7 in Come Substitution

7/21/2019 7 in Come Substitution

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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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A deeper analysis of price changes

Break down effects of price change How to understand response to price change

What is fair compensation for a price change


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Price change - income effect

m=p1x1+p2x2p1 falls to p

1

Can buy more x1 atany x2

Similar to increase inincome

Income effect:Change in demanddue to

increased/decreasedbuying power fromprice change


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Price change - substitution effect

m=p1x1+p2x2p1 falls to p

1

Cost of good 1 interms of good 2changes

p1p2

> p1p2

Substitution effect:Change in demanddue solely to change

in relative prices

S b i i ff WARP I ff Sl k id i E l A li i


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Compensated demand - 1

Create imaginarybudget line

Slopep1p2

Through original(x

1, x

2)

m =p1

x1

+p2x

2

m=m m=x1

p1

S b tit ti ff t WARP I ff t Sl t k id tit E l A li ti


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Compensated demand - 2

Find choice onimaginary budget line

Label new optimalchoice (xC

1 , xC

2)

(xC1 , xC2) is calledcompensated demand

m:compensation for

change inpurchasing power



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Computing substitution effect

Substitution effect:xs

1=

x1(p

1, p2, m

) x1(p1, p2, m)

Isolate xdue top1p2



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Direction of the substitution effect

When p1

, xS1

0

xS

1 >0Old indifference curve crossesnew budget line

xS1

= 0 at kink, or at cornersolution



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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

Old budget: chose (x1 , x

2 )over blue dashed line

Substitution effect: Mustnot choose blue dashed line

over (x

1 , x

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



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Example: substitution effect

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

1= 2.5



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Substitution effect RP Income effect Slutsky identity Examples pplications



1= 2.5

x1

= 20



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y y p pp



1= 2.5

x1

= 20

m = 110, xC1

= 22



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1= 2.5

x1

= 20

m = 110, xC1

= 22

Subs effect: xS1

= 2



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Measuring the income effect

Income effect: totalchange minus subseffect

Label

x1

(p

1, p2, m) =x

1

Income effect: xn1

=x1(p

1, p2, m)

x1(p

1, p2, m

)

xn1

=x1

xC1



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Direction of the income effect

Income effect Start with

compensateddemand

m

x1 normalxn

1 >0 for p1


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Inferior good

x1 is inferiorx2 is normal



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Giffin good

x1 is inferior, and Giffin goodx2 is normal



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Example: income effect


1= 2.5

xC1

= 22



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1= 2.5

xC1

= 22

x1

= 24



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1= 2.5

xC1

= 22

x1

= 24

Inc effect: xn1

= 2



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Slutsky identity

Slutsky identityEffect of p1:

Total Subst Income

x1 = xs1

+ xn1

(

) (

) (

) Normal(?) () (+) Inferior



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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

x1m

>0 x1p1


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Example: Perfect substitutes

u(x1, x

2

) =x1

+x2

m= 1000, p1= 0.5, p2= 1p1

= 2

x1

= 2000

m = 4000, xC1

= 0, xS1

=2000

x1

= 0

xn1

= 0



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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+x2m= 120, p1= 1, p2= 2

p1

= 2



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Application 1: Gas tax

1974: volatile oil prices Goals

1. reduce gasoline demand2. not harm consumers

Proposal:

Charge per gallon tax t Rebate average revenue R=tx

Example: Cobb-Douglas utilityu(x1, x2) =x

y1, px=p, py = 1Assume everyone has same preferences, income



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Application 1: Gas tax

Budget lines cross atwith-tax-choice

WARP says no-taxpreferred to tax

Goals1. reduce gasolinedemand: yes

2. not harm consumers:no

Same analysis holds forsubsidizing goods Set t


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Application 2: Social Security

Social security payments increase to keep up with inflation

Find average price increase for average consumers choice

Cost Of Living Adjustment (COLA)

Use Slutsky compensated demand



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Use Slutsky compensated demand Easy:

All prices 10% m by 10%



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Use Slutsky compensated demand Easy:

All prices 10% m by 10%

Dilemma: p1 10%, p1 0%

0%

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7 in Come Substitution