Anthony spends his income on fishing lures (L ) and guitar picks (6 ). while a package of guitar picks costs $1. Assume that Anthony has $30 to spend and his utility function can be represented as U(L,G) = L 0.5 G 0.5 a. Given the utility function in this problem, use calculus to derive the marginal utilities of fishing lures and guitar picks. b. Using the tangency condition, what is the MRSL,G? c. What is the optimal number of lures and guitar picks for Anthony to purchase? d. At the optimal bundle, how much utility does Anthony have? e. If the price of guitar picks doubles to $2, how much income must Anthony have to maintain the same level of utility? f. Suppose that the prices of lures and guitar picks are $2 and $1 (as they were before the price change) and that Anthony still has only $30 to spend. Use a Lagrangian to solve his constrained utility maximization problem and confirm that the answer is the same as that to part (c). 3. Now suppose that the prices of lures and guitar picks are $2 and $2 (as they were after the price change). Assuming Anthony wants to maintain the same level of utility as he did before the price change with the lowest possible expenditure, write a statement of Anthony's constrained optimization problem. Use a Lagrangian to solve Anthony's expenditure-minimization problem and confirm that the answer is the same as that to part (e). Again suppose that the prices of lures and guitar picks are $2 and $2 (as they were after the price change). This time, however, assume that Anthony still has only $30 to spend. Use a Lagrangian to solve his constrained utility-maximization problem.
Anthony spends his income on fishing lures (L ) and guitar picks (6 ). while a package of guitar picks costs $1. Assume that Anthony has $30 to spend and his utility function can be represented as U(L,G) = L 0.5 G 0.5 a. Given the utility function in this problem, use calculus to derive the marginal utilities of fishing lures and guitar picks. b. Using the tangency condition, what is the MRSL,G? c. What is the optimal number of lures and guitar picks for Anthony to purchase? d. At the optimal bundle, how much utility does Anthony have? e. If the price of guitar picks doubles to $2, how much income must Anthony have to maintain the same level of utility? f. Suppose that the prices of lures and guitar picks are $2 and $1 (as they were before the price change) and that Anthony still has only $30 to spend. Use a Lagrangian to solve his constrained utility maximization problem and confirm that the answer is the same as that to part (c). 3. Now suppose that the prices of lures and guitar picks are $2 and $2 (as they were after the price change). Assuming Anthony wants to maintain the same level of utility as he did before the price change with the lowest possible expenditure, write a statement of Anthony's constrained optimization problem. Use a Lagrangian to solve Anthony's expenditure-minimization problem and confirm that the answer is the same as that to part (e). Again suppose that the prices of lures and guitar picks are $2 and $2 (as they were after the price change). This time, however, assume that Anthony still has only $30 to spend. Use a Lagrangian to solve his constrained utility-maximization problem.
Chapter10: Consumer Choice Theory
Section: Chapter Questions
Problem 8P
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Focus on parts e-h
Transcribed Image Text:6. Anthony spends his income on fishing lures (L ) and guitar picks (G ). Lures are priced at $2,
while a package of guitar picks costs $1. Assume that Anthony has $30 to spend and his
utility function can be represented as U(L,G) = L 0.5 GO5.
a. Given the utility function in this problem, use calculus to derive the marginal utilities of
fishing lures and guitar picks.
b. Using the tangency condition, what is the MRSL,G?
c. What is the optimal number of lures and guitar picks for Anthony to purchase?
d. At the optimal bundle, how much utility does Anthony have?
e. If the price of guitar picks doubles to $2, how much income must Anthony have to
maintain the same level of utility?
f. Suppose that the prices of lures and guitar picks are $2 and $1 (as they were before the
price change) and that Anthony still has only $30 to spend. Use a Lagrangian to solve his
constrained utility maximization problem and confirm that the answer is the same as
that to part (c).
g. Now suppose that the prices of lures and guitar picks are $2 and $2 (as they were after
the price change). Assuming Anthony wants to maintain the same level of utility as he
did before the price change with the lowest possible expenditure, write a statement of
Anthony's constrained optimization problem. Use a Lagrangian to solve Anthony's
expenditure-minimization problem and confirm that the answer is the same as that to
part (e).
h. Again suppose that the prices of lures and guitar picks are $2 and $2 (as they were after
the price change). This time, however, assume that Anthony still has only $30 to spend.
Use a Lagrangian to solve his constrained utility-maximization problem.
codar
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