Consider a worker who consumes one good and has a preference for leisure. She maximizes the tility function u(x, L) = xL, where à represents consumption of the good and L represents eisure. Suppose that this worker can choose any L = [0, 1], and receives income w(1 – L); w epresents the wage rate. Let p denote the price of the consumption good. In addition to her vage income, the worker also has a fixed income of y ≥ 0. (a) Write down the utility maximization problem for this consumer. Solution: The problem is max L s.t. px ≤w(1-L)+y. z>0,L= [0,1] The budget constraint may also be written with equality since preferences are monotone. (b) Find the Marshallian demands for the consumption good and leisure. Solution: Using FOCs will find the maximum since preferences are Cobb-Douglas (and therefore conver). Dividing the FOCs L = Xp and x = Xw gives wL = pr. Substituting into the budget constraint and checking the restriction L = [0, 1], we get and r(p, w, y) L(p, w, y) = v(p, w, y) HIN 2p P A if y

Consider a worker who consumes one good and has a preference for leisure. She maximizes the tility function u(x, L) = xL, where à represents consumption of the good and L represents eisure. Suppose that this worker can choose any L = [0, 1], and receives income w(1 – L); w epresents the wage rate. Let p denote the price of the consumption good. In addition to her vage income, the worker also has a fixed income of y ≥ 0. (a) Write down the utility maximization problem for this consumer. Solution: The problem is max L s.t. px ≤w(1-L)+y. z>0,L= [0,1] The budget constraint may also be written with equality since preferences are monotone. (b) Find the Marshallian demands for the consumption good and leisure. Solution: Using FOCs will find the maximum since preferences are Cobb-Douglas (and therefore conver). Dividing the FOCs L = Xp and x = Xw gives wL = pr. Substituting into the budget constraint and checking the restriction L = [0, 1], we get and r(p, w, y) L(p, w, y) = v(p, w, y) HIN 2p P A if y

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter17: Capital And Time

Section: Chapter Questions

Problem 17.1P

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