slutsky compensation

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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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OSU Econ 804 Slutsky EquationsFall 93/94/95/01/02/03 - 2 - Hajime Miyazak

SLUTSKY2003Draft.do

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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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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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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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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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

slutsky compensation

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