Consumer theory practice problems

5/27/2018 Consumer theory practice problems

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Consumer Theory

Very Basic Question Set Series

001 Microeconomics www.ecopoint.in

ECOPOINT MA Economics Entrance Online Coaching Page 1

Preferences

(1)[Preferences] Consider bundle X , consisting of 6 cans of beans and 4 boxes of

cereal. Use just the assumption of monotonicity (more is better) to determine whether

each of the following bundles are

more preferred to bundle X, or less preferred to bundle X, or preference cannot be determined without more information. Briefly

justify your answers.

a. Bundle A , consisting of 6 cans of beans and 3 boxes of cereal.

b. Bundle B , consisting of 7 cans of beans and 4 boxes of cereal.

c. Bundle C , consisting of 10 cans of beans and 2 boxes of cereal.

d. Bundle D , consisting of 4 cans of beans and 6 boxes of cereal.

e. Bundle E , consisting of 8 cans of beans and 5 boxes of cereal.

(2) [Preferences] Suppose the bundles X and Y are equally preferred, where bundle

X consists of 5 units of energy and 8 units of food, while bundle Y consists of 11 units

of energy and 6 units of food. Use the assumption of diminishing marginal rate of

substitution in consumption (MRSC) to determine whether the following bundles are

more preferred to either bundle X or bundle Y, or. less preferred to either bundle X or bundle Y.

Briefly explain your answers. [Hint: You may find it useful to plot bundles X, Y, A, and B

carefully on a graph.]

a. Bundle A , consisting of 8 units of energy and 7 units of food.

b. Bundle B , consisting of 17 units of energy and 4 units of food.

(3) [Utility functions] Suppose a person has the utility function U(q1, q2) = q1q21/2 ,

where q1 denotes the quantity of food the person enjoys and q2 denotes the quantity of

clothing. Rank the following bundles from most preferred to least preferred.

a. Bundle A , consisting of 10 units of food and 16 units of clothing.

b. Bundle B , consisting of 7 units of food and 25 units of clothing.

c. Bundle C , consisting of 13 units of food and 9 units of clothing.


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(4) [Utility functions] Suppose a person has the utility function U(q1, q2) = 5q1 + 3q2 .

Assume q1and q2are positive quantities.

a. Find formulas for the marginal utilities MU1and MU2.

b. Determine whether this utility function satisfies the assumption of monotonicity

(more is better). Explain your reasoning. [Hint: Determine whether the

marginal utilities are positive.]

c. Find a formula for the marginal rate of substitution in consumption (MRSC) of

good 2 for good 1. [Hint: This is the absolute value of the slope of the

indifference curve, when good 1 is on the vertical axis and good 2 is on the

horizontal axis.]

d. Determine whether this utility function satisfies the assumption of diminishing

MRSC. Explain your reasoning. [Hint: According to the formula for the MRSC,

does it diminish as q1decreases and q2increases?]

(5) [Utility functions] Suppose a person has the utility function U(q1, q2) = (q1-5) q22 .

Assume q1> 5 and q2> 0 .








horizontal axis.]




(6) [Utility functions] Suppose a person has the utility function U(q1, q2)

= - (3/q1) (5/q2). Assume q1and q2are positive quantities.



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horizontal axis.]




(7) [Utility functions] Suppose a person has the utility function U(q1, q2)

= 3 q11/2+ 2 q2

1/2. Assume q1and q2are positive quantities.








horizontal axis.]




(8) [Utility functions] Consider the utility function U(q1, q2) = q13/4q2

1/4.

a. Find a formula for the marginal rate of substitution in consumption (MRSC) of

good 2 for good 1.

b. Find three different utility functions that yield exactly the same MRSC formula as

your answer to part (a). Check your answers by finding the MRSC formulas in

each case.


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(9) [Utility functions, finance] In portfolio theory, the utility of investors is often

modeled as a function of the expected rate of return (R) of their investment portfolio and

the risk associated with that portfolio. Risk is measured as standard deviation (). Atypical utility function might be U(R,) = R 0.03 2.

a. Find formulas for the marginal utilities MURand MU.

b. We usually assume that "more is better" for consumer, and therefore that the

marginal utilities should be positive. Explain why it makes sense for MU R to be

positive and for MUto be negative in this situation.


for R. [Hint: This is the absolute value of the slope of the indifference curve,

when R is on the vertical axis and is on the horizontal axis.]

Budget And Choice

(1) [Budget line] Suppose a consumer has $50 to spend on hamburgers and minipizzas

this month. Hamburgers cost $3 and minipizzas cost $4. Consider this consumer s

budget line.

a. Let q1 denote the number of hamburgers and q2 denote the number of minipizzas.

Give an equation for the consumers budget line.

b. Which of the following bundles are just affordable? Which are affordable withmoney left over? Which are not affordable?

Bundle (i), consisting of 5 hamburgers and 5 minipizzas. Bundle

(ii) consisting of 10 hamburgers and 5 minipizzas. Bundle (iii)

consisting of 6 hamburgers and 8 minipizzas. Bundle (iv)

consisting of 8 hamburgers and 8 minipizzas.

(2) [Budget line] Suppose a consumer has $80 to spend on movie tickets and video

rentals this month. Movie tickets cost $5 and video rentals cost $4. Consider this

consumers budget line.

a. Let q1 denote the number of movie tickets and q2 denote the number of videorentals. Give an equation for the consumers budget line.

b. Compute the intercept of the budget line on themovie ticket axis.


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c. Compute the intercept of the budget line on thevideo rental axis.

d. Find the slope of the budget line when movie tickets are on the vertical axis and

video rentals are on the horizontal axis.

(3) [Budget line] Consider the impact on a consumers budget constraint of each

scenario below. Indicate whether the impact is a parallel shift in the budget line, a

rotation of the budget line, or no change in the budget line. Also indicate whether the

budget line moves closer to the origin or farther away from the origin.

a. Income increases by 20%.

b. The price of one good increases by 20%.

c. The price of both goods increase by 20%.

d. The price of one good increases by 20% and income simultaneously increases by

20%.

e. The prices of both goods increase by 20% and income simultaneously increases

by 20%.

(4) [Kinked budget line] Suppose oranges cost $4 per pound for the first 5 pounds,

but, due to a special discount program, additional oranges cost only $1 per pound.

Assume that apples always cost $2 per pound and that the consumer has $30 income.

Plot the consumers budget constraint. [Hint: This budget constraint has a kink where

the quantity of oranges equals 5 pounds.]

(5) [Kinked budget line] Suppose a consumer can enjoy a reduced price on food by

paying an up-front annual membership fee at a discount food store. The consumer has a

total income of $1000. Without a discount, the usual price of food is $2 per unit. If the

consumer pays a fee of $200, then the price of food is only $1 per unit. The price of

other goods is always $1 per unit.

a. Give an equation for the consumers budget line without the discount. Sketch the

budget line or describe it in words. Compute the intercepts. Compute the slope

when food is on the horizontal axis.

b. Give an equation for the consumers budget line with the discount. [Hint: Treat

the membership fee as a loss of income.] Sketch the budget line or describe it in

words. Compute the intercepts. Compute the slope when food is on the

horizontal axis.


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(6) [Kinked budget line] Suppose a consumer enjoy a frequent-customer discount on

DVD rentals. The first five rentals in a month cost $5 each. Additional rentals cost only

$3 each. The consumer has an entertainment budget of $100 per month and other

entertainment goods cost $1 each. Note that this change in price after the fifth rental

causes a kink in the budget line. Consider the graph of this budget line with DVD rentals

on the horizontal axis.

a. What is the maximum number of other goods this consumer could afford, if they

never rented DVDs?

b. What is the maximum number of video rentals this consumer could afford, if they

spent their entire budget on DVD rentals?

c. Compute the coordinates of the kink point. [Hint: How many other goods could

the consumer afford if they purchased five DVD rentals?]

d. Compute the slope of the budget line, with DVD rentals on the horizontal axis,

when the consumer rents fewer than five DVDs.

e. Compute the slope of the budget line, with DVD rentals on the horizontal axis,

when the consumer rents more than five DVDs.

f. Sketch the budget line or describe it in words.

g. From your sketch of the budget line, do you think anyone would ever choose to

rent exactly five DVDs per month? Why or why not?

(7) [Choice] Suppose the price of beans is p1=$5 and the price of potatoes is p2=$2.

Find equations for the tangency conditions for each of the following consumers. Which

consumers will make the same choices if they have the same income? Explain your

reasoning. [Hint: Compare the tangency conditions for these consumers.]

a. Anne, whose utility function is U(q1,q2) = q12q2.

b. Bill, whose utility function is U(q1,q2) = q12/3q2

1/3.

c. Carol, whose utility function is U(q1,q2) = (q1-2)(q2-1)

(8) [Choice] Suppose a consumer has $150 to spend on food and clothing. Food costs

$4 per unit and clothing costs $5 per unit. The consumers utility function is U(q1,q2) =q1

2q2, where q1denotes the quantity of food and q2denotes the quantity of clothing.

a. Give an equation for the consumers budget line.


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b. Give a formula for the consumers marginal rate of substitution in consumption

(MRSC) of clothing for food. [Hint: This is the slope of the consumers

indifference curve when food is on the vertical axis and clothing is on the

horizontal axis.]

c. Compute the quantities of food and clothing that this consumer will choose.

(9) [Choice] Suppose a consumer has $310 to spend on energy and other goods.

Energy costs $3 per unit and other goods cost $2 per unit. The consumer s utility

function is U(q1,q2) = q1(q2-5), where q1 denotes the quantity of energy and q2 denotes

the quantity of other goods.



(MRSC) of other goods for energy. [Hint: This is the slope of the consumers

indifference curve when energy is on the vertical axis and other goods is on the

horizontal axis.]

c. Compute the quantities of energy and other goods that this consumer will choose.

(10)[Choice] Suppose the consumer in the previous problem enjoys an increase in

income to $370. There is no change in prices. Compute the quantities of energy and

other goods that this consumer will now choose.

(11)[Choice] Suppose a consumer has $210 to spend on health care and other goods.

Health care costs $9 per unit and other goods cost $8 per unit. The consumer s utilityfunction is U(q1,q2) = (5/q1) (10/q2), where q1denotes the quantity of health care and q2

denotes the quantity of other goods.



(MRSC) of other goods for health care. [Hint: This is the slope of the

consumers indifference curve when health care is on the vertical axis and other

goods is on the horizontal axis.]

c. Compute the quantities of health care and other goods that this consumer will

choose

(12) [Choice] Suppose a consumer has $60 to spend on food and other goods. Food

costs $2 per unit and other goods cost $4 per unit. The consumers utility function is


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U(q1,q2) = q11/2 + q2

1/2, where q1 denotes the quantity of food and q2 denotes the

quantity of other goods.



(MRSC) of other goods for food. [Hint: This is the slope of the consumers

indifference curve when food is on the vertical axis and other goods is on the

horizontal axis.]

c. Compute the quantities of food and other goods that this consumer will choose.

(13) [Choice with corner solutions] Suppose a consumer has income of $60 to spend

on sodapop. This consumer has utility function U(q1,q2) = q1 + 2q2 , where q1 denotes

the number of small bottles of sodapop and q2 denotes the number of large bottles of

sodapop consumed.

a. Draw the consumer's indifference curves when U = 10, 20, or 30.

b. On the same graph, but in a different color, draw the consumer's budget line when

p1 = $2 and p2 = $3. How many large bottles and how many small bottles will

this consumer choose? [Hint: Calculus is useless for this problem because the

solution is not a tangency. Instead, just study your graph.]

c. On the same graph, but in a different color, draw the consumer's budget line when

p1 = $2 and p2 = $6. How many large bottles and how many small bottles will

this consumer choose?

(14) [Choice, finance] Suppose an investor has the utility function

U(R,) = R0.032.




According to the Capital Asset Pricing Model, if there is a risk-free asset with a return of 4

percent, and if the market return is 10 percent with a standard deviation of 20 percent, then

the investor faces a constraint of R = 4 + (10-4)/20 or R = 4 + 0.3 .

b. Compute the slope of the constraint, dR/d.


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c. Compute the values of R and that this investor will choose. [Hint: Set your

answer to (a) equal to your answer to (b) and solve for . Then insert this into

the constraint to find R.]

In reality, the investor cannot purchase R and directly. Instead, the investor purchases

the risk-free asset and the market portfolio. The investor's R and are then weighted

averages of the values for the two assets. In particular, let w equal the fraction of the

investor's wealth invested in the risk-free asset, and thus (1-w) equal the fraction of the

investor's wealth invested in the market portfolio. Then

R = 4 w + 10 (1-w) and = 0 w + 20 (1-w).

d. Compute w, the fraction of wealth that the investor will devote to the risk-free

asset, and (1-w), the fraction that the investor will devote to the market portfolio

(15) [Choice, finance] Suppose an investor has the utility function U(R,) =

R0.012.




According to the Capital Asset Pricing Model, if there is a risk-free asset with a return of 4

percent, and if the market return is 10 percent with a standard deviation of 20 percent, then

the investor faces a constraint of R = 4 + (10-4)/20 or R = 4 + 0.3 .

b. Compute the slope of the constraint, dR/d.

c. Compute the values of R and that this investor will choose. [Hint: Set your

answer to (a) equal to your answer to (b) and solve for . Then insert this into

the constraint to find R.]

In reality, the investor cannot purchase R and directly. Instead, the investor purchases

the risk-free asset and the market portfolio. The investor's R and are then weighted

averages of the values for the two assets. In particular, let w equal the fraction of the

investor's wealth invested in the risk-free asset, and thus (1-w) equal the fraction of the

investor's wealth invested in the market portfolio. Then

R = 4 w + 10 (1-w) and = 0 w + 20 (1-w).

d. Compute w, the fraction of wealth that the investor will devote to the risk-free

asset, and (1-w), the fraction that the investor will devote to the market portfolio.


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Budget Constraint, Homogeneity and Demand Functions

1. [Budget constraint] Assume there are only two goods and that the demand for good#1 is given by . Substitute this into the equation for the budget line to find the

formula for the demand for good #2.

2. [Budget constraint] If a consumer always spends one-fourth of her or his income onhousingregardless of income, the price of housing, or the price of other goodsthen what

must be the demand function for housing?

3. [Budget constraint and homogeneity] Consider whether the following functions mightbe legitimate demand functions for an individual consumer.

a. Is the budget constraint satisfied by this demand system? (Assume there are only

two goods.) Show your work, step by step.

b. Are these functions homogeneous of degree zero in income and prices? Show

your work, step by step.

4. [Budget constraint and homogeneity] Consider whether the following functions

might be legitimate demand functions for an individual consumer.





5. [Budget constraint and homogeneity] Consider whether the following functions

might be legitimate demand functions for

an individual consumer.






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6. [Homogeneity] Which the following functions are homogeneous of degree zero?

Show your work, step by step.

7. [Homogeneity] Are the following functions homogeneous of degree zero in income

and prices? Show your work, step by step.

8. [Finding demand functions] Suppose a consumer has utility function U(q1,q2) =

q11/3q2

2/3and has income I. The price of good #1 is p1and the price of good #2 is p2.

a. Give the equation for the consumers budget line.


(MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumers

indifference curve when good #1 is on the vertical axis and good #2 is on the

horizontal axis.]

c. Find an expression for the consumers demand for good #1 ( q1* ) as a function of

p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p2/p1. Solve thisequation for q2. Substitute the resulting expression in the budget line and solve

for q1.]

d. Find an expression for the consumers demand for good #2 ( q2* ) as a function of

p1, p2, and I .


q11/4q2

3/4and has income I. The price of good #1 is p1and the price of good #2 is p2.


b. Give a formula for the consumers marginal rate of substitution in consumption(MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumers


horizontal axis.]


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p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p2/p1. Solve this

equation for q2. Substitute the resulting expression in the budget line and solve

for q1.]


p1, p2, and I .


(q1-15) q22and has income I. The price of good #1 is p1and the price of good #2 is p2.



(MRSC) of good #2 for good #1. [Hint: This is the |slope| of the consumers indifference curve when good #1 is on the vertical axis and good #2 is on the

horizontal axis.]


p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p2/p1. Solve this


for q1.]

d. Find an expression for the consumersdemand for good #2 ( q2* ) as a function of

p1, p2, and I .

11. [Finding demand functions] Suppose a consumer has utility function U(q1,q2) =(q1-5)(q2-4) and has income I. The price of good #1 is p1 and the price of good #2 is

p2.





horizontal axis.]


p1 , p2, and I . [Hint: Begin by setting the MRSC equal to p2/p1. Solve thisequation for q2. Substitute the resulting expression in the budget line and solve

for q1.]


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p1, p2, and I .


q11/2 + q2

1/2 and has income I. The price of good #1 is p 1 and the price of good #2 is

p2.





horizontal axis.]

c. Find an expression for the consumers demand for good #1 ( q1* ) as a function ofp1 , p2, and I . [Hint: Begin by setting the MRSC equal to p2/p1. Solve this


for q1.]


p1, p2, and I .

13. [Finding demand functions] The following three utility functions must yield

exactly the same demand functions:

Explain why, without solving explicitly for the demand functions.

14. [Expansion path] Suppose a consumer has the utility function U(q1,q2) = -(1/q1)

(1/q2) and faces prices p1=$2 and p2 = $3. What is the equation for the consumers

income-expansion path?

15. [Properties of demand functions] Suppose the purported demand function for

good #1 is supposed to be given by q1* = (1/2) I p1-2/3p2

-1/3.

a. Is this function homogeneous of degree zero in income and prices? Why or whynot?

b. Find an expression for q1*/I. Is good #1 a normal good or an inferior good?

Why?


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c. Find an expression for q1*/p1. Is good #1 an ordinary good or a Giffen good?

Why?

d. Find an expression for q1*/p2. Are goods #1 and #2 complements or

substitutes? Why?


good #1 is supposed to be given by q1* = 5 I p1-4/5p2

1/5.

a. Is this function homogeneous of degree zero in income and prices? Why or why

not?

b. Find an expression for q1*/I. Is good #1 a normal good or an inferior good?

Why?

c. Find an expression for q1*/p1. Is good #1 an ordinary good or a Giffen good?

Why?

d. Find an expression for q1*/p2. Are goods #1 and #2 complements or

substitutes? Why?


good #1 is supposed to be given by q1* = 3 (p1*)-1/2(I*) , where p1* = (p1/CPI), I* =

(I/CPI), and CPI = an index of consumer prices. Note that p2does not appear in this

function, except through the CPI.

a. Is this function homogeneous of degree zero in income and prices? Why or whynot?

b. Find an expression for q1*/I. [Hint: Use the chain rule.] Is good #1 a normal

good or an inferior good? Why?

c. Find an expression for q1*/p1. [Hint: Use the chain rule.] Is good #1 an

ordinary good or a Giffen good? Why?


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Slutsky and Hicksian Substitution Effects

(1) [Total effect of price change] Suppose a consumer has the demand function q1* =

I/p1 (p2/p1) . Suppose initially that income is $1000, the price of good #1 is $20, and

the price of good #2 is $5.

a. Calculate exactly the change in quantity demanded as the price of good #1 rises

from $20 to $21.

b. Find a formula for the partial derivative of q1* with respect to p1.

c. Compute the value of the partial derivative of q1* with respect to p1 when

income is $1000, the price of good #1 is $20, and the price of good #2 is $5.

d. Use the approximation formula (6.1) to calculate the change in quantity demanded

as the price of good #1 rises from $20 to $21.

(2) [Slutsky substitution effect] The graph below shows a consumer's response to a

rise in the price of energy. The consumer's income remains constant at $30.

a. What was the old price of energy, according to the old budget line?

b. How much energy was purchased with the old budget line?

c. What is the new price of energy, according to the new budget line?


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d. Did the Slutsky substitution effect of the energy price change cause the consumer

to buy more or less energy? How much more or less?

e. Did the income effect of the energy price change cause the consumer to buy more

or less energy? How much more or less?

f. Did the total effect of the energy price change cause the consumer to buy more or

less energy? How much more or less?

(3) [Slutsky substitution effect] The graph below shows a consumer's response to a

fall in the price of clothing. The consumer's income remains constant at $20.

a. What was the old price of clothing, according to the old budget line?

b. How much clothing was purchase with the old budget line?

c. What is the new price of clothing, according to the new budget line?

d. Did the Slutsky substitution effect of the clothing price change cause the

consumer to buy more or less clothing? How much more or less?

e. Did the income effect of the clothing price change cause the consumer to buy

more or less clothing? How much more or less?

f. Did the total effect of the clothing price change cause the consumer to buy more

or less clothing? How much more or less?


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(4) [Slutsky equation] Suppose a consumer's monthly demand for telephone calls (in

minutes) is given (approximately) by q1= 50 1000 p1+ 0.5 I. Monthly income is

originally I=$1,000 and the price of telephone minutes is p1=$0.05. Then the price of

telephone minutes rises to p1=$0.10. Compute the following:

a. the original amount purchased (q1).

b. the total effect (q1) of the price change.

c. the income adjustment required to keep the old bundle affordable.

d. the income effect of the price change.

e. the substitution effect of the price change.

(5) [Slutsky equation] Suppose at a particular point, the partial derivatives of aconsumers weekly demand for gasoline take the following values. The partial derivative

with respect to the price of gasoline is q*/p = -4. The partial derivative with respect

to the consumers (weekly) income is q*/I = 0.05. The consumer currently buys 20

gallons of gasoline per week.

a. Is gasoline a normal good or an inferior good for this consumer? Why?

b. Is gasoline an ordinary good or a Giffen good for this consumer? Why?

c. Compute the approximate total change in the amount of gasoline purchased if the

price rose by $0.50 (fifty cents).

d. Compute the income-effect component of this change.

e. Compute the substitution-effect component of this change.

(6)[Slutsky equation] Suppose the price of gasoline rose by fifty cents per gallon, but the

government awarded tax credits to compensate for the increase. In particular, if the

average consumer previously bought 500 gallons per year, then each consumer would be

given a tax credit equal to $250. Would consumption of gasoline by the average consumer

increase, decrease, or stay the same? Explain your answer using an indifference curve

diagram, if possible.

(7)[Slutsky equation] Suppose the government is concerned that poor people are having

trouble paying their electric power bills. Assume the price of electricity is currently $0.10

per kilowatt hour and the typical poor family uses about 2000 kilowatt-hours per month.

Thus, the typical poor family pays about $200 per month for electricity. The government is


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considering two alternative programs to help poor people, programs which to non-

economists might seem equivalent.

Lump-sum payment: Poor families would continue to pay the rate of $0.10 per kilowatthour, but the government would mail a check for $100 per month to all poor families that

could be applied toward their utility bills.

Rate subsidy: Poor families would enjoy a reduced electricity rate of $0.05 per kilowatthour and the government would pay electricity companies the difference.

Now consider two alternative formulas for the change in electricity use by poor families.

a. Is formula (i) positive or negative? Why?

b. Is formula (ii) positive or negative? Why?

c. Which program would cause the change in electricity consumption given by

formula (i)? Why? [Hint: See equation 6.4.]

d. Which program would cause the change in electricity consumption given by

formula (ii)? Why?

e. Which program would cause the larger increase in electricity usage by poor

familiesthe lump-sum payment or the rate subsidy? Why?

f Which program would be more costly for the government? Why?

Elasticity

(1) [Price elasticity] Suppose a consumer always spends a total of $100 on CDs every

year, no matter what the price and no matter what her income.

a. Find a formula for this person's demand function q = f(p).

b. Compute this person's price elasticity of demand.


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(2) [Price and income elasticities] Suppose a consumer always spends 25% of his

income on housing, no matter what the price of housing and no matter what his income.

a. Find a formula for this person's demand function q = f(p,I).

b. Compute this person's price elasticity of demand.

c. Compute this person's income elasticity of demand.

(3) [Price and income elasticities] Suppose the consumer has the particular utility

function U = q1q22and faces budget constraint I = p1q1+ p2q2.

a. Find the consumers demand function for good 1.

b. Is the consumers own-price elasticity of demand for good #1 constant? If so,what is its value?

c. Is the consumers cross-price elasticity of demand for good #1 with respect to the

price of good #2 constant? If so, what is its value.

d. Is the consumers income elasticity of demand for good #1 constant? If so, what

is its value?

(4) [Price elasticity of demand] Indicate for each of the following demand functions

whether the function has a constant price elasticity of demand. If the price elasticity is

constant, give its value.

(5) [Price elasticity and revenue] Determine whether the following statement is true,

false, or uncertain, and justify your answer using the concept of elasticity. If College X

raises tuition, it will get more tuition revenue.


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To answer the next two problems, use the following definition:

Price elasticity of demand = (% change in quantity) (% change in price), and the

following formula, derived from the approximation formula for products: % change in

revenue (% change in price) + (% change in quantity)

(6) [Price elasticity and revenue] Refer to the information in the previous box.

Suppose the price of gasoline rises by 10% and the elasticity of demand for gasoline is

known to be -0.4 . Assume income and other prices do not change.

a. Will the quantity of gasoline demanded increase or decrease? By how much?

b. Will the total the total amount of money spent by consumers on gasoline increase

or decrease? By approximately how much?

(7) [Price elasticity and revenue] Refer to the information in the previous box.

Suppose a company believes that the elasticity of demand for its product is 1.5, and

consider what would happen if it decreased its price by 2%. Assume income and other

prices do not change.

a. Will the quantity sold increase or decrease? By how much?

b. Will the total the total amount of revenue generated by the product increase or

decrease? By approximately how much?

(8) [Income elasticity of demand] Determine whether the following statement is true,

false, or uncertain, and justify your answer.If a persons income elasticity of demand

for clothing is one, then as the persons income rises, the fraction of income spent on

clothing remains constant.

(9) [Computing income elasticity of demand] The U.S. government's Consumer

Expenditure Survey reports the following figures. *

Low-income

consumers

High-income

consumers

Total annual expenditures $30 thousand $50 thousand

Expenditure on eggs $38 $42

Expenditure on car rentals $200 $440


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Consumer Theory




Assume that total annual expenditures equal consumer income. Assume that all consumers

face the same prices; this implies that the percent change in the quantity of a good is equal

to the percent change in the expenditure on that good.

a. Compute the income elasticity of demand for eggs using the arc-elasticity

formula.

b. Compute the income elasticity of demand for car rentals using the arc-elasticity

formula.

c. Compute the income elasticity of demand for eggs using the difference-in-

logarithms formula

d. Compute the income elasticity of demand for car rentals using the difference-in-

logarithms formula.

(10) [Income elasticity of demand] Indicate for each of the following demand functions

whether the function has a constant income elasticity of demand. If the income elasticity

is constant, give its value.

(11) [Income elasticity and budget share] Refer to the information in the previous box.

Suppose the income elasticity of demand for travel is 2.5. Now suppose income rises by

2%.

a. Will the amount of travel demanded increase or decrease? By how much?

b. Will spending on travel, as a fraction of a consumer's total budget, increase or

decrease? By approximately how much?

To answer the next two use the following definition:

Income elasticity of demand = (% change in quantity) (% change in income).

and the following formula, derived from the formula for ratios:% change in budget share (% change in quantity) - (% change in income)


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Consumer Theory




(12) [Income elasticity and budget share] Refer to the information in the previous box.

Suppose the income elasticity of demand for toothpaste is 0.4. Now suppose income

rises by 5%.

a. Will the amount of toothpaste demanded increase or decrease? By how much?

b. Will spending on toothepaste, as a fraction of a consumer's total budget, increase

or decrease? By approximately how much?

(13) [Demand elasticities] Suppose a typical consumer is believed to have the

following demand function for electricity: q1* = 50 p1-0.6p2

0.1I0.5. Here, p1denotes the

price of electricity, p2denotes the price of natural gas, and I denotes the consumers

income.

a. Is this function homogeneous of degree zero in income and prices? Why or why

not?

b. Find the price elasticity of demand for electricity. Is electricity an ordinary good

or a Giffen good? Why?

c. Find the income elasticity of demand for electricity. Is electricity a normal good

or an inferior good? Why?

d. Find the cross price elasticity of demand for electricity with respect to the price of

natural gas. Are electricity and natural gas substitutes, complements, or unrelated

goods why?

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Consumer theory practice problems