Consider a worker who consumes one good and has a preference for leisure. She maximizes the utility function u(x, L) = xL, where a represents consumption of the good and L represents leisure. Suppose that this worker can choose any L = [0, 1], and receives income w(1 – L); w represents the wage rate. Let p denote the price of the consumption good. In addition to her wage 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 xL s.t. px ≤ w(1-L) + y. x>0,LE[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 convex). Dividing the FOCs L = Xp and x = Xw gives wL = px. Substituting into the budget constraint and checking the restriction L = [0, 1], we get and x(p, w, y) L(p, w, y): = = | 1 w+y if y

Consider a worker who consumes one good and has a preference for leisure. She maximizes the utility function u(x, L) = xL, where a represents consumption of the good and L represents leisure. Suppose that this worker can choose any L = [0, 1], and receives income w(1 – L); w represents the wage rate. Let p denote the price of the consumption good. In addition to her wage 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 xL s.t. px ≤ w(1-L) + y. x>0,LE[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 convex). Dividing the FOCs L = Xp and x = Xw gives wL = px. Substituting into the budget constraint and checking the restriction L = [0, 1], we get and x(p, w, y) L(p, w, y): = = | 1 w+y 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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