Question $1.(15$ points$)$ Alice is a student at UT Austin who consumes two goods: internet services $(gdx$ and books $($good $y.$ Her preferences are represented by the Cobb$-$Douglas utility function $u(x,y)=x^{\frac{1}{2}}{y}^{\frac{1}{2}}.$ Alice's weekly income is I. The prices of internet services and books are $p_x,p_y,$ respectively.

$($a$)$ Find the Marshallian demand functions, $x(p_x,p_y,I)$ and $y(p_x,p_y,I)$ 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 $p_x=$$$1,p_y=$$$1,$ and $I=$$$100?$ What is the highest utility level, $U^{**},$ that Alice can achieve if $p_x=$$$1,p_y=$$$1,$ and $I=$$$100?$

Recently, due to high demand, the price of the Internet services has increased to $$2.$ The price of books remains $p_y=$$$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.$

$2$

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

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