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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1ab2968-d288-4fd8-b87c-74963c459231%2F860962d4-ff62-4e0e-b86a-1259c0f4850f%2Fnjkgi5n_processed.png&w=3840&q=75)
Transcribed Image Text: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.
Expert Solution
Step 1
Given
Consumer utility function:
Here x represents the consumption of a good and L denotes leisure.
The wage rate is w and the price of good x is p.
Cost of leisure = wage earned by working
Also
Fixed income of consumer =y
A rational consumer chooses x and L such that the total utility gets maximized.
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