Introductory Microeconomics (ES10001)

Topic 2: Consumer Theory

Introductory Microeconomics (ES10001) Topic 2:

Consumer Theory

Calculating Income and Substitution Effects Outline:

1.

1. 2. 3. 4.

Introduction The Slutsky Equation The Total Change in Demand Example – Calculating Income and Substitution Effects

Introduction

We have seen that a change in price exerts both an income effect and a substitution effect and that these may work with each other, as in the case of Normal goods, or against each other, as in the case of Inferior and Giffen goods. We now examine these effects more formally via the Slutsky Equation

2.

The Slutsky Equation

Consider Figure 1 following:

x2 Original Choice

m p2

Final Choice

A

x2

C

x2′

B I1

I0 0

x1

m p1

x1′

m′ p1′

I2

m p1′

x1

Figure 1: Sltutsky Compensating Variation (Price Fall)

Thus in Figure 1 we have a fall in the price of good 1 from p1 to p1′ < p1 , a pivot outwards of the

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Introductory Microeconomics (ES10001)

Topic 2: Consumer Theory

budget line and a change in the consumer’s optimal bundle from A to B. The fall in the price of good 1 has impacted on the consumer in two ways: First, there is the substitution effect whereby the fall in the economic rate of substitution (ERS) between the two goods means that the consumer does not have to sacrifice as many units of good 2 for additional units of good 1. Second, there is an income effect whereby the fall in the price of good 1 changes leads to an increase in the purchasing power (i.e. real income) of the consumer. In terms of Figure 1, we measure the substitution effect from A-B and the income effect from B-C. Under the Slutsky decomposition, the substitution effect is found by adjusting the consumer’s income following the price change such that the consumer’s original consumption bundle is affordable. Thus, if the price of a good falls we have to reduce money income and vice versa. By how much do we need to adjust money income? Let m′ denote the amount of money income that will make the original consumption bundle affordable at the new price of good 1 vis:

p1′x1 + p2 x2 = m′

(1)

Since x = ( x1 , x2 ) is affordable at both ( p1 , p2 , m ) and ( p1′, p2 , m′ ) , then we also have:

p1 x1 + p2 x2 = m

(2)

Subtracting (2) from (1) implies:

( m′ − m ) = x1 ( p1′ − p1 ) ⇒ Δm = x1Δp1

(3)

where Δm = m′ − m and Δp1 = p1′ − p1 . Note that the change in income and the change in price will always move in the same direction. For example, if the consumer is originally consuming 5 units of good 1, and the price of good 1 rises by £2 per unit, then money income must be increased by 5*£2 = £10 for the consumer to be able to continue to consume his original bundle (i.e. the price of good 2 has not changed). If the price of good 1 fell by £2 per unit, then money income must be decreased by 5*£2 = £10 for the consumer to be able to continue to consumer his original bundle. The Substitution Effect The substitution effect from A to B measures how the consumer ‘substitutes’ one good for the other when a price changes but purchasing power remains constant. It is calculated as: Δx1s = x1 ( p1′, m′ ) − x1 ( p1 , m )

(4)

That is, the consumer’s demand for good 1 when he faces the new price of good 1, p1′ , and has the level of money income, m′ , that would permit him to purchase his original bundle, x = ( x1 , x2 ) , less his demand for good 1 when he has his original level of money income, m , and faces the original price of good 1, p1 . The substitution effect is sometimes called the change in

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Introductory Microeconomics (ES10001)

Topic 2: Consumer Theory

the consumer’s ‘compensated demand’ for a good. The idea is that the consumer is just being ‘compensated’ for a change in price. The Income Effect The income effect from B to C measures the change in the demand for good 1 when we change income from m to m′ , where m′ is the amount of income that enables the consumer to purchase the original consumption bundle, and we hold prices constant at p1′, p2 . It is calculated as:

(

)

Δx1n = x1 ( p1′, m ) − x1 ( p1′, m′ )

(5)

Note that when the price of a good decreases, we need to decrease income in order to keep purchasing power constant. If the good is a normal good, then the decrease in income will lead to a decrease in demand, If the good is an inferior good, then the decrease in income will lead to an increase in demand. Thus, the income effect can be positive or negative depending on whether the good is a normal good or an inferior good Sign of the Substitution Effect The substitution effect is always negative since the change in demand due to the substitution effect is always opposite to that of the change in price. The theory of revealed preference implies that the change in demand for the good due to the substitution effect must be negative. That is, if p1 > p1′ then we must have x1 p1′, m′ ≥ x1 p1 , m such that Δx1s ≥ 0 . Similarly, assuming convex

(

)

(

)

preferences and invoking indifference curve analysis suggests that if p1 > p1′ then we must have

(

)

(

)

x1 p1′, m′ > x1 p1 , m such that Δx1s > 0 . We generally say that the ‘own price substitution effect is negative’ since the change in demand due to the substitution effect is opposite to the change in price: if the price increases then the substitution effect leads to a fall in demand, and vice versa.

3.

The Total Change in Demand

The total change in demand for good 1, Δx1 , is the change in demand due to the change in price holding income constant: Δx1 = x1 ( p1′, m ) − x1 ( p1 , m )

(6)

We have seen how this change can be broken up into two changes: the substitution effect and the income effect. Thus:

Δx1 = Δx1s + Δx1n ⇒

(7)

x1 ( p1′, m ) − x1 ( p1 , m ) = ⎡⎣ x1 ( p1′, m′ ) − x1 ( p1 , m ) ⎤⎦ + ⎡⎣ x1 ( p1′, m ) − x1 ( p1′, m′ ) ⎤⎦ Note that the Slutsky equation is actually an identity – it is true for all values of p1 , p1′ , m and m′ . To be sure, the first and the forth term on the right-hand side cancel such that the right-hand side is identically equal to the left-hand side. Note that when the price of a normal good rises, we 3

Introductory Microeconomics (ES10001)

Topic 2: Consumer Theory

have:

Δx1 = Δx1s +Δx1n (−)

(−)

(8)

(−)

That is, if good 1 is a normal good then Δx1n = ⎡⎣ x1 ( p1′, m ) − x1 ( p1′, m′ ) ⎤⎦ < 0 since a rise in price, p′ > p , implies m′ > m . In this case, the substitution and income effects reinforce each other such that demand falls. For an inferior good, however, we have:

Δx1 = Δx1s +Δx1n (? )

(−)

(9)

(+)

That is, if good 1 is an inferior good then Δx1n = ⎡⎣ x1 ( p1′, m ) − x1 ( p1′, m′ ) ⎤⎦ > 0 since a rise in price, p′ > p , implies m′ > m . If the second term on the right-hand side – the income effect – is sufficiently large, then the total change in demand could be positive. This is the case of a Giffen good – the increase in price has reduced the consumer’s purchasing power to such an extent that the consumer increases his consumption of the inferior good. It is apparent from equations (8) and (9) that Giffen goods must be inferior, but inferior goods need not be Giffen – see Figures 2-4 following.

x2

A B

I0

0

x1

C I1

I2

x1′′

x1′

Substitution Income

Figure 2: Normal Good

4

x1

Introductory Microeconomics (ES10001)

Topic 2: Consumer Theory

x2

C A I2 I0

B I1

0

x1′

x1

x1

x1′′

Substitution Income

Figure 3: Inferior Non-Giffen Good

x2

C I2 A

I0

B I1

0

x1′ x1

x1

x1′′

Substitution Income

Figure 4: Inferior Good

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Introductory Microeconomics (ES10001)

4.

Topic 2: Consumer Theory

Example - Calculating Income and Substitution Effects

Assume the consumer has the following demand function for milk:

x1 = 10 +

m 10 p1

(10)

The consumer’s income is originally £120 per week and the price of milk is £3 per quart. Thus, his demand for milk will be 10 + £120/(10*£3) = 14 quarts per week. Now suppose that the price of milk falls to £2 per quart. Then his demand at the new price will be 10 + £120/(10*£2) = 16 quarts per week. The total change in demand is +2 quarts per week. We decompose this total change as follows: Substitution Effect Recall from (4) above that the substitution effect is defined as: Δx1s = x1 ( p1′, m′ ) − x1 ( p1 , m )

(11)

Thus we need to calculate how much we would have to change the consumer’s income by to make the original consumption of 14 quarts of milk a week just affordable when the price of milk is £2 a quart. We therefore apply the formula set out in equation (3) vis:

Δm = x1Δp1 = 14 ( £2 − £3) = −£14

(12)

Thus, we would have to reduce the consumer’s income by £14 to hold his purchasing power constant following the fall in the price of milk. Thus: m ′ = m + Δm = £120 − £14 = £106

(13)

Note that if money income were £106, then the consumer could purchase his original bundle of goods. His original bundle was 14 units of good 1, which originally cost him £42, implying that he spent £120 - £42 = £78 on other goods. With an income of £106, he can still purchase 14 units of good 1 at the lower price of £2 per unit, thereby spending £28 on good 1 and leaving £106 £28 = £78 to purchase his original quantity of other goods. The consumer’s demand for milk when he faces the new price of £2 per quart and has £106 income is: x1 ( p1′, m ′ ) = x1 ( £2, £106 ) = 10 +

£106 = 15.3 10 ⋅ £2

(14)

Thus the substitution effect is:

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Introductory Microeconomics (ES10001)

Topic 2: Consumer Theory

Δx1s = x1 ( p1′, m ′ ) − x1 ( p1 , m ) ⇒ Δx1s = x1 ( £2, £106 ) − x1 ( £3, £120 )

(15)

⇒ Δx1s = 15.3 − 14 = 1.3 Holding his purchasing power constant, the fall in the price of milk has induced the consumer to substitute towards milk by 1.3 quarts. Income Effect Recall from (5) above that the income effect is defined as: Δx1n = x1 ( p1′, m ) − x1 ( p1′, m′ )

(16)

Thus in terms of our example, we have: Δx1n = x1 ( £2, £120 ) − x1 ( £2, £106 ) = 16 − 15.3 = 0.7

(17)

Thus milk is a normal good for this consumer – the fall in the price of milk has induced a positive income effect of 0.7 quarts per week.

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