Chapter Eight
Slutsky Equation
What Are We Doing in this Chapter?
We take a further look at the
comparative statics of demand
function with respect to prices;
We divide the changes in demand
due to price changes into two
effects:
–Substitution effect;
–Income effect?
Why? Deeper economic intuition.
Effects of a Price Change
What happens when a commodity’s
price decreases?
–Substitution effect,the commodity
is relatively cheaper,so
consumers substitute it for now
relatively more expensive other
commodities.
Effects of a Price Change
–Income effect,the consumer’s
budget of $y can purchase more
than before,as if the consumer’s
income rose,with consequent
income effects on quantities
demanded.
Effects of a Price Change
x1
Lower price for commodity 1
pivots the constraint outwards.
Consumer’s budget is $y.x
2
y
p2
Effects of a Price Change
x1
Lower price for commodity 1
pivots the constraint outwards.
Consumer’s budget is $y.x
2
y
p2
y
p
'
2
Now only $y’ are needed to buy the
original bundle at the new prices,
as if the consumer’s income has
increased by $y - $y’.
The Idea behind the Slutsky Equation
Divide the total changes to demand
from a price change into
1) pure substitution effect;
2) an income effect.
Pure Substitution Effect
,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?”
I.e.,changes in demand without
changes in real income
Pure Substitution Effect Only
x2
x1
x2’
x1’
Pure Substitution Effect Only
x2
x1
x2’
x2’’
x1’ x1’’
Pure Substitution Effect Only
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.
Pure Substitution Effect Is
Always Negative!x
2
x1
x2’
x2’’
x1’ x1’’
(x1’’,x2’’) must always be on the
right-hand-side of (x1’,x2’) by WRSP!
And Now The Income Effect
x2
x1
x2’
x2’’
x1’ x1’’
(x1’’’,x2’’’)
And Now The Income Effect
x2
x1
x2’
x2’’
x1’ x1’’
(x1’’’,x2’’’)
The income effect is
(x1’’,x2’’)? (x1’’’,x2’’’).
The Overall Change in Demand
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’’’).
Question,What is the Contribution of
Slutsky?
Answer:
1) A nice way to divide the total
changes in demand;
2) Show that the pure substitution
effect is always negative.
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.
Slutsky’s Effects for Normal Goods
x2
x1
x2’
x2’’
x1’ x1’’
(x1’’’,x2’’’)
Good 1 is normal because
higher income increases
demand
x2
x1
x2’
x2’’
x1’ x1’’
(x1’’’,x2’’’)
Good 1 is normal because
higher income increases
demand,so the income
and substitution
effects reinforce
each other,
Slutsky’s Effects for Normal Goods
Since both the substitution and
income effects increase demand
when own-price falls,a normal
good’s ordinary demand curve
slopes down.
The Law of Downward-Sloping
Demand therefore always applies to
normal goods.
Slutsky’s Effects for 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
oppose each other when an income-
inferior good’s own price changes.
Slutsky’s Effects for Income-Inferior
Goods
x2
x1
x2’
x1’
Slutsky’s Effects for Income-Inferior
Goods
x2
x1
x2’
x2’’
x1’ x1’’
Slutsky’s Effects for Income-Inferior
Goods
x2
x1
x2’
x2’’
x1’ x1’’
The pure substitution effect is as for
a normal good,But,….
Slutsky’s Effects for Income-Inferior
Goods
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,
Slutsky’s Effects for Income-Inferior
Goods
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,
Slutsky’s Effects for Income-Inferior
Goods
x2
x1
x2’
x2’’
x1’ x1’’
(x1’’’,x2’’’)
The overall changes to demand are
the sums of the substitution and
income effects.
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
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 extremely income-
inferior goods.
Summary,The Key to This
Chapter
To be able to find the,auxiliary” choice
–It is the choice at the new price but
with compensated income
–Once this is found,we can calculate
the pure substitution effect and
income effect