] = r as for all incomes.
Graphs
First we plot the static ATR and GNR for values of a real net wage index from 1 till 10. Figure 31 plots the paths for various marginal tax rates: 10%, 20%, ..., and even 70%, all assuming x = B = 1. These plots show the point made earlier, that the ATR is close to the marginal rate at already low income values, e.g. 2 or 3 times subsistence.
Figure 31: Average tax, in statics,
for various marginal tax rates
We might interprete static Figure 31 in a dynamic way. Take B[0] = x[0] = 1, j = 1. We may take a theoretical example. If you have a period of 35 years, then a real growth of 2% per annum would suffice to double incomes. So in the standard unrefined analysis, the tax creep in 35 years would cause incomes to be taxed at average rates close to the marginal rate. [100]
The more refined analysis for the minimum wage takes account of the multiplier effect. First of all, if real subsistence doubles from B[0] = 1 to B[35] = 2 B[0], the gross minimum wage would be M = (2 - ½) / ½ = 3, and hence we should look in Figure 31 at index 3 instead of index 2. This issue however is a bit more complex, since when rwi = 2, rsi is not 2 but 1.7.
In Figure 32 we compare the standard ATR and the dynamic ATRM. We regard only one marginal rate (a 50% rate) and a ‘peg average’ W[0] = 2 B[0] or h = 0.5. It appears that the dynamic ATRM is steeper and higher than the static ATR. However, the difference is not that big. Note though that we would want an average tax rate of 0 for the minimum wage (subsistence) instead of something close to 30%.