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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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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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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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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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x2
x1
x2
x1
x2
x1
x2
x1
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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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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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x2
x1
x2
x1
A decrease in p1 causes
quantity demanded of
good 1 to fall.
x2
x1
x2
x1x1
x2
A decrease in p1 causes
quantity demanded of
good 1 to fall.
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x2
x1
x2
x2
x1 x1x1
x2
Substitution effect
Income effect
A decrease in p1 causes
quantity demanded of
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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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.