13. Consider a consumer that spends her income on two goods: good X and good Y. Her utility function is U(x, y) = x^1/2 y^1/2. \text { a. Find the

marginal utility of X function }(\partial U / \partial x) \text { for this consumer. } \text { b. Find the marginal utility of Y function }(\partial U / \partial y) \text { for this consumer. } c. Letting I, Px, and Py denote the exogenously determined values of income, the price of good X, and the price of good Y, write down the matrix equation, (Ax = b) that relates the utility d. Show that there is a unique utility-maximizing bundle for each set of values for the exogenous variables. e . Use Cramer's rule to find the function that relates the optimal quantity of x (x*) to itsе.exogenous determinants. \text { f. Derive } \partial x^{*} / \partial I \text { and } \partial x^{*} / \partial P_{x} \text { and determine their signs. } maximizing bundle, x =[x* y*]to its exogenous determinants (I, Px, and Py). Hint: the optimal bundle (x*, y*) will solve a system of two linear equations: (1) I = PxX* + PyY* and (2) MU/Px = MUy/Py.