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)

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

Computing substitution effect

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

′)−x1(p1, p2,m)

I Isolate ∆x due to∆p1

p2

Substitution effect WARP Income effect Slutsky identity Examples Applications

Direction of the substitution effect

When p1 ↓, ∆xS1 ≥ 0

∆xS1 > 0Old indifference curve crossesnew budget line

∆xS1 = 0 at kink, or at cornersolution

Substitution effect WARP Income effect Slutsky identity Examples Applications

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)**

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

Inferior good

x1 is inferiorx2 is normal

Substitution effect WARP Income effect Slutsky identity Examples Applications

Giffin good

x1 is inferior, and Giffin goodx2 is normal

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

Slutsky identity

Slutsky identityEffect of ↑ p1:

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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 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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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 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

Substitution effect WARP Income effect Slutsky identity Examples Applications

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

Substitution effect WARP Income effect Slutsky identity Examples Applications

Hicks compensated demand

Substitution effect WARP Income effect Slutsky identity Examples Applications

Application 2: Social Security

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