1 slutsky equation. 2 slutsky’s identity let be consumer’s demand for good i when price of good...

1

Slutsky Equation

2

Slutsky’s Identity

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 =

ip

m),p(x ii

m),(px ii

ip

m),(px-m),p(x iiii

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 =

3

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

∆xi = can be rewritten as

∆xi = +or denote

∆xi = ∆xis

+ ∆xin

where ∆xis

= substitution effect and ∆xin = income effect

]m),(px - )m,p([x iiii ])m,p(x- m),p([x iiii ′′′

m′

m),(px-m),p(x iiii

4

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)

Denote ∆m = and ∆pi =

∆m = ∆pi

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

m),(px ii

m-m

m-m

ii p-p

5

Slutsky’s Identity 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 = ∆xi

m

6

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.

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

7

Effects of a Price Change

x2

x1

Original choice

Consumer’s budget is $m.

2pm

8


x1

Lower price for commodity 1pivots the constraint outwards

x2

2pm

New Constraint:purchasing power is increased at new relative prices

9


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

x2

2pm

2pm'

Imagined Constraint: Income is adjusted to keep purchasing power constant

10


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 increasedmore income is needed to buy the original

bundle then “real income” is decreased

12

Real Income Changes

x1

x2

Original budget constraint and choice

New budget constraint

13

Real Income Changes

x1

x2

Less income is needed tobuy original bundle.Hence, ……………………..

14

Real Income Changes

x1

x2


New budget constraint

15

Real Income Changes

x1

x2

More income is needed tobuy original bundle.Hence, ………………………

16

Absence of Money illusion

If money income and prices increase (or decrease) by the same proportion, e.g. double

→ budget constraint and consumer’s choice remain unchanged

Real Income Changes

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 consumer’s income is adjusted so that, at the new prices, she can only just buy the original bundle?”

18

Budget Constraints and Choices

1x

x2

x1


Original Indifference Curve

2x

19

x2

x1

New budget constraint when relative price of x1 is lower


1x

2x

20

x2

x1

Imagined budget constraint


1x

2x

21

1x

2x

x2

x1

Imagined Budget Constraint, Indifference Curve, and Choice


2x

1x

22

Pure Substitution Effect Only

1x

2x

2x

x2

x1

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

pure substitution effect

1x

23

The Income Effect

1x

2x

2x

x2

x1

( , )

The income effect is ( , )( , )

2x 1x

1x

24

Total Effect

1x

2x

2x

The change in demand due to lower p1 is the sum of the income and substitution effects,

x2

x1

( , )

( , )( , )

1x

2x 1x

25

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.

26


Good 1 is normal because .………………………………………….

x2

x1

( , )

1x

2x

2x

1x

2x 1x

27

Slutsky’s Effects for Normal Goodsx2

x1

( , )

so the income and substitution effects ………… each other

Total

Effect 1x

2x

2x

1x

2x 1x

28

When pi decreases, ∆pi is negative (─)

∆pi → ∆xi = ∆xis + ∆xi

n

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

Alternatively,

∆xi ∆xis

∆xim xi(pi, m)

∆pi ∆pi ∆m ( ) ( ) ( ) x ( )


= ─

29

When pi decreases, ∆pi is positive (+)


n

(+) ( ) ( ) ( )both substitution and income effects decrease demand when own-price rises.

Alternatively,

∆xi ∆xis

∆xim xi(pi, m)

∆pi ∆pi ∆m ( ) ( ) ( ) x ( )


= ─

30

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…………

31

Slutsky’s 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 good’s own price changes.

32

x2

x1

Slutsky’s Effects for Income-Inferior Goods

1x

2x

33

Slutsky’s Effects for Income-Inferior Goodsx2

x1

( , )

Good 1 is income-inferior because………………………………………………………………………

1x

2x

2x

1x

2x 1x

34


Substitution and Income effects ……….. each other

x2

x1

( , )

Total

Effect 1x

2x

2x

1x

2x 1x

35

When pi decreases, ∆pi is negative (─)


n

(─) ( ) ( ) ( )substitution effect increases demand while income effect reduces demand

Alternatively,

∆xi ∆xis

∆xim xi(pi, m)

∆pi ∆pi ∆m ( ) ( ) ( ) x ( )

= ─


36

When pi decreases, ∆pi is positive (+)


n

(+) ( ) ( ) ( )both substitution and income effects decrease demand when own-price rises.

Alternatively,

∆xi ∆xis

∆xim xi(pi, m)

∆pi ∆pi ∆m ( ) ( ) ( ) x ( )

= ─


37


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

∆xi

∆pi

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

Slutsky’s Effects for Giffen Goods

1x

x2

x1

Income effect …………Substitution effect.

1xSubstitution effectIncome effect

2x

1x

2x

2x

40

Slutsky’s Effects for Giffen Goodsx2

x1

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

Total

Effect1x

2x

1x

2x

41

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 Giffen goods.

42

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 given just enough money to be on the same indifference curve.

Hick’s Substitution Effect: the change in demand when utility is kept constant.

43

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

m),(px-m),p(x iiii

] m),(p x- u)),pe(,p([x iiiii ′′

] u)),pe(,p(x- m),p([x iiiii ′′′

Hick’s Income and Substitution Effects

u),pe( i′

ip′] m),(p x- u)),pe(,p([x iiiii ′′] u)),pe(,p(x- m),p([x iiiii ′′′

44

x2

x1

New budget constraint when p1 falls

Original budget constraint

Original choice


New choice

1x

2x

45

2x

x2

x11x


1x

2x

1x

2x

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

Income Effect

46

2x

x2

x11x


1x

2x

1x

2x

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

47

Demand Curves Marshallian (Ordinary) Demand

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

Slutsky Demandshows Slutsky substitution effect when own price changes holding …………………… constant

Hicksian (Compensated) Demandshows Hick substitution effect when own price changes holding ……….. constant

48

2x 1x

S1x

x2

x1

Comparison: Hick and Slutsky Substitution Effects when own price falls

1x

2x……… budget constraint

……….. budget constraint

H1x …… Substitution

………. Substitution

49

Demand Curves for Normal Good when

Own Price Falls

x1

p1

1p

1p′

S1x1x H

1x 1x

…………… Demand

………. Demand

……….. Demand

1 slutsky equation. 2 slutsky’s identity let be consumer’s demand for good i when price of good...

Documents