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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.

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28 Sep 2019
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Related questions

PLEASE HELP WITH ONLY THE BOLDED QUESTION THANK YOU

Consider the market for new, single-family homes in New Orleans. The general demand function for new housing in New Orleans is estimated to be Qd =15-2P+0.05M+0.10R where Qd is the monthly quantity demanded, P is the price per square foot, M is average monthly income in New Orleans, and R is the average monthly rent for a three-bedroom apartment in New Orleans. Qd is measured in units of 1,000 square feet per month. 1. New housing in New Orleans is a(n) __________________ (normal, inferior) good. How can you tell from the general demand function? 2. New housing and three-bedroom apartments are _________________ (substitutes, comple- ments) in New Orleans. How can you tell from the general demand function? 3. If average monthly income is $1,500 and the monthly rental rate for three-bedroom apart- ments is $700, then the demand function for new housing in New Orleans is Qd = _____________________________________. 4. Graph the demand curve for new housing in New Orleans on the axes provided below. Label the demand curve D0. 5. What happens to demand when the following changes occur? a. b. c. d. e. The price of the commodity rises. Income decreases and the commodity is normal. Income decreases and the commodity is inferior. The price of a complement good increases. The price of a substitute good decreases. The general supply function for new housing in New Orleans is estimated to be Qs =96+2P-10PL -4PK where P is the price per square foot of new housing in New Orleans, PL is the average hourly wage rate for construction workers, and PK is the price of capital (as measured by the average rate of interest paid on loans to home builders). Qs is measured in units of 1,000 square feet per month. 6. Does it make sense for PL and PK to have negative coefficients in the general supply func- tion? Explain why or why not. 7. If the average hourly wage rate for construction workers is $10 per hour and the average rate of interest on loans to builders is 9 percent (i.e., PK = 9), then the supply function for new housing is Qs = ______________________. 8. Graph the supply curve for new housing in the graph below. Label supply S0. 9. Other things remaining the same, what would happen to the supply of a particular commodity if the following changes occur? a. The price of the commodity increases. b. Government regulators outlaw the use of a technology that enables the good to be produced at a significantly lower cost. c. The prices of inputs used to produce the commodity decrease. d. The price of a commodity that is a substitute in production increases. e. Managers expect the price of the good to fall in the near future. f. As the cost of capital rises, firms in the industry stop purchasing plant and equip- ment, which allows depreciation to diminish the productive capacity in the indus- try. 10. Solve mathematically for equilibrium price and quantity. Show your work: PE = $__________ per square foot. QE = __________ square feet per month (in 1,000s). 11. Do your supply and demand curves intersect at PE and QE found in question 8 above? Should they? 12. In equilibrium found in question 8, compute total consumer, producer, and social surpluses: CS = $____________ PS = $____________ SS = $____________ 13. Suppose New Orleans suffers a serious recession that causes average monthly income to fall from $1,500 to $1,100 per month. If other things remain the same, the demand for new hous- ing in New Orleans is now: Qd = _______________________. Plot this new demand curve in the figure below. Label it D'. 14. Suppose that because of the recession in New Orleans, the wage rate for construction workers falls to $8 per hour. If other things remain the same, the supply of new housing in New Orle- ans is now: Qs = _______________________. Plot this new supply curve in the figure. Label the new supply curve S'. 15. After income falls to $1,100 and wages fall to $8, new equilibrium price and quantity are PE = $__________ per square foot QE = __________ square feet per month (in 1,000s) 16. Determine the effect upon equilibrium price and quantity sold if the following events occur in a particular market: a. The price of a substitute good (in consumption) decreases. b. Consumers’ income increases and the good is inferior. c. Government regulators decide to outlaw a cost-reducing technological process in order to protect the environment. d. The price of a complement good (in production) increases. e. The price of inputs used to produce the good decrease. f. Consumers expect that the price of the good will fall in the near future. g. It is widely publicized that consumption of the good is helpful in prevent- ing cancer. h. The price of a substitute good (in production) decreases. 17. Suppose that a pair of events from problem 15 occur simultaneously. For each of the pairs of events indicated below, perform a qualitative analysis to predict the direction of change in either the equilibrium price or the equilibrium quantity. Explain why the change in one of these two variables is indeterminate. a. Both a and h in problem 15 occur simultaneously. b. Both b and e in problem 15 occur simultaneously. c. Both d and g in problem 15 occur simultaneously. d. Both f and c in problem 15 occur simultaneously.

THE PARTS IN BOLD ARE MARKED AS THE PARTS NEEDED HELP. ATTACHED IS THE ENTIRE ASSIGNMENT FOR ONLY PURPOSE THAT ALL INFORMATION IS INCLUDED. IF YOU COULD PLEASE ONLY HELP WITH PART 15 AND 17. PE=? QE=? AND PART 17 SITUATIONS OCCURING SIMULTANEOUSLY! THANK YOU!

Consider the market for new, single-family homes in New Orleans. The general demand function for new housing in New Orleans is estimated to be

Qd =15-2P+0.05M+0.10R

where Qd is the monthly quantity demanded, P is the price per square foot, M is average monthly income in New Orleans, and R is the average monthly rent for a three-bedroom apartment in New Orleans. Qd is measured in units of 1,000 square feet per month.

New housing in New Orleans is a(n) __________________ (normal, inferior) good. How can you tell from the general demand function?

New housing and three-bedroom apartments are _________________ (substitutes, comple- ments) in New Orleans. How can you tell from the general demand function?

If average monthly income is $1,500 and the monthly rental rate for three-bedroom apart- ments is $700, then the demand function for new housing in New Orleans is

Qd = _____________________________________.

Graph the demand curve for new housing in New Orleans on the axes provided below. Label

the demand curve D0.

What happens to demand when the following changes occur?

a. b. c. d. e.

The price of the commodity rises.
Income decreases and the commodity is normal. Income decreases and the commodity is inferior.

The price of a complement good increases. The price of a substitute good decreases.

The general supply function for new housing in New Orleans is estimated to be Qs =96+2P-10PL -4PK

where P is the price per square foot of new housing in New Orleans, PL is the average hourly wage rate for construction workers, and PK is the price of capital (as measured by the average rate of interest paid on loans to home builders). Qs is measured in units of 1,000 square feet per month.

Does it make sense for PL and PK to have negative coefficients in the general supply func- tion? Explain why or why not.

If the average hourly wage rate for construction workers is $10 per hour and the average rate of interest on loans to builders is 9 percent (i.e., PK = 9), then the supply function for new housing is

Qs = ______________________.

Graph the supply curve for new housing in the graph below. Label supply S0.

Other things remaining the same, what would happen to the supply of a particular commodity if the following changes occur?

The price of the commodity increases.

Government regulators outlaw the use of a technology that enables the good to be

produced at a significantly lower cost.

The prices of inputs used to produce the commodity decrease.

The price of a commodity that is a substitute in production increases.

Managers expect the price of the good to fall in the near future.

As the cost of capital rises, firms in the industry stop purchasing plant and equip-

ment, which allows depreciation to diminish the productive capacity in the indus- try.

Solve mathematically for equilibrium price and quantity. Show your work:

PE = $__________ per square foot.
QE = __________ square feet per month (in 1,000s).

Do your supply and demand curves intersect at PE and QE found in question 8 above? Should they?

In equilibrium found in question 8, compute total consumer, producer, and social surpluses:

CS = $____________ PS = $____________ SS = $____________

13. Suppose New Orleans suffers a serious recession that causes average monthly income to fall from $1,500 to $1,100 per month. If other things remain the same, the demand for new hous- ing in New Orleans is now:

Qd = _______________________. Plot this new demand curve in the figure below. Label it D'.

Suppose that because of the recession in New Orleans, the wage rate for construction workers falls to $8 per hour. If other things remain the same, the supply of new housing in New Orle- ans is now:

Qs = _______________________.
Plot this new supply curve in the figure. Label the new supply curve S'.

After income falls to $1,100 and wages fall to $8, new equilibrium price and quantity are PE = $__________ per square foot
QE = __________ square feet per month (in 1,000s)

Determine the effect upon equilibrium price and quantity sold if the following events occur in a particular market:

The price of a substitute good (in consumption) decreases.

Consumers’ income increases and the good is inferior.

Government regulators decide to outlaw a cost-reducing technological

process in order to protect the environment.

The price of a complement good (in production) increases.

The price of inputs used to produce the good decrease.

Consumers expect that the price of the good will fall in the near future.

It is widely publicized that consumption of the good is helpful in prevent-

ing cancer.

The price of a substitute good (in production) decreases.

17. Suppose that a pair of events from problem 15 occur simultaneously. For each of the pairs of events indicated below, perform a qualitative analysis to predict the direction of change in either the equilibrium price or the equilibrium quantity. Explain why the change in one of these two variables is indeterminate.

Both a and h in problem 15 occur simultaneously.

Both b and e in problem 15 occur simultaneously.

Both d and g in problem 15 occur simultaneously.

Both f and c in problem 15 occur simultaneously.

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