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Questions: 1. Prof. Shiell has gone to the cafeteria for lunch. He must choose the amount of soup ($) and salad (V), given his budget B and prices P, and Pv. His preferences are characterized by the utility function U(S, V) = alnS + B InV where a + B = 1. (a and Bare utility parameters.) Assume for this problem that he spends all of his income B and that he consumes positive amounts of both goods. a.) (4) Set up the Lagrangian for this problem and derive the first-order optimality conditions. (You do not need to solve.) b.) (4) The first-order conditions give a system of three equations in three unknowns - S, V, and the Lagrangian multiplier. Suppose you wish to determine whether the solution values of these variables can be written as functions of the parameters - i.e. functions of P., P., B, a, and B. What are the three conditions of the Implicit Function Theorem which you would need to check? (Define your terms in the three conditions without actually checking them.)c.) (17) Check the second-order conditions to ensure the solution characterized by the first- order conditions is in fact a (constrained) maximum. Is there any relationship between the implicit function theorem and the second-order conditions?d.) (9) Without actually solving for his optimal consumption choice, determine whether soup is a normal good for Prof. Shiell, i.e. whether - as dB > 0 . (You may assume the conditions of the Implicit Function Theorem are met.)