### Document

```Income and Substitution Effects
The Law of Demand:
Slope of budget
line from px/py to
steeper px’/py
y
x
p x
px/py
U1
U2
px’/py
px/py
xb
xa
Qd falls from xa to xb
x
x*=x(px,py,M)
xb
xa
Qd falls from xa to xb
x
x
p x
Decompose
into
Income and Substitution effects.
• Substitution Effect: Change in Qd caused by
change in px/py , but holding utility constant.
• Income Effect: Change in Qd caused by change
in purchasing power resulting from price
change, but holding px/py.
Substitution Effect
y
px/py
U1
px’/py
px/py
xb xc xa
x
x*=x(px,py,M)
xb xc xa
Substitution effect: How Qd changes as a result of the price change, even
when utility can be held constant (Qd from xa to xc)
x
Substitution Effect
• Utility maximization requires a tangency (MRS
= px/py) be maintained.
• Because of diminishing MRS, an increase in
px/py means the tangency will be where x* is
lower and a decrease in px/py means the
tangency will be where x* is higher.
• Substitution effect is consistent with the law
of demand.
Income Effect
y
px/py
U1
px’/py
px/py
xb xc
x
x*=x(px,py,M)
xb xc xa
Income effect: How Qd changes as a result ONLY of the change in purchasing
power resulting from a price change -- holding the price ratio at the new level,
px’/py – (Qd from xc to xb)
x
Income Effect
• By isolating the change in purchasing power (but
leaving the price ratio unchanged), the income
effect looks exactly like the change resulting from
a change in income.
• Normal goods, increase in price means decrease
in purchasing power, so income effect is negative
– reinforces the law of demand.
• Inferior goods, increase in price means decrease
in purchasing power, so income effect is positive
– runs counter to the law of demand.
Substitution and Income Effects
(Inferior Good)
y
px/py
U1
px’/py
px/py
xc xb
xa
x
x*=x(px,py,M)
xc xb
xa
Income effect: After a substitution effect from xa to xc the individual feels poorer
and because it is an inferior good, the income effect is positive (Qd from xc to xb)
Overall the change in consumption conforms to the law of demand unless the
good is inferior and the income effect so large that it overwhelms the substitution
effect. Goods for which this occurs are called Giffen goods.
x
Giffen Good
• Case where x  0
p x
• Why so rare?
• To be Giffen
– Inferior
– Large income effect (to overwhelm the substitution
effect) – meaning expenditure must be a
substantial portion of income
• Goods that we spend a lot on tend to be
normal.
Ordinary or Compensated Steeper?
px/py increases to px’/py
px/py
Normal Good
Compensated steeper
px/py
Inferior Good
Ordinary Steeper
x*=x(px,py,M)
xc*=xc(px,py,Ū)
px’/py
px’/py
x*=x(px,py,M)
px/py
xb
xc xa
px/py
x
xc*=xc(px,py, Ū)
xc xb xa
SE
SE
IE
IE
x
Typical Inferior vs Giffen
px/py increases to px’/py
px/py
Inferior Good
Ordinary Steeper
Giffen Good
Positive slope
px/py
x*=x(px,py,M)
x*=x(px,py,M)
px’/py
px’/py
px/py
xc*=xc(px,py, Ū) px/py
xc xb xa
x
xc*=xc(px,py, Ū)
xa xb
xc
SE
SE
IE
IE
x
Elasticity – Substitution Effect
• Demand will be more inelastic if the elasticity of
substitution, σ, is smaller – smaller substitution
effect.
y
Ub
Ua
x
Elasticity – Income Effect
• But holding σ constant, a normal good will have
the more elastic demand as the income effect
reinforces the substitution effect. For an inferior
good, the income effect works against the
substitution effect.
• Goods that are small portions of budget will
tend to have very small income effects.
Slutsky Equation
• What happens to purchases of good x change
when px changes?
x
p x
• Ideally we want to decompose the change into
x into the two components:
– Substitution effect: the curvature of the utility
function -- substitutability between goods
– Income effect: the magnitude and direction of the
effect of a change in purchasing power.
Slutsky Equation
• The equation that decomposes the
substitution and income effects:
x
x

p x p x
x
x
M
UU
Slutsky Derivation (Modern)
• At the optimal bundle we are at the intersection
of the Marshallian and Hicksian demand curves:
x c (p x , p y , U)  x *  x(p x , p y , M)
Where income = M is the minimum income
required to acheive utility = U.





So if: M  E*  E* p x , p y , U ,we can define M  M* p x , p y , U
Then:

x*  x p x , p y , M* p x , p y , U
And we can set up the following identity:


x c (p x , p y , U)  x p x , p y , M * p x , p y , U


x (p , p , U)  x  p , p , M  p , p , U  
c
*
x
y
x
y
x
y
And we can differentiate each side w.r.t. p x :
x c (p x , p y , U)
p x
And since



x(p x , p y , M)
p x

M* p x , p y , U  E* p x , p y , U
x (p x , p y , U)
c
p x


M

x(p x , p y , M)
p x


*
x(p x , p y , M) M p x , p y , U
p x

*

E
px , py , U
x(p x , p y , M)
M
p x
By Shepard's Lemma,
x c (p x , p y , U)
p x

x(p x , p y , M)
p x

x(p x , p y , M )
M

x c (p x , p y , U)

At the Optimal Bundle
• Rearrange to get:
x(p x , p y , M)
p x

x c (p x , p y , U)
p x
 x (p x , p y , U)
c
x(p x , p y , M)
M
And since we are at an optimum where M and U such that:
x c (p x , p y , U)  x(p x , p y , M)
Yielding:
x(p x , p y , M)
p x

x c (p x , p y , U)
p x
 x(p x , p y , M)
x(p x , p y , M)
M
One last troubling variable
• We have:

x p x , p y , M
p x
  x (p , p , U)  x
c
x
y
p x
p , p , M
x
y
• But we need
x(p x , p y , M)
p x
UU
x c (p x , p y , U)
p x

x p x , p y , M
M

At the Optimal Bundle


Substitute: U  V p x , p y , M into
*

x c p x , p y , U
p x
So:


x p x , p y , M
p x


x c p x , p y , U
p x

x c p x , p y , V* p x , p y , M
p x


x c p x , p y , U
p x


   x  p , p , M 
x
y
p x
x
p , p , M
x
UU

x p x , p y , M

I
y

Becomes by substituting the indirect utility function in for U :

x p x , p y , M
p x
  x  p , p , M 
x
p x

y
 x px , py , M
UU


x p x , p y , M
M

Slutsky Equation
Substitution Effect
x  p x , p y , M 
p x

x  p x , p y , M 
p x
 x  px , py , M 
x  p x , p y , M 
M
UU
Always negative because of convexity of preferences.
Income Effect
x  p x , p y , M 
p x

x  p x , p y , M 
p x
 x(p x , p y , M )
UU
 x

Positive if good is inferior 
 0
 M

 x

Negative if good is normal 
 0
 M

x  p x , p y , M 
M
Own-Price Slutsky
• Decomposition:
partial derivative
of the ordinary
demand
for x w.r.t. px
x(p x , p y , M)
p x

x(p x , p y , M)
p x
x
Take
, then substitute in
p x
x
p x
UU
Ordinary
demand for x
c
U  V(p x , p y , M) to get
 x(p x , p y , M)
UU
x(p x , p y , M)
M
partial derivative
of the ordinary demand
for x w.r.t. M
A Slutsky Decomposition Example
• We can demonstrate the decomposition of a
price effect using the Cobb-Douglas example
studied earlier
U  xy
.5
• The Marshallian demand function for good x was
2M
x  px , py , M  
3p x
• With a total effect of a change in px
x  p x , p y , M  2M

p x
3p 2x
Substitution Effect
Hicksian demand:
x  px , py , U  
c
U
2/3
 2p 
1/3
y
, and
1/3
x
p
x  p x , p y , U 
c
p x

U
Indirect utility:
U  V  px , py , M  
2M
3
2
3p x  3p y 
1
2
Substitution effect:
x  p x , p y , M 
p x
UU
3


2
2M



1 

2 
3p
3p
x
y



4
3
3p x
2/3
 2p 
1/3
y

2M
9p 2x
2/3
 2p 
1/3
y
3p 4/3
x
Income Effect
x  p x , p y , M 
2M
2
x(p x , p y , M) 
and

3p x
M
3p x
The product is
x  px , py , M  
x  p x , p y , M 
M
2M 2
4M

 2
3p x 3p x 9p x
Slutsky Equation
x  p x , p y , M 
p x
2M
3p 2x

p x
Effect
2M
9p 2x

Effect

2M
9p 2x
 x  px , py , M 
UU
 Substitution
Total
6M
9p 2x

x  p x , p y , M 

4M
9p 2x
Income
Effect

4M
9p 2x
x  p x , p y , M 
M
If you only have Marshallian demand
equations…
• You can get the total and income effects from
them, and then add them to get the
substitution effect.
x  p x , p y , M 
p x
6M
9p 2x
Total
Effect
+
+
 x  px , py , M 
4M
9p 2x
Income
Effect
x  p x , p y , M 
I


2M
9p 2x
= Substitution
Effect
x  p x , p y , M 
p x
UU
Cobb-Douglas Slutsky
IE   x
x
4M
 2
M
9p x
TE=
SE 
x 2M

p x 3p 2x
x
p
UU

2M
9p2x
Cross Price Effects,
x
p y
• Out analysis of cross-price effects in a twogood world is limited as spending more on x,
necessarily means spending less on y and viceversa.
• Yet, we can use the two good world to define
terms and gain an intuitive understanding.
Net Substitutes
• Net effect, limit analysis to the substitution effect:
– py rises, (px/py falls), Qd of x rises
y
SE
x
p y
0
UU
y1
y2
U1
SE
x1
x2
x
Net Compliments
• Net effect, limit analysis to the substitution effect:
– py rises, (px/py falls), Qd of x falls
y
!
x
p y
0
UU
Cannot be represented in a two good world!!!
With only two goods, they must be net
substitutes.
In a multi-good world, it is possible for x to be a
net substitute for y, but a net compliment of z.
y1
U1
x1
x
Substitutability with Many Goods
• Demand for Bacon, Eggs, Cereal, etc.
p bacon rises
bacon
0
p bacon
eggs
 0, net compliments
p bacon
cereal
 0, net substitutes
p bacon
Gross Compliments
• Gross effect, both income and substitution effect:
– py rises, x* falls, both goods normal.
– When the price of y rises, the substitution effect is to consume less y and
more x
– Because of the larger income effect, individuals buy less of both x and y.
y
SE
IE
x
0
p y
y1
yc
y2
U1
SE
I E
x2
x1 xc
U2
x
Gross Substitutes
• Gross effect, both income and substitution effect:
– py rises, Qd of x falls, y normal, x inferior.
– When the price of y rises, the substitution effect is to consume less y and
more x.
– Because x is inferior, the income effect reinforces the substitution.
y
x
0
p y
SE
y1
yc
IE
U1
y2
SE
U2
IE
x1 xc x2
x
Gross Effects
• Is the status (normal vs. inferior) the
determining feature?
• No. You can have gross substitutes even if x is
inferior, so long as the income effect is small.
Gross Substitutes
• Gross effect, both income and substitution effect:
– py rises, Qd of x rises, both x and y normal.
– When the price of y rises, the substitution effect is to consume less y and
more x.
– While x is normal, the magnitude of the income effect is smaller than the
substitution effect. y
x
0
p y
SE
y1
yc
IE
U1
y2
SE
U2
IE
x1x2 xc
x
Asymmetry of the Gross Definitions
• The gross definitions of substitutes and
complements are not symmetric
– it is possible for x to be a gross substitute for y
(when the price of y changes) and at the same time
for y to be a gross complement of x (when the price
of x changes).
Asymmetry of the Gross Definitions
• Suppose that the utility function for two goods
is given by
U(x,y) = ln x + y
• Setting up the Lagrangian
L = ln x + y + (M – pxx – pyy)
Asymmetry of the Gross Definitions
• We get the following FOCs:
Lx = 1/x - px = 0
Ly = 1 - py = 0
Lλ = M - pxx - pyy = 0
• Manipulating the first two equations, we get
pxx = py
Asymmetry of the Gross Definitions
•Inserting this into the budget constraint, we can
find the Marshallian demand for x and y
x
py
px
; y
M  py
py
•The cross price effects are not symmetric
x
1
 ;
p y px
y
0
px
Cross-Price Slutsky
• We’ll skip the derivation, but here it is:
x
x

p y p y
UU
substitution
effect (+)
x
 y
M
income effect
(-) if x is normal
combined effect
(ambiguous)
Cross-Price Slutsky
• Cross Slutsky decomposition:
partial derivative
of the ordinary
demand
for x w.r.t. py
x
x

p y p y
UU
x
, then substitute in
p y
U  V  p x , p y , M  to get
x
p y
partial derivative
of the ordinary demand
for x w.r.t. M
Ordinary
demand for y
c
Take
x
 y
M
UU
A Slutsky Decomposition Example
• We can demonstrate the decomposition of a
price effect using the Cobb-Douglas example
studied earlier
0.5
U  xy
• The Marshallian demand function for good x was
2M
x(p x , p y , M) 
3p x
• With a total cross price effect of a change in px
x(p x , p y , M)
0
p y
Remember this Graph?
Qd for y was unaffected by
the change in px
y
0
p x
IE   x
x
4M
 2
M
9p x
TE=
SE 
x 2M

p x 3p 2x
x
p
UU

2M
9p2x
And the effect on x of a change in py
y
TE=
p y
SE 
x
p
IE   y
y
M
UU
x
0
p y
Qd for x was unaffected by
the change in py
Substitution Effect
Hicksian demand:
1
 2p y 
x  px , py , U   U 
 , and
 px 
2
c
3
3
x c  p x , p y , U 
p y
1

U  V  px , py , M  
2M
2
3 px  py 
3
1
2
2
Substitution effect:
3


2
1
2M

2 3 3
1 
 2
2 
3
p
p


x
y



x  p x , p y , M 
p y
1
UU
3
2
3p x p y 3
2
3

2 U
1
2M
9p x p y
3
2
3
2
3p x p y 3
Indirect utility:
3
3
Income Effect
x  p x , p y , M 
M
2
y  px , py , M  
and

3p y
M
3p x
The product is
y  px , py , M  
x  p x , p y , M 
M
M 2
2M


3p y 3p x 9p y p x
Cross-Price Slutsky Decomposition
x  p x , p y , M 
p y
0


x  p x , p y , M 
UU
p y
2M
9p x p y
Total  Substitution
Effect
Effect

 y  px , py , M 
2M
9p x p y
 Income
Effect
x  p x , p y , M 
M
Slutsky Equation Via
Comparative Statics
• Using Utility Maximization and Expenditure
Minimization
• Yes, Rockin’ it Old School
Comparative Statics:
Differentials of U-max FOC w.r.t. px
• Remember
 0

 p x
 p
 y
p x
U xx
U yx
• Using cofactors
0
x p y
p x
 U xy
  
 p 
p y   x   x 
  x   
U xy   
   

p
x
 0 
U yy  
 y 
 p 
 x
Substitution effect < 0
0
p y
p x
U xy
p y 0 U yy
p y U yy
p y U yy
x


x
p x
H
H
H
Income effect
Assume SOC hold and H  2U x U y U xy  U x 2 U yy  U y 2 U xx  0
Comparative Statics:
Differentials of U-max FOC w.r.t. px
• Simplify
Substitution effect < 0
0
x p y
p x
 U xy
0
p y
p x
U xy
p y 0 U yy
p y U yy
p y U yy
x


x
p x
H
H
H
Assume H  0
• To get this
p y 2
p x U yy  p y U xy
x

x
p x
H
H
Income effect
Building the Income Effect:
Differentials of U-max FOC w.r.t. M
• Remember
 0

 p x
 p
 y
p x
U xx
U yx
  


p y   M   1
  x   
U xy  •
 0
 


M

U yy   y   0 


 M 
• Using cofactors
0
1 p y
p x
0
U xy
x p y

M
0
U yy
H
Assuming H  0
p x U xy


 p y U yy

   1
H


p x
p y
H  p x
U xx
U xy
p y
U yx
U yy
0
Look Familiar?


  p x U yy  p y U xy

H


Combine
x
p x
x
M
p y 2
p x U yy  p y U xy
x

x
p x
H
H
x
M
• Yielding
p y
x
x

x
p x
M
H
2
Save this for later!!!
Comparative Statics:
Differential of E-Min FOC w.r.t px
• Remember
 0

U x
U
 y
  
 p 
U y   x   0 
  x c   
U xy   
   1
 p x   0 

U yy   c   
y


 p x 
U x
U xx
U yx
U x
U y
H min   U x
U xx
U xy
U y
U yx
U yy
0
• By cofactors
0
U x
U y
0
1 U xy
U y 0
x

p x
H
U yy
c
U y
0
 1
U y
H
min
Assuming SOC hold and H
min
U yy

min
 1
  U y 
H
min
2

U y2
H
0
min
2U x U y U xy  U x 2 U yy  U y 2 U xx
0
• Note that
H  2U x U y U xy  U x 2 U yy  U y 2 U xx  0
H
min
• Now take
 2U x U y U xy  U x 2 U yy  U y 2 U xx  0
0
U x
1 U xy
x c  U y 0

p x
H
Multiply by
• Yielding
U yy
min

U y2
H
0
min
1
, now  H min  H  0
1
x c  U y

0
p x
H
2
U y
0
x c  U y

0
p x
H
2
• Recall that
=
Uy
py
, and therefore, U y   py
• By substitution
x c* p y

p x
H
• All green
2
x c* p y

p x
H
2
SLUTSKY!!!
• Go back to this
x p y
x

x
p x
M
H
2
• Substitute
x c p y

p x
H
2
• To get
x x c
x

x
p x p x
M
• As before, we need to substitute in to get
x
x

p x p x
x
x
M
UU
```