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Finance and Economics Discussion Series: Accessible Versions of Figures in 001031*

Capital Taxation with Entrepreneurial Risk

Vasia Panousi

Figure 1: Lorenz Curves for Wealth and Consumption

This figure plots the Lorenz curves for the model's aggregate wealth and consumption distributions. The Lorenz curve for wealth is on panel (a) on the left. The Lorenz curve for consumption is on panel (b) on the right. The model produces results in the right direction, in that the distribution of wealth is much more unequal than the distribution of consumption. The model's Gini coefficient for wealth, conditional on wealth being positive, is 0.62. The model's Gini coefficient for consumption is 0.15.

Figure 2: Wealth Distribution for Entrepreneurs and Laborers

This figure presents the model's conditional wealth distributions over entrepreneurs and laborers. On the horizontal axis is wealth normalized by mean annual income in the economy. On the vertical axis are frequencies. The solid line represents entrepreneurs, and the dashed line laborers. Consistent with the data, the distribution of wealth for the population of entrepreneurs displays a fatter tail than the one for laborers. This is due to the random-walk component that the uninsurable investment risk introduces into entrepreneurial wealth. Furthermore, the entrepreneurial wealth distribution is shifted to the right, and it has lower frequencies at lower levels of wealth. This is due to the higher mean return of the total entrepreneurial portfolio. Finally, the distributions of wealth for both groups have significant mass of people with wealth higher than fifty times mean income. In the model, the laborers at the right tail of the wealth distribution are former successful entrepreneurs.

Figure 3: Steady State and Capital-Income Taxation

This figure shows the behavior of the steady-state aggregates and welfare with respect to the capital-income tax, which is on the horizontal axis. Capital (top left panel (a)) and output (top right panel (b)) are inversely-U shaped with respect to the capital-income tax, and they reach a maximum when τK = 0.4. The same is true for employment, the capital-labor (capital per work-hour) ratio, and output per work-hour (not shown). The net interest rate (middle left panel (c)) increases with the tax, and that it tends to the discount rate, β = 0.024, as τK → 1. This is the demonstration of the precautionary saving motive mentioned in section 4: when the capital-income tax increases, the effective volatility of risk facing an entrepreneur decreases, which reduces the demand for precautionary saving, and therefore increases the interest rate. The risk premium (middle right panel (d)) reinforces this interpretation of the capital-income tax as providing insurance: when the tax increases, the precautionary saving motive becomes weaker, and therefore entrepreneurs are satisfied with a lower risk premium. The fraction of wealth held by entrepreneurs in the economy (bottom left panel (e)) is decreasing in the capital-income tax. This results from the combination of the weaker precautionary saving motive, and the fall in the risk premium. Aggregate welfare (bottom right panel (f)) is maximized at τK = 0.7, whether for entrepreneurs (solid line), laborers (dashed line), or the economy as a whole (dotted line). This is because of the combined direct insurance effect of the tax, through the reduction in σ (1-τK), and the effect of the tax on aggregates.

Figure 4: Robustness Checks

This figure presents robustness checks with respect to risk aversion, γ, and volatility, σ. The coefficient of relative risk aversion is on the horizontal axis of the left panel (a). The standard deviation of idiosyncratic investment risk is on the horizontal axis of on the right panel (b). On the vertical axis for both panels is the tax that maximizes the steady-state capital stock. When either the volatility of risk increases or risk aversion increases, the tax that maximizes the steady-state capital stock increases. These comparative statics also indicate that the main result of the paper is robust to the wide range of empirically plausible values of σ ∈ (0,1) and of γ ∈ (2,20]. In particular, for the low value of σ = 0.15, the capital-income tax that maximizes the steady-state capital stock is positive for all γ>2, and it is actually zero when γ = 2.

Figure 5: Dynamics of Incomplete vs. Complete Markets: Eliminating the Capital-Income Tax

This figure plots the impulse responses of the aggregate variables when the capital-income tax is eliminated. The solid line indicates incomplete markets. The dashed line indicates complete markets. The top left panel shows the behavior of labor supply, the top right panel the behavior of aggregate output, the middle left panel the behavior of consumption, the middle right panel the behavior of labor productivity, the bottom left panel the behavior of the investment-output ratio, and the bottom right panel the behavior of the net interest rate. Under complete markets, a permanent (unanticipated) tax cut leads to an immediate negative jump in consumption and an immediate positive jump in investment. Capital slowly increases and converges to a higher steady-state value, while consumption is initially lower and increases over time. In other words, the long-run increase in investment requires an initial period of lower consumption, which in turn allows for an immediate increase in investment as well. By contrast, under incomplete markets, the exact opposite is the case. In light of the main mechanism of the paper, investment decreases in the long run. This allows for an immediate increase in consumption, and therefore necessitates a fall in current investment. In particular, the investment-output ratio falls by more than 3 percentage units.

Figure 6: Welfare Implications of Eliminating the Capital-Income Tax

This figure presents the welfare implications of abolishing the capital-income tax for entrepreneurs (solid line) and laborers (dashed line). Panel (a) on the left shows the welfare implications for the current generation, taking into account the entire transitional dynamics of the economy towards the new steady-state, and panel (b) on the right shows the welfare implications for the generations alive at the new steady state. Financial wealth normalized by annual mean income is on the horizontal axis, and the compensating differentials are on the vertical axis. A negative number on the vertical axis indicates an agent who benefits from the reform: the agent would have to be paid to be indifferent between the old regime and the regime initiated by the impact of the policy change, hence the agent prefers the new regime with the zero capital-income tax. The left panel shows that current-generation poor agents, whether entrepreneurs or laborers, prefer the zero capital-income tax regime. As wealth increases, both entrepreneurs and laborers prefer the positive capital-income tax regime. Finally, the mean cost of eliminating the tax is higher for the middle-class agents than for the very rich. The right panel shows that, in the long run, both types of agents and at all wealth levels prefer the steady state with the positive tax, the rich less so than the poor, and the entrepreneurs less so than the laborers.

Figure 7: Immediate (SR) vs Long Run (LR): Human Wealth and Saving Returns

This figure explains the reasons for the cross-sectional differences in the welfare implications of eliminating the capital-income tax. The top row present the behavior of aggregate variables in the short run or current generation, and the bottom row the behavior of the same variables in the long run. The left panel is the response of human wealth, the middle panel is the response of the net interest rate, and the right panel is the response of the risk-adjusted return to saving. On the horizontal axis for all panels is the tax rate of the policy reform. In the short run, the decrease in the capital-income tax from τK = 0.25 to τK = 0 increases the demand for precautionary saving, and therefore leads to a fall in the interest rate. For poor agents, whether entrepreneurs or laborers, human wealth constitutes a significant part of total wealth, and hence they benefit from the elimination of the tax. Furthermore, poor agents do not benefit much from insurance directly, since they invest little or nothing in the risky asset. In the long run, the elimination of the capital-income tax increases the demand for precautionary saving, and it therefore leads to a fall in the interest rate. This fall in the interest rate reduces steady-state wealth and capital accumulation. It turns out that the fall in the steady-state capital stock dominates the fall in the interest rate, so that in the end steady-state human wealth falls. This adversely affects poor agents of all types, since human wealth represents a big part of their total wealth. Because the risk-adjusted return for entrepreneurs increases when the capital-income tax is eliminated, the cost of the policy change is not as high for an entrepreneur as it is for a laborer at any given level of wealth.

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