Comparative Statics Slutsky Equation

Comparative Statics Slutsky Equation

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Slutsky Equation

Perbandingan Statis

Perbandingan 2 kondisi ekuilibrium yang terbentuk dari perbedaan nilai parameter dan variabel eksogen

dan variabel eksogen

Contoh: Perbandingan 2 keputusan konsumen (consumer’s behaviour) dengan perubahan harga

Secara umum, harga dan jumlah permintaan berbanding terbalik.

Artinya, jika harga naik maka permintaan akan turun.

Tetapi ternyata ada permintaannya yang

Tetapi ternyata ada permintaannya yang jumlahnya menurun dengan menurunnya harga barang tersebut, yang dikenal

dengan GIFFEN GOOD

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Slutsky’s Identity

Let be consumer’s demand for good i when price of good i is p_i and income is m

holding other prices constant Similarly for x (p′,m)

m) ,

(p x_{i i}

Let be consumer’s demand for good i when price of good i is p_i and income is m

holding other prices constant Similarly for

Similarly for

If the price of good i changes from to

Total change in demand denoted by

∆x_i =

pi

m) ,

p ( x_{i i}′

p′i

m) ,

(p x

- m) ,

p (

x_{i i}′ _{i i}

Similarly for

If the price of good i changes from to

Total change in demand denoted by

∆x_i =

Slutsky’s 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

∆x =

m′

m) ,

(p x

- m) ,

p ( x ′

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∆x_i =

can be rewritten as

∆x_i = +

or denote

∆x_i = ∆x_i^s + ∆x_iⁿ

where ∆xis = substitution effect and ∆xin = income effect

] m) ,

(p x

- ) m , p (

[x_{i i}′ ′ _{i i} [x_i(p′_i,m)- x_i(p′_i,m′) ] m)

, (p x

- m) ,

p (

x_i ′_{i i i}

Slutsky’s 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)

m - m′

purchasing power is constant)

Denote ∆m = and ∆p_i =

∆m = ∆p_i

This is the amount of money that should be given to the consumer to hold purchasing power constant

m) ,

(p x_{i i}

m -

m′ p′_i -p_i

Slutsky’s Identity

In terms of the rates of change, we can write Slutsky’s Identity as

∆x_i ∆x_i^s ∆x_i^m x_i(p_i, m)

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∆x_i ∆x_i ∆x_i x_i(p_i, m)

∆p_i ∆p_i ∆m where ∆x_iⁿ = ∆x_i^m

Dampak Perubahan Harga

Apa yang terjadi jika harga satu barang turun?

Efek Substitusi: barang tersebut menjadi relatif murah. Konsumen akan

meningkatkan konsumsinya terhadap meningkatkan konsumsinya terhadap barang yang menjadi relatif murah.

Efek Pendapatan (Income effect): Dengan budget yang tetap, konsumen dapat

mengkonsumsi lebih banyak barang, seolah-olah pendapatnya meningkat.

Effects of a Price Change

x₂

Original choice

Consumer’s budget is $m.

p2

m

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x₁

Effects of a Price Change

Harga turun untuk komoditi 1

Menyebabkan pergeseran batasan (pivot) x₂

p2

m

New Constraint:

Daya beli (purchasing power) naik

x

Daya beli (purchasing power) naik pada rasio harga yang baru

Effects of a Price Change

Sekarang budget yang dibutuhkan hanya sebesar $m' agar konsumen dapat mempertahankan tingkat

konsumsinya x₂

p2

m

p m'

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x₁ p2

Imagined Constraint: Income is adjusted to keep purchasing power constant

Effects of a Price Change

Perubahan jumlah permintaan akibat perubahan ‘extra’ income ($m - $m') merupakan income effect akibat

perubahan harga.

perubahan harga.

Slutsky menemukan bahwa perubahan jumlah permintaan selalu merupakan

penjumlahan dari pure substitution effect dan income effect.

Real Income Changes

Menurut Slutsky pada harga yang baru,

less income is needed to buy the original bundle then “real income” is increased

more income is needed to buy the original

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more income is needed to buy the original bundle then “real income” is decreased

Real Income Changes

x₂

x

Original budget constraint and choice New budget constraint

Real Income Changes

x₂

Less income is needed to buy original bundle.

Hence, ………..

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x₁

Hence, ………..

Real Income Changes

x₂

x

Original budget constraint and choice New budget constraint

Real Income Changes

x₂

More income is needed to buy original bundle.

Hence, ………

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x₁

Hence, ………

Absence of Money illusion

If money income and prices increase (or decrease) by the same proportion,

Real Income Changes

(or decrease) by the same proportion, e.g. double

→ budget constraint and consumer’s choice remain unchanged

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

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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?”

Budget Constraints and Choices

x₂

Original budget constraint and choice

x2

x

^x

Original Indifference Curve

x₂

New budget constraint

when relative price of x₁ is lower

Budget Constraints and Choices

x2

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x₁

x

1

x₂

Budget Constraints and Choices

x2

x

Imagined budget constraint

x

x2

x₂

Imagined Budget Constraint,

Budget Constraints and Choices

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x

1 ^x₁

Imagined Budget Constraint, Indifference Curve, and Choice

x′2

x′1

Efek Substitusi (

bundle yang sama)

x2

x₂ Lower p₁ makes good 1 relatively cheaper and causes a substitution from good 2 to good 1.

( , ) →→→→ ( , ) is the pure substitution effect

x

x′2

x pure substitution effect

x′

The Income Effect (

bundle beda

)

x2

x₂

( , )

The income effect is

→

→

→→ ( , ) ( , )

x ′′2

x ′′1

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x

1

x′2

x₁ ( , )_{x ′′1} x ′′₂

x′1

Total Effect

x2

The change in demand due to lower p₁ is the sum of the

income and substitution effects,

→

→

→

→ x₂

( , )

( , ) ( , )

x ′′2

x ′′1

x

x′2

x ( , )

x′

x ′′2

x ′′1

Slutsky’s Effects for Normal Goods

Umumnya barang bersifat normal (i.e.

demand naik dengan pertambahan income).

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income).

The substitution and income effects

reinforce each other when a normal good’s own price changes.

Slutsky’s Effects for Normal Goods

Good 1 is normal because .……

……….

x₂

( , )

x2

x ′′2

x ′′1

x ( , )

x

x′2

x′

x ′′2

x ′′1

Slutsky’s Effects for Normal Goods

x₂

( , )

so the income and substitution effects

………… each other

x2

x ′′2

x ′′1

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x₁ ( , )

Total

Effect

x

₁ x′2

x′1

x ′′2

x ′′1

When p_i decreases, ∆p_i is negative (─)

∆p_i → ∆x_i = ∆x_i^s + ∆x_iⁿ (─) ( ) ( ) ( )

both substitution and income effects

Slutsky’s Effects for Normal Goods

both substitution and income effects increase demand when own-price falls.

Alternatively,

∆x_i ∆x_i^s ∆x_i^m x_i(p_i, m)

∆p_i

= ─

∆p_i ∆m

When p_i decreases, ∆p_i is positive (+)

∆p_i → ∆x_i = ∆x_i^s + ∆x_iⁿ (+) ( ) ( ) ( )

both substitution and income effects

Slutsky’s Effects for Normal Goods

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both substitution and income effects

decrease demand when own-price rises.

Alternatively,

∆x_i ∆x_i^s ∆x_i^m x_i(p_i, m)

∆p_i ∆p_i ∆m

( ) ( ) ( ) x ( )

= ─

In both cases, a change is own price

results in an opposite change in demand

∆x_i

∆p

Slutsky’s Effects for Normal Goods

is always…………

∆p_i

→ 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 Income-Inferior Goods

Tetapi ada barang yang bersifat income- inferior (i.e. Permintaan berkurang

dengan bertambahnya higher income).

Efek substitusi sama dengan barang

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Efek substitusi sama dengan barang normal tetapi efek income berlawanan arah.

Dengan demikian, efek substitusi dan efek income berlawanan arah jika harga barang inferior berubah.

x₂

Slutsky’s Effects for Income-Inferior Goods

x2

x

x

Slutsky’s Effects for Income-Inferior Goods

x₂

( , )

Good 1 is income- inferior because

………

………

………

x2 x ′′²

x ′′1

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x₁

………

x

1

x′2

x′1

Slutsky’s Effects for Income-Inferior Goods

Substitution and Income effects

……….. each other x₂

( , )

x2 x ′′²

x ′′1

……….. each other

x Total

Effect

x

x′2

x′

When p_i decreases, ∆p_i is negative (─)

∆p_i → ∆x_i = ∆x_i^s + ∆x_iⁿ (─) ( ) ( ) ( )

substitution effect increases demand while

Slutsky’s Effects for Income-Inferior Goods

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substitution effect increases demand while income effect reduces demand

Alternatively,

∆x_i ∆x_i^s ∆x_i^m x_i(p_i, m)

∆p_i ∆p_i ∆m

( ) ( ) ( ) x ( )

= ─

When p_i decreases, ∆p_i is positive (+)

∆p_i → ∆x_i = ∆x_i^s + ∆x_iⁿ (+) ( ) ( ) ( )

both substitution and income effects

Slutsky’s Effects for Income-Inferior Goods

both substitution and income effects

decrease demand when own-price rises.

Alternatively,

∆x_i ∆x_i^s ∆x_i^m x_i(p_i, m)

∆p_i

= ─

∆p_i ∆m

Slutsky’s Effects for Income-Inferior Goods

In general, substitution effect is greater than income effect.

Hence, ∆x_i is usually positive when p_i decreases.

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Hence, ∆x_i is usually positive when p_i decreases.

and ∆x_i is usually negative when p_i increases.

That is is ………..

and Demand Curve slopes downward

∆x_i

∆p_i

Giffen Goods

In rare cases of extreme income-inferiority, the income effect may be larger in size

than the substitution effect, causing than the substitution effect, causing

quantity demanded to fall as own-price rises.

Such goods are called Giffen goods.

Slutsky’s Effects for Giffen Goods

x₂

Income effect …………

Substitution effect.

x ′′2

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x ′′

1

x′

_{1 x1}

Substitution effect Income effect

x

1

x2

x′2

Slutsky’s Effects for Giffen Goods

x₂

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

x ′′2

Total

x ′′ x

x

x2

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

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and an income effect thus explains why the Law of Downward-Sloping Demand is violated for Giffen goods.

Hick’s Income and Substitution Effects

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 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.

Total change in demand when price changes

∆x_i =

can be rewritten as

∆x

_i

=

m) ,

(p x

- m) ,

p (

x_{i i}′ _{i i}

] m) ,

(p x

- u)) ,

p e(

, p (

[x_i ′_i ′_{i i i}

Hick’s Income and Substitution Effects

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∆x

_i

= +

Where is minimum income needed to achieve the original utility u at price

= substitution effect

= income effect

] m) ,

(p x

- u)) ,

p e(

, p (

[x_i ′_i ′_{i i i}

] u)) ,

p e(

, p ( x - m) ,

p (

[x_i ′_{i i} ′_i ′_i

u) , p e( ′_i

p′i

] m) ,

(p x - u)) ,

p e(

, p (

[x_i ′_i ′_{i i i}

] u)) ,

p e(

, p ( x - m) , p (

[x_i ′_{i i} ′_i ′_i

x₂

New budget constraint when p₁ falls Original choice

Hick’s Income and Substitution Effects

x2

x Original budget constraint

New choice

x

x₂

Hick’s Income and Substitution Effects

x2

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

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x′2

x₁

x′1

x

1 2

x ′′

1

x ′′2

Income Effect

x₂

Hick’s Income and Substitution Effects

x2

As before, Substitution and Income effects

……….. each other

x′2

x′ x

x

2

x ′′

x ′′2

Demand Curves

Marshallian (Ordinary) Demand

shows the quantity actually demanded when own price changes holding ……….. constant

Slutsky Demand

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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

x′2 x′₁

x₂

Comparison: Hick and Slutsky Substitution Effects when own price falls

x2 ……… budget constraint

……….. budget constraint

xS ^x

x

2

xH

Demand Curves for Normal Good when Own Price Falls

p₁

p

1 ……… Demand

………. Demand

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x₁

p ′

1

S

x1

x

1 _x^H₁ x ′′₁

……….. Demand