13. Consider Bill and Ted, the two citizens in the country of Adventureland desc

Question

13. Consider Bill and Ted, the two citizens in the country of Adventureland described in Problem 9 from Chapter 1. Suppose that Bill and Ted have the same utility function U(Y) = Y/2, where Yis 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 Rawlsian social welfare function b. How would your answer change if the utility function was instead U(Y) = YAS? c. Suppose that Bill and Ted instead have dif- ferent utility functions: Bills utility is given by U)1/4Y2, and Ted's is given by U"(Y) = Y/2. (This might happen for exam- ple, because Bill has significant disabilities and therefore needs more income to get the same level of utility.) How would a Rawlsian rank the three tax policies now? Effects of Redistributive Policies in Adventureland 0% 25% 40% Bll's pre tax income Bil's taxes Bill's net income Ted's pre-tax income Ted's transfer payment Ted's net income S1000 $800 $400 0 200 160 1000 600 240 120 120 120 0 200 160 120 320 280

Explanation / Answer

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 income ($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 =

1,0001/2+ 1201/2 42.58

Utilitarian social welfare with a 25% tax =

6001/2 + 3201/2 42.38

Utilitarian social welfare with a 40% tax =

2401/2 + 2801/2 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 =

1,0001/5 + 1201/5 6.59.

utilitarian social welfare with 25% tax =

6001/5 + 3201/5 6.76.

utilitarian social welfare with 40% tax =

2401/5 + 2801/5 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 1201/2 10.95 and Bill has utility 1/4* 1,0001/2 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 40% taxes, the policies that make Bill the best and worst off, respectively.

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