chapter 8: slutsky decomposition - econ 3023 …d.umn.edu/~watanabe/econ3023fl12/docs/sluho.pdf ·...

Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks

Econ 3023 Microeconomic Analysis

Chapter 8: Slutsky Decomposition

Instructor: Hiroki WatanabeFall 2012

Watanabe Econ 3023 8 Slutsky Decomposition 1 / 59


1 Introduction

2 Decomposing Effects

3 Giffen is Income-Inferior

4 Examples

5 Hicks

6 Now We Know



1 IntroductionOverviewBackground



4 Examples

5 Hicks

6 Now We Know



Overview

Chpt 6: Categories and definitions.Chpt 8: Price change → demand.Chpt 14: Price change → welfare.



Overview

Discussion 1.1 (Orange vs Tomato)

Listen to NPR News Clip .Orange juice is not a required staple forpeople to live. So they will turn to otherthings and that will bring down priceseventually.

When the price of OJ goes up, do people1 increase consumption of tomato juice to substitute

away from OJ, or2 can they possibly consume less tomato juice?



Overview

Exogenous Variable φC(p,m) Î φC(p,m) È

m Î normal income inferiorpC Î Giffen LODpT Î substitute complement


http://www.marketplace.org/topics/business/orange-juice-prices-rise-thanks-perfect-storm


Overview

ΔC denotes change in C.If C = 5 and then C becomes 2 after the pricechange (or income change), ΔC = −3.

Δ denotes change in�

CT

�

.

If T = 3 and then T becomes 8 after the pricechange (or income change),

Δ =�

−35

�

.



Overview

Split the price change in three progressive stages:1 Original: O2 Transitional: T3 Final: F



Background

Consider (pOC, pT) = (100,1) and (pFC, pT) = (1,1).

ΔC has two components:1 Income effect, C Î : some money left after

purchasing OC

2 Substitution effect, C Î : Liz has to give up onlyone slice for a cup of tea.



Background

∗Tresponds to the change in pC as well:

1 Income effect, T Î2 Substitution effect, T È

In total, T Î or È ?



Background

Definition 1.2 (Income & Substitution Effect)1 Income effect is a change in demand Δ due to

having more purchasing power.2 Substitution effect is a change in demand ΔS

due to change in relative price.3 Total effect Δ is the sum of the above:

Δ := Δ + ΔS.



Background

To resolve ambiguity with tomato juice consumptionin Discussion 1.1

we want to separate two effects from each other.Recall the change in parameters:

1 Income change:2 Price change:



1 Introduction

2 Decomposing EffectsEliminating Income EffectSlutsky DecompositionExample 1: Cobb-Douglas


4 Examples

5 Hicks

6 Now We Know



Eliminating Income Effect

Goal: Eliminate the income effect (change inpurchasing power) and isolate the substitutioneffect.Idea: You can compensate the income to suppressthe change in purchasing power.Then let Liz solve her UMP with compensatedincome.




0 2 4 6 8 10 12 14 160

2

4

6

8

10

12

14

16

Cheese xC (slices)

Tea

xT (

cups

)

BLO

BLT

BLF

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2

4

6

8

10

12

14

16

Cheese xC (slices)

Tea

xT (

cups

)




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Cheese xC (slices)

Tea

xT (

cups

)

BLO

BLT

BLF

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4

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Tea

xT (

cups

)




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Tea

xT (

cups

)

BLO

BLT

BLF

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Tea

xT (

cups

)




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xT (

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Tea

xT (

cups

)




Adjust (compensate) Liz’s income so that she canstill buy her original bundle O under the newprice (pF

C, pT) (pOC↘p

FC).

Give her a hypothetical budget

mT := pFCOC+ pT

OT.

Notice the difference:

mT = pFCOC+ pT

OT< pO

COC+ pT

OT=m.

O just uses up m under p = (pOC, pT).

The same O costs less (mT <m) under p = (pFC, pT).




We’ll take away m−mT dollars from m.Liz: I don’t feel any change in my purchasingpower. With income adjustment, I still use up myincome to purchase O regardless of the price.Or equivalently, change (pO

C, pT ,m)→ (pFC, pT ,m

T)reserves Liz’s purchasing power since she canafford O in both environments.Setting income at mT suppresses the income effectand isolates the substitution effect.




We solve two problems and compare the result:1 UMPO:

max(C, T) s.t. pOCC + pTT =m

with the solution O =

φC(pOC, pT ,m)

φT(pOC, pT ,m)

.

2 Transition: UMPT3 UMPF:

max(C, T) s.t. pFCC + pTT =m

with the solution F =

φC(pFC, pT ,m)

φT(pFC, pT ,m)

.




Problem 2.1 (UMPO)

max(C, T) s.t. pOCC + pTT =m

O = (φC(pOC, pT ,m), φT(pOC , pT ,m)).

↓ pure substitution effect ΔS

Problem 2.2 (UMPT)

max(C, T) s.t. pOCC + pTT =mT . a

T = (φC(pOC, pT ,mT), φT(pO

C, pT ,mT)),

amT := pFCOC+ pTOT .

↓ pure income effect Δ

Problem 2.3 (UMPF)

max(C, T) s.t. pFCC + pTT =m

F = (φC(pFC, pT ,m), φT(pFC, pT ,m))



Slutsky Decomposition

Substitution effect ΔS is given by

ΔS = φ(pFC, pT ,m

T)− φ(pOC, pT ,m).

Income effect Δ is given by

Δ = φ(pFC, pT ,m)− φ(pFC, pT ,m

T).

Total effect Δ is given by

Δ = φ(pFC, pT ,m)− φ(pOC , pT ,m)

= Δ + ΔS.



Slutsky Decomposition

Definition 2.4 (Slutsky Decomposition)

Slutsky decomposition splits the total effect into twoparts:

Δ = Δ + ΔS.



Example 1: Cobb-Douglas

Example 2.5 (Cobb-Douglas Utility Function)

Suppose Liz consumes cheesecakes C and tea T .Initial price of pO

C= 2 was slashed in half to pF

C= 1.

pT = 1 and m = 16 throughout. Her preference isrepresented by

(C, T) = CT ,

whose MRS at (C, T) is−TC

. What are ΔS and Δ?




Steps:1 UMPO: Find O.2 Find mT := pF

COC+ pTT .

3 UMPT : Find T .4 UMPF: Find F.5 Compute Δ := Δ + ΔS.




1 UMPO:

max(C, T) = CT s.t. pOCC + T =m.

MRS at (C, T) is−TC

.O = (4,8).




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Cheese xC (slices)

Tea

xT (

cups

)

Indifference Curves

BLO

BLT

BLF

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Tea

xT (

cups

)




2 Find mT .O = (4,8).mT := pF

C4+ 8 = 12.

3 UMPT :

max(C, T) = CT s.t. pFCC + T =mT .

T = (6,6).

ΔS = T − O =�

66

�

−�

48

�

=�

2−2

�

.




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Cheese xC (slices)

Tea

xT (

cups

)

Indifference Curves

BLO

BLT

BLF

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Cheese xC (slices)

Tea

xT (

cups

)




4 UMPF:

max(C, T) = CT s.t. pFCC + T =m.

F = (8,8).

Δ = F − T =�

88

�

−�

66

�

=�

22

�

.

5 Slutsky decomposition:

Δ = Δ + ΔS =�

22

�

+�

2−2

�

=�

40

�

.




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xT (

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BLT

BLF

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Proposition 2.6 (Cobb-Douglas Utility Functionand Cross-Price Effect)

Cross-price effect is absent for Cobb-Douglas utilityfunction.

Proof.Income effect and substitution effects cancel eachother out (see above). �



1 Introduction


3 Giffen is Income-InferiorFactNormal GoodIncome-Inferior Good

4 Examples

5 Hicks

6 Now We Know



Fact

Fact 3.1 (Sign of Substitution Effect)

Substitution effect ΔSCagainst pC È is positive if

preferences are convex.

If purchasing power remains the same, decrease inpC results in increase in C. 1

1Note

LOD says pC È implies C Î .The fact above says pC È implies C Î if thepurchasing power remains the same.



Normal Good

For a normal good with convexity,



Normal Good

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xT (

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Normal Good

Suppose pC È .Slutsky decomposition:

ΔC = ΔC︸︷︷︸

+ by normality

+ ΔSC︸︷︷︸

+ by Fact 3.1

.

Conclude: pC È ⇒ ΔC Î .Conclude: a normal good satisfies LOD.



Income-Inferior Good

For an income-inferior good with convexity,




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Cheese xC (slices)

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xT (

cups

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Suppose pC È .Slutsky decomposition:

ΔC = ΔC︸︷︷︸

- by inferiority

+ ΔSC︸︷︷︸

+ by Fact 3.1

.

Can not conclude: pC È ⇒ ΔC Î .Can not conclude: an income-inferior goodsatisfies LOD.




In case of extreme income-inferiority, income effectmay be larger in magnitude than the substitutioneffect,causing C È with pC È .Conclude: an income-inferior cheesecake may beGiffen.Recall the definition: Giffen good: pC È ⇒ C È .




Conversely, if a cheesecake is Giffen,

ΔC︸︷︷︸

−

= ΔC︸︷︷︸

?

+ ΔSC︸︷︷︸

+ by Fact 3.1

.

ΔChas to be negative.

Conclude: Giffen cheesecake has to beincome-inferior.




Proposition 3.2 (Income-Inferior Good)

m-wiseC

pC-wise

normal

LOD

income-inferior

Giffen




Proof.See above. �

RemarkThere is no such thing as a normal Giffen cheesecake.



1 Introduction



4 ExamplesExample 2: Perfect ComplementsExample 3: Perfect SubstitutesExample 4: Quasi-Linear Preferences

5 Hicks

6 Now We Know



Example 2: Perfect Complements

0 25 50 75 100 125 1500

25

50

75

100

125

150

25

50

75

100

125

Left Shoes xL

Rig

ht S

hoes

xR

Indifference Curves

BLO

BLT

BLF

0 25 50 75 100 125 1500

25

50

75

100

125

150

25

50

75

100

125

Left Shoes xL

Rig

ht S

hoes

xR



Example 2: Perfect Complements

Rotation does not change ∗: ΔS = T − O =�

00

�

.

Δ = Δ +�

00

�

.



Example 3: Perfect Substitutes

0 1 2 3 4 5 60

6

12

18

24

3 6

912

1518

2124

27 30 33 3639

4245

4851

5457

Six−Packs x6

Bot

tles

x 1

Indifference Curves

BLO

BLT

BLF

0 1 2 3 4 5 60

6

12

18

24

3 6

912

1518

2124

27 30 33 3639

4245

4851

5457

Six−Packs x6

Bot

tles

x 1



Example 3: Perfect Substitutes

T is already F.

Δ =�

00

�

+ ΔS (all the change is due to

substitution effect).



Example 4: Quasi-Linear Preferences

0 2 4 6 8 100

2

4

6

8

10

2

3 5

6

7 8 910

1213

1415

Composite Goods xC

Tea

xT

Indifference Curves

BLO

BLT

BLF

0 2 4 6 8 100

2

4

6

8

10

2

3 5

6

7 8 910

1213

1415

Composite Goods xC

Tea

xT




MRS is constant at any given T : MRS(1,2) =MRS(30,2) = MRS(32170984872109874,2).i.e., one basket is worth the same amount of tearegardless of the number of baskets.Baskets represent Liz’s income less expenditure ontea.She won’t get bored with cash regardless of theamount of cash.




Parallel shifts do not change T .φT(pF

C, pT ,mT) = φT(pF

C, pT ,m).

ΔT =�

00

�

+ ΔST.

Income effect for tea is absent for quasi-linearpreferences.



1 Introduction



4 Examples

5 Hicks

6 Now We Know



An alternative way to compensate (adjust) incomelevel for Slutsky decomposition.Slutsky: adjust m to guarantee that O is availableunder pO

C.

Hicks: adjust m in the way that guarantees (O)under pO

C.

When ΔpC is small, the Slutsky substitution effect isalmost the same as the Hicksian substitution effect.



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Cheese xC (slices)

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xT (

cups

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Indifference Curves

BLO

BLT (Hicks)

BLF

BLT (Slutsky)

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1 Introduction



4 Examples

5 Hicks

6 Now We Know



Decomposing total change in quantity demanded.Property of Giffen goods.Alternative compensation.



Index

Cobb-Douglas UtilityFunction, 25convex, 35cross-price effect, 33Δ, see total effectΔ, see income effectΔS, see substitutioneffectGiffen good, 42income effect, 9–11, 22income-inferior good, 39normal good, 36

perfect complements, 47perfect substitutes, 49purchasing power, 11quasi-linear preferences,53Slutsky decomposition, 24,55substitution effect, 9–11,22Hicksian –, 55

total effect, 11


chapter 8: slutsky decomposition - econ 3023 …d.umn.edu/~watanabe/econ3023fl12/docs/sluho.pdf ·...

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