3.1. Economy

The economy is populated with two types of agents: entrepreneurs and workers. We normalize the measure of entrepreneurs to 1 and the measure of workers by $\lambda$ . We interpret $\lambda$ as the worker–entrepreneur ratio. There is a continuum of individuals for each type indexed by $i$ and $j$ . Each worker is endowed with $\frac{1}{\lambda }$ unit of time so that the aggregate labor endowment is 1. We denote $u_{t}$ as instantaneous utility. Each agent maximizes her expected lifetime utility subject to her own budget constraint and borrowing constraints. The aggregate productivity, $z_{t}$ , affects the production of each firm in the economy following an AR(1) process,

(2) \begin{equation} \log z_{t}=\rho _{z}\log z_{t-1}+{\varepsilon }_{t}^{z}, \end{equation}

where ${\varepsilon }_{t}^{z}$ is normally distributed, ${\varepsilon }_{t}^{z}\sim \mathrm{\mathcal{N}}(0,{\sigma }_{z}^{2})$ , and the autoregressive parameter, $\rho _{z}\in [0,1)$ .

3.2. Entrepreneurs

We assume that entrepreneurs only care about consumption. We specify a CRRA utility function $u\left({c}_{t}^{i}\right)=\frac{({c}_{t}^{i})^{1-\gamma }}{1-\gamma }$ , where $\gamma$ denotes the risk aversion degree. Each entrepreneur owns a private firm. At $t$ , an entrepreneur can invest capital, ${k}_{t}^{i}$ , in her own firm but not in other private firms. She can buy or sell risk-free government bonds, ${b}_{t}^{i}$ , as an alternative financial asset. Entrepreneurs do not work and receive asset gains as their unique source of income. Let ${r}_{t}^{i}$ and $R_{t}$ denote the return on the invested capital and the interest rate of bonds, and ${\tau }_{t}^{k}$ , the capital income tax rate. Effective wealth consists of the gross income from capital and bond holdings net of income taxation. An entrepreneur is also taxed by consumption taxes at the rate of ${\tau }_{t}^{c}$ . The budget constraint and nonnegativity constraints are as follows:

(3) \begin{align} (1+{\tau }_{t}^{c}){c}_{t}^{i}+{k}_{t}^{i}+{b}_{t}^{i}&=[(1-{\tau }_{t}^{k})({r}_{t}^{i}-\delta )+1]{k}_{t-1}^{i}+R_{t-1}{b}_{t-1}^{i}, \\[5pt] {c}_{t}^{i} \geq 0\; & \textrm{and}\; k^{i}_{t} \geq 0. \nonumber \end{align}

Entrepreneurs are allowed to borrow in the bond market, but the borrowing amount must fulfill the no-Ponzi condition,

(4) \begin{equation} \lim _{T\rightarrow \infty }{\mathbb{E}}_{t}\frac{{b}_{t-1+T}^{i}}{{\prod }_{s=0}^{T-1}R_{t-1+s}}=0. \end{equation}

3.3. Firms

Each firm hires labor in a competitive labor market, employs its owner’s capital, and produces consumption goods. We name the firm run by Entrepreneur $i$ also as $i$ . We assume the neoclassical production technology,

(5) \begin{equation} {y}_{t}^{i}=F({k}_{t-1}^{i},{n}_{t}^{i},{A}_{t}^{i})=({A}_{t}^{i})^{\alpha }({k}_{t-1}^{i})^{\alpha }({n}_{t}^{i})^{1-\alpha }, \end{equation}

which exhibits constant returns to scale with respect to $k$ and $n$ . Firm-level productivity specific to firm $i, {A}_{t}^{i}$ affects the final output, ${y}_{t}^{i}$ . ${A}_{t}^{i}$ consists of two components, an idiosyncratic investment risk, ${e}_{t}^{i}$ , and aggregate productivity, $z_{t}$ : $\log {A}_{t}^{i}=\log z_{t}-\frac{{\sigma }_{e}^{2}}{2}+{e}_{t}^{i}$ , where ${e}_{t}^{i}$ is independently and identically distributed among firms and across time while $z_{t}$ captures business cycles. The idiosyncratic risk, ${e}_{t}^{i}$ , is modeled as a normal, ${e}_{t}^{i}\sim \mathrm{\mathcal{N}}(0,{\sigma }_{e}^{2})$ , where ${\sigma }_{e}^{2}$ represents the variance of idiosyncratic investment risks. The correction term, $-\frac{{\sigma }_{e}^{2}}{2}$ , renders the average productivity of private firms equivalent to aggregate productivity. We define the profit of Firm $i$ as the revenue net of the labor cost,

(6) \begin{equation} {\pi }_{t}^{i}({k}_{t-1}^{i},{n}_{t}^{i},{A}_{t}^{i})={y}_{t}^{i}-w_{t}{n}_{t}^{i}, \end{equation}

where $w_{t}$ represents the wage rate at $t$ . A competitive labor market ensures a universal wage rate in each period that only depends on the aggregate productivity and aggregate allocations.

The assumption of i.i.d. idiosyncratic investment risks may seem restrictive. Nevertheless, it is necessary for tractability and may not be so unrealistic given that DeBacker et al. (Reference DeBacker, Heim, Panousi, Ramnath and Vidangos>2012) find that business income is much less persistent than labor income and that it is characterized by higher probabilities of extreme upward or downward mobility. For instance, conditional on leaving the starting business income decile, a household faces a 52% probability of moving to either of the two immediately adjacent deciles in a year. More strikingly, households starting at the lowest decile of the business income distribution face a 12% probability of transitioning to decile 8 or higher, whereas the corresponding probability is essentially zero for labor income. Hence, business income may transition from low to high amounts within a relatively short time, making the i.i.d. shock a reasonable abstraction.

3.4. Workers

Worker $j$ has preferences in consumption that are specified as $u\left({c}_{t}^{j}\right)=\frac{({c}_{t}^{j})^{1-\gamma }}{1-\gamma }$ . We assume that a worker shares the same curvature of consumption as an entrepreneur. Workers supply their labor in the competitive labor market, work in firms owned by entrepreneurs, and consume all earnings as hand-to-mouth agents. This hand-to-mouth assumption is not so unrealistic given that quite a number of households hardly own wealth other than a house. Workers provide identical working hours but differ in their labor efficiency, ${e}_{t}^{j}$ . The labor efficiency is independently and identically distributed across workers. We assume that idiosyncratic labor efficiency follows a persistent stochastic process,

(7) \begin{equation} \log {e}_{t}^{j}=\rho _{w}\log {e}_{t-1}^{j}+{\varepsilon }_{t}^{j}, \end{equation}

where $\rho _{w}\in [0,1)$ denotes the autocorrelation of labor efficiency and ${\varepsilon }_{t}^{j}$ is normally distributed, ${\varepsilon }_{t}^{j}\sim \mathrm{\mathcal{N}}(0,{\sigma }_{w}^{2})$ .

A worker’s personal labor income depends on the idiosyncratic labor efficiency and the wage rate. A worker is taxed by consumption taxes and labor income taxes, whose tax rates are ${\tau }_{t}^{c}$ and ${\tau }_{t}^{n}$ , respectively. The assumption of hand-to-mouth implies that the aggregate consumption and income share of workers do not depend on the income distribution of workers. The budget constraint for Worker $j$ is

(8) \begin{equation} \left(1+{\tau }_{t}^{c}\right){c}_{t}^{j}=\left(1-{\tau }_{t}^{n}\right)w_{t}\frac{1}{\lambda }{e}_{t}^{j}. \end{equation}

3.5. Government

Each period, the government finances government spending, $g_{t}$ , by levying the proportional taxes $\{{\tau }_{t}^{c},{\tau }_{t}^{k},{\tau }_{t}^{n}\}$ and issuing bonds $\{B_{t}\}$ ,

(9) \begin{equation} {\tau }_{t}^{c}\left(\int _{i}{c}_{t}^{i}+\int _{j}{c}_{t}^{j}\right)+\int _{i}{\tau }_{t}^{k}\left({r}_{t}^{i}-\delta \right){k}_{t-1}^{i}+\int _{j}{\tau }_{t}^{n}w_{t}\frac{1}{\lambda }{e}_{t}^{j}+B_{t}=g_{t}+R_{t-1}B_{t-1}, \end{equation}

where $B_{t}$ denotes the total amount of bonds issued at $t$ and paid off at $t+1$ .

The current paper intends to analyze the effect of the cyclical behavior of fiscal policy conditional on its long-run level. Thus, we assume that the government adjusts its policy over the cycles and sets rules for ${\tau }_{t}^{c}, {\tau }_{t}^{k}, {\tau }_{t}^{n}$ , and government spending as a share of output, $\textit{gshar}e_{t}$ , following Fernández-Villaverde et al. (Reference Fernández-Villaverde, Guerrón-Quintana, Kuester and Rubio-Ramírez>2015):

(10) \begin{equation} \begin{aligned} {\Omega} _{t}-\overline{{\Omega} }= \rho _{{\Omega} }\left({\Omega} _{t-1}-\overline{{\Omega} }\right)+m_{Y{\Omega} }\log \left(\frac{Y_{t-1}}{\overline{Y}}\right)&+m_{B{\Omega} }\left(\frac{B_{t-1}}{Y_{t-1}}-\overline{\textit{Bshare}}\right) +\exp \left(\sigma _{{\Omega},t}\right){\varepsilon }_{t}^{{\Omega} },\\[5pt] & \;\; {\varepsilon }_{t}^{{\Omega} }\sim \mathrm{\mathcal{N}}(0,1),{\enspace}t=1,2,\ldots, \end{aligned} \end{equation}

where $\textit{Bshare}$ represents the debt-to-output ratio and ${\Omega}$ denotes the four policy instruments, $\{\tau ^{c},\tau ^{k},\tau ^{n},\textit{gshare}\}$ ; terms with a bar above indicate steady-state levels. This specification embraces time-varying volatility,

(11) \begin{equation} \sigma _{{\Omega},t}=\left(1-\rho _{{\sigma _{{\Omega} }}}\right)\sigma _{{\Omega} }+\rho _{{\sigma _{{\Omega} }}}\sigma _{{\Omega},t-1}+\left(1-{\rho }_{\sigma _{{\Omega} }}^{2}\right)^{\frac{1}{2}}\eta _{{\Omega} }u_{{\Omega} },\,u_{{\Omega} }\sim \mathrm{\mathcal{N}}\left(0,1\right),{\enspace}t=1,2,\ldots \end{equation}

It applies a “volatility shock” and $\eta _{{\Omega} }$ is the standard deviation of the volatility shock to the instrument.

$m_{Y{\Omega} }$ and $m_{B{\Omega} }$ denote the responses of fiscal policy instruments to output and to debt, respectively. For instance, $m_{Y{\tau ^{k}}}\gt 0$ and $m_{B{\tau ^{k}}}\gt 0$ when these parameters are calibrated in the US’ case, which shows a cut in the capital income tax rate when facing low output and a low debt level. A change in $m_{Y{\Omega} }$ reflects a different degree—or even a different direction—of the responses of fiscal policy instruments over the business cycles.

Notice that labor income tax only affects workers’ income and consumption while capital income tax affects the welfare of both types of agents as indicated in the empirical result: it directly reduces the capital income of entrepreneurs and indirectly influences the wage and the consumption of workers by containing capital accumulation.

3.6. Timing

Every period, aggregate shocks and idiosyncratic risks hit the economy. The government then announces fiscal policy based on the current state. After observing the shocks and policy, workers supply labor while firms optimally choose the labor demand. Firms take labor and predetermined capital as inputs in the production of consumption goods. Entrepreneurs consume and save in forms of capital and bonds after they receive asset income and pay taxes. Workers pay labor income taxes and consume afterwards.

3.7. Stationarity

The current model would imply a nonstationary distribution of wealth and income. We add an exogenous death probability, $Pr_{d}$ , to all entrepreneurs to guarantee that every entrepreneur has limited income. Specifically, the death shock is assumed to happen after every entrepreneur makes her own decision each time. Some entrepreneurs die, while the same number of newborns enter into the economy. Every newborn inherits the average amount of risky and risk-free assets previously owned by the dead. We will show in the later section that entrepreneurs make saving and portfolio choices independent of income distribution, making these decisions only as functions of the aggregate state. Therefore, this assumption does not change the original choices before death and facilitates computation.

4.1. Equilibrium Definition

Denote $K_{t-1},B_{t-1},z_{t},{\Omega} _{t}$ as the aggregate state, $S_{t}$ and ${k}_{t-1}^{i},{b}_{t-1}^{i},{e}_{t}^{i}$ as the individual state for entrepreneur i, ${s}_{t}^{i}$ .

For expository purposes, we will mostly drop the state in each variable for the rest of the current paper and only use the expression with the state when necessary.

4.2. Individual Behavior

Because Firm $i$ chooses employment, ${n}_{t}^{i}$ , after observing the shock and after the capital stock is determined, ${n}_{t}^{i}$ is the only control variable to maximize profit. By constant returns to scale, the optimal firm employment and the maximized profit are linear in capital, as in Angeletos (Reference Angeletos>2007):

(12) \begin{equation} {n}_{t}^{i}=\left(\frac{1-\alpha }{w_{t}}\right)^{\frac{1}{\alpha }}{A}_{t}^{i}{k}_{t-1}^{i}=n\left({A}_{t}^{i},w_{t}\right){k}_{t-1}^{i}, \end{equation}

(13) \begin{equation} {\pi }_{t}^{i}=\left[\alpha \left(\frac{1-\alpha }{w_{t}}\right)^{\frac{1}{\alpha }-1}{A}_{t}^{i}\right]{k}_{t-1}^{i}=r\left({A}_{t}^{i},w_{t}\right){k}_{t-1}^{i}. \end{equation}

The firm experiences linear returns to investment by adjusting its employment according to a linear function to capital. Notice that the return to capital, ${r}_{t}^{i}=r({A}_{t}^{i},w_{t})$ , is expressed by firm-specific productivity, ${A}_{t}^{i}$ , and the economy-wide wage, $w_{t}$ , which can be further expressed by the aggregate productivity and total capital stock.

We denote the effective wealth of Entrepreneur $i$ in period $t$ by

(14) \begin{equation} {x}_{t}^{i}\equiv \left[(1-{\tau }_{t}^{k})(r({A}_{t}^{i},w_{t})-\delta )+1\right]{k}_{t-1}^{i}+R_{t-1}{b}_{t-1}^{i}. \end{equation}

We rewrite the budget constraint as $(1+{\tau }_{t}^{c}){c}_{t}^{i}+{k}_{t}^{i}+{b}_{t}^{i}={x}_{t}^{i}$ .

We characterize an entrepreneur’s behavior following Angeletos (Reference Angeletos>2007) in the following lemma, the proof of which is provided in Appendix A:

Lemma. Given prices and fiscal policy, the optimal consumption, capital stock in the private firm, and bond holdings are linear in effective wealth,

(15) \begin{equation} (1+{\tau }_{t}^{c}){c}_{t}^{i}=\nu _{t}{x}_{t}^{i}, \end{equation}

(16) \begin{equation} {k}_{t}^{i}=(1-\nu _{t})\phi _{t}{x}_{t}^{i}, \end{equation}

(17) \begin{equation} {b}_{t}^{i}=(1-\nu _{t})(1-\phi _{t}){x}_{t}^{i}, \end{equation}

where the marginal propensity to consume out of effective wealth, $\nu _{t}$ , and the share of private equity in the portfolio, $\phi _{t}$ , are two variables that depend solely on the current aggregate states, $S_{t}$ , and satisfying,

(18) \begin{equation} \phi _{t}=\mathop{\mathrm{arg\;max}}\limits_{\phi \in [0,1]}{\mathbb{C}\mathbb{E}}_{t}\left\{\phi _{t}[(1-{\tau }_{t+1}^{k})(r({A}_{t+1}^{i},w_{t+1})-\delta )+1]+(1-\phi _{t})R_{t}\right\}, \end{equation}

(19) \begin{equation} \begin{aligned} \frac{{\nu }_{t}^{-\gamma }}{(1+{\tau }_{t}^{c})^{1-\gamma }} & =\beta \left(1-\nu _{t}\right)^{-\gamma }{\mathbb{E}}_{t}\left\{\frac{{\nu }_{t+1}^{-\gamma }}{(1+{\tau }_{t+1}^{c})^{1-\gamma }}\left\{\phi _{t}\left[\left(1-{\tau }_{t+1}^{k}\right)\left(r\left({A}_{t+1}^{i},w_{t+1}\right)\right.\right.\right.\right.\\[5pt] & {\quad}\left.\left.\left.\left.-\delta \vphantom{{A}_{t+1}^{i}} \right)+1 \vphantom{1-{\tau }_{t+1}^{k}}\right]+(1-\phi _{t})R_{t}\right\}^{1-\gamma }\vphantom{\frac{{\nu }_{t+1}^{-\gamma }}{(1+{\tau }_{t+1}^{c})^{1-\gamma }}}\!\right\}, \end{aligned} \end{equation}

where ${\mathbb{C}\mathbb{E}}$ represents the certainty equivalent of an entrepreneur.

Define the value function for Entrepreneur $i$ as $V({x}_{t}^{i})$ ,

(20) \begin{equation} V\left({x}_{t}^{i}\right)=\frac{{\nu }_{t}^{-\gamma }({x}_{t}^{i})^{1-\gamma }}{(1-\gamma )(1+{\tau }_{t}^{c})^{1-\gamma }}. \end{equation}

The entrepreneur makes the optimal portfolio choice, $\phi _{t}$ , by maximizing risk-adjusted portfolio returns that are expressed by the certainty equivalent, ${\mathbb{C}\mathbb{E}}$ , of the portfolio return given the saving choice, $(1-\nu _{t})$ . If the return to capital is surely greater than the return to bonds, all savings are invested in capital. The uncertainty of the capital return, however, induces the entrepreneur to divide investment into both assets to ensure an interior solution to the portfolio problem. A change in the capital tax rate directly influences the portfolio choice since government bonds are free from taxation. Given the portfolio choice, the entrepreneur chooses the marginal propensity to consume according to the intertemporal condition to maximize lifetime utility. Because of i.i.d. idiosyncratic investment risks and the homogeneity of production and utility functions, entrepreneurs make saving and portfolio decisions independent of wealth distribution. Hence, the marginal propensity to consume and the share of private equity in the portfolio only depend on the aggregate state.

The hand-to-mouth worker consumes all after-tax labor income every period, implying that the consumption path is completely characterized by the budget constraint of the worker.

4.3. General Equilibrium

Recall that aggregate productivity, $z_{t}$ , and idiosyncratic risks, ${e}_{t}^{i}$ , are orthogonal. Aggregate labor demand and firm profit are given by ${N}_{t}^{D}=\tilde{n}(w_{t},z_{t})K_{t-1}$ and ${\Pi} _{t}=\tilde{r}(w_{t},z_{t})K_{t-1}$ , where

(21) \begin{equation} \tilde{n}\left(w_{t},z_{t}\right)\equiv {\int }_{\!\!\!-\infty }^{\infty }n\left({A}_{t}^{i},w_{t}\right)\,\mathrm{d}\mathrm{\mathcal{F}}\left({e}_{t}^{i}\right)=\left(\frac{1-\alpha }{w_{t}}\right)^{\frac{1}{\alpha }}z_{t}, \end{equation}

and

(22) \begin{equation} \tilde{r}\left(w_{t},z_{t}\right)\equiv {\int }_{\!\!\!-\infty }^{\infty }r\left({A}_{t}^{i},w_{t}\right)\,\mathrm{d}\mathrm{\mathcal{F}}\left({e}_{t}^{i}\right)=\alpha \left(\frac{1-\alpha }{w_{t}}\right)^{\frac{1}{\alpha }-1}z_{t}. \end{equation}

${\mathcal{F}}(\!\cdot\!)$ represents the distribution function of i.i.d. idiosyncratic investment risks. Thus, $\tilde{n}(w_{t},z_{t})$ denotes the average labor employed by a firm and $\tilde{r}(w_{t},z_{t})$ , the return to capital. Let ${N}_{t}^{S}=\int _{j}\frac{1}{\lambda }{e}_{t}^{j}=1$ represent the aggregate labor supply; labor market clearing requires that ${N}_{t}^{D}=\tilde{n}(w_{t},z_{t})K_{t-1}=1$ . The wage is expressed as $w_{t}=w(K_{t-1},z_{t})=(1-\alpha )(z_{t}K_{t-1})^{\alpha }$ . Further algebra shows that $r_{t}=\tilde{r}(w_{t},z_{t})=\alpha {z}_{t}^{\alpha }{K}_{t-1}^{\alpha -1}$ . We write the aggregate output as $Y_{t}={\Pi} _{t}+w_{t}{N}_{t}^{S}=r_{t}K_{t-1}+w_{t}={z}_{t}^{\alpha }{K}_{t-1}^{\alpha }$ . Aggregate allocations are also independent of wealth distribution because consumption, private investment, and bond holdings are linear in individual wealth and because idiosyncratic risks in every period are i.i.d. across firms and periods. The general equilibrium is determined by the following equations. We denote ${C}_{t}^{E}$ as the aggregate consumption of entrepreneurs and ${C}_{t}^{W}$ , of workers.

(23) \begin{equation} {C}_{t}^{E}+{C}_{t}^{W}+K_{t}+g_{t}={z}_{t}^{\alpha }{K}_{t-1}^{\alpha }+(1-\delta )K_{t-1}, \end{equation}

(24) \begin{equation} (1+{\tau }_{t}^{c}){C}_{t}^{W}=(1-{\tau }_{t}^{n})(1-\alpha ){z}_{t}^{\alpha }{K}_{t-1}^{\alpha }, \end{equation}

(25) \begin{equation} (1+{\tau }_{t}^{c}){C}_{t}^{E}=\nu _{t}\{[(1-{\tau }_{t}^{k})(r_{t}-\delta )+1]K_{t-1}+R_{t-1}B_{t-1}\}, \end{equation}

(26) \begin{equation} K_{t}=(1-\nu _{t})\phi _{t}\{[(1-{\tau }_{t}^{k})(r_{t}-\delta )+1]K_{t-1}+R_{t-1}B_{t-1}\}, \end{equation}

(27) \begin{equation} B_{t}=(1-\nu _{t})(1-\phi _{t})\{[(1-{\tau }_{t}^{k})(r_{t}-\delta )+1]K_{t-1}+R_{t-1}B_{t-1}\}, \end{equation}

These five equations, together with (18) and (19), solve for the seven variables ${C}_{t}^{E}, {C}_{t}^{W}, K_{t}, B_{t}, R_{t}, \nu _{t}$ , and $\phi _{t}$ , given the current aggregate state and fiscal policy.

5.1. Computation Process

We apply the second-order approximation to simulate a time series of equilibrium at the aggregate level over the business cycle starting from a certain state. We are able to accomplish it because every entrepreneur makes exactly identical saving and portfolio decisions regardless of their individual wealth. Thus, we can always find a representative entrepreneur to study the behavior of all entrepreneurs. Our model employs the exact aggregation as in the framework of Angeletos (Reference Angeletos>2007) in the setting with aggregate risks. This modeling approach with idiosyncratic investment risks computes the aggregate variables without considering wealth distribution, making it simple to study the dynamics of the aggregate economy over the business cycles with heterogeneous agents.

The model calculates the dynamics of entrepreneurs’ income distribution in a simple way. We divide aggregate allocations into the individual level. The next period’s capital and bonds held by entrepreneur $i$ can be calculated as a product of saving choice, portfolio choice, and effective wealth as in (16) and (17). The first two components are obtained from the time series of the general equilibrium. We separate the last part into idiosyncratic capital returns, interest rate of bonds, and amounts of capital and bonds held from the last period. The time series of equilibrium contains values for the interest rate and the average returns to capital. From (44) in Appendix B, we get the idiosyncratic returns to capital by inputting a stochastic process of idiosyncratic risks. The amount of assets forms a recursion given the idiosyncratic risks and asset prices at the aggregate level.

We create 2000 grids for the number of entrepreneurs, and, by assumption, $2000\lambda$ is the number of workers. We equally allocate the steady-state level of capital and bonds to 2000 entrepreneurs as the initial allocation. Because individual returns to capital are idiosyncratic to entrepreneurs, lucky entrepreneurs accumulate more and more assets so that the gap of the capital income rises over time. Yet the assumption of random death prevents infinite inequality. We record the income of every agent, either an entrepreneur or a worker, order all values, and then compute quantile statistics. We run a simulation with 500 periods and drop the first 250 periods to guarantee stability.

5.2. Calibration

A numerical analysis is required to discuss the welfare change and evaluate fiscal policy. This subsection first describes the details of calibrating the model.

We assume that a period corresponds to a quarter. We use some common parameter values in the literature on business cycles. The capital share of output, $\alpha$ , is assumed to be 0.34 and the risk aversion degree, $\gamma$ , 2. We compute the Solow residuals using quarterly data from the first quarter of 1973 to the last quarter of 2011.Footnote ² Then, we estimate the autoregressive parameter, $\rho _{z}$ , and the standard deviation of the technology process, $\sigma _{z}$ .

The probability of death, $Pr_{d}$ , for entrepreneurs is intended to match the working lifetime of an entrepreneur or the lasting time of a private firm. However, the statistic is unclear and hard to measure. We assume the probability of death to be 0.00625 for an expected working lifetime of 40 years. We calibrate the discount rate, $\beta$ , and the depreciation rate, $\delta$ , to produce the annual public-holding debt-to-GDP ratio of 37.60% and capital to GDP ratio of 2.7, as in the USA. We assume the worker–entrepreneur ratio to be 5 to match the estimated proportion of self-employed or business owners in the USA as reported by Cagetti and De Nardi (Reference Cagetti and De Nardi>2006).

It is challenging to find an exact measurement of idiosyncratic investment risks. DeBacker et al. (Reference DeBacker, Heim, Panousi, Ramnath and Vidangos>2012) find that the standard deviation of uninsurable idiosyncratic income risks from privately held businesses accounts for 45% of the average business income.Footnote ³ Therefore, we calibrate our model to match their finding that the cross-sectional standard deviation of an individual firm’s return is 45% of the average return. Notice that a number of macroeconomic studies, such as the one of Sandri (Reference Sandri>2014), apply the volatility of firm returns as 50% of the mean return. Thus, we ensure that our calibration is close to their choice and that our results are comparable.

We set the autocorrelation of idiosyncratic labor income risks and the standard deviation equal to 0.99 and 0.0097. All parameter values are quarterly counterparts of the estimation from Storesletten et al. (Reference Storesletten, Telmer and Yaron>2004).

We apply the same parameter values for the policy rule estimated by Fernández-Villaverde et al. (Reference Fernández-Villaverde, Guerrón-Quintana, Kuester and Rubio-Ramírez>2015). All the tax rates have positive correlations with output and debt, while the government spending share is negatively correlated with output and debt.

All the above parameter values for the baseline calibration are included in Table 2.

Table 2. Baseline calibration

5.3. Simulated income distribution

The income of Entrepreneur $i$ is defined as the asset gains, ${I}_{t}^{i}=({r}_{t}^{i}-\delta ){k}_{t}^{i}+(R_{t}-1){b}_{t}^{i}$ , while the income of Worker $j$ is the labor income, $\frac{1}{\lambda }w_{t}{e}_{t}^{j}$ . For comparison, we take the US income distribution from the CPS and compute the means of income shares over time, as shown in Table 3. The next column reports the means of income shares of different income groups obtained from the simulation. Our model fits the data in the sense that it generates considerable income inequality, even though the rich obtain less income and the poor share more compared with the data.

Table 3. Income inequality: level and cyclicality

Note: All the values of income shares are means computed using corresponding data from the Current Population Survey (CPS) from 1947 to 2013. The CPS provides the statistics of households only since 1967, so we pick up the income share of families from 1947 to 1966.

The last two columns of Table 3 report the correlations with output of the income shares owned by income groups in the data and in our model. We choose the same data from the CPS as for the level of income inequity and choose seasonally adjusted real GDP as measured by 2009 billion dollars during 1947 and 2013. We detrend the shares, ratios, and the logarithm of real GDP by using the HP-filter and calculate the correlations. For the model, we pick the time series of income shares of the corresponding income groups and obtain the correlations with output over the business cycles.

Our model with the baseline calibration quantitatively matches the correlation with output of income shares except for the 20–40% group and the 40–60% group. Our model reflects the small procyclicality for the bottom 20% and the small countercyclicality for the 60–80% and 80–95% groups, and it precisely duplicates the cyclical behavior of the income share of the top income group. It is still very challenging to generate decent cyclical behaviors of income distribution even after the emergence of many heterogeneous agent models. Our model provides a tractable and efficient framework to simulate the realistic dynamics of income distribution.

6.1. Objective and method of policy experiments

As the welfare criterion, we adopt the ex-ante utilitarian social utility in which the social planner sums up the lifetime individual utility across agents,Footnote ⁴

(28) \begin{align} {U}_{0}^{\textrm{SP}} & =\int_{i}{\mathbb{E}}_{0}\left[\sum\limits_{t=0}^{\infty }\beta ^{t}\frac{({c}_{t}^{i})^{1-\gamma }}{1-\gamma }\right]+\int _{j}{\mathbb{E}}_{0}\left[\sum\limits_{t=0}^{\infty }\beta ^{t}\frac{({c}_{t}^{j})^{1-\gamma }}{1-\gamma }\right] \nonumber \\[5pt] & =\int _{i}V\left({x}_{0}^{i}\right)+\int _{j}{\mathbb{E}}_{0}\left[\sum\limits_{t=0}^{\infty }\beta ^{t}\frac{({c}_{t}^{j})^{1-\gamma }}{1-\gamma }\right]. \end{align}

We assume that the initial conditions of capital and bond holdings are identical under different policy specifications, making the value function of entrepreneurs maps one-to-one with the marginal propensity to consume. Moreover, since the risk aversion degree, $\gamma$ , is larger than one in the following calibration, the lifetime utility for entrepreneurs increases with the marginal propensity to consume. In other words, more consumption conditional on certain effective wealth heightens the welfare of entrepreneurs.

The government searches for the proper direction and degree of responses of fiscal policy instruments over the business cycles by changing the responses of policy variables to output and comparing welfare in each scenario. We compute the consumption equivalent variation in the percentage of the aggregate lifetime utility across entrepreneurs and workers as the responses of capital income tax rate and labor income tax rate to output range from −0.02 to 0.08 and the responses of consumption tax rate and the government spending share range from −0.01 to 0.01 at the interval of 0.001.Footnote ⁵ The base to compute the consumption equivalent variation is the lifetime utility for each type of agents with the baseline calibration representing the US’ actual fiscal policy. The economy starts from a fixed state at which state variables are evaluated at unconditional means with the baseline calibration. The government considers the effect of a transition from the baseline policy on welfare.

6.2. Welfare conflict between entrepreneurs and workers

This subsection elaborates on the analysis of the welfare change of different groups as fiscal policy changes over the business cycles. The policy experiment shows that entrepreneurs and workers favor different responses of capital and labor income taxes in the qualitative sense, but this conflict fails to happen in the choice of consumption tax and government spending.

We simultaneously modify four parameter values that indicate the respective responses of capital income tax rate, labor income tax rate, consumption tax rate, and government spending to output. Table 4 shows the preferred policy mixes of workers, entrepreneurs, and all agents, respectively, and the welfare change in terms of the consumption equivalent variation.

Table 4. Welfare comparison of the change in all fiscal policy instruments (in percentage)

Note: The values are welfare changes in terms of consumption equivalent variation compared with the baseline policy combination ( $m_{Y{\tau ^{n}}}=0.04, m_{Y{\tau ^{k}}}=0.043, m_{Y{\tau ^{c}}}=0.001, m_{\textit{Ygshare}}=-0.004$ ).

Entrepreneurs and workers have diverse tastes in the cyclical behaviors of capital and labor income taxes but prefer the same consumption tax and government spending policies. In particular, workers maximize their welfare with a procyclical capital income tax and with a slightly countercyclical labor income tax, while the entrepreneurs’ welfare is optimized by the countercyclical capital income tax and the strongly countercyclical labor income tax. When one group reaches its maximal welfare, the interest of the other group is compromised by the certain fiscal policy. Workers are even worse off by 0.1%, which is a large change if we realize that only the cyclical properties instead of the long-run level of tax rates are modified. Although we leave the detailed analysis of the mechanism to the next subsection, the results clearly indicate that because two groups have different income sources, the timing choice of income taxes matters for their welfare. Thus, the welfare conflict takes place when the government chooses the responses of income tax rates to output.

On the contrary, the timing choice of consumption tax and government spending fails to cause such a welfare conflict. These two groups favor the same policies of consumption tax and government spending, a countercyclical consumption tax, and a countercyclical government spending, because the two policy instruments affect each group in the same way.

Recall that our empirical result presents the relationship between the responses of fiscal policy instruments to output and the welfare of different groups of people. The model generates the welfare conflict that people with different income sources even have qualitatively diverse preferences in the change of fiscal policy over the business cycles, in line with the empirical finding.

6.3. Mechanism

This subsection explains why workers and entrepreneurs prefer different responses of fiscal policy to output. Because both groups only care about consumption, more and smoother consumption results in higher welfare. However, these two goals may not be obtained at the same time. If this situation happens, one effect will outweigh the other, leading to either a rise or drop in welfare.

According to Table 4, workers favor a procyclical capital tax policy that magnifies the effect of fluctuations and a slightly countercyclical labor tax policy that cushions productivity shocks. The average capital income tax rate rises, and the mean level of government bonds also increases when compared with the baseline. The two changes lead to less capital stock. As a result, the wage decreases, driving the labor income to diminish. Yet the consumption of workers on average increases because of the lower average labor income tax rate and consumption tax rate. Entrepreneurs own less effective wealth because the majority of effective wealth, the capital, shrinks. Thus, they consume less.

The volatility of the consumption of workers decreases by 13% compared with the baseline policy. Workers enjoy more and smoother consumption, so they are better off than in the baseline case. Entrepreneurs see a lower mean consumption and lower volatility of consumption. Because the mean effect dominates the fluctuation effect, entrepreneurs are worse off.

The policy mix where the labor and capital income tax rates are strongly positively correlated with the output maximizes the welfare of entrepreneurs. With this policy, the average capital tax rate increases and the bonds decrease. Theoretically, the former policy prevents entrepreneurs from investing in capital, while the latter promotes capital accumulation. The average capital stock increases because of the stronger crowd-in effect and the mean wage rises thanks to higher capital. The labor income tax rate and the consumption tax rate experience an increase in their means. Entrepreneurs have more consumption because of more asset income, while workers see a decrease in their consumption because of a greater tax burden. This policy results in a lower volatility of the consumption of entrepreneurs but a high volatility of consumption of workers. Therefore, entrepreneurs are better off because of more and smoother consumption, yet workers are worse off because their consumption is lower and more turbulent.

Notice that capital income tax discourages entrepreneurs from accumulating capital, which affects both the effective wealth of entrepreneurs and the wage. Thus, capital income tax influences the consumption of entrepreneurs and workers. On the contrary, labor taxation only has an effect on workers’ consumption because only workers have labor income. The difference in income sources results in different consequences of income taxes on the welfare of two groups.

In contrast, agents with different income sources in our model are affected in the same way by consumption tax and government spending. These two policy instruments do not distinguish income sources, thus both entrepreneurs and workers prefer the same.

6.4. Time to modify fiscal policy

Chari et al. (Reference Chari, Christiano and Kehoe>1994) investigate optimal fiscal policy over the business cycle. They find that the correlation of an optimal labor income tax rate is positive with a technology shock while the correlation of an optimal capital income tax rate is negative in their baseline model with uncontingent debt and state-contingent taxes. In addition, when risk aversion reaches to a high level, the labor tax rate turns to be procyclical. They argue that it is efficient for these shocks that affect the government budget constraint to be absorbed mainly by the tax on private assets. Farhi (Reference Farhi>2010) discusses the same problem of optimal policy in incomplete markets with non-state-contingent debt and non-state-contingent taxes. He also finds that capital taxes have a large positive spike following a transition from the good state to the bad state and claims that capital taxes help buffer the impact of shocks on the government budget. The labor tax rate in his model drops with a low productivity.

The policy that maximizes the social welfare in our model features a 0.006 percentage-point decrease in the labor tax rate and a 0.037 percentage-point decrease in the capital tax rate compared to the baseline policy when the output drops by one percent. Basically, the government chooses a tradeoff between both groups. The result on the capital tax rate differs from the findings of Chari et al. (Reference Chari, Christiano and Kehoe>1994) and Farhi (Reference Farhi>2010). The capital tax maximizing social welfare varies in the same qualitative pattern as the one favoring entrepreneurs, which matches the observation that certain benefit groups may affect the government decisions.

Since the utilitarian social utility weighs less the equality compared to many other welfare criteria, the government possibly specifies a capital tax whose rate increases in the recession if the government puts sufficient emphasis on workers.

6.5. Comparison of inequality

We compare the extent of inequality among different fiscal policy instruments in this subsection. We choose three typical measurements of inequality, the Gini index, the 90/50 ratio, and the 50/20 ratio. Then, we decompose the inequality into two components—inequality within groups and among groups—by computing the Theil index. The policy that maximizes the welfare of workers reaches a level of inequality close to the lowest. But the differences of inequality among policies are small.

We modify the policy parameters in the baseline model to generate the dynamics of income distribution with different policies. The three inequality measurements are obtained from taking the mean of the corresponding indices of the generated distribution in each period and are presented in Table 5. We adopt the Gini index as the main criterion of inequality to determine the policy that results in the minimal inequality.

Table 5. Comparison of inequality

Note: The policy minimizing inequality has the policy parameters as $m_{Y{\tau ^{n}}}=0.01, m_{Y{\tau ^{k}}}=-0.02, m_{Y{\tau ^{c}}}=0.01, m_{\textit{Ygshare}}=-0.01$ .

The policy that minimizes inequality is the combination of a procyclical labor tax, a countercyclical capital tax, a countercyclical consumption tax, and countercyclical government spending ( $m_{Y{\tau ^{n}}}=-0.006, m_{Y{\tau ^{k}}}=0.036, m_{Y{\tau ^{c}}}=0.01, m_{\textit{Ygshare}}=-0.01$ ). This policy qualitatively differs from the actual policy and other counterfactual policies that maximize the welfare of one group or all agents. However, the level of inequality shown by the Gini index is relatively close to the one resulting from the policy maximizing workers’ welfare. Recall that workers outnumber entrepreneurs and moreover, workers have lower average income. When the policy caters to the benefit of workers, it certainly improves the welfare of relatively poor agents, thus the income distribution becomes more equal.

The policy that maximizes the social welfare leads to a level of inequality close to the level generated by the policy maximizing workers’ welfare. Since the utilitarian social utility regards every agent as equal, the policy preferred by the social planner is partial to the larger group, the group of workers. But the income distribution is still more unequal than the one resulting from the policy maximizing workers’ welfare, because the planner also considers the welfare of rich entrepreneurs. Although the policy that emphasizes equity seems to contradict the one that optimizes the social welfare in the qualitative cyclicality of policy instruments, the difference of resulted income distribution is small. Thus, there is no substantial conflict between the policy maximizing the social welfare and the one minimizing inequality. The policy preferred by the social planner can guarantee the equity to a certain extent. If the government weighs more the group of workers or the poor in the social utility, then the inequality will improve, even closer to the minimal level of inequality.

7.1. Starting from a crisis

When a country faces a recession—or even worse, a crisis—the government will most likely adjust its fiscal policy to stimulate the economy. Because the decision is made at the time when the output is lower than the normal level (the long-run mean), the transition cost may change the above result of the welfare conflict. This subsection aims to conduct a policy experiment under a crisis to see if the welfare conflict exists. The result confirms the finding.

Economists are used to applying a decline in the output to describe a crisis. The last financial crisis witnessed the US’ GDP drop from the preceding quarter by 8.9% in the third quarter of 2008. Thus, we reduce the output level by 10% from its long-run mean level to represent a large crisis as the starting point of the new simulation. Notice that the long-run targets to compute the output gap and the debt gap do not change. The base to compute the consumption equivalent variation is the lifetime utility for each type of agents with the baseline policy and the same starting point.

The result is presented in Table 6, indicating that the welfare conflict also happens in the case of low output. Workers and entrepreneurs split on preferred choices of implementing the capital income tax: workers advocate for a procyclical policy while entrepreneurs support the opposite. Both groups show qualitatively identical preferences in the time choice of the other three policy instruments. More importantly, choosing the cyclical behavior of fiscal policy at the time of the crisis results in a larger divergence between the welfare change of workers and entrepreneurs.

Table 6. Welfare change when a crisis happens

Note: The values for the welfare changes are percentage changes compared with the baseline case.

7.2. Long-run and short-run changes of fiscal policy

We have shown that short-run changes of income taxes lead to a welfare conflict. This subsection explores how the long-run and short-run changes of fiscal policy together affect the welfare of the two groups. The results show that entrepreneurs and workers experience different welfare changes when the government modifies the responses of fiscal policy to the output; besides, the change in the long-run levels aggravates the conflict in choosing the fiscal policy responses to the output.

From the beginning of the last financial crisis (the third quarter of 2008), the average effective capital and labor income tax rates and the consumption tax rate in the USA have decreased compared with the average levels before the crisis. The government spending share almost remains unchanged, while the debt rises dramatically. Accordingly, we recalibrate the model to capture the change in the long-run level of fiscal policy. We set the discount rate, $\beta$ , and the depreciation rate, $\delta$ , to produce the annual public-holding debt-to-GDP ratio, 62.68% and capital to GDP ratio, 2.7. Then, we compute the steady-state level of the output in this case. We list the parameter values that have been adjusted in Table 7.

Table 7. Recalibration

Table 8. Welfare change when both long-run and short-run properties change

Note: The values for the welfare changes are percentage changes compared with the baseline case.

We input the recalibrated parameters into the model and repeat the policy experiment in the last section. The base to compute the consumption equivalent variation is the lifetime utility for each type of agents with the newly calibrated parameters.

Table 8 confirms that the welfare conflict remains if the government changes the long-run and short-run properties of fiscal policy simultaneously. Workers favor a slightly countercyclical labor income tax and a procyclical capital income tax. With this policy combination, entrepreneurs suffer a welfare loss of −0.02 $\%$ , more than two times greater than the loss in the baseline case. Entrepreneurs prefer a strongly countercyclical labor income tax and a countercyclical capital income tax, with which workers also experience a welfare loss. In addition, the welfare gap between these two groups increases in each scenario compared with the experiment with only a change in the cyclical behavior.