## Sydney Afriat

Print publication date: 2014

Print ISBN-13: 9780199670581

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199670581.001.0001

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# 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
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.

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

$Display mathematics$

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

$Display mathematics$

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

$Display mathematics$

and the budget vector is

$Display mathematics$

with components

$Display mathematics$

in the budget spaces $Bi=Ωni$. Then we have

$Display mathematics$

that is,

$Display mathematics$
(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

$Display mathematics$

so

$Display mathematics$

and

$Display mathematics$

consequently

$Display mathematics$
1.1

A demand correspondence is any DB × C such that

$Display mathematics$

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

$Display mathematics$

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

$Display mathematics$

with budget vectors

$Display mathematics$

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

$Display mathematics$

From this is obtained the chain-vector

$Display mathematics$

Then there is the cyclical consistency condition

$Display mathematics$
(p.137)

which denies the possibility of a seminegative cycle,

$Display mathematics$

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

$Display mathematics$

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

$Display mathematics$

equivalently,

$Display mathematics$

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

$Display mathematics$

Hence

$Display mathematics$

and so

$Display mathematics$
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

$Display mathematics$

equivalently,

$Display mathematics$
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,

$Display mathematics$
(p.138)

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

$Display mathematics$
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

$Display mathematics$

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,

$Display mathematics$

for which

$Display mathematics$

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

$Display mathematics$

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

$Display mathematics$

defines compatibility between the demand and the utility.

The further condition

$Display mathematics$
(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

$Display mathematics$

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,

$Display mathematics$

Also,

$Display mathematics$

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,

$Display mathematics$

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

$Display mathematics$

for which

$Display mathematics$

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

$Display mathematics$
(p.140)

where now

$Display mathematics$

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

$Display mathematics$

which assures that

$Display mathematics$

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

$Display mathematics$

so that

$Display mathematics$

while

$Display mathematics$

Moreover, with λr > 0, we have

$Display mathematics$

which shows that

$Display mathematics$

and hence that

$Display mathematics$

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

# 3 Classical separability

Let

$Display mathematics$
(p.141)

be classical utility functions, so that

$Display mathematics$
3.1

also is classical.

Let

$Display mathematics$

be support gradients of ɸ0, Φ, ɸ at

$Display mathematics$

where

$Display mathematics$

Then

$Display mathematics$
3.2

where

$Display mathematics$

Then, by 3.1, with G0r > 0,

$Display mathematics$
3.3

and, by comparison with 3.2,

$Display mathematics$
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

$Display mathematics$
3.5

where

$Display mathematics$

Similarly, for a separate equilibrium at x0r under the budget $σ0r−1u0r$,

$Display mathematics$
3.6

where

$Display mathematics$
(p.142)

By 3.2, line 1, and 3.6,

$Display mathematics$

in particular, for x0 = x0s,

$Display mathematics$

so that

$Display mathematics$
3.7

By 3.4, 3.5 and 3.6,

$Display mathematics$
3.8

From this, with 3.3 and 3.5,

$Display mathematics$

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

$Display mathematics$

that is,

$Display mathematics$

with reference to 1.2, that is,

$Display mathematics$

that is,

$Display mathematics$
3.9

Combining this with 3.7,

$Display mathematics$
3.10

which implies

$Display mathematics$
3.11
(p.143)

Theorem 3.1. $CH0)1⇒ML0)1$

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

$Display mathematics$

$Display mathematics$
$Display mathematics$

We now have

$Display mathematics$

A question with a possibly negative answer is whether

$Display mathematics$

# 4 Construction

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

$Display mathematics$
4.1

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

$Display mathematics$

that is,

$Display mathematics$

Then with

$Display mathematics$

we have

$Display mathematics$
4.2
(p.144)

Now let

$Display mathematics$
4.3

so, by ML0 in 3.7,

$Display mathematics$

and then with

$Display mathematics$

we have

$Display mathematics$
4.4

Now by 4.1, 4.3 and 4.4,

$Display mathematics$

Therefore let

$Display mathematics$
4.5

Then

$Display mathematics$
4.6

while

$Display mathematics$
4.7

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

$Display mathematics$

that is,

$Display mathematics$
(p.145)

that is,

$Display mathematics$
4.8

Hence with

$Display mathematics$
4.9

and

$Display mathematics$
4.10

it follows that

$Display mathematics$
4.11

By 4.6, and then by 4.11

$Display mathematics$

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

$Display mathematics$

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

$Display mathematics$

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

$Display mathematics$

where

$Display mathematics$

and

$Display mathematics$
(p.146)

Bibliography

Bibliography references:

Aczel, J. (1966): Functional Equations and their Applications. New York: Academic Press.

Afriat, S. N. (1953): Utility separability and budget decentralization. Communication to Robert H. Strotz, Department of Applied Economics, Cambridge (September).

—— (1964): The Construction of Utility Functions from Expenditure Data. Cowles Foundation Discussion Paper No. 144 (October), Yale University. First World Congress of the Econometric Society, Rome, September 1965. International Economic Review 8, 1 (1967), 67–77.

—— (1969a): The Construction of Cost-efficiencies and Approximate Utility Functions from Inconsistent Expenditure Data. New York Meeting of the Econometric Society, December.

—— (1969b): The Construction of Separable Utility Functions from Expenditure Data. Department of Economics, University of North Carolina, Chapel Hill.

—— (1970): The Theory of International Comparisons of Real Income and Prices. In International Comparisons of Prices and Output, Proceedings of the Conference at York University, Toronto, 1970, edited by D. J. Daly. National Bureau of Economic Research, Studies in Income and Wealth Volume 37, New York, 1972. (Ch. I, 13–84).

—— (1973): On a System of Inequalities in Demand Analysis: an Extension of the Classical Method. International Economic Review 14, 2 (June), 460–72.

—— (1976): Combinatorial Theory of Demand. London: Input-Output Publ. Co.

—— (1980): Demand Functions and the Slutsky Matrix. Princeton University Press. (Princeton Studies in Mathematical Economics, 7)

—— (1981): On the Constructibility of Consistent Price Indices Between Several Periods Simultaneously. In Essays in Theory and Measurement of Demand: in honour of Sir Richard Stone, edited by Angus Deaton. Cambridge University Press, 1981. 133–61.

—— (1987a): Logic of Choice and Economic Theory. Oxford: Clarendon Press.

—— (1987b): Lagrange multipliers. In Eatwell et al.

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—— (1968a): The structure of utility functions. Revue of Economic Studies 35 (4), 104, 367–90.

—— (1968b): Conditions for additive separability. Econometrica 36, 605–9.

—— (1970a): Two-stage budgeting. Department of Economics, University of North Carolina, Chapel Hill.

—— (1970b): The concavity of additive utility functions. Department of Economics, University of North Carolina, Chapel Hill.

—— (1987): Separability. In Eatwell et al.

—— (1995): Separability and Aggregation. Collected Works of W. M. Gorman, Volume I, Edited by C. Blackorby and A. F. Shorrocks. Oxford: Clarendon Press. (p.147)

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—— (1957b): A note on the interrelation of subsets of independent variables of a continuous function with continuous first derivatives. Bull. Am. Math. Soc. 53, 343–56.

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—— (1959): The Utility Tree: A Correction and Further Appraisal. Econometrica 27, 482–8.

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