Question $1.(35$ points$)$ Bob consumes only two goods: cigars and

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

tity $of$ whisky $(measuredin$ 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 $is{p}_x,$

and the price $of$ one liter $of$ whisky $is{p}_y.$

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

$x(p_x,p_y,I)$ and $y(p_x,p_y,I),by$ writing down Bob's optimization

problem and solving $it$ with the Lagrange method.

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

$p_x=$$$4,p_y=$$$28,$ and $I=$$$168?$ What $is$ the highest utility level,

$U^{*},$ that Bob can achieve $if{p}_x=$$$4,p_y=$$$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 first policy $isto$ tax only whisky $(asitis$ more expensive

than cigars$)at$ the rate $t=$$$4p_y^{\text{'}}=$$$32$ per liter $ofwhisky{p}_x=$$$4,$ and $I=$$$168.$

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

That $is$ goods prices will become $p_x^{\text{'}}=4(1+)$ and

$p_y^{\text{'}}=28(1+).$ Assume that the tax rate $is$ selected $so$ that

the second policy generates the government the same revenue

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

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

The lump sum tax equals the tax revenue generated $by$ the

first policy. With this policy, the goods prices are $p_x=$$$4$ and

$p_y=$$$28.$

$(c)$ How much $of$ each good will Bob consume with the first 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 first two policies

policy will make Bob least happy? Explain why.

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

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

least happy? Explain why.