ECONOMICS DEPARTMENT

IN PRODUCTION FUNCTIONS

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 = S_{i}[f(*l _{i}*)]. 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)

subject to the constraint that

S

The first order conditions are that:

(2)

where v is the Lagrangian multiplier. This means that marginal labor productivity is the same in all the plants so the level of labor input is the same in all the plants. Thus the optimal allocation of firm input L is to divide it equally among all n plants.

The firm production from the n plants is nf(L/n). The
firm production function is:

(3)

Since n is a discrete variable the first order conditions are not as simple as if n were a continuous variable. Nevertheless for each value of L the optimal n is clearly defined. Suppose the optimal n is known for a given value of L, say n*. Then consider a labor input of sL. The optimal value of n for kL is simply sn* and the per plant labor input is the same as for a labor input of L and hence

(4)

Thus the firm production function has constant returns to scale.

If n is considered a continuous variable then the first order condition
for a maximum with respect to n is:

(5)

f(L/n) - [f'(L/n)(L/n)] = 0.

This is achieved where:

(6)

i.e., where marginal labor productivity equals average labor productivity and hence average labor productivity is a maximum. This means all plants are operated at a labor input of

(7)

This is obviously a constant returns to scale production function.

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)

the integral of f(

subject to the constraint that:

the integral of

This lead to the first order condition that the levels operation at all times are equal. Thus for a given T the level of production is thus:

(9)

The optimal choice of T requires the satisfaction of the first order condition:

(10)

f(L/T)- f(L/T)(L/T) = 0.

This is achieved where marginal labor productivity is equal to average labor productivity and thus average labor productivity is a maximum. Therefore the plant is always operated at a labor input of

(11)

Again this is obviously a constant returns to scale production function.

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.