Geoscience Reference
In-Depth Information
Table 14.1
Values of
preference functions
(0,0)
(1,0)
(0,1)
(1,1)
ˆ
A
0
1
2
3
ˆ
B
0
2
1
3
(
a*
>
1
), plus the constraints (
14.1a
) and (
14.1b
), and in addition:
x
A
,x
B
,y
A
,y
B
ð
:
Þ
0
1
14
5c
The solution, on the pay-off frontier, is known to be a strong equilibrium (on this
concept, see Friedman (
1977
), p. 168).
A question that could be raised is why not use a classical Edgeworth
box-diagram as a representation? The answer is that the way the problem was set
up here gives clearer insights into the workings of the model hypothezised, as the
previous and following reasonings show.
14.2.2 Non-Linear Case
Suppose now first the constraints to be non-linear; a reasonable assumption (
A
4
*)
is
that the absolute values of their slopes decrease as a function of the importance of
the issue (
x, y
) withheld. Figure
14.2
illustrates this, and also generates a possible
interesting case:
There is now a perfectly feasible
area
for negotiation, despite the fact that
α
and
ʲ
do not reach the extreme coordinates (
0,1
) and (
1,0
); indeed, all points belonging
to the closed area around
C
satisfy Eqs. (
14.1a
) and (
14.1b
). Full remittance,
however [solutions
A(0,1)
and
B(1,0)
], can once more only be attained if the
non-linear constraints reach those points.
As to the preference functions, some of them exclude rhe attainability of those
extreme points; such is the case of a Cobb-Douglas specification, as it has to “jump”
from a given isoquant for strictly positive values of
x
and
y
to a point with a zero
coordinate; quadratic functions—and as will be seen below, extended Cobb-
Douglas functions—do not suffer from that drawback.
Nevertheless, fractional solutions might
reveal
the presence of Cobb-Douglas
specifications, hence the following considerations on the case of Fig.
14.2
. Be it first
noticed that in the case of linear preference functions under the conditions of
Sect.
14.2
,A
and
B
will move out from their initial endowments until they hit the
limit of the feasible region, but nothing guarantees that conditions (
14.1a
) and
(
14.1b
) are in fact satisfied, moreover the separate optima not being necessarily
points on the diagonal; a simple graphical argument can show this. Under certain
circumstance—to be studied hereafter, to wit the introduction of a “relative
price”—the solution will lie on the diagonal, the point of shortest Euclidean
distance to
C
determining the relative positions of
A
and
B
.
If both
A
and
B
are driven by Cobb-Douglas specifications, the same problem
will arises, but the analytical computation of the (fractional) solutions is easier.
Suppose
A
to pick a point below the diagonal;
B
will then be stuck in the
