Slutsky Equation

1
Slutsky Equation
2
Slutskys Identity

• Let be consumers demand for good
  i when price of good i is pi and income is m
  holding other prices constant
• Similarly for
• If the price of good i changes from to
• Total change in demand denoted by
• ?xi

• Let be consumers demand for good
  i when price of good i is pi and income is m
  holding other prices constant
• Similarly for
• If the price of good i changes from to
• Total change in demand denoted by
• ?xi

3
Slutskys Identity

• Now let be the new level of income such
  that the consumer is just able to buy the
  original bundle of goods
• Total change in demand
• ?xi
• can be rewritten as
• ?xi
• or denote
• ?xi ?xis ?xin
• where ?xis substitution effect and ?xin
  income effect

4
Slutskys Identity

• Note that is the amount of the change
  in money income such that the consumer is just
  able to buy the original bundle of goods (i.e.
  purchasing power is constant)
• Denote ?m and ?pi
• ?m ?pi
• This is the amount of money that should be given
  to the consumer to hold purchasing power constant

5
Slutskys Identity

• In terms of the rates of change, we can write
  Slutskys Identity as
• ?xi ?xis ?xim xi(pi, m)
• ?pi ?pi ?m
• where ?xin ?xim

6
Effects of a Price Change

• What happens when a commoditys price decreases?
• Substitution effect the commodity is relatively
  cheaper, so consumers substitute it for now
  relatively more expensive other commodities.
• Income effect the consumers budget of m can
  purchase more than before, as if the consumers
  income rose, with consequent income effects on
  quantities demanded.
• Vice versa for a price increase

7
Effects of a Price Change
Consumers budget is m.
x2
Original choice
x1
8
Effects of a Price Change
x2
Lower price for commodity 1 pivots the constraint
outwards
New Constraint purchasing power is increased at
new relative prices
x1
9
Effects of a Price Change
x2
Now only m' are needed to buy the original
bundle at the new prices, as if the consumers
income hasincreased by m - m'.
x1
Imagined Constraint Income is adjusted to keep
purchasing power constant
10
Effects of a Price Change

• Changes to quantities demanded due to this
  extra income (m - m') are the income effect
  of the price change.
• Slutsky discovered that changes to demand from a
  price change are always the sum of a pure
  substitution effect and an income effect.

11
Real Income Changes

• Slutsky asserted that if, at the new prices,
• less income is needed to buy the original bundle
  then real income is increased
• more income is needed to buy the original bundle
  then real income is decreased

12
Real Income Changes
x2
Original budget constraint and choice
New budget constraint
x1
13
Real Income Changes
x2
Less income is needed to buy original
bundle. Hence, ..
x1
14
Real Income Changes
x2
Original budget constraint and choice
New budget constraint
x1
15
Real Income Changes
x2
More income is needed to buy original
bundle. Hence,
x1
16
Real Income Changes

• Absence of Money illusion
• If money income and prices increase (or
  decrease) by the same proportion, e.g. double
• ? budget constraint and consumers choice remain
  unchanged

17
Pure Substitution Effect

• 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 the original bundle?

18
Budget Constraints and Choices
x2
Original budget constraint and choice
Original Indifference Curve
x1
19
Budget Constraints and Choices
x2
New budget constraint when relative price of x1
is lower
x1
20
Budget Constraints and Choices
x2
Imagined budget constraint
x1
21
Budget Constraints and Choices
x2
Imagined Budget Constraint, Indifference Curve,
and Choice
x1
22
Pure Substitution Effect Only
x2
Lower p1 makes good 1 relativelycheaper and
causes a substitutionfrom good 2 to good 1.
( , ) ? ( , ) is the
pure substitution effect
x1
23
The Income Effect
x2
The income effect is ?
( , )
( , )
( , )
x1
24
Total Effect
x2
The change in demand due to lower p1 is the sum
of the income and substitution effects,
?
( , )
( , )
( , )
x1
25
Slutskys 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 goods own price changes.

26
Slutskys Effects for Normal Goods
x2
Good 1 is normal because . .
( , )
x1
27
Slutskys Effects for Normal Goods
x2
so the income and substitution effects
each other
( , )
Total Effect
x1
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Slutskys Effects for Normal Goods

• When pi decreases, ?pi is negative (-)
• ?pi ? ?xi ?xis ?xin
• (-) ( ) ( ) ( )
• both substitution and income effects increase
  demand when own-price falls.
• Alternatively,
• ?xi ?xis ?xim xi(pi, m)
• ?pi ?pi ?m
• ( ) ( ) ( ) x ( )

-
29
Slutskys Effects for Normal Goods

• When pi decreases, ?pi is positive ()
• ?pi ? ?xi ?xis ?xin
• () ( ) ( ) ( )
• both substitution and income effects decrease
  demand when own-price rises.
• Alternatively,
• ?xi ?xis ?xim xi(pi, m)
• ?pi ?pi ?m
• ( ) ( ) ( ) x ( )

-
30
Slutskys Effects for Normal Goods

• In both cases, a change is own price results in
  an opposite change in demand ?xi
• ?pi
• ? a normal goods ordinary demand curve slopes
  down.
• The Law of Downward-Sloping Demand therefore
  always applies to normal goods.

is always
31
Slutskys Effects for Income-Inferior Goods

• Some goods are income-inferior (i.e. demand is
  reduced by higher income).
• The pure substitution effect is as for a normal
  good. But, the income effect is in the opposite
  direction.
• Therefore, the substitution and income effects
  oppose each other when an income-inferior goods
  own price changes.

32
Slutskys Effects for Income-Inferior Goods
x2
x1
33
Slutskys Effects for Income-Inferior Goods
x2
Good 1 is income-inferior because

( , )
x1
34
Slutskys Effects for Income-Inferior Goods
x2
Substitution and Income effects .. each other
( , )
Total Effect
x1
35
Slutskys Effects for Income-Inferior Goods

• When pi decreases, ?pi is negative (-)
• ?pi ? ?xi ?xis ?xin
• (-) ( ) ( ) ( )
• substitution effect increases demand while
  income effect reduces demand
• Alternatively,
• ?xi ?xis ?xim xi(pi, m)
• ?pi ?pi ?m
• ( ) ( ) ( ) x ( )

-
36
Slutskys Effects for Income-Inferior Goods

• When pi decreases, ?pi is positive ()
• ?pi ? ?xi ?xis ?xin
• () ( ) ( ) ( )
• both substitution and income effects decrease
  demand when own-price rises.
• Alternatively,
• ?xi ?xis ?xim xi(pi, m)
• ?pi ?pi ?m
• ( ) ( ) ( ) x ( )

-
37
Slutskys Effects for Income-Inferior Goods

• In general, substitution effect is greater than
  income effect.
• Hence, ?xi is usually positive when pi decreases.
  
• and ?xi is usually negative when pi increases.
• That is is ..
• and Demand Curve slopes downward

38
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 called Giffen goods.

39
Slutskys Effects for Giffen Goods
x2
Income effect Substitution effect.
x1
Substitution effect
Income effect
40
Slutskys Effects for Giffen Goods
x2
A decrease in p1 causes quantity demanded of
good 1 to fall.
Total Effect
x1
41
Slutskys Effects for Giffen Goods

• 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 Giffen
  goods.

42
Hicks Income and Substitution Effects

• Previously, we learn
• Slutskys Substitution Effect the change in
  demand when purchasing power is kept constant.
• Hick proposed another type of Substitution Effect
  where consumer is given just enough money to be
  on the same indifference curve.
• Hicks Substitution Effect the change in demand
  when utility is kept constant.

43
Hicks Income and Substitution Effects

• Total change in demand when price changes
• ?xi
• can be rewritten as
• ?xi
• Where is minimum income needed to
  achieve the original utility u at price
• substitution effect
• income effect

44
Hicks Income and Substitution Effects
x2
New budget constraint when p1 falls
Original choice
New choice
Original budget constraint
x1
45
Hicks Income and Substitution Effects
x2
Substitution Effect is optimal choice found on
the original indifference curve using the new
relative prices
x1
Income Effect
46
Hicks Income and Substitution Effects
x2
As before, Substitution and Income effects
.. each other
x1
47
Demand Curves

• Marshallian (Ordinary) Demand
• shows the quantity actually demanded when own
  price changes holding .. constant
• Slutsky Demand
• shows Slutsky substitution effect when own price
  changes holding constant
• Hicksian (Compensated) Demand
• shows Hick substitution effect when own price
  changes holding .. constant

48
Comparison Hick and Slutsky Substitution Effects
when own price falls
x2
.. budget constraint
budget constraint
x1
Substitution
. Substitution
49
Demand Curves for Normal Good when Own Price
Falls
p1
Demand
. Demand
.. Demand
x1