4. (24 points) Consider a quasi-linear utility function, U(X, Y) = X^(1/2) + Y, with P_x and P_y being the price of X and Y respectively.
a. (6 pts) Find the demand functions for X* and Y* for both the interior and the corner solutions.
b. (4 pts) Ceteris paribus, how does X* change when the price of X increases? When income Increases? Show your work.
c. (6 pts) Now, let P_y = $1, I = 24, and Px increases from $0.5 to $2. Decompose the effect of this price increase into the substitution effect and the income effect, and calculate their magnitude.