u. [47]
When all incomes grow as fast
Before we continue it is useful, however, to first clarify a formal property for the Bentham tax function.
Property (13.3e): For the Bentham tax function: There is equal growth of gross and net income, if and only if exemption is indexed on either.
Note: The distinction between (13.3d) and (13.3e) is that the former indexes x[0] on P only, and the latter indexes x[0] and B[0] on wi = P rwi.
Corrollary: Under (13.3e): If the income distribution remains the same (all incomes grow with the same rate) then also the average income, y = W grows at the same rate, and then also the net income distribution remains the same, and then the ratio of net average to subsistence remains the same too. Note: Western nations thus could wisely index subsistence and exemption on gross average income.
Note: These relations seem obvious enough, but actually proving it turned out to be a bit tedious.
Proof: Denote y[+1] = (1+gr) y = g y for growth rate gr, and Net[y[+1] = n Net[y] (both g and n one period indices).
Net income with the Bentham tax is Net[y[+1] = g y - r (g y - X) with X the new exemption. This should be equal to n Net[y] = n (y - r (y - x)). Thus n is defined by:
g y - r (g y - X) = n (y - r (y - x))