B.3 Marie has preferences over two goods, cake q₁ and bread q2. She chooses quantities to consume so as to best satisfy these preferences subject to the budget constraint p1q1 + P292 y where p₁ and P2 are prices and y is total budget. Suppose that Marie's preferences are represented by utility function u(q1, 92) = 91 + In (bq1 +92) where b≥ 0 is a preference parameter. Assume that p1/bp2 ≥ y/ (P1 - bp2) ≥ 1. (a) Show that Marie's indifference curves are downward sloping and that her weakly preferred sets are convex for all possible values of b. (b) Show that her Marshallian demand for cake gi is Y fi (y, P1, P2) 1 P1-bp2 and find her Marshallian demand for bread, f2 (y, P1, P2). Discuss the shape of Engel curves for the two goods. (c) Explain why Marshallian demand curves for normal goods slope down. Are there any values of b for which either cake or bread could be a Giffen good for Marie? Discuss. (d) Find the form of the indirect utility function and expenditure function and hence establish expressions for the Hicksian demands for the two goods.
B.3 Marie has preferences over two goods, cake q₁ and bread q2. She chooses quantities to consume so as to best satisfy these preferences subject to the budget constraint p1q1 + P292 y where p₁ and P2 are prices and y is total budget. Suppose that Marie's preferences are represented by utility function u(q1, 92) = 91 + In (bq1 +92) where b≥ 0 is a preference parameter. Assume that p1/bp2 ≥ y/ (P1 - bp2) ≥ 1. (a) Show that Marie's indifference curves are downward sloping and that her weakly preferred sets are convex for all possible values of b. (b) Show that her Marshallian demand for cake gi is Y fi (y, P1, P2) 1 P1-bp2 and find her Marshallian demand for bread, f2 (y, P1, P2). Discuss the shape of Engel curves for the two goods. (c) Explain why Marshallian demand curves for normal goods slope down. Are there any values of b for which either cake or bread could be a Giffen good for Marie? Discuss. (d) Find the form of the indirect utility function and expenditure function and hence establish expressions for the Hicksian demands for the two goods.
Micro Economics For Today
10th Edition
ISBN:9781337613064
Author:Tucker, Irvin B.
Publisher:Tucker, Irvin B.
Chapter6: Consumer Choice Theory
Section6.A: Indifference Curve Analysis
Problem 3SQP
Related questions
Question
![B.3 Marie has preferences over two goods, cake q₁ and bread q2. She chooses quantities to consume so
as to best satisfy these preferences subject to the budget constraint p1q1 + P292 y where p₁ and
P2 are prices and y is total budget.
Suppose that Marie's preferences are represented by utility function u(q1, 92) = 91 + In (bq1 +92)
where b≥ 0 is a preference parameter. Assume that p1/bp2 ≥ y/ (P1 - bp2) ≥ 1.
(a) Show that Marie's indifference curves are downward sloping and that her weakly preferred sets
are convex for all possible values of b.
(b) Show that her Marshallian demand for cake gi is
Y
fi (y, P1, P2)
1
P1-bp2
and find her Marshallian demand for bread, f2 (y, P1, P2). Discuss the shape of Engel curves
for the two goods.
(c) Explain why Marshallian demand curves for normal goods slope down. Are there any values of
b for which either cake or bread could be a Giffen good for Marie? Discuss.
(d) Find the form of the indirect utility function and expenditure function and hence establish
expressions for the Hicksian demands for the two goods.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e4258f6-9fde-49fd-a4ec-53b01e867f7b%2Fc24d8820-2bbd-4501-b7b5-b810a0eb73b5%2Fk93gg5q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:B.3 Marie has preferences over two goods, cake q₁ and bread q2. She chooses quantities to consume so
as to best satisfy these preferences subject to the budget constraint p1q1 + P292 y where p₁ and
P2 are prices and y is total budget.
Suppose that Marie's preferences are represented by utility function u(q1, 92) = 91 + In (bq1 +92)
where b≥ 0 is a preference parameter. Assume that p1/bp2 ≥ y/ (P1 - bp2) ≥ 1.
(a) Show that Marie's indifference curves are downward sloping and that her weakly preferred sets
are convex for all possible values of b.
(b) Show that her Marshallian demand for cake gi is
Y
fi (y, P1, P2)
1
P1-bp2
and find her Marshallian demand for bread, f2 (y, P1, P2). Discuss the shape of Engel curves
for the two goods.
(c) Explain why Marshallian demand curves for normal goods slope down. Are there any values of
b for which either cake or bread could be a Giffen good for Marie? Discuss.
(d) Find the form of the indirect utility function and expenditure function and hence establish
expressions for the Hicksian demands for the two goods.
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