Journal
of Mathematical
Economics
11 (1983) 261-266.
North-Holland
A THEOREM ON THE ADDITIVITY OF THE QUASI-CONCAVE
CLOSURE OF AN ADDITIVE CONVEX FUNCTION*
Uzi SEGAL
Nufleld College, Oxford OX1 INF, UK
Received
September
In this paper a necessary and sufficient
of an additive convex function is given.
1982, accepted
condition
January
for the additivity
1983
of the quasi-concave
closure
1. Introduction
Most of utility theory depends on the assumption that the preference
relation is convex, i.e., that for all x the set {ylykx} is a convex set.
However, even if we do not assume that the preference relation is convex,
some of the results of utility theory still hold. To obtain these results, it is
convenient to define the ‘convexed’ relation 2‘ in the following way [see
Hildenbrand (1974)]: Given a relation 2 define the convexed relation 2’ by
=>[ XE C onv{wlwkz}], where ConvA means
x&‘y iff Vz [yEConv{wlw~z}]
the convex hull of the set A. However, not all the properties of the
unconvexed relation hold for the convexed one as well. In this paper it will
be assumed that the relation 2 can be represented by an additive convex
function of the form 1 ui(xi) (x,, . . . , x, E R), and it will be shown under what
conditions the relation 2’ can be represented by an additive function as well.
Note that since the utility function is not a quasi-concave one, we shall not
be able to use the technique which was used by Houthakker (1960) to show
the conditions under which both direct and indirect utility functions are
additive.
2. Definitions and propositions
Definitions
1. R”,={(x,
,..., x,)ER”Ix~~O ,..., x,20},
R”, + = {(x 1,..., x,)ER”Ix~>O ,..., x,>O}.
*This research was supported
Gorman for useful discussion.
03044068/83/$3.00
0
by the A.V. and J.E. Posnansky
1983, Elsevier
Science Publishers
Trusts.
I am also thankful
B.V. (North-Holland)
to T.
262
U. Segal, An additive convex function
2.
A function U: R: +R is quasi-concave (quasi-convex) if for all x, YE R;
and AE [O, 11, U(Ax + (1 - 1)~) L min {U(x), U(y)} ( 5 max {U(x), U(y)}).
3.
Let U:R” + +R be a continuous strictly increasing function (i.e., [for
all i, Xi Lyi, and, for some i, Xi >yJ*U(x)>
U(y)). The quasi-concave
closure of U will be denoted by U* and is defined by U*(x,, . . ., x,)
=min {max {U(yl, . . .,~~)lCUi(yi-~i)=O}l(~,,
. . ., a,)~ R: +}.
Remark.
The min and the max in the last definition
continuity and the monotonicity assumptions.
4.
The function U: R”, +R is a transformation
and h:R+R
there are ur,..., u,:R++R
is justified by the
of an additive function if
such that U(X~,...,X,,)
= h(C ui(xi)).
5.
The set of the indifference curves of a function
R}.
H(x r,. . . ,x,)E R; IU(x,,. . .,x,)=a}la~
U: R;+R
is the set
Propositibns
1. A convex function is quasi-convex.
such that, for all i, Ui is convex, then U is
2.
If U(Xl,..., X,)=CUi(Xi)
convex.
3.
If U represents
relationk”.
4.
The indifference curves of the quasi-concave closure of a quasi-convex
function are intersections of R: with hyperplanes.
5.
the relation
If U is a transformation
2,
then
U*
represents
the convexed
of an additive function, then, for all i #j # k # i,
(a/ax,)((aU/axi)/(XJ/axj))~o.
6.
If U = h 0 V, then there exists h* such that U* = h* 0 V*.
3. The theorem
Theorem.
Let u 1,. . . , u, : R + +R be strictly increasing differentiable convex
functions. The quasi-concave closure of the function U(x,, . . .,x,) =cui(xi)
is a
transformation of an additive function if and only if there are u: R+R and
positive d,, . . . , d, such that
w
Proof.
1,.
.
.)
Xn)=CU(diXi).
We shall prove first that the condition is a sufficient one. Let
ccording to the assumptions on U, u is convex and
U(x l,...,xn)=zu(dixi)..A
U. Segal, An additive convex function
strictly
increasing.
Obviously,
each indifference
263
curve of U* is of the form
Given
a point XE R”, it is easy to verify that x~H(xd~q)
and that
u*(x l,...,Xn)=U(Cdixi).
As for the necessity of the condition.
Let U(x,, . . . ,x,) =cui(xi)
be such
that, for all i, ui is a strictly increasing
differentiable
convex function
and
assume that U*(x,, . . . , x,) = h(~ui(xi)).
Assume first that u,(O) =. . . = u,(O) = 0
and define wi=u;roui.
Obviously,
for all i, W,(O) =O, Wi is a strictly
increasing differentiable
function and
U(x,, . . ., xn)
=C”l(wi(xi))
=“l(xl)
+
i$2
ul(wi(xi))~
Define
Yj(z)
=(wi(z),O,f ** O),
j=l,
9
=(O
o,goO),
)...)
j=i,
)...)
0, w,: ‘wyyz)), 0,. . . O),
=(O,...,
)
U(yi(z)) =. . . = U(yL(z)) hence
of the set {y~(z)}jnzI for some
Let x=(x,,0 ,..., O,xi,O ,...,
and x=~~Lx~~~(z). Obviously,
Xi =( 1 -c~~)z. We obtain that
j${Li}.
each indifference curve of U* is the convex hull
iE{l,...,n}
and ZER+.
0), ~20 and (c(~,..., c(,)ER;,
such that ccli=l
for j$(l, i}, olj =O, hence x1 =a,w,(z)
and
~1~=(z-xi)/z,
hence
zwi(z)-xiwi(z)-xxlz=o.
(1)
For all i #j denote
Zij(Xi,Xj)'U*(O,.
..,O,Xi,O,.
u*(x
r,
.
. .,x,,) = h(CUi(xi))
see Debreu
j(X;,X[i)
=
Zi
.,O,Xj,O,.
..,O).
and it is easy to verify that
[zij(Xi,XS)=Zij(X:,Xj)
Zi
.
A Zij(Xi,Xji)=Zij(XF,Xj)]*
j(X;l,
XJ);
(1959). Define
Ai={(O ,..., O,xi,O ,..., O)eR’!+).
264
U. &gal, An additive convex function
On thse se’t L\ A,.U* =U,. hence
Let a,flzO. zli(wi(a),O)=z,i(O,cr),
Zli(Wi(B),O)=Z,i(O,p), hence
In other words,
U*(Wi(p)pO,.
.
.,O,
Cty0, * * * ~0)~ U*(Wi(OI),Oy
and, by using eq. (l), we obtain that
two following equations:
ZWi(Z)
-
aWi(Z)
-ZWi(B)
there
exists
*
a.
yOy~,O,.
f*90)~
z=z(ac,
fl),
which solves the
=O,
(2)
ZWi(Z)-PWi(Z)-ZWi(a)=O.
Differentiating
these two equations implicitly yields
aZ
zwi(a)
wi(z)
aol=Wi(Z)-Wi(~)+(Z-a)w~(z)=Wi(Z)-Wi(cI)+(Z-~)W~(Z)’
aZ
WP)
wi(z)
ag=Wi(z)-Wi(j?)+(Z-CI)W:(Z)
=wi(z)-wi(c()+(z-~)w:(z)’
hence
Wi(Z)/ZW:(B)
=zw{(E)/w~(z)
or
(Wi(Z))’-z~w~(N)w~(/?)
=O.
As can be seen from (2), ~(a, 0) = 01,hence, for B = 0, we obtain
The s6nii~on ti ‘r?msr?iS5eren’ix$equakon js
Wi(ct)= tip/( 1+ Cic(),
and, since wi is an increasing function, di ~-0 and ci 20. Define
and we obtain that
U(X
1,.
.
.)
X,) =CU,(diXi/(
1+
CiXi)).
d, =
1, c1 =O,
U. Segal, An additive convex function
265
Let ~20 and define xi =z/(di -c~z). For all i, Ul(dixi/(l +cixi)) =
each indifference curve of U* is a set of the form
Us
hence
{(%Zl(4-c,z), . . . f O&Z/(&-Cnz))ICai= l, (cI1,.. *>am)E RT}>
and the value of U* over this set is ul(z). Let (x,, . . .,x,) E R’!+.
&iZ/(di
-CiZ)
=Xi(di -ciz)/z>
=Xi*cli
and, since Ccq = 1, we obtain that
1 +CCiXi,
CdiXt/z=
hence
u*(x 1,. . .,X,)=~1(~)=ul(CdiXi/(l
By Proposition
+CciXi))*
5 we obtain that, for all i # j # k # i,
(dic,-d,ci)(dj(l
+CClXl)-Cj Cd,xl)
-(djCk-dkCj)(dt(l
+CC,X,)
-ciCdlx,)~O.
Let x1=.,. =xn =x, and denote d=xd,,
c=Cct,
thus
Differentiating with respect to x yields
and we obtain that
ck = cd,/d.
c,=O and d,=l,
hence c=O and, therefore, c,=...=c,=O,
thus
U(x,, . . . 7xn) =Cu,(dixi)*
If U(0,. . . , 0)# 0, define
V(x
1,‘“,
By Proposition
X,)=U(X1,...,Xn)-U(O,...,O)=C(Ui(Xi)-Ui(O)).
6, V* is also a transformation
of an additive function,
U. Segal, An additive convexfunction
266
hence, by what has been proved already,
Define u(x) = u(x) +xu,(O)/n,
Remark. We actually
b 1,..., b,ER, such that
and obtain that
proved
that
there
are u: R-R,
dl, . . . , d, > 0 and
Ui(Xi) = U(dixi) + bi.
References
Debreu, G., 1959, Topological methods in cardinal utility theory, in: K.J. Arrow, S. Sarlin and P.
SupIpes, e&s., Matnemaficd~ m&to& ‘m ‘he so&h sktences ‘~&i&orb ihitver~try “rress,
Stanford, CA).
Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press,
Princeton, NJ).
Houthakker, H.S, t960, Additive preferences, Econometrica 28,. 244-257.