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 =
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

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

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 = ∆xim

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
Consumer’s budget is $m. x2
Original choice x1

8. Effects of a Price Changex2
Lower price for commodity 1
pivots the constraint outwards
New Constraint:
purchasing power is increased
at new relative prices x1

9. Effects of a Price Changex2
Now only $m' are needed to buy the
original bundle at the new prices, as if the consumer’s income has increased by $m - $m'. x1
Imagined Constraint: Income is adjusted to keep purchasing power constant

10. Effects of a Price Change
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 increased
more income is needed to buy the original bundle then “real income” is decreased

12. Real Income Changes x2 Original budget constraint and choice
New budget constraint x1

13. Real Income Changes x2 Less income is needed to buy original bundle.
Hence, …………………….. x1

14. Real Income Changes x2 Original budget constraint and choice
New budget constraint x1

15. Real Income Changes x2 More income is needed to buy original bundle.
Hence, ……………………… x1

16. Real Income Changes 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

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 Choicesx2
Original budget constraint and choice
Original Indifference Curve x1

19. Budget Constraints and Choicesx2
New budget constraint
when relative price of x1 is lower x1

20. Budget Constraints and Choicesx2
Imagined budget constraint x1

21. Budget Constraints and Choicesx2
Imagined Budget Constraint,
Indifference Curve, and Choice x1

22. Pure Substitution Effect Onlyx2
Lower p1 makes good 1 relatively cheaper and causes a substitution from good 2 to good ( , )  ( , ) is the pure substitution effect x1

23. The Income Effect x2
The income effect is 
( , )
( , )
( , ) x1

24. Total Effect x2 The change in demand due to lower p1 is the sum of the
income and substitution effects, 
( , )
( , )
( , ) x1

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. Slutsky’s Effects for Normal Goodsx2
Good 1 is normal because .……
…………………………………….
( , ) x1

27. Slutsky’s Effects for Normal Goodsx2
so the income and
substitution effects
………… each other
( , )
Total
Effect x1

28. Slutsky’s Effects for Normal Goods
When pi decreases, ∆pi is 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 ( )
= ─

29. Slutsky’s Effects for Normal Goods
When pi decreases, ∆pi is 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 ( )
= ─

30. Slutsky’s Effects for Normal Goods
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. Slutsky’s Effects for Income-Inferior Goodsx2 x1

33. Slutsky’s Effects for Income-Inferior Goodsx2
Good 1 is income-inferior because
………………………………………………………………………
( , ) x1

34. Slutsky’s Effects for Income-Inferior Goodsx2
Substitution and Income effects
……….. each other
( , )
Total
Effect x1

35. Slutsky’s Effects for Income-Inferior Goods
When pi decreases, ∆pi is 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 ( )
= ─

36. Slutsky’s Effects for Income-Inferior Goods
When pi decreases, ∆pi is 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 ( )
= ─

37. Slutsky’s Effects for Income-Inferior Goods
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 Goodsx2
Income effect …………
Substitution effect. x1
Substitution effect
Income effect

40. Slutsky’s Effects for Giffen Goodsx2
A decrease in p1 causes quantity demanded of good 1 to fall.
Total
Effect x1

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. Hick’s Income and Substitution Effects
Total change in demand when price changes
∆xi =
can be rewritten as +
Where is minimum income needed to achieve the original utility u at price
= substitution effect
= income effect

44. Hick’s Income and Substitution Effectsx2
New budget constraint when p1 falls
Original choice
New choice
Original budget constraint x1

45. Hick’s Income and Substitution Effectsx2
Substitution Effect is optimal choice found on the original indifference curve using the new relative prices x1
Income Effect

46. Hick’s Income and Substitution Effectsx2
As before, Substitution and Income effects ……….. each other x1

47. Demand Curves 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

48. Comparison: Hick and Slutsky Substitution Effects when own price fallsx2
……….. budget constraint
……… budget constraint x1
…… Substitution
………. Substitution

49. Demand Curves for Normal Good when Own Price Fallsx1