A country has two citizens, Bill and Ted.

Bill has a private legal business. He earns $$100$ per hour. At a tax rate of $0\%,$ Bill

works $20$ hours. At a $25\%$ tax rate, he chooses to work $16$ hours, and at a $40\%$ tax

rate, he would work only $8$ hours per week.

Ted works a manufacturing job. He works $20$ hours per week and earns $$12$ 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.

Effects of Redistributive Policies

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

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

$($i$)$ Is either policy proposal more efficient than the status quo of no taxes? Why?

$($ii$)$ Is either tax$-$transfer proposal less than optimal? Explain.$($b$)$ Suppose that Bill and Ted have the same utility function $u(y)=\sqrt[\phantom{2}]{y},$ where $y$ is the

consumption $($which is equal to net income$).$

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

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

$($c$)$ How would your answer change to $($b$)$ if the utility function is instead $u(y)=y^{\frac{1}{5}}?$

Answer both $(2$a$)$ and $(2$b$),$ and explain why.

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