Suppose that a family's tax liability equaled its income multiplied by one-half minus $10,000. Under this system, some families would pay taxes to the government, and some families would receive money from the government through a "negative income tax."
a. Consider families with pre-tax incomes of $0, $10,000, $20,000, $30,00, and $40,000. Make a table showing pre-tax income, taxes paid to the government or money received form the government, and after-tax income for each family.
b. What is the marginal tax rate in this system? What is the maximum amount of income at which a family receives money from the government?
c. Now suppose that the tax liability equals its income multiplied by one-quarter minus $10,000. What is the marginal tax rate in this new system? What is the maximum amount of income at which a family receives money from the government?
d. What is the main advantage of each of the tax schedules discussed here?

Now the first tax schedule discussed in the start of this problem would have the advantage, I believe, of having the government paying out to the family for the pre-tax incomes of $0 and $10,000. For the second tax schedule discussed in part c the advantage, I believe, is having a 0% marginal tax rate. Are these two statements accurate?

I gather from your description the tax formulas are:
TaxA = .50*Y - 10000.
TaxB = .25*Y - 10000.
(where Y is pretax income)
After-tax income is simply Y-TaxA or Y-TaxB.

a) use simple algebra. Plug in the pre-tax income values and calculate tax.
b) use simple algebra. marginal tax rate is the (change in tax)/(change in income). It will be .50 = 50%.
c) the marginal tax rate under B is .25
d) TaxA raises more tax revenue, pays less to low income folks, etc. TaxB raises less revenue, pays more to low income folks, has a lower marginal tax rate, etc.


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