Slutsky Equation Explaining Income and Substitutione Effects of a Price Change
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Transcript of Slutsky Equation Explaining Income and Substitutione Effects of a Price Change
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Price Change: Income and Substitution Effects
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THE IMPACT OF A PRICE CHANGE
Economists often separate the impact of a price change into two components:
– the substitution effect; and
– the income effect.
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THE IMPACT OF A PRICE CHANGE
The substitution effect involves the substitution of good x1 for good x2 or vice-versa due to a change in relative prices of the two goods.
The income effect results from an increase or decrease in the consumer’s real income or purchasing power as a result of the price change.
The sum of these two effects is called the price effect.
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THE IMPACT OF A PRICE CHANGE
The decomposition of the price effect into the income and substitution effect can be done in several ways
There are two main methods:
(i) The Hicksian method; and
(ii) The Slutsky method
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THE HICKSIAN METHOD Sir John R.Hicks (1904-1989) Awarded the Nobel Laureate in
Economics (with Kenneth J. Arrrow) in 1972 for work on general equilibrium theory and welfare economics.
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THE HICKSIAN METHOD
X2
X1
Ea
I1
xa
Optimal bundle is Ea, on indifference curve I1.
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THE HICKSIAN METHOD
X2
X1
I1
xa
Ea
A fall in the price of X1
The budget line pivots out from P P*
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THE HICKSIAN METHOD
X2
X1
Eb
I1
I2
xa xb
Ea
The new optimum is Eb on I2.
The Total Price Effect is xa to xb
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THE HICKSIAN METHOD
To isolate the substitution effect we ask….
“what would the consumer’s optimal bundle be if s/he faced the new lower price for X1 but experienced no change in real income?”
This amounts to returning the consumer to the original indifference curve (I1)
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THE HICKSIAN METHOD
X2
X1
Eb
I1
I2
xa xb
Ea
The new optimum is Eb on I2.
The Total Price Effect is xa to xb
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THE HICKSIAN METHOD
X2
X1
I1
I2
xa xb
EaEb
Draw a line parallel to the new budget line and tangent to the old indifference curve
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THE HICKSIAN METHOD
X2
X1
Ec I1
I2
xa xc xb
EaEb
The new optimum on I1 is at Ec. The movement from
Ea to Ec (the increase in quantity demanded from
Xa to Xc) is solely in response to a change in
relative prices
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THE HICKSIAN METHOD
X2
X1
I1
I2
Substitution Effect
EaEb
Ec
This is the substitution effect.
Xa Xc
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THE HICKSIAN METHOD
To isolate the income effect … Look at the remainder of the total
price effect This is due to a change in real
income.
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THE HICKSIAN METHOD
X2
X1
I1
I2
Income Effect
EaEb
Ec
The remainder of the total effect is due to a change
in real income. The increase in real income is
evidenced by the movement from I1 to I2
XcXb
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THE HICKSIAN METHOD
X2
X1
I1
I2
xa xc xb
Sub Effect
IncomeEffect
EaEb
Ec
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HICKSIAN ANALYSIS and DEMAND CURVES
22111 xpxpM
P1
P1*
22111 xpxpM A
A
B
B
C
Hicksian Demand Curve (A & C)
Marshallian Demand Curve (A & B)
P
X1
X1
P
A fall in price from p1 to p1
*
C
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HICKSIAN ANALYSIS and DEMAND CURVES
Hicksian (compensated) demand curves cannot be upward-sloping (i.e. substitution effect cannot be positive)
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THE SLUTSKY METHOD Eugene Slutsky (1880-1948) Russian economist expelled from the
University of Kiev for participating in student revolts.
In his 1915 paper, “On the theory of the Budget of the Consumer” he introduced “Slutsky Decomposition”.
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THE SLUTSKY METHOD
X2
X1
Ea
I1
xa
Optimal bundle is Ea, on indifference curve I1.
![Page 21: Slutsky Equation Explaining Income and Substitutione Effects of a Price Change](https://reader033.fdocuments.net/reader033/viewer/2022061103/541162357bef0a856f8b45fb/html5/thumbnails/21.jpg)
THE SLUTSKY METHOD
X2
X1
I1
xa
Ea
A fall in the price of X1
The budget line pivots out from P P*
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THE SLUTSKY METHOD
X2
X1
Eb
I1
I2
xa xb
Ea
The new optimum is Eb on I2.
The Total Price Effect is xa to xb
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THE SLUTSKY METHOD Slutsky claimed that if, at the new prices,– less income is needed to buy the original
bundle then “real income” has increased–more income is needed to buy the
original bundle then “real income” has decreased
Slutsky isolated the change in demand due only to the change in relative prices by asking “What is the change in demand when the consumer’s income is adjusted so that, at the new prices, s/he can just afford to buy the original bundle?”
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THE SLUTSKY METHOD
To isolate the substitution effect we adjust the consumer’s money income so that s/he change can just afford the original consumption bundle.
In other words we are holding purchasing power constant.
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THE SLUTSKY METHOD
X2
X1
Eb
I1
I2
xa xb
Ea
The new optimum is Eb on I2.
The Total Price Effect is xa to xb
![Page 26: Slutsky Equation Explaining Income and Substitutione Effects of a Price Change](https://reader033.fdocuments.net/reader033/viewer/2022061103/541162357bef0a856f8b45fb/html5/thumbnails/26.jpg)
THE SLUTSKY METHOD
X2
X1
Eb
I1
I2
xa xb
Ea
Draw a line parallel to the new budget line which passes through the point
Ea.
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THE SLUTSKY METHOD
X2
X1
Eb
I3
I2
xa xb
Ea
The new optimum on I3 is at Ec. The
movement from Ea to Ec is the
substitution effect
Ec
xc
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THE SLUTSKY METHOD
X2
X1
Eb
I3
I2
xa
Ea
The new optimum on I3 is at Ec. The
movement from Ea to Ec is the
substitution effect
Ec
xc
Substitution Effect
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THE SLUTSKY METHOD
X2
X1
Eb
I3
I2Ea
The remainder of the total price effect is the Income Effect.
The movement from Ec to Eb.
Ec
xc
Income Effect
xb
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THE SLUTSKY METHOD for NORMAL GOODS
Most goods are normal (i.e. demand increases with income).
The substitution and income effects reinforce each other when a normal good’s own price changes.
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THE SLUTSKY METHOD for NORMAL GOODS
X2
X1
Eb
I3
I2Ea
The income and substitution effects
reinforce each other.
Ec
xc xbxa
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Since both the substitution and income effects increase demand when own-price falls, a normal good’s ordinary demand curve slopes downwards.
The “Law” of Downward-Sloping Demand therefore always applies to normal goods.
THE SLUTSKY METHOD for NORMAL GOODS
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THE SLUTSKY EQUATION
Let
be the original budget constraint
and let
22111 xpxpM
22112 xpxpM
represent the budget constraint after the Slutsky compensating variation in income has been carried out.
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THE SLUTSKY EQUATION
X2
X1
Ea
xa
22111 xpxpM
22112 xpxpM
Demand for x1 is
Mppxx d ,, 211 M2 < M1
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THE SLUTSKY EQUATIONM2 - M1
11111
11112
111112
2211221112
2211221112
-
- M
- M
- M
- M
pppaspxM
ppxMM
xpxpMM
xpxpxpxpMM
xpxpxpxpMM
M=x1p1
gives the change in money income needed to consume the original bundle of goods (at EA)
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THE SLUTSKY EQUATION
1211211 ,,,, MppxMppxx dd
The demand curve holding M constant is given by
which is the change in demand for x1 due to the change in its own price, holding M and the price of x2 constant
(1)
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THE SLUTSKY EQUATION
,,,, 121221 MppxMppxx dds
The change in demand due to the Slutsky substitution effect is given by
(3)
,,,, 221121 MppxMppxx ddm
The income effect is given by
(2)
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THE SLUTSKY EQUATION
121221 ,,,, MppxMppxx dds
221121 ,,,, MppxMppxx ddm
1211211 ,,,, MppxMppxx dd
ms xxx 1
Given
Claim
Show this by substituting equations (1), (2) and (3) into equation (4)
(1)
(2)
(4)
(3)
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THE SLUTSKY EQUATION
ms xxx 1
Divide across by p1
111
1
p
x
p
x
p
x ms
Recall11 pxM
so 11 )( xMp
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THE SLUTSKY EQUATION
Substituting
11 )( xMp
111
1
p
x
p
x
p
x ms
Gives1
11
1 xM
x
p
x
p
x ms
THE
SLUTSKY EQUATION
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THE SLUTSKY METHOD: INFERIOR GOODS
Some goods are (sometimes) inferior (i.e. demand is reduced by higher income).
The substitution and income effects “oppose” each other when an inferior good’s own price changes.
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THE SLUTSKY METHOD: INFERIOR GOODS
X2
X1
Eb
I3
I2
Ea
The substitution effect is as per usual. But, the
income effect isin the opposite
direction.
Ec
xcxbxa
xa to xc
xc to xb
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GIFFEN GOODS
In rare cases of extreme inferiority, the income effect may be larger in size than the substitution effect, causing quantity demanded to rise as own price falls.
Such goods are Giffen goods. Giffen goods are very inferior goods.
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THE SLUTSKY METHOD for INFERIOR GOODS
X2
X1
Eb
I3
I2
Ea
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 falls. Ec
xb xcxaxa to xc
xc to xb
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SLUTSKY’S EFFECT FOR GIFFEN GOODS
Slutsky’s 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 very inferior goods.
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DECOMPOSITION of TOTAL PRICE EFFECT: PERFECT COMPLEMENTS
I1 I2 No substitution effect
X2
X1
A=C
BOriginal Budget Constraint
New Budget Constraint
A fall in the price of X1
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DECOMPOSITION of TOTAL PRICE EFFECT PERFECT SUBSTITUTES
?