Predictions of Utility Theory
About the Nature of Demand
References:
Varian: Ch 8 Slutsky Equation
(Especially Appendix to Chapter 8)
Suplementary References: Deaton &
Muelbaur, Consumer behaviour and
Predictions of Utility Theory About
the Nature of Demand
• Given the axioms (or properties) 1-7 we
have assumed about the utility function,
these imply certain things about the demand
function.
• We can now want to derive a set of core
properties we would expect any demand
function we estimated to exhibit
Testable Predictions and the
Theory
• We can then test the demand functions we
derived and ask if they exhibit these
properties.
• If they don’t then we either have a problem
with our data, or we have used the wrong
functions or estimation method
• OR (more seriously),
• with
our theory!
Properties of Demand Function:
No. 3
The Pure Substitution effect is
Negative
(Strictly it is Never Positive)
Start with: The SLUTSKY EQUATION
xD = x (P, Py, M)
We know that a change in the price of x leads to a change
in demand for the good which we have identified as a
combination of a substitution and income effect.
Furthermore, the theory we have developed suggests that
this substitution effect is always negative. This is a
prediction of our theory. How could we test it, since it
relies on an unmeasurable quantity –the substitution
effect?
In particular, how could we measure this in the case of
the Slutsky and Hicksian demand curves we have seen?
The SLUTSKY EQUATION
xD = x (Px, Py, M)
First, although this is known as the Slutsky equation, we
will look at the Hicksian case where we compensate the
consumer for the change in prices by placing her back on
her original indifference curve.
Units of good Y
Income and substitution effects: normal good
h
f
I1
I2
B2
QX3
B1
I3
I4
I5
I6
QX1
Units
X
Units of Good X
Units of good Y
Income and substitution effects: normal good
g
h
f
I1
I2
B2
QX3
QX2
Income
effect
B2a
B1
I3
I4
I5
I6
QX1
Substitution
effect
Units
X
Units of Good X
The SLUTSKY EQUATION
xD = x (Px, Py, m)
We know that for a change in the price of x,
Overall effect = Substitution effect (U held constant)
+ income effect
Thus:
x D x

Px Px
But M= Pxx, + Py,y
and
x m
 
U
m Px
So, xD = x (Px, Py, Pxx + Pyy)
m
 x
Px
x D
x

Px
Px
So,
x D
x

Px
Px
x D x

Px Px
Substitution
Effect
x
Px

U
x m
 
U
m Px

U
x
 x

m
x
x
m
holding U constant
x
x
x
 
U
Px
m
Income Effect
Called the Hicks - Slutsky Decomposition
The Slutsky Equation
or
The Hicks - Slutsky Decomposition
x
Px
x
x
x
 
U
Px
m
D
This says that the pure substitution effect is a
combination of the price effect and the income
effect.
While we cannot observe the variable on the
LHS, we can observe everything on the RHS.
So we can test the prediction that the pure x
x
, x and
substitution effect is negative by measuring
Px
m
First Testable Prediction:
• The Pure Substitution Effect is always
Negative (never positive)
x
Px
x D
x
x
 
U
Px
m
The Slutsky Equation
or
The Hicks - Slutsky Decomposition
x D x

Px Px

U
x
x
m
If it is true that the pure substitution effect
is always negative, then we know from the
expression above that as long as the good
is normal
(that is, mi
implies x i)
Then the demand curve slopes down.
The SLUTSKY EQUATION
xD = x (Px, Py, M)
Technically, as we have already said, this measures the
Hicksian effect rather than the Slutsky effect, but we can
write a similar expression for Slutsky.
Recall, with Slutsky we compensate the consumer for the
change in prices by allowing him to purchase the original
bundle.
Units of good Y
Income and substitution effects: Hicks
(Solid line)
g
h
f
I1
I2
B2
QX3
Income
effect
QX2
B2a
B1
I3
I4
I5
I6
QX1
Substitution
effect
Units
X
Units of Good X
Units of good Y
Income and substitution effects: Slutsky
(Solid line)
g’
h
f
I1
I2
B2
QX3
Income
effect
QX2
B2a
B1
I3
I4
I5
I6
QX1
Substitution
effect
Units
X
Units of Good X
To represent this we only have to make a
minor qualification to the equation:
x
x

Px Px
D
Substitution
Effect
x
Px
 
m
x
x
x
Px
m

m
x
x
m
Income Effect
holding m constant
Hence called the
Hicks - Slutsky Decomposition
To represent this we only have to make a
minor qualification to the equation:
x
x

Px Px
D
Substitution
Effect
x
Px
 
m
x
x
x
Px
m

m
x
x
m
Income Effect
holding m constant
Hence called the
Hicks - Slutsky Decomposition
To represent this we only have to make a
minor qualification to the equation:
x
x

Px Px
D
Substitution
Effect
x
Px
 
m
x
x
x
Px
m

m
x
x
m
Income Effect
holding m constant
Hence called the
Hicks - Slutsky Decomposition
Hicks - Slutsky Decomposition
• In practice for very small changes in prices the
Hicksian and Slutsky effects are essentially the
same.
• For the most part, the terms we use to test the
theory are derivatives of an estimated function and
the Slutsky and Hicksian effects are synonymous
• However, if we are looking at compensating
government tax changes for example then the
difference is important.
Note:
Main text in Varian uses the terminology,  xs , and
discusses the concept in terms of  changes. You
can use this if you want, but prefer to use the
derivative terms as in the appendix to the chapter.
x D x

Px Px
x
_ x
m
m
The Varian (appendix) expression is (The two goods are x1
and x2) :
_
_
_
_
x1 ( P1 , P2 , m) x1 ( P1 , P2 , x1 , x 2 )
x1 ( P1 , P2 , m)

 x1
P1
P1
m
Finally, we can write
The Hicks - Slutsky Decomposition in Elasticity Form
x
Px
Px x
x D Px
x D
x
x
 
U
Px
m
Px x D Px m x
 D x
 
D
U
x Px x
m m
Finally, we can write
The Hicks - Slutsky Decomposition in Elasticity Form
x
Px
Px x
x D Px
x D
x
x
 
U
Px
m
Px x D Px m x
 D x
 
D
U
x Px x
m m
Finally, we can write
The Hicks - Slutsky Decomposition in Elasticity Form
x
Px
x D
x
x
 
U
Px
m
Px x
x D Px
Px x D Px m x
 D x
 
D
U
x Px x
m m
Px x
x D Px
Px x D Px x m x

 
D
U
x Px
m x D m
Finally, we can write
The Hicks - Slutsky Decomposition in Elasticity Form
x
Px
x D
x
x
 
U
Px
m
Px x
x D Px
Px x D Px m x
 D x
 
D
U
x Px x
m m
Px x
x D Px
Px x D Px x m x

 
D
U
x Px
m x D m
 xp
x

U
  xp x  s x x
Slutsky Summary:
• Although the Pure Substitution is
unobservable the Slutsky Equation tell us
that we can test whether it is negative (not
positive) by checking the magnitude of
three observable phenomenon:
– the elasticity of demand for x,
– the share of x in expenditurte
– and the income elasticity of demand for x.
Summaryof Properties
• This section has identified three properties
of demand functions (there are others):
1.The Adding-Up Condition
2. The Cournot Condition
3. The Non-Positive Pure Substitution Effect
Summary of Course So far:
• So after a lot of hard work we have identified 7
axioms we would like to assume preferences
follow to generate well behaved demand
functions.
• As a consequence we have identified three
properties we would like these demand curves to
obey
• If demand functions fit these properties are
assumptions about preferences are reasonable.
• Do They?
Well….., yes and no