Thus the average individual would not be unharmed by the introduction of tax arbitrage. Even if she does not engage in asset trade, she would be harmed by the tax increase and thus suffer a fall in utility. The distribution of Л over income groups would therefore look like the dashed curve in Figure 1, showing a utility loss in the middle of the income distribution.
On the other hand, we can not rule out that the introduction of tax arbitrage leads to a budget surplus rather than a deficit. This would happen if the reduction of marginal tax rates for people with high wage rates leads to such a large increase in the labor supply among those people that it outweighs the fall in labor supply among those with low wage rates.12 Then the tax base would increase, and the government could reduce taxes. Thus the average individual would gain from the introduction of tax arbitrage, as would everybody else. The distribution of utility gains would then look like the dotted curve in Figure 1.
How likely are these different cases? This is of course an empirical question. In any case, it can be shown that for the particular parameterization employed above (logarithmic preferences, a constant-progressivity tax schedule), the introduction of tax arbitrage will always necessitate a tax increase. Thus, if the distribution F(w) is continuous, the people in the middle of the distribution will always lose. This worker exists. What happens if this is not the case? Clearly, removal of people who do not exploit the scope for tax arbitrage, but pay additional taxes, increases the probability that there will be only winners from the introduction of tax arbitrage.
Consider an economy with two equally sized groups of workers, low-ability workers with a wage rate wL = \-k and high-ability workers with a wage rate wH = 1 + к. Solving the model for this setup, and for logarithmic preferences and a constant v, we can compute the utility gains AL and AH for low- and high-income earners from going from a regime with no tax arbitrage to a regime where arbitrage is possible. The results for the case of a = 1 in (v, k) space is shown in Figure 2.