Chapter 8 Slutsky Equation

1
Chapter 8Slutsky Equation
2
Introduction

• Previously, we analyzed the effect of changes in
  prices and income on a consumers demand.
• In this chapter, we want to further analyze the
  price change effect. Specifically, we will
  decompose the effect into substitution effect and
  income effect.

3
Effects of a Price Change

• What happens when a commoditys price decreases?
• Substitution effect change in demand due to the
  change in the rate of exchange between the two
  goods (this commodity becomes relatively cheaper
  while the other good becomes relatively more
  expensive).
• Income effect change in demand due to the
  increase in the consumers purchasing power.

4
Effects of a Price Change
x2
Consumers budget is y.
Original choice
x1
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Effects of a Price Change
x2
Consumers budget is y.
Lower price for commodity 1 pivots the constraint
outwards.
x1
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Effects of a Price Change
x2
Consumers budget is y.
Lower price for commodity 1 pivots the constraint
outwards.
Now only y are needed to buy the original
bundle at the new prices, as if the
consumers income has increased
by y - y.
x1
7
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.

8
Effects of a Price Change

• Changes to quantities demanded due to the change
  in relative prices, keeping income just enough to
  buy the original bundle, are the (pure)
  substitution effect of the price change.
• Changes to quantities demanded due to the change
  in real income are the income effect of the
  price change.

9
Effects of a 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.

10
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 consumers income
  is adjusted so that, at the new prices, she can
  only just afford the original bundle?

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Pure Substitution Effect Only
x2
x2
x1
x1
12
Pure Substitution Effect Only
x2
x2
x1
x1
13
Pure Substitution Effect Only
x2
x2
x1
x1
14
Pure Substitution Effect Only
x2
x2
x2
x1
x1
x1
15
Pure Substitution Effect Only
x2
x2
x2
x1
x1
x1
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Pure Substitution Effect Only
x2
Lower p1 makes good 1 relativelycheaper and
causes a substitutionfrom good 2 to good 1.
(x1,x2) ? (x1,x2) is the
pure substitution effect.
x2
x2
x1
x1
x1
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Pure Substitution Effect

• Substitution effect is always negatively related
  to the price change.
• Note that the portion of the grey compensated
  budget line below x1 is inside the budget set
  of the original budget, thus these bundles should
  be less preferred than the original bundle.
• As a result, the consumer must choose a point at
  or more than x1 with the compensated budget. As
  a result, the substitution effect is positive for
  a price decrease.

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Income Effect
x2
(x1,x2)
x2
x2
x1
x1
x1
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Income Effect
x2
The income effect is (x1,x2) ?
(x1,x2).
(x1,x2)
x2
x2
x1
x1
x1
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The Overall Change in Demand
x2
The change to demand due to lower p1 is the sum
of the income and substitution effects,
(x1,x2) ? (x1,x2).
x2
(x1,x2)
x2
x1
x1
x1
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Slutskys 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 goods own price changes.

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Slutskys Effects for Normal Goods
x2
Good 1 is normal becausehigher income
increasesdemand
(x1,x2)
x2
x2
x1
x1
x1
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Slutskys Effects for Normal Goods
x2
Good 1 is normal becausehigher income
increasesdemand, so the income
and substitution
effects reinforce each
other.
(x1,x2)
x2
x2
x1
x1
x1
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Slutskys Effects for Normal Goods

• Since both the substitution and income effects
  increase demand when own-price falls, a normal
  goods ordinary demand curve slopes down.
• The Law of (Downward-Sloping) Demand therefore
  always applies to normal goods.

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Slutskys Effects for Inferior Goods

• Some goods are inferior (i.e. demand is reduced
  when income is higher).
• The substitution and income effects oppose each
  other when an inferior goods own price changes.

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Slutskys Effects for Inferior Goods
x2
x2
x1
x1
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Slutskys Effects for Inferior Goods
x2
x2
x1
x1
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Slutskys Effects for Inferior Goods
x2
x2
x1
x1
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Slutskys Effects for Inferior Goods
x2
x2
x2
x1
x1
x1
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Slutskys Effects for Inferior Goods
x2
The pure substitution effect is as fora normal
good. But, .
x2
x2
x1
x1
x1
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Slutskys Effects for 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
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Slutskys Effects for Inferior Goods
The pure substitution effect is as for a normal
good. But, the income effect is in the
opposite direction. Good 1 is
inferior because an
increase in income
causes demand to
fall.
x2
(x1,x2)
x2
x2
x1
x1
x1
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Slutskys Effects for Inferior Goods
x2
The overall changes to demand arethe sum of the
substitution and
income effects.
(x1,x2)
x2
x2
x1
x1
x1
34
Giffen Goods

• In rare cases of extreme inferiority, the income
  effect may be larger than the substitution
  effect, causing quantity demanded to fall as
  own-price rises.
• Such goods are Giffen goods.

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Slutskys Effects for Giffen Goods
x2
A decrease in p1 causes quantity demanded of
good 1 to fall.
x2
x1
x1
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Slutskys Effects for Giffen Goods
x2
A decrease in p1 causes quantity demanded of
good 1 to fall.
x2
x2
x1
x1
x1
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Slutskys Effects for Giffen Goods
x2
A decrease in p1 causes quantity demanded of
good 1 to fall.
x2
x2
x2
x1
x1
x1
x1
Substitution effect
Income effect
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Slutskys Effects for Giffen Goods

• Giffen good can only result when the income
  effect of an inferior good is so strong that it
  dominates the pure substitution effect.
• This may be possible for poor households where
  the low-quality necessity has taken up a large
  portion of expenditure.
• This case is very rare, even if exists, so we
  have confidence that the Law of Demand almost
  always holds.

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Mathematical Treatment

• If we denote m as the income required to obtain
  the original bundle at the new prices, so that
  mp1 x1 p2 x2 and mp1 x1 p2 x2 .
• Thus the change in real income ism m (p1
  p1 ) x1
• Or

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Mathematical Treatment

• The substitution effect is
• The Income effect is
• Total Effect (Slutsky identity)

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Perfect Complements
42
Perfect Substitutes
43
Quasi-Linear Preference
44
Summary

• In this chapter, a decomposition of price effect
  on quantity demanded is introduced.
• Substitution effect effect of change of price
  holding real income constant.
• Income effect effect of change in real income.
• For normal goods, both effects are negative
  w.r.t. a price rise.
• For inferior goods, sub. effect is negative, but
  income effect is positive w.r.t. a price rise.
• Giffen goods can only be inferior goods with very
  strong income effect.