Consumer Theory Properties of Consumer Demand

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Topic 3 Part IV
Consumer Theory Properties of Consumer Demand
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Properties of Consumer Demand

• Demand function xi(p, y), i 1, , n is
  homogeneous of degree zero in all prices and
  income
• xi(tp, ty) t0xi(p, y) xi(p, y) for all t gt0
• Absence of money illusion If both, prices and
  income increase in the same proportion, the
  budget set does not change, and therefore, no
  change in demand behavior can occur

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Properties of Consumer Demand

• Demand function xi(p, y), i 1, , n satisfies
  budget balancedness
• p x(p, y) y for all (p, y)
• Because u(.) is strictly increasing, x(p, y) must
  exhaust
• the consumers income. Otherwise, the consumer
  should
• afford to purchase strictly more of every good
  and strictly
• increase her utility

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Changes in Income Inferior and Normal Goods

• Inferior goods A good xi for which ?xi/?y lt 0
  over some range of income variation is an
  inferior good in that range
• Normal goods A good xi for which ?xi/?y ? 0 over
  some range of income variation is a normal or
  non-inferior good in that range

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Total Effect of a Change in Price Substitution
and Income Effects

• The total effect (TE) of a reduction in the price
  of a good generates two effects
• (Hicksian Decomposition)

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Substitution Effect

• Substitution Effect (SE) the good becomes
  relatively cheaper compared to other goods.
  Because all goods are desirable, even if the
  consumers total command (purchasing power) over
  goods were unchanged, we would expect the
  consumer to substitute the relative cheaper good
  for the now relatively more expensive ones

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Income Effect

• Income Effect (IE) when the price of a good
  decreases, the consumers total command
  (purchasing power) over goods increases, allowing
  the consumer to increase her purchase of all
  goods if she wants this

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Total Effect of a Change in Price Substitution
and Income Effects

• The utility-maximization behavior suggests that,
  for normal goods, a fall in the price of a good
  leads to an increase in the quantity purchased
• The substitution effect causes more to be
  purchased as the individual moves along an
  indifference curve
• The income effect causes more to be purchased
  because the price decline increases the
  purchasing power, thereby permitting a movement
  to a higher indifference curve

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Total Effect of a Change in Price Substitution
and Income Effects

• For inferior goods, substitution and income
  effects work in opposite directions and no
  definite prediction can be made
• Giffens paradox if the income effect of a price
  is strong enough, the change in price and the
  resulting change in the quantity demanded move in
  the same direction

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Slutsky Equation

• The Slutsky equation summarizes the total effect
  of a change in the price of a good
• Let x(p,y) be the consumers Marshallian demand
  system and u be the level of utility the
  consumer achieves at prices p and income y

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Slutsky Equation

• Then,
• TE SE
  IE
• ?xi(p,y)/?pi ?xhi(p,u)/?pi - xi(p,y)
  ?xi(p,y)/?y
• for all i 1, 2,, n

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Generalized Slutsky Equation

• ?xi(p,y)/?pj ?xhi(p,u)/?pj - xj(p,y)
  ?xi(p,y)/?y
• for all i, j 1, 2,, n

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Restrictions on the Substitution Terms

• Given that ?e(p,u)/?pi xhi(p,u) and that e(p,
  u) is concave in p, our theory tells us quite a
  bit about substitution terms
• We will use this knowledge and the Slutsky
  equation to generate theoretical restrictions on
  the observable Marshallian demand

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Restrictions on the Substitution Terms
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Hessian Matrix of Expenditure Function and
Substitution Matrix ?

• e11 e12 e1n ?xh1/?p1 ? xh1/?p2 ? xh1/?pn
  
• e21 e22 e2n ?xh2/?p1 ? xh2/?p2 ? xh2/?pn
• H . . .
  . ?
• en1 en2 enn ?xhn/?p1 ?
  xhn/?p2 ? xhn/?pn

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Restrictions on the Substitution Terms
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Restrictions on the Substitution Terms
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Slutsky Matrix s and Substitution Matrix ?

• s11 s12 s1n ?xh1/?p1 ? xh1/?p2 ? xh1/?pn
  
• s21 s22 s2n ?xh2/?p1 ? xh2/?p2 ? xh2/?pn
• s . .
  . . ?
• sn1 sn2 snn ?xhn/?p1 ?
  xhn/?p2 ? xhn/?pn
• where
• sij ? ?xi(p,y)/?pj xj(p,y) ?xi(p,y)/?y
  ?xhi(p,u)/?pj
• for all i, j 1, 2,, n

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Slutsky Matrix s

• Then, the Slutsky matrix inherits all the
  characteristics of the substitution matrix ?
• Symmetry
• sii ? 0

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Testing the Theory of Demand or Applying it
Empirically

• Requirements on consumer demand
• Homogeneity
• Budget balancedness
• Associated Slutsky matrix s be symmetric
• These requirements provide a set of restrictions
  on allowable values for the parameters in any
  empirically estimated Marshallian demand system
  if that system is to be viewed as belonging to a
  price-taking, utility-maximizing consumer

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Demand Elasticities and Income Shares

• Let xi(p,y) be the consumers Marshallian demand
  for good i. Then,
• Income elasticity of demand for good i measures
  the percentage change in the quantity of i
  demanded per 1 percent change in income
• ?i ? ?xi(p, y)/?y y/xi(p, y)

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Demand Elasticities and Income Shares

• Price elasticity of demand for good i measures
  the percentage change in the quantity of i
  demanded per 1 percent change in the price pj
• ?ij ? ?xi(p, y)/?pj pj/xi(p, y)
• i j ? (own) price elasticity
• i ? j ? cross-price elasticity

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Demand Elasticities and Income Shares

• Income share is the proportion of the consumers
  income spent on purchases of good i
• si ? pixi(p, y)/y
• so that
• si ? 0 and ? si 1

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Aggregation in Consumer Demand

• Let xi(p,y) be the consumers Marshallian demand
  for good i. Given budget balancedness, the
  following relations must hold among income
  shares, price, and income elasticities of demand

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Aggregation in Consumer Demand

• Engel aggregation
• ? si ?i 1
• The weighted average on income elasticities for
  all goods that a person buys must be one

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Aggregation in Consumer Demand

• Cournot aggregation
• ? si ?ij - sj , j 1, , n
• Consider the case of 2 goods
• s1 ?11 s2 ?21 - s1 (1)
• s1 ?12 s2 ?22 - s2 (2)
• Equation (1) shows that the size of the
  cross-price effect of a change in the price of
  good 1 on the quantity demanded of good 2 is
  restricted because of the budget constraint,
  i.e., directown-price effect cannot be totally
  offset by cross-price effect