Chapter Eight
Slutsky Equation
Slutsky 方程
Effects of a Price Change
?What happens when a commodity?s
price decreases?
–Substitution effect (替代效应）, the
commodity is relatively cheaper,
so consumers substitute it for now
relatively more expensive other
commodities.
Effects of a Price Change
–Income effect （ 收入效应）, 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
x2
x1
Original choice
Consumer?s budget is $y.
y
p2
Effects of a Price Change
x1
Lower price for commodity 1
pivots the constraint outwards.
Consumer?s budget is $y.x
2y
p2
Effects of a Price Change
x1
Lower price for commodity 1
pivots the constraint outwards.
Consumer?s budget is $y.x
2y
p2y
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?.
Effects of a Price Change
?Changes to quantities demanded due
to this ?extra? income are the income
effect of the price change.
Effects of a 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.
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
Real Income Changes
x1
x2
Original budget constraint and choice
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint; real
income has risen
Real Income Changes
x1
x2
Original budget constraint and choice
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint; real
income has fallen
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
consumer?s income is adjusted so
that,at the new prices,she can only
just buy the original bundle?”
?Use WARP to determine the direction
of change.
Pure Substitution Effect Only
x2
x1
x2?
x1?
Pure Substitution Effect Only
x2
x1
x2?
x1?
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.
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???).
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
x2
x1
x2?
x1?
A decrease in p1 causes
quantity demanded of
good 1 to fall.
Slutsky’s Effects for Giffen Goods
x2
x1
x2?
x1?x1???
x2???
A decrease in p1 causes
quantity demanded of
good 1 to fall.
Slutsky’s Effects for Giffen Goods
x2
x1
x2?
x2??
x1? x1??x1???
x2???
Substitution effect
Income effect
A decrease in p1 causes
quantity demanded of
good 1 to fall.
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.
The Slutsky Equation
1 1 2 1 1 2 1 2 1 1 2
1
11
(,,) (,,,) (,,)sx p p m x p p x x x p p m
x
p p m
? ? ?
??
? ? ?
(-) (+) if normal
(-) if inferior
(-)
(?)
Remaining Topics
? Examples of decomposition
– Perfect complements preferences
– Perfect substitute preferences
– Quasi-linear preferences
? Applications
– Rebating a tax
– Real time pricing
? Hicksian substitution effect
Quasi-linear
preferences
Rebating A Tax
?Original budget:
px + y = m
?Taxing x and rebate the tax,at
optimal consumption (x?,y?):
(p+t)x? + y? = m + tx?
px? + y? = m
?(x?,y?) is on the original budget
Rebating a tax based
on final consumption
Rebate a Tax on Initial
Consumption
?Original budget,at optimal
consumption (x,y):
px* + y* = m
?Taxing x and rebate the tax tx*,the
budget is:
(p+t)x + y = m + tx*
?(x*,y*) is on the budget
Rebating a
tax based on
initial
consumption
Slutsky Equation
Slutsky 方程
Effects of a Price Change
?What happens when a commodity?s
price decreases?
–Substitution effect (替代效应）, the
commodity is relatively cheaper,
so consumers substitute it for now
relatively more expensive other
commodities.
Effects of a Price Change
–Income effect （ 收入效应）, 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
x2
x1
Original choice
Consumer?s budget is $y.
y
p2
Effects of a Price Change
x1
Lower price for commodity 1
pivots the constraint outwards.
Consumer?s budget is $y.x
2y
p2
Effects of a Price Change
x1
Lower price for commodity 1
pivots the constraint outwards.
Consumer?s budget is $y.x
2y
p2y
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?.
Effects of a Price Change
?Changes to quantities demanded due
to this ?extra? income are the income
effect of the price change.
Effects of a 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.
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
Real Income Changes
x1
x2
Original budget constraint and choice
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint; real
income has risen
Real Income Changes
x1
x2
Original budget constraint and choice
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint
Real Income Changes
x1
x2
Original budget constraint and choice
New budget constraint; real
income has fallen
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
consumer?s income is adjusted so
that,at the new prices,she can only
just buy the original bundle?”
?Use WARP to determine the direction
of change.
Pure Substitution Effect Only
x2
x1
x2?
x1?
Pure Substitution Effect Only
x2
x1
x2?
x1?
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.
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???).
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
x2
x1
x2?
x1?
A decrease in p1 causes
quantity demanded of
good 1 to fall.
Slutsky’s Effects for Giffen Goods
x2
x1
x2?
x1?x1???
x2???
A decrease in p1 causes
quantity demanded of
good 1 to fall.
Slutsky’s Effects for Giffen Goods
x2
x1
x2?
x2??
x1? x1??x1???
x2???
Substitution effect
Income effect
A decrease in p1 causes
quantity demanded of
good 1 to fall.
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.
The Slutsky Equation
1 1 2 1 1 2 1 2 1 1 2
1
11
(,,) (,,,) (,,)sx p p m x p p x x x p p m
x
p p m
? ? ?
??
? ? ?
(-) (+) if normal
(-) if inferior
(-)
(?)
Remaining Topics
? Examples of decomposition
– Perfect complements preferences
– Perfect substitute preferences
– Quasi-linear preferences
? Applications
– Rebating a tax
– Real time pricing
? Hicksian substitution effect
Quasi-linear
preferences
Rebating A Tax
?Original budget:
px + y = m
?Taxing x and rebate the tax,at
optimal consumption (x?,y?):
(p+t)x? + y? = m + tx?
px? + y? = m
?(x?,y?) is on the original budget
Rebating a tax based
on final consumption
Rebate a Tax on Initial
Consumption
?Original budget,at optimal
consumption (x,y):
px* + y* = m
?Taxing x and rebate the tax tx*,the
budget is:
(p+t)x + y = m + tx*
?(x*,y*) is on the budget
Rebating a
tax based on
initial
consumption
