a unified perspective on the slutsky equation - Bishop's University

A UNIFIED PERSPECTIVE
ON THE SLUTSKY EQUATION:
A PEDAGOGICAL NOTE 1
Ambrose Leung and Robert Sproule
2
Department of Economics, Bishop’s University, Lennoxville, Québec, J1M 1Z7, Canada
Abstract: More than three decades ago, Cook (1972) observed that the traditional, Hicks-
Allen approach to the Slutsky Equation “is very tedious and non-intuitive”, and then
proceeded to present a “one-line proof” of the Slutsky decomposition, based on the novel
use of the expenditure function. While Cook (1972) did show a way around the tedium,
he did so for only one of several cases [viz., the Hicksian formulation of the substitution
effect under an infinitesimally-small price change]. This paper presents a new way or
stratagem that is more flexible than the one employed in Cook (1972). Ours can
accommodate a finite versus an infinitesimally-small price change, and a Hicksian versus
Slutsky formulation of the substitution effect. The unity offered by this added flexibility
has obvious pedagogical value.
Keywords:
JEL classification:
Hicks-type decomposition, Slutsky-type decomposition,
compensating variation, pedagogy
A2, D1
June 15, 2005
1 The authors acknowledge the contribution of useful comments by Marianne Vigneault, and assume
responsibility for all remaining errors and omissions.
2 Corresponding author: E-mail rsproule@ubishops.ca, and phone 819-822-9600, extension 2480.
1

I. Introduction
The Slutsky Equation has a long and venerated history in microeconomics. It was first
articulated by Eugene Slutsky (1915) ninety years ago, and was revisited in such classics
as Hicks and Allen (1934), and Hicks (1946). Today, the Slutsky Equation is a staple of
most modern microeconomics textbooks [e.g., Luenberger (1995), Roberts and Schulze
(1976), Takayama (1993), and Varian (1992 and 2003)].
More than three decades ago, Cook (1972) observed that the traditional, Hicks-Allen
approach to the Slutsky Equation “is very tedious and non-intuitive”, and then proceeded
to present a “one-line proof” of the Slutsky decomposition, based on the novel use of the
expenditure function. While Cook (1972) did show a way around the tedium, he did so
for only one of several cases [viz., the Hicksian formulation of the substitution effect
(SE) under an infinitesimally-small price change].
This paper presents a new way or stratagem that is more flexible than the one employed
in Cook (1972). Ours can accommodate a finite versus an infinitesimally-small price
change, and a Hicksian versus a Slutsky formulations of the SE.
The rest of this paper is organized as follows. Section II outlines our analysis of the
Slutsky Equation under the Hicksian formulation of the SE. Section III outlines our
analysis of the Slutsky Equation under the Slutsky formulation of the SE. Both Sections
II and III offer a general analysis, and a function-specific example, under both a finite
and an infinitesimally-small price change. Summary comments are offered in Section IV.
2

II. The Hicksian Formulation of the Substitution Effect
Without loss of generality, let U = U( x1, x2)
denote any utility function, where x 1 and
x 2 denote two goods. Let x1 = x1( p1, p2, m)
and x2 = x2( p1, p2, m)
denote the
associated ordinary demand functions, where p 1 and p 2 denote the unit prices of x 1 and x 2 ,
and m denotes income.
Definition 1: If U = U( x1, x2)
, and if p1 goes to p1+ ∆p1 where ∆p1 ≠ 0, then the
following statement applies:
x1( p1+∆p1, p2, m) − x1( p1, p2, m)
∆p
≡
1
x1( p1+∆ p1, p2, m+∆m) − x1( p1, p2, m)
∆p
1
−
x1( p1+∆ p1, p2, m+∆m) − x1( p1+∆p1, p2, m) ∆m
.
∆m
∆p
1
(1)
If a conceptually meaningful constraint is placed on the value of ∆m, then Definition 1
may serve as the basis of the Slutsky decomposition. For example, if ∆m is set equal to
the compensating variation (CV) associated with ∆p 1 , then the initial utility level will be
identical to the utility level associated with (∆p 1, ∆m). To see how this applies to
Definition 1, consider the following proposition, which is preceded by two remarks:
Remark 1: Within the context of the Slutsky Equation, the left-hand side of Equation (1)
is termed the total effect (TE), viz.,
x1( p1+∆p1, p2, m) − x1( p1, p2, m)
∆p
1
=
TE
3

Remark 2: Let V( p1, p2, m ) denote the indirect utility function. After Takayama (1993,
p. 627), the CV is the money metric which solves the following equation,
V( p , p , m) = V( p +∆ p , p , m+ CV)
. 3
1 2 1 1 2
Proposition 1: If U = U( x1, x2)
, and if ∆m is set equal to the CV associated with ∆p 1 ,
then:
x1( p1+∆ p1, p2, m+∆m) − x1( p1, p2, m) |
∆ m=
CV
= Substitution Effect ( SE )
∆p
1
and
−
x1( p1+∆ p1, p2, m+∆m) − x1( p1+∆p1, p2, m) ∆m
. |
∆ m = CV
∆m
∆p
1
=
Income Effect
( IE)
Proof: Definition 1. •
Next, consider the following proposition:
Proposition 2: If U = U( x1, x2)
, and if ∆m is set equal to the CV associated with ∆p 1 ,
and if ∆p 1 goes to zero (viz., ∆p 1 0), then
∂ x ( p , p , m) x ( p , p , u) x ( p , p , m)
= ∂ − x ( p , p , m) .
∂
1 1 2 1 1 2 1 1 2
1 1 2
∂p1 ∂p1
∂m
which is the common form of the Slutsky Equation. 4
3 In words, the CV is defined as the amount of “money we would have to give the consumer after the price
change to make him as well off as he was before the price change” [Varian (2003, p. 255)]. Note that the
core idea is that income is altered so as to preserve utility in the face of a price change.
4 For example, see Cook (1972).
4

Proof: If ∆m is set equal to the CV associated with ∆p 1 in Definition 1, and if ∆p 1 0,
then it follows (from Remark 1 and Proposition 1) that, 5
TE :
x1( p1+∆p1, p2, m) −x1( p1, p2, m) ∂x1( p1, p2, m)
→
∆p
∂p
1 1
x ( p +∆ p , p , m+∆m) − x ( p , p , m) ∂x ( p , p , u)
SE : | m CV
→
IE
−
1 1 1 2 1 1 2 1 1 2
∆ =
∆p1 ∂p1
x ( p +∆ p , p , m+∆m) − x ( p +∆p , p , m)
∆m
1 1 1 2 1 1 1 2
: . |
∆ m = CV
∆m
∆p1
in that if ∆p 1 0, then:
→ −
∂x ( 1
p , 1
p , ) 2
m . x ( , , )
1
p1 p2
m
∂m
∆m
∆p
∂m
→ =
∂p
1 1
x ( p , p , m)
1 1 2
by virtue of Shephard’s Lemma. 6 In summary, if ∆m is set equal to the CV associated
with ∆p 1 , and if ∆p 1 0, then
∂ x ( p , p , m) x ( p , p , u) x ( p , p , m)
= ∂ − x ( p , p , m) .
∂
1 1 2 1 1 2 1 1 2
1 1 2
∂p1 ∂p1
∂m
which is the Slutsky decomposition. •
To add detail to the above, consider the Cobb-Douglas utility function,
U x x
α 1−α
= (
1) .(
2)
instead of the arbitrary utility function, U U x1 x2
= ( , ).
Let x1
m
= α and x2
p
1
= (1 − α) m denote the associated ordinary demand functions.
p
2
5 In the equation, “SE”, x1( p1, p2, u)
is termed a compensated demand function.
6 See Varian (1992, pp. 74-75).
5

a 1−a
Lemma 1: If U( x , x ) = ( x ) .( x ) , then the associated indirect utility function is:
1 2 1 2
a
−
⎛ a ⎞ ⎛1−
a⎞
V( p1, p2, m) = m. ⎜ ⎟ . ⎜ ⎟
p p
⎝ 1⎠ ⎝ 2 ⎠
a 1−a
m
m
Proof: If U( x1, x2) = ( x1) .( x2)
, then x = 1
a. and x2
(1 a). .
p
= − p
Likewise,
1 a
1 2
a 1−a ⎛am . ⎞ ⎛(1 − a).
m⎞
V( x1, x2) = ( x1) .( x2) = ⎜ ⎟ . ⎜ ⎟
⎝ p1 ⎠ ⎝ p2
⎠
a
1−a
a
⎝ 1⎠ ⎝ 2 ⎠
1−a
⎛ a ⎞ ⎛1−
a⎞
= m. ⎜ ⎟ . ⎜ ⎟ = V( p1, p2, m) .
p p
•
a 1−a
Lemma 2: If U( x , x ) = ( x ) .( x ) , then if p 1 goes to p 1 + ∆p 1 , then
1 2 1 2
CV
a
⎛⎛
p +∆p
⎞
= m
−
⎜
⎜ ⎟
p
⎝⎝
1 ⎠
⎞
⎟
⎠
1 1
. 1 .
Proof: Using the indirect utility function [Lemma 1],
V( p , p , m) = V( p +∆ p , p , m+
CV)
1 2 1 1 2
a 1−a a 1−a
⎛ a ⎞ ⎛1−a⎞ ⎛ a ⎞ ⎛1−a⎞
. ⎜ ⎟ . ⎜ ⎟ . ⎜ ⎟ . ⎜ ⎟
p p p +∆p p
⇔ m = ( m+
CV)
⎝ 1⎠ ⎝ 2 ⎠ ⎝ 1 1 ⎠ ⎝ 2 ⎠
⎛ 1 ⎞ ⎛ 1 ⎞
. ⎜ ⎟ . ⎜ ⎟
p p +∆p
⇔ m = ( m+
CV)
⇔
1 1
a
⎝ 1⎠ ⎝ 1 1 ⎠
a
⎛ p +∆p
⎞
m.
⎜ ⎟ = m + CV
⎝ p1
⎠
⎛ p +∆p
⎞
CV = m.
⎜ ⎟ − m
⎝ p1
⎠
⇔
1 1
a
a
a
⎛⎛
p +∆p
⎞ ⎞
= m. − 1 .
⎜
⎜ ⎟
p ⎟
⎝⎝
1 ⎠ ⎠
⇔
1 1
CV
•
6

a 1−a
Proposition 3: If U( x , x ) = ( x ) .( x ) , if p 1 goes to p 1 + ∆p 1 , and if ∆m is set equal
1 2 1 2
to the CV associated with ∆p 1 , then
TE :
SE
x1( p1+∆p1, p2, m) −x1( p1, p2, m)
∆p
m m
α − α
p +∆p p
1 1
x ( p +∆ p , p , m+∆m) − x ( p , p , m)
1 1 1 2 1 1 2
: |
∆ m = CV
∆p1
=
1 1 1
∆p
=
m+∆m m
α − α
p +∆p p
1 1 1
∆p
1
|
∆ m = CV
IE
x ( p +∆ p , p , m+∆m) − x ( p +∆p , p , m)
1 1 1 2 1 1 1 2
: −
|
∆ m = CV
∆p1
= −
m+ ∆m m
α − α
p +∆ p p +∆p
1 1 1 1
∆p
1
|
∆ m = CV
a 1−a
Example 1: If U( x , x ) = ( x ) .( x ) , if m = 100, p 1 = 1, and p 1 + ∆p 1 = 2, and if α =
1/2, then by Lemma 2,
1 2 1 2
CV
a
⎛
1/2
⎛ p1+∆p
⎞ ⎞ ⎛
1
⎛2
⎞ ⎞
( )
= m. − 1 = 100 . −1
⎜
⎜ ⎟ ⎜ ⎟
⎝ p
⎜
1 ⎠ ⎟ ⎝1
⎠ ⎟
⎝
⎠ ⎝ ⎠
= 100. 2 − 100 = 141.42 − 100 = 41.42
Furthermore, by Proposition 3, CV = 41.42 = ∆ m ,
TE
m m
α − α
p +∆p p
1 1 1
= = = −
∆p
1
1 100 1 100
. − .
2 2 2 1
1
25,
7

SE
m+∆ m m
α − α
1 100 + 41.42 1 100
. − .
p +∆p p
|
2 2 2 1
14.645,
1 1 1
=
∆ m=
CV
= = −
∆p1
1
and
IE
= −
m+∆m m
α − α
p +∆ p p +∆p
1 1 1 1
∆p
1
|
∆ m = CV
1 100 + 41.42 1 100
. − .
= −
2 2 2 2
= − 10.355.
1
III. The Slutsky Formulation of the Substitution Effect
Varian (1992 and 2003) has noted that there are two alternative formulations of the SE:
(a) the Hicksian formulation, in which utility is held fixed, and (b) the Slutsky
formulation, in which purchasing power is held constant. Clearly, the discussion
presented to this point concerns the former.
It should be noted that the approach outlined in Section II can be modified to
accommodate the Slutsky formulation of the SE. Just as Definition 1 serves the starting
point for our development of the Hicksian formulation of the SE, Definition 1 can serve
as a starting point for the development of the Slutsky formulation. In particular, in our
development of the Hicksian formulation, we set ∆m equal to the CV associated with ∆p 1
[Proposition 1]. To capture the notion of holding purchasing power constant in the
Slutsky formulation, one would set ∆m equal to ∆p 1 .x 1 , where x 1 denotes the initial
equilibrium value. Thus, one would reformulate Proposition 1 by writing:
Proposition 4: If U = U( x1, x2)
, and if ∆m is set equal to ∆p 1 .x 1 associated with ∆p 1 ,
then:
x1( p1+∆ p1, p2, m+∆m) − x1( p1, p2, m) |
∆ m =∆ p 1.
x 1
∆p
1
=
SE
8

and
x ( p +∆ p , p , m+∆m) − x ( p +∆p , p , m) ∆m
. | m p x
1 1 1 2 1 1 1 2
−
∆ =∆ 1.
=
1
∆m
∆p1
Proof: Definition 1. •
IE
Proposition 5: If U( x1, x2) = x1.
x2, and if ∆m is set equal to ∆p 1 .x 1 associated with ∆p 1 ,
then
TE :
x1( p1+∆p1, p2, m) −x1( p1, p2, m)
∆p
1 m 1 m
. − .
2 p +∆p 2 p
1 1
=
1 1 1
∆p
SE
x ( p +∆ p , p , m+∆m) − x ( p , p , m)
1 1 1 2 1 1 2
: |
∆ m =∆ p 1.
x 1
∆p1
=
1 m+∆m 1 m
. − .
2 p +∆p 2 p
1 1 1
∆p
1
|
∆ m =∆ p . x
1 1
IE
x ( p +∆ p , p , m+∆m) − x ( p +∆p , p , m)
1 1 1 2 1 1 1 2
: −
|
∆ m =∆ p 1.
x 1
∆p1
= −
1 m+ ∆m 1 m
. − .
2 p +∆ p 2 p +∆p
1 1 1 1
∆p
1
|
∆ m =∆ p . x
1 1
m
where ∆ m = ∆ p1. x1 = ∆ p1. since x1
2 p
1
m
= .
2 p
1
Example 2: If U( x1, x2) = x1.
x2, if m = 100, p 1 = 10, and p 1 + ∆p 1 = 5, then by
Proposition 5, x1
m 100
= = = 5 , ∆ m = ∆ p1. x1
= ( − 5).(5) = − 25,
2p
2.10
1
1 m 1 m
. − . 1 100 1 100
. − .
2 p1+∆p1 2 p1
TE : =
2 5 2 10
= −1
∆p
−5
1
9

1 m+∆m 1 m
. − . 1 100 − 25 1 100
. − .
p p p
SE : |
2
1+∆
1
2
1
2 5 2 10
∆ m=∆p1.
x
=
1
∆p1
−5
7.5 − 5
= = − 0.5
−5
1 m+∆m 1 m
. − . 1 100 − 25 1 100
. − .
p p p p
IE : −
|
2
1+∆ 1
2
1+∆
1
2 5 2 5
∆ m=∆p1.
x
= −
1
∆p1
−5
7.5 − 10
= − = − 0.5
−5
Remark 3: The case of an infinitesimally-small price change follows immediately from
∆m
∂m
Proposition 2. As before, as ∆p 1 0, then → = x1( p1, p2, m)
[Shephard’s
∆p
∂p
1 1
Lemma]. This is as Mosak (1942) has shown: the Hicksian and Slutsky formulations of
the SE are identical under an infinitesimally-small price change [Cornes (1992, p. 102)].
IV. Conclusion
The present paper offers a stratagem that is more flexible than the one employed in
Cook’s (1972) terse proof of the Slutsky Equation.
Our claim of added flexibility comes from two facts. One, Cook’s (1972) stratagem
accommodates one case, the Hicksian formulation of the SE under an infinitesimallysmall
price change. Two, the stratagem outlined in this paper can accommodate a finite
versus an infinitesimally-small price change, and a Hicksian versus a Slutsky formulation
of the SE.
The unity offered by this added flexibility has obvious pedagogical value.
10

References
Cook, P. (1972), “A 'one-line' proof of the Slutsky decomposition,” American Economic
Review 62, 139.
Cornes, R. (1992), Duality and Modern Economics (Cambridge, UK: Cambridge
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Hicks, J. (1946), Value and Capital, 2 nd edition (Oxford: Claredon Press).
Hicks, J., and R. Allen (1934), “A reconsideration of the theory of value,” Economica 1,
54-76 and 196-219.
Mosak, J. (1942), “On the interpretation of the fundamental equation of value theory,” in
O. Lange et al. (eds.), Studies in Mathematical Economics and Econometrics, (Chicago,
Illinois: University of Chicago Press).
Luenberger, D.G. (1995), Microeconomic Theory (New York, New York: McGraw-Hill).
Roberts, B., and D.L. Schulze (1976), Modern Mathematics and Economic Analysis
(New York, New York: W.W. Norton).
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Stigler, eds., as “On the theory of the budget of the consumer,” in Readings in Price
Theory (London: Allen & Unwin, 1953).]
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University of Michigan Press).
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Norton).
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York, New York: W.W. Norton).
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