# Finding Utility-Maximizing Demand for Goods

Let

U(x,y) = x * (y^2)

1. Derive the indirect utility function as a function of px, py and M, where px and py

are respectively the prices of the two goods x and y, and where M is the consumer's

income.

2. Now calculate the level consumption of both goods and the level of utility achieved

by this consumer if prices and income are as follows: px = 2; py = 3 ;M = 9

3. Now set up the dual of this problem: minimize expenditure subject to the level of

utility that you calculated in part 2 and with prices px and py. Find the expression

for the expenditure function.

https://brainmass.com/economics/utility/finding-utility-maximizing-demand-for-goods-42822

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Let

1. Derive the indirect utility function as a function of px, py and M, where px and py

are respectively the prices of the two goods x and y, ...

#### Solution Summary

Solution provides detailed explanation of how to find utility-maximizing demand for goods given a specific utility function. It also shows how to calculate the expenditure function. While a specific functional form for utility function is used , the steps are detailed enough for a student comfortable with basic calculus to repeat the steps for a different utility function.

Economics Questions

Economics

Question 1

Raju has $100 a week to spend on itunes and cases of beer. The price of an itune album is $10; the price of a case of beer is $20. He faces the following indifference curves:

a. What is the relative price of an album in terms of cases of beer?

b. Calculate the opportunity cost of consuming a case of beer.

c. Calculate the equation for Raju's budget line (placing cases of beer on the left

hand side of the equation).

d. Draw a graph of Raju's budget constraint with albums on the x-axis.

e. What is the slope of Raju's budget constraint?

f. What quantities of albums and cases of beer does Raju buy?

Question 2

Now suppose the price of an itune album rises to $20 and the price of beer and Raju's income remain the same.

a. What quantities of itune albums and beer does Raju now buy?

b. Find two points on Raju's demand curve for itune albums.

c. Find the substitution effect of a price change. Show graphically and estimate the

change in the quantity of itune albums due to the substitution effect.

d. Find the income effect of the price change. Show graphically and estimate the

change in the quantity of itune albums due to the income effect.

e. Is an itune album a normal good or an inferior good for Raju. Explain.

QUESTION 3: Answer these 5 mc questions.

Use the following information to answer questions 1-2.

Figure6.3

1. Jan spends all her income on CDs and candy bars. Figure 6.3 shows Jan's budget constraint and her indifference curves. At point D, her valuation of a CD, in terms of candy bars, is ____ the relative price of a CD, measured in candy bars. Jan therefore ____ her consumption of CDs.

A. higher than; reduces

B. higher than; increases

C. lower than; increases

D. lower than; reduces

E. equal to; does not change

2. Jan spends all her income on CDs and candy bars. Figure 6.3 shows Jan's budget constraint and her indifference curves. At which points is the marginal rate of substitution higher than the relative price ratio?

A. A and D

B. D and E

C. F and G

D. D and F

E. E and G

Page 1

Figure 6.4

3. Figure 6.4 shows an increase in income for Bill, who buys only meat and potatoes. Which of the following is correct?

A. Meat is an inferior good, potatoes are a normal good.

B. Meat and potatoes are both normal goods.

C. Meat and potatoes are both inferior goods.

D. Whether meat and potatoes are inferior goods is indeterminate.

E. Meat is a normal good, potatoes are an inferior good.

4. Fran buys only CDs and candy bars. For a given level of utility, the number of candy bars Fran is willing to give up to obtain another CD is smaller, the higher the number of CDs she already has.

When Fran's indifference curves are drawn with CDs on the horizontal axis, this means that, along one of her indifference curves, as she increases her consumption of CDs, the indifference curve

A. becomes flatter.

B. eventually intersects another of Fran's indifference curves.

C. becomes vertical.

D. begins to slope upward.

E. becomes steeper.

5. Jan buys only CDs and candy bars. Along one of Jan's indifference curves, as she increases her consumption of CDs,

A. her valuation of an extra candy bar, measured in candy bars, falls.

B. the number of candy bars she must give up to buy another CD falls.

C. the number of CDs she must give up to buy another candy bar falls.

D. her valuation of an extra CD, measured in candy bars, falls.

E. her valuation of an extra CD, measured in candy bars, rises.

SHORT ANSWER PART #1

Sasha has $60 allocated to pizza and beer.

Crud E Pizza is $6.

Cases of Natural Light are $12 a piece.

The following table gives utility-maximizing combinations of pizza and cases of beer that fall on Sasha's budget constraint.

Pizza Beer

QUANTITY Total Utility of Pizza Total Utility of Beer QUANTITY (cases)

0 0 225 5

2 50 220 4

4 80 210 3

6 100 180 2

8 105 120 1

10 105 0 0

1. If we assume cases of beer is on the y axis of Sasha's budget constraint, what is the y intercept? The x intercept? The slope? Draw his budget constraint.

2. Calculate the marginal utility and the marginal utility per dollar spent columns for cases of beer and pizzas.

Pizza Cases of Beer

Pizzas Marginal Utility of pizza Marginal Utility per Dollar Spent Cases of Beer Marginal Utility of Beer Marginal Utility per Dollar Spent

0 0 0 5

2 30 4

4 15 3

6 8 2

8 3 1

10 1 0

3. What is the utility maximizing choice? In terms of MU/$. Explain carefully.

4. Why not consume 6 pizzas and 2 cases of beer? Explain carefully in terms of MU/$

5. Why not consume 10 pizzas and 5 cases of beer?

SHORT ANSWER PART#2

Assume the following demand schedule for cases of mystery meat:

DEMAND SUPPLY

PRICE QUANTITY (tons) PRICE QUANTITY

(tons)

350 24 350 42

315 27 315 39

280 30 280 36

245 33 245 33

210 36 210 30

175 39 175 27

140 42 140 24

1. Where does the market clear?

Now, assume the government places a $70 tax per ton of mystery meat. Assume (correctly) that demand for mystery is relatively price inelastic.

2. Show, like we did in class, GRAPHICALLY, the amount the demanders and suppliers of mystery meat pay of the tax.

? Make sure you LABEL your graph for full credit. Include new equilibrium, any applicable shifts, the $ amount the buyer must pay of the tax and the $ amount the seller must pay.

3. Who pays more of the tax? Why?

4. Assume the elasticity of demand for mystery meat is now zero. When would we have zero elasticity? How does your answer to part b change? Show graphically.

5. Assume there are an infinite number of substitutes for mystery meat. How does your answer change?

SHORT ANSWER PART #3

You, fresh out of ECO211N, decide to manufacture "Econ is Great!" shirts to economics professors (who are a bunch of nerds).

The market for stupid t-shirts for poorly-dressed economics professors is saturated, so you are a price taker.

Quantity (Q) T-shirts per Day Price

Total Revenue (TR) Total Cost (TC) Profit Marginal Revenue

Marginal Cost

ATC

0 8

1 18

2 30

3 45

4 62

5 82

6 120 112

7 135

8 162

9 194

10 230

11 275

Fill in the blanks in the following graph. The time-saving student will copy and paste this into excel.

1. Where is your profit-maximizing level of output? How much are maximum profits?

2. How much are fixed costs?

3. If production is at 7 shirts a day, should you increase, decrease or not change production? Explain using MR and MC.

4. If production is at 5 shirts a day, should you increase, decrease or not change production? Explain using MR and MC.

See attached file for full problem description.

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