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
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
T20,L=[0,1]
and
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 = px. Substituting
into the budget constraint and checking the restriction L = [0, 1], we get
L s.t. px ≤w(1-L)+y.
x(p, w, y):
L(p, w, y):
w+y if y<w
2p
v(p, w, y):
²0
Р
w+y
-{(*) (+
Р
Y
2w
otherwise
c) Find the indirect utility as a function of p, w, and y.
Solution: Substituting the answer from part (b) into the utility function gives
if y < w
otherwise.
2) (+) if y < w
otherwise.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1ab2968-d288-4fd8-b87c-74963c459231%2F062961c4-0168-4dd1-884a-706e367b0d16%2Fhy1ecf8_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
T20,L=[0,1]
and
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 = px. Substituting
into the budget constraint and checking the restriction L = [0, 1], we get
L s.t. px ≤w(1-L)+y.
x(p, w, y):
L(p, w, y):
w+y if y<w
2p
v(p, w, y):
²0
Р
w+y
-{(*) (+
Р
Y
2w
otherwise
c) Find the indirect utility as a function of p, w, and y.
Solution: Substituting the answer from part (b) into the utility function gives
if y < w
otherwise.
2) (+) if y < w
otherwise.
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