Let us first derive the real subsistence index rsi again, but now for the nonlinear tax. Recall the definitions of Book III. Let B = rsi P B[0] with B[0] subsistence in the base year. Let exemption x be adjusted for inflation with index P, then x = P x[0], with x[0] the exemption in the base year that now may differ from subsistence in the base year B[0]. Let also c be indexed on inflation as c = P c[0]. Let the average wage index be W = P rwi W[0], with W[0] the average wage in the base year. Let h = x[0] / W[0] and f = c[0] / W[0].
rsi = Net[W] / Net[W[0] / P =
which for f = 0 reduces to the Bentham-rsi deduced in Book III. For the limit, in general, we find:
which is normally below 1. Denote the denominator as F, and note that W[0] F = Net[W[0] or F = 1 - ATR[W[0].
We use these properties for the following theorem.