Long-run Wealth Tax Simulator

How does this work?

For the complete analytical details, you can read the paper this simulator is based on: “Uncovering the Dynamics of the Wealth Distribution” by Thomas Blanchet (2022)  

Below is a brief explanation.

Modelisation of wealth accumulation

I assume that people’s wealth \(w_{it}\) follows a continuous time drift-diffusion process characterized by:

\[\mathrm{d} w_{it} = \mu(w_{it})\,\mathrm{d}t + \sigma(w_{it})\,\mathrm{d}B_{it}\]

where the parameters \(\mu(w_{it})\) and \(\sigma(w_{it})\) are directly estimated so as to reproduce the trajectory of the wealth distribution since the 1980s.

Introduction of a wealth tax: mechanical effect

Now, introduce a wealth tax \(\tau(w)\). Ignoring behavioral responses for the moment, the dynamic of wealth is now:

\[\mathrm{d} w_{it} = [\mu(w_{it}) - \tau(w_{it})]\,\mathrm{d}t + \sigma(w_{it})\,\mathrm{d}B_{it}\]

Blanchet (2022)    shows that if the original steady-state distribution of wealth was \(f(w)\), then the steady-state distribution of wealth with the tax is:

\[g(w) \propto f(w)\exp\left\{-\int_{-\infty}^w \frac{2\tau(s)}{\sigma^2(s)}\,\mathrm{d}s\right\}\]

Behavioral effects

First, I assume that, in response to a marginal tax rate \(\tau'(w)\), people only report a fraction \(\alpha(w)=[1-\tau'(w)]^\varepsilon\) of their wealth.

Then, I assume that, in response to a tax with marginal rate \(\tau'(w)\) on wealth, people increase their consumption by a factor \(\gamma(w)=[1 - \tau'(w)]^{-\eta}\).

Finally, I allow for changes to the wealth mobility parameter \(\sigma(w_{it})\), which I rewrite \(\beta\sigma(w_{it})\) with \(\beta=1\) by default.

Behavioral responses take the form a change to the drift term, which is analogous to a change in the effective rate of the wealth tax. The formula for the mechanical effect of the wealth tax can therefore be directly extended to account for behavioral effects. It becomes:

\[g(w) \propto f(w)\exp\left\{-\int_{-\infty}^w \frac{2\tau(s)\alpha(s)}{\beta^2\sigma^2(s)}\,\mathrm{d} s -\int_{-\infty}^w \frac{2c(s)[1 - \gamma(s)]}{\beta^2\sigma^2(s)}\,\mathrm{d} s\right\}\]

where \(c(w)\) is an estimate of average consumption calibrated to match the evolution of wealth inequality since the 1980s.

Practical implementation

This simulator uses a recent estimate of the wealth distribution in the United States from Saez, and Zucman (2022)    as the baseline long-run wealth distribution. This constitutes a reasonable approximation given that Blanchet (2022)    finds that the current wealth distribution is close to its steady state. This data takes the form of simplified microdata. I use discretized versions of the formulas above and use them to reweight the microdata appropriately.

When including a lump-sum rebate, an additive term \(\bar{\tau}\) is substracted from the tax. The value is of that term is set equal to the amount that is raised from the tax using a simple iterative algorithm: we start by estimating the revenue raised without the rebate, set the rebate equal to that amount, recalculate the effect on the distribution with the rebate, recalculate the new amount of tax raised, recalculate the rebate, and repeat until convergence.

To construct the last graph (tax revenue as a function of the average marginal tax rate), I alter the tax schedule provided by the user as follows. For any bracket \(k\) with marginal rate \(\tau_k\), and for any average marginal tax rate \(\bar{\tau}\), I define \(\tau^*_k = \tau_k + \phi(\bar{\tau})(1-\tau_k)\) where \(\phi(\bar{\tau})\) is chosen so that the average marginal tax rate of the tax schedule defined by the user’s thresholds and the marginal rates \((\tau_1, \dots, \tau_k)\) is equal to \(\bar{\tau}\).