part 2c. individual demand functions 3. slutsky equations
TRANSCRIPT
Part 2C. Part 2C. Individual Demand FunctionsIndividual Demand Functions
3.3.Slutsky EquationsSlutsky EquationsSlutsky EquationsSlutsky EquationsSlutsky Slutsky 方程式方程式
OwnOwn--Price EffectsPrice Effects
yy
OwnOwn Price EffectsPrice Effects A A SlutskySlutsky DecompositionDecomposition CrossCross--Price EffectsPrice Effects CrossCross--Price EffectsPrice Effects Duality and the Demand Concepts Duality and the Demand Concepts
12014.11.20
OwnOwn Price EffectsPrice EffectsOwnOwn--Price EffectsPrice Effects
QQ Wh t h t h f d h hQ: Q: What happens to purchases of good x change when px changes?
x/px
Differentiation of the F O Cs from utilityDifferentiation of the F.O.Cs from utility maximization could be used.H thi h i b dHowever, this approach is cumbersome and provides little economic insight.
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The Identity b/w The Identity b/w MarshallianMarshallian & Hicksian & Hicksian Demands:Demands:Since x* = x(px, py, I) = hx(px, py, U) y y
Replacing I by the EF, e(px, py, U), and U by givesg
x(px, py, e(px, py, )) = hx(px, py, )
Diff ti ti b ti t h
hx x e
Differentiation above equation w.r.t. px, we have
x
x x x
hx x ep e p p
xp
x
= constantx Up
= hx = x
x x x I
= constantx x U
xp p I
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x x x
= constantx x U
xp p I
S.E.( – )
I.E.( ?)
The S E is always negative h
Th L f D d Th L f D d h ld
The S.E. is always negative as long as MRS is diminishing.
0x
x
hp
The Law of Demand The Law of Demand holds as long as x is a normal goodnormal good. 0 0
x
x xI p
If x is a GiffenGiffen goodgood, hen x must be an inferiorinferior goodgood
0 0x xp I
hen x must be an inferiorinferior goodgood. xp I
E/px = hx = xA $1 i i i ditA $1 increase in px raises necessary expenditures by x dollars. 4
Compensated Demand ElasticitiesCompensated Demand ElasticitiesCompensated Demand ElasticitiesCompensated Demand ElasticitiesThe compensated demand function: h (p , p , U)The compensated demand function: hx(px, py, U)
dh Compensated OwnCompensated Own--Price Elasticity of DemandPrice Elasticity of Demand
x
x x xh p
dhh h pe d h
,x xh px x x
x
dp p hp
Compensated CrossCompensated Cross--Price Elasticity of DemandPrice Elasticity of Demand
xdh
,x y
x
yx xh p
ph he dp p h
y y x
x
dp p hp
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OwnOwn--Price Elasticity form of the Price Elasticity form of the SlutskySlutsky EquationEquationyy yy qq
xhx xxI
x xp p I
x x x xp h p px x Ix
x x
xp x p x I x I
h Ie e s e , , ,x x xx p h p x x Ie e s e
where Expenditure share on x.xx
p xsI
I
The Slutsky equation shows that the t d d t d icompensated and uncompensated price
elasticities will be similar ifthe share of income devoted to x is small
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the share of income devoted to x is small.the income elasticity of x is small.
A Slutsky DecompositionA Slutsky DecompositionA Slutsky DecompositionA Slutsky Decomposition
Example: Example: CobbCobb Douglas utility functionDouglas utility function Example: Example: CobbCobb--Douglas utility functionDouglas utility functionU(x,y) = x0.5y0.5
1 I 1 IThe Marshallian Demands: 12 x
Ixp
0 5 0 5
12 y
Iyp
The IUF: 0.5 0.5
0.5 0.5
1 1( , , )2 2 2x y
x x x y
I I Ip p Ip p p p
x x x yp p p p
The EF: 0.5 0.5( , , ) 2x y x ye p p p p
The Hicksian Demands:0.5pe 0.5pe0.5y
xx x
pehp p
0.5
xy
y y
pehp p
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The Slutsky Decomposition: The Slutsky Decomposition: y py p
1 0x IT E 2. . 0
2x x
T Ep p
0.5 0.5
1.5 1.5 0.5 0.5 2
1 1 1. . 02 2 2 4
y yx p ph I IS Ep p p p p p
2 2 2 4x x x x y xp p p p p p
1 1 1 1I I 2
1 1 1 1. . 02 2 4x x x
x I II E xI p p p
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Numerical Example: Numerical Example: CobbCobb--Douglas utility functionDouglas utility functionpp g yg yU(x,y) = x0.5y0.5
Let $1 $4 I $8
The Marshallian Demands:
Let px = $1, py = $4, I = $8
1 4Ix 1 1Iy The Marshallian Demands: 4
2 x
xp
The IUF: 0.5 0.5( ) 4 1 2p p I
12 y
yp
The IUF: ( , , ) 4 1 2x yp p I
The EF: ( , , ) 8x ye p p I x y
The Hicksian Demand for x:0.5 0.5
0.5 0.5
4 2 41
yx
ph
p
0.5 0.5
0.5 0.5
1 2 14
xy
php
1xp 4yp
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Suppose that px : $1 $4pp px
The Marshallian Demands:1 8' 12 4
x 1 8' 12 4
y 2 4
The IUF: 0.5 0.5( , , ) 1 1 1x yp p I
h l i 0 5 0 5The real income: 0.5 0.5' ( , , ') 2 4 1 2 16x ye e p p
The Hicksian Demand for x:The Hicksian Demand for x:0.5
0.5
4 2 24xh
0.5
0.5
4 2 24yh
4 0.54y
The Slutsky Decomposition: . . : 1 4 3T E x
. . : 2 4 2xS E h x
. . . . . . ( 3) ( 2) 1I E T E S E 10
Figure: Figure: The Slutsky Decomposition The Slutsky Decomposition
px : $1 $4
y
px
. . : 1 4 3T E x
2 4 2S E h4
. . : 2 4 2xS E h . . . . . . ( 3) ( 2) 1I E T E S E
I
2
I I = –2y
IC1
1IC0
xS.E.I.E.
1 42 8
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Figure: Figure: The Slutsky Decomposition The Slutsky Decomposition
px : $1 $4
px
px
. . : 1 4 3T E x
2 4 2S E h4
. . : 2 4 2xS E h
. . . . . . ( 3) ( 2) 1I E T E S E . . . . . . ( 3) ( 2) 1I E T E S E
xhx
1
xS.E.I.E.
1 42
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CC P i Eff tP i Eff tCrossCross--Price EffectsPrice Effects The identity b/w The identity b/w MarshallianMarshallian & & HicksianHicksian The identity b/w The identity b/w MarshallianMarshallian & & HicksianHicksian
Demands:Demands:x(p p e(p p )) = h (p p )x(px, py, e(px, py, )) = hx(px, py, )
Diff ti ti b ti t h
hx x e
Differentiation above equation w.r.t. py, we have
x
y y y
hx x ep e p p
xp
x
= constanty Up
= hy = y
x x x I
= constanty y U
yp p I
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CrossCross--Price Elasticity form of the Price Elasticity form of the SlutskySlutskyyy yyEquationEquation
hx x x
y y
hx xyp p I
y y yx
y y
p p phx x Iyp x p x I x I
y yp p
, , ,y x yx p h p y x Ie e s e
where Expenditure share on y.yy
p ys
I
I
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Definition: Definition: Gross SubstitutesGross SubstitutesTwo goods are (gross) substitutes(gross) substitutes if one good may replace the other in use. i.e., ifreplace the other in use. i.e., if
0i
j
xp
e.g, tea & coffee, butter & margarinejp
Definition: Definition: Gross ComplementsGross ComplementsTwo goods are (gross) complements(gross) complements if they are usedTwo goods are (gross) complements (gross) complements if they are used together. i.e., if
0ix 0i
jp
e.g., coffee & cream, fish & chips15
FigureFigure: : Gross Substitutes Gross Substitutes
When the price of y falls the
y
When the price of y falls, the substitution effect may be so large that the consumer purchases less xy
In this case we call x and y gross gross
and more y.
In this case, we call x and y gross gross substitutes.substitutes.y1
y0 U1x/py > 0
x x
U0
xx1 x0
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FigureFigure: : Gross ComplementsGross Complements
When the price of y falls the
y
When the price of y falls, the substitution effect may be so small that the consumer purchases more x and y pmore y.
In this case we call x and y gross gross
y1
In this case, we call x and y gross gross complements.complements.
y0
U1Ux/py < 0
xx
U0
xx1x0
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Definition: Definition: Net SubstitutesNet SubstitutesTwo goods are net substitutesnet substitutes if
h
constant
or 0i
j U
xp
0i
j
hp
Definition: Definition: Net ComplementsNet Complements
constantj U
ppTwo goods are net complements net complements if
xh
constant
or 0i
j U
xp
0i
j
hp
Note: The concepts of net substitutes and net substitutes and ll tt f l l b tit ti ff tb tit ti ff tcomplemencomplements ts focuses solely on substitution effectssubstitution effects.
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x x x
= constanty y U
yp p I
S.E.( + )
I.E.( ?)
The S E is always positive 0xh
If i l dl d I E 0
The S.E. is always positive if DMRS and n = 2.
0x
yp
If x is a normal goodnormal good, I.E. < 0. The combined effect is ambiguous. 0x
S.E. > |I.E.| Gross Substitutes
S E < |I E | Gross Complements
yp
0xS.E. < |I.E.| Gross Complements 0
yp
If x is an inferiorinferior goodgood both S E > 0 If x is an inferiorinferior goodgood, both S.E. > 0 and I.E. >0 Gross Substitutes 0
y
xp
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Case of Many Goods (Case of Many Goods (nn > > 22))y (y ( ))The Generalized Slutsky Equation is:
x x x
=constant
i i ij
j j U
x x xxp p I
When n > 2, hi/pj can be negative.i.e., xi and xj can be net complementsnet complements.If the utility function is quasi-concave, then the the crosscross--netnet--substitution effectssubstitution effects are symmetricsymmetric. i.e., yy ,
ji
j i
hhp p
j ip p Proof:Proof:
2j jii
e p he ph e j jii
j j i j i i
pp ep p p p p p
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Asymmetry of the Gross CrossAsymmetry of the Gross Cross--Price Effects Price Effects y yy yThe gross definitions of substitutes and complements are not symmetric.complements are not symmetric. It is possible for xi to be a substitute for xj and at the same time for x to be a complement of xthe same time for xj to be a complement of xi.
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D lit d th D d C t D lit d th D d C t UMP EMP
“Dual” ProblemDuality and the Demand Concepts Duality and the Demand Concepts
Slutsky Equation*
xhx xxI* ( , , ) x yx x p p I
R ’ Id tit Shephard’s
( , , )x x yh p p U x xp p I*
x
y y
hx xyp p I
( , , )x x yh p p U
Roy’s Identity Shephard sLemma
( , , )
x yx p p I
( , , ) ( , , ( , , ))x y x x y x yx p p I h p p p p I
x
ep
xp
I( , , ) ( , , ( , , ))x x y x y x yh p p U x p p e p p U
* ( , , ) x yU p p I * ( , , ) x ye e p p U* ( , , ( , , )) x y x ye e p p p p I I( ( ))x y x yp p p p* ( , , ( , , )) x y x yU p p e p p U U
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