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Slutsky Equation

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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 =

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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

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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

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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

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**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

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**Effects of a Price Change**

Consumer’s budget is $m. x2 Original choice x1

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**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

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**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

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**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.

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**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

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**Real Income Changes x2 Original budget constraint and choice**

New budget constraint x1

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**Real Income Changes x2 Less income is needed to buy original bundle.**

Hence, …………………….. x1

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**Real Income Changes x2 Original budget constraint and choice**

New budget constraint x1

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**Real Income Changes x2 More income is needed to buy original bundle.**

Hence, ……………………… x1

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**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

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**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?”

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**Budget Constraints and Choices**

x2 Original budget constraint and choice Original Indifference Curve x1

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**Budget Constraints and Choices**

x2 New budget constraint when relative price of x1 is lower x1

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**Budget Constraints and Choices**

x2 Imagined budget constraint x1

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**Budget Constraints and Choices**

x2 Imagined Budget Constraint, Indifference Curve, and Choice x1

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**Pure Substitution Effect Only**

x2 Lower p1 makes good 1 relatively cheaper and causes a substitution from good 2 to good ( , ) ( , ) is the pure substitution effect x1

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

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**Total Effect x2 The change in demand due to lower p1 is the sum of the**

income and substitution effects, ( , ) ( , ) ( , ) x1

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**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.

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**Slutsky’s Effects for Normal Goods**

x2 Good 1 is normal because .…… ……………………………………. ( , ) x1

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**Slutsky’s Effects for Normal Goods**

x2 so the income and substitution effects ………… each other ( , ) Total Effect x1

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**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 ( ) = ─

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**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 ( ) = ─

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**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…………

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**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.

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**Slutsky’s Effects for Income-Inferior Goods**

x2 x1

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**Slutsky’s Effects for Income-Inferior Goods**

x2 Good 1 is income-inferior because ……………………………………………………………………… ( , ) x1

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**Slutsky’s Effects for Income-Inferior Goods**

x2 Substitution and Income effects ……….. each other ( , ) Total Effect x1

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**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 ( ) = ─

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**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 ( ) = ─

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**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

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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.

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**Slutsky’s Effects for Giffen Goods**

x2 Income effect ………… Substitution effect. x1 Substitution effect Income effect

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**Slutsky’s Effects for Giffen Goods**

x2 A decrease in p1 causes quantity demanded of good 1 to fall. Total Effect x1

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**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.

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**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.

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**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

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**Hick’s Income and Substitution Effects**

x2 New budget constraint when p1 falls Original choice New choice Original budget constraint x1

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**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

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**Hick’s Income and Substitution Effects**

x2 As before, Substitution and Income effects ……….. each other x1

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**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

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**Comparison: Hick and Slutsky Substitution Effects when own price falls**

x2 ……….. budget constraint ……… budget constraint x1 …… Substitution ………. Substitution

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**Demand Curves for Normal Good when Own Price Falls**

x1

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