It is perhaps not widely enough appreciated
among economists that the concept of a production function for a firm is
quite different from the concept of a production function for a plant. The
differenceis that for a firm there is an optimizing choice of the number
of plants. Even for a single plant the firm may make an optimizing choice
of the proportion of the year the plant is operated. As a result of the
optimization the production function for the firm will generally exhibit
constant returns to scale even if the plant production function does not.
In order to present the argument in its simplest form, let us consider first
a production function that depends only upon the labor input. Suppose
f(l) is the production function for a single plant. It is presumed
to have the general form shown below and average labor productivity reaches
a peak at l*.
The company production from n plant, Q, is the sum of the productions; i.e.,
Q = Si[f(li)]. The number of plants is a choice
of the firm in the long run, but here let us suppose n is given and determine
how to optimally allocate labor amongst the n plants.
The optimization problem is to maximize
(1)
The firm production from the n plants is nf(L/n). The
firm production function is:
(3)
If n is considered a continuous variable then the first order condition
for a maximum with respect to n is:
(5)
Now consider the special case in which the firm has only a single plant
and can choose the proportion of the year it is operated and the level of
operation while is operating. Let f(l) be the instantaneous
production function. The optimization problem involved in obtaining
the firm level production function is:
(8)
For the case of two inputs, labor and capital one must consider the average
and marginal productivity for bundles of inputs. Let the plant production
function be f(l,k). For an input combination (L,K) consider the
scale of operation of a plant of s where one unit of s is the bundle
(1, k), where k=K/L. This means that the scale level happens to be equal to
the labor input, but the scale represents the level of input of a bundle of
labor and capital and not just labor alone. To obtain the maximum production
for a given ratio of capital to labor one needs to find the level s* where
f(s,sk)/s is a maximum. Then the number of plants is S/s* where S=L and
the firm level production function is then:
(12)
Thus if the inputs are scaled up by a factor g there is just an increase in the number of plants by a factor of g and the output is increased by a factor of g. Thus the firm level production function has constant returns to scale for an capital ratio k.