Intermediate Micro Lecture 7 - Georgetown...

Substitution effect WARP Income effect Slutsky identity Examples Applications

Slutsky decomposition:Substitution and income effects

Intermediate Micro

Lecture 7

Chapter 8 of Varian (and chapter 7, briefly)


A deeper analysis of price changes

I Break down effects of price change

I How to understand response to price change

I What is fair compensation for a price change


Price change - income effect

m = p1x1 + p2x2p1 falls to p′1

I Can buy more x1 atany x2

I Similar to increase inincome

I Income effect:Change in demanddue toincreased/decreasedbuying power fromprice change


Price change - substitution effect

m = p1x1 + p2x2p1 falls to p′1

I Cost of good 1 interms of good 2changes

I p1p2

>p′1p2

I Substitution effect:Change in demanddue solely to changein relative prices


Compensated demand - 1

I Create imaginarybudget line

I Slope − p′1

p2I Through original

(x∗1 , x∗2 )

I m′ = p′1x∗1 + p2x

∗2

I ∆m = m′ −m =x∗1∆p1


Compensated demand - 2

I Find choice onimaginary budget line

I Label new optimalchoice (xC1 , xC2 )

I (xC1 , xC2 ) is calledcompensated demand

I ∆m:compensation forchange inpurchasing power


Computing substitution effect

I Substitution effect:∆x s1 =x1(p′1, p2,m

′)−x1(p1, p2,m)

I Isolate ∆x due to∆p1

p2


Direction of the substitution effect

When p1 ↓, ∆xS1 ≥ 0

∆xS1 > 0Old indifference curve crossesnew budget line

∆xS1 = 0 at kink, or at cornersolution


Weak axiom of revealed preference

WARP: If (x1, x2) is chosen over(y1, y2) among one set ofoptions, it can not be that(y1, y2) is chosen over (x1, x2)among a different set of options

I Old budget: chose (x∗1 , x∗2 )

over blue dashed line

I Substitution effect: Mustnot choose blue dashed lineover (x∗1 , x

∗2 )

**The substitution effect never makes the consumer worse off(maybe no better off)**


Example: substitution effect

u(x1, x2) = x1x2m = 120, p1 = 3, p2 = 1p1 falls to p′1 = 2.5

I x∗1 = 20

I m′ = 110, xC1 = 22

I Subs effect: ∆xS1 = 2


Measuring the income effect

I Income effect: totalchange minus subseffect

I Labelx1(p′1, p2,m) = x∗∗1

I Income effect: ∆xn1 =x1(p′1, p2,m)−x1(p′1, p2,m

′)

I ∆xn1 = x∗∗1 − xC1


Direction of the income effect

I Income effectI Start with

compensateddemand

I ↑ mI x1 normal ⇔

∆xn1 > 0 for ∆p1 < 0

I x1 inferior ⇔∆xn1 < 0 for ∆p1 < 0


Inferior good

x1 is inferiorx2 is normal


Giffin good

x1 is inferior, and Giffin goodx2 is normal


Example: income effect

u(x1, x2) = x1x2m = 120, p1 = 3, p2 = 1p1 falls to p′1 = 2.5

I xC1 = 22

I x∗∗1 = 24

I Inc effect: ∆xn1 = 2


Slutsky identity

Slutsky identityEffect of ↑ p1:

Total Subst Income∆x1 = ∆x s1 + ∆xn1(−) (−) (−) Normal(?) (−) (+) Inferior


Law of demand

Law of demand: If demand for a good increases when incomeincreases, then the demand for that good must decrease when itsprice increases

I ∂x1∂m > 0⇒ ∂x1

∂p1< 0

I Corollary: All Giffen goods are inferior


Example: Perfect substitutes

u(x1, x2) = x1 + x2m = 1000, p1 = 0.5, p2 = 1p′1 = 2

I x∗1 = 2000

I m′ = 4000, xC1 = 0,∆xS1 =−2000

I x∗∗1 = 0

I ∆xn1 = 0


Other examples

Example: Find the substitution and income effects of the followingprice change on good 2.

u(x1, x2) = min{x1, x2}m = 200, p1 = 1, p2 = 1

p′2 = 3

Example: Find the substitution and income effects of the followingprice change on good 1.

u(x1, x2) =√x1 + x2

m = 120, p1 = 1, p2 = 2p′1 = 2


Application 1: Gas tax

1974: volatile oil pricesI Goals

1. reduce gasoline demand2. not harm consumers

I Proposal:I Charge per gallon tax tI Rebate average revenue R = tx

Example: Cobb-Douglas utilityu(x1, x2) = xαy1−α, px = p, py = 1

Assume everyone has same preferences, income


Application 1: Gas tax

I Budget lines cross atwith-tax-choice

I WARP says no-taxpreferred to tax

I Goals

1. reduce gasolinedemand: yes

2. not harm consumers:no

I Same analysis holds forsubsidizing goods

I Set t < 0


Application 2: Social Security

Social security payments increase to keep up with inflation

I Find average price increase for average consumer’s choice

I Cost Of Living Adjustment (COLA)

I Use Slutsky compensated demand

I Easy:I All prices ↑ 10%I ↑ m by 10%

I Dilemma:I p1 ↑ 10%, p1 ↑ 0%I 0% <↑ m10%I Slutsky makes recipients better off



Social security payments increase to keep up with inflation

I Find average price increase for average consumer’s choice

I Cost Of Living Adjustment (COLA)

I Use Slutsky compensated demandI Easy:

I All prices ↑ 10%I ↑ m by 10%

I Dilemma:I p1 ↑ 10%, p1 ↑ 0%I 0% <↑ m10%I Slutsky makes recipients better off


Hicks compensated demand

I Slutsky compensated demand: Demand when new budget linegoes through old choice

I Hicks compensated demand: Demand when new budget line

is tangent to indifference curve through old choice, (xh1 , xh2 )

I ∆m so utility is unchanged with new price

I Hicks substitution effect: ∆xh1 = xh1 − x∗1


Hicks compensated demand



Applying Hicks compensation to Social SecurityI Pros

I ↓ growth in gov’t spending (and taxes!)I COLA accurately reflects price growth

I ConsI Retirees incomes don’t ↑I COLA based on average personI Retirees spend more on healthcare, (%-wise)I ↑ phealthcare very big

Intermediate Micro Lecture 7 - Georgetown...

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