1. 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. a. Find the marginal utility of good X function (au/ax) for this consumer. b. Find the marginal utility of good Y function (au/ay) 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 maximizing bundle, x = 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) MUX/Px = MUy/Py. d. Show that there is a unique utility-maximizing bundle for every particular combination of I, Px, and Py. e. Use Cramer's rule to find the function that relates the optimal quantity of good X (x*) to its exogenous determinants. f. Derive ax*/a1 and determine its sign. g. Is good X a "normal good" for this consumer? Explain.