# Demand Functions and the Slutsky Matrix. (PSME-7)

SYDNEY N. AFRIAT
Pages: 286
https://www.jstor.org/stable/j.ctt7zv530

1. Front Matter
(pp. i-vi)
2. Preface
(pp. vii-x)
(pp. xi-2)
4. Introduction
(pp. 3-24)

The Slutsky theory is a familiar topic in economics. Also, it has its own mathematical interest. But it has no value for a heavy matter such as the empirical foundation of classical utility as Slutsky and others thought. Slutsky even imagined that the immateriality of the order of utility differentiations, instead of being merely a consequence of the continuity of second derivatives, expressed indifference to the order of consumption (whether the main course comes before the dessert orvice versa). Pareto’s notion (1897, pp. 251, 270, 539 ff.) was similar, as can be gathered from Stigler (1965, p. 124), though...

5. CHAPTER I Slutsky’s Problem and the Coefficients
(pp. 25-41)

With Ω as the nonnegative numbers, Ωn, Ωnare the nonnegative row and column vectors withnelements. Then anyp∈Ωn,x∈Ωnhave a productpx∈Ω, expressing the value of any quantitiesxof somengoods at the pricesp. There is a symmetrical relation, or duality, between Ωnand Ωnin that each represents the space of nonnegative homogeneous linear functions defined in the other. They are distinguished aspriceandquantity spaces, respectively.

Given any pricesp∈Ωn>and an amount of moneyM∈Ω, the bundles of goodsx∈Ωnthat cost that money those prices are those under...

6. CHAPTER II McKenzie’s Method
(pp. 42-94)

The connection between demand and utility appearing in the Slutsky theory is based on a relation between a demand function and a utility function. But this relation can be represented more basically in terms of a relation between a single demand and a utility function. This basic relation will now be restated in various ways, suited to different developments. Also, variants of it will be shown that put it in a further perspective.

For any utility functionϕn→Ω and a reflexive and transitive utility orderR⊂Ωn×Ωn, the functionrepresentsthe order if

xRyϕ(xϕ(y).

Obviously, any utility function represents a unique...

7. CHAPTER III Symmetry and Negativity
(pp. 95-110)

Lett=ϕ(x) be a utility function determining utilityt∈Ω as a function of commodity amountsx∈Ωn, and let the gradientg=ϕx, exist and be continuous.

Letfbe a demand function that hasϕas a utility function. This meansfandϕtogether satisfy the condition

(H*)x=f(u) . ⇔ .uy⫹1,yxϕ(y)<ϕ(x)

But, as seen in Chapter II, this implies, for alluandx=f(u),

ux=1,uy=1⇒ϕ(y)⫹ϕ(x)

and also

ux=1,ϕ(y)=ϕ(x)⇒uy⫺1.

That is,

(1.1)For all u and x=f(u),y=x is a maximum of ϕ(y)subject to the constraint ux=1,

(1.2)For all u and x=f(u),y=x is a minimum...

8. CHAPTER IV Utility Contours and Profiles
(pp. 111-141)

Autility profile, for a demand functionfwith a utility functionϕ, specifies utility level as a function of position on an expansion locus. Thus ifψis the derived budget-utility function, then, for the expansion locus associated with pricespthe profile is given by the valueψ(M-1p) considered as a function ofM. This is the utility at the pointf(M−1p) corresponding to expenditureMat the pricesp, because of the conditionϕ(x)=ψ(u) holding for alluandx=f(u).

Themarginal utility of moneyat the level of expenditureM, at pricesp, is

∂ψ(M−1P)/∂M=1/P

and...

9. CHAPTER V De Finetti and Convexification
(pp. 142-164)

Any utility functionϕrepresents a utility orderR, where

xRyϕ(x)⫺ϕ(y)

Utility functions areequivalentif they represent the same utility order. A necessary and sufficient condition for utility functionsϕ,ϕ*to be equivalent is thatϕ*=w(ϕ), wherewis an increasing function.

If a utility function belongs to a given demand function, then, it follows that any equivalent utility function does as well. It will now be seen that if two continuously differentiable utility functions belong to the same demand function, then they must be equivalent. If the demand function is given as differentiable, this equivalence is...

10. CHAPTER VI Slutsky and Samuelson
(pp. 165-185)

Aregular pathin the budget space is described by a functionu(t)∈Ωndefined forxt⫹1 with continuous derivative$\dot{u}(t)=du(t)/dt.$

The extremities

u0=u(0),u1=u(1)

are distinguished as the initial and final points. It is anintegral pathof a demand functionf(u) if$\dot{u}x=0\quad \text{for all}\quad t,$wherex=f(u) andu=u(t).

Aprojectionof a regular pathu=u(t) is any other regular pathv=v(t) of the formv=ρ−1uwhereρ>0,ρ=ρ(t) having a continuous derivative$\dot{\rho }$. The points of the paths are in perspective along rays, so the paths are projections of each other from the origin.

It will be considered now...

11. CHAPTER VII Transitivity and Integrability
(pp. 186-201)

An integral path of a demand function that projects a line segment will be called alinear integral path. Points in the budget space that are the extremities of such a path have the relation oflinear integral connection, determined with respect to the demand function. With this relation denotedL,u0Lu1asserts it holds betweenu0,u1.

Since a linear integral path projects a line segment, it must project the line segment joining its extremitiesu0,u1described byu=u(t) where

u=u0+t(u1u0) (0⫹t⫹1).

Its points, being projections of these points, have the form

ρ−1u=ρ−1(1−t)u0+ρ−1tu1,

so they lie in the...

12. CHAPTER VIII Slutsky and Frobenius
(pp. 202-219)

LetuRnbe given as a nonvanishing function ofxD, whereDRn. This defines avector fieldinD. Let it be continuously differentiable,Dbeing open and the elements ofuhaving continuous partial derivatives

uij=∂ui/∂xj

with respect to the elements ofxat any point inD.

A functionϕ(x)∈R(xNdefined in an open setNRwith partial derivatives

ϕj=∂ϕ/∂xj

is anintegralof the vector field inNif there exists a functionλ(x)∈R(xN), theintegrating factor, such that

λ≠0,ϕj=λuj

for alljand allxN.

Ifϕhas second derivatives, these will be denoted...

13. CHAPTER IX Slutsky, Finally
(pp. 220-239)

The condition for a utilityϕfunction to belong to a demand functionfis that, for any budgetuthe commodity bundlex=f(u) determined by the demand function is the unique maximum ofϕ(x) subject to the constraintux⫹1. In other words

x=f(u) . ⇔ .uy⫹1,yxϕ(y)<ϕ(x), (1.1)

which is equivalent to

x=f(u) . ⇔ .uy⫹1,ϕ(y)⫺ϕ(x)⇔y=x. (1.1)′

A consequence of this condition is

x=f(u)⇒ϕ(x)=max[ϕ(y):uy⫹1], (1.2)

showing that the function

ψ(u)=max[ϕ(y):uy⫹1] (1.3)

is defined for all budgetsuand is such that

x=f(u)⇒ϕ(x)=ψ(u). (1.4)

From the definition (1.3),

uy⫹1⇒ϕ(y)⫹ψ(u). (1.5)

Also, with (1.2) and (1.3), (1.1) becomes equivalent to...

14. Bibliography
(pp. 240-264)
15. Index
(pp. 265-269)
16. Back Matter
(pp. 270-270)