V.

Income and Substitution Effects

D Individual demand function

• From the FOC, X = dX (PX , PY , I), Y = dY (PX , PY , I), λ = λ(PX , PY , I)

βI
• With Cobb-Douglas utility function, X = αI
PX
and Y = PY
.
D Homogeneity of demand function

• Homogeneous of degree zero

• X = dX (tPX , tPY , tI)

– No change in the budget constraint if prices and income double.

· (tPX )X + (tPY )Y = tI is equivalent to PX X + PY Y = I

D Normal goods and inferior goods
∂X
• ∂I
>0

– X= αI
PX
in the above is a normal good
• See figures 5.1 and 5.2
D Changes in a good’s price

• See figures 5.3 and 5.4

• Substitution effect

– Real income (utility level) fixed

– When the price increases, consumption decreases
• Income effect

– Relative price fixed

– When the price increases, real income decreases so that consumption decreases if the
good is normal
• For a normal good, both effects are of the same sign.

– If the good is an inferior good, both effects go in the opposite direction.

– Giffen’s paradox

· If the income effect dominates, consumption may increases.

D Individual demand curve

26
 • See figure 5.5.
• Given X = dX (PX ; PY , I), the position of dX depends on the values of PY and I.

– Factors such as changes in tastes and weather also shifts the demand curve.
– In general, ∂dX
∂PX
< 0 unless the good is a Giffen good.
D Compensated demand curve

• See figure 5.7.

• Substitution effect

– Main reason ordinary demand curve is downward-sloping

• Income effect may cause the demand curve to be upward-sloped
• Separate the income effect to get the compensated demand

– Its like compensating nominal income to make real purchasing power unchanged
• Can be obtained from expenditure minimization.
• Compensated demand or Hicksian demand

– Ordinary demand: Marshallian demand

D Relationship between the indirect utility function and the expenditure function

• U = V (PX , PY , I) and I = E(PX , PY , U ) are inverse functions of each other

• Solve for I from U = V (PX , PY , I), which gives

I = V −1 (PX , PY , U) = E(PX , PY , U).

• Solve for U from I = E(PX , PY , U), which gives

U = E −1 (PX , PY , I) = V (PX , PY , I).

D Relationship between compensated and ordinary demand curves

• See figure 5.7

• Ordinary demand curve is flatter if the good is normal
• Ordinary demand curve is estimatable
• Theoretically, compensated demand curve is superior
• The Hicksian demand can be derived from the Marshallian demand

– H(PX , PY , U) = X(PX , PY , I) where U = V (PX , PY , I)).

βI
– Example: Cobb-Douglas demand when α + β = 1 is X = αI
PX
, Y = PY
and V =
27
 α β
β
M(PX , PY )I α+β where M(PX , PY ) = α
PX PY
. Suppose that α = β = 12 .
Then, M = √ 1
2 PX PY
and E(PX , PY , U ) = U
M
, which gives H = U PY
PX

– PY appears in the Hicksian demand while not in the Marshallian demand because the
former depends on the relative price
D Shephard lemma

• Given PX∗ , PY∗ and U ∗ , suppose that H(PX∗ , PY∗ , U ∗ ) and K(PX∗ , PY∗ , U ∗ ) minimize the ex-
penditure. Define B(PX , PY , U ∗ ) = E(PX , PY , U ∗ )−PX H(PX∗ , PY∗ , U ∗ )−PY K(PX∗ , PY∗ , U ∗ )
∗ ,P ∗ ,U ∗ )
∂B(PX
Note that B(PX , PY , U ∗ ) ≤ 0 and B(PX∗ , PY∗ , U ∗ ) = 0. Therefore, ∂PX
Y
=
∗ ,P ∗ ,U ∗ )
∂E(PX
∂PX
Y
− H(PX∗ , PY∗ , U ∗ ) = 0. Since this is true for any PX∗ , PY∗ and U ∗ , we have
∂E(PX , PY , U)
H(PX , PY , U) = .
∂PX
• More directly, ∂E
∂PX
∂H
= H + PX ∂PX
∂K
+ PY ∂PX
. But, from the FOC, PX ∂P
∂H
X
∂K
+ PY ∂PX
=
∂H
µ U1 ∂PX
∂K
+ U2 ∂PX
, which is zero since utility is fixed.
∂(PX H+PY K+µ(U −U (H,K)))
• This is in fact the Envelope Theorem, ∂E
∂PX
= ∂PX
= H.
D Roy’s identity
∂(U(X,Y )+λ(I−PX X−PY Y ))
• From the Envelope Theorem, ∂V
∂PX
= ∂PX
= −λX and ∂V
∂I
=
∂(U(X,Y )+λ(I−PX X−PY Y ))
∂I
= λ so that
∂V (PX , PY , I)/∂PX
X(PX , PY , I) = − .
∂V (PX , PY , I)/∂I
D Slutsky equation

• Given PX , PY and I, let U ≡ V (PX , PY , I). Differentiate both sides of H(PX , PY , U) ≡

∂H(PX ,PY ,U )
X(PX , PY , E(PX , PY , U)) with respect to PX . ∂PX
= ∂X(PX ,PY∂P ,E(PX ,PY ,U ))
X
+
∂X(PX ,PY ,E(PX ,PY ,U )) ∂E(PX ,PY ,U ) ∂X(PX ,PY ,E(PX ,PY ,U )) ∂X(PX ,PY ,E(PX ,PY ,U ))
∂I ∂PX
= ∂PX
+ ∂I
H(PX , PY , U).
Then, since H(PX , PY , U) ≡ X(PX , PY , I), or
∂X(PX , PY , I) ∂H(PX , PY , U ) ∂X(PX , PY , I)
= − X(PX , PY , I)
∂PX ∂PX ∂I
– ∂H
∂PX
: Substitution effect
– − ∂X
∂I
X: Income effect

· If price of X increases by one, your money roughly decreases by X so that have

to reduce you consumption of X by ∂X
∂I
X.
• Example

28
 – Cobb-Douglas utility function with α = β = 1
2
√
U PY
– Since X = I
2PX
and H = U PY
, ∂X
PX ∂PX
= − 2PI 2 , ∂X
∂PX |U :const
= ∂H
∂PX
=− √ 3.
X 2( PX )
Since V = X 1/2
Y 1/2
= 1√ I
2 pX pY
, ∂H
∂PX
= − 4PI 2 . ∂X
∂I
X = 1 I
2PX 2PX
= I
4PX2 .
X

D Elasticity
∆B/B ∂B A d ln B
• eB,A = ∆A/A
= ∂A B
= d ln A

– The ratio of percentage changes

– Pure number
• Price elasticity of demand
∂Q P
– eQ,P = ∂P Q

– Elastic, unit elastic, inelastic

– R = P Q(P ). Taking logarithm, ln R = ln P + ln Q. Total differentials of both sides
dQ
are dR
R
= dP
P
+ Multiplying
⎧ Q
. P
dP
, we have eR,P = 1 + QQP = 1 + eQ,P
⎨ inelastic
· If eQ,P ( −1 or unit elastic , eR,P ( 0
⎩ elastic
· An increase in agricultural product price increases total expenditure on it.
• Income elasticity of demand
∂Q I
– eQ,I = ∂I Q

· Normal and inferior goods

• Cross-price elasticity
∂Q P
– eQ,P = ∂P Q

· Substitutes and complements

D Relationship between elasticities

• Differentiate the budget constraint with respect to I.

PX X ∂X I
– PX ∂X
∂I
+ PY ∂Y
∂I
= 1, which can be rewritten as I ∂I X
+ PYI Y ∂Y I
∂I Y
= 1. Therefore
sX eX,I + sY eY,I = 1

· A convex combination of income elasticities is one

· If some good has an income elasticity less than one, there must be a good that
has an income elasticity is greater one:

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 • Differentiate the budget constraint with respect to PX .

– PX ∂P
∂X
X
∂Y
+X+PY ∂P X
= 0, which can be rewritten as PXX ∂P
∂X PX X
X I
+ PXI X + PYX ∂P
∂Y PY Y
X I
=
0. Therefore sX eX,PX + sY eY,PX = −sX

· Direct price effect, usually negative, is not totally overwhelmed.

• Slutsky equation in elasticities

– eX,PX = eSX,PX − sX eX,I where eSX,PX = ∂X PX

∂PX X
: compensated price elas-
U =const
ticity

· If eSX,PX = 0, then eX,PX is proportional to income elasticity.

· If sX is small, compensated and uncompensated price elasticities are approxi-
mately equal.
• Homogeneity

– Euler’s theorem

· Let f (X, Y ) be homogeneous of degree r. Then, f (tX, tY ) ≡ tr f (X, Y ). Dif-

∂f (tX,tY ) ∂f (tX,tY )
ferentiating both sides with respect to t, we have ∂X
X + ∂Y
Y =
∂f (X,Y )
rtr−1 f (X, Y ). This is true for all t especially when t = 1. Then, ∂X X +
∂f (X,Y )
∂Y
Y = rf (X, Y ). For example, if f (X) = X 2 , r = 2. Then f (X)X =
2X · X = 2 · X 2 = rf (X).
∂f ∂f
· If f (X, Y ) is homogeneous of degree zero, then ∂X
X + ∂Y
Y =0
– ∂X
P
∂PX X
+ ∂X
∂PY
PY + ∂X
∂I
I = 0, which gives eX,PX + eX,PY + eX,I = 0
D Examples

• Linear demand

– Q = a + bP where a > 0 and b < 0.

· eQ,P = b (Q−a)/b
Q
= Q−a
Q

– P =a +bQ
1 P 1 P P
· eQ,P = b Q
= b (P −a )/b
= P −a

• Constant elasticity demand function

– Q = aP b
d ln Q
· ln Q = ln a + b ln P so that d ln Q = bd ln P or eP,Q = d ln P
=b

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D Revealed preference

• Utility maximization implies some restrictions on observed behaviors

– Suppose that we observe a person chooses A = (XA , YA ) when prices and income
level are (PXA , PYA , I A ) and B = (XB , YB ) when prices and income level are (PXB , PYB , I B ).
Suppose further that A = (XA , YA ) and B = (XB , YB ) are affordable at (PXA , PYA , I A ),
that is,
PXA XB + PYA YB ≤ PXA XA + PYA YA = I A
Then A is said to be revealed preferred to B.
• Axiom of revealed preference:

– If A is revealed preferred to B, B is never revealed preferred to A.

PXA XB + PYA YB ≤ PXA XA + PYA YA ⇒ PXB XA + PYB YA > PXB XB + PYB YB .

· If not, the person does not maximize his utility because at (PXA , PYA , I A ) he
chooses A even if he could have chosen B,which means that utility is greater
from A than from B. But at (PXB , PYB , I B ), he selects B even if A is affordable.
• Substitution effect

– Suppose that A and B are indifferent.

PXA XA + PYA YA ≤ PXA XB + PYA YB

PXB XB + PYB YB ≤ PXB XA + PYB YA .

· If PXA XA + PYA YA > PXA XB + PYA YB , at (PXA , PYA ) A costs more. Yet the person
chooses A. It must be that he obtains more utility from A than from B, if he is
maximizing utility.
– The second inequality is rewritten as −PXB XA −PYB YA ≤ −PXB XB −PYB YB . Adding
both inequalities gives

(PXA − PXB )(XA − XB ) + (PYA − PYB )(YA − YB ) ≤ 0.

Suppose that PYA = PYB .

PXA > PXB ⇒ XA ≤ XB .

Since A and B are indifferent, this represents the substitution effect.

D Consumer surplus

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• Suppose that initially prices and income are PX0 , PY and I. The government is thinking
of increasing price of X from PX0 to PX1 .
Case 1: A person does not mind moving to the new situation although he never accepts
a utility reduction. If the government wants to increase the price without resistance, it
should compensate him by

∆0 = −(I − E(PX1 , PY , U0 ))
= −(E(PX0 , PY , U0 ) − E(PX1 , PY , U0 )) > 0.

∆0 = E(PX1 , PY , U0 ) − E(PX0 , PY , U0 )
1
PX
= dE(PX , PY , U0 )
0
PX
1
PX
= H(PX , PY , U0 )dPX
0
PX
∂E(PX ,PY ,U0 )
because H(PX , PY , U0 ) = ∂PX
.

– This is the area to the left of the Hicksian demand curve corresponding to U0

· See figure 5.8

– ∆0 is called the compensation variation:
• Case 2: A person hates moving to the new situation although he does not resist a utility
reduction. If the government can be bribed not to increase the price, he is willing to do
so by

∆1 = −(E(PX0 , PY , U1 ) − I)
= −(E(PX0 , PY , U1 ) − E(PX1 , PY , U1 )).

∆1 = E(PX1 , PY , U1 ) − E(PX0 , PY , U1 )
1
PX
= H(PX , PY , U1 )dPX .
0
PX

– This is the area to the left of the Hicksian demand curve corresponding to U1 .

· See figure 5.9

– ∆1 is called the equivalent variation
• The absolute value of ∆1 is less than ∆0 because the corresponding Hicksian demand is
smaller at U1 .

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• Note that

∆0 = E(PX1 , PY , U0 ) − I
= E(PX1 , PY , U0 ) − E(PX1 , PY , U1 )

∆1 = I − E(PX0 , PY , U1 )
= E(PX0 , PY , U0 ) − E(PX0 , PY , U1 ).

• For practical purpose, consumer surplus is used to measure the welfare effect.
1
PX
∆= X(PX , PY , I)dPX .
0
PX

– See figure 5.10.

· Note that H(PX0 , PY , U0 ) = X(PX0 , PY , I) and H(PX1 , PY , U1 ) = X(PX1 , PY , I).

· V (PX0 , PY , I) = U0 and V (PX1 , PY , I) = U1
– Consumer surplus is a good approximation if the income effect is small.

33