1. Consider a consumer with Cobb-Douglas utility function U(x,y) = x¹/³.y2/3 Assume that the consumer faces a price of $1 for y, and a total income of $100. The price of a decreases from $4 to $2. We next analyze the substitution and income effect of this price change. (a) Find the optimal consumption bundle at the initial price of $4. Label it bundle A. (b) Find the optimal consumption bundle at the final price of $2. Label it bundle C. (c) What is the total effect of the price change? (d) We next seek to disentangle the total effect you found in part (c) into the substitution and income effects. In order to do that, let us start by finding the decomposition bundle. Label it bundle B. [Hint: Recall that the decomposition bundle must satisfy two conditions: (1) it must generate the same utility level as the initial bundle A; and (2) we must have a that the slope of the consumer's indifference curve, MRS, coincides with the new price ratio.] (e) Write the amount of a that this individual consumes on bundles A, B and C. What is the increase in consumption of good 1 due to the substitution effect? What is due to the income effect? (f) Using the sign of the income effect, what can you say about a? Is it a normal, or an inferior good?
1. Consider a consumer with Cobb-Douglas utility function U(x,y) = x¹/³.y2/3 Assume that the consumer faces a price of $1 for y, and a total income of $100. The price of a decreases from $4 to $2. We next analyze the substitution and income effect of this price change. (a) Find the optimal consumption bundle at the initial price of $4. Label it bundle A. (b) Find the optimal consumption bundle at the final price of $2. Label it bundle C. (c) What is the total effect of the price change? (d) We next seek to disentangle the total effect you found in part (c) into the substitution and income effects. In order to do that, let us start by finding the decomposition bundle. Label it bundle B. [Hint: Recall that the decomposition bundle must satisfy two conditions: (1) it must generate the same utility level as the initial bundle A; and (2) we must have a that the slope of the consumer's indifference curve, MRS, coincides with the new price ratio.] (e) Write the amount of a that this individual consumes on bundles A, B and C. What is the increase in consumption of good 1 due to the substitution effect? What is due to the income effect? (f) Using the sign of the income effect, what can you say about a? Is it a normal, or an inferior good?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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part D E F
![1. Consider a consumer with Cobb-Douglas utility function
U(x, y) = x¹/³.y2/3
Assume that the consumer faces a price of $1 for y, and a total income
of $100. The price of a decreases from $4 to $2. We next analyze the
substitution and income effect of this price change.
(a) Find the optimal consumption bundle at the initial price of $4.
Label it bundle A.
(b) Find the optimal consumption bundle at the final price of $2.
Label it bundle C.
(c) What is the total effect of the price change?
(d) We next seek to disentangle the total effect you found in part (c)
into the substitution and income effects. In order to do that, let
us start by finding the decomposition bundle. Label it bundle
B. [Hint: Recall that the decomposition bundle must satisfy two
conditions: (1) it must generate the same utility level as the
initial bundle A; and (2) we must have a that the slope of the
consumer's indifference curve, MRS, coincides with the new price
ratio.]
(e) Write the amount of r that this individual consumes on bundles
A, B and C. What is the increase in consumption of good 1 due
to the substitution effect? What is due to the income effect?
(f) Using the sign of the income effect, what can you say about x?
Is it a normal, or an inferior good?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5fafdf3-0520-45b0-b021-a69422b83770%2F4982741c-1b3b-4263-9973-a574b215eed1%2Fhklpzow_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Consider a consumer with Cobb-Douglas utility function
U(x, y) = x¹/³.y2/3
Assume that the consumer faces a price of $1 for y, and a total income
of $100. The price of a decreases from $4 to $2. We next analyze the
substitution and income effect of this price change.
(a) Find the optimal consumption bundle at the initial price of $4.
Label it bundle A.
(b) Find the optimal consumption bundle at the final price of $2.
Label it bundle C.
(c) What is the total effect of the price change?
(d) We next seek to disentangle the total effect you found in part (c)
into the substitution and income effects. In order to do that, let
us start by finding the decomposition bundle. Label it bundle
B. [Hint: Recall that the decomposition bundle must satisfy two
conditions: (1) it must generate the same utility level as the
initial bundle A; and (2) we must have a that the slope of the
consumer's indifference curve, MRS, coincides with the new price
ratio.]
(e) Write the amount of r that this individual consumes on bundles
A, B and C. What is the increase in consumption of good 1 due
to the substitution effect? What is due to the income effect?
(f) Using the sign of the income effect, what can you say about x?
Is it a normal, or an inferior good?
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