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Chapter 8 Slutsky Equation Introduction Previously, we analyzed the effect of changes in prices and income on a consumer’s 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. 2 Effects of a Price Change What happens when a commodity’s 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 consumer’s purchasing power. 3 Effects of a Price Change x2 y p2 Consumer’s budget is $y. Original choice x1 4 Effects of a Price Change x2 y p2 Consumer’s budget is $y. Lower price for commodity 1 pivots the constraint outwards. x1 5 Effects of a Price Change x2 Consumer’s budget is $y. y p2 y' p2 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 consumer’s income has increased by $y - $y’. x1 6 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. 7 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. 8 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. xi x x s i n i 9 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 afford the original bundle?” 10 Pure Substitution Effect Only x 2 x2’ x1’ x1 11 Pure Substitution Effect Only x 2 x2’ x1’ x1 12 Pure Substitution Effect Only x 2 x2’ x1’ x1 13 Pure Substitution Effect Only x 2 x2’ x2’’ x1’ x1’’ x1 14 Pure Substitution Effect Only x 2 x2’ x2’’ x1’ x1’’ x1 15 Pure Substitution Effect Only x 2 Lower p1 makes good 1 relatively cheaper and causes a substitution from good 2 to good 1. (x1’,x2’) (x1’’,x2’’) is the pure substitution effect. x2’ x2’’ x1’ x1’’ x1 16 Pure Substitution Effect Substitution effect is always negatively related to the price change. Note that the portion of the grey compensated budget line below x’1 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 x’1 with the compensated budget. As a result, the substitution effect is positive for a price decrease. 17 Income Effect x2 (x1’’’,x2’’’) x2’ x2’’ x1’ x1’’ x1 18 Income Effect x2 The income effect is (x1’’,x2’’) (x1’’’,x2’’’). (x1’’’,x2’’’) x2’ x2’’ x1’ x1’’ x1 19 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’’’). (x1’’’,x2’’’) x2’ x2’’ x1’ x1’’ x1 20 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. 21 Slutsky’s Effects for Normal Goods x2 Good 1 is normal because higher income increases demand (x1’’’,x2’’’) x2’ x2’’ x1’ x1’’ x1 22 Slutsky’s Effects for Normal Goods x2 Good 1 is normal because higher income increases demand, so the income and substitution (x1’’’,x2’’’) effects reinforce each other. x2’ x2’’ x1’ x1’’ x1 23 Slutsky’s Effects for Normal Goods Since both the substitution and income effects increase demand when own-price falls, a normal good’s ordinary demand curve slopes down. The Law of (Downward-Sloping) Demand therefore always applies to normal goods. 24 Slutsky’s 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 good’s own price changes. 25 Slutsky’s Effects for Inferior Goods x2 x2’ x1’ x1 26 Slutsky’s Effects for Inferior Goods x2 x2’ x1’ x1 27 Slutsky’s Effects for Inferior Goods x2 x2’ x1’ x1 28 Slutsky’s Effects for Inferior Goods x2 x2’ x2’’ x1’ x1’’ x1 29 Slutsky’s Effects for Inferior Goods x2 The pure substitution effect is as for a normal good. But, …. x2’ x2’’ x1’ x1’’ x1 30 Slutsky’s Effects for Inferior Goods The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction. (x1’’’,x2’’’) x2 x2’ x2’’ x1’ x1’’ x1 31 Slutsky’s Effects for Inferior Goods x2 x2’ x2’’ The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction. Good 1 is (x1’’’,x2’’’) inferior because an increase in income causes demand to fall. x1’ x1’’ x1 32 Slutsky’s Effects for Inferior Goods x2 The overall changes to demand are the sum of the substitution and income effects. (x ’’’,x ’’’) 1 x2’ 2 x2’’ x1’ x1’’ x1 33 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. 34 Slutsky’s Effects for Giffen Goods x2 A decrease in p1 causes quantity demanded of good 1 to fall. x2’ x1’ x1 35 Slutsky’s Effects for Giffen Goods x2 A decrease in p1 causes quantity demanded of good 1 to fall. x2’’’ x2’ x1’’’ x1’ x1 36 Slutsky’s Effects for Giffen Goods x2 A decrease in p1 causes quantity demanded of good 1 to fall. x2’’’ x2’ x2’’ x1’’’ x1’ x1’’ Substitution effect Income effect x1 37 Slutsky’s 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. 38 Mathematical Treatment If we denote m’ as the income required to obtain the original bundle at the new prices, so that m’=p’1 x1 + p2 x2 and m=p1 x1 + p2 x2 . Thus the change in real income is m’– m = (p’1 – p1 ) x1 Or m p1 x1 39 Mathematical Treatment The substitution effect is x1s x1 ( p'1 , m' ) x1 ( p1 , m) The Income effect is x x1 ( p1 ' , m) x1 ( p1 ' , m' ) n 1 Total Effect (Slutsky identity) x1 x1 ( p1 ' , m) x1 ( p1 , m) x x s 1 n 1 40 Perfect Complements 41 Perfect Substitutes 42 Quasi-Linear Preference 43 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. 44