1. Suppose that Bobby has a utility function U(X,Y)= Xay¹-a and an income of I. Px and Py are the prices of X and Y respectively. a. Set up Bobby's utility maximization problem subject to her budget constraint. Use the Lagrange multiplier method to solve for her Marshallian demand function Xm = X(I, PX, Py). b. Solve for and her uncompensated own-price elasticity of demand ex,px SX SPX SX c. Solve for and her income elasticity of demand ex,1. SI Now solve for her Hicksian demand using the following steps: d. Write down her total expenditure E(X,Y). e. Set up her expenditure minimization problem subject to a constant level of utility U using the Lagrange multiplier method. f. Use the Lagrange multiplier method to solve for her Hicksian demand function X₁ = X(U, Px, Py). g. Solve for- SX and her compensated own-price elasticity of demand ex,px SPX U=U Next, assume that a = 0.25, I = $240, Px = $12, Py = $15. Assume that X and Y are perfectly divisible goods. h. Use the Marshallian demand function you found in part a. to solve for the quantities of X and Y she consumes. i. Bobby has the opportunity to join a discount club that allows her to purchase any quantity of Y at $12 per unit, while Px remains unchanged. The fee for joining the club is $60. Should she pay the fee and join the discount club? Support your answer by comparing the utilities from joining the club and not joining the club.
1. Suppose that Bobby has a utility function U(X,Y)= Xay¹-a and an income of I. Px and Py are the prices of X and Y respectively. a. Set up Bobby's utility maximization problem subject to her budget constraint. Use the Lagrange multiplier method to solve for her Marshallian demand function Xm = X(I, PX, Py). b. Solve for and her uncompensated own-price elasticity of demand ex,px SX SPX SX c. Solve for and her income elasticity of demand ex,1. SI Now solve for her Hicksian demand using the following steps: d. Write down her total expenditure E(X,Y). e. Set up her expenditure minimization problem subject to a constant level of utility U using the Lagrange multiplier method. f. Use the Lagrange multiplier method to solve for her Hicksian demand function X₁ = X(U, Px, Py). g. Solve for- SX and her compensated own-price elasticity of demand ex,px SPX U=U Next, assume that a = 0.25, I = $240, Px = $12, Py = $15. Assume that X and Y are perfectly divisible goods. h. Use the Marshallian demand function you found in part a. to solve for the quantities of X and Y she consumes. i. Bobby has the opportunity to join a discount club that allows her to purchase any quantity of Y at $12 per unit, while Px remains unchanged. The fee for joining the club is $60. Should she pay the fee and join the discount club? Support your answer by comparing the utilities from joining the club and not joining the club.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Transcribed Image Text:1. Suppose that Bobby has a utility function U(X,Y)= Xay¹-a and an income of I. Px and Py
are the prices of X and Y respectively.
a. Set up Bobby's utility maximization problem subject to her budget constraint. Use the
Lagrange multiplier method to solve for her Marshallian demand function Xm = X(I, PX,
Py).
b. Solve for and her uncompensated own-price elasticity of demand ex,px
SX
SPX
SX
c. Solve for and her income elasticity of demand ex,1.
SI
Now solve for her Hicksian demand using the following steps:
d. Write down her total expenditure E(X,Y).
e. Set up her expenditure minimization problem subject to a constant level of utility U using
the Lagrange multiplier method.
f.
Use the Lagrange multiplier method to solve for her Hicksian demand function X₁ = X(U,
Px, Py).
g. Solve for-
SX
and her compensated own-price elasticity of demand ex,px
SPX U=U
Next, assume that a = 0.25, I = $240, Px = $12, Py = $15. Assume that X and Y are
perfectly divisible goods.
h.
Use the Marshallian demand function you found in part a. to solve for the quantities of X
and Y she consumes.
i.
Bobby has the opportunity to join a discount club that allows her to purchase any quantity
of Y at $12 per unit, while Px remains unchanged. The fee for joining the club is $60.
Should she pay the fee and join the discount club? Support your answer by comparing the
utilities from joining the club and not joining the club.
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