Effects of a Price Decrease
• Can be broken down into two components
– Income effect
•
•
•
•
Separating Income and
Substitution Effects
When the price of one goods falls, w/ other constant;
Effectively like increase in consumer’s real income
Since it unambiguously expands the budget set
Income effect on demand is positive, if normal good
– Substitution effect
ECON 370: Microeconomic Theory
• Measures the effect of the change in the price ratio;
• Holding some measure of ‘income’ or well being constant
• Consumers substitute it for other now relatively more
expensive commodities
• That is, Substitution effect is always negative
Summer 2004 – Rice University
Stanley Gilbert
– Two decompositions: Hicks, Slutsky
Hicks and Slutsky Decompositions
Hicks Substitution and Income Effects
• Due to Sir John Hicks (1904-1989; Nobel 1972)
• Hicks
– To get Substitution Effect: Hold utility constant and find
bundle that reflects new price ratio
– Substitution Effect = change in demand due only to this
change in price ratio (movement along IC)
– Income Effect = remaining change in demand to get
back to new budget constraint (parallel shift)
– Substitution Effect: change in demand, holding utility
constant
– Income Effect: Remaining change in demand, due to m
change
• Slutsky
– Substitution Effect: change in demand, holding real
income constant
– Income Effect: Remaining change in demand, due to m
change
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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1
Hicks Decomposition Graphically
x2
Hicks Decomposition Graphically 2
x2
Given a drop in Price:
x2 ´
Given a drop in Price:
• Insert an “imaginary” budget line
tangent to original IC and parallel to
new budget line
x2 ´
x1 ´
x1 ´
x1
x1
Substitution Effect
Income Effect
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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Slutsky Decomposition Graphically
Slutsky Substitution and Income Effects
x2
• Due to Eugene Slutsky (1880-1948)
Given a drop in Price:
– To get Substitution Effect: Hold purchasing power
constant
• (that is, adjust income so that the consumer can exactly afford
the original bundle)
x2 ´
– and find bundle that reflects new price ratio
– Substitution Effect = change in demand due only to this
change in price ratio (movement along IC)
– Income Effect = remaining change in demand to get
back to new budget constraint (parallel shift)
x1 ´
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
x1
8
2
Slutsky Decomposition Graphically 2
x2
Signs of Substitution and Income Effects
• Sign of Substitution Effect is unambiguously negative as
long as Indifference Curves are convex
Given a drop in Price:
• Insert an “imaginary” budget line
through the original bundle…
• Income effect may be positive or negative
– That is, the good may be either normal or inferior
x2 ´
• For Normal goods, the income effect reinforces the
substitution effect
• For Inferior goods, the two effects offset
• For Giffen Goods
x1 ´
– Remember, the Income effect is Negative
– And the income effect is greater than the substitution effect
x1
Substitution Effect
Income Effect
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
Slutsky’s Effects for Normal Goods
x2
Slutsky’s Effects for Inferior Goods
x2
From Before…
• Since Substitution Effect and
Income Effect reinforce each
other…
x2 ´
x1 ´
In this case:
• Since Substitution Effect and
Income Effect offset each
other…
x2 ´
• This is a Normal Good
• This is an Inferior Good
x1 ´
x1
Substitution Effect
x1
Substitution Effect
Income Effect
Econ 370 - Ordinal Utility
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Income Effect
11
Econ 370 - Ordinal Utility
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3
Slutsky’s Effects for Giffen Goods
x2
Mathematics of Slutsky Decomposition
• We seek a way to calculate mathematically the
Income and Substitution Effects
In this case:
• Since Income Effect
completely cancels the
Substitution Effect
• Assume:
–
–
–
–
• This is a Giffen Good
x2 ´
Income: m
Initial prices: p10, p2
Final prices: p11, p2
Note that the price of good two, here, does not change
• Given the demand functions, demands can be
readily calculated as:
x1 ´
Substitution Effect
Income Effect
– Initial demands: xi0 = xi( p10, p2, m)
– Final demands: xi1 = xi( p11, p2, m)
x1
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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Slutsky Mathematics (cont)
Hicks’ Mathematics
• We need to calculate an intermediate demand that
holds buying power constant
• The only difference is between Hicks’ and Slutsky
is in the calculation of the intermediate demand
• Let ms the income that provides exactly the same
buying power as before at the new price
• Let mh the income that provides exactly the same
utility as before at the new price
– If u0 is initial utility level, then
– Thus: mh solves u0 = u( x1(p11, p2, mh), x2(p11, p2, mh))
– Thus: ms = p11x10 + p2x20
• The demand associated with this income is:
• The demand associated with this income is:
– xis = xi( p11, p2, ms) = xis( p11, p2, x10, x20)
– xih = xi( p11, p2, mh) = xih( p11, p2, u0)
• Finally we have:
– Substitution Effect:
– Income Effect:
Econ 370 - Ordinal Utility
• Finally we have:
SE = xis – xi0
IE = xi1 – xis
– Substitution Effect:
– Income Effect:
15
Econ 370 - Ordinal Utility
SE = xih – xi0
IE = xi1 – xih
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4
Calculating the Slutsky Decomposition
Calculating the Slutsky Decomposition 2
⎤
⎡ p1
s
Since m = ⎢α 0x + (1 − α )⎥ m
⎦
⎣ px
Assume that u = xαy1 – α
x =α
So the demand functions are:
Initial Price is px0
x0 = α
s
m =
x1 = α
+ py y =
y = (1 − α )
m
py
We get:
⎤
ms
m ⎡ p1
m
m
x s = α 1 = α 1 ⎢α 0x + (1 − α )⎥ = α 2 0 + α (1 − α ) 1
px
px ⎣ px
px
px
⎦
Final Price is px1
m
p 0x
p1x x 0
m
px
p1xα
m
p1x
y 0 = y1 = y = α
m
py
or
Finally, we get:
(
SE = x s − x 0 = αx 0 + (1 − α )x1 − x 0 = (1 − α ) x1 − x 0
⎡ p1x
⎤
m
m
(
)
+
1
−
α
p
=
y
⎢α 0 + (1 − α )⎥ m
0
p
px
y
⎣ px
⎦
Econ 370 - Ordinal Utility
x s = αx 0 + (1 − α )x1
[
] (
IE = x1 − x s = x1 − αx 0 + (1 − α )x1 = α x1 − x 0
17
Calculating the Hicks Decomposition
)
)
Econ 370 - Ordinal Utility
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Calculating the Hicks Decomposition 2
α
⎛ p1 ⎞
m h = ⎜⎜ 0x ⎟⎟ m
⎝ px ⎠
We need to calculate mh, so
Since
Substituting our demand functions back into utility we get:
We get:
1−α
α
⎛ m ⎞ ⎜⎛
m⎞
u = xα y1−α = ⎜ α
⎟ ⎜ (1 − α ) ⎟⎟
py ⎠
⎝ px ⎠ ⎝
α
⎛α ⎞
Then mh solves: ⎜⎜ 1 ⎟⎟
⎝ px ⎠
1−α
⎛
⎞
⎜1 − α ⎟
⎜ p ⎟
⎝ y ⎠
α
⎛α ⎞
=⎜ ⎟
⎝ px ⎠
α
⎛α ⎞
m h = ⎜⎜ 0 ⎟⎟
⎝ px ⎠
xh = α
1−α
⎛
⎞
⎜1 − α ⎟
⎜ p ⎟
⎝ y ⎠
m
Econ 370 - Ordinal Utility
( ) ( )1−α
1−α
⎛
⎞
⎜1 − α ⎟
⎜ p ⎟
⎝ y ⎠
m
Finally, we get:
⎛ p1 ⎞
m h = ⎜⎜ 0x ⎟⎟ m
⎝ px ⎠
s
19
0
Econ 370 - Ordinal Utility
1⎛
⎜
α
p1 ⎞
SE = x − x = x ⎜ 0x ⎟⎟ − x 0
⎝ px ⎠
α
or
α
mh
m ⎛ p1 ⎞
m
= α 1 ⎜⎜ x0 ⎟⎟ = α
1
0 α 1
px
px ⎝ px ⎠
px px
s
1
1⎛
⎜
α
p1 ⎞
IE = x − x = x − x ⎜ 0x ⎟⎟
⎝ px ⎠
1
20
5
Demand Curves
Demand Curves (cont)
• We have already met the Marshallian demand
curve
• We mentioned before that with Giffen Goods, the
Marshallian demand curve slopes upward
– It was demand as price varies, holding all else constant
• However,
• There are two other demand curves that are
sometimes used
– Since the substitution effect is always negative, Then
– Both the Slutsky and Hicks Demands always slope
downward—even with Giffen Goods
• Slutsky Demand
– Change in demand holding purchasing power constant
– The function xis = xi( p11, p2, ms) we just defined
• Hicks Demand
– Change in demand holding utility constant
– The function xih = xi( p11, p2, mh) we just defined
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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6