The marginal rate on the marginal dollar can be approximated as T[y + $1] - T[y] so that the common tax payer will have no problem in determining it. The proper formula itself is not too simple. At y = x it starts with the value r x / (c + x) and in the limit it equals r. For the whole range:
(29.1)
Note that the tax function can be transformed into a linear format consisting of income, average tax and a constant:
Tax[y] = r.y - r.x - c.Tax[y] / y = a1.y + a2 + a3.ATR[y]
Colignatus (1992) used this relation for a simple linear least square estimation that neglects the error on the average on the right hand side, using 1988 Dutch data for 12 selected income levels. The result was:
(in 1988 $)
The equation can be plotted for two ranges, (H1) for a low income range till $25 thousand to show the curvature, and (H2) for a wider income range till $250 thousand to show the straightness in the limit. In a plot, the 45-degree line is usefully added to allow visualisation of net income. Since the Dutch estimate has a high marginal rate in the limit of 57.2 %, we add US-alike lines (U1) and (U2) with a r = 40 % limit. The two ranges are plotted in Figure 25.