1. Chapter Eight
Slutsky’s Equation

2. Effects of a Price Change
What happens when the price of a commodity decreases?
First, the commodity becomes relatively cheaper, so consumers substitute towards it and away from now relatively more expensive other commodities. This is the substitution effect of the price change.

3. Effects of a Price Change
Second, the consumer’s budget of $y can purchase larger bundles than before. It is as if the consumer received an increase in income.
The consequent changes in quantities demanded are the income effects of the price change.

4. Effects of a Price Change
Consumer’s budget is $y. x2
Original choice x1

5. Effects of a Price Change
Consumer’s budget is $y. x2
Lowered price for commodity 1
pivots outwards the budget constraint. x1

6. Effects of a Price Change
Consumer’s budget is $y. x2
Lowered price for commodity 1
pivots outwards the budget constraint.
Now only $y’ are needed to buy the original bundle at the new prices. It is as if the consumer’s income has increased by $y - $y’. x1

7. Effects of a Price Change
The change to quantities demanded due to this ‘extra’ income is the income effect of the price change.

8. Effects of a Price Change
Slutsky’s insight was that the effects on quantities demanded of any price change can always be decomposed into a pure substitution effect and an income effect.
The overall change in quantities demanded due to a price change is the sum of the substitution and income effects.

9. Pure Substitution Effect
Slutsky isolated the change in quantities demanded due only to the change in relative prices by asking “What is the change in quantities demanded when the consumer’s income is adjusted so that, at the new prices, she can only just buy the original bundle?”

10. Pure Substitution Effect Onlyx2
x2’
x1’ x1

11. Pure Substitution Effect Onlyx2
x2’
x1’ x1

12. Pure Substitution Effect Onlyx2
x2’
x1’ x1

13. Pure Substitution Effect Onlyx2
Lowering p1 makes good 1 relatively cheaper and causes a substitution from good 2 to good 1. The change from (x1’,x2’) to (x1’’,x2’’) is the pure substitution effect.
x2’
x2’’
x1’
x1’’ x1

14. And Now The Income Effectx2
The income effect is the change from (x1’’,x2’’) to (x1’’’,x2’’’).
x2’
(x1’’’,x2’’’)
x2’’
x1’
x1’’ x1

15. The Overall Change in Demandx2
The overall effect on demands of the change in p1 is the sum of the income and substitution effects. This is the change from (x1’,x2’) to (x1’’’,x2’’’).
x2’
(x1’’’,x2’’’)
x2’’
x1’
x1’’ x1

16. Slutsky’s Effects for Normal Goodsx2
Good 1 is normal because an
increase to income causes demand to rise.
x2’
(x1’’’,x2’’’)
x2’’
x1’
x1’’ x1

17. Slutsky’s Effects for Normal Goodsx2
Good 1 is normal because an
increase to income causes demand to rise. So the income and substitution effects reinforce each other.
x2’
(x1’’’,x2’’’)
x2’’
x1’
x1’’ x1

18. Slutsky’s Effects for Normal Goods
Since both the substitution and income effects increase demand when own-price decreases, the ordinary demand curve for a normal good must slope downwards.
The Law of Downward-Sloping Demand therefore always applies to normal goods.

19. Slutsky’s Effects for Income-Inferior Goods
Some goods are income-inferior; that is, demand is reduced by an increase in income.
Slutsky showed that, for income-inferior goods, the substitution and income effects oppose each other when a good’s own price changes.

20. Slutsky’s Effects for Income-Inferior Goodsx2
The pure substitution effect is as for a normal good. But, ….
x2’
x2’’
x1’
x1’’ x1

21. Slutsky’s Effects for Income-Inferior Goods
The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction. x2
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’ x1

22. Slutsky’s Effects for Income-Inferior Goods
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. x2
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’ x1

23. Slutsky’s Effects for Income-Inferior Goodsx2
The overall changes to demand are the sums of the substitution and income effects.
(x1’’’,x2’’’)
x2’
x2’’
x1’
x1’’ x1

24. Giffen Goods
In the rare case of extreme income-inferiority, the income effect may be larger in size than the substitution effect, causing quantity demanded to decrease as own-price decreases.
Such goods are called Giffen goods.

25. Slutsky’s Effects for Giffen Goodsx2
A decrease in p1 causes a decrease in the quantity demanded of good 1.
x2’
x1’ x1

26. Slutsky’s Effects for Giffen Goodsx2
A decrease in p1 causes a decrease in the quantity demanded of good 1.
x2’’’
x2’
x1’’’
x1’ x1

27. Slutsky’s Effects for Giffen Goodsx2
A decrease in p1 causes a decrease in the quantity demanded of good 1.
x2’’’
x2’
x2’’
x1’’’
x1’
x1’’ x1
Substitution effect
Income effect

28. 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.