Journal
of Mathematical
Economics
14 (1985) 129-134.
North-Holland
ON THE SEPARABILITY
OF THE QUASI CONCAVE
CLOSURE
OF AN ADDITIVELY
SEPARABLE FUNCTION
Uzi
SEGAL
University of Toronto, Toronto MS8 IA1 Canada
Received
May
1983, iinal version
This paper gives a characterization
closure also is additively separable.
of the additively
accepted
separable
March
1985
functions
whose
quasi
concave
1. Introduction
Let 2 be a binary relation on R’!+. The convexed relation 2 is defined by
xky
iff Vz[y~conv{w:w~z}]*[x~conv{w:w~z)]
[see Hildenbrand
(1974)]. Starr (1969), first formally presented this relation, which was used by
him and by others in equilibrium analysis of markets with non-convex
preference relations [see for example Shaked (1976)]. The question arises
which properties of the unconvexed relation hold also for the convexed one.
Segal (1983) proved that if the relation 2 on R’!+ can be represented by an
additively separable convex function of the form U(x,, . . . , x,) =I ui(xi) where
are all convex functions, then the convexed relation 2 can be
Ul,.
. . > u,
represented by an additively separable function if and only if there are U:
R+R,
d, ,..., d,>Q, and b, ,..., b, E R such that for every i, Ui(xi) = u(d,x,) + bi.
Let U be a representation of 2. Denote by U” the quasi concave closure
of U, the function which represents 2 and which is equal to U where
possible (for an exact definition see section 2 below). This paper gives a full
characterization of all the additively separable functions whose quasi concave
closure is a transformation of an additively separable function. The above
conditions are necessary and sufficient ones if U has first order derivatives
and if it is not quasi concave, even if it is not a convex function. In the
general case these conditions are not necessary. Necessary and sufficient
conditions for this case are presented in section 5.
2. Definitions and propositions
The following notations will be used: R”, = {x =(x1, . . . , x,):x1 2 0,. . . , x, 2 0},
x,>O},
A”={x~R!+:~x~=l),
and A’;=A”nR”,+.
R”++=(x~R”:x~>0,...,
03044068/85/$3.30
0
1985, Elsevier
Science Publishers
B.V. (North-Holland)
130
U. Segal, Separability of the quasi concave closure
A function U:R: +R is quasi concave (quasi convex) if for all x, PER;
and 0 5 2 5 1, U(Ax + (1 - 1)~) 2 min {U(x), U(y)} ( smax {U(x), U(y)>). Let
U:R’L,+R be a continuous and strictly increasing function [i.e., if for every
i, Xi~yi, and for some i, xi>yi, then U(x) > U(y)]. U* denotes the quasi
concave closure of U and is defined by U*(x) =inf {sup{ U(y):a(y-xx) =O>:
and”).
By the properties of U it follows that on the axes U* = U. If
then
the inf in the definition
of U” is obtained,
XER:.,
and it is obtained in a vector EAT (which is unique when n = 2), hence on
:CIE A: >. This function is unique,
RF,> U*(x)=min(max{U(y):a(y-x)=0}
and it is easy to verify that if U represents the relation 2, then U* represents
the convexed relation 2.
Henceforth, Ti = [ri, $1 denotes a closed segment on the ith axis.
3. The function 17
This section discusses the case of two variables (n=2). Let U(x,, x2) be a
strictly increasing function, and assume that U* is a transformation of an
additively separable function, that is, U*(x,, x2) = h(u:(x,) +$(x2)). Let A(U) =
(x: U*(x) # U(x)) (this set is relatively open in Rt).
Theorem 1. Let x E A(U). There exists a rectangle around x (bounded or
unbounded) on which both u: and u4 are linear, and such that on its boundary
(if exists) U* = U.
Proof: Let CIbe the vector in 4: such that U*(x) =max{ U(y):a(y-xx) =0}
and let H(x, a) = {y: cc(y-x) = O}. There is a segment L cH(x, a) around x on
which U* # U. I first prove that for every ZE L, U*(z)= U*(x). Let ZE L
and assume that U*(z) =max{ U(y):fi(y-z) =O>, where P#K Obviously,
&x-z)>O,
otherwise max{U(y):cc(y-z)=O)=max(U(y):cl(y-x)=O}=U*(x)
<max{U(y):B(y-x)=O}~max{U(y):/?(y-z)=O}=U*(z),
in contradiction
to the definition of U*.
It is easy to verify that there is a point w~H(x, a) such that U*(x) = U(w)
and bw </?z. U*(z) =max{U(y):P(y-z)
=0} >max{U(y):B(y-w)=O)ZU(w)=
m
U*(x)=max{U(y):cr(y-x)=0)
. contradiction to the definition of U*, hence
for every z E L, U*(z) = U*(x).
There is a (closed) rectangle C= r1 x TZ around x on which the indifference curves of U* are straight lines. I now prove that on C both u: and
u; are linear. It follows first that for every y=(y,, y,)~Int C, u: is differentiable at y, iff uz is differentiable at y,. Since an increasing function is almost
everywhere differentiable, these two functions are differentiable on C.
Let M and N be two indifference curves of U* (see fig. 1). Since U* is a
transformation of an additively separable function, points 1 and 6 are on the
U. Segal, Separability
of the quasi concave closure
131
“2
Fig. 1
5
same indifference curve of U*. The slope of this curve, which is a straight
line, equals the value of ur’/ur’ at point 1. On the segment 2-7 this ratio is
linear in x1, hence u:(xr) =a& +a,~, +a,. Similarly, one can prove that
ut’(x,) = b/(cx, + d), hence on C, u4 is a linear function, or the log of a linear
function. Since on C the indifference curves of U* are straight lines, both UT
and ut are linear.
Assume now that C is maximal in the sense that there is no other
rectangle C’ 1C on which u: and u2 are linear. Obviously, on the axes
U* = U. Assume that there is a point y E R: + on the boundary of C such that
U*(y) # U(y). There is a rectangle PI x Yz around y on which UT and ur are
linear. Since (rI+r2)
n(Yl
x F2)#&
one obtains that on (r, u Yr) x (r, u r;)
UT and ul are linear, in contradiction to the maximality of C. Q.E.D.
4. The function U: The differentiable case
Let x E A( U), and let C = rI x r2 be the maximal rectangle whose existence
has been proved in theorem 1. On each existing side of C which does not lie
on the axes, the indifference curves of U* and U are tangent to each other.
Hence, on each such side u’& is fixed, and the corresponding ui is linear.
132
U. Segal, Separability
of the quasi concave closure
Since on C UT and UT are linear, and since on the boundary of C, U* = U,
it follows that ai is linear on r1 iff Z.Qis linear on TZ. However, if both of
them are linear, then on C, U* = U. We assumed that U*(x)# U(x), hence
C= R:, that is, U* is a transformation of the function a.x for some a~ A:.
It is easy to verify that in this case, for every p, q, r, s 2 0 such that U(p, 0) =
U(0, q) = U(r, s), air + CQS1 a,p = a2q (that is, the indifference curve passing
through (p, 0) and (0, q) lies above the segment connecting these two points).
By the same technique used by Segal (1983), one can prove that there are
u:R++R,
d,, d,>O, and b,, b,ER, such that Ui(Xi)=u(dixi)+bi, i=l,2.
Obviously, this is also a sufficient condition. These results are summarized
in the following theorem.
Theorem 2. Let ul, u2: R, -rR be strictly increasing, diflerentiable functions.
The quasi concave closure of the function U(x,, x2) =u1(x1)+u2(xJ
is a
transformation of an additively separable function, tff U is quasi concave, or
alternatively, if for every p, q, r, s > 0 such that U(p, 0) = U(0, q) = U(r, s),
ps>q(p-r),
and if there are u:R.+R,
d,,d,>O,
and b,,b,ER
such that
ui(xi) = u(dixi) + b,, i = 1,2.
Remark.
Functions satisfying the second condition of this theorem are not
necessarily quasi convex. Counter-examples can easily be found.
5. The function ZJzThe non-differentiable case
be the maximal rectangle whose
Let XEA(U) and let C(x)=r,xr,
existence has been proven in theorem 1. As has been proved already, Ui on Ti
is non-linear, i= 1,2. Let ~EA( U) such that y1 E r1 but y, # TZ. There is a
maximal rectangle C(y) =Pr x r; around y on which UT and uz are linear
and such that on its boundary U* = U. Obviously, r1 = P1 and TXA rl, = 8.
Denote G,=u{C(x):x~(r~xR+)nA(U)).
If ClG,#r,
xR+, then there
exists a rectangle rr x Pi on which U* = U, that is, ul(xl) +u,(x,) = h(uT(x,)
+$(x2)).
h solves this functional equation only if it is a linear function [see
Aczel (1966)], and since on rl, UT is linear, u1 too is linear, a contradiction,
hence Cl G, =rr x R,.
Similarly, let G, = u {C(x):x E (R, x r,) A A(U)}, and it follows that Cl Gz =
R, x Tz. It thus follows that u1 and u2 are piecewise linear, and if y, is a
kink of UT, or yZ is a kink of u;, then u*(y,, yZ)= U(y,, y2). Moreover, if
between two kinks of UT, u1 is linear, or between two kinks of ~2, u2 is linear,
then A(U) = 8.
Let C = r1 x TZ = [yr, y’J x [y2, y;] be a maximal rectangle as in theorem 1.
Assume that on Ti, @(xi) = aixi + bi, i = 1,2, and let c = a1/a2. Assume, without
loss of generality, that U(y,, y;) = U(6, yJ, where Sjy;. Let &=6-y,.
On
U. Segal, Separability of
the quasi
concave
closure
133
[yl, S], u,(x,)=u,(c~~+K)+k,
on r2, u~(x,)=u,(c-‘x,+L)+1,
and on
[y; -8, y’J, ul(xl) =u,(cx, +K) +k’. Moreover, there exists M such that for
every x1 e[yl, 7; -81, ul(xl +~‘)=u,(x,)+
M. One can now prove that if
~=(y; -y&i
-yl) is an irrational number, then u1 on r1 is a linear
function. If E is a rational number, and on rl, u1 is not linear, then there exist
a minimal 5>0 and M, such that for every x1 E [yl, y; -51, ul(xl + 5) =
ul(xl) + M, and for every x2 E [y2, y)2-c<], u2(x2 + ct) =4(x2) + M. 5 does not
depend on x2, ~~(0) exists, and since V* is quasi concave, the ratio c is
decreasing with x2. It thus follows that there are sequences 0 = y: < yi < ... ,
d: 1 d: 2 . . ., Mi, M& . . , and ml, m:, . . . such that dJ;(yj2- y’,- ‘) = 5, and such
that on [y’,- ‘, ~$1, u2(x2) = u,(dJlx, + n/l’,) +mi. The same argument holds for
ul. These results are summarized in the following theorem.
Theorem 3. The quasi concave closure of the function V(x,,x,)=u,(x,)
+u2(xJ
is a transformation of an additively separable function, iff V satisfies
one of the two conditions of Theorem 2, or alternatively, if there are 5 >O,
u:[O,t]-R,
O=$<y!<...,
d!zd:z...,
and M!,M?,...
such that the segment &[yk
1 , yi]
j + M{
is equal to the segment [0,41, and such that on
[$‘,$J,
Li(xi)=u(d{xi+M$+(j-l)u({),
i=l,2, j=l,2,....
6. The case of n variables
Theorem 4.
Theorems 2 and 3 can be extended to functions of n variables.
ProoJ:
Let i#j and let (a, b)‘j=(O, . . . . O,s,O, . . . . O,F,O ,..., 0). For every
define
V’j:R:+R
Obviously,
by Vij(xl, x2) = V((x,, x,)‘j).
(V*)‘jz(V’j)*
(as before, V* denotes the quasi concave closure of V), and if
x1x2 = 0, then (V*)‘j=( Vij)*. Let (x1, x2) E R: +, and let (CQ,az) be the vector
in A: such that (V’j)*(x,, x,)=sup{V’j(y,,
~,):(a,, a,).((~~, y,)-(x,, xJ)=O}.
For every E> 0 define CC’
E A?+ by
V:R;-tR
c$=(l _&)@I,
k= i,
=(l -&)cQ,
k =j,
= e/(n - 2),
k$ {Li).
It is easy to verify that lim,,,sup{V(y):cr”(y-(x,,x,)ii)=O}
s(V’j)*(x,,
Hence (V*)‘js (U’j)*, and therefore, ( U*)ij= (U’j)*.
Let U(X)=CUi(Xi)
such that V*(x) = h(CuT(xJ).
On the axes V*
hence if for every i, ui is a linear transformation of u:, then h is linear.
every i, Ui is the same linear transformation of u*, and V*= V. It
follows that if A(V) # 8, i.e., if V is not quasi concave, then there exists i
x2).
= V,
For
thus
such
134
U.
Segal,
Separability of the quasi concave closure
that ui is not a linear transformation of ur. Since for every j# i, (U*)ij=
(w)*, and since (U*)‘j is a transformation
of an additively separable
function, it follows that (Vi’,* # Vii, otherwise ui is a linear transformation of
UT [see Aczel (1966)]. Theorems 2 and 3 apply now for i and every j, hence
theorem 4. Q.E.D.
References
Aczel, J., 1966, Lectures on functional equations and their applications (Academic Press, New
York).
Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press,
Princeton, NJ).
Segal, U., 1983, A theorem on the addivity of the quasi concave closure of an additive convex
function, Journal of Mathematical Economics 11, 261-266.
Shaked, A., 1976, Absolute approximations to equilibrium in markets with non-convex
oreferences. Journal of Mathematical Economics 3, 185-196.
Stair, R., 1964, Quasi equilibria in markets with non convex preferences, Econometrica 37,
25-38.