The utility idea has had a long history in economics, especially in the explanation of demand and in welfare economics. In a comprehensive survey and critique of the Slutsky theory and the pattern to which it belongs in the economic context, S. N. Afriat offers a resolution of questions central to its main idea, including sufficient conditions as well.
Originally published in 1980.
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Demand Functions and the Slutsky Matrix
By Sydney N. Afriat
PRINCETON UNIVERSITY PRESSCopyright © 1980 Princeton University Press
All rights reserved.
Slutsky's Problem and the Coefficients
1.1. Demand functions
With Ω as the nonnegative numbers, Ωn, Ωn are the nonnegative row and column vectors with n elements. Then any p [member of] Ωn, x [member of] Ωn have a product px [member of] Ω, expressing the value of any quantities x of some n goods at the prices p. There is a symmetrical relation, or duality, between Ωn and Ωn in that each represents the space of nonnegative homogeneous linear functions defined in the other. They are distinguished as price and quantity spaces, respectively.
Given any prices p [member of] Ωn and an amount of money M [member of] Ω, the bundles of goods x [member of] Ωn that cost that money at those prices are those under the budget constraint px = M. Provided M >0, the same constraint can be stated ux = 1, where u [member of] Ωn given by u = M-1p is the budget vector specifying the constraint. Such vectors describe the budget space that is a replica of the price space Ωn.
With any bundle x and budget u, x is said to be within or on u according as
ux ≤ 1 or ux = l.
With u = M-1p, these relations signify that the cost of x at prices p is at most, or equals M.
Any function F of prices p and money M that depends on these just to the extent of the budget constraint they determine, that is on u = M-1p, is a function just of the ratios and must have the property
F(p, M) = F(αp, αM)(α > 0).
By taking α = M-1, one finds this gives
F(p, M) = f(u),
f(u) = F(u, 1).
Such is the case for a demand function F for which any x = F(p, M) determined by the demand function is also determined as the unique maximum of some function under the constraint px = M. This property is usually, as now, incorporated into the definition of a demand function. It can be understood also as expressing the "absence of monetary illusion," meaning nothing significant is changed if all money amounts, including prices, change in proportion, as they would, for instance, when just the unit of account is changed.
Accordingly, a demand function is defined as a function x = F(p, M) with arguments p [member of] Ωn, M [member of] Ω and values x [member of] Ωn having the properties
(i) pF(p, M) = M
(ii) F(αp, αM) = F(p, M) for α [member of] Ω.
Such a function is understood to determine quantities x purchased with an expenditure M when the prices are p. Property (i) is the budget identity, requiring the cost px to coincide with the expenditure M, and (ii) is the absence of monetary illusion, by which if prices and expenditure change in the same ratio so the budget constraint they determine remains unchanged, and then so are the quantities chosen unchanged. From such a function F is derived a function
f(u) = F(u, 1) (1.1)
that, according to (i), has the property
(iii) uf(u) = 1.
Then, according to (ii), F is recovered from f as
F(p, M) = f(M-1 p). (1.2)
To distinguish the two types of demand function F and f, any function F with properties (i) and (ii) defines a standard demand function, and any function f with property (iii) is a normal demand function. For any standard function F, its normal equivalent is the normal function f derived from it by (1.1). Also, any normal function f has a standard equivalent function, the standard function F derived from it by (1.2).
The normal functions, which give an alternative specification of the standard functions since they are in a one-to-one correspondence with them, are simpler. Many developments here will depend on dealing directly with a normal function rather than the corresponding standard function. But it needs to be recognized, nevertheless, that usual discussions of demand functions have been in terms of a standard function F rather than the alternative normal f.
In view now as the subject of all discussion is a function x=f(u) with arguments u [member of] Ωn and values x [member of] Ωn subject to the constraint ux = 1. In other words, a function is considered that determines a single commodity bundle x under every expenditure budget. At the same time x can also be considered a function F of p [member of] Ωn, M [member of] Ω that for M > 0 is given by (1.2). While this definition of demand functions is often suitable in the Slutsky theory, here the center of considerations, it is necessary to understand the definition more narrowly as involving only positive budget vectors, and subject to the requirement that the values all be positive commodity bundles. In other words, there is a function x = f(u) with arguments u > 0 and values x > 0. This redefinition is now simply accomplished by understanding that everywhere Ω denotes the positive numbers instead of the nonnegative numbers as in the broader context. While it is possible to extend features of the Slutsky theory to accommodate zero prices and quantities, the resulting intricacies do not belong to the original subject, nor do they suit the methods and material of this book. The foregoing definitions are gathered together as follows.
Definition 1.3.Let Ωnand Ωn be the spaces of positive row and column vectors with n elements, so that any p [member of] Ωn, x [member of] Ωn determine px [member of] Ω.
A standard demand function is any function
x = F(p, M),
with arguments p [member of] Ωn, M [member of] Ω and value x [member of] Ωn, having the properties
(i) pF(p, M) [equivalent to] M
(ii) F(p, M) [equivalent to] F(M-1p, 1).
A normal demand function is any function
x = f(u),
with argument u [member of] Ωn and value x [member of] Ωn, having the property
(iii) uf(u) [equivalent to] 1.
The standard and normal demand functions are in a one-to-one correspondence, where any standard F corresponds to the normal f given by
f(u) = F(u, 1)
and any normal f corresponds to the standard F given by
F(p, M) = f(M-1p).
With x given as a differentiable function of u, the elements xi have partial derivatives with respect to the elements uj, denoted
xij = [partial derivative]xi/[partial derivative]uj
and forming the elements of the derivative matrix xu. Then with u = M-1p as a function of p and M, x becomes a function of p and M. The derivatives, by the chain rule, are then given by
xp = xuu'p, xM = xuu'M, (2.1)
where u' is the transpose of u, xp being the matrix with elements [partial derivative]xi/[partial derivative]pj, xM the vector with elements [partial derivative]xi/[partial derivative]M, u'p the matrix with elements [partial derivative]uif/[partial derivative]pj and u'M the vector with elements [partial derivative]ui/[partial derivative]M. But from ui = piM-1 follows
[partial derivative]ui/[partial derivative]pj = δijM-1
δij being the Kronecker δ, for which δii = 1 and δij = 0 for i ≠ j; that is,
u'p = 1M-1, (2.2)
1 being the unit matrix. Also,
[partial derivative]ui/[partial derivative]M = -piM-2
u'M = -u'M-1. (2.3)
xp = xuM-1 (2.4)
xM = -xuu'M-1 (2.5)
The relations for derivatives that have been obtained are stated in the following theorem.
Theorem 2.6.Let a differentiable demand function be given alternatively in the standard form
(i) x = F(p, M)
and in the normal form
(ii) x = f(u),
(iii) u = M-1p.
The standard function (i) has a matrix xp of derivatives [partial derivative]xi/[partial derivative]pj and a vector xM of derivatives [partial derivative]xi/[partial derivative]M; the normal function (ii) has a matrix xu of derivatives [partial derivative]xi/[partial derivative]uj, and there is a matrix u'p and vector uM of derivatives obtained similarly from (iii). Then there are the formulae
(iv) Mup' = 1, MuM = -u
(v) xpM = xu, xMM = -xuu'.
1.3. Expansion paths
The path described by x = F(p, M) as M varies while p remains fixed is the expansion path for prices p. Because F(αp, αM) = F(p, M), any prices having the same ratios determine the same such path; in other words the path depends just on the ray through p. In terms of the function f, the expansion paths are described by f(ρ-1u) as ρ varies while u remains fixed.
For x = f(u), the expression denoted
[??] = [partial derivative]f(ρ-1u)/[partial derivative]ρ|ρ=1
will be called the expansion derivative of x, corresponding to the budget vector u. It is itself a function just of u.
[partial derivative]f(ρ-1u)/[partial derivative]ρ = f'(ρ-1u)(-u' ρ-2),
f' denoting the derivative matrix of f, by the substitution p = 1 the expression
[??] = -xuu'
is obtained. Then from Theorem 2.6, an alternative expression obtained from considering x = F(p, M) is
[??] = xMM.
Hence the next theorem follows.
Theorem 3.1.For a differentiable demand function given in the standard form x = F(p, M) (px = M) and the equivalent normal form x = f(u)(ux = 1), the formulae
[??]i = M[partial derivative]Fi(p, M)/[partial derivative]M
[??]i = [partial derivative]fi(ρ-1u)/[partial derivative]ρ|ρ=1
are identical. They define the expansion derivatives, which form the elements
[??]i = ([partial derivative]xi/[partial derivative]M)M
of the vector
[??] = xMM
identical with the vector
[??] = -xuu'
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
xu being the matrix of derivatives
xij = [partial derivative]xi/[partial derivative]uj.
Just as x has an expansion derivative considered as a function of u and then also of p and M where u = M-1p, so similarly does any other function of u. In particular, the expansion derivative of u is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
or alternatively with u = M-1p,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Generally, any function defined in the budget space, denoted t = ψ(u)(u [member of] Ωn, t [member of] Ω), has an expansion derivative, given by
[??] = tu[??],
and because, as just seen, [??] = -u, this is also
[??] = -tuu'.
Also, given a function t = φ(x) (x [member of] Ωn, t [member of] Ω) in the commodity space and a demand function x = f(u), let ψ be the function in the budget space defined by
ψ(u) = φ(f(u)),
φ(x) = ψ(u) for x = f(u).
This function ψ derived from φ by means of the demand function f can be called the function induced by f in the budget space from the function φ in the commodity space. By referring to the induced function, produced by a given differentiable demand function, one can say that also every differentiable function in the commodity space has an expansion derivative. Thus, t is a first function of x and then also a function of u, since x is a function of u. If φ and f are differentiable, then so is ψ with gradient
(i) ψu = φx, xu.
Now, as a function of u, the function t directly has an expansion derivative, given by
(iii) [??] = ψu[??]' = -ψuu',
since [??] = -u. Alternatively, with t a function of x and x a function of u, by means of the chain rule,
(iii) [??] = φx, [??] = -φx, xuu',
since [??] = -xuu', according to Theorem 3.1. But, because of (i), (ii) and (iii) give the same expression for i.
It should be remarked that if the demand function f is invertible, that is,
f(u)=f(v) only if u = v,
then the inverse demand function in the same way induces a function in the commodity space from any function in the budget space. Moreover, this is in such a way as to produce a one-to-one correspondence between functions defined in the commodity space and in the budget space, where with any corresponding pair each is the function induced from the other. This operation of deriving a function in one space from a function in the other is given a further elaboration later when it is joined with the relation between direct and indirect utility functions.
Excerpted from Demand Functions and the Slutsky Matrix by Sydney N. Afriat. Copyright © 1980 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- FrontMatter, pg. i
- Preface, pg. vii
- Contents, pg. xi
- Introduction, pg. 1
- Chapter I. Slutsky’s Problem and the Coejjicients, pg. 25
- Chapter II. Mckenzie's Method, pg. 42
- Chapter III. Symmetry and Negativity, pg. 95
- Chapter IV. Utility Contours and Profiles, pg. 111
- Chapter V. De Finetti and Convexification, pg. 142
- Chapter VI. Slutsky and Samuelson, pg. 165
- Chapter VII. Transitivity and Integrability, pg. 186
- Chapter VIII. Slutsky and Frobenius, pg. 202
- Chapter IX. Slutsky, Finally, pg. 220
- Bibliography, pg. 240
- Index, pg. 265