Notes on separable preferences

JOURNAL

OF ECONOMIC

THEORY

33, 309-321

(1984)

Notes on Separable

Preferences*

KING-TIM

MAK

Department of Quantitative Methods. College of Business Administration, University of Illinois at Chicago, Box 4348, Chicago, Illinois 60680 Received

December

20, 1982;

revised

December

19, 1983

Examples are constructed to show that Gorman’s important basic theorem for overlapping strictly separable sectors does not hold if the sectors are only nonstrictly separable. This is done with the help of a new definition for the notion of nonstrict separability of utility preferences. The new definition is equivalent to that of Bliss; but it clarifies the relatiokhip between the geometry and representation of preferences with separable sectors. The gained geometric insight leads to the conjecture that if all sectors are strictly essential, then German’s Basic Theorem would hold. Journal of Economic Literature Classification Number: 022.

1. INTRODUCTION

The concept of separability, first introduced by Sono [ 121 and Leontief [9], plays an important role in both consumption and demand theory. For a comprehensive survey of the applications of separability, see Blackorby, Primont, and Russell [ l]. In the modern theory of separable consumption preferences, distinction is made between strict separability (as defined by Debreu [5], Koopmans [lo], Stigum [ 131, and Gorman [8]) versus the more general notion of nonstrict separability (introduced by Bliss [3]). As we shall see, the theory of strict separability is much further developed than that of nonstrict separability. Common to both theories, the most fundamental question is the following. Given that two separable sectors of a preference overlap, what do we know about the structure of the preference? If the answer to this question is known, then the implications of any given collection of separable sectors on a preference can be analyzed by considering how they overlap. It is well known that if two strictly separable sectors overlap, then the preference can be represented by a utility function which is additively separable in the sectors corresponding to the intersection and set differences * The suggestions

of an anonymous

referee

greatly

improved

the presentation

of this paper.

309 0022-0531184

53.00

Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

310

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MAK

of the two overlapping sectors. This important result had been established by Gorman [7, 81, Debreu [5], and Koopmans [lo] under assumptions of varying generality.’ While Debreu and Koopmans were mainly concerned with deriving the additively separable representation, Gorman was concerned with showing the structure of strict separability under the most general condition.2 Gorman’s basic theorem asserts that if two strictly separable sectors overlap, then their intersection, set differences, symmetric difference, and union are also strictly separable. 3 By repeated applications of the basic theorem, Gorman was able to analyze the structure of any preference with a given collection of strictly separable sectors. Indeed, he gave a unique decomposition for the collection via a utility tree. This result, together with the additively separable representation, completely characterizes the structure of utility functions which represent preferences with strictly separable sectors. The weaker notion of nonstrict separability was introduced by Bliss [3] to encompass more general classes of preferences, including those of Leontieftype. Little, however, is known about the structure of nonstrict separability. Specifically, it is not known whether the intersection and the set differences of two overlapping nonstrictly separable sectors are also nonstrictly separable. Without this knowledge, it is impossible to characterize the structure or the functional forms of utility functions representing nonstrictly separable preferences. The purpose of this paper is to lay the groundwork for developing a theory of nonstrict separability analogous to that of Gorman for strict separability. We begin by giving a new definition for nonstrict separability (Section 2). It is equivalent to that formulated by Bliss but is analytically more tractable. Next (Section 3), we relate the geometric properties of separable preferences with their functional representations. This, incidentally, clarifies the connection between strict and nonstrict separability (cf. Blackorby et al. [2], i In 171. Gorman worked with utility functions which are differentiable. In 15, 8. IO] only continuity of the preferences is assumed. For overlapping sectors PU Q and Q U R, Debreu [ 5 1 assumed that P. Q. R are strictly separable and that P U Q. Q U R, P U R are also strictly separable. Koopmans [ 10 1 assumed that P. Q, R, PU Q, and Q U R are all strictly separable. Gorman 18 ] only assumed that the overlapping sectors PU Q and Q U R are strictly separable. * Koopman was ultimately interested in studying the structure of consumption preferences over time and in deriving the (additively separable) discounted utility function (see [Ill). He assumed directly, for expediency, that sectors P, Q, R are strictly separable. Only Gorman was concerned with whether the strict separability of P, Q, and R follows from that of PU Q and QUR. 3 Gorman first showed the strict separability of the intersection and the set differences. This result enabled him to use the theory of functional equations to derive the additively separable representation, which in turn is used to show the strict separability of the symmetric difference and the union.

SEPARABLE

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PREFERENCES

Fare and Primont [6]). The substantive result of this paper (Section 4) shows that, in general, Gorman’s basic theorem does not hold for nonstrictly separable sectors. This negative result, however, points positively (in (Section 5) to a regularity condition under which Gorman’s basic theorem could be established. 2. DEFINITION We are interested in the structure of utility preferences each of which is given as a continuous complete ordering4 2 on a product space s=s, xs,x *** xs,,

(2.1)

each Si is topologically separableand arc-connected.’

(2.2)

where

The corresponding strict preference and indifference relationships are denoted by > and -, respectively.6 Denote by l-2 : = (1, 2,..., n)

(2.3)

the set of elementary sectors; and write a consumption vector as

xi E St.

x : = (x, 9x~,**‘Y x,) = (Xi)i,ca2

(2.4)

For a sector A c B define xA : = (Xi)iEA 9

s, : = n si,

x:=0-A.

(2.5)

iEA

To simplify notation, when it is clear what sector A is, we write a consumption vector X = (X, , Xx) simply as (x, y) with x E S, and y f S,. Given a fixed subvector (called a reference) y E S,- off sector A, S, clearly inherits from a preference 2 a conditional ordering. If given somey E S,- not all elements in S, are indifferent, we say that sector A is essential; if given every, strictly essential. Since nonessential sectors affect nothing, they may be ignored. Therefore, we assumethat all sectors are essential.

(2.6)

’ An ordering > is complete if it is transitive and reflexive, and if x, y E S. either x 8 .v or y > x, or both. It is continuous if ( y E Six > y \ and ( y E S( y > x) are both closed for all x E s. ’ A set Si is fopobgically separable if it has a countable dense subset, such as the rationals among the reals. It is arc-connected if all xi, x; E S, are joined by a continuous arc in S,. 6Forx,YES,x>Yifx>ybuty#x;x-yifx>.vandy>x.

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MAK

In Sections 4 and 5, we will see that the stronger condition of strict essentiality appears to be necessary for studying the structure of preferences. The basic question on the structure of a utility preference is how the conditional orderings on S,, A E 0, depend on what happens off sector A. For instance, whether the preference on food consumption is influenced by the consumption of clothing; whether the preference on today’s consumption is contingent upon the consumption tomorrow. The traditional notion of strict separability, defined below, assumes that the dependence is null: sector A E R is strictly separable (in 2) if the conditional orderings on S, are the same for every reference y E S,-.

(2.7)

On the other hand, sector 2 clearly bears upon the conditional orderings on S, if for some x’, x” in S, conditioning on two different references y’, y” in Sz results in a reversal of strict preference: (x’, y”) < (x”, y”).

(x’, Y’) > (x”, Y’h

(2.8)

This observation motivates the following definition: sector A c f2 is nonstrictly separable (in 2) if for all x’, x” in S, that (x’, y*) > (x”, y*) for some y* implies (x’, y) 2 (x”, y) for all y in S,-. (2.9) That is, whatever happens off sector A, the superiority of x’ over x” is never reversed, although under some reference vector off A they may become indifferent. Note that the preference on a nonstrictly separable sector A does depend on what happens off it. The dependence, however, is well behaved-no reversal or “crossing” of conditional preferences is allowed. To illustrate this point further, we define for each reference y’ E Srsthe conditional level sets WI

Y’) : = {x E S,l(X,Y’)

z (X’,Y’)l,

allx’ ES,.

(2.10)

(2.11) PROPOSITION. Sector A G R is nonstrictly separable if and onij) if for all (x’, y’), (x’!, y”) in S, X S, L(x’I y’)sL(x”l

y”)

or

L(x’ ( y’) 2 L(x” / y”);

(2.12)

that is, the conditional level sets are nested. Proof: Suficiency. Suppose sector A G R is not nonstrictly separable. Then for some x’, x” in S,and y’, y” in S,- the strict preferences in (2.8) hold. That is, x’ @L(x”I y”) and x” @L(x’ 1y’); or condition (2.12) does not hold.

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Necessity. Suppose for some (x’, y’) and (x”, y”) the nestedness condition (2.12) does not hold. Then there exist x* E L(x’ 1y’) - L(x” ] y”) and x0 E L(x” 1y”) - L(x’ 1y’). Explicitly, (i)

(x*, y’) Z (x’, u’>,

(ii)

(x*, y”) -< (x”, y”);

(iii)

(x0, y”) 2 (x”, y”),

(iv)

(x0, y’) < (x’, y’).

Items (i) and (iv) imply (x*, y’) > (x0, y’). If sector A were nonstrictly separable, then it would follow that (x”, y”) 2 (x’,y”). But this contradicts the fact that (x’,y”) > (x*,y”) which follows from (ii) and (iii). Hence.4 is not nonstrictly separable. 1 In contrast, Sector A is strictly separable if and only if for all y’, y” E S,and XE S,, L(x]y’)=L(x]y”). Condition (2.12) on the nestednessof the conditional level sets was originally used by Bliss [31 to introduce nonstrict separability. According to Proposition (2.1 l), our defintion is equivalent to his. Because our definition is given directly in terms of preference orderings rather than their geometric properties, it is analytically more tractable.

3. REPRESENTATION

AND GEOMETRY

In this section we review the well-known representation theorems for separable preferences. Our purpose here is to relate the geometric properties of separable (strict and nonstrict) preferences with their representations by utility functions. Applying a well known theorem of Debreu [4] to a preference satisfying (2.1) and (2.2) yields (3.1) LEMMA. There exists a continuous utility function U defined on S such that lJ(X’) > U(X”) if and only if X’ > X”, all X’, X” E S. Suppose sector A c 52 is nonstrictly separable. On S,d define an ordering & .J as follows: for all x’, x” E S, , x’ > Ax” if (X’, y) > (x”, y) for all y E S,-.

(3.2)

Using the nestednesscondition (2.12) Bliss [3 ] showed that 2 A is a complete continuous ordering on S, .’ Then, again by the theorem of Debreu, there exists a continuous function g which represents>A ; i.e., for all ’ Bliss assumed the argument.

the preference

2 is monotone.

Monotonicity,

however,

is not essential

to

314

KING-TIM

MAK

x’, x” in S, , g(x’) > g(Y) if and only if x’ +A x”. Let S, be the image of g. Define a function G on S, x S,- so that U(x, Y) = G( g(x)> Y>

all

(x, y) E S, X ST.

(3.3)

The continuity of U and g readily imply that G is continuous. Note that for every fixed reference y E S, the conditional ordering on S, (incidentally represented by the function U(. , y)) is, according to (3.2), coarser than >A. Since g represents >A, it follows immediately that G is nondecreasing in the image of g. If sector A is actually strictly separable, then +A is exactly the same as the conditional ordering on S, induced by each reference y E S,-. Hence, for all y E S,-, g is a strictly monotone transform of U(. , y); thus G must be increasing in g. Formally, we state (3.4) THEOREM (Gorman [8], Bliss [3]). Sector A G Q is nonstrictzy (strictly) separable tf and only tf the underlying preference can be represented as (3.3) where G, g are continuous and G is nondecreasing (increasing) in the image of g. Verification of the “if” part of the theorem is straightforward, hence omitted. In applications, it is common to endow a postulated preference with separable structure by specifying its representation in the form (3.3). According to Theorem (3.4), whether strict or nonstrict separability is actually postulated depends on if G is specified to be strictly or nonstrictly monotone in g. The analysis of nonstrict separability is difficult because of the lack of strict monotonicity-the technique of taking inverses is no longer applicable. To aid developing a theory for nonstrict separability, we seek further insight to its geometry. For this purpose, define for each reference y E S,- and ,U in the image of U(. , y) the conditional isoquant. Q01Y>:=fxES,lU(x,Y)=~u.

(3.5)

If x E Q@l y) and every neighborhood of x contains some x* with U(X*, y) > ,u, then we say that x is in the upper boundary &tt) y) of the isoquant. The lower boundary Q@ 1y) of the isoquant is defined analogously. In more conventional jargons, the boundaries of the conditional isoquants are precisely those elements in S, which are conditionally local-nonsatiated (cf. the definition of nonsatiation in [2] and [6]). The next proposition relates the conditional isoquants to the function g. (3.6) PROPOSITION. Suppose sector A E R is separable (strictly or nonstrictly) and the function g represents the ordering > A, If for some

SEPARABLE PREFERENCES

315

reference y E S,- and utility level ,u both x’ and x” belong to the same upper or lower boundary of Q@ 1y), then g(x’) = g(x”). Moreover, both x’ and x” are nonsatiated with respect to the function g. Proof Suppose x’ and x” both belong to the upper boundary Q(u ] y). Assume contrapostitively that g(x”) < g(x’). Since g is continuous, there exists a neighborhood N of x” such that every x E N yields g(x) < g(x’). Then, by the monotonicity of G in g in the representation (3.3), for all x E N u(x, Y) = G(g(x), y> < G( dx’), y> = W’, Y> = ~1. This means x” cannot be in the upper boundary Q@ 1y); contradicting our hypothesis. Reversing the role of x’ and x”, we can show that assuming g(x’) < g(x”) will also lead to a contradiction. Hence g(x’) = g(x”). This conclusion can be established in the same manner had x’ and x” both belonged to the lower boundary Q@] y). The second assertion follows directly from the monotonicity of G in g and the definition of the boundaries. m Suppose sector A c a is indeed nonstrictly separable and the representation (3.3) of the underlying preference has g representing the ordering +A. Then for some reference y * E S,- the function G(. , y *) is nondecreasing but not increasing. This implies there exist some x’, x” in S, with g(x’) < g(x”) but G(g(x’), y*) = G(g(x”), y*). Thus, conditioned on the reference y*, x’ and x” are in the same conditional isoquant. On the other hand, since g represents > A and g(x’) < g(x”), there exists some other reference y* * E S,such that (x’, y* *) < (x”, y**). Here, x” does not belong to the same conditional (on y**) isoquant as x’. The point we are raising is that in the case of nonstrict separability, the conditional isoquants containing a particular element, say x’ above, can vary in size. This is in sharp contrast to the case of strict separability. There, for every element in the strictly separable sector, the conditional isoquants containing it are all identical. In view of the above discussion, and recalling that the boundaries of the conditional isoquants are local-nonsatiated, the following proposition due to Fare and Primont [6] is transparent. (3.7) PROPOSITION. Suppose sector A G B is nonstrictly separable, then it is actually strictly separable tf and only if the utility function U being localnonsatiated at x’ when conditioned on some y* E S, implies x’ is localnonsatiated when conditioned on all y E S, . More important for our purpose, the variability of the conditional isoquants gives us insight to why Gorman’s basic theorem does not hold for nonstrict separability. This will be discussedin Section 5.

h4?!33/?

Y

316

KING-TIM

4. GORMAN'S

MAK

BASIC THEOREM

In this section, we shall construct examples to show that the important theorem of Gorman is in general not true for nonstrict separability. First, we state formally (4.1) GORMAN'S BASIC THEOREM. Suppose two strictly separable sectorsA and B overlap; i.e., A n B, A - B, and B -A are all nonempty. Then sector A n B is strictly separable. Furthermore, if B -A or A - B is strictly essential, then A - B, B -A, A A B = (A - B) U (B -A), and A U B are all strictly separable. Note that the strict separability of the intersection A n B does not presuppose any strict essentiality. Our first example shows that this cannot be the case for nonstrict separability. It will be convenient to change notation slightly. For overlapping sectors A and B, write x for XAPB, y for X,.,ns, and z for X,-, ; also write S, for SA-B, S, for S,,,, and S, for S,-,. (4.2) EXAMPLE. Suppose S = S, X S, X S, : = [0,5] X ([0, 11 X [0, lo]) x [0,2]. Let U be a utility function on S with the form u(x, Y, z) = G( g(x, Y), z> where for all x E [0, 51 and y = (y,, y2) E [0, I] x [0, lo], g(x,Y,,Y*)

:=x + OJY, +y,

if x + 0.5y, + y, < 2,

:=2

if 22and Y, +y, f z ,< 3,

:=y,t.vz+z-1 /2

i

FIG. 2.

Conditional

ify,+y,+z>3; X=0

v,

isoquants

of (I with

references

x = 0 and 5.

318

KING-TIM

and H is given by H(x,P):=x+P

MAK

if

x +/? Q 2,

:=2

if

2 U(0, 0, z = 0) = 0, U(0, 5, z = 1) = 3 < U(O,5, z = 0) = 4. To see that the symmetric difference A A B = (A - B)U (B -A) is not nonstrictly separable, consider (x’, z’) : = (0.5, 0) and (x”, z”) : = (0.5, 1). Conditioning on the y reference values of 0 then 5 yields U(O.S,JJ = 0,O) = 0.5 < U(O.5,y = 0, 1) = 1.5, U(O.5,y=5,0)=4.5

> U(O.5,y=5,

1)=3.5;

i.e., the strict preferences of (x’, z’) versus (x”. z”) reverse order.

a

320

KING-TIM 5. INTERPRETATION

MAK AND CONJECTURE

In this last section, we interpret why Gorman’s basic theorem fails for nonstrict separability, and conjecture a stronger regularity condition under which it might hold. We shall focus on the failure of the intersection of overlapping sectors to be nonstrictly separable. Refer back to Example (4.2) where sector A spansthe vector (x, y) and B spans (u, z). We identified y’, y” E S,, and references (x*, z *), (x0, z”) such that U(x*,y’,z*)=G(g(x*,y’),z*)

< G(g(x*,y”),z*)=

U(x*,y”.z*),

U(x”, y’, z”) = G( g(x”, y’), z”) > G( g(x”, y”), z”) = U(x’, y”, z”>

(5.la) (5. lb)

thereby showed that A n B is not nonstrictly separable. Now consider conditioning on the reference (x*, z”) instead. By the nonstrict separability of A, (5.1.a) implies u(x*,y’,

z”) < U(X*,y”,

z”).

(5.2)

In fact, equality must hold in (5.2). This is because if otherwise, that U(x*, y’, z”) < U(x*, y”, z”) will imply (by the nonstrict separability of B) U(x”, y’, z”) < U(x’, y”, z’); contradicting (5.1.b). That is to say, conditioned on z”, (x*, v’) and (x*, y”) be1ong to the same conditional isoquant in S, ; but conditioned on z*, they do not. Had sector A been strictly separable, strict inequality would prevail in (5.2). Then the conditional isoquants in S, are identical but (5.lb) cannot hold. Hence, it is precisely the variability in the size of the conditional isoquants that allows the intersection A n B not to be nonstrictly separable. In both Examples (4.2) and (4.3) sectors A -B and B -A are not strictly essential. We now give, very loosely and intuitively, an interpretation of the role of strict essentiality which will motivate our conjecture. Conditioning on (x*, z*) and (x0, z’), the inequalities (5.la) and (5.lb) assert that A n B is not non-strictly separable and that y’ and y” belong to different conditional isoquants. Moreover, because of the reversal in the direction of the strict inequalities, the “curvature” of the conditional isoquants containing y’ “reverses shape” as we progressively vary the reference from (x*, z*) to (x0, z”). A perhaps appropriate analogy to this situation is an onion with two cores: as we move from one core of the onion to the other the curvature of its layers change from concave to convex. Let us imagine that the layers of such an onion correspond to the isoquants of a preference ordering. Then moving from (x*, z*) to (x0, z”) is like taking a sectional cut of the onion and going from one core to the other. Moving thus, we should encounter a conditional isoquant (a layer of the sectioned onion)

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321

whose elements are all indifferent (the layer is flat); i.e., some sector is not strictly essential. Hence, by imposing strict essentiality on sectors A - B, A ~7 B, and B - A, we hope to rule out the joint occurrence of the inequalities (5.la) and (5.lb), thereupon show the nonstrict separability of A n B. Going one step further, we hazard

(5.3) CONJECTURE. Suppose nonstrictly separable sectors A and B overlap and A -B, A n B, and B -A are strictly essential. Then sectors A n B, A - B, B -A, A A B, and A U B are all nonstrictly separable. Rigorous

verification

of this conjecture will be the subject of another paper.

REFERENCES 1. C. BLACKORBY, D. PRIMONT, AND R. R. RUSSELL, “Duality, Separability, and Functional Structure: Theory and Economic Applications,” North-Holland-American Elsevier, New York, 1978. 2. C. BLACKORBY, D. PRIMONT, AND R. R. RUSSELL, Separability vs. functional structure: A characterization of their differences, J. Econom. Theory 15 (1977), 135-144. 3. C. J. BLISS. “Capital Theory and the Distribution of Income,” North-Holland-American Elsevier, New York, 1975. 4. G. DEBREU, Representation of a preference ordering by a numerical function, in “Decision Processes” (Thrall, Coombs, and Davis, Eds.). pp. 159-165. Wiley. New York, 1954. 5. G. DEBREU. Topological methods in cardinal utility theory, in “Mathematical Methods in the Social Sciences” (Arrow, Karlin, and Suppes, Eds.), pp. 16-26, Stanford Univ. Press, 1960. 6. R. F~CRE AND D. PRIMONT. Separability vs. strict separability: A further result, J. Econom. Theory 25 (1981), 455-460. 7. W. M. GORMAN, Condition for additive separability, Econometrica 36 (1968), 605-609. 8. W. M. GORMAN, The structure of utility functions, Reu. Econom. Stud. 35 (1968), 367-390. 9. W. W. LEONTIEF, Introduction to a theory of the internal structure of functional relationships, Econometrica 15 (1947), 361-373. 10. T. C. KOOPMANS, Representation of preference orderings with independent components of consumption, in “Decision and Organization,” (McGuire and Radner, Eds.), pp. 57-78, North-Holland, Amsterdam, 1972. 11. T. C. KOOP~~ANS. Representation of preference orderings over time, in “Decision and C);;;;isation,” (M c G uxe and Radner, Eds), pp. 79-100. North-Holland, Amsterdam, 12. M. SONO, The effect of price changes on the demand and supply of separable goods, Infernal. Econom. Rev. 2 (1961), 239-271 (translated from the original 1945 article in Japanese). 13. B. P. STIGUM,On certain problems of aggregation, Infernat. Econom. Rev. 8 (1967), 349-367.