QUESTION FOUR Masters students only. (12 pts). Suppose a consumer with income m has a (strictly quasiconcave, twice differen- tiable) utility function over two goods, u ( 91, 92), priced at p1 and p2, respectively. (i) (1 pts). Write the utility maximization problem of the consumer, using the standard budget constraint. (ii) (1 pts). Define the corresponding LaGrangian function for this problem. (iii) (3 pts). Write a set of three first-order conditions for maximizing this LaGrangian. (iv) (3 pts). Letting the utility-maximizing (qi, q2, *) all be functions of m, implicitly differentiate these first-order conditions with respect to m. (v) (4 pts). Using conditions derived in parts (iii) and (iv), prove that the indirect utility function, v ( p, m) = ulq1(p, m), 92(p, m) ), is (weakly) increasing in nominal income m