If B = 2H (i.e. home maintenance gets the minimum wage), then for y = B,
= 1/3. This means that the partner remains exempt from taxes as long as spillover is limited to a third of income. Interestingly, at that point also the taxable income of the out partner is yo = (B - H) /
= 3 H so that he does not pay taxes either (since x = B + H = 3H here).
Above relationships show that individual taxation is possible that takes into account household spillover effects. For us the issue is primarily interesting for complications about subsistence. We find that there are no great complications, and we thus will further neglect the issue of partners.
Differential indexation
With subsistence indexed on income and taxes indexed on inflation, there is differential indexation, and due to the tax structure there is a multiplier increase in the minimum wage. Required gross minimum M shows a relative rise compared to other incomes, and it rises faster than both net minimum B and the general level of income Y/LE. In Figure 10 (in Book III), when we subtract the inflation component from x, B and M, then differential indexation shows up as: x stays fixed, B moves with the income density, M moves to the right, and M, as the intersection of the subsistence and tax lines, moves up more speedily. If productivity in the lower earnings scales doesn’t rise faster than general productivity or income, then ever more people grow unemployed.
For all clarity we shall prove this. This chapter uses the specific tax function (chapter 39 will give a proof independent of form). First we will show that M grows faster than B, and then we will show that M grows faster than productivity too, causing unemployment.