Question

2. Consider two citizens, Bill and Ted, who live in the fictional country of "Econoland". Bill works as a lawyer and eams $50 per hour. Ted works at Meijer and earns $6 per hour. Both Bill and Ted have identical utility functions equal tou=y"(which is the same as U=JT)where yìs take-home pay (which also equals consumption, if neither Bill nor Ted save). The govemment of Econoland is concemed about i President of Econoland wants you to evaluate three policy options that tax Bill and redistribute the proceeds to Ted. Ifthe current policy of taxing Bill at 0% and redistributing nothing to Ted is left in place, Bill will choose to work 20 hours. If the government taxes Bill at a rate of 25%, Bill will chose to work 16 hours. If the government taxes Bill at a rate of 40%, Bill will choose to work 12 hours. Ted will choose to work 20 hours a week no matter what the tax rate is. As a result, the What is Bill's take-home pay and utility at each possible tax rate? What is Ted's? a. b. Rank the three policy proposals using a Utilitarian social welfare function. Rank them using a Rawlsian social welfare function. c. How would your answer to part b. change if the utility fumction was d. Suppose a more realistic representation of the problem is to assume a utility function of 7' 3 for Bill and -Y12 for Ted. This is because Bill has to support a family with his income, and thus needs a higher level of income to get on the same level of utility. However, Ted is just a high school student living at home and working at Meijer as a summer job for ome extra on the same level of utility. How does the Rawlsian social welfare function rank the three policy proposals now? money. Thus, he needs less income than Bili to get

Answer 1

a. Bill's take home pay at each possible tax rate are as follows:

For policy-1, I.e, the policy of taxing Bill at 0%,

Bills Take home pay will be $50$\times$20 = $1000 and Biil's take home utility will be U=(1000)^1/2 = 31.62

For Policy -2: Before taxation, Bill will earn $50$\times$16 =$800.

Since he pay 25% as government taxes, therefore, Bill Pay $800$\times$25%=$200 to the Government.

Therefore in that case Bills Home pay earning will be $(800-200)=$600

and Bill's Take home Utility will be U=(600)^1/2 = 28.28

In the last case, Before taxation Bill will earn $50$\times$12 =$600.

Since he pay 40% as government taxes, therefore, Bill Pay $600$\times$40%=$240 to the Government.

Therefore in that case Bills Home pay earning will be $(600-240)=$360

and Bill's Take home Utility will be U=(360)^1/2 =18.97

For Ted,

In the first case the government redistribute nothing to Ted,

Therefore Ted's Take home Money will be = $6$\times$20 = $120

and his Utility will be = (120)^1/2 = 10.95

If the government takes the second policy then the government's earning from Bill is $200. this amount is redistributed to Ted and will be added to Ted's income . Then Ted earn $120 + $200 =$320.

The Utility of Ted will be = (320)^1/2= 17.88

Similarly for third case Ted'd income = (His Original Earning + Redistributable earning from Government) = $120 +$ 240 =$360.

and the utility of Ted's take home income = (360)^1/2 = 18.97

b. The Utilitarian Social welfare function is derived from the formula

W=   $cjpXknOwWlQAAAABJRU5ErkJggg==$

Where i= 1,2,    … n, is the number of individuals and U is the Utility.

In the first proposal,

The utility of Bill is U_B= 31.62, and that of Ted is U_t=10.95

Thus Social welfare is W₁= $h+Mfc5v0pjWTVLUAAAAASUVORK5CYII=$ = $j9OXToQAAAAASUVORK5CYII=$ =21.285

Similarly, for second proposal W₂= = 23.08

For Proposal 3,   W₃= $MGGDzR6axRgAAAAASUVORK5CYII=$ = 18.97

Definitely, the ranking will be W2>W1>W3

The Rawlsian Social welfare is measure by the utility of worst-off individual.

in the first case the the worst off individual is Ted and his utility is 10.95

In the second case the utility of worst off individual is same and the utility is 17.98

In third case the defined utility is 18.97.

Therefore the ranking of social welfare will be W₃$>$W₂$>$W₁