$50m, which in the benchmark calibration corresponds to a linear tax rate of 12%. The tax government rebates the tax lump sum every year, so the dynamics of wealth are now:

$$dw_{it} = (\mu(w_{it}) - \tau(w_{it}) + \bar{\tau})dt + \sigma_t(w_{it})dB_{it}$$

where $\bar{\tau}$ is the average tax revenue. The steady-state wealth distribution is now characterized by:

$$f^*(w) \propto \theta(w)[\lambda(w)]^{\bar{\tau}}f(w) \quad \text{where} \quad \lambda(w) = \exp\left\{2 \int_{-\infty}^{w} \frac{1}{\sigma^2(s)} ds\right\}$$

and where $\theta(w)$ is defined similarly as in the no-rebate case (and, in the current example, accounts for behavioral responses.) The value $\bar{\tau}$ must satisfy the equation:

$$\bar{\tau} - \frac{\int_{-\infty}^{+\infty} \tau(w)\theta(w)[\lambda(w)]^{\bar{\tau}}f(w) dw}{\int_{-\infty}^{+\infty} \theta(w)[\lambda(w)]^{\bar{\tau}}f(w) dw} = 0$$

which can be solved numerically. Figure C.6 shows the results from this exercise. It reports the wealth shares of four brackets (the bottom 50%, the middle 40%, the next 9%, and the top 1%) and four scenarios (no wealth tax, wealth tax without rebate, wealth tax with rebate, broader wealth tax with rebate). The wealth distribution remains highly unequal in all cases: the bottom 50% has virtually no wealth. Still, the wealth tax operates a significant amount of redistribution. The wealth tax alone primarily redistributes away from the top 1% to the benefit of the middle 40% and the next 9%. But it does not affect the bottom 50%. With the rebate, the bottom 50% more than doubles its share, but from a very low baseline (0.5%, to 1.3%). The tax with a broader base (all wealth above $1m) increases it a bit more, to 3.6%.

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