Chapter 8Slutsky Equation

1
Chapter 8 Slutsky Equation

• Course Microeconomics
• Text Varians Intermediate Microeconomics

2
Introduction

• In Chapter 6, we talk about how demand changes
  when price and income change individually.
• In this chapter, we want to further analyze how
  the change in price changes the demand.
• In particular, we decompose the change in
  quantity demanded due to price change into
  substitution effect and income effect.

3
Effects of a Price Change

• What happens when a commoditys price decreases?
• Substitution effect the commodity is relatively
  cheaper, so consumers use more of it, instead of
  other commodities, which are now relatively more
  expensive.
• Income effect the consumers budget of m can
  purchase more than before, as if the consumers
  income rose, with consequent income effects on
  quantities demanded.

4
Effects of a Price Change
Consumers budget is m.
x2
Original choice
x1
5
Effects of a Price Change
Consumers budget is y.
x2
Lower price for commodity 1 pivots the constraint
outwards.
x1
6
Effects of a Price Change
Consumers budget is m.
x2
Lower price for commodity 1 pivots the constraint
outwards.
Now only m are needed to buy the original
bundle at the new prices, as if the
consumers income has increased
by m -- m.
x1
7
Real Income Changes

• Slutsky asserted that if, at the new prices,
• If less income is needed to buy the original
  bundle then real income is increased
• If 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 Only
x2
x2
x1
x1
11
Pure Substitution Effect Only
x2
x2
x1
x1
12
Pure Substitution Effect Only
x2
x2
x1
x1
13
Pure Substitution Effect Only
x2
x2
x2
x1
x1
x1
14
Pure Substitution Effect Only
x2
x2
x2
x1
x1
x1
15
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 yellow 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, and
  as a result, the substitution effect is positive
  for a price decrease.

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And Now The Income Effect
x2
(x1,x2)
x2
x2
x1
x1
x1
18
And Now The 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
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
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, so the income
and substitution
effects reinforce each
other.
(x1,x2)
x2
x2
x1
x1
x1
22
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.

23
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.

24
Slutskys Effects for Income-Inferior Goods
x2
x2
x1
x1
25
Slutskys Effects for Income-Inferior Goods
x2
x2
x1
x1
26
Slutskys Effects for Income-Inferior Goods
x2
x2
x1
x1
27
Slutskys Effects for Income-Inferior Goods
x2
x2
x2
x1
x1
x1
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Slutskys Effects for Income-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 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
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Slutskys Effects for Income-Inferior Goods
x2
The overall changes to demand arethe sums of the
substitution and
income effects.
(x1,x2)
x2
x2
x1
x1
x1
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Giffen Goods

• In rare cases of extreme income-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
34
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 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.

36
Mathematical Treatment

• If we denote m as the income required to obtain
  the original bundle at the new prices, so
  thatmp1 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

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Mathematical Treatment

• In terms of derivative (or rate of change)
• Which is known as the Slutsky Equation.
• (This is just a rough presentation. The tools
  need for formal derivations is not covered in
  this class.)

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Perfect Complements
40
Perfect Substitutes
41
Quasi-Linear Preference
42
Hicks Substitution Effect

• Slutskys method of decomposition is not the only
  reasonable way.
• Hicks proposed another way of holding real
  income constant.
• Instead of compensating him to be able to buy
  back the original bundle, Hicks method
  compensates the consumer to buy back a bundle
  that gives him the same utility as before.

43
Hicks Substitution Effect
44
Hicks Substitution Effect

• Hicks Substitution Effect is also negative,
  because of the convex preference. (It can also be
  shown by revealed preference.)
• The nominal income required to maintain the
  utility constant is less than the one required to
  buy back the same bundle. It implies a larger
  income effect for a price decrease, but a smaller
  income effect for a price increase.

45
Income Tax vs. Quantity Tax

• If government wants to impose tax to support
  public expenditure, or to punish consumption of
  a good, say, due to pollution, various means can
  be used.
• Here, given the revenue are the same in
  equilibrium, how do the effects of income tax and
  quantity tax on good 1 differ?(Note Income and
  tax rates are regarded as given for consumers.
  The tax rate is adjusted so that at equilibrium
  the tax revenue is the same.)

46
Income Tax vs. Quantity Tax
47
Income Tax vs. Quantity Tax

• Income tax corresponds to an inward shift of
  budget line.
• Quantity tax corresponds to an inward rotation of
  the budget line.
• When the revenues are held the same for
  comparison, the budget line for the income tax
  must pass through the optimal point for quantity
  tax. (Note The tax revenue is tx1, where x1
  is the optimal quantity under quantity tax.)

48
Income Tax vs. Quantity Tax

• With the same tax revenue, the utility level
  attained is higher with income tax than with the
  quantity tax.
• But quantity tax has a stronger effect in
  reducing the consumption of good 1 than income
  tax.

49
Quantity Tax with Rebate

• Consider a similar case. Now a tax is imposed to
  reduce consumption of certain good, but at the
  same time, an equivalent amount is rebated (given
  back) to the consumer.
• Again, consumer has to take the rebate and tax
  rate constant for his decision.

50
Quantity Tax with Rebate
51
Quantity Tax with Rebate

• Note that the new consumption bundle must be on
  the original budget line, because in equilibrium,
  the tax amount and rebate are the same.
• The consumer has become worse off after this
  quantity tax and rebate program.

52
Summary

• In this chapter, a decomposition of price effect
  on quality demand 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.