Question $1.(15$ points$)$ Alice is a student at UT Austin who con$-$

sumes two goods: internet services $($good x$)$ and books $($good y$).$

Her preferences are represented by the Cobb$-$Douglas utility function

u$($x$,$ y$)=$ x$1/2$y$1/2.$ Alice$$s weekly income is I. The prices of internet

services and books are px$,$ py$,$ respectively.

$($a$)$ Find the Marshallian demand functions, x$($px$,$ py$,$ I$)$ and y$($px$,$ py$,$ I$)$

by writing down Alice$$s optimization problem and solving it with

the Lagrange method.

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

px $=$ $$1,$ py $=$ $$1,$ and I $=$ $$100?$ What is the highest utility level,

U $,$ that Alice can achieve if px $=$ $$1,$ py $=$ $$1,$ and I $=$ $$100?$

Recently, due to high demand, the price of the Internet services

has increased to $$2.$ The price of books remains py $=$ $$1.$ UT

has decided to give a transfer to Alice so that she can recover her

initial welfare. In order to determine the transfer UT has hired

three consultants who have made the following suggestions:

$$ Consultant A The transfer should allow the student to get

her initial level of utility.

$$ Consultant B The transfer should allow the student to buy

her initial bundle.

$$ Consultant C UT should give her a transfer of $$20.$

$($c$)$ Find the amount of the transfer implied by consultant A $,$ given that Alice's current income is $\backslash ($ I$=100\backslash ).$

$($d$)$ Find the amount of the transfer implied by consultant B$,$ given that Alice's current income is $\backslash ($ I$=100\backslash ).$ Is Alice better off if the government follows the advice of Consultant A or Consultant B$?$

$($e$)$ Determine whether Alice is better or worse off from Consultant C$\text{'}$s suggestion than before the price increase, given that Alice's current income is $\backslash ($ I$=100\backslash ).$