Suppose the economy consists of three households. The income of the richest hous

Question

Suppose the economy consists of three households. The income of the richest household is 10,000, the income of the second household is 6,400, and the income of the poorest household is 625. The utility of each household is given by the utility function = , where U represents the utility of the household and X represents the income of the household (after any taxes or transfers).

(3 points) Assume there are no taxes or government transfers in this economy. Use both the Utilitarian and Rawlsian (maximin) functions to find the values of social welfare based on the economy described above. Briefly explain why the values differ.

(4 points) Suppose the government of this economy decides to begin a redistribution program. They will tax the high-income household at 19%, and the middle-income household at 9.75%. All of the money that is generated from tax revenue will be given as an income transfer to the low-income household.

Determine how this will impact the utility of each household.

Determine how this will impact social welfare (using each of the welfare

functions from part a).

Summarize briefly how this redistribution policy has impacted household utility

and social welfare (using each type of social welfare function).

(4 points) Suppose that instead of the utility function ( = ) used in parts a and b, the utility of each household is given by: = .

Recalculate the values of social welfare before and after redistribution using this new utility function.

Explain how this has changed how we might evaluate redistribution policies. Then explain why the different utility functions might lead to different answers about the effects of redistribution on welfare. Which utility function represents a more realistic relationship between income and utility?

(4 points) For this part of the problem, please focus on the Utilitarian social welfare function. Suppose that there are costs of running the government redistribution program. This means that only 50% of the tax revenue generated is redistributed to the low-income household (the rest is used to pay for the costs of collecting taxes, running the transfer program, etc.).

Using each of the household utility functions mentioned in part c, calculate social welfare before and after the redistribution program.

Discuss the findings, and the implications this has for redistribution programs in general.

Explanation / Answer

Utilitarian Welfare Function is the sum of all the welfare functions in the Economy

In the given case, we have (10,000)1/2 + (6400)1/2 + (625)1/2 = 100 + 80 + 25 = 205 utils

Rawlsian Utility function: min(Utility) = 25

There is the difference in the two utilities because, in the utilitarian case, the sum of the utilities is taken into consideration whereas in Rawlsian case, the min. value of the Utility is taken into consideration.

2). The income of highest income group after taxation: 10000*(81/100) = 8100

New utility: 81001/2 = 90 utils

The income of the second highest income group after taxation: 6400*(90.25/100) = 5776

New Utility: 57761/2 = 76 utils

Income transferred = 1900 + 624 = 2524

Income of the lowest income group person after transfer: 625 + 2524 = 3149

New Utility = 56.11 utils

New Utilitarian Welfare Function: 90 + 76 + 56.11 = 222.11

Rawlsian Case: min(utility): 56.11

Thus, we can see that the total utilites have increased in both the cases after income transfer.

If, the utility function is given by U=X,

Then the total utility using Utilitarian Function is 10000+6400+625 = 17,025

Utility after income transfer = 8100+5776+3149 = 17,025

Rawlsian before income transfer: 625

Rawlsian after income transfer: 625+624+1900 = 3149

Thus, there is only a change in Rawlsian utility function.

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