Consider a worker who consumes one good and has a preference for leisure. She maximizes theutility function u(x, L) = xL, where a represents consumption of the good and L representsleisure. Suppose that this worker can choose any L € [0, 1], and receives income w(1 – L); wrepresents the wage rate. Let p denote the price of the consumption good. In addition to herwage income, the worker also has a fixed income of y ≥ 0.(a) Write down the utility maximization problem for this consumer.Solution: The problem ismax x L 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 (andtherefore convex). Dividing the FOCs L= Xp and x = Xw gives wL = px. Substitutinginto the budget constraint and checking the restriction L = [0, 1], we getandx(p, w, y)L(p, w, y) ===1w+y if y

Answered: Consider a worker who consumes one good…

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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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 x L 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<w
2p
Y
+2
2w
otherwise
if y< w
otherwise.

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The term "utility" in economics basically refers to the satisfaction or pleasure experienced after consuming a certain good or service. A gauge of customer preferences for a variety of goods and services is the utility function. However, determining the usefulness or level of satisfaction that customers experience is imprecise and vague. The Spanish term meaning useful, utils, is the unit used to quantify utility.

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