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

The solution rests, on the derivation of market boundaries along which
consumers are indifferent as between alternative suppliers because their
delivered prices are equal.
'Take a simple case involving two production sites only, the
boundary will be the locus of the points in space where the following
'equation holds:
(1)
Pa 1. CaZa 4. La 2 Pb CbZb 4. Lb .
where P a and P b are the prices or costs of productions at the points A and B
respectively,
C a and C b are the carrying costs per ton-mile from A and B,
Z a and Zb are 'the 'distances over which the product is carried from
A and B respectively,
L a and L b are the costs of loading and unloading a ton of commodity
produced at A and B.
If one assumes that C a = Cb , and L a = Lb , equation (1) becomes:
(2) , Ca (Za - Zb ) = Pb = Pa .
Let us define P b - P a i CabZab' where Z is the distance between A and B, and
ab
C is a coefficient such that the identity holds. Equation (2) becomes:
ab
(3)
(Z a - Z b ) = C ab /C a . Z ab .
As P b - P a is a constant value, so are Z ab and C ab . The cost of transportation
C a is also a constant, while Z a and Z b are variables. Equation (3) is therefor*
the equation of a hyperbola, defined as the locus of the points for which the
difference between the distances which separate them from two fixed points is
equal to a constant. Here the two fixed points are A and B. The position and
the shape of this hyperbola will depend on the right-hand-side of the equation,
i.e., on the value of the ratio C ab /Ca and on the distance Zab . Diagram 1
illustrates the various possibilities: If P b - Pa = 0, i.e., if CabZab = 0,
then Z a = Z b' and the boundary is a perpendicular straight line midway between
the two production points. If CabZab is greater than zero, and if the ratio
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