Geoscience Reference
In-Depth Information
x
B
¼
p
˃
ð
14
:
9
Þ
x
A
þ
py
A
¼
1
ð
14
:
10
Þ
x
B
þ
py
B
¼
p
ð
14
:
11
Þ
x
A
þ
x
B
¼
1
ð
14
:
12
Þ
y
A
þ
y
B
¼
1
ð
14
:
13
Þ
one of the equations being redundant. The solutions are then:
x
A
o
¼
ˁ
ð
14
:
14
Þ
x
B
o
ˁ
ð
:
Þ
¼
1
14
15
y
A
o
¼
˃
ð
14
:
16
Þ
y
B
o
¼
1
˃
ð
14
:
17
Þ
¼
˃
1
x
B
o
y
A
o
p
ð
1
ˁ
Þ
=
>
<
1
ð
14
:
18
Þ
provided they lie on the feasible domain; the “price” is clearly only equal to one if
˃
¼
, which was already hinted at in the discussion of the first solution concept.
The new variable
p
plays indeed the role of an exchange rate of
y
for
x
; for instance
p
1
ρ
1
implies that
A
exchanges more units of
x
than he would receive units of
y
.This
reminiscence of a Walrasian “umpire' leads up to the possible role of a third “good
offices” body, persuading (in the case studied,
A
) to accept the “price” given the
relative preferences of
A
and
B
.
A mixed linear non-linear solution is also possible, but the solution, as in the
double linear preference function case, is analytically less tractable.
It has already be said about the Cobb-Douglas specification that
(1,0)-(0,1)
solutions are excluded as a result of maximising behaviour, even asymptotically,
as “inferior” isoquants are nearer to the axes than 'superior” ones, without any of
them ever cutting those axes. A modified specification might include the extreme
points, e.g.:
>
1
ˁ
Þ
ˁ
y
A
þ ʸ
ˆ
A
¼
ð
x
A
þ
q
ð
Þ
ð
14
:
19
Þ
giving as “diagonal” optimal values:
Þ
1
x
A
o
ˁ ʸ
ð
1
þ
2
ʸ
!
¼
0, y
A
¼
1
ð
14
:
20a
Þ
1
1
1
1
Þ
1
ʸ
ð
1
þ
2
ʸ
< ˁ <
þ ʸ
þ
2
ʸ
!
x
A
o
¼
þ
2
ʸ
ˁ ʸ
Þ
1
ˁ
ʸ
ð
y
A
o
¼
ð
1
þ
2
ʸ
14
:
20b
Þ
Þ
1
x
A
o
1, y
A
o
ˁ
ð
1
þ ʸ
Þ
ð
1
þ
2
ʸ
!
¼
¼
0
ð
14
:
20c
Þ
