Question

13. Consider HBill and Ted, the two citizens in the from Chapter 1. Suppose that Bill and Ted have consumption (which is equal to net income). a. Rank the three tax policies discussed in Prob- lem 9 from Chapter 1 for a utilitarian social welfare function. Rank the three for a Ra sian social welfare function. b. How would your answer change if the utility function was instead U(Y) YA/5?

Answer 1

a.The utility function is increasing in income. Rawlsian social welfare is therefore equal to the utility of the individual with lower income. For 0% and 25% tax rates, Ted has the lower incomes ($120 and $320, respectively). For a 40% tax rate, Bill has the lower in-come ($240). Since $320 > $240 > $120, Rawlsian social welfare is highest under the 25% tax rate and lowest under the 40% tax rate. To compute utilitarian social welfare, we compare:

Utilitarian social welfare with a 0% tax = 1,000^1/2 + 120^1/2≈42.58

Utilitarian social welfare with a 25% tax = 600^1/2 + 320^1/2 ≈42.38

Utilitarian social welfare with a 40% tax = 240^1/2 + 280^1/2≈32.33

We see that the 0% tax rate is best.

b. This change does not affect the order of tax rates according to the Rawlsian social welfare function. To compute social welfare for the utilitarian social welfare function we compare:

utilitarian social welfare with 0% tax = (1000)^1/5 + 120^1/5≈6.59.

utilitarian social welfare with 25% tax = 600^1/5 + 320^1/5 ≈6.76.

utilitarian social welfare with 40% tax = 240^1/5 + 280^1/5≈6.08.

We see that the 25% tax rate is best and the 40% tax rate is the worst.

c.

Since the two have different utility functions, it is no longer easy to see who is better off under each situation. Under the 0% tax policy, we see that Ted has utility 120^1/2≈10.95 and Bill has utility 1,000^1/2  / 4 ≈7.91.

We see that Bill is worse off under this policy

Since the other two tax policies make Bill worse off and Ted better off than the 0% policy,Bill’s utility will be used to compute Rawlsian social welfare. Rawlsian social welfare is highest with 0% taxes and lowest with 50% taxes, the policies that make Bill the best and worst off, respectively.