economics 102 lecture 7 income and substitution effects rev

8/22/2019 Economics 102 Lecture 7 Income and Substitution Effects Rev

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Lecture 7: Theory of the Consumer

Income and Substitution Effects

The substitution effect

The income effect

The Slutsky identity

The Law of Demand

Examples

The Hicks substitution effect

Compensated demand curves


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There are two sorts of effects of a pricechange:

Rate of exchange between the two goods arealtered

Total purchasing power of your income hasbeen altered

Substitution effect change in demand

due to the change in the rate of exchangebetween the two goods

Income effect the change in demand dueto having more purchasing power.

x2

x1

Original choice

Consumers budget is y.

y

p2


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x1

Lower price for commodity 1

pivots the constraint outwards.

Consumers budget is y.x2

y

p2

x1

Lower price for commodity 1

pivots the constraint outwards.

Consumers budget is y.x2

y

p2

y

p

'

2

Now only y are needed to buy the

original bundle at the new prices,

as if the consumers income has

increased by y - y.

Changes to quantities demanded due to

this extra income are the income effect of

the price change


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We analyze each effect in two steps: Let relative prices change and adjust money income in order

to hold purchasing power constant

Let purchasing power adjust while holding relative pricesconstant

Analysis can proceed by pivot-shift of the budgetline.

Pivotthe budget line around the original demandedbundle so that relative prices have changed (i.e., same

slope as the new budget line) Shiftthis budget line outward so that the purchasing

power changes slope stays constant but purchasingpower has changed

Slutsky isolated the change in demand due

only to the change in relative prices by

asking What is the change in demand when

the consumers income is adjusted so that,

at the new prices, she can only just buy theoriginal bundle?


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x2

x1

x2

x1

x2

x1

x2

x1


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x2

x1

x2

x1

x2

x1

x2

x2

x1 x1


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x2

x1

x2

x2

x1 x1

x2

x1

x2

x2

x1 x1

Lower p1 makes good 1 relatively

cheaper and causes a substitution

from good 2 to good 1.


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x2

x1

x2

x2

x1 x1

Lower p1

makes good 1 relatively

cheaper and causes a substitution

from good 2 to good 1.

(x1,x2) (x1,x2) is the

pure substitution effect.

Economic interpretation of the pivoted line: thesubstitution effect

While relative prices are the same as in the finalbudget line, the money income is associated with itis different since the intercepts are different.

Purchasing power is held constant since the originalbundle of goods is just as affordable at the newpivoted line


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How much should money income be adjusted in

order to keep the old bundle just as affordable?

Lety amount of money income that will just

make the original consumption bundle

affordable,

is affordable at both and

Therefore:

),( '2'

1 xx),,( 21 ypp),,( '2

'

1 ypp

2211

221

'

1

'

''

''

xpxpy

xpxpy

The change in money income necessary to make theold bundle affordable at the new prices is just theoriginal amount of consumption of good 1 times thechange in prices. This is derived by subtracting thesecond equation from the first:

Change in income and change in price will movein the same direction. If the price goes up, then we have to raise income to

keep the same bundle affordable.

If prices go down, then income should be reduced tokeep the old bundle of goods affordable

11

1

'

11

'

']['

pxy

ppxyy


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1

The movement from bundle X to X is known asthe substitution effect. It indicates how theconsumer substitutes one good for the other whena price changes but purchasing power remainsconstant

the substitution effect, , is the change indemand for good 1 when the price of good 1changes to at the same time that money

changes to y:

sx1

'

1p

),(),( 11''

111 ypxypxxs

To estimate the substitution effect, we use theconsumers demand function to calculate choices at

Example: Suppose a demand function of the form:

At income = P120 and p1 = 3, original demand will be 14.

Suppose price falls to 2, the new demand will be 16. Totalchange in demand is 2.

),(and),( 1''

1 ypyp

1

110

10p

yx


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To calculate substitution effect, calculate how much

income would have to change in order for the

original bundle to be affordable:

The level of money income to keep purchasing

power constant is:

The consumers demand at this income and new

price is:

The substitution effect is:

14)32(1411 Ppxy

10614120' yyy

3.15)106,2(),( 1''

11 xypx

3.1143.15)120,3()106,2(111 xxx

s

The substitution effect is sometimes

called the change in compensated

demand. The consumer is being

compensated for a price change by

having his income changed so that he is

able to afford his old consumption

bundle.


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1

Shift movement is the income effect. This

change moves us from the pointX to X.

Change income while keeping prices fixed at the

new prices.

The income effect, , is the change in

demand for good 1 when we change income

from y to y, holding the price of good 1 fixed

at p1.

nx1

x2

x1

x2

x2

x1 x1

(x1,x2)


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1

x2

x1

x2

x2

x1 x1

(x1,x2)

The income effect is

(x1,x2) (x1,x2).

x2

x1

x2

x2

x1 x1

(x1,x2)

The change to demand due to

lower p1 is the sum of the

income and substitution effects,

(x1,x2) (x1,x2).


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Income effect can operate either way:

If good is normal, decrease in income

will lead to a decrease in demand

If good is inferior, decrease in income

will lead to an increase in demand.

Calculating the income effect:

In the previous example we saw that:

Thus the income effect is:

3.15)106,2(),(

16)120,2(),(

1''

11

1

'

11

xypx

xypx

7.0)106,2()120,2( 111 xxxn


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The substitution effect is negative If the price of good 1 decreases, then the change in the demand

for good 1 must be nonnegative: If

If the consumer is choosing the best bundle that he canafford, then X must be preferred to all of the bundles on thepart of the pivoted budget line that lies inside the originalbudget set. This means that the optimal choice on the pivotedbudget line must not be one of the bundles that lieunderneath the original budget line.

Optimal choice should be X or something to the right of X.This means that the new optimal choice must implyconsuming at least as much as good 1 originally.

0xthat,so),,(),(then,, s1111'

111

'

1 ypxypxpp

x2

x1

x2

x2

x1 x1


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The Slutsky identity The total change in demand, , is the change in

demand due to the change in price holdingincome constant:

This change is broken down into thesubstitution and income effects:

This is the Slutsky identity

x

)(),(11

'

111ypxypxx

)],(),([)],(),([),(),(''

11

'

1111

''

1111

'

11

111

ypxypxypxypxypxypx

xxxns

The effect of a change in price can be positive ornegative. The substitution effect is always negative, the income

effect can go either way.

For normal goods, the income and substitution effects goin the same direction so that the effects reinforce each

other. For inferior goods, it may happen that the income effect

outweighs the substitution effect so that the total changein demand associated with a price increase is positive. Thisis the perverse Giffen good result.

A Giffen good must be an inferior good but an inferiorgood need not necessarily be a Giffen good. For a Giffengood, the income effect is of the the opposite sign and hasto be large enough to outweigh the substitution effect.


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Most goods are normal (i.e. demand

increases with income).

The substitution and income effects

reinforce each other when a normal goods

own price changes.

x2

x1

x2

x2

x1 x1

(x1,x2)

Good 1 is normal because

higher income increases

demand


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1

x2

x1

x2

x2

x1 x1

(x1,x2)

Good 1 is normal because

higher income increasesdemand, so the income

and substitution

effects reinforce

each other.

Since both the substitution and income

effects increase demand when own-price

falls, a normal goods ordinary demand

curve slopes down.

The Law of Downward-Sloping Demandtherefore always applies to normal goods.


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1

Some goods are income-inferior (i.e.

demand is reduced by higher income).

The substitution and income effects oppose

each other when an income-inferior goods

own price changes.

x2

x1

x2

x1


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2

x2

x1

x2

x1

x2

x1

x2

x1


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2

x2

x1

x2

x2

x1 x1

x2

x1

x2

x2

x1 x1

The pure substitution effect is as for

a normal good. But, .


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2

x2

x1

x2

x2

x1 x1

(x1,x2)

The pure substitution effect is as for a normal good. But, the income

effect isin the opposite direction.

x2

x1

x2

x2

x1 x1

(x1,x2)

The pure substitution effect is as for a normal good. But, the income

effect is

in the opposite direction. Good 1 is

income-inferior

because an

increase to income

causes demand to

fall.


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2

x2

x1

x2

x2

x1 x1

(x1,x2)

The overall changes to demand arethe sums of the substitution and

income effects.

In rare cases of extreme income-inferiority,

the income effect may be larger in size than

the substitution effect, causing quantity

demanded to fall as own-price rises.

Such goods are Giffen goods.


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2

x2

x1

x2

x1

A decrease in p1 causes

quantity demanded of

good 1 to fall.

x2

x1

x2

x1x1

x2



good 1 to fall.


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2

x2

x1

x2

x2

x1 x1x1

x2

Substitution effect

Income effect



good 1 to fall.

Slutskys decomposition of the effect of a

price change into a pure substitution effect

and an income effect thus explains why the

Law of Downward-Sloping Demand is

violated for extremely income-inferiorgoods.


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Expressing the identity in terms of rates of change

Define the negative of the income effect as:

The Slutsky identity then becomes:

Dividing by , results in:

nmxypxypxx 1

'

11

''

111 ),(),(

msxxx

111

1p

1

1

1

1

1

1

p

x

p

x

p

xms

The first term on the RHS is the rate of changein demand when prices changes and incomeis adjusted to keep the old bundle affordable.

Working on the second term, recall that:

Substituting into the last term results in:

1

111 :therefore,x

yppxy

11

1

1

1

1x

y

x

p

x

p

xms


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Interpretation of the terms:

The substitution effect: the rate of change in demand as

price changes, adjusting income in order to keep the old

bundle affordable:

The income effect: rate of change in demand holding

prices fixed and letting income change:

Income effect is composed of two pieces, how demand

changes as income changes times the original level of

demand

1

'

11

''

11

1

1 ),(),(

p

ypxypx

p

xs

1'

'

11

''

111

1 ),(),(x

yy

ypxypxx

y

xm

If the demand for a good increases when incomeincreases, then the demand for that good mustdecrease when its price increases.

This follows directly from the Slutsky equation. If the

demand increases when income increases, we have anormal good. If we have a normal good, then thesubstitution effects and the income effect reinforce eachother, and an increase in price will unambiguouslyreduce demand.


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Perfect complements: Substitution effect is zero. Thechange in demand is due entirely to the income effect

Perfect substitutes: Entire change in demand is due tothe substitution effect. There is no shifting left to do.

Quasilinear preferences: Entire change in demand isdue to the substitution effect, the income effect iszero.

x1

x2

Total effect = income effect

Old budget line

New budget line


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2

Final budget line

Total effect = Substitution Effect

Original budget line

Total effect = substitution effect

x2

x1


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The Hicksian substitution effect

Keeps utility constant instead of keeping purchasing powerconstant.

Instead of pivoting the budget line, we roll the budget linearound the indifference curve through the originalconsumption bundle such that the consumer has a budgetline with the same relative prices as the new one but with adifferent income.

The purchasing power that he has in this instance will besufficient to purchase a bundle that is just indifferent to hisoriginal bundle.

Hicks substitution effect gives the consumer just enough moneyto get back to his old indifference curve.

Hicks substitution effect is also negative just like the Slutskysubstitution effect.

x2

x1

x2

x1

Substitution

effectIncome effect

Final budget

Original budget


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Total change in demand is still

equivalent to the income plus

substitution effect, except that it is the

Hicks substitution effect that is

considered. It can be shown that for

small changes in prices, the two

substitution effects are identical.

We have analyzed how quantity demanded changes as pricechanges in three different situations:

Holding income fixed (the standard case) Holding purchasing power fixed ( the slutsky substitution effect) Holding utility fixed (Hicks substitution effect)

We can draw the relationship between price and quantityholding any of these three variables fixed, giving rise to threedemand curves: the standard or ordinary, Slutsky and Hicksian

demand curve. Slutsky and Hicksian demand curves are always downward

sloping. The ordinary demand curve is always downward sloping for

normal goods

Hicksian demand curve is also called the compensated demandcurve. This is since the consumers income is adjusted as pricechanges to keep him with the same level of utility, i.e., theconsumer is not made worse off or better off.

economics 102 lecture 7 income and substitution effects rev

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