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The Index Number ProblemConstruction Theorems$

Sydney Afriat

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The Construction of Separable Utility Functions from Expenditure Data

The Construction of Separable Utility Functions from Expenditure Data

Chapter:
(p.133) 4 The Construction of Separable Utility Functions from Expenditure Data
Source:
The Index Number Problem
Author(s):

S. N. Afriat

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199670581.003.0010

Abstract and Keywords

Everything in utility theory depends on separability, either explicitly or implicitly. It is made explicit with the idea of a subgroup of goods producing a separate utility on their own which is unaffected by other goods—and by all else besides. Then it becomes a matter of the internal structure of utility. Afriat (1953) marked the decentralization effect on demand behaviour, taken up by Strotz (1957) with a departure amended by Gorman (1959). W. W. Leontief (1957a,b) approached structure for differentiable utility functions by means of conditions on the derivatives. Goldman and Uzawa (1964) deal with the utility belonging to a demand function by reference to the Slutsky coefficients. Further developments not dependent on differentiability have been shown by W. M. Gorman (1968), who has dealt most extensively with separability in this and several articles ranging from (1959) to several of 1968, 1970 and later, including (1987). The purpose in 1969 had been to deal with the matter having reference to a scheme of demand data and compatible utilities, by the finite methods. A main part of the treatment applies to the neglected matter of function construction based on separability models.

Keywords:   consumption, demand, standard of living, Afriat's Theorem, utility structure, Gorman

1969 UNC Chapel Hill USA

EXP version 1999

Word & MathType 2006

University of Siena

The original of this paper, issued from UNC Chapel Hill in 1969, was submitted for publication in the International Economic Review as a successor to the paper Afriat (1967) (for the now familiar “Afriat‘s Theorem” here being developed further for structured utility). The kind editor, Phebus Dhrymes, informed me they were willing to publish it, and only asked that I make the effort to make it more understandable.

Those were early days of mathematical wordprocessing and any call for a revision was not to be taken lightly. In any case I could not see how to respond to his request. The earlier paper, which makes the starting point for the approach, was not then assimilated, as it is now, even with a textbook representation and the name “Afriat’s Theorem”. Now by mention of that Theorem it should be easy to tell anyone what this paper is about, but it is a surprise that it could not be known earlier, even just by asking Gorman, who made structured utility his dedication after it was born in the DAE coffee break mentioned below.

The paper got put aside though not abandoned, and in 1999 the math wordprocessor EXP (invented by Pembroke Cambridge classmate Walter Smith) was put in service for doing something further about it. But the paper disappeared from view until recovered again in a recent excavation of neglected works.

A recurrent misleading report from a prominent dedicated professional is that the paper was not resubmitted to IER, but was published elsewhere, despite that it remained unpublished, even up to its provision herewith.

It fits to tell of the start of our subject. In 1953, my first year at the Department of Applied Economics (DAE), Cambridge, at a coffee break out in the open on (p.134) a sunny morning on benches round a large table, I had just been introduced to utility and all that and was talking idly to a colleague on my left and remarked about a subgroup of goods producing a separate utility on their own as if together making a single good.

The stranger on my right, Strotz, a guest from the USA, turned to me and with odd intensity asked that I repeat exactly what I had just said. I did that more or less.

Later I learnt from Gorman about Strotz’s absurd paper that he had issued without blame to myself. I reported to Gorman about the encounter and he also in his response to Strotz spared me from connection with nonsense that initiated a voluminous stream about “structure of utility” to which he himself contributed, becoming a main authority.

After my invisible contribution for the invention of the subject, in 1953, I can mention another, nothing else but this paper itself, 1969, up to this point still invisible even though now it seems it may be no longer.

Introduction

The universe cannot be dealt with in one stroke and so a bit has to be broken off and treated as if the rest did not matter. Statements are offered as true on condition of “other things being equal”, as if one should know when they are. Ricardo pioneered that way of arguing in economics.

Another escape from the influence of other things is “separability”, where things do not have to be equal because they do not matter. When it is not stated explicitly this assumption is often implicitly understood, and then regarded as being not worth mentioning.

Everything in utility theory depends on separability, either explicitly or implicitly. It is made explicit with the idea of a subgroup of goods producing a separate utility on their own which is unaffected by other goods—and by all else besides. Then it becomes a matter of the internal structure of utility. Afriat (1953) marked the decentralization effect on demand behaviour, taken up by Strotz (1957) with a departure amended by Gorman (1959).

W. W. Leontief (1957a, 1957b) approached structure for differentiable utility functions by means of conditions on the derivatives. Goldman and Uzawa (1964) deal with the utility belonging to a demand function by reference to the Slutsky coefficients. Further developments not dependent on differentiability have been shown by W. M. Gorman (1968a), who has dealt most extensively with separability in this and several articles ranging from (1959) to several of 1968, 1970 and later, including (1987).

The purpose in the 1969 submission to IER had been to deal with the matter having reference to a scheme of demand data and compatible utilities, by the (p.135) finite methods. A main part of the treatment applies to the neglected matter of function construction based on separability models. Here is early ‘non-parametric’, then hard to understand, since then commonplace.

1 Expenditure data

With Ω as the non-negative numbers, and a number n of goods, B = Ωn is the budget space (non-negative row vectors, order n) and C = Ωn the commodity space (column vectors). Then any pB , xC determine pxΩ for the value of the commodity bundle x at the prices p.

A demand element is any (p, x) ∈ B × C for which px > 0, showing a commodity bundle x demanded at prices p, the expenditure being M = px. Then

u=M1p,

is the associated budget vector. The demand element (u, x) is such that ux = 1, so it is a normal demand element, in this case the normalization of (p, x).

The goods may be distinguished in two classes, a 0-class of some n0 goods, and a 1-class of n1 goods, where n0 + n1 = n. Then

x=[x0x1]

is the partition of a vector x of quantities of the n goods into component vectors xi (i = 0, 1) for the goods in classes 0, 1. These lie in commodity spaces Ci=Ωni . The price vector has a conformable partition

p=[p0p1],

and the budget vector is

u=[u0u1],

with components

ui=M1pi(i=0,1)

in the budget spaces Bi=Ωni. Then we have

px=p0x0+p1x1,

that is,

M=M0+M1,
(p.136)

where Mi = pixi is the expenditure on goods in class i (i = 0, 1), and M is the total expenditure on all goods. The expenditure shares for the two classes are

σi=Mi/M=pixi/px=uixi,

so

σ0+σ1=u0x0+u1x1=ux=1

and

Mi1pi=Mi1MM1pi=σi1ui,

consequently

(pixi)1pi=(uixi)1ui(i=0,1)
1.1

A demand correspondence is any DB × C such that

pDxpx>0.

For the case of a normal demand correspondence EB × C,

uExux=1,

and this is the normalization of D if its elements are normalizations of the elements of D.

Consider a finite demand correspondence D, say with k elements, so these may be listed as

(pr,xr)B×C(r=1,,k),

with budget vectors

ur=(prxr)1pr.

With the partition of the goods into classes any term that, with the class index i, refers to a single demand (p, x) may now occur with the additional index r.

The cross-coefficient from r to s is

Drs=(prxr)1pr(xsxr)=urxs1=u0rx0s+u1rx1s1

From this is obtained the chain-vector

Drijks=(Dri,Dij,,Dks).

Then there is the cyclical consistency condition

KDrijkrODrijkr=O,
(p.137)

which denies the possibility of a seminegative cycle,

KDrijkrO.

This simplifies the stricter condition of Houthakker, distinguished as strict cyclical consistency,

K*DrijkrOxr=xi=

usual for dealing with a single-valued demand function. A part of this condition is

DrsrOxr=xs,

equivalently,

prxsprxr˄xrxspsxr>psxs,

which is Samuelson’s condition (the “Weak Axiom”, a part of Houthakker’s “Strong Axiom”) put in its usual form.

Similarly for goods confined to the classes i = 0, 1

Dirs=(pirxir)1pir(xisxir)=σir1uirxis1

Hence

Drs=(u0rx0sσ0r)+(u1rx1sσ1r)=σ0rD0rs+σ1rD1rs

and so

Drs=σ0rD0rs+σ1rD1rs
1.2

A multiplier-level solution for the given demands is given by any positive numbers (λr, ɸr) which are a solution of the system of homogeneous linear inequalities

λrDrsφsφr,

equivalently,

(ML)Drsφsφrλr0.
1.3

The existence of such multipliers and levels defines the condition of multiplier-level consistency. There is the theorem that multiplier-level consistency is equivalent cyclical consistency, that is,

MLK.
(p.138)

Similarly with restriction of goods to class i = 0, 1 there is the system

(MLi)Dirsφisφirλir0
1.4
to determine multipliers and levels (λir, ɸir).

2 Gossen’s Law

A utility function is any ɸ : CΩ, and a utility order is any RC × C which is reflexive, xRx, and transitive, xRyRzxRz. A utility function ɸ represents the utility order R given by

xRyφ(x)φ(y).

A utility order so representable by a numerical function is a special case which can be distinguished as numerical. Such an order is necessarily complete, ~ xRyyRx. Usually we deal with a general utility R, partitioned by its symmetric and antisymmetric parts,

E=RR,P=RR¯,

for which

EP=O,EP=R.

Because R is an order, E is an equivalence, and P is a strict order, irreflexive and transitive. With R as a preference relation, E expresses indifference, and P strict preference.

For a normal demand element (u, x) and utility order R, consider the two conditions

Huy1xRy,HyRxuy1.

The first expresses that, with the criterion provided by R, the demand of x under the budget u is cost-effective, x being as good as any y attainable at no greater cost. The second condition shows the demand to be cost-efficient, any y which is as good as x costing at least as much. The combination

HHH

defines compatibility between the demand and the utility.

The further condition

H*uy1,yxxPy
(p.139)

defines strict compatibility between a normal demand (u, x) and a strict order P, or an order R with strict part P, or a utility function ɸ that represents R. It is immediate that

H*H,

and in general the strict compatibility H* is also more restrictive, unless applied to a correspondence already restricted to be a function.

The conditions H′ and H″ are in general independent. However, conditions on R can produce a relation between them. For instance,

(i)if the sets xRare closed, then HH.

Also,

(ii)if y>xxRy,then  HH

the hypothesis here expressing that a smaller bundle cannot be as good as a larger one, or insatiability. Hence, in particular, if R is represented by a continuous semi-increasing function, the conditions become equivalent,

HH.

But it is in any case most simple and satisfactory to deal with both of these conditions, treating them as independent, with no prior assumptions at all about R.

Let ɸ(x) be a classical utility function, non-decreasing and concave, to be compatible with any given demand (p, x) with normalization (u, x), or now with all the elements of the given finite demand correspondence. Since the function is concave, by the support theorem for concave functions, at any point xr it has a support function

φr(x)=φr+gr(xxr),

for which

φ(x)φr(x)for all x,φ(xr)=φr(xr),

in which case ɸr = ɸ(xr), and gro since ɸ is non-decreasing.

By the differentiability theorem for concave functions, the support gradient gr is unique if and only if ɸ is differentiable at xr, with differential gradient g(xr), and then the differential and support gradients coincide, g(xr) = gr. But generally the support gradient gr is indeterminate in the closed convex set of support gradients of ɸ at xr.

Gossen’s Law, that preference and price directions coincide in equilibrium, would require

gr=λrurfor some λr,
(p.140)

where now

λr=grxrsince  urxr=1.

This can apply when ɸ is differentiable at xr, and at an interior point where xr > o. It simply provides first order Lagrange conditions, in any case only of local significance, and for an ambiguous stationarity, possibly a maximum or a minimum or neither. Otherwise the required condition, as comes also from Kuhn-Tucker theory, is that

grλrur,grxr=λr,

which assures that

(H)φr=max{φ(x):urx1},

which represents maximum utility for the cost, or cost-effectiveness H′, now instead an unambiguous global maximum. For then

gr(xxr)λr(urx1)

so that

urx1gr(xxr)0φ(x)φr,

while

urxr=1,φ(xr)=φr.

Moreover, with λr > 0, we have

urx<1gr(xxr)<0φ(x)<φr,

which shows that

φ(x)φrurx1,

and hence that

(H)1=min{urx:φ(x)φr},

which represents minimum cost for the utility, or cost-efficiency H″.

3 Classical separability

Let

φ0(x0)(x0C0)and Φ(φ0,x1)(φ0Ω,x1C1)
(p.141)

be classical utility functions, so that

(H0)1)φ(x0,x1)=Φ(φ0(x0),x1)(x0C0,x1C1)
3.1

also is classical.

Let

g00r,Gr=(G0r,G1r),gr=(g0r,g1r)

be support gradients of ɸ0, Φ, ɸ at

x0r,(φ0r,x1r),(x0r,x1r),

where

φ0r=φ0(x0r),G1r=g1r.

Then

φ0(x0)φ0r+g00r(x0x0r)Φ(φ0,x1)φr+G0r(φ0φ0r)+G1r(x1x1r)φ(x0,x1)φr+g0r(x0x0r)+g1r(x1x1r)
3.2

where

φr=Φ(φ0r,x1r).

Then, by 3.1, with G0r > 0,

φ(x0,x1)φr+G0r(φ0(x0)φ0(x0r))+G1r(x1x1r)φr+G0rg00r(x0x0r)+G1r(x1x1r)
3.3

and, by comparison with 3.2,

g0r=G0rg00r,
3.4

in other words, if G0r, g00r are support gradients as stated, then so is this value for g0r determined from them.

For general equilibrium at xr = (x0r, x1r) under the budget ur = (u0r, u1r) the conditions, shown in Section 2, are

g0rλru0r,g1rλru1r,
3.5

where

g0rx0r+g1rx1r=λr.

Similarly, for a separate equilibrium at x0r under the budget σ0r1u0r,

g00rλ0rσ0r1u0r,
3.6

where

g00rx0r=λ0r.
(p.142)

By 3.2, line 1, and 3.6,

φ0(x0)φ0r+λ0rσ0r1u0r(x0x0r),

in particular, for x0 = x0s,

φ0sφ0r+λ0rσ0r1u0r(x0sx0r)=φ0r+λ0rD0rs

so that

(ML0)D0rsφ0sφ0rλ0r0.
3.7

By 3.4, 3.5 and 3.6,

G0r=λrσ0rλ0r.
3.8

From this, with 3.3 and 3.5,

φ(x0,x1)φr+λrσ0rλ0r(φ0(x0)φ0r)+λru1r(x1x1r).

In particular, for x0 = x0s, x1 = x1s,

φsφr+λrσ0rλ0r(φ0sφ0r)+λru1r(x1sx1r),

that is,

λrσ0rλ0r(φ0sφ0r)+λrσ1rD1rsφsφr,

with reference to 1.2, that is,

λrσ0rλ0r(φ0sφ0r)+λr(Drsσ0rD0rs)φsφr,

that is,

Drsφsφrλrσ0r(D0rsφ0sφ0rλ0r).
3.9

Combining this with 3.7,

(ML0)1)Drsφsφrλrσ0r(D0rsφ0sφ0rλ0r)0
3.10

which implies

(ML0,1)Drsφsφrλr0.
3.11
(p.143)

Theorem 3.1. CH0)1ML0)1

That is, classical consistency based on the stated utility model 0)1 implies the multiplier-level consistency condition associated with that model.

In the next section we have the converse.

It has been seen, as obvious, that

ML0)1ML0˄ML0,1.

To this can be added, as already familiar,

K0H0ML0CH0,
K0,1H0,1ML0,1CH0,1.

We now have

H0,1CH0,1

A question with a possibly negative answer is whether

H0)1CH0)1.

4 Construction

With any solution of the conditions ML0)1 given in 3.10, let

φr(x0,x1)=φr+λr(u0r(x0x0r)+u1r(x1x1r))
4.1

so that, by the consequence ML0,1 of 3.11,

φr(x0s,x1s)φs,φs(x0s,x1s)=φs,

that is,

minrφr(x0s,x1s)=φs.

Then with

φ(x0,x1)=minrφr(x0,x1)

we have

φ(x0s,x1s)=φs.
4.2
(p.144)

Now let

φ0r(x0)=φ0r+λ0rσ0ru0r(x0x0r)
4.3

so, by ML0 in 3.7,

φ0r(x0s)φ0s,φ0s(x0s)=φ0s,

and then with

φ0(x0)=minrφ0r(x0)

we have

φ0(x0s)=φ0s.
4.4

Now by 4.1, 4.3 and 4.4,

φr(x0,x1)=φr+λr(u0r(x0x0r)+u1r(x1x1r))φr+λr(σ0rλ0r(φ0(x0)φ0r)+u1r(x1x1r)).

Therefore let

Φr(φ0,x1)=φr+λr(σ0rλ0r(φ0φ0r)+u1r(x1x1r))
4.5

Then

φr(x0,x1)Φr(φ0(x0),x1),
4.6

while

φs(x0s,x1s)=φs=Φs(φ0s,x1s)=Φs(φ0(x0s),x1s).
4.7

A restatement of the part of our assumption ML0)1 shown in 3.9 is that

λr(σ0rλ0r(φ0φ0r)+σ1rD1rs)φsφr,

that is,

φr+λr(σ0rλ0r(φ0sφ0r)+u1r(x1sx1r))φs,
(p.145)

that is,

Φr(φ0(x0s),x1s)φs.
4.8

Hence with

Φ(φ0,x1)=minrΦr(φ0,x1),
4.9

and

φ(x0,x1)=Φ(φ0(x0),x1)
4.10

it follows that

φ(x0s,x1s)=φs.
4.11

By 4.6, and then by 4.11

φ(x0,x1)φr(x0,x1)for all x,φ(x0r,x1r)=φr(x0r,x1r).

Thus ɸr is a support of ɸ at (x0r, x1r), and so (g0r, g1r) is a support gradient there, where

g0r=λru0r,g1r=λru1r,

that is, Gossen’s Law is satisfied. Hence (x0r, x1r) is a point of equilibrium for ɸ under the budget (u0r, u1r) (r = 1,…, k).

Under the hypothesis ML0)1, a function ɸ which is admitted by the data and has the considered separability structure in terms of classical functions has now been exhibited, so as to provide the conclusion CH0)1, and prove the following:

Theorem 4.1

ML0)1CH0)1,

and given any multipliers λr, λ0r > 0 and levels ɸr, ɸ0r satisfying the multiplier-level consistency condition ML0)1, a compatible classical utility on the separability model 0)1 is given by

φ(x0,x1)=Φ(φ0(x0),x1)

where

φ0(x0)=minr{φ0r+λ0rσ0ru0r(xrx0r)}

and

Φ(φ0,x1)=minr{φr+λr(σ0rλ0r(φ0φ0r)+u1r(x1x1r))}.
(p.146)

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