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Microeconomics Lecture Supplements Department of EconomicsHajime Miyazaki OHIO STATE UNIVERSITYslutsk95.usc Fall 93/94/95/01/02File Name: SLUTSKY2003Draft.doc [email protected]
SLUTSKY COMPENSATION AND DECOMPOSITION
Throughout this note, a consumer obeys WARP (Weak Axiom of
Revealed Preference) and a typical budget set takes the form B(p, m) =
{x | px m }. For reasons stated in the preceding lecture note, we
assume income exhaustion; unless otherwise noted; all consumption
choices are made on the budget line p x = m. If the income constraint is
not binding, a comparative statics of a changing income on a consumption
choice become obtuse. Since any price change involves an effectivechange in the purchasing power of money income, the income exhaustion
assumption will give us a better handle on decomposing the gross effect of
a price change into pure price effect and implied income effect. .
In what follows, as in almost all textbooks, illustrations are
exclusively for the case of a two-dimensional commodity space. It is
useful and instructive to remember that the dimension of commodity
vectors in the real market economy is very large. Theoretically, the
dimension of the vector space in a general equilibrium setting is even
larger than the real world economy. This is because duality theory
x2
0
x1
B(p, m)
x =x(p, m)
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assumes a complete contingent markets over time and space. While
intuition and insights gained from two-dimensional analyses are often
generalizable to higher dimensions, there are some notable lacuna between
the two and higher dimensional. For example, in the case of Slutsky
equationsand revealed preference, two dimensional analysis induce
strong results that are not extendable to three dimensions1.
To simplify our two dimensional illustrations further, we fix (or
normalize) p2 = 1 and allow only p1 to change. Further, we first draw
diagrams for a case of only p1 increasing, which is technically identical
to a case in which the relative price p1/p2 increases. The next diagram
shows such a price change; pchanges to p+ p = (p1+ p1,p2) where
p1 > 0 and p2 = 0. The case of a decrease in p1 as well as other
cases of a price change are left to the reader. Definitions, algebraic proofs
and theorem statements are stated for a general case.
1Notably, WARP is equivalent to SARP (Strong Axiom of Revealed Preference) in thetwo dimensional commodity space, but not at all so in higher dimensions. The Slutskymatrix is automatically symmetric in two dimensions but not necessarily so in higherdimensions.
0
x2
x1
x =x(p, m)
(p1/p2) (p1+ p1)/p2
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SLUTSKY COMPENSATION
Let x =x(p, m) be a commodity bundle that a consumer buys
when the price vector is p with income m. Suppose that the price vector
changes to p+ p while the consumers money income m remains
unchanged. This ceteris paribus change in the price vector affects the
consumers purchasing power of m. The consumer may no longer be able
to buy x under the new price vector unless the money income is
increased. Of course, the price change may be favorable and the
consumers purchasing power might have increased.
For the sake of exposition, let us adopt a scenario that the initial
bundle xo becomes too expensive to buy. To buy the initial bundle x
under a new price vector, the consumer needs at least (p + p)x of
money income. The extra income that the consumer needs is at least (p+
p)xopx = px. Let us define the Slutsky compensation mS as
the minimum monetary subsidy that enables the consumer to buy the
initial bundle x under a new price vector. Thus,
mS = px = px(p, m).
If the consumers purchasing power increased after a price change,
the new budget set will include x, and there is no need for a subsidy.
Still, we can ask for a hypothetical scenario in which we tax the
consumers increased purchasing power. What would be the maximum
tax that we can levy without making the consumer worse off than before
the price change? One way to measure this maximum levy is to ask how
much the consumer can give up and still able to purchase the initial vector
x. The change in the value of the consumption vector x is (p+ p)x
px = (p)x. The current scenario is that this px is negative;x is
cheaper than before by the amount equal to px. In this scenario we
should define the Slutsky compensation as the maximum tax that can be
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levied on the consumer while enabling him/her to buy x. Thus, |mS| =
|px|, and in the present scenario, mS = px < 0.
The purpose of Slutsky compensation mS is to adjust the
consumers money income so that he/she can havejust enough to buy
back x after a price change. Since the change in the value of the
consumption vector x is (p+ p)xpx = (p)x, the requisite
money income change is given by mS = px. This mS can be
either positive or negative depending on p. The Slutsky compensation
is positive, or a subsidy, if x becomes too expensive to buy. The
Slutsky compensation is negative if x becomes cheaper, and is like an
income tax that takes away |px| from the consumer.
The standard definition ofSlutsky compensationis
mS = px
where p = (p + p) p. If a price change occurs only in the price of
the i-th commodity, p = (0, , pi, , 0). Consequently, for an
isolated price change, the Slutsky Compensation is
mS = px = (pi)(xi)
In the diagram below, p = (p1, 0) and ms = (p1)x1 .
x2
m+ mS
0 x1
x
m
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SLUTSKY SUBSTITUTION EFFECT
Let the price vector changes from p to p+ p, and the consumer
be compensated by mS = px. Let the consumers budget change
from B(p, m) to the compensated budget B(p+ p, m+ mS)2. By
construction,x is in B(p+ p, m+ mS) as well as in B(p, m). The
consumer, given the compensated budget, however, need not choose x
again. Let xS be the consumers choice in B(p+ p, m+ mS). If xS
x,xS is revealed preferred tox. We call xS the Slutsky-compensated
demand. The Slutsky substitutioneffect refers to xS xS x as it
measures how a consumer substitutes xS for x in response to a price
change. This is illustrated in the diagram below.
The above diagram illustrates that in response to a ceteris paribus
rise in the price of commodity 1, the consumers demand for commodity 1
2Since m+ mS = (p+ p)x, the compensated budget B(p+ p, m+ mS) can beexpressed as B(p+ p, (p+ p)x). Because x = x(p, m), the compensated budget setcan also be expressed as B(p+ p, (p+ p)x(p, m)).
m+ mS
0
x2
x1
xS
x
m
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decreases given the compensated budget. That is, p1xS
1 = (+) () < 0
in the diagram. A (relatively) higher p1 induces the consumer to
purchase less of x1 and more of x2 in the two-commodity world.
Generally, a ceteris paribuschange in the price vector should induce a
less consumption of a commodity that has become relatively more
expensive. This negative relationship between the price change and the
Slutsky compensated demand can be generalized to pxS 0 as a
major consequence of WARP. If the consumers choice is unique in every
budget set B(p, m), the substitution effect under WARP is strictly
negative p xS < 0 if xS 0.
LEMMA(Law of Slutsky Compensated Demand or Law of Negative
Slutsky Substitution Effect): Assume that a consumer obeys WARP*,
income-exhaustive, and that a consumers choice is unique in any given
budget set B(p, m). Then, pxS 0, and the strict inequality holds
whenever xS 0.
The Slutsky compensation guarantees that xand xS are both in
B(p+ p, m+ mS). The scenario is that in B(p+ p, m+ mS) the
consumer chose xS rather than x, establishing the revealed preference
of xS over x. To consider first the case of xS 0, let us suppose xS
x. In the initial budget set B(p, m),however, the consumer chooses
x, not xS. Thus, by invoking WARP* given xS x, we conclude that
xSwas outside B(p, m). If xS were available in B(p, m), the consumer
would have again chosen xS over x. Hence,pxS > m. If the consumer
is income exhaustive,pxS > m =px.
Summing up, we have two inequalities3:
3The first inequality follows from (p+ p)xS = m+ m = (p+ p)x, which saysthat xS is purchasable with m+ mS and that x is on the budget line (p+p)x = m+ mS by the construction of Slutsky compensation. Since px = m, the second
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(p+ p)xS = (p+ p)x
pxS > px.
These two inequalities can be re-written as
(p+ p)xS = (p+ p)x
pxS < px.
Adding up each side, we obtain
pxS < px
which can then be rearranged as
p(xSx) < 0 .
By definition, xS = xSx. Thus,
pxS < 0 .
The only case not yet covered is when xS = x. In that case xS = x,
however, xS = 0, and then pxS = 0 follows immediately.
The above lemma is derived under the dual assumption of income
exhaustion and uniqueness of consumption choice. It is instructive to
understand modifications that becomes due once either assumption is
relaxed. Suppose that income exhaustion holds but the consumers choice
is a genuine correspondence, that is, d(p, m) is multi-valued or non
singleton set. WARP now says that with respect to B(p, m), either xs
was not affordable or xs (as well as x) also belonged to d(p, m). To the
extent that all choices must be income-exhausting, if xs is in d(p, m), pxs
= m must hold. By WARP, if xs is not in d(p, m),xs must be
unaffordable, i.e.,pxS > m. Hence, WARP implies pxs m. Since by
income exhaustion m = px, the second inequality in the above lemmaneed only be replaced by a weak inequality as pxS px. Consequently,
the lemma holds with weak inequality, pxS 0 even when xS 0.
inequality says that xS was not purchasable when x was, namely xS was outside thebudget set B(p, m).
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Corollary: Assume that a consumers demand correspondence is income-
exhaustive and obeys WARP. Then, pxS 0 where xS is the
differenece between any two vectors x in d(p, m) and xS in d(p+ p,
m+ mS).
A demand correspondence allows the possibility of both xS and
x belonging to d(p, m) as well as to d(p+ p, m+ mS). But, income
exhaustion forces the desired inequalities we need to establish the
nonpositive Slutsky substitution effect. If the consumers choice is not
income exhaustive, the case of demand correspondence may result in a
choice of xS such that pxS < pxo, thus invalidating the inequality
required for the proof. Readers are asked to provide examples in which
Slutsky substitution effects fail to be nonpositive due to income non-
exhaustion for a demand correspondence.
The following discourse illustrates the importance of income
exhaustion even when the consumers demand choice is summarized by a
demand function. Let us maintain the assumption that the consumers
choice is unique for each B(p, m), but not necessarily income-exhaustive.
If the budget is not exhausted, a slight change in the price vector can keep
x affordable with the same money income. When px < m, for a
sufficiently small p, we can guarantee (p+ p)x < m. Since the
consumer can buy the original bundle after the price change, there is no
need to give a Slutsky compensation. The new budget set is B(p+ p, m)
to which x again belongs. If xSdifferent from x is chosen, then by
WARP*, we must conclude that xS
was unaffordable previously, that is,px
S> m and m > px. Thus, the second inequality in the above lemma
is met as pxS > px. But, there is no guarantee that the first inequality is
also met. Because income exhaustion is not assumed, it is quite possible
that m (p+ p)xS > (p+ p)x, the derivation of the desired
inequality is not assured. Readers, try to provide two-commodity
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examples in which the substitution effect can be positive if no such
compensation mechanism is adopted.
To carry out the Slutsky compensation analysis in a non income-
exhaustive case, therefore, we still insist that the consumer be given just
enough income to buy the initial bundle. Such a budget adjustment is
consistent with the approach we have taken for the case of a price change
that induced an increase in the purchasing power of money income. We
shall require that the Slutsky compensated budget set be B(p+ p,mS)
where mS = (p+ p)x, or more compactly expressed,
B(p+ p,(p+ p)x).
The implied Slutsky compensation adjustment mS in this case is less
than p x: mS = mS m = (p+ p)x m = (px m) + p x
< p x because px m < 0 by the assumed non-exhaustion of
income.
We summarize the above qualifications regarding the validity of
the negative Slutsky substitution effect in the following table. The entry,
(valid), for the non income-exhaustion cumdemand function case means
that the Slutsky compensation need to be modified accordingly.
D-FUNCTION D-CORRESPONDENCE
income exhaustion valid valid
income non exhaustion (valid) not valid
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REMARK1: Consider two budget situations B(p, m) and B(p, m),
and let x(p, m) be a demand vector in B(p, m). As (p, m) changes to (p,
m), we define the generalized Slutsky compensation as
mS = px(p, m) m,
and the Slutsky compesated demand becomes
x(p, m + mS) = x(p,px(p, m))
where m + mS = px(p, m). The Slutsky substitution effect is
xS = x(p,px(p, m)) x(p, m)
We then have the Law of Slutksy compensated demand
(p p)[x(p,px(p, m)) x(p, m)] 0,
a restatement of pxS 0 where p = p p.
REMARK 2: Let B(p, m) and B(p, m) be two distinct budget
situations, and let x(p, m) be a demand vector in B(p, m) and x(p, m) a
demand vector in B(p, m). Suppose that px(p, m) = m. It means that
the (new) income m is just adequate enough to purchase the intial bundle
x(p, m) under the (new) price vector p. We can thus regard B(p, m) as
the Slutsky-compensated budget set when the price changes from p to p.
The implied Slutsky compensation is mS = m m because it is equal
to px(p, m) m. Hence,x(p, m) = x(p,px(p, m)) is the Slutsky
compensated demand and the substitution effect is xS = x(p, m)
x(p, m). We can express the Law of Slutsky compensated demand as (p
p) [x(p, m) x(p, m)] 0.
Theorem: Assume that a consumers demand correspondence is income-
exhaustive and obeys pxS 0 for all price changes that are
accompanied by Slutsky compensation. Then, the demand
correspondence obeys WARP.
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The logic behind this theorem is another important lemma that WARP
holds for demand correspondence if and only if WARP holds for Slutsky
compensated demand correspondence. A Slutsky compensated demand
correspondence is defined by d(p,px) or d(p, m) subject to m = px.
Thus, xS = x(p+ p, (p+ p)x) x(p,px) which is same as x(p+
p, m+mS) x(p, m) because mS = px and m = px.
INCOME EFFECT
Whenever a price vector changes, with the money income m
remaining constant, the consumers purchasing power also changes. To
capture the substitution effect, we need to neutralize the effect of
purchasing power change as a result of a price change. The Slutsky
compensation is designed just to achieve this end. It is also useful to
know the pure effect of a purchasing power change. The pure income
effect is an effect of a change in money income while the price vector
remains constant. In terms of the budget set geometry, because prices are
held fixed, the budget lines make parallel shifts as money income changes.
An increase in money income shifts the budget line outward, while adecrease in money income shifts the budget line inward.
Income effectis defined as a change in consumption (x) in
response to a ceteris paribusmoney-income change (m). Component
wise we can express it as xi/m. The diagram below indicates possible
hypothetical changes in the consumers demand vector as income
increases4.
4A more detailed exposition of the income effect is given in the appendix of thepreceding lecture note Budget Sets, Demand and Revealed Preference.
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SLUTSKY DECOMPOSITION
As a starting point, let a consumer choose x in the initial budget
set B(p, m) = {x | px m}. In the new budget set B(p+ p, m) = {x |
(p+ p)x m}, let the consumer chooses xG. Our task is to decompose
the consumers move from x to xG into the income effectand
substitution effect. To isolate the substitution effect, we need to neutralize
the effect of purchasing power change due to the price change. That is,
we must adjust the consumers income in the manner of Slutsky
compensation. Having benchmarked the Slutsky substitution vector under
a new price vector, we then identify the income effect as we restore the
consumers income to m while maintaining the new price vector.
SLUTSKY DECOMPOSITION:
gross price effect = substitution effect + income effect.
To facilitate the ease of diagrammatic illustration of the Slutsky
decomposition, we again normalize p2 = 1 and change only p1. In what
follows in diagrams, we consider only the case of p1 increasing, which is
essentially identical to the case in which the relative price p1/p2
increases. In all diagrams in this section,p changes to p+ p = (p1+
x
x2
x1
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p1,p2) where p1 > 0 and p2 = 0. Because p1 has increased, with
the same money income m, the consumers purchasing power has
decreased. In other words, the consumer cannot buy back xo under the
new price vector unless the money income is increased. The Slutsky
compensation mS is the minimum monetary subsidy that enables the
consumer to buy the same initial bundle xo under a new price vector.
Step 1: The Slutsky substitution term measures thepureeffect of
price on the consumers consumption substitution as the consumerspurchasing power itself is unaffected by the Slutsky compensation. The
movement from x to xS in the diagram measures the consumers
response to a price change p while simultaneously compensated by
mS. The vector difference, or the located vector, xS = xSx, is the
Slutsky substitution effect.
0
x2
x1
xG
x
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Step 2: The movement from xS to xG in the diagram measures
the consumers response as mS is taken away while price vector is
maintained at p + p. The vector difference, or the located vector, xI =
xG x
S is the income effect, or the pure effect of the ceteris paribus
income reduction.
The Slutsky decomposition is the vector equation given by
xG x
= (xG x
S) + (xS x)
which we rearrange as
x1
x2
0
xG
x
xs
0 x1
x2
xG
x
xs
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xG x
= (xS x) + (xG xS)
Although the diagrammatic exposition has been limited to an isolated
price change in p1, exactly the same decomposition works for any general
price change. Any p will induce the same vector formula as above.
Compactly stated, the Slutsky decomposition is
The Slutsky decomposition is often expressed in terms of a ceteris
paribuschange in the i-th commodity price. This means that we measure
the effect of an isolated pi change on consumption decisions. To
express such a ceteris paribuseffect, we can divide the above vector
formula by pi, so that the Slutsky decomposition becomes
i
I
i
S
i
G
p
x
p
x
p
x
+
=
.
0 x1
x2
xG
x
xs
xG
xS
xG
xI x
S
x
x x xG S I
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It is desirable to convert the income effect term from xI/pi into
the form expressed in terms of an income change. To repeat, any price
change pi induces a change in the consumers purchasing power of a
given money income. To compensate for this change in purchasing power
of money income, the consumer was given (pi)xi = mS.The income
effect measured in the Slutsky Equation is the effect of taking away mS
from the consumer. The vector xI = xGxS traces the consumption
change as the consumer gives up |mS|. Thus, the correct sign of Slutsky
compensation is given by mS = (pi)xi or pi = (xi)/ mS. We
thus obtain
S
I
ii
S
i
G
m
xx
p
x
p
x
=
.
This is the vector version of Slutsky Equation. Note that these are vectors
in n-dimensions. Component wise, the Slutsky Equation can be expressed
component-wise as
x
p
x
px
x
m
j
G
i
j
S
ii
j
I
S=
Scomp
(i,j = 1, , n) .
where Scomp is a shorthand to say that it is a comparative static after the
Slutsky compensation.
The analytic geometry of our decomposition method is transparent,
and the derived formula is valid for finite as well infinitesimal changes.
When a price change is infinitesimal, we can replace by , and obtain
S
Ij
ii
Sj
i
Gj
m
xx
p
x
p
x
=
o
Scomp
(i,j = 1, , n).
The derivative version, which is the about the only way almost all
textbooks express the Slutsky equation, is stated without superscripts G,S
and I.
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m
xx
p
x
p
xjjj
iii
=
o
Scomp
(i,j = 1, , n).
In the next section, we demonstrate this version by taking derivatives of
an identity equation that translates a Slutsky compensated demand as a
Marshallian demand.
CALCULUS DERIVATION OF THE SLUTSKY EQUATION
The foregoing derivation of Slutsky equation has relied on Slutsky
compensation and vector additions. The primary use of Slutsky
compensation is to let a consumer have just enough income to buy back
the initial vector after a price change. That is, the Slutsky compensated
demand xS(p,xo) and the Marshallian demand x(p, m) are identical as
long as we maintain m = pxo for all price vectors for the Marshallian
demand.
xS(p,xo) x(p, m) subject to m = pxo,
that is,
xS(p,xo) x(p,pxo).
Differentiate both sides with respect to pi to derive for each j = 1, , n,
i
j
i
j
i
Sj
p
m
m
x
p
x
p
x
+
=
where o
o
iii
xp
px
p
m=
=
)(.
Thus, we obtain
m
xx
p
x
p
x ji
i
j
i
Sj
+
=
o
which we rearrange as
m
xx
p
x
p
x ji
i
Sj
i
j
=
o
That is,
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),(),(),( mpm
xxxp
p
xmp
p
x ji
i
Sj
i
j
=
oo where m = pxo.
At an initial price-income pair (po,xo), we have xS(p,xo) = xo = x(po,
mo) as well as m = pxo. If we evaluate the Slutsky Equation at (po,xo),
we get
),(),(),(
=
mp
m
xxxp
p
xmp
p
x ji
i
Sj
i
j oo
Starting with an arbitrary price-income pair, therefore, we have
),(),(),( mpm
xxxp
p
xmp
p
x ji
i
Sj
i
j
=
SLUTSKY EQUATION WITH AN INITIAL ENDOWMENT VECTOR
Suppose that a consumer is endowed with = (1,, i,, n).
This consumer can sell all of its initial endowment vector for p = (p1,,
pi,,pn) to obtain money income p , which is the market valuation of
the consumers initial endowment. In addition to an initial commodity
endowment , the consumer may also have a separate money income m.
The consumers total income is p + m, and the consumers budget set
becomes
B(p, ) = {x| p x p + m} = {x| p(x ) m}.
Under the proviso of budget exhaustion, the consumer buys a commodity
vector on the budget hyperplane p(x ) = m.
Once again, by the same vector decomposition as before we obtain
x x xG S I
. It is imperative that we compute the correct
amount of Slutsky compensation for this consumer to buy back exactly the
initial bundle x. Since we assume budget exhaustion, after the price
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change the consumer can afford x if and only if the compensation mS
meets the budget equality: (p + p)(x ) = m + mS. Thus5,
mS = p (x ).
When the price change entails only a ceteris paribuschange in pi, the
compensation needed to effect the Slutsky substitution demand becomes
mS = (pi) (xi i).
To effect the income effect that moves the consumer from xS to xG, we
need to take away this take away mS from the consumer. We can then
substitute 1/pi = (xi i)/mS in the Slutsky decomposition under
the ceteris paribusprice change: i
I
i
S
i
G
p
x
p
x
p
x
+
=
, and we obtain
S
I
iii
S
i
G
m
xx
p
x
p
x
=
)( .
Component-wise, the derived Slutsky decomposition is
S
Ij
iii
Sj
i
Gj
m
xx
p
x
p
x
=
)(
comp
(j = 1, , i, , n) .
The calculus derivation of the above Slutsky Equation is asfollows. The Slutsky compensated demand xS(p,xo) with initial
endowment is then same as the Marshallian demand x(p, m) subject to m
= p(xo ). That is,
xS(p,xo) x(p, m) subject to m = p(xo )
Thus, setting the Slutsky compensated demand as
xS(p,xo) x(p,p(xo )),
we can differentiate both sides of the equation with respect to pi and
obtain for each j = 1, , n,
i
j
i
j
i
Sj
p
m
m
x
p
x
p
x
+
=
where ii
ii
xp
px
p
m=
=
oo )(
.
5By the assumed budget exhaustion, the consumer buys xin the initial budget conditionby meeting p(x ) = m.
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OSU Econ 804 Slutsky EquationsFall 93/94/95/01/02/03 - 20 - Hajime Miyazak
SLUTSKY2003Draft.do
The upshot is that
m
xx
p
x
p
x jii
i
j
i
Sj
+
=
)( o
which we can rearrange as
m
xx
p
x
p
x jii
i
Sj
i
j
=
)( o .
In a full expression,
),()(),(),( mpm
xxxp
p
xmp
p
x jii
i
Sj
i
j
=
oo
where m = p(xo ).
If we let (p, m) be an initial price-income pair as (p, m), we
have xS(p,xo) = x = x(p, m) as well as m = p(xo ). More
concisely stated, the initial condition is
xS(p,xo) = x = x(p,p(xo )).
Slutsky equation evaluated at (p, m) is
),()(),(),( ooooooo mpm
xxxp
p
xmp
p
xjii
i
S
j
i
j
=
.
Given an arbitrary initial condition (p, m), we can thus express the Slutsky
equation as
),()(),(),( mpm
xxxp
p
xmp
p
x jii
i
Sj
i
j
=
where m = p(x ).
The set up of initial endowment is an indispensable component of
a general equilibrium analysis, especially of a pure exchange economy.
The Edgeworth exchange economy cannot be analyzed without having
consumers with initial endowment vectors. But, the setup of initially
endowed consumers is a useful framework in many applied economic
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SLUTSKY2003Draft.do
EXERCISES
The diagrammatic exposition has so far assumed that p2 = 1 and
considered only cases of price increase in p1. There are a number of
exercises that are straightforward, but that may facilitate the understanding
of, and the potential applications of, the Slutsky decomposition. As
immediate exercises, try to duplicate the diagrammatic analysis for p1
decreases, i.e., p1 < 0.
Also, it will make a good practice to reconstruct the analysis while fixing
p1 = 1 and changing p2. This exercise pivots the budget line around
(m, 0) on the horizontal axis. As such, it has applications to leisure-
income choice, and can be generalized to the case of a consumer with an
initial endowment vector. Readers, try to depict the case of p2 increase
as well as p2 decrease while p1 is fixed (or normalized) at 1.
0
x2
x1
x
Gx
x
S
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OSU Econ 804 Slutsky EquationsFall 93/94/95/01/02/03 - 23 - Hajime Miyazak
SLUTSKY2003Draft.do
Case: p2 > 0
Case: p2 < 0
Finally, readers are encouraged to analyze the case of both prices
changing. It amounts to the case of a relative price change
0
x
m/p2
m
xG
x
S
(m+mS)/p2
m/ (p2+ p)
m+ mS
0
x
m/p2
m
xS
(m+mS)/p2
m/ (p2+ p2)
m + mS
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OSU Econ 804 Slutsky EquationsFall 93/94/95/01/02/03 - 24 - Hajime Miyazak
Case: p1/p2 decreases
Case: p1/p2 increases
0
x
m/p2
m/p1
x
G
x
S
0
x
m/p2
m/p1
xS
xG