In the following we first derive the formulas and then give plots for the average tax rate (ATR) and the gross-to-net ratio (GNR). The latter ratio may better express the effect on the gross minimum wage. We find that the ATR and the GNR at the minimum rise faster than for other incomes, since the minimum itself moves faster than those other incomes. For ease of exposition we use the Bentham tax.
Formulas
The average tax rate (ATR) and the gross to net ratio (GNR) are:
ATR[y] = Bentham[y] / y = r (1 - x / y)
GNR[y] = y / (y - Bentham[y]) = y / ( (1 - r) y + r x) = 1/ (1 - r + r x/y)
Examples work best. Let subsistence B be exempt from taxation so that x = B, and let the marginal tax rate be 50%. The average tax rate (ATR) of a subsistence worker then is 0, and the gross to net ratio (GNR) is 1. At twice subsistence, the tax is 50% (2 B - B ) = B / 2, and thus the average tax is 25% and the gross to net ratio of 4/3. In the limit, i.e. when exemption has been reduced to a negliglible proportion, then the average tax equals the marginal rate of 50% while the gross-to-net ratio is 2.
Next, notice two points. First, the formulas by themselves do not quite show how quickly the limit values are approached. To answer this question we can best look at some graphs. Secondly, these examples are static, i.e. at one point in time for different incomes. Thus, when we make graphs, then we can use a static index, and compare an income level 1 to an income ten times as large. In dynamics, i.e. when incomes rise, things are a bit complicated.
In dynamics, and concerning the current practice of adjusting exemption for inflation, we can take exemption as constant, and look at real incomes (adjusted for inflation). It seems as if we can take the formulas and graphs of the statics case, and compare real incomes regardless of the time. However, in dynamics, ‘minimum income’ is not just ‘income’ but is a mechanism. The concept of M is that it picks out one income as the minimum, but it can pick that income at a different rate of growth depending upon the mechanism. The interaction between indexation, net subsistence, the tax parameters cause a multiplier effect. Before we make plots we have to develop on this.
Let us first regard a general formula for dynamics, and see that it seems as if there were no difference with the formula for the statics case. Let exemption x be adjusted for inflation with index P, then x = P x[0]. Here we assume that x[0] can differ from subsistence in the base year B[0]. Let y be adjusted for the real level of income, with index rwi, too; then y = P rwi y[0]. Define f = x[0] / y[0]. Then:
ATR[y] = r (1 - x / y) = r (1 - x[0] / (y[0] rwi)) = r (1 - f / rwi) = ATRwi[f, rwi]