intermediate micro lecture 7 - georgetown...
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![Page 1: Intermediate Micro Lecture 7 - Georgetown Universitystudent.georgetown.edu/jm755/Econ101/Slides/7IncomeSubstitution.pdf · Substitution e ectWARPIncome e ectSlutsky identityExamplesApplications](https://reader031.vdocuments.us/reader031/viewer/2022020411/5ababc8d7f8b9a441d8c0068/html5/thumbnails/1.jpg)
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)
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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
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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
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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
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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
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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
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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
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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
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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)**
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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
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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
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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
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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
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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
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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
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Substitution effect WARP Income effect Slutsky identity Examples Applications
Inferior good
x1 is inferiorx2 is normal
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Substitution effect WARP Income effect Slutsky identity Examples Applications
Giffin good
x1 is inferior, and Giffin goodx2 is normal
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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
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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
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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
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Substitution effect WARP Income effect Slutsky identity Examples Applications
Slutsky identity
Slutsky identityEffect of ↑ p1:
Total Subst Income∆x1 = ∆x s1 + ∆xn1(−) (−) (−) Normal(?) (−) (+) Inferior
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Substitution effect WARP Income effect Slutsky identity Examples Applications
Hicks compensated demand
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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