# The Construction of Separable Utility Functions from Expenditure Data

# The Construction of Separable Utility Functions from Expenditure Data

# 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* = *Ω _{n}* is the

*budget space*(non-negative row vectors, order

*n*) and

*C*=

*Ω*the

^{n}*commodity space*(column vectors). Then any

*p*∈

*B*,

*x*∈

*C*determine

*px*∈

*Ω*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

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 *n*_{0} goods, and a 1-class of *n*_{1} goods, where *n*_{0} + *n*_{1} = *n*. Then

is the partition of a vector *x* of quantities of the *n* goods into component vectors *x _{i}* (

*i*= 0, 1) for the goods in classes 0, 1. These lie in commodity spaces ${C}_{i}={\Omega}^{{n}_{i}}$ . The price vector has a conformable partition

and the budget vector is

with components

in the budget spaces ${B}_{i}={\Omega}_{ni}$. Then we have

that is,

where *M _{i}* =

*p*is the expenditure on goods in class

_{i}x_{i}*i*(

*i*= 0, 1), and

*M*is the total expenditure on all goods. The expenditure shares for the two classes are

so

and

consequently

A *demand correspondence* is any *D* ⊂ *B* × *C* such that

For the case of a *normal demand correspondence E* ⊂ *B* × *C*,

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

with budget vectors

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

The *cross-coefficient* from *r* to *s* is

From this is obtained the *chain-vector*

Then there is the *cyclical consistency condition*

which denies the possibility of a seminegative cycle,

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

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

equivalently,

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

Hence

and so

A *multiplier-level solution* for the given demands is given by any positive numbers (*λ _{r}, ɸ_{r}*) which are a solution of the system of homogeneous linear inequalities

equivalently,

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,

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

*λ*).

_{ir}, ɸ_{ir}# 2 Gossen’s Law

A *utility function* is any *ɸ* : *C* → *Ω*, and a *utility order* is any *R* ⊂ *C* × *C* which is reflexive, *xRx*, and transitive, *xRyRz* ⇒ *xRz*. A utility function *ɸ represents* the utility order *R* given by

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, ~ *xRy* ⇒ *yRx*. Usually we deal with a general utility *R*, partitioned by its symmetric and antisymmetric parts,

for which

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

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

defines *compatibility* between the demand and the utility.

The further condition

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

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,

Also,

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,

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 *x _{r}* it has a support function

for which

in which case *ɸ _{r}* =

*ɸ*(

*x*), and

_{r}*g*≥

_{r}*o*since

*ɸ*is non-decreasing.

By the differentiability theorem for concave functions, the support gradient *g _{r}* is unique if and only if

*ɸ*is differentiable at

*x*, with differential gradient

_{r}*g*(

*x*), and then the differential and support gradients coincide,

_{r}*g*(

*x*) =

_{r}*g*. But generally the support gradient

_{r}*g*is indeterminate in the closed convex set of support gradients of

_{r}*ɸ*at

*x*.

_{r}Gossen’s Law, that preference and price directions coincide in equilibrium, would require

where now

This can apply when *ɸ* is differentiable at *x _{r}*, and at an interior point where

*x*>

_{r}*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

which assures that

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

so that

while

Moreover, with *λ _{r}* > 0, we have

which shows that

and hence that

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

# 3 Classical separability

Let

be classical utility functions, so that

also is classical.

Let

be support gradients of *ɸ*_{0}, *Φ, ɸ* at

where

Then

where

Then, by 3.1, with *G*_{0r} > 0,

and, by comparison with 3.2,

in other words, if *G*_{0r}, *g*_{00r} are support gradients as stated, then so is this value for *g*_{0r} determined from them.

For general equilibrium at *x _{r}* = (

*x*

_{0r},

*x*

_{1r}) under the budget

*u*= (

_{r}*u*

_{0r},

*u*

_{1r}) the conditions, shown in Section 2, are

where

Similarly, for a separate equilibrium at *x*_{0r} under the budget ${\sigma}_{0r}^{-1}{u}_{0r}$,

where

By 3.2, line 1, and 3.6,

in particular, for *x*_{0} = *x*_{0s},

so that

By 3.4, 3.5 and 3.6,

From this, with 3.3 and 3.5,

In particular, for *x*_{0} = *x*_{0s}, *x*_{1} = *x*_{1s},

that is,

with reference to 1.2, that is,

that is,

Combining this with 3.7,

which implies

** Theorem 3.1**. $C{H}_{0)1}\Rightarrow M{L}_{0)1}$

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

To this can be added, as already familiar,

We now have

A question with a possibly negative answer is whether

# 4 Construction

With any solution of the conditions *ML*_{0)1} given in 3.10, let

so that, by the consequence *ML*_{0,1} of 3.11,

that is,

Then with

we have

Now let

so, by *ML*_{0} in 3.7,

and then with

we have

Now by 4.1, 4.3 and 4.4,

Therefore let

Then

while

A restatement of the part of our assumption *ML*_{0)1} shown in 3.9 is that

that is,

that is,

Hence with

and

it follows that

By 4.6, and then by 4.11

Thus *ɸ _{r}* is a support of

*ɸ*at (

*x*

_{0r},

*x*

_{1r}), and so (

*g*

_{0r},

*g*

_{1r}) is a support gradient there, where

that is, Gossen’s Law is satisfied. Hence (*x*_{0r}, *x*_{1r}) is a point of equilibrium for *ɸ* under the budget (*u*_{0r}, *u*_{1r}) (*r* = 1,…, *k*).

Under the hypothesis *ML*_{0)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 *CH*_{0)1}, and prove the following:

*Theorem 4.1*

and given any multipliers *λ _{r}, λ*

_{0r}> 0 and levels

*ɸ*

_{r}, ɸ_{0r}satisfying the multiplier-level consistency condition

*ML*

_{0)1}, a compatible classical utility on the separability model 0)1 is given by

where

and

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