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If F and B are two goods and I is the income, using the substitution method, derive the demand functions F(pf,pb,I)

and B(pf,pb,I) for the Cobb-Douglas utility function U = FaBb.

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Darryn D'Souza
Darryn D'SouzaLv10
28 Sep 2019

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Related questions

Consider the following individual utility functions:

Adam Smith: U(x,y) = xy

Jeremy Bentham: U(x,y) = xayb (Cobb-Douglas utility function)

Alfred Marshall: U(x,y) = ln x + ln y

John M. Keynes: U(x,y) = x + yb (Quasi-linear utility function)

Joan Robinson: U(x,y) = aX + bY (linear; perfect substitutes)

Find demand functions for x and y for Smith, Bentham, Marshall, and Keynes. Assume that each man has income I and that the prices of x and y are px and py respectively.

This question explores the quasi-linear utility function. Consider Thomas who has preferences over food, QF, and clothing, QC. His preferences are represented by

U (QF,QC) = [alpha]QF^(1/3) + QC

where [alpha] > 0. Let pF, pC, and I represent the prices of food, clothing, and Thomas's budget for food and clothing.

i. Using the Lagrangian method, solve mathematically for the demand function for each good. Make sure to show each step of your work.

ii. Determine whether food and clothing are normal or inferior goods for Thomas. Prove your answer mathematically.

iii. Are food and clothing substitutes or complements for Thomas? Prove your answer mathematically.

iv. Now assume that [alpha] = 3, and I = $1,224.

1. If pf = $3, and pc = $12 what is Thomas's demand for food and clothing?

2. If the price of food goes down to pf = $4/3, what happens to Thomas's demand for food and clothing?

3. If Thomas’ income falls to $996, what happens to quantity consumed of each good?

4. Graph Thomas's inverse demand curve (i.e price on y-axis, quantity on x-axis) for food when his income is I = $996 and the price of clothing is pc = $12. Be as precise as possible, labeling the exact value of any important points on the graph.

1.The Phillie Phanatic always eats his ballpark frank in a special way; he uses a foot-long hot dog together with precisely half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. He only gains utility/satisfaction from these 4 items, any extra of a single item without the others is worthless.

What can we say about the relationship among these goods for the Phillie Phanatic and the shape of his indifference curve with respect to these goods?

2. Consider a person with convex indifference curves. When she optimally chooses a consumption bundle of a particular amount of fruit on the Y-axis, and vegetables on the X-axis, her marginal rate of substitution equals:

A. 1

B. PV/PF (The price of vegetables divided by the price of fruit)

C. PF/PV (The price of fruit divided by the price of vegetables)

D. 0

3. A consumer has a budget for food (F) and shelter (S). With income M, and prices PF and PS, the budget constraint for this consumer may be represented as:

A. PSS + PFF = M

B. F = M/PF - (PS/PF)S

C. S = M/PS - (PF/PS)F

D. All of the above

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