Q2 Z2,(Q2) Z2(Q2) - Institute for Water Resources - U.S. Army

Z b as the distance in miles between any producing point and B,
as the fixed distance in miles between A and B.
For the time being, let us make an abstraction from all handling costs,
including the cost of transferring from trucks to barges. The boundary, or
indifference locus for this problem will be given by the following equation:
(Li)
which becomes
( 5 )
CZ :CZ b + CZ
ta t w ab
Z -z = C/C • Z
a b
ab.
w t
This equation is similar to equation (3) of the spatial competition model.
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The shape and the position of the boundary will depend on the same conditions
as in the former problem:
If C = C, Z a = Z b' and the frontier will be a perpendicular straight line .
.
midway between A and B.
If C > 0 and C> 0, then
w , t
if Cw /Ct = 1, Z a - Zb = Zab , the boundary degenerates into .
a simple line from B to the right in the axis of A-b; if
C /C > 1, all farmers will ship directly by truck to A; if
w t
0 < C w /C t < 1, the boundary is a hyperbola with vertex to
the right of the mid-point and curving around B.
These cases are the most interesting, although one could conceive
of cases where either C w or Ct would be smaller than zero. These would be cases
•
where one of the two modes is subsidized. The conditions for position and shape
of the boundary would be similar: if C w /Ct < 1, all grain producers will ship
to B and use the waterway;
if -I< C w /C t < 0, the boundary will be a hyperbola with
vertex to the left of the mid-point between A and B, and
curving around A.
These cases are in Diagram 2, which is similar to Diagram 1. Note that, as
above, the distance between the mid-point and the vertex is equal to C w /2Ct • Z.
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