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which matches the relative preference elasticities for x and y; anyhow, interior
solutions remain possible. In fact, the resulting “demand equations” are of the well-
known “linear expenditure” type.
Still another possible solution concept is of a Cournot-oligopoly nature. The
preference equations then are e.g. of type (14.6), plus the conditions (
14.1a
) and
(
14.1b
), leading up to the following solutions:
Þ
1
x
A
o
¼
ˁˁþ ˃
ð
ð
14
:
21a
Þ
Þ
1
x
B
o
¼
˃ˁþ ˃
ð
ð
14
:
21b
Þ
1
y
A
o
¼ 1
ˁ
ð
Þ
½
2
ˁ þ ˃
ð
Þ
ð
14
:
21c
Þ
y
B
o
1
¼
ð
1
˃
Þ
½
2
ˁ þ ˃
ð
Þ
ð
14
:
21d
Þ
These solutions have this time an
implicit
“price”:
1
x
B
o
y
A
o
p
¼
=
¼
˃
½
2
ˁ þ ˃
ð
Þ ˁ þ ˃
ð
Þ
ð
1
ˁ
Þ
ð
14
:
22
Þ
ρ
˃
¼
which is only equal to one for
1; again the solution needs in general a
feasible region exceeding the diagonal to the south-west, and using specification
(
14.19
) allows of obtaining exact or constrained binary
(0-1)
solutions.
It should finally be mentioned that the interior equilibria are not of the strong
type (again Friedman
1977
, p.168), except possibly for the mixed linear/non-linear
case and the extended Cobb-Douglas preference functions; a further remark is that
in the interior cases the existence of different strategies leads to multiple possible
equilibria, implying a variety of cases in terms of values of the outcomes for both
parties concerned.
+
14.2.3 Multiple Dimensions
A first case is that of more than two issues, some of them being commanded by
A
,
the others by
B.
Take the case of
A
handling only one,
B
two; Figs.
14.1
and
14.2
have then to be generalised to
3
dimensions, as illustrated by Fig.
14.3
:
A possibility exists that, even if in the (
x,y
)-direction the program is feasible,
B
imposes a constraint on
z
(see the line ending up at
z
, the
conjunction of both conditions excluding “symmetrical” solutions with respect to
the centre of gravity, so even if the (
x,y
)-problem might have a solution, the
generalised (
x,y,z
)-problem has (provisionally: see Sect.
14.2.4
) none. This example
shows, on the one hand that behind the scenes “implicit” constraints may lurk, and
have to be laid bare, on the other that issues are to be approached globally (see
Paelinck and van Gastel (
1991
), p.74).
The above multidimensional specification might also be used for the case of
three (or more) agents, each handling one (or several) of the issues; the analysis is
similar.
ʳ
) and
A
imposes
y
¼
