CHAPTER 8: Slutsky Equation
TRUE/FALSE
1. A Giffen good must be an inferior good.
2. If a good is an inferior good, then an increase in its price will increase the demand for it.
3. The compensated demand function refers to the demand function of someone who is adequately paid
for what he or she sells.
4. The Slutsky substitution effect measures the movement between two points on the same indifference
curve.
5. In the case of homothetic preferences, the entire change in demand from a price change is due to the
substitution effect.
6. If two goods x and y are perfect complements, then if the price of x falls, the entire change in the
demand for x is due to the income effect.
7. If the Engel curve slopes up, then the demand curve slopes down.
8. A rational consumer prefers more of good x to less. If the price of good x rises and the prices of all
other goods remain constant, then the consumer must necessarily demand less of x.
9. When the price of a good rises and income remains constant, there is a substitution effect on demand
but there cannot be an income effect.
10. Ivan spends his entire income on two goods. One of them is a Giffen good. If the price of the Giffen
good rises, demand for the other good must fall.
11. An increase in the price of a Giffen good makes the people who consume that good better off.
12. Jessica’s preferences for peanut butter and jelly are represented by the utility function U(p, j) =
min2p, 5j. If prices and income change, but her old consumption bundle lies somewhere on her new
budget line, she will not change her consumption.
13. Jimmy’s utility function is U(a, b) = ab, where a is his consumption of apples and b is his consumption
of bananas. If prices and income change in such a way that Jimmy’s old consumption lies on his new
budget line, then Jimmy will not change his consumption bundle.
14. Suppose a consumer has strictly convex preferences and her Engel curve for a good is a vertical line
for some range of income. In that same income range, her demand curve for the good slopes down.
15. John purchases two goods, x and y. Good x is an inferior good for some range of income. There must
be another range of income for which good x is a normal good.
16. A consumer has the utility function U(x, y) = x + 2y1/2. The price of good x is 2 and the price of good y
is 1. The consumer’s income is 20. If the price of good y rises to 2, then entire change in demand for y
is due to the substitution effect.
17. The Hicks version of the substitution effect of a price change measures the change in a consumer’s
demand if the consumer’s income were changed just enough so the consumer would remain on the
same indifference curve as before the price change.
MULTIPLE CHOICE
1. Cindy consumes goods x and y. Her demand for x is given by x(px, m) = 0.05m −5.15px. Now her
income is $419, the price of x is $3, and the price of y is $1. If the price of x rises to $4 and if we
denote the income effect on her demand for x by DI and the substitution effect on her demand for x by
DS, then
a.
DI = −0.28 and DS = −0.52.
b.
DI = −0.28 and DS = −4.88.
c.
DI = −0.52 and DS = −0.52.
d.
DI = 0 and DS = −2.00.
e.
None of the above.
2. Cindy consumes goods x and y. Her demand for x is given by x(px, m) = 0.04m −4.24px. Now her
income is $322, the price of x is $2, and the price of y is $1. If the price of x rises to $3 and if we
denote the income effect on her demand for x by DI and the substitution effect on her demand for x by
DS, then
a.
DI = 0 and DS = −2.00.
b.
DI = −0.18 and DS = −0.52.
c.
DI = −0.52 and DS = −0.52.
d.
DI = −0.18 and DS = −4.06.
e.
None of the above.
3. Walt considers x and y to be perfect substitutes. They originally cost $10 and $9 respectively. His
income is $720. One day the price of x drops to $8.
a.
The income effect increases the quantity of y by 90.
b.
The substitution effect increases the quantity of y by 80.
c.
The substitution effect increases the quantity of x by 90.
d.
The income effect increases the quantity of x by 80.
e.
None of the above.
4. Ernest’s income elasticity of demand for natural gas is 0.4. His price elasticity of demand for natural
gas is −0.3, and he spends 10% of his income on natural gas. What is his substitution price elasticity?
a.
−.26
b.
−.34
c.
.20
d.
−.12
e.
None of the above.
5. Suppose that bananas are a normal good and Woody is currently consuming 100 bananas at a price of
10 cents each.
a.
His Slutsky compensated demand curve going through this point is steeper than his
ordinary demand curve.
b.
His ordinary demand curve going through this point is steeper than his Slutsky
compensated demand curve.
c.
His ordinary demand curve is steeper to the left and his Slutsky compensated demand
curve is steeper to the right of this point.
d.
Whether his ordinary demand curve or his Slutsky compensated demand curve is steeper
depends on whether his price elasticity is greater than 1.
e.
None of the above.
6. The following can be said about the income and substitution effects of a price increase on the demand
for a good whose price rose:
a.
The former is always positive and the latter is always negative.
b.
Both can be either positive or negative.
c.
While the latter is always negative, the former can be either positive or negative.
d.
While the former is always negative, the latter can be either positive or negative.
e.
The former can at times be negative, but it will never overwhelm the latter.
7. In 2000, Bruce spent his income on two goods, x and y. Between 2000 and 2001, the price of good x
rose by 8% and the price of good y rose by 8%. In 2001, Bruce bought the same amount of x as he
bought in 2000, but he bought more of good y than he had bought in 2000.
a.
y is a normal good.
b.
y is an inferior good.
c.
x is an inferior good.
d.
Nothing can be said about inferiority or superiority, since we don’t know what happened
to income.
e.
Bruce is acting irrationally, since the relative prices of x and y did not change.
8. When the price of x rises, Marvin responds by changing his demand for x. The substitution effect is the
part of this change that represents his change in demand
a.
holding the prices of substitutes constant.
b.
if he is allowed to substitute as much x for y as he wishes.
c.
if his money income is held constant when the price of x changes.
d.
if the prices of all other goods are held constant.
e.
None of the above.
9. Polly consumes crackers and fruit. The price of fruit rose and the price of crackers stayed constant.
The income effect on Polly’s demand is
a.
zero because Polly’s income didn’t change.
b.
the change in Polly’s demand if her income is decreased by the change in the price of fruit
times her old consumption of fruit.
c.
the change in Polly’s demand if her income is decreased by the total amount she used to
spend on fruit.
d.
the change in Polly’s demand if her income is increased by the amount she used to spend
on fruit.
e.
the change in Polly’s demand if her income is increased by the change in the price of fruit
times the amount she used to buy.
10. Waldo consumes only apples and bananas and bananas are an inferior good for him. The price of
apples increases, but there is an increase in his income that keeps him on the same indifference curve
as before. (Waldo has convex preferences, and he prefers more to less of either good.)
a.
After the change, Waldo will buy more bananas and fewer apples.
b.
After the change, Waldo will buy fewer bananas and more apples.
c.
After the change, Waldo will buy more of both goods.
d.
After the change, Waldo will buy fewer of both goods.
e.
We would need to know his utility function to determine whether any of the above
statements are true.
11. Charlie consumes apples and bananas. His utility function is U(XA, XB) = xAx2B. The price of apples is
$1 the price of bananas is $2, and his income is $30 per week. If the price of bananas falls to $1
a.
Charlie demands fewer apples and more bananas.
b.
the substitution effect of the fall in banana prices reduces his apple consumption, but the
income effect increases his apple consumption by the same amount.
c.
the substitution effect of the fall in the price of bananas reduces his banana consumption
but the income effect increases his banana consumption by enough so that his banana
consumption rises.
d.
the income used to calculate the substitution effect is higher than his original income,
since the change made Charlie better off.
e.
More than one of the above is true.
12. Rob consumes two goods, x and y. He has an allowance of $50 per week and is not endowed with
either of the goods. If the price of good x increases and his substitution and income effects change
demand in opposite directions,
a.
good x must be a Giffen good.
b.
good x must be an inferior good.
c.
WARP is violated.
d.
good x must be a normal good.
e.
There is not enough information to judge whether good x is a normal or inferior good.
13. Ben consumes two goods and his utility function is U(x1, x2) = x21x42. The price of good 2 does not
change and his income does not change, but the price of good 1 decreases.
a.
The income effect is zero, since his income remained constant.
b.
The substitution effect on the demand for good 2 is zero, since the price of good 2 did not
change.
c.
The substitution effect reduces the demand for good 2, and since the income effect is zero,
the demand for good 2 falls.
d.
The substitution effect of the price change reduces the demand for good 2 and increases
the demand for good 1.
e.
More than one of the above statements is true.
14. Herbie consumes two goods and his utility function is U(x1, x2) = x31x42. The price of good 2 does not
change and his income does not change, but the price of good 1 decreases.
a.
The substitution effect of the price change reduces the demand for good 2 and increases
the demand for good 1.
b.
The substitution effect reduces the demand for good 2, and since the income effect is zero,
the demand for good 2 falls.
c.
The substitution effect on the demand for good 2 is zero, since the price of good 2 did not
change.
d.
The income effect is zero, since his income remained constant.
e.
More than one of the above statements is true.
15. Charlie’s utility function is xAxB. The price of apples used to be $1 per unit and the price of bananas
was $2 per unit. His income was $40 per day. If the price of apples increased to $2.25 and the price of
bananas fell to $1.25, then in order to be able to just afford his old bundle, Charlie would have to have
a daily income of
a.
$57.50.
b.
$116.
c.
$28.75.
d.
$86.25.
e.
$230.
16. Charlie’s utility function is xAxB. The price of apples used to be $1 per unit and the price of bananas
was $2 per unit. His income was $40 per day. If the price of apples increased to $1.50 and the price of
bananas fell to $1.75, then in order to be able to just afford his old bundle, Charlie would have to have
a daily income of
a.
$23.75.
b.
$47.50.
c.
$96.
d.
$71.25.
e.
$190.
17. Charlie’s utility function is xAxB. The price of apples used to be $1, the price of bananas used to be $2,
and his income used to be $40. If the price of apples increased to $6 and the price of bananas stayed
constant, the substitution effect on Charlie’s apple consumption would reduce his consumption by
a.
16.67 apples.
b.
5 apples.
c.
8.33 apples.
d.
13.33 apples.
e.
None of the above.
18. Charlie’s utility function is xAxB. The price of apples used to be $1, the price of bananas used to be $2,
and his income used to be $40. If the price of apples increased to $5 and the price of bananas stayed
constant, the substitution effect on Charlie’s apple consumption would reduce his consumption by
a.
4 apples.
b.
13 apples.
c.
8 apples.
d.
16 apples.
e.
None of the above.
19. Neville from your workbook has a friend named Peregrine. Peregrine has the same demand function
for claret as Neville, namely q = .02m − 2p, where m is income and p is price. Peregrine’s income is
$6,500 and he initially had to pay a price of $50 per bottle of claret. The price of claret rose to $60.
The substitution effect of the price change
a.
reduced his demand by 20.
b.
increased his demand by 20.
c.
reduced his demand by 14.
d.
reduced his demand by 26.
e.
reduced his demand by 24.
20. Neville from your workbook has a friend named Cedric. Cedric has the same demand function for
claret as Neville, namely q = .02m − 2p, where m is income and p is price. Cedric’s income is $6,000
and he initially had to pay a price of $40 per bottle of claret. The price of claret rose to $60. The
substitution effect of the price change
a.
reduced his demand by 24.
b.
reduced his demand by 40.
c.
reduced his demand by 56.
d.
increased his demand by 40.
e.
reduced his demand by 34.
21. Goods 1 and 2 are perfect complements and a consumer always consumes them in the ratio of 2 units
of good 2 to 1 unit of good 1. If a consumer has an income of $300 and if the price of good 2 changes
from $5 to $6, while the price of good 1 stays at $1, then the income effect of the price change
a.
is 6 times as strong as the substitution effect.
b.
does not change the demand for good 1.
c.
accounts for the entire change in demand.
d.
is exactly twice as strong as the substitution effect.
e.
is 5 times as strong as the substitution effect.
22. Goods 1 and 2 are perfect complements and a consumer always consumes them in the ratio of 2 units
of good 2 to 1 unit of good 1. If a consumer has an income of $120 and if the price of good 2 changes
from $3 to $4, while the price of good 1 stays at $1, then the income effect of the price change
a.
is 4 times as strong as the substitution effect.
b.
does not change the demand for good 1.
c.
is exactly twice as strong as the substitution effect.
d.
accounts for the entire change in demand.
e.
is 3 times as strong as the substitution effect.
23. Suppose that Agatha has $465 to spend on tickets for her trip. She intends to spend the entire amount
$465 on tickets and prefers traveling first class to traveling second class. She needs to travel a total of
1,500 miles. Suppose that the price of first-class tickets is $.40 per mile and the price of second-class
tickets is $.10 per mile. How many miles will she travel by second class?
a.
450
b.
600
c.
225
d.
550
e.
150
24. Suppose that Agatha has $825 to spend on tickets for her trip. She intends to spend the entire amount
$825 on tickets and prefers traveling first class to traveling second class. She needs to travel a total of
1,500 miles. Suppose that the price of first-class tickets is $.60 per mile and the price of second-class
tickets is $.30 per mile. How many miles will she travel by second class?
a.
125
b.
400
c.
350
d.
250
e.
83.33
25. Maude thinks delphiniums and hollyhocks are perfect substitutes, one for one. If delphiniums currently
cost $5 per unit and hollyhocks cost $6 per unit and if the price of delphiniums rises to $8 per unit,
a.
the income effect of the change in demand for delphiniums will be bigger than the
substitution effect.
b.
there will be no change in the demand for hollyhocks.
c.
the entire change in demand for delphiniums will be due to the substitution effect.
d.
1/3 of the change will be due to the income effect.
e.
2/3 of the change will be due to the income effect.
26. Maude thinks delphiniums and hollyhocks are perfect substitutes, one for one. If delphiniums currently
cost $2 per unit and hollyhocks cost $3 per unit and if the price of delphiniums rises to $7 per unit,
a.
there will be no change in the demand for hollyhocks.
b.
the income effect of the change in demand for delphiniums will be bigger than the
substitution effect.
c.
1/5 of the change will be due to the income effect.
d.
the entire change in demand for delphiniums will be due to the substitution effect.
e.
4/5 of the change will be due to the income effect.
27. Carlos consumes only two goods, apples and bananas. His utility function is given by U(x, y) = a3b2.
He is endowed with wa apples and wb bananas. Unaware that prices are about to change, Carlos buys
the quantities of apples and bananas that maximize his utility subject to his budget constraint. After he
has made his purchases but before he has eaten them, the relative prices change. Carlos is then free to
make further trades at the new relative prices if he wishes.
a.
Carlos will definitely be better off after the price change.
b.
Carlos will definitely be worse off after the price change.
c.
Carlos will be better off after the price change if the price of the good for which he is a net
seller rises relative to the price of the other good.
d.
Carlos will be better off after the price change if the price of the good for which he is a net
buyer rises relative to the price of the other good.
e.
Carlos’s utility will not be affected by the price change.
28. Carlos consumes only two goods, apples and bananas. His utility function is given by U(x, y) = minx,
2y. He is endowed with wa apples and wb bananas. Unaware that prices are about to change, Carlos
buys the quantities of apples and bananas that maximize his utility subject to his budget constraint.
After he has made his purchases but before he has eaten them, the relative prices change. Carlos is then
free to make further trades at the new relative prices if he wishes.
a.
Carlos will definitely be better off after the price change.
b.
Carlos will definitely be worse off after the price change.
c.
Carlos will be better off after the price change if the price of the good for which he is a net
seller rises relative to the price of the other good.
d.
Carlos will be better off after the price change if the price of the good for which he is a net
buyer rises relative to the price of the other good.
e.
Carlos’s utility will not be affected by the price change.
29. Gladys loves music and spends her money only on tapes and compact discs. She is always willing to
trade 2 tapes for 1 compact discs. Originally, music stores sold compact disc for $9 each and tapes for
$5 each. Then the price of compact discs fell to $8 each. The change in her consumption of compact
discs that resulted was
a.
entirely due to the substitution effect.
b.
entirely due to the income effect.
c.
partly due to the income effect and partly due to the substitution effect.
d.
due to the income effect for low incomes and the substitution effect for high incomes.
e.
There was no change in her demand for compact discs.
PROBLEM
1. A taxpayer says, “Sure I pay a lot of income tax, but I don’t mind because I get back just as much
money as I pay in.” Assuming that his facts are correct, explain why the taxpayer’s reasoning is faulty.
Use a diagram to show that an income tax can make a person worse off even if he is rebated an amount
of money equal to what he paid in.
2. Use a diagram to prove that in case there are two goods, the substitution effect of an increase in the
price of good x reduces the demand for good x.
3. Draw two different diagrams, one illustrating the Slutsky version of income and substitution effects
and the other illustrating the Hicks version of income and substitution effects. How do these two
notions differ?
4. What conditions ensure that the quantity of a good demanded increases as its price falls? Explain your
answer, using diagrams.
5. Suggest at least one reason why it might be worth the trouble it takes to learn how to decompose the
effects of a price change into an income effect and a substitution effect.