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

production relationship, e.g., a man-hour is not always constant in
quality even when defined rather narrowly. However, barge and towboat
services seem particularly troublesome in this regard. These problems
exist quite apart from the problems associated with using ton-miles as
a measure of output. The ton-mile, being the product of two terms, is
not an unambiguous unit in which to measure output. Nevertheless, it
is a commonly used measure of output in transportation.
Both production and planning functions were estimated by Howe.
For the production functions, he used monthly time-series data of
three firms. For the planning functions, combined cross-section and
annual time-series data for six firms were employed. Howe assumed
log-linear production and planning functions. The inputs of the pro-
duction function were surrogates for barge and towboat services, and the
inputs of the planning function were barge and towboat stocks as well as
time. Output in both cases was cargo ton-miles. Howe assumed a demand
function facing the firm, and with this and the production (planning)
function, he maximized a profit function subject to the production
(planning) constraint. From the first-order conditions for a maximum,
input demand equations were obtained. However, Howe argued that an
adjustment lag for the barge and towboat inputs should be assumed for
the production function model because "we have omitted stochastic ele-
ments from our model and (most important) because the preceding period
(month) always leaves a legacy of geographical distribution of equip-
ment...."8 The introduction of the adjustment lags resulted in lagged
values of the barge and towboat inputs appearing in the input demand
equations. Finally two equations were added to each model: (1) a
24