Question

7. The country of Tinyland has two citizens, Bill and Ted.

Bill owns a business. He earns $50 per hour. When no tax, Bill works 20 hours. At

a 25% tax rate he works 16 hours, and at a 40% tax rate he works only 8 hours per

week.

Ted has a manufacturing job. He works 20 hours per week and earns $6 per hour,

regardless of the tax rate.

The government is considering imposing an income tax of either 25% or 40% on Bill and

using the revenues to make transfer payments to Ted. The following table summarizes the

three possible policies.

0% 25% 40%

Bill

Pre-Tax Income $1,000 $800 $400

Taxes 0 $200 $160

Net Income $1,000 $600 $240

Ted

Pre-Tax Income $120 $120 $120

Transfer Payment 0 $200 $160

Net Income $120 $320 $280

(a) Suppose that Bill and Ted care only about net income, i.e., they utility function is

linear in net income y, u(y) = y.

(i) Does either policy proposal more ecient than the status quo of no taxes? Why?

(ii) Are either policy proposal obviously less than optimal? Explain.

(b) Suppose that Bill and Ted have the same utility function u(y) = y1=2 where y is

consumption (which is equal to net income).

(i) Rank the three tax policies for a utilitarian social welfare function.

(ii) Rank the three tax policies for a Rawlsian social welfare function.

Show the detail of your computation/argument for your conclusion.

(c) How would your answer change to (a) if the utility function is instead u(y) = y1=5.

Answer both (b.i) and (b.ii), and explain why.

Answer 1

Answer -

Under the 0% tax rate policy:

Under this policy, it is given that Bill’s net income, YB, is 1000 and Ted’s net income,

YT , is 120.

We can calculate Bill’s utility, UB , using his utility function:

UB(YB)=YB

⇒UB(YB)=1000= 31.62.

Similarly, we can calculate Ted’s utility, UT , using his utility function:

UT(YT )=YT

⇒UT(YT )=120=10.95.

Using a utilitarian social welfare function we can calculate the level of social welfare under this tax policy:

SW =UB(YB)+UT(YT )

⇒SW = 31.62 +10.95 = 42.57 .

Under the 25% tax rate policy:

Under this policy, it is given that Bill’s net income, YB, is 600 and Ted’s net income, YT , is 320.

We can calculate Bill’s utility, UB , using his utility function:

UB(YB)=YB

⇒UB(YB)= 600= 24.49 .

Similarly, we can calculate Ted’s utility, UT , using his utility function:

UT(YT )=YT

⇒UT(YT )= 320=17.89 .

Using a utilitarian social welfare function we can calculate the level of social welfare under this tax policy:

SW =UB(YB)+UT(YT )

⇒SW = 24.49 +17.89 = 42.38.

Under the 40% tax rate policy:

Under this policy, it is given that Bill’s net income, YB, is 240 and Ted’s net income, YT , is 280.

We can calculate Bill’s utility, UB , using his utility function:

UB(YB)=YB

⇒UB(YB)= 240=15.49 .

Similarly, we can calculate Ted’s utility, UT , using his utility function:

UT(YT )=YT

⇒UT(YT )= 280=16.73.

Using a utilitarian social welfare function we can calculate the level of social welfare under this tax policy:

SW =UB(YB)+UT(YT )

⇒SW =15.49 +16.73 = 32.22.

Ranking of the Tax Policies according to the level of social welfare using a utilitarian social welfare function:

Ranks   Social Welfare Levels Tax Policies

1st (best) 42.58 0%

2nd 42.38 25%

3rd 32.23   40%