Technical Appendix - The Welfare Cost of Tax Rate Uncertainty

Proof of Optimal Leisure Formula¶

Consider the optimization problem:

\max_{L} U(L, w(1-\tau)(T-L) + v)

For Cobb-Douglas utility $U(L,C) = L^\alpha C^\beta$ , the first-order condition is:

\alpha L^{\alpha-1}C^\beta = \beta L^\alpha C^{\beta-1} w(1-\tau)

Simplifying:

\frac{\alpha C}{\beta L} = w(1-\tau)

Substituting the budget constraint $C = w(1-\tau)(T-L) + v$ :

\frac{\alpha[w(1-\tau)(T-L) + v]}{\beta L} = w(1-\tau)

Solving for L:

L^* = \frac{\alpha[w(1-\tau)T + v]}{w(1-\tau)(\alpha + \beta)}

Let $\tau \sim F(\tau)$ be the distribution of tax rates. Under certainty, expected utility is:

EU_c = \int U(L^*(\tau), C^*(\tau)) dF(\tau)

Under uncertainty, agents choose based on $E[\tau]$ :

EU_u = \int U(L^*(E[\tau]), C(\tau, L^*(E[\tau]))) dF(\tau)

The deadweight loss equals:

DWL = EU_c - EU_u

By Taylor expansion around $E[\tau]$ :

DWL \approx \frac{1}{2}\text{Var}(\tau) \cdot U''(\tau)|_{\tau=E[\tau]}

The effect of uncertainty on optimal tax rates:

\frac{\partial \tau^*}{\partial \sigma_\tau} = -\frac{\partial^2 W/\partial \tau \partial \sigma}{\partial^2 W/\partial \tau^2} < 0

This confirms that optimal tax rates decrease with uncertainty.