Economics A consumer’s demands x, y for two different goods are chosen to maximize the utility function U (x,y) = √x + √y (x ≥ 0, y ≥ 0) subject to the budget constraint px+qy = m (where p,q,m > 0). (a) Find the utility-maximizing demands for both goods, as well as the Lagrange multiplier λ, all as functions of the three variables (p,q,m). Simplify all expressions as much as possible. (b) Find the maximized utility U*(p, q, m). Simplify the expression as much as possible (c) Show that (dU/dm)* is the Lagrangian multiplier λ.
Economics A consumer’s demands x, y for two different goods are chosen to maximize the utility function U (x,y) = √x + √y (x ≥ 0, y ≥ 0) subject to the budget constraint px+qy = m (where p,q,m > 0). (a) Find the utility-maximizing demands for both goods, as well as the Lagrange multiplier λ, all as functions of the three variables (p,q,m). Simplify all expressions as much as possible. (b) Find the maximized utility U*(p, q, m). Simplify the expression as much as possible (c) Show that (dU/dm)* is the Lagrangian multiplier λ.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Economics
A consumer’s demands x, y for two different goods are chosen to maximize the utility function U (x,y) = √x + √y (x ≥ 0, y ≥ 0) subject to the budget constraint px+qy = m (where p,q,m > 0).
(a) Find the utility-maximizing demands for both goods, as well as the Lagrange multiplier λ, all as functions of the three variables (p,q,m). Simplify all expressions as much as possible.
(b) Find the maximized utility U*(p, q, m). Simplify the expression as much as possible
(c) Show that (dU/dm)* is the Lagrangian multiplier λ.
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