i,0} - 1 = \delta \left( 0 \right)\left( {\omega_{i}^{*} - 1} \right),} \\ \end{array} $$

(24b)

and

$$ \begin{array}{*{20}c} {\delta \left( 0 \right) \equiv \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l^{*} } \right)}}{{\left( {1 - l^{*} } \right)\left( {1 + b^{*} } \right)\left( {a_{33} - \mu } \right)}}\left( {1 - \frac{l\left( 0 \right)}{{l^{*} }}} \right) + 1.} \\ \end{array} $$

(25b)

Therefore, using Eqs. (24a)–(25b), we obtain the following relation:

$$ \begin{array}{*{20}c} {\omega_{i} \left( t \right) - 1 = \frac{\delta \left( t \right)}{{\delta \left( 0 \right)}}\left( {\omega_{i,0} - 1} \right).} \\ \end{array} $$

(26)

Using the facts

$$ s_{\omega }^{2} = \mathop \int \limits_{0}^{1} \left( {\omega_{i} \left( t \right) - 1} \right)^{2} {\text{d}}i = \left( {\delta \left( t \right)} \right)^{2} \mathop \int \limits_{0}^{1} \left( {\omega^{*} - 1} \right)^{2} {\text{d}}i $$

and

$$ s_{\omega ,0}^{2} = \mathop \int \limits_{0}^{1} \left( {\omega_{i,0} - 1} \right)^{2} di = \left( {\delta \left( 0 \right)} \right)^{2} \mathop \int \limits_{0}^{1} \left( {\omega^{*} - 1} \right)^{2} {\text{d}}i, $$

Equation (26) denotes that standard deviation of relative wealth, $s_{\omega } = \sqrt {s_{\omega }^{2} }$ can be represented as follows:

$$ \begin{array}{*{20}c} {s_{\omega } = \frac{\delta \left( t \right)}{{\delta \left( 0 \right)}} s_{\omega ,0} .} \\ \end{array} $$

(27)

Therefore, the standard deviation of relative wealth is denoted as a constant rate of initial standard deviation of relative wealth. The dynamics of $\omega_{i} \left( t \right)$ are determined as follows. First, $l^{*}$ and $b^{*}$ are determined simultaneously by Eqs. (13) and (14), respectively. $b\left( 0 \right)$, $l\left( 0 \right)$, and $s_{\omega ,0}$ are given, then we obtain $s_{\omega }$ from Eq. (27)⁹. Using Eqs. (24a) and (24b), we obtain

$$ \begin{aligned} \omega_{i} (t) - \omega_{i}^{*} & = \frac{\delta (t) - 1}{{\delta (0) - 1}}(\omega_{i,0} - \omega_{i}^{*} ) \\ & = \frac{{l^{*} - l(t)}}{{l^{*} - l(0)}}(\omega_{i,0} - \omega_{i}^{*} ) = e^{\mu t} (\omega_{i,0} - \omega_{i}^{*} ). \\ \end{aligned} $$

(28)

Equation (28) shows that relative wealth, $\omega_{i} \left( t \right)$ also converges to the steady state. When $\omega_{i}^{*}$ is determined, we can describe the dynamics of the leisure time of agent $i$. Using Eqs. (22′), (23) and (24b), we can describe agent $i$’s allocation of leisure time:

$$ \begin{aligned} z_{i} - 1 & = \left( {1 - \frac{\eta }{{l^{*} \left( {1 + \eta } \right)}}} \right)\left( {\omega_{i}^{*} - 1} \right) \\ & = \frac{{\omega_{i,0} - 1}}{\delta \left( 0 \right)}\left( {1 - \frac{\eta }{{l^{*} \left( {1 + \eta } \right)}}} \right), \\ \end{aligned} $$

(29)

and

$$ \begin{array}{*{20}c} {l_{i} \left( t \right) = \left( {\frac{{\omega_{i,0} - 1}}{\delta \left( 0 \right)}\left( {1 - \frac{\eta }{{l^{*} \left( {1 + \eta } \right)}}} \right) + 1} \right)l\left( t \right).} \\ \end{array} $$

(29′)

Equation (29′) denotes that wealthier agents choose more leisure time than do their poorer counterparts.

Income distribution, leisure, and the public-debt-to-private-capital ratio

In this section, we discuss the mechanism yielding income inequality. We introduce the relative income of agent $i$, who has wealth $W_{i}$. The representative agent’s total income is

$$ \begin{array}{*{20}c} {Y_{i} = \left( {1 - \tau_{r} } \right)r\left( l \right)W_{i} + \left( {1 - \tau_{w} } \right)\left( {1 - l_{i} } \right)w\left( l \right)K,} \\ \end{array} $$

(30)

while the economy-wide average income is

$$ \begin{array}{*{20}c} {Y = \left( {1 - \tau_{r} } \right)r\left( l \right)W + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)w\left( l \right)K.} \\ \end{array} $$

(30′)

By defining the relative income as $y_{i} \equiv Y_{i} /Y$, we yield the following:

$$ \underbrace {{y_{i} - 1}}_{{\text{total income inequality}}} = \underbrace {{\frac{{\left( {1 - \tau_{r} } \right)r\left( l \right)W}}{{\left( {1 - \tau_{r} } \right)r\left( l \right)W + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)w\left( l \right)K}}\left( {\omega_{i} - 1} \right)}}_{{\text{Interest income inequality}}}\underbrace {{ - \frac{{\left( {1 - \tau_{w} } \right)w\left( l \right)K}}{{\left( {1 - \tau_{r} } \right)r\left( l \right)W + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)w\left( l \right)K}}\left( {l_{i} - l} \right)}}_{{\text{wage income inequality}}} $$

(31a)

In Eq. (31a), the first term denotes interest income inequality, while the second term denotes wage income inequality. Summing up these two income inequalities, we yield total income inequality, $y_{i} - 1$. Using Eq. (29′), relative income can be shown as

$$ y_{i} = \omega_{i} - \frac{{1 + \left( {\frac{1}{\delta \left( t \right)}\left( {1 - \frac{\eta }{{l^{*} \left( {1 + \eta } \right)}}} \right) - 1} \right)l}}{{\left( {1 - \tau_{r} } \right)r\left( l \right)\left( {1 + b} \right) + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)w\left( l \right)}}\left( {1 - \tau_{w} } \right)w\left( l \right)\left( {\omega_{i} \left( t \right) - 1} \right). $$

Namely, we can rewrite Eq. (31a) as follows:

$$ \begin{array}{*{20}c} {y_{i} - 1 = \left( {1 - \theta } \right)\left( {\omega_{i} \left( t \right) - 1} \right),} \\ \end{array} $$

(31b)

where

$$ \theta = \frac{{\left[ {1 + \left( {\frac{1}{\delta \left( t \right)}\left( {1 - \frac{\eta }{{l^{*} \left( {1 + \eta } \right)}}} \right) - 1} \right)l} \right]\left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l} \right)}}{{\left( {1 - \tau_{r} } \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right)\left( {1 + b} \right) + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)}}. $$

In Eq. (31b), using fact that $\mathop \int \limits_{0}^{1} \left( {y_{i} - 1} \right)^{2} di = s_{y}^{2}$ and

$\mathop \int \limits_{0}^{1} \left( {\omega_{i} - 1} \right)^{2} di = s_{\omega }^{2}$, we obtain

$$ \begin{array}{*{20}c} {s_{y} = \left( {1 - \theta } \right)s_{\omega } ,} \\ \end{array} $$

(32)

where $s_{y}$ denote standard deviation of relative income, while $s_{\omega }$ denote standard deviation of relative wealth. Equation (32) stands for that income inequality is proportional to wealth inequality. We can confirm as $0 < 1 - \theta < 1$, which means that wealth inequality is more extensive than income inequality. Now, using Eqs. (27) and (32), we can represent as follows:

$$ \begin{array}{*{20}c} {s_{y} = \left( {1 - \theta } \right)\frac{\delta \left( t \right)}{{\delta \left( 0 \right)}} s_{\omega ,0} .} \\ \end{array} $$

(33)

Equation (33) shows that initial wealth distribution, the time allocation to either leisure or to labor supply, and the public-debt-to-private-capital ratio determine the income inequality. In Eq. (32), letting $t \to \infty$, we obtain

$$ \begin{array}{*{20}c} {s_{y}^{*} = \left( {1 - \theta^{*} } \right)\left( {\omega_{i}^{*} - 1} \right) = \left( {1 - \theta^{*} } \right)s_{\omega }^{*} ,} \\ \end{array} $$

(34)

where

$$ \theta^{*} = \frac{{\left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l} \right)}}{{\left( {1 + \eta } \right)\left[ {\left( {1 - \tau_{r} } \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right)\left( {1 + b} \right) + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right]}}. $$

Equation (34) shows steady-state income inequality. That is, income inequality is determined by the wealth inequality and the reward of production factors, $L$ and $K$.

Numerical examples

In this section, we perform a numerical simulation to gain further insights for the analysis of conducting fiscal reform by changing policy parameter value. Here, we specify production technology as Cobb–Douglas $Y=A{L}^{\alpha }K$. The benchmark parameters are summarized in Table 1. First, following García-Peñalosa and Turnovsky [15], we set $\alpha =0.6$, $\rho =0.04$, $\gamma =-2$, $\eta =$ $1.75$. Second, the income tax rate and parameters that determine the primary surplus are obtained from empirical analyses. McDaniel [24] estimates the effective tax rate in the United States and shows that the average income tax rate is in the range of 0.1 to 0.2. We employ the mid-value, ${\tau }_{w}={\tau }_{r}=0.15$. Additionally, we employ the fiscal stability coefficient and parameter standing for policy space, $\beta =0.586$ and $\phi =-0.658$, respectively, from Fincke and Greiner [13] (Table 15).¹⁰ Finally, we set $A=0.7$, which yields a growth rate of around 2% (OECD average). We assume that the initial period is in the benchmark’s steady state and compare the effect of the policy shock. The results of the numerical simulation are summarized in Tables 2 and 3.¹¹

Table 1 Parameter values

Table 2 Growth and the distribution of wealth and income (short run)

Table 3 Growth and the distribution of wealth and income (long run)

The first line of Table 2 shows the benchmark case. Average leisure is $l=0.753$, and the public-debt-to-private-capital ratio is $b=0.395$. These values yield a growth rate of $\psi =2.10\mathrm{\%}$. We set the standard deviation of relative wealth to 1. The standard deviation of relative income becomes 0.238. Therefore, as shown in Eq. (32), wealth is more concentrated on a few people than income.

Increasing $\beta$

As emphasized throughout this paper, the government should conduct fiscal policy while satisfying the intertemporal budget constraint and NPG conditions. In this paper, the government has a sustainable fiscal policy by setting the growth rate of public debt to be the same as the equilibrium growth rate following the fiscal rules of Bohn [8]. Here, we examine the policy effects of the rise in $\beta$. The increase in $\beta$ means that the government reduces the public-debt-to-GDP ratio by fiscal consolidation with institutional reform attaining the budgetary target.

Table 2 describes the short-run (instantaneous) effect of fiscal reform. Following Turnovsky and García-Peñalosa’s [35] numerical simulation, we calibrate the linearizing system as an approximation. Of course, the results of the linearized approximation yield errors. However, the appearing a zero root problem makes numerical simulation of the global saddle path difficult. We also do an examination of a robustness check by conducting Trimborn et al.’s [32] algorithm in Eqs. (13) and (14). The errors are not so significant. Therefore, the calibration of the linearized system is appropriate and not misleading as an approximation.¹²

When $\beta$ is raised, leisure increases immediately. At the same time, the public-debt-to-private-capital ratio is a state variable, which can be seen as same to the initial steady-state value. Intuitively, raising $\beta$ declines the marginal product of private capital, and reduces the long-run return of wealth, namely the return of consumption. This effect reduces labor supply and increases leisure time in the short run. Because of the lowering of labor supply, the output-to-private-capital ratio decline, while the public-debt-to-GDP ratio rise. In this model, the source of inequality is only wealth heterogeneity, and the decline of the return of wealth decreases income inequality.¹³ Table 3 shows the long-run (steady-state) effect of fiscal reform. Raising $\beta$ induces higher growth rate and smaller income inequality than the initial benchmark case. First, retrenching the public debt promotes private capital accumulation and enhances growth. Then, this raises the productivity of labor, and agents decrease leisure time. Wealthier agents’ return of capital become larger than poorer, and wealth inequality increases. However, it induces to lower income inequality by increasing the marginal product of labor and decreasing the marginal product of capital. This mechanism is yielded by wealth heterogeneity and heterogeneous labor supply induced by wealth heterogeneity. The resource allocation with fiscal reform influences income distribution between production factors, and this reallocation impacts income inequality alleviation. Moreover, the public-debt-to-GDP ratio has decreased compared to this ratio of the benchmark case.

This numerical simulation shows a trade-off between short and long-run effects for economic growth, while the government can attain two goals, namely accelerating economic growth and reducing income inequality, simultaneously by decreasing the government-debt-to-GDP ratio in the long run.

Concluding remarks

In this study, we have extended the endogenous growth model of García-Peñalosa and Turnovsky [15], which has flexible leisure and heterogeneous initial wealth, and have examined the relationships among public debt, economic growth, and income inequality. In doing so, we have considered fiscal policies that affect economic growth and income inequality under sustainable public finance, adopting the fiscal rule of Bohn [8].

The analysis can be facilitated using the property of the homogeneity of the utility function (RCTD model). The dynamics of aggregate variables are determined independently in terms of the distributional aspect, and the measure of income inequality is determined after the dynamics of the aggregate variables are determined. Unlike Azzimonti et al. [2], this study shows that public debt accumulation affects income inequality, but that income inequality does not affect public debt accumulation.

This model has transitional dynamics, unlike García-Peñalosa and Turnovsky’s [15] model. Even in this case, a BGP exists, showing that it is a locally stable saddle point. When there is a policy shock, the distribution of wealth, which is exogenous in García-Peñalosa and Turnovsky’s [15], is determined endogenously. Therefore, income inequality is decided by endogenous wealth distribution, the public-debt-to-private-capital ratio, and the time allocation either to labor supply or to leisure.

We have conducted a numerical simulation to consider the effect of shock in policy parameter on the economic growth rate and the measure of income inequality. As a result, it is clarified that there is a trade-off between short and long-run effects for economic growth. At the same time, the government could increase the economic growth rate and reduce income inequality by conducting fiscal policy with strict fiscal discipline in the long run. This result suggests that it is crucial for fiscal policy design to raise the economic growth rate and improve the income gap.

Finally, we describe three points concerning the remaining problems of this paper. First, in this paper, it is assumed that government spending affects neither utility nor production. However, government spending should be considered as a public good and productive public expenditure to gain more practical implications, as analyzed by Barro [4] and Futagami et al. [14]. Second, different results may be obtained considering the heterogeneity of wealth and skill endowments. García-Peñalosa and Turnovsky [16] show that the positive relationship between initial wealth endowments and leisure time cannot be obtained by relying on the Ramsey-type neoclassical growth model with skill heterogeneity. Therefore, assumably, the endogenous growth model with stabilized public debt and heterogeneous skills can also obtain different results. Finally, this work analyzes only that situation when the interest rate exceeds the growth rate ($r>\psi$). However, recent US data have shown that interest rates tend to be below growth rates ($r<\psi$) (e.g., Blanchard [7], Mian et al. [26]). As noted in Blanchard [7] and Mian et al. [26], the fiscal sustainability condition is changed in this case. If the fiscal rules change, the economic growth rate and income inequality also vary. Considering these standpoints, we have many essential study directions to analyze the relationships among public debt, economic growth, and income inequality.

Appendices

Appendices

A1: Derivation of aggregate variables

Dividing Eq. (5b) by Eq. (5a), agent $i$’s consumption-to-wealth ratio is

$$ \begin{array}{*{20}c} {\frac{{C_{i} }}{{W_{i} }} = (1 - \tau_{w} )w\left( l \right)\frac{{l_{i} }}{\eta }\frac{K}{{W_{i} }}.} \\ \end{array} $$

(A1)

Using Eqs. (4) and (A1), the growth rate of agent $i$’s wealth is

$$ \begin{aligned} \frac{{\dot{W}_{i} }}{{W_{i} }} & = \left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)\left( {1 - l_{i} } \right)w\left( l \right)\frac{K}{{W_{i} }} - \frac{{C_{i} }}{{W_{i} }} \\ & = \left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)w\left( l \right)\frac{K}{{W_{i} }}\left( {\left( {1 - l_{i} } \right) - \frac{{l_{i} }}{\eta }} \right). \\ \end{aligned} $$

(A2)

Using the time derivative of Eq. (5a) and combining it with Eq. (5c), the shadow value’s condition leads to

$$ \begin{array}{*{20}c} {\left( {\gamma - 1} \right)\frac{{\dot{C}_{i} }}{{C_{i} }} + \eta \gamma \frac{{\dot{l}_{i} }}{{l_{i} }} = \frac{{\dot{\lambda }_{i} }}{{\lambda_{i} }} = \rho - \left( {1 - \tau_{r} } \right)r\left( l \right).} \\ \end{array} $$

(A3)

Equation (A3) implies that the growth rate of the shadow value of wealth is equal across all agents. The time differentiation of Eq. (A1) implies

$$ \begin{array}{*{20}c} {\frac{{\dot{C}_{i} }}{{C_{i} }} - \frac{{\dot{l}_{i} }}{{l_{i} }} = \frac{{w^{\prime}\left( l \right)l}}{w\left( l \right)}\frac{{\dot{l}}}{l} + \frac{{\dot{K}}}{K}.} \\ \end{array} $$

(A4)

Equation (A4) has a shape in which the right side is independent on index $i$. Here, we consider a different individual $h$. From Eqs. (A3) and (A4), we obtain

$$ \left( {\gamma - 1} \right)\left( {\frac{{\dot{C}_{i} }}{{C_{i} }} - \frac{{\dot{C}_{h} }}{{C_{h} }}} \right) + \eta \gamma \left( {\frac{{\dot{l}_{i} }}{{l_{i} }} - \frac{{\dot{l}_{h} }}{{l_{h} }}} \right) = 0. $$

$$ \left( {\frac{{\dot{C}_{i} }}{{C_{i} }} - \frac{{\dot{C}_{h} }}{{C_{h} }}} \right) - \left( {\frac{{\dot{l}_{i} }}{{l_{i} }} - \frac{{\dot{l}_{h} }}{{l_{h} }}} \right) = 0. $$

Therefore, this implies that

$$ \begin{array}{*{20}c} {\frac{{\dot{C}_{i} }}{{C_{i} }} = \frac{{\dot{C}_{h} }}{{C_{h} }};\frac{{\dot{l}_{i} }}{{l_{i} }} = \frac{{\dot{l}_{h} }}{{l_{h} }}.} \\ \end{array} $$

(A5)

Equation (A5) shows that the growth rate of consumption and that of leisure are equal across all individuals. Aggregating Eq. (A1), the economy-wide consumption-to-wealth ratio is

$$ \begin{array}{*{20}c} {\frac{C}{W } = \left( {1 - \tau_{w} } \right)w\left( l \right)\frac{l}{\eta }\frac{K}{W}.} \\ \end{array} $$

(A1′)

Using Eq. (A2), we obtain the growth rate of economy-wide wealth as

$$ \begin{aligned} \psi & \equiv \frac{{\dot{W}}}{W} = \left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)w\left( l \right)\frac{K}{W} - \frac{C}{W} \\ & = \left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)w\left( l \right)\frac{K}{W}\left( {\left( {1 - l} \right) - \frac{l}{\eta }} \right). \\ \end{aligned} $$

(A2′)

Therefore, the growth rates of the average consumption and average leisure are equal to that of individual $i$'s consumption and leisure, respectively:

$$ \begin{array}{*{20}c} {\frac{{\dot{C}_{i} }}{{C_{i} }} = \frac{{\dot{C}}}{C};\frac{{\dot{l}_{i} }}{{l_{i} }} = \frac{{\dot{l}}}{l}.} \\ \end{array} $$

(A6)

By rewriting Eq. (A′2) and combining Eq. (6′), the average economy-wide private capital growth rate can be represented as

$$ \begin{array}{*{20}c} {\frac{{\dot{K}}}{K} = \left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)w\left( l \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta \frac{B}{K} + \phi f\left( {1 - l} \right).} \\ \end{array} $$

(A7)

By inserting Eqs. (A7), (A4), and (A6) into Eq. (A3), the average economy-wide leisure growth rate becomes

$$ \begin{array}{*{20}c} {\frac{{\dot{l}}}{l} = \frac{{\rho - \left( {1 - \tau_{r} } \right)r\left( l \right) - \left( {\gamma - 1} \right)\left[ {\left( {1 - \tau_{r} } \right)r\left( l \right) + \left( {1 - \tau_{w} } \right)w\left( l \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta \frac{B}{K} + \phi f\left( {1 - l} \right)} \right]}}{{\left( {\gamma - 1} \right)\left( {\frac{{w^{\prime}\left( l \right)l}}{w\left( l \right)} + 1} \right) + \eta \gamma }}.} \\ \end{array} $$

A2: Proof of Proposition1

If a BGP exists, then from Eqs. (13), (14), (15a), and (15b), the system

$\dot{b}$ = 0:

$$ - \left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l} \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right)b - \beta (1 + b)b - (1 + b)\phi f(1 - l) = 0, $$

$\dot{l}$ = 0:

$$ \begin{aligned} \frac{1}{H\left( l \right)} & \left[ {\frac{{\left( {1 - \tau_{r} } \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right) - \rho }}{1 - \gamma } - \left( {1 - \tau_{r} } \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right)} \right. \\ & \quad \quad \quad \quad \quad \left. { - \left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l} \right)\left( {\left( {1 - l} \right) - \frac{l}{\eta }} \right) - \beta b - \phi f\left( {1 - l} \right)} \right] = 0 \\ \end{aligned} $$

must intersect at least one point in the $l$ − $b$ planes. Obtaining the solution $b$ leads to the following:

$$ \begin{aligned} \left. b \right|_{{\dot{b} = 0}} & = \frac{1}{2\beta }\left[ { - \left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l} \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta + \phi f\left( {1 - l} \right)} \right)} \right. \\ & \quad \left. {\quad \quad \quad \quad \pm \sqrt {\left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l} \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta + \phi f\left( {1 - l} \right)} \right)^{2} - 4\beta \phi f\left( {1 - l} \right)} } \right], \\ \end{aligned} $$

(A8)

$$ \begin{aligned} \left. b \right|_{{\dot{l} = 0}} & = \frac{1}{\beta }\left[ {\frac{{\left( {1 - \tau_{r} } \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right) - \rho }}{1 - \gamma } - \left( {1 - \tau_{r} } \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right)} \right. \\ & \quad \quad \quad \quad - \left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l} \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) - \phi f\left( {1 - l} \right). \\ \end{aligned} $$

(A9)

$\left. b \right|_{{\dot{b} = 0}}$ is a real number, and the value in the root must be positive. We assume that $\phi \le 0$, which shows that the government has an automatic stabilizing or discretionary policy, which is consistent with the estimates of recent empirical studies.

Deriving the value of $b$ in the case of $l = 0$ and $l = 1$ shows that

$$ \left. b \right|_{{\dot{b} = 0,l = 0}} = \frac{1}{2\beta }\left[ { - \left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right) + \beta + \phi f\left( 1 \right)} \right) \pm \sqrt {\left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right) + \beta + \phi f\left( 1 \right)} \right)^{2} - 4\beta \phi f\left( 1 \right)} } \right], $$

(A8′)

$$ \begin{array}{*{20}c} {\left. b \right|_{{\dot{b} = 0,l = 1}} = 0, - 1,} \\ \end{array} $$

(A8″)

$$ \left. b \right|_{{\dot{l} = 0,l = 0}} \begin{array}{*{20}c} { = \frac{1}{\beta }\left[ {\frac{{\left( {1 - \tau_{r} } \right)\left( {f\left( 1 \right) - f^{\prime}\left( 1 \right)} \right) - \rho }}{1 - \gamma } - \left( {1 - \tau_{r} } \right)\left( {f\left( 1 \right) - f^{\prime}\left( 1 \right)} \right) - \left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right) - \phi f\left( 1 \right)} \right],} \\ \end{array} $$

(A9′)

$$ \begin{array}{*{20}c} {\left. b \right|_{{\dot{l} = 0,l = 1}} = + \infty .} \\ \end{array} $$

Since $b$ must be positive, from Eqs. (A8′) and (A8″), we must consider only the case as follows:

$$ \left. b \right|_{{\dot{b} = 0,l = 0}} = \frac{1}{2\beta }\left[ { - \left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right) + \beta + \phi f\left( 1 \right)} \right) + \sqrt {\left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right) + \beta + \phi f\left( 1 \right)} \right)^{2} - 4\beta \phi f\left( 1 \right)} } \right], $$

(A9″)

$$ \left. b \right|_{{\dot{b} = 0,l = 1}} = 0. $$

We can always see that

$$ - \left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right) + \beta + \phi f\left( 1 \right)} \right) < \sqrt {\left( {\left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right) + \beta + \phi f\left( 1 \right)} \right)^{2} - 4\beta \phi f\left( 1 \right)} $$

and

$$ \left. b \right|_{{\dot{b} = 0,l = 0}} > 0. $$

The conditions which satisfy the existence of a BGP is represented as follows:

$$ \left. b \right|_{{\dot{b} = 0,l = 0}} > \left. b \right|_{{\dot{l} = 0,l = 0}} . $$

With calculation, we obtain the condition of $\left. b \right|_{{\dot{b} = 0,l = 0}} > \left. b \right|_{{\dot{l} = 0,l = 0}}$ as follows:

$$ \beta > \frac{{\left[ {\rho - \left( {1 - \tau_{r} } \right)\left( {f\left( 1 \right) - f^{\prime}\left( 1 \right)} \right)\gamma + \left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right)\left( {1 - \gamma } \right) + \phi f\left( 1 \right)\left( {1 - \gamma } \right)} \right]\left( {\rho - \left( {1 - \tau_{r} } \right)\left( {f\left( 1 \right) - f^{\prime}\left( 1 \right)} \right)\gamma } \right)}}{{\left[ {\rho - \left( {1 - \tau_{r} } \right)\left( {f\left( 1 \right) - f^{\prime}\left( 1 \right)} \right)\gamma + \left( {1 - \tau_{w} } \right)f^{\prime}\left( 1 \right)} \right]\left( {1 - \gamma } \right)}} > 0. $$

(A10)

Therefore, if $\beta$ satisfies inequalities (A10), there exists at least one intersection in the range $l \in \left( {0,1} \right)$ and $b \ge 0$.

If Eq. (A8) is monotonic decreasing function, and Eq. (A9) is monotonic increasing function, we can obtain the uniqueness of the BGP. We assume a production function as CES. The elasticity of substitution is defined as

$$ \sigma \equiv \frac{{F_{L} F_{K} }}{{FF_{LK} }} = \frac{{ - f^{\prime}\left( {1 - l} \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right)}}{{f\left( {1 - l} \right)f^{\prime\prime}\left( {1 - l} \right)\left( {1 - l} \right)}}, $$

and rewriting this equation, we obtain

$$ \begin{array}{*{20}c} {f^{\prime\prime}\left( {1 - l} \right) = - \frac{{f^{\prime}\left( {1 - l} \right)\left( {f\left( {1 - l} \right) - \left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)} \right)}}{{f\left( {1 - l} \right)\left( {1 - l} \right)\sigma }}.} \\ \end{array} $$

(A11)

Moreover, the share of labor income is defined as

$$ \alpha \equiv \frac{{\left( {1 - l} \right)f^{\prime}\left( {1 - l} \right)}}{{f\left( {1 - l} \right)}}, $$

and rewriting this equation, we obtain

$$ \begin{array}{*{20}c} {f^{\prime}\left( {1 - l} \right) = \frac{{\alpha f\left( {1 - l} \right)}}{1 - l}.} \\ \end{array} $$

(A12)

Then, inserting Eqs. (A12) into (A10a), we obtain

$$ \beta > \frac{{\left[ {\rho - \left( {\left( {1 - \tau_{r} } \right)\left( {1 - \alpha } \right)\gamma - \left( {\left( {1 - \tau_{w} } \right)\alpha + \phi } \right)\left( {1 - \gamma } \right)} \right)f\left( 1 \right)} \right]\left( {\rho - \left( {1 - \tau_{r} } \right)\left( {1 - \alpha } \right)f\left( 1 \right)\gamma } \right)}}{{\left[ {\rho - \left( {\left( {1 - \tau_{r} } \right)\left( {1 - \alpha } \right)\gamma - \left( {1 - \tau_{w} } \right)\alpha } \right)f\left( 1 \right)} \right]\left( {1 - \gamma } \right)}} > 0. $$

(A10a)

Differentiating Eqs. (A8) and (A9) respect to $l$, using Eqs. (A11) and (A12), we obtain

$$ \begin{aligned} \frac{{\partial \left. b \right|_{{\dot{b} = 0}} }}{\partial l} & & \quad = \frac{{\alpha f\left( {1 - l} \right)}}{1 - l}\frac{1}{2\beta } \\ \times \left[ { - \left( {\left( {1 - \tau_{w} } \right)\left( {\frac{1 - \alpha }{\sigma }\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) - \left( {\frac{1 + \eta }{\eta }} \right)} \right) - \phi } \right)} \right. \\ & \quad \quad \left. { + \frac{{\left( {\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta + \phi f\left( {1 - l} \right)} \right)\left( {\left( {1 - \tau_{w} } \right)\left( {\frac{1 - \alpha }{\sigma }\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) - \frac{1 + \eta }{\eta }} \right) - \phi } \right) + 2\beta \phi }}{{\sqrt {\left( {\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) + \beta + \phi f\left( {1 - l} \right)} \right)^{2} - 4\beta \phi f\left( {1 - l} \right)} }}} \right] \\ \end{aligned} $$

and

$$ \frac{{\partial \left. b \right|_{{\dot{l} = 0}} }}{\partial l} = \frac{{\alpha f\left( {1 - l} \right)}}{1 - l}\frac{1}{\beta }\left[ { - \left( {\left( {1 - \tau_{r} } \right)\left( {1 - l} \right)\frac{\gamma }{1 - \gamma } + \left( {1 - \tau_{w} } \right)\left( {1 - l} \right)\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right)} \right)\frac{{\left( {1 - \alpha } \right)}}{{\left( {1 - l} \right)\sigma }} + \left( {1 - \tau_{w} } \right)\left( {\frac{1 + \eta }{\eta }} \right) + \phi } \right]. $$

Therefore, if $l$ is satisfying following condition, we obtain $\partial \left. b \right|_{{\dot{b} = 0}} /\partial l < 0$ and $\partial \left. b \right|_{{\dot{l} = 0}} /\partial l > 0$:

$$ \left( {1 - \tau_{w} } \right)\left( {\frac{1 - \alpha }{\sigma }\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) - \left( {\frac{1 + \eta }{\eta }} \right)} \right) > \phi > \left( {1 - \tau_{r} } \right)\frac{\gamma }{1 - \gamma }\frac{1 - \alpha }{\sigma } + \left( {1 - \tau_{w} } \right)\left( {\frac{1 - \alpha }{\sigma }\left( {1 - \frac{l}{{\left( {1 - l} \right)\eta }}} \right) - \left( {\frac{1 + \eta }{\eta }} \right)} \right). $$

(A10b)

As we established, there exists a unique BGP with conditions (A10a) and (A10b).

A3: Proof of Proposition 2

Linearizing Eqs. (13) and (14) around the steady state, we obtain

$$ \begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} {\dot{l}} \\ {\dot{b}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {l - l^{*} } \\ {b - b^{*} } \\ \end{array} } \right),} \\ \end{array} $$

(A13)

where

$$ a_{11} = \frac{{G_{l} \left( {l^{*} ,b^{*} } \right)}}{{H\left( {l^{*} } \right)}} = \frac{{\alpha f\left( {1 - l^{*} } \right)\left[ {\left( {1 - \tau } \right)\left( {\frac{1 - \alpha }{\sigma }\left( { - \frac{1}{1 - \gamma } + \frac{l}{{\eta \left( {1 - l} \right)}}} \right) + \frac{1 + \eta }{\eta }} \right) + \phi } \right]}}{{\left( {1 - l^{*} } \right)H\left( {l^{*} } \right)}} > 0, $$

(A14a)

$$ a_{12} = \frac{{G_{b} \left( {l^{*} ,b^{*} } \right)}}{{H\left( {l^{*} } \right)}} = - \frac{\beta }{{H\left( {l^{*} } \right)}} < 0, $$

(A14b)

$$ a_{21} = \frac{{\alpha f\left( {1 - l^{*} } \right)b^{*} }}{{1 - l^{*} }}\left[ { - \left( {1 - \tau } \right)\left( {\frac{1 - \alpha }{\sigma }\left( {1 - \frac{{l^{*} }}{{\left( {1 - l^{*} } \right)\eta }}} \right) - \frac{1 + \eta }{\eta }} \right) + \left( {1 + b^{* - 1} } \right) \phi } \right] < 0, $$

(A14c)

$$ \begin{array}{*{20}c} {a_{22} = - \beta b^{*} + \phi f\left( {1 - l^{*} } \right)b^{* - 1} < 0.} \\ \end{array} $$

(A14d)

Setting the Jacobian of Eq. (A13) as $J$, the determinant of the Jacobian is

$$ \begin{aligned} \det J & = a_{11} a_{22} - a_{21} a_{12} \\ = \frac{{\alpha f\left( {1 - l^{*} } \right)}}{{\left( {1 - l^{*} } \right)H\left( {l^{*} } \right)}}\left[ {\left( {\left( {1 - \tau } \right)\left( {\frac{1 + \eta }{\eta } + \frac{{\left( {1 - \alpha } \right)l^{*} }}{{\left( {1 - l^{*} } \right)\sigma \eta }}} \right) + \phi } \right)\left( { - \beta b^{*} + b^{* - 1} \phi f\left( {1 - l^{*} } \right)} \right) + \left( { - \left( {1 - \tau } \right)\frac{1 - \alpha }{\sigma } + b^{* - 1} \phi } \right)\beta b^{*} } \right] < 0. \\ \end{aligned} $$

As it is established that $\det J < 0$, the steady state is a saddle point.

A4: Derivation of the dynamics of relative wealth

Linearizing Eq. (21) around steady state $l^{*} , b^{*}$ and $\omega_{i}^{*}$, we obtain

$$ \begin{aligned} \dot{\omega }_{i} \left( t \right) & = \frac{{1 - \tau_{w} }}{{1 + b^{*} }}\left[ {f^{\prime}\left( {1 - l^{*} } \right)\left( {\omega_{i}^{*} - z_{i} } \right)\left( {1 + \frac{1}{\eta }} \right) - f^{\prime\prime}\left( {1 - l^{*} } \right)\left( {1 - \omega_{i}^{*} + \left( {\omega_{i}^{*} - z_{i} } \right)l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right)} \right]\left( {l - l^{*} } \right) \\ & \quad + \frac{{\left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l^{*} } \right)}}{{\left( {1 + b^{*} } \right)^{2} }}\left( { - 1 + \omega_{i}^{*} - \left( {\omega_{i}^{*} - z_{i} } \right)l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right)\left( {b - b^{*} } \right) \\ & \quad + \frac{{\left( {1 - \tau_{w} } \right)f^{\prime}\left( {1 - l^{*} } \right)}}{{1 + b^{*} }}\left( { - 1 + l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right)\left( {\omega_{i} - \omega_{i}^{*} } \right). \\ \end{aligned} $$

(A15)

By inserting Eqs. (A11) and (A12) into Eq. (A15), we obtain

$$ \begin{aligned} \dot{\omega }_{i} \left( t \right) & = \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l^{*} } \right)}}{{\left( {1 - l^{*} } \right)\left( {1 + b^{*} } \right)}}\left[ {\left( {\omega_{i}^{*} - z_{i} } \right)\left( {1 + \frac{1}{\eta }} \right) + \frac{1 - \alpha }{{\left( {1 - l^{*} } \right)\sigma }}\left( {1 - \omega_{i}^{*} + \left( {\omega_{i}^{*} - z_{i} } \right)l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right)} \right]\left( {l - l^{*} } \right) \\ & \quad + \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l} \right)}}{{\left( {1 - l} \right)\left( {1 + b^{*} } \right)^{2} }}\left( { - 1 + \omega_{i}^{*} - \left( {\omega_{i}^{*} - z_{i} } \right)l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right)\left( {b - b^{*} } \right) \\ & \quad + \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l} \right)}}{{\left( {1 - l} \right)\left( {1 + b^{*} } \right)}}\left( { - 1 + l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right)\left( {\omega_{i} - \omega_{i}^{*} } \right). \\ \end{aligned} $$

(A15′)

Therefore, from Eq. (A15′), the stable adjustment path of Eq. (21) is found to be

$$ \begin{array}{*{20}c} {\omega_{i} \left( t \right) = \frac{{a_{31} \left( {l\left( 0 \right) - l^{*} } \right) + a_{32} \left( {b\left( 0 \right) - b^{*} } \right)}}{{\mu - a_{33} }}e^{\mu t} + \omega_{i}^{*} ,} \\ \end{array} $$

(A16)

where using Eq. (19b), we obtain

$$ a_{31} = \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l^{*} } \right)}}{{\left( {1 + b^{*} } \right)\left( {1 - l^{*} } \right)}}\left[ {\left( {\omega_{i}^{*} - z_{i} } \right)\left( {1 + \frac{1}{\eta }} \right) + \frac{1 - \alpha }{{\left( {1 - l^{*} } \right)\sigma }}\left( {1 - \omega_{i}^{*} + \left( {\omega_{i}^{*} - z_{i} } \right)l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right)} \right], $$

$$ a_{32} = \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l} \right)}}{{\left( {1 - l} \right)\left( {1 + b^{*} } \right)^{2} }}\left( { - 1 + \omega_{i}^{*} - \left( {\omega_{i}^{*} - z_{i} } \right)l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right), $$

$$ a_{33} = \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l} \right)}}{{\left( {1 + b^{*} } \right)\left( {1 - l^{*} } \right)}}\left( { - 1 + l^{*} \left( {1 + \frac{1}{\eta }} \right)} \right) > 0. $$

Setting $t = 0$ in Eq. (A16) yields

$$ \begin{array}{*{20}c} {\omega_{i,0} = \frac{{a_{31} \left( {l\left( 0 \right) - l^{*} } \right) + a_{32} \left( {b\left( 0 \right) - b^{*} } \right)}}{{\mu - a_{33} }} + \omega_{i}^{*} .} \\ \end{array} $$

(A17)

Using Eqs. (22′) and (23), we obtain

$$ z_{i} = \frac{{\eta \left( {1 - \omega_{i}^{*} } \right)}}{{l^{*} \left( {1 + \eta } \right)}} + \omega_{i}^{*} . $$

Inserting this equation into Eq. (A17) leads to

$$ \begin{array}{*{20}c} {\omega_{i,0} = \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l^{*} } \right)\left( {1 - \omega_{i}^{*} } \right)\left( {l\left( 0 \right) - l^{*} } \right)}}{{\left( {1 - l^{*} } \right)l^{*} \left( {1 + b^{*} } \right)\left( {a_{33} - \mu } \right)}} + \omega_{i}^{*} .} \\ \end{array} $$

(A17′)

When $l\left( 0 \right)$ and $\omega_{i,0}$ are given, $l^{*}$ and $b^{*}$ are obtained in the steady state. When $z_{i}$ is determined, Eq. (A17′) represents the distribution of wealth in the steady state. By inserting Eq. (A17′) into Eq. (A16), the stable adjustment path of the relative wealth is found to be

$$ \begin{array}{*{20}c} {\omega_{i} \left( t \right) = \frac{{\left( {1 - \tau_{w} } \right)\alpha f\left( {1 - l^{*} } \right)}}{{\left( {1 - l^{*} } \right)\left( {1 + b^{*} } \right)\left( {a_{33} - \mu } \right)}}\left( {1 - \frac{l\left( 0 \right)}{{l^{*} }}} \right)\left( {\omega_{i}^{*} - 1} \right)e^{\mu t} + \omega_{i}^{*} .} \\ \end{array} $$

(A16′)

A5: Supplementary material

Numerical simulation code is available on https://dataverse.harvard.edu/dataverse/harvard.