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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
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
1 Introduction
2 Decomposing Effects
3 Giffen is Income-Inferior
4 Examples
5 Hicks
6 Now We Know
Watanabe Econ 3023 8 Slutsky Decomposition 2 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
1 IntroductionOverviewBackground
2 Decomposing Effects
3 Giffen is Income-Inferior
4 Examples
5 Hicks
6 Now We Know
Watanabe Econ 3023 8 Slutsky Decomposition 3 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Overview
Chpt 6: Categories and definitions.Chpt 8: Price change → demand.Chpt 14: Price change → welfare.
Watanabe Econ 3023 8 Slutsky Decomposition 4 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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?
Watanabe Econ 3023 8 Slutsky Decomposition 5 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Overview
Exogenous Variable φC(p,m) Î φC(p,m) È
m Î normal income inferiorpC Î Giffen LODpT Î substitute complement
Watanabe Econ 3023 8 Slutsky Decomposition 6 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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
�
.
Watanabe Econ 3023 8 Slutsky Decomposition 7 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Overview
Split the price change in three progressive stages:1 Original: O2 Transitional: T3 Final: F
Watanabe Econ 3023 8 Slutsky Decomposition 8 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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.
Watanabe Econ 3023 8 Slutsky Decomposition 9 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Background
∗Tresponds to the change in pC as well:
1 Income effect, T Î2 Substitution effect, T È
In total, T Î or È ?
Watanabe Econ 3023 8 Slutsky Decomposition 10 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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.
Watanabe Econ 3023 8 Slutsky Decomposition 11 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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:
Watanabe Econ 3023 8 Slutsky Decomposition 12 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
1 Introduction
2 Decomposing EffectsEliminating Income EffectSlutsky DecompositionExample 1: Cobb-Douglas
3 Giffen is Income-Inferior
4 Examples
5 Hicks
6 Now We Know
Watanabe Econ 3023 8 Slutsky Decomposition 13 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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.
Watanabe Econ 3023 8 Slutsky Decomposition 14 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income Effect
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Cheese xC (slices)
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xT (
cups
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Cheese xC (slices)
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xT (
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Watanabe Econ 3023 8 Slutsky Decomposition 15 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income Effect
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Tea
xT (
cups
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BLF
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4
6
8
10
12
14
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Cheese xC (slices)
Tea
xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 16 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income Effect
0 2 4 6 8 10 12 14 160
2
4
6
8
10
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Cheese xC (slices)
Tea
xT (
cups
)
BLO
BLT
BLF
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4
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10
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Cheese xC (slices)
Tea
xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 17 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income Effect
0 2 4 6 8 10 12 14 160
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6
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Tea
xT (
cups
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BLT
BLF
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Cheese xC (slices)
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xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 18 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income Effect
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).
Watanabe Econ 3023 8 Slutsky Decomposition 19 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income Effect
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.
Watanabe Econ 3023 8 Slutsky Decomposition 20 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income 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)
.
Watanabe Econ 3023 8 Slutsky Decomposition 21 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Eliminating Income Effect
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))
Watanabe Econ 3023 8 Slutsky Decomposition 22 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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.
Watanabe Econ 3023 8 Slutsky Decomposition 23 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Slutsky Decomposition
Definition 2.4 (Slutsky Decomposition)
Slutsky decomposition splits the total effect into twoparts:
Δ = Δ + ΔS.
Watanabe Econ 3023 8 Slutsky Decomposition 24 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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?
Watanabe Econ 3023 8 Slutsky Decomposition 25 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
Steps:1 UMPO: Find O.2 Find mT := pF
COC+ pTT .
3 UMPT : Find T .4 UMPF: Find F.5 Compute Δ := Δ + ΔS.
Watanabe Econ 3023 8 Slutsky Decomposition 26 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
1 UMPO:
max(C, T) = CT s.t. pOCC + T =m.
MRS at (C, T) is−TC
.O = (4,8).
Watanabe Econ 3023 8 Slutsky Decomposition 27 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
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xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 28 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
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
�
.
Watanabe Econ 3023 8 Slutsky Decomposition 29 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
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Cheese xC (slices)
Tea
xT (
cups
)
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BLO
BLT
BLF
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Cheese xC (slices)
Tea
xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 30 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
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
�
.
Watanabe Econ 3023 8 Slutsky Decomposition 31 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
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112
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Cheese xC (slices)
Tea
xT (
cups
)
Indifference Curves
BLO
BLT
BLF
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8
10
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96
112
128
144
160
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224
240
Cheese xC (slices)
Tea
xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 32 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 1: Cobb-Douglas
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). �
Watanabe Econ 3023 8 Slutsky Decomposition 33 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
1 Introduction
2 Decomposing Effects
3 Giffen is Income-InferiorFactNormal GoodIncome-Inferior Good
4 Examples
5 Hicks
6 Now We Know
Watanabe Econ 3023 8 Slutsky Decomposition 34 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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.
Watanabe Econ 3023 8 Slutsky Decomposition 35 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Normal Good
For a normal good with convexity,
Watanabe Econ 3023 8 Slutsky Decomposition 36 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Normal Good
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Cheese xC (slices)
Tea
xT (
cups
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BLO
BLT
BLF
0 2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
16
Cheese xC (slices)
Tea
xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 37 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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.
Watanabe Econ 3023 8 Slutsky Decomposition 38 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Income-Inferior Good
For an income-inferior good with convexity,
Watanabe Econ 3023 8 Slutsky Decomposition 39 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Income-Inferior Good
0 2 4 6 8 10 12 14 160
2
4
6
8
10
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Cheese xC (slices)
Tea
xT (
cups
)
BLO
BLT
BLF
0 2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
16
Cheese xC (slices)
Tea
xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 40 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Income-Inferior Good
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.
Watanabe Econ 3023 8 Slutsky Decomposition 41 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Income-Inferior Good
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 .
Watanabe Econ 3023 8 Slutsky Decomposition 42 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Income-Inferior Good
Conversely, if a cheesecake is Giffen,
ΔC︸︷︷︸
−
= ΔC︸︷︷︸
?
+ ΔSC︸︷︷︸
+ by Fact 3.1
.
ΔChas to be negative.
Conclude: Giffen cheesecake has to beincome-inferior.
Watanabe Econ 3023 8 Slutsky Decomposition 43 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Income-Inferior Good
Proposition 3.2 (Income-Inferior Good)
m-wiseC
pC-wise
normal
LOD
income-inferior
Giffen
Watanabe Econ 3023 8 Slutsky Decomposition 44 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Income-Inferior Good
Proof.See above. �
RemarkThere is no such thing as a normal Giffen cheesecake.
Watanabe Econ 3023 8 Slutsky Decomposition 45 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
1 Introduction
2 Decomposing Effects
3 Giffen is Income-Inferior
4 ExamplesExample 2: Perfect ComplementsExample 3: Perfect SubstitutesExample 4: Quasi-Linear Preferences
5 Hicks
6 Now We Know
Watanabe Econ 3023 8 Slutsky Decomposition 46 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 2: Perfect Complements
0 25 50 75 100 125 1500
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150
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Left Shoes xL
Rig
ht S
hoes
xR
Indifference Curves
BLO
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BLF
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75
100
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Left Shoes xL
Rig
ht S
hoes
xR
Watanabe Econ 3023 8 Slutsky Decomposition 47 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 2: Perfect Complements
Rotation does not change ∗: ΔS = T − O =�
00
�
.
Δ = Δ +�
00
�
.
Watanabe Econ 3023 8 Slutsky Decomposition 48 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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
Watanabe Econ 3023 8 Slutsky Decomposition 49 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 3: Perfect Substitutes
T is already F.
Δ =�
00
�
+ ΔS (all the change is due to
substitution effect).
Watanabe Econ 3023 8 Slutsky Decomposition 50 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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
Watanabe Econ 3023 8 Slutsky Decomposition 51 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 4: Quasi-Linear Preferences
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.
Watanabe Econ 3023 8 Slutsky Decomposition 52 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Example 4: Quasi-Linear Preferences
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.
Watanabe Econ 3023 8 Slutsky Decomposition 53 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
1 Introduction
2 Decomposing Effects
3 Giffen is Income-Inferior
4 Examples
5 Hicks
6 Now We Know
Watanabe Econ 3023 8 Slutsky Decomposition 54 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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.
Watanabe Econ 3023 8 Slutsky Decomposition 55 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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Cheese xC (slices)
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xT (
cups
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Indifference Curves
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BLT (Hicks)
BLF
BLT (Slutsky)
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Cheese xC (slices)
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xT (
cups
)
Watanabe Econ 3023 8 Slutsky Decomposition 56 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
1 Introduction
2 Decomposing Effects
3 Giffen is Income-Inferior
4 Examples
5 Hicks
6 Now We Know
Watanabe Econ 3023 8 Slutsky Decomposition 57 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
Decomposing total change in quantity demanded.Property of Giffen goods.Alternative compensation.
Watanabe Econ 3023 8 Slutsky Decomposition 58 / 59
Intro Decomposing Effects Giffen is Income-Inferior Examples Hicks
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
Watanabe Econ 3023 8 Slutsky Decomposition 59 / 59