(a) Suppose the indirect utility function v(p, m) is differentiable. Prove that dv(p, m) av (p, m) 0. Σ am + m Pi api (b) Consider a differentiable quasi-linear utility function u(x) = x₁ + p(x2, X3), where x₁ can take on any value (including a negative value). Use the first-order conditions of the UMP and EMP for an interior solution to prove that ox, (p, m)/ap, = h,(p, u)/ap, for i = 2 and 3. (c) Prove that the profit function 7 (p, w) is convex in input prices w.

Ann's utility function is U = q1q2/(q1 + q2). Solve for her optimal values of q1 and q2 as a function of p1, p2 and Y.

Anton, Betsy and Catherine are three college friends who regularly go the movies. At the movies, they can purchase skittles (x) and junior mints (y). The table below display the total utility each of them get from bundles of these two snacks (x, y). Bundle A = (1,1) B = (1,2) C = (1,3) D = (1,4) E = (1,5) Anton's Anton's Utility MUY (2) (1) 10 14 16 17 17.5 Betsy's Utility (3) 10 10 9 8 7 Betsy's MU, (4) Catherine's Utility (5) 10 12 15 19 24 Catherine's MUY (6) a) Yesterday the three friends went to watch the Barbie movie and they purchased the same bundle, bundle B = (1,2), with 1 bag of skittles and 2 bags of junior mints. Can you say who of the three friends experienced the highest utility from consuming bundle B? Explain. b) As you may have noticed, all bundles in the table contain 1 bag of skittles while they differ in the number of bags of junior mints. Fill in columns (2), (4) and (6) in the table by computing the marginal utility each friend receives from choosing a bundle…

Assume that a person’s utility depends on two products, x and y. The utility function is given by U(x, y) = (x + 2)^2(y + 3)^3. Find the marginal utility of x and marginal utility of y.