1. [15 pts. in total; 10 pts. and 5 pts.] Suppose that Miles' utility function is represented by
$U = \sqrt{X \cdot Y}$. Note that X and Y are the goods that Miles can consume, which are in the
number of units. Miles faces the following budget constraint: $I = P_xX + P_yY$, where I = $192
is Miles' total income; $P_x = 8$ and $P_y = 16$ are the unit prices of X and Y, respectively.
Assume that, at any time, Miles spends his entire income on these two goods only.
A. Compute the optimal levels of X and Y that Miles should consume. [Hint. One way to
answer this question is as follows: From the budget line, obtain an expression
(equation) of X as a function of Y. Insert this expression into the utility function. Take
the first derivative of U with respect to Y, equate the result to zero (0) and then solve for
$Y^*$. Insert this value into the obtained expression for X and solve for $X^*$. Feel free to use
any other method you may be comfortable with.).
B. Compute the total utility at the optimal levels of X and Y.
