# Appendix One: Mathematics of a Negative Income Tax

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### Appendix One: Mathematics of a Negative Income Tax

What is an NIT? An NIT is an income transfer mechanism that assures each family an income above a given floor. It awards low income families the given floor and takes back a portion of their earnings up to a breakeven level. Let G be the family income floor or guarantee level, Y be the family's earned income, 0 < t(Y) < 1 be the negative tax or take-back rate, P be the payment that the family receives, and T be the family's net income after the negative tax.

The simplest form of NIT fixes t(Y), at say t. Such an NIT sets P = G - tY for Y < [G/t] and P = 0 for Y>_ [G/t]. Thus for Y < [G/t] the family's after-tax income T is Y + P = Y + (G - tY) = G + (1 - t)Y, i.e., the government gives a guarantee of G and takes back tY of earned income. Y = [G/t] is the breakeven level of earned income, i.e., the level of earned income at which payment ceases. For Y  [G/t] the family receives no payment. Table 9 (a) illustrates how such an NIT works for G = \$500 per month and t = 0.5.

INSERT TABLE 9 HERE

A more complicated form of NIT disallows the first E of family earnings and taxes only the remaining (Y - E), say at t. Such an NIT sets P = G for Y - E<_ 0, P = G - t(Y - E) for 0 < Y - E < [G/t], and P = 0 for Y - E >_ [G/t]. Thus for Y - E<_ 0, the family's after-tax income T is Y + P = Y + G. For 0 < Y - E < [G/t] the family's after-tax income T is Y + P = Y + [G - t(Y - E)] = G + E + (1 - t)(Y - E), i.e., the government adds a guarantee of G to the first E of earnings and takes back t(Y - E) of earnings above E. Y = E + [G/t] is the breakeven level of earned income, i. e, the level of earned income at which payment ceases. For Y >_ E + [G/t] the family receives no payment. Table 9 (b) illustrates how such an NIT works for G = \$500 per month, E _ \$100 per month, and t =0.5.