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Slutsky Equation

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Slutsky Equation

- Let be consumer’s 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 consumer’s 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=

- 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

- 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

- In terms of the rates of change, we can write Slutsky’s Identity as
∆xi ∆xis∆xim xi(pi, m)

∆pi∆pi∆m

where ∆xin = ∆xim

- 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.
- Income effect: the consumer’s budget of $m can purchase more than before, as if the consumer’s income rose, with consequent income effects on quantities demanded.

- Vice versa for a price increase

Consumer’s budget is $m.

x2

Original choice

x1

x2

Lower price for commodity 1

pivots the constraint outwards

New Constraint:

purchasing power is increased

at new relative prices

x1

x2

Now only $m' are needed to buy the

original bundle at the new prices, as if the consumer’s income hasincreased by $m - $m'.

x1

Imagined Constraint: Income is adjusted to keep purchasing power constant

- 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.

- 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

x2

Original budget constraint and choice

New budget constraint

x1

x2

Less income is needed to

buy original bundle.

Hence, ……………………..

x1

x2

Original budget constraint and choice

New budget constraint

x1

x2

More income is needed to

buy original bundle.

Hence, ………………………

x1

- Absence of Money illusion
If money income and prices increase (or decrease) by the same proportion, e.g. double

→ budget constraint and consumer’s choiceremain unchanged

- 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?”

x2

Original budget constraint and choice

Original Indifference Curve

x1

Budget Constraints and Choices

x2

New budget constraint

when relative price of x1 is lower

x1

x2

Imagined budget constraint

x1

Budget Constraints and Choices

x2

Imagined Budget Constraint,

Indifference Curve, and Choice

x1

x2

Lower p1 makes good 1 relativelycheaper and causes a substitutionfrom good 2 to good 1. ( , ) ( , ) is thepure substitution effect

x1

x2

The income effect is

( , )

( , )

( , )

x1

x2

The change in demand due to

lower p1 is the sum of the

income and substitution effects,

( , )

( , )

( , )

x1

- 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.

x2

Good 1 is normal because .……

…………………………………….

( , )

x1

x2

so the income and

substitution effects

………… each other

( , )

Total

Effect

x1

- When pidecreases, ∆piis 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 ( )

= ─

- When pidecreases, ∆piis 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 ( )

= ─

- In both cases, a change is own price results in an opposite change in demand ∆xi
∆pi

→ a normal good’s ordinary demand curve slopes down.

- The Law of Downward-Sloping Demand therefore always applies to normal goods.

is always…………

- 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 good’s own price changes.

x2

x1

x2

Good 1 is income-inferior because

………………………………………………………………………

( , )

x1

x2

Substitution and Income effects

……….. each other

( , )

Total

Effect

x1

Slutsky’s Effects for Income-Inferior Goods

- When pidecreases, ∆piis 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 ( )

= ─

Slutsky’s Effects for Income-Inferior Goods

- When pidecreases, ∆piis 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 ( )

= ─

∆xi

∆pi

- In general, substitution effect is greater than income effect.
- Hence, ∆xi is usually positive when pidecreases.
and ∆xi is usually negative when piincreases.

- That is is …………………..
and Demand Curve slopes downward

- 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.

x2

Income effect …………

Substitution effect.

x1

Substitution effect

Income effect

x2

A decrease in p1 causes quantity demanded of good 1 to fall.

Total

Effect

x1

- 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 Giffen goods.

- Previously, we learn
Slutsky’s 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.
- Hick’s Substitution Effect: the change in demand when utility is kept constant.

Hick’s 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

Hick’s Income and Substitution Effects

x2

New budget constraint when p1 falls

Original choice

New choice

Original budget constraint

x1

Hick’s Income and Substitution Effects

x2

Substitution Effect is optimal choice found on the original indifference curve using the new relative prices

x1

Income Effect

Hick’s Income and Substitution Effects

x2

As before, Substitution and Income effects ……….. each other

x1

- 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

Comparison: Hick and Slutsky Substitution Effects when own price falls

x2

……….. budget constraint

……… budget constraint

x1

…… Substitution

………. Substitution

p1

…………… Demand

………. Demand

……….. Demand

x1