1 FOOD STAMPS Consider a person who has $750 per month in income and has preferences for food (F) and all other goods (X) represented by a utility function, U(F, X) = F0.2 X0.8. To simplify analysis, let's assume that both the price of food and the price of all other goods is $1 per unit such that F and X represent expenditure on food and all other goods, respectively. a. Set up a Lagrangian and derive the consumer's demand functions for food and all other goods. As you are solving, make sure you can explain each step in words. b. Now solve for the demand functions using the “substitution approach" whereby we sub- stitute the constraint directly into the objective function. Show that the resulting demand functions are the same as in part (a). c. What is the consumer's optimal bundle given the prices and income? How much utility does the consumer derive from this bundle?
1 FOOD STAMPS Consider a person who has $750 per month in income and has preferences for food (F) and all other goods (X) represented by a utility function, U(F, X) = F0.2 X0.8. To simplify analysis, let's assume that both the price of food and the price of all other goods is $1 per unit such that F and X represent expenditure on food and all other goods, respectively. a. Set up a Lagrangian and derive the consumer's demand functions for food and all other goods. As you are solving, make sure you can explain each step in words. b. Now solve for the demand functions using the “substitution approach" whereby we sub- stitute the constraint directly into the objective function. Show that the resulting demand functions are the same as in part (a). c. What is the consumer's optimal bundle given the prices and income? How much utility does the consumer derive from this bundle?
Microeconomics A Contemporary Intro
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Chapter6: Consumer Choice And Demand
Section: Chapter Questions
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![1 FOOD STAMPS
Consider a person who has $750 per month in income and has preferences for food (F) and
all other goods (X) represented by a utility function, U(F, X) = F⁰.2 X0.8. To simplify analysis,
let's assume that both the price of food and the price of all other goods is $1 per unit such that
F and X represent expenditure on food and all other goods, respectively.
a. Set up a Lagrangian and derive the consumer's demand functions for food and all other
goods. As you are solving, make sure you can explain each step in words.
b. Now solve for the demand functions using the "substitution approach" whereby we sub-
stitute the constraint directly into the objective function. Show that the resulting demand
functions are the same as in part (a).
c. What is the consumer's optimal bundle given the prices and income? How much utility
does the consumer derive from this bundle?
Now assume that the government introduces a Food Stamps program that gives this con-
sumer $250 that she can only spend on food.
d. Draw both her old and new budget constraints with F on the x-axis. How does the food
stamp program incentivize more food consumption? In other words, do you think that the
individual will consume more or less food? Why? [Note: The budget constraint under Food
Stamps should have a kink.]
e. What is the individual's optimal choice under the food stamp program? Will the individual
locate on the kink? Why or why not? What is the individual's utility at her optimum under the
food stamps program? How does it compare to her utility prior to the food stamp program?
[Note: You cannot use the standard technique here because the slope of the budget constraint
is not defined at the kink. You can, however, figure out what side of the kink the consumer
would prefer by evaluating her marginal rate of substitution at the kink.]
f. Now suppose that instead of food stamps the government gives each consumer $250 in
cash. What bundle does the consumer choose now? What is the utility level associated with
the optimal bundle under the cash transfer program?
g. A reporter from the Boston Globe calls you and asks you for a short opinion on whether or
not the food stamp program should be replaced with a program that provides cash transfers.
Based on your analysis, how would you answer the reporter?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf5b15e2-e6d6-41f9-adf4-8004644211b0%2Fd3bb87c6-ec48-46d3-8982-c83b99aba32d%2Fq8o4i5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1 FOOD STAMPS
Consider a person who has $750 per month in income and has preferences for food (F) and
all other goods (X) represented by a utility function, U(F, X) = F⁰.2 X0.8. To simplify analysis,
let's assume that both the price of food and the price of all other goods is $1 per unit such that
F and X represent expenditure on food and all other goods, respectively.
a. Set up a Lagrangian and derive the consumer's demand functions for food and all other
goods. As you are solving, make sure you can explain each step in words.
b. Now solve for the demand functions using the "substitution approach" whereby we sub-
stitute the constraint directly into the objective function. Show that the resulting demand
functions are the same as in part (a).
c. What is the consumer's optimal bundle given the prices and income? How much utility
does the consumer derive from this bundle?
Now assume that the government introduces a Food Stamps program that gives this con-
sumer $250 that she can only spend on food.
d. Draw both her old and new budget constraints with F on the x-axis. How does the food
stamp program incentivize more food consumption? In other words, do you think that the
individual will consume more or less food? Why? [Note: The budget constraint under Food
Stamps should have a kink.]
e. What is the individual's optimal choice under the food stamp program? Will the individual
locate on the kink? Why or why not? What is the individual's utility at her optimum under the
food stamps program? How does it compare to her utility prior to the food stamp program?
[Note: You cannot use the standard technique here because the slope of the budget constraint
is not defined at the kink. You can, however, figure out what side of the kink the consumer
would prefer by evaluating her marginal rate of substitution at the kink.]
f. Now suppose that instead of food stamps the government gives each consumer $250 in
cash. What bundle does the consumer choose now? What is the utility level associated with
the optimal bundle under the cash transfer program?
g. A reporter from the Boston Globe calls you and asks you for a short opinion on whether or
not the food stamp program should be replaced with a program that provides cash transfers.
Based on your analysis, how would you answer the reporter?
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