Journal
of Mathematical
Economics
23 (1994) 295-303.
A sufficient condition
separable functions
North-Holland
for additively
Uzi Segal
University of Toronto, Toronto, Ont., Canada and California Institute of Technology, Pasadena
CA, USA
Submitted
July 1992, accepted
March
1993
This paper presents a set of sufficient conditions
under which a completely separable function on
an open S c RN is additively separable. The new condition is that different connected elements of
intersections
of parallel-to-the-axes
hyperplanes
with the domain S are intersected
by common
indifference surfaces.
Key words: Completely
JEL
separable
functions;
Additively
separable
functions
classification: C60
1. Introduction
Additively separable functions on RN are of the form I;= 1 Ui(Xi). These
functions have the obvious property that they are completely separable. That
is, the induced orders on the (N - 1)-dimensional sets {(x,, . . . , xN): xi0 = c}
are independent of c. A natural question is whether the opposite is also true,
that is, whether complete separability of a function V implies additive
separability. Debreu (1954) proved this to be the case whenever the domain
of I/ is a product of intervals rcl x ... x rcNc RN. Wakker (1989) extended this
result to ordered cones (i.e., sets in RN where xi 2..* 2x, 20). He also
pointed out the importance of the requirement that indifference surfaces are
connected. In Segal (1991) I showed that one can relax the assumption that
the domain S is an ordered cone and that it is sufficient to require that
intersections of the domain of the function with parallel-to-the-axes hyperplanes are connected sets. Chateauneuf and Wakker (1993) extended these
results to general product spaces. For further references, see this last paper.
See also Blackorby et al. (1978) for a survey of applications of additively
separable functions in economics.
In this paper I offer a further relaxation of the above mentioned
requirements. Intersections
of parallel-to-the-axes
hyperplanes with the
Correspondence to: Professor
Humanities
and Social Sciences,
Uzi Segal, California
Institute of Technology,
228-77, Pasadena,
CA 91125, USA.
0304-4068/94/%07.00
0 1994 Elsevier
SSDI 0304-4068(93)EOO62-Z
Science B.V. All rights reserved
Division
of the
296
V. Segal, Additively separable functions
domain do not have to be connected, as long as connected elements of such
intersections can be tied up through the indifference surfaces of V (see next
section for details and section 3 for a proof). In section 4 I discuss the
necessity of this condition and present some relevant examples. Topological
definitions and claims are taken from Dugundji (1966).
2. Definitions
Let S be connected open subset of RN, Nz 3, and let I/ be a strictly
monotonic function on it. That is, [vi~{i,...,N)
yiz:xi and 3i~(l,...,N)
such that yi>xi]jV(y)=I/(y,,...,y~)>V(x)=V(X,,...,X~).
For XES, let
Z(X)= {YES: V(y) = V(x)} be the indzfirence surface of V through X. We
assume throughout that all indifference surfaces of I/ are connected subset
of RN.
Let pi
be the projection of S on the ith axis,
pi
= {Xi:3(X,, . . . ) Xi,. . . ) XN)ES}.
Since S is an open connected subset of RN, x(S) is an open interval,
i=l , . . . ,N. In the sequel, the term open box means a set of the form
nF= 1 Ji, where Ji IS
. a nonempty bounded open interval in pi,
i = 1,. . . , N.
Dejinition 1. The function V: S+ R is called completely
every (X1,...,Xi-l,Xi,Xi+l,...,XN),
(yl,...,yi-l,Xi,yi+l,...,yN),
yi,xi+l,...,xN),
and
WI,...,
(yl,...,yi-l,yi,yi+l,...,YN)
in
separable
if for
(x1,**.,xi-19
S,
Xi-1,Xi,Xi+l,...,XN)~V(yI,...,yi-1,Xi,yi+l,...,YN)
eV(xt,...,
Xi-l,Yi,Xi+lr...,XN)~I/(yl,...,yi-l,Yi,yi+l,...,yN)’
Definition 2. The function I/: S+ R is called additively separable if there
exist continuous and strictly increasing functions Ui:zi(S) -+ R, i = 1,. . . , N,
and a strictly increasing
function
c: Rng(czv= Ui(.)) + [w such that
w
ere
Rng(~~l
ui( *)) = {tE b!: 3(x1,.
. . ,xN) E RN
Ui(Xi)),
h
V(XI,*..,XN)=C(C?=I
such that t=Cy= 1 pi>.
Definition 3. The function V: S --, Iw is called locally separable if, for every
XES, there is neighborhood M(x) of x on which the function V is additively
separable.
The set S is open. Therefore, for every x ES, there is an open box R such
that x E R c S. It thus follows by Debreu (1960) that complete separability
implies local separability. Consequently, if one wishes to show that on a
certain domain complete separability implies additive separability, it is
U. &gal,
291
Additively separable functions
sufficient to prove that local separability implies additive separability there.
One should note, however, that local separability is strictly weaker than
complete separability. The function in Example 1 is locally separable, but not
completely separable. This distinction becomes relevant in the proof of
Lemma 2 below.
Example 1.
Let
B=((x,,x~,x3)ER3:O<X1<15,
O<x2<8,
C={(x,,x2,x3)~R3:0<x1<5,
Let S=AuBuC
6<x2<8,
and define V:S-tR
vxI?x,>x,)=
x1 +x,+x,,
x +x +2x
1
2
x,20,
x,+Z!x,<lO},
by
(x,,
3,
-l<x,<O},
x2,
~3)
(X1,X2,X3)EC
E A
u
13
’
This function is locally separable, but not completely separable. For example,
1/(1,7,4)=V(11,1,4)=16,but
1/(1,7,3)=14while
V(11,1,3)=15
0.
For every i~{l,...,N)
and c E R, let S(i, c) = {(x1,. . . , xN) ES: Xi= c}. Since
S is open, S(i,c) is relatively open in RN- ’ . We say that the connected set
T E S is a maximal connected element of S(i, c) if there is no connected set T’
such that Ts 7” G S(i, c). Let d be the set of maximal connected elements of
S(i,c). That is,
1. S(i,c)=u,,,T;
and
2. for every T E d, T is a maximal connected element of S(i, c).
The set d is countable. To see this, observe that if the intersection of two
connected sets is not empty, then their union is connected. Since S(i,c) is
(relatively) open, each point in it has a (connected) neighborhood around it
in S(i,c). Therefore, a maximal connected subset of S(i,c) is open. It is well
known that RN-i contain at most countable disjoint nonempty open sets.
Hence d==(T’,T’,...
>. For TE&‘, define
d’(T)
s&(T)=
= (T’E d: there exists x e T and yo T’ such that V(x) = I’(y)},
u
J@(T’),
i-Ed’-‘(T)
298
U. Segal, Additively separable functions
d(T)=(J&zP(T).
I
Lemma
1.
T’~sz?(T)o
Proof.
TE&(T’).
T’ E JZ?(T) c> 31 such that
Ti’, . . . , Til
such
that
T”=T
x2.1
1.2
2,2
x2.1-1
l_~~xxl:~xZ:“.;‘Ti~,
TESS?(T’),
By definition there are
T” = T’,
and
and
there
are
‘*’ such that x’s1 E Til, x~*‘E Til, for every 25 rns
a;lxd V(xZ.m)= V(Xl.m+l),
m=l,...,
1- 1. Therefore,
hence TE&(T’).
T’ E J#( T).
0
The set & can thus be partitioned into equivalence classes dl,. . . where
T, T’ belong to the same equivalence class d, if and only if Tsd( T’) [and
T’EJ~(T)].
For a given i and c, denote this set of equivalence classes by
W(i,c) and denote its cardinality by I%?(i,c)j. In the sequel we will be
interested in functions and sets where jW(i,c)l= 1, that is, in cases where for
every T and T’, T’ E s4( T).
3. Sufficient conditions
This section presents sufficient conditions under which a completely
separable function must also be additively separable. These conditions are
weaker than those discussed in Sepal (1991). Necessity is discussed in the
next section.
Let V be a continuous and strictly monotonic completely separable (hence
locally separable) function on S, an open subset of IWN. Let x* =
(x:,...,x$...,
x;E) and y*=(y: ,..., xc ,..., yf) in S such that V(x*) = V(y*)
and let R”’ and RY* be two open boxes in S such that X*E RX’ and y*eRY*.
By local separability it follows that for z = x*, y*, V on R” is given by
(1)
V(xI,...,+)=(for some strictly monotonic
functions c”’ and cy*.
Lemma 2. Under the above conditions, there are a* > 0, b*, and E* > 0, such
that for every xi0 E(X$ -E*, xi*, + E*), $(xio)
= a*@~(xi,,)
+ b*.
Proof.
ei=(ei,...,
z=x*,y*,
Suppose, without loss of generality, that i0 # 1. For ie (1,. . . , IV}, let
et), where ej equals 1 if i = j and 0 otherwise. Let E*,sk,‘>O,
k= 1,2, such that
1. z+(-l)kek~‘e’+(-l)ke*eio~R(z),z=x*,y*,
k=l,2; and
2. V(x* + (- l)k.sk,x*el)= V(y* +( - l)ksk*Y*e’),k = 1,2.
299
V. Segal, Additively separable functions
For
let
E’={w=(w,,...,
w,)ERN: wr EIZ1-&l*=,zl +@],
z =x*, y*,
WioEIZio-E*,Zio+E*], and Wi=Zi, iE{l,...,N}\{l,iO)).
By condition
1
above, E’ c R(z), z=x*, y*. By condition 2, continuity, and monotonicity it
follows that there is a strictly increasing, continuous, and onto function
x:+E’*~]
such
that
for
WOE
P: cv:-s ‘,“,~:+E~,~*]_,[x:--E~*~,
[y:-~r*~*, y:+.~~*~], V(w,, yf,. .., y;l;)= V(p(w,),xz,. . .,xg). Since the function
V
is
completely
separable,
it
follows
that
for
every
and ~~~[y~-&~*,y~+s~~~*],
wiOE Cx$,-s*,x$+s*]=[yz-s*,y$+s*]
v(w19YF. * *vwio9*. *TY;G)= V(P(W1),Xf,~
e-7 wio,*- .~~f)*
(2)
(This last equation requires complete, and not only local, separability.)
The functions 5”’ and ly* are strictly monotonic and therefore have an
inverse. It follows therefore by eq. (1) and (2) that on E”
uY;(wl) +4:
=r’*-’
(wi,) +
ix*
[
1
i#l.io
C(Yf)
U;‘(p(Wl))+U~(Wi~)+
(
C
i#l,io
U”(X*)
)I
.
This last equation implies that [y*-’ o [“* is affine [see AczCl (1966)]. That is,
ly*-’ 0 C”‘(x)= KC+ /I. By the monotonicity of l”’ and cP, u > 0. Denote a* = c(,
and b*=aCi+i, u;*(x~)-Cigiou~*(Y~)+~
[ recall that p(xr)= y:] to obtain
that for every Xi0E [X:-E*,
X$ + E*], ut(xi,) = a*u~(xi,)
+ b*.
0
Theorem 1.
1.
2.
3.
4.
Let N 2_3 and let (S, V) satisfy the following conditions:
The set S is an open and connected subset of [WN.
For every i and c, I%(i,c)l= 1.
The function V on S is continuous and strictly monotonic.
All indifference surfaces of V are connected subsets of IWN.
Then the function
separable.
V is completely separable if and only if it is additively
Remark. This theorem is a weaker version of Theorem 1 in Segal (1991)
where the second condition requires that for every i and c, S(i,c) is a
connected set.
Proof.
If V is additively separable, then it is of course completely separable.
Suppose therefore that it is completely separable. Let x0 =(xy, . . . , xFO,.. . , xg),
yO=(yY ,...) xi )...) yi) ES and let Rx0 and RYObe two open boxes in S such
that x”~RXO and y”~RyO. Since V is completely separable, it follows by
U. Segal, Additively separable functions
300
Debreu (1960) that on R’ it is equal to &Jxr= I uf(xi)), z = x0, y”. Next I show
that there are a>O, b, and s>O, such that for every xi0 E(x~-c,x~ +E),
z&(xJ = uz&x,) + b.
If x0 and y” belong to the same connected element of S(io,xFO), then the
proof of the above claim is the same as the proof of Lemma 1 in Segal
(1991). Suppose therefore that TX0 and TYo are different maximal connected
elements of S(i,,xt)
such that x0 E TX0 and y” E Tyo. Assume first that
TYo E d’( TX’). Let x* E TX0 and y* E TYo such that V(x*)= V(y*). Let RX* and
RY” be two open boxes in S such that x* ER”’ and y* ERY’. Let V on R’ be
given by [‘(~~= I uf(xi)), z=x*, y*. By Lemma 1 in Segal (1991) and Lemma
2 above, there are a2 >O, b’, and E’> 0, z = x0, x*, y*, such that for every
Xi0 E [X~ -EXa, X~ + Exe],
U~(Xio)=UxoU~(XtD)
+ b”“,
for
every
xio E
[x;
- EX*, x; +&X*1,
Ut(X.)
= ax* uf~(x,)
+ b”*,
and for every xi0 E Lx,“,sY*,X: + E’*], Use
= uy*Ut(xt,)
+ by*. Let E= min{eXO,.E?‘,sY*>, u = cPu~*u~,
and b = uY*ax*b”” + uY*b”’ + by*. It follows that for every xi0 E[X: &,X~ +E],
U~~(Xio)=UU~~(Xi,)
+b.
By finite induction, such E, a, and b exist whenever TYo E d’( TX’). Since for
every i and c, )%(i,c)) = 1, it follows that for every x0, y” E S(i,, x:) there exists
1 such that TYo EJ&( TX’). Therefore, for every x0 =(xy,. . .,xE, . . .,x$), y” =
yjl,)E S there are a > 0, b, and s>O, such that for every
(YL...&...,
xi0 E (xFO-E, x: + E), u{i (xi,) = UUic,“(X,)
+ b. This property is proved in Segal
(1991) under the assumption that for every i and c, the set S(i,c) is a
connected subset of RN. Moreoever, this connectedness assumption is used in
no other place in the proof of Theorem 1 there. Therefore, we can now use
the rest of the proof in Segal (1991) (i.e., Lemmas 24) to complete the proof
of Theorem 1.
0
4. Necessity of the conditions
A natural question is whether the second condition in Theorem 1 [i.e., that
for every i and c, 148(i,c) ( = l] can be replaced by a weaker condition. First,
notice that this condition may be implied, for some sets S, by the rest of the
conditions of the Theorem.
Example 2.
Let
A={(xl,x,,x,)EW3:O<X1<3,
B=((x1,x2,x3)~(W3:x1<3,
C={(x,,x,,x3)~lR3:0c~1<3,
0<x,<3,0<x3<1},
x2)0,
x1--x,>2,
O<x,<3,
lSx,<Z?},
xi-x2<1,
lsx,<2}.
Let S= A u B u C. For every CE(O,3), S(l, c) and S(2, c) are connected sets
and for every CE(O, l), S(3,c) is connected. For CE [1,2), S(3,c) is discon-
U. Segal, Additively separable functions
301
netted. However, if V is a strictly monotonic function, then for every
c* E [l, 2) and for every XE S(3,c*) n B there exists a point YE S(3,c*) n C
0
such that V(x) = V(y). Therefore, for CE Cl, 2), IV(3, c)( = 1.
The next example shows that Theorem
not satisfied.
Example 3.
1 may hold even if condition
2 is
Let
B={(x1,x2,x3)~R3:x1
+x2+x3=6,
x,23).
Let S=A\B. For all the pairs (i, c), except for the pairs (3,~) where c>,3, the
set S(i,c) is connected. Let the completely additive function Y satisfy all the
conditions of Theorem 1, except for condition 2. This is possible. For
I prove next that such a function V
example, let V(x1,x2,x3)=xI
+x,+x,.
must also be additively separable. The first step is to extend V to a
continuous function V* on A.
Let (x7, XT,x$) E B and let {(xt”, XT,xijn)>Fzi c A, such that (xij’, xf, x’j”) --f
(xf, x& x:), i= 1,2. Assume, without loss of generality, that for every it, i= 1,2,
and j=l,3,
Let a* = 2 + 3 max {XT,xf, x;} and let a, = f min {XT,xz, x;}. Since V is monotonic, we can assume, without loss of generality, that for every n,
V(a,, a,, a,) s I/(x$“, xl, xi”) 5 V(a*, a*, a*), i = 1,2. Assume
first
that
Therelim,, m V(xy”,x;,x$“) exists and equals 0’. I prove next that #=8’.
fore all converging subsequences of V(X$“,X~,X$~) have the same limit and
the sequences converge.
Suppose that 0’ < 02. Since V is continuous and strictly monotonic, there is
a point (w, xz, w) ES such that 0 = V(w, xt, w) E(8i, 0’). Therefore, there exists
E> 0 such that for n 2 n* (without loss of generality, n* = l), V(x:*“, xz, xi,“) s
V(w-.s,xT,w--s)
and V(x:*“,xt,x $“)>= V(W+E,X~, w+E). Since V is completely separable and by eq. (3), it follows that these two last inequalities are
also satisfied when x2 is replaced by x:/2. Since (x:,x~/~,x~)E A, these new
inequalities violate the continuity of V. Define the value of the function V* at
the point (XT,x$, XT) as this common limit and let V* E V on S.
Let, as before, (xT,xt,xJ) EB and let (xl,x”,,x”,) +(x:,xf,xj)
such that for
every n, (x7, xl, x;) E S. Suppose, for example, that
V(xl, x;, x;) +
V*(x:, xr, x:) -{
for some 5 >O. By the continuity of V on S and the
302
V. Segal, Additively separable functions
construction
of V* on B, there are q>O and oz~(O,+) such that
V(x: - V,xt, x3 - q) = V*(x:, XT,xs) -at. For a sufliciently large n, x; > x7 -q,
x”,>x$-?I,
V(xl, x1, x:) < V*(xf, xr, xs) - 3(/4, and by the continuity of V
on S, V(x: - q, x;, xf - V)> V(x: - V,x,*,xf - q) - 5/4 > I/*(x:, xr, xf) - 35/4, a
contradiction with the assumption that V on S is strictly monotonic.
Finally, let (xl, x”,, xJ) + (xl, xt, xs) E B such that for every n, (xl, x”,, x;) E B.
Let (xl’, XT, x7) ES satisfy
1.
1V(xy, x”,l, x;l) - V*(x;,
2.
paragraph),
d((xl’, xy’, x”,l),(xl, xz, ~“3))52-”
RN.
x;, x’$)( s2-”
(this is possible by the preceding
where d( *, .) is the Euclidean metric on
Obviously, (xl’, x”;, x”;) 4(x:, xr, xf) and lim V(xy, x”;, x7) = lim V*(x;, xl, x;).
By the last paragraph, this common limit exists and equals V*(x:,x;,xf).
This establishes the continuity of I/*.
The function V* is clearly monotonic. Also, it follows by continuity that it
is completely separable. All the conditions of Debreu’s (1960) are satisfied,
hence V* is additively separable. Since on S, V* = V, so is I/.
0
A variant of this last example shows that even a combination of the
second and fourth conditions in Theorem 1 is not a necessary condition for
the theorem. Let A be as in Example 3 and let
C={(x,,x,,x,)&3:x,+x,=3,x3=3}.
Let S = A \ C. If (%?(3,3)(= 2, then there must be a disconnected indifference
surface.
Let a*=inf{I’(x,,x,,3):x,+x2>3}
and
let a,=sup{V(x,,
x2, 3): xi +x, < 3). By the monotonicity of V,a* sol*. Let /?E [cI*, cl*]. By
the continuity and monotonicity of V there are x,,x,,x~,x;
and E>O such
surface
The
indifference
that
V(x,,x,,3+&)=V(X;,X;,3-&)=~.
{(xi, x2, x3) ES: I/(x1, x2, x3) = /3} is therefore disconnected. Nevertheless, similarly to the proof of Example 3, it can be shown here as well that a
completely separable function I/ satisfying all the rest of the conditions of
Theorem 1, must also be additively separable.
Example 3 seems to suggest that the second condition of Theorem 1 will
become necessary if we require in addition that S=Int(Cl(S)). This conjecture is false, as is demonstrated by the following example.
Example 4.
Let A and B be as in Example 3. For
C”={(x,,x,,x,)~R~:x~
3+ l/2”sx3
54-
+x,+x,
l/2”},
=6-
n=
1,. . . , co let
1/2”;x, 2 1/2”,x, 2 l/2”,
U. Se@,
D”={(x1,x2,xJ)E
D=
(j
303
Additively separable functions
lR3:d((x1,x2,x3), (x;,x;,x;))s
l/4” for some
D".
n=l
Let S=A\(B u D). Clearly S=Int(Cl(S)) and S(i,c) is connected except for
S(3, c) for c 2 3 where 1%?(3,
c)( may equal 2. It nevertheless follows, as in
Example 3, that a completely separable function on S is also additively
separable there.
0
References
Aczel, J., 1966, Lectures on functional
equations
and their applications
(Academic Press, New
York).
Blackorby, C., D. Primont and R.R. Russell, 1978, Duality, separability,
and functional structure:
Theory and economic applications
(North-Holland,
New York).
Chateauneuf,
A. and P. Wakker, 1991, From local to global additive representation,
Journal of
Mathematical
Economics 22, no. 6.
Debreu, G., 1954, Representation
of a preference
ordering by a numerical
function, in: R.M.
Thrall, C.H. Coombs and R.L. Davis, eds., Decision processes (Wiley, New York).
Debreu, G., 1960, Topological
methods in cardinal utility theory, in: K.J. Arrow, S. Karlin and
P. Suppes, eds., Mathematical
methods
in the social sciences (Stanford
University
Press,
Stanford, CA).
Dugundji, J., 1966, Topology (Allyn and Bacon, Boston, MA).
Segal, U., 1991, Additively separable
representations
on non-convex
sets, Journal of Economic
Theory 56, 89-99.
Wakker,
P.P., 1989, Continuous
subjective
expected
utility with nonadditive
probabilities,
Journal of Mathematical
Economics
18, l-27.