4. [This question gives us some practice using calculus to think about a utility's properties, and to solve the consumer's problem.] Suppose the consumer's utility is given by u(x) = −(x1 − 1)² – (x2 − 1)², and their income and prices are given by (m, p₁, p2) = (3,2,2). In what follows, you may take it for granted that the consumer has some optimal solution to the consumer's problem. (a) Given r e X, write a formula for the marginal utilities MU₁(x) and MU₂(x). (b) Show that these preferences are not increasing on X(p, m). (c) Show that the conusmer will optimally spend all their money. [Hint: The previous question's first part may be useful.] (d) Explain why the consumer's preferences are convex. (e) What is the consumer's optimal bundle from their budget set?

4. [This question gives us some practice using calculus to think about a utility's properties, and to solve the consumer's problem.] Suppose the consumer's utility is given by u(x) = −(x1 − 1)² – (x2 − 1)², and their income and prices are given by (m, p₁, p2) = (3,2,2). In what follows, you may take it for granted that the consumer has some optimal solution to the consumer's problem. (a) Given r e X, write a formula for the marginal utilities MU₁(x) and MU₂(x). (b) Show that these preferences are not increasing on X(p, m). (c) Show that the conusmer will optimally spend all their money. [Hint: The previous question's first part may be useful.] (d) Explain why the consumer's preferences are convex. (e) What is the consumer's optimal bundle from their budget set?

Principles of Microeconomics (MindTap Course List)

8th Edition

ISBN:9781305971493

Author:N. Gregory Mankiw

Publisher:N. Gregory Mankiw

Chapter21: The Theory Of Consumer Choice

Section: Chapter Questions

Problem 5CQQ

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