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

(2)marginal productivity of the boat input is always decreasing,
(3)marginal productivities of both inputs eventually become negative.
, .
The first two results are not surprising. It is a standard assump-
tion of production theory that beyond some point the marginal produc-
tivities of inputs decrease. This behavior of marginal productivity
-
is usually referred to as the law of diminishing marginal productivity. 4
In the case of barge tows, there appears to be no stage of increasing .
returns for either input. The negativity of the boat-input marginal
productivity is probably due to the fact that too large a boat in a
given waterway can draw water from under the barge flotilla causing'
the latter to sink lower in the water thereby increasing resistance
substantiatly. The negativity of the barge-input marginal productiv-
ity probably results from the increase in flotilla size relative to,
the fixed width of the waterway. .
In addition to the above results, Howe examined the substitution
possibilities of both inputs by means of isoquants. The isoquants
exhibited the usual convexity to the origin. Moreover, the spacing
of the isoquants indicated decreasing returns to scale for the process.
Howe next derived unit cost curves (average linehaul cost per ton
mile) for the tow. These were obtained by attaching factor prices to
the boat and barge inputs. With horsepower fixed, varying the barge
input permitted the derivation of a "short-run" cost curve. A family
. . .,
of such curves was presented. The short-run curves possessed the
typically assumed U-shape. The implied envelope curve was quite flat
over a wide range of output rates, "implying a fairly broad range of
tow makeups capable of producing output at essentially the same aver-
age cost." 5
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