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Slutsky’s Equation Chapter 8 – Consumer’s Theory Effects of a Price Change • What happens when a commodity s price decreases? • Substitution effect: the commodity is relatively cheaper compared to others, so consumers substitute it for other commodities. • Income effect: the consumer can now purchase more than before, and is, in this sense, richer in real terms. The consumer shoud tend to consume more of all goods. Effects of a Price Change x2 Original choice x1 Effects of a Price Change x2 Lower price for commodity 1 pivots the constraint outwards. x1 Effects of a Price Change x2 A lower income would be needed to buy the original bundle at the new prices. x1 Real Income Changes x2 Original budget constraint and choice x1 Real Income Changes x2 Original budget constraint and choice New budget constraint x1 Real Income Changes x2 Original budget constraint and choice New budget constraint: real income has increased x1 Real Income Changes x2 Original budget constraint and choice x1 Real Income Changes x2 Original budget constraint and choice New budget constraint x1 Real Income Changes x2 Original budget constraint and choice New budget constraint: real income has fallen x1 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 buy the original bundle? Pure Substitution Effect !" !′" !′# !# Pure Substitution Effect !" !′" !′# !# Pure Substitution Effect !" !′" Income adjustment !′# !# Pure Substitution Effect !" Optimal choice with adjusted income !′" !′′" !′# !′′# !# Pure Substitution Effect !" Lower p1 makes good 1 relatively cheaper and causes a substitution from good 2 to good 1. !′" (!′&# − ! &# ; !′& " − ! & " ) = Pure Substitution Effect !′′" !′# !′′# !# And Now The Income Effect !" !′" !′′" !′# !′′# !# And Now The Income Effect !" !′" (!′′′# ; !′′' " ) !′′" !′# !′′# !# And Now The Income Effect The income effect is: !" (!′′′# −! ''# ; !′′′" − !′′" ) !′" (!′′′# ; !′′' " ) !′′" !′# !′′# !# Total Effect Decomposed !" The total change in demand is the sum of the income effect and the substitution effect: !′" (!′′′# −! '# ; ! ''' " − ! ' " ) = = (!′′′# −! ''# ; !′′′" − ! '' " ) + + (!′′# −! '# ; !′′" − !′" ) + (!′′′# ; !′′' " ) !′′" !′# !′′# !# Total Effect Decomposed The total change in demand is the sum of the income effect and the substitution effect: !′′′# = (!′′′# −! ''# ) + (! ''# − ! '# ); !′′′" = (!′′′" −! '' " ) + (! '' " − ! ' " ). !" !′" (!′′′# ; !′′' " ) !′′" !′# !′′# !# 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. Good 1 is normal, and its demand goes up with income. !" !′" (!′′′# ; !′′' ") When a good is normal, Income and substitution effects reinforce each other !′′" !′# !′′# !# Slutsky s Effects for Normal Goods • Since both the substitution and income effects increase demand when own-price falls, a normal good s demand curve slopes down. • The Law of Downward-Sloping Demand therefore always applies to normal goods. Slutsky s Effects for Income-Inferior Goods • Some goods are income-inferior (i.e. demand is reduced by higher income). • The substitution and income effects go in opposite directions when an income-inferior good s own price changes. Slutsky s Effects for Income-Inferior Goods !" !′" !′$ !$ Slutsky s Effects for Income-Inferior Goods !" !′" !′$ !$ Slutsky s Effects for Income-Inferior Goods !" !′" !′$ !$ Slutsky s Effects for Income-Inferior Goods !" !′" !′′" !′$ !′′$ !$ Slutsky s Effects for Income-Inferior Goods The pure substitution effect is, as usual, positive… !" !′" !′′" !′$ !′′$ !$ Slutsky s Effects for Income-Inferior Goods The pure substitution effect is, as usual, positive…but the income effect goes in the opposite direction (inferior good) !" !′" !′′" !′$ !′′$ !$ Slutsky s Effects for Income-Inferior Goods !" (!′′′$ ; !′′' " ) The pure substitution effect is, as usual, positive…but the income effect goes in the opposite direction (inferior good) !′" !′′" !′$ !′′$ !$ Giffen Goods • In rare cases of extreme income-inferiority, the income effect may be larger in size than the substitution effect, causing quantity demanded to fall as own-price rises. • Such goods are Giffen goods. Slutsky s Effects for Giffen Goods !" !′" !′# !# Slutsky s Effects for Giffen Goods !" !′′′" !′" !′′′# !′# !# Slutsky s Effects for Giffen Goods !" !′′′" !′" !′′′# !′# !# Slutsky s Effects for Giffen Goods !" !′′′" !′" !′′" !′′′# !′# !′′# Substitution effect !# Slutsky s Effects for Giffen Goods !" !′′′" !′" !′′" !′′′# !′# !′′# !# Income Effect (strong and negative) Slutsky s Effects for Giffen Goods !" A decrease in the price of good 1 is followed by a reduction in the demand for good 1. !′′′" The Law of Demand fails for strongly inferior goods. !′" !′′" !′′′# !′# !′′# !# Income Effect (strong and negative) A More Formal Treatment Let us write the new income m’ needed to support the old allocation at new prices, and compare it with the actual income m: The net transfer to go from m to m’ is given by: The substitution effect is then defined as: The income effect is then the complement to the final choice: We obtain the total effect decomposed: Consider an increase in the price of good 1. - If 1 is a normal good: - If 1 is an inferior good: Rates of Change - Define the negative of the income effect: - Write: - In rates of change: Remember now that: from which: We obtain (Slutsky Equation): Perfect Complements Perfect Substitutes Quasi-linear Preferences A Numerical Problem Let: U(x1, x2 ) = x1 x2 p1 = p2 = 1 p1 ' = 2 m = 100 We obtain: MU1 (x1, x2 ) = x2 MU 2 (x1, x2 ) = x1 MRS(x1, x2 ) = x2 x1 Let us compute the optimal choice at original prices: MRS(x1, x2 ) = x2 p1 = = 1 → x1 = x2 x1 p2 x1 + x2 = 100 → x1 + x1 = 100 → x1* = 50 = x *2 Let us now compute the optimal choice at new prices: MRS ( x1 , x2 ) = x2 p1 = = 2 ® 2 x1 = x2 x1 p2 2 x1 + x2 = 100 ® 2 x1 + 2 x1 = 100 ® x'1 = 25; x'2 = 50. Let us now compute the substitution effect: * * m' = p '1 x1 + p2 x2 = 2 x* + x* 2 = 150 x s 2 p' 1 MRS ( x 1 , x 2 ) = s = = 2 ® 2 x s1 = x s 2 x1 p2 s s 2 x s1 + x s 2 = 150 ® 2 x s1 + 2 x s1 = 150 ® x s1 = s * dx s1 = x1 - x1 = 75 2 ; x 2 = 75. 2 75 25 - 50 = - . 2 2 As expected, consumption of 1 decreases (50>75/2) and consumption of 2 increases (75>50). Let us now compute the income effect: n s dx1 = x'1 - x1 = 25 - 75 25 =2 2 An applied problem • Suppose Paul is living in Rome, consuming Italian and French food, with convex preferences represented by the utility function: U(xI , xF ) • Suppose Paul’s boss wants to convince him to move to the Paris offices, and is considering what increase in salary would convince Paul to move. Obviously, the boss just wants to give Paul the salary that would serve this purpose, not a Euro more… j p Let i be the price of food i in country j. Assume that: pII < pIF pFF < p II 1) Argue graphically that if Paul’s preferences are strictly convex, then his boss can convince him to move at a salary which is lower than the salary Paul would need to maintain his Italian eating habits in France. 2) Argue graphically that Paul’s boss would need to pay Paul enough to maintain his Italian eating habits in France if Paul’s preferences are of the perfect complements type