# Chapter 8 - Slutsky’s Equation

```Slutsky’s Equation
Chapter 8 – Consumer’s Theory
Effects of a Price Change
• What happens when a commodity s price decreases?
• Substitution effect: the commodity is relatively cheaper compared
to others, so consumers substitute it for other commodities.
• Income effect: the consumer can now purchase more than before,
and is, in this sense, richer in real terms. The consumer shoud tend
to consume more of all goods.
Effects of a Price Change
x2
Original choice
x1
Effects of a Price Change
x2
Lower price for commodity 1
pivots the constraint outwards.
x1
Effects of a Price Change
x2
A lower income would be needed to buy
the original bundle at the new prices.
x1
Real Income Changes
x2
Original budget constraint and choice
x1
Real Income Changes
x2
Original budget constraint and choice
New budget constraint
x1
Real Income Changes
x2
Original budget constraint and choice
New budget constraint: real income has increased
x1
Real Income Changes
x2
Original budget constraint and choice
x1
Real Income Changes
x2
Original budget constraint and choice
New budget constraint
x1
Real Income Changes
x2
Original budget constraint and choice
New budget constraint: real income has fallen
x1
Pure Substitution Effect
• Slutsky isolated the change in demand due only to the change in
What is the change in demand when the consumer s income is
adjusted so that, at the new prices, she can only just buy the original
bundle?
Pure Substitution Effect
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Pure Substitution Effect
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Pure Substitution Effect
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Pure Substitution Effect
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Optimal choice with
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Pure Substitution Effect
!"
Lower p1 makes good 1 relatively cheaper and
causes a substitution from good 2 to good 1.
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(!′&# − ! &# ; !′& " − ! & " ) = Pure Substitution
Effect
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And Now The Income Effect
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And Now The Income Effect
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And Now The Income Effect
The income effect is:
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Total Effect Decomposed
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The total change in demand is the sum
of the income effect and the
substitution effect:
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= (!′′′# −! ''# ; !′′′" − ! '' " ) +
+ (!′′# −! '# ; !′′" − !′" ) +
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Total Effect Decomposed
The total change in demand is the sum of the income
effect and the substitution effect:
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Slutsky s Effects 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.
Good 1 is normal, and its demand goes
up with income.
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When a good is normal,
Income and substitution
effects reinforce each other
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Slutsky s Effects for Normal Goods
• Since both the substitution and income effects increase demand
when own-price falls, a normal good s demand curve slopes down.
• The Law of Downward-Sloping Demand therefore always applies to
normal goods.
Slutsky s Effects for Income-Inferior Goods
• Some goods are income-inferior (i.e. demand is reduced by higher
income).
• The substitution and income effects go in opposite directions when
an income-inferior good s own price changes.
Slutsky s Effects for Income-Inferior Goods
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Slutsky s Effects for Income-Inferior Goods
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Slutsky s Effects for Income-Inferior Goods
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Slutsky s Effects for Income-Inferior Goods
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Slutsky s Effects for Income-Inferior Goods
The pure substitution effect is, as usual,
positive…
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Slutsky s Effects for Income-Inferior Goods
The pure substitution effect is, as usual,
positive…but the income effect goes in the
opposite direction (inferior good)
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Slutsky s Effects for Income-Inferior Goods
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The pure substitution effect is, as usual,
positive…but the income effect goes in the
opposite direction (inferior good)
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Giffen Goods
• 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.
Slutsky s Effects for Giffen Goods
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Slutsky s Effects for Giffen Goods
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Slutsky s Effects for Giffen Goods
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Slutsky s Effects for Giffen Goods
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Substitution effect
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Slutsky s Effects for Giffen Goods
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Income Effect (strong and negative)
Slutsky s Effects for Giffen Goods
!"
A decrease in the price of good 1 is
followed by a reduction in the demand
for good 1.
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The Law of Demand fails for strongly
inferior goods.
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Income Effect (strong and negative)
A More Formal Treatment
Let us write the new income m’ needed to support the old allocation at
new prices, and compare it with the actual income m:
The net transfer to go from m to m’ is given by:
The substitution effect is then defined as:
The income effect is then the complement to the final choice:
We obtain the total effect decomposed:
Consider an increase in the price of good 1.
- If 1 is a normal good:
- If 1 is an inferior good:
Rates of Change
- Define the negative of the income effect:
- Write:
- In rates of change:
Remember now that:
from which:
We obtain (Slutsky Equation):
Perfect Complements
Perfect Substitutes
Quasi-linear Preferences
A Numerical Problem
Let:
U(x1, x2 ) = x1 x2
p1 = p2 = 1
p1 ' = 2
m = 100
We obtain:
MU1 (x1, x2 ) = x2
MU 2 (x1, x2 ) = x1
MRS(x1, x2 ) =
x2
x1
Let us compute the optimal choice at original prices:
MRS(x1, x2 ) =
x2 p1
=
= 1 → x1 = x2
x1 p2
x1 + x2 = 100 → x1 + x1 = 100 → x1* = 50 = x *2
Let us now compute the optimal choice at new prices:
MRS ( x1 , x2 ) =
x2 p1
=
= 2 ® 2 x1 = x2
x1 p2
2 x1 + x2 = 100 ® 2 x1 + 2 x1 = 100 ® x'1 = 25; x'2 = 50.
Let us now compute the substitution effect:
*
*
m' = p '1 x1 + p2 x2 = 2 x* + x* 2 = 150
x s 2 p' 1
MRS ( x 1 , x 2 ) = s =
= 2 ® 2 x s1 = x s 2
x1
p2
s
s
2 x s1 + x s 2 = 150 ® 2 x s1 + 2 x s1 = 150 ® x s1 =
s
*
dx s1 = x1 - x1 =
75 2
; x 2 = 75.
2
75
25
- 50 = - .
2
2
As expected, consumption of 1 decreases (50>75/2) and consumption of 2
increases (75>50).
Let us now compute the income effect:
n
s
dx1 = x'1 - x1 = 25 -
75
25
=2
2
An applied problem
• Suppose Paul is living in Rome, consuming Italian and French
food, with convex preferences represented by the utility function:
U(xI , xF )
• Suppose Paul’s boss wants to convince him to move to the Paris
offices, and is considering what increase in salary would convince
Paul to move. Obviously, the boss just wants to give Paul the salary
that would serve this purpose, not a Euro more…
j
p
Let i be the price of food i in country j.
Assume that:
pII < pIF
pFF < p II
1) Argue graphically that if Paul’s preferences are strictly convex, then his boss
can convince him to move at a salary which is lower than the salary Paul
would need to maintain his Italian eating habits in France.
2) Argue graphically that Paul’s boss would need to pay Paul enough to maintain
his Italian eating habits in France if Paul’s preferences are of the perfect
complements type
```