Slutsky Equation

Slutsky’s Identity Let be consumer’s demand for good i when price of good i is pi and income is m holding other prices constant Similarly for If the price of good i changes from to Total change in demand denoted by ∆xi = Let be consumer’s demand for good i when price of good i is pi and income is m holding other prices constant Similarly for If the price of good i changes from to Total change in demand denoted by ∆xi =

Slutsky’s Identity Now let be the new level of income such that the consumer is just able to buy the original bundle of goods Total change in demand ∆xi = can be rewritten as ∆xi = + or denote ∆xi = ∆xis + ∆xin where ∆xis = substitution effect and ∆xin = income effect

Slutsky’s Identity Note that is the amount of the change in money income such that the consumer is just able to buy the original bundle of goods (i.e. purchasing power is constant) Denote ∆m = and ∆pi = ∆m = ∆pi This is the amount of money that should be given to the consumer to hold purchasing power constant

Slutsky’s Identity In terms of the rates of change, we can write Slutsky’s Identity as ∆xi ∆xis ∆xim xi(pi, m) ∆pi ∆pi ∆m where ∆xin = ∆xim

Effects of a Price Change What happens when a commodity’s price decreases? Substitution effect: the commodity is relatively cheaper, so consumers substitute it for now relatively more expensive other commodities. Income effect: the consumer’s budget of $m can purchase more than before, as if the consumer’s income rose, with consequent income effects on quantities demanded. Vice versa for a price increase

Effects of a Price Change Consumer’s budget is $m. x2 Original choice x1

Effects of a Price Change x2 Lower price for commodity 1 pivots the constraint outwards New Constraint: purchasing power is increased at new relative prices x1

Effects of a Price Change x2 Now only $m' are needed to buy the original bundle at the new prices, as if the consumer’s income has increased by $m - $m'. x1 Imagined Constraint: Income is adjusted to keep purchasing power constant

Effects of a Price Change Changes to quantities demanded due to this ‘extra’ income ($m - $m') are the income effect of the 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.

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

Real Income Changes x2 Original budget constraint and choice New budget constraint x1

Real Income Changes x2 Less income is needed to buy original bundle. Hence, …………………….. x1

Real Income Changes x2 Original budget constraint and choice New budget constraint x1

Real Income Changes x2 More income is needed to buy original bundle. Hence, ……………………… x1

Real Income Changes Absence of Money illusion If money income and prices increase (or decrease) by the same proportion, e.g. double → budget constraint and consumer’s choice remain unchanged

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?”

Budget Constraints and Choices x2 Original budget constraint and choice Original Indifference Curve x1

Budget Constraints and Choices x2 New budget constraint when relative price of x1 is lower x1

Budget Constraints and Choices x2 Imagined budget constraint x1

Budget Constraints and Choices x2 Imagined Budget Constraint, Indifference Curve, and Choice x1

Pure Substitution Effect Only x2 Lower p1 makes good 1 relatively cheaper and causes a substitution from good 2 to good 1. ( , )  ( , ) is the pure substitution effect x1

The Income Effect x2 The income effect is  ( , ) ( , ) ( , ) x1

Total Effect x2 The change in demand due to lower p1 is the sum of the income and substitution effects,  ( , ) ( , ) ( , ) x1

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.

Slutsky’s Effects for Normal Goods x2 Good 1 is normal because .…… ……………………………………. ( , ) x1

Slutsky’s Effects for Normal Goods x2 so the income and substitution effects ………… each other ( , ) Total Effect x1

Slutsky’s Effects for Normal Goods When pi decreases, ∆pi is negative (─) ∆pi → ∆xi = ∆xis + ∆xin (─) ( ) ( ) ( ) both substitution and income effects increase demand when own-price falls. Alternatively, ∆xi ∆xis ∆xim xi(pi, m) ∆pi ∆pi ∆m ( ) ( ) ( ) x ( ) = ─

Slutsky’s Effects for Normal Goods When pi decreases, ∆pi is positive (+) ∆pi → ∆xi = ∆xis + ∆xin (+) ( ) ( ) ( ) both substitution and income effects decrease demand when own-price rises. Alternatively, ∆xi ∆xis ∆xim xi(pi, m) ∆pi ∆pi ∆m ( ) ( ) ( ) x ( ) = ─

Slutsky’s Effects for Normal Goods In both cases, a change is own price results in an opposite change in demand ∆xi ∆pi → a normal good’s ordinary demand curve slopes down. The Law of Downward-Sloping Demand therefore always applies to normal goods. is always…………

Slutsky’s Effects for Income-Inferior Goods Some goods are income-inferior (i.e. demand is reduced by higher income). The pure substitution effect is as for a normal good. But, the income effect is in the opposite direction. Therefore, the substitution and income effects oppose each other when an income-inferior good’s own price changes.

Slutsky’s Effects for Income-Inferior Goods x2 x1

Slutsky’s Effects for Income-Inferior Goods x2 Good 1 is income-inferior because ……………………………………………………………………… ( , ) x1

Slutsky’s Effects for Income-Inferior Goods x2 Substitution and Income effects ……….. each other ( , ) Total Effect x1

Slutsky’s Effects for Income-Inferior Goods When pi decreases, ∆pi is negative (─) ∆pi → ∆xi = ∆xis + ∆xin (─) ( ) ( ) ( ) substitution effect increases demand while income effect reduces demand Alternatively, ∆xi ∆xis ∆xim xi(pi, m) ∆pi ∆pi ∆m ( ) ( ) ( ) x ( ) = ─

Slutsky’s Effects for Income-Inferior Goods When pi decreases, ∆pi is positive (+) ∆pi → ∆xi = ∆xis + ∆xin (+) ( ) ( ) ( ) both substitution and income effects decrease demand when own-price rises. Alternatively, ∆xi ∆xis ∆xim xi(pi, m) ∆pi ∆pi ∆m ( ) ( ) ( ) x ( ) = ─

Slutsky’s Effects for Income-Inferior Goods In general, substitution effect is greater than income effect. Hence, ∆xi is usually positive when pi decreases. and ∆xi is usually negative when pi increases. That is is ………………….. and Demand Curve slopes downward ∆xi ∆pi

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 called Giffen goods.

Slutsky’s Effects for Giffen Goods x2 Income effect ………… Substitution effect. x1 Substitution effect Income effect

Slutsky’s Effects for Giffen Goods x2 A decrease in p1 causes quantity demanded of good 1 to fall. Total Effect x1

Slutsky’s Effects for Giffen Goods Slutsky’s decomposition of the effect of a price change into a pure substitution effect and an income effect thus explains why the Law of Downward-Sloping Demand is violated for Giffen goods.

Hick’s Income and Substitution Effects Previously, we learn Slutsky’s Substitution Effect: the change in demand when purchasing power is kept constant. Hick proposed another type of Substitution Effect where consumer is given just enough money to be on the same indifference curve. Hick’s Substitution Effect: the change in demand when utility is kept constant.

Hick’s Income and Substitution Effects Total change in demand when price changes ∆xi = can be rewritten as + Where is minimum income needed to achieve the original utility u at price = substitution effect = income effect

Hick’s Income and Substitution Effects x2 New budget constraint when p1 falls Original choice New choice Original budget constraint x1

Hick’s Income and Substitution Effects x2 Substitution Effect is optimal choice found on the original indifference curve using the new relative prices x1 Income Effect

Hick’s Income and Substitution Effects x2 As before, Substitution and Income effects ……….. each other x1

Demand Curves Marshallian (Ordinary) Demand shows the quantity actually demanded when own price changes holding ……….. constant Slutsky Demand shows Slutsky substitution effect when own price changes holding …………………… constant Hicksian (Compensated) Demand shows Hick substitution effect when own price changes holding ……….. constant

Comparison: Hick and Slutsky Substitution Effects when own price falls x2 ……….. budget constraint ……… budget constraint x1 …… Substitution ………. Substitution

Demand Curves for Normal Good when Own Price Falls x1