$(35$ points$)$ Bob consumes only two goods: cigars and

whisky. Let x denote the number of cigars, and y denote the quan$-$

tity of whisky $($measured in liters$)$ that Bob consumes every month.

Bob's preferences are represented by the Cobb$-$Douglas utility function

u$($x;y$)=$ x

$2$

y$.$ Bob's monthly income is I. The price of one cigar is px$,$

and the price of one liter of whisky is py$.$

$($a$)$ Find the Marshallian demand functions for cigars and whisky,

x$($px;py;I$)$ and y$($px;py;I$),$ by writing down Bob's optimization

problem and solving it with the Lagrange method.

$($b$)$ What is Bob's demand for each of the goods at

px $=$ $$4,$ py $=$ $$28,$ and I $=$ $$168?$ What is the highest utility level,

U

$,$ that Bob can achieve if px $=$ $$4,$ py $=$ $$28,$ and I $=$ $$168?$

The government realizes that Bob has unhealthy habits and

wants to tax him. The government considers the following three

policies.

The rst policy is to tax only whisky $($as it is more expensive

than cigars$)$ at the rate t $=$$$4($which means that e ectively,

Bob will be paying p

$0$

y $=$$$32$ per liter of whisky$).$ With this

policy px $=$ $$4,$ and I $=$ $$168.$

The second policy will tax both goods at the same rate $0<$

$<1.$ That is goods prices will become p

$0$

x $=4(1+)$ and

p

$0$

y $=28(1+).$ Assume that the tax rate is selected so that

the second policy generates the government the same revenue

as the rst one. Bob's income income remains I $=$ $$168.$

$2$

The third policy will impose a lump sum tax on Bob's income.

The lump sum tax equals the tax revenue generated by the

rst policy. With this policy, the goods prices are px $=$ $$4$ and

py $=$ $$28.$

$($c$)$ How much of each good will Bob consume with the rst tax policy?

What will be the tax revenue collected by the government?

$($d$)$ What is the tax rate in the second policy? How many cigars and

how much whisky will Bob consume? Which of the rst two policies

policy will make Bob least happy? Explain why.

$($e$)$ How many cigars and how much whisky will Bob consume if he

pays the lump sum tax? Which of the three policies will make Bob

least happy? Explain why.