The theory of the firm and industry equilibrium

Martin J. Osborne
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2.2 Allocating output between two plants

A firm can produce output in one or both of two plants. The variable cost functions in the plants are VC1 and VC2. The firm wants to produce y units in total, and must decide how much to produce in each plant. As a cost-minimizer, the firm chooses the outputs y1 and y2 of the two plants to solve the problem
miny1,y2 VC1(y1) + VC2(y2) subject to y1 + y2 = y.
Isolate y2 from the constraint to obtain the equivalent problem
miny1 VC1(y1) + VC2(y − y1).
If a positive output is produced in both plants at the solution of this problem, this solution satisfies the condition that the derivative of VC1(y1) + VC2(y − y1) is zero, or SMC1(y1) − SMC2(y − y1) = 0 (recalling that SMC is the derivative to VC). Thus if (y*1, y*2) is the solution of the problem, we have
SMC1(y*1) = SMC2(y*2).
In words, if the firm produces a positive output in each plant, it allocates that output so that the marginal cost of production in each plant is the same.

Intuitively, if the marginal cost of production is higher in one plant than another and the firm is producing a positive output in the plant with the higher marginal cost, then the firm can reduce its cost by transferring one unit of output from the plant with the higher marginal cost to the plant with the lower marginal cost. If the marginal cost is an increasing function of output in each plant, then after the transfer of the unit of output the difference between the marginal costs will be less than it was originally. So long as the difference is positive, the firm can reduce its costs further by transferring output to the plant with the lower marginal cost. Only when the marginal costs in both plants are the same, or the output in the plant with the higher marginal cost is zero, can the firm not reduce its cost any further.

Thus at the cost-minimizing allocation of output

  • either the firm produces a positive output in each plant and the marginal cost in each plant is the same
  • or the firm produces all its output in one plant, and the marginal cost in this plant is no greater than the marginal cost at the output of zero in the other plant.
Example
A firm owns two plants, with VC functions
VC1(y1) = 3y2
1
VC2(y2) = y2
2.
How should it allocate output between the two plants if it wants to produce 80 units of output?

It should choose the allocation of output that makes the marginal cost in each plant the same, if there is such an allocation. First find the marginal costs:

SMC1(y1) = 6y1
SMC2(y2) = 2y2.
Thus the firm wants to choose y1 and y2 such that y1 + y2 = 80 and
6y1 = 2y2,
or
6y1 = 2(80−y1),
or
8y1 = 160,
or
y1 = 20 and y2 = 60.
Example
A firm owns two plants, with VC functions
VC1(y1) = 2y1
VC2(y2) = y2.
How should it allocate output between the two plants if it wants to produce y units of output?

The marginal cost functions in the two plants are

SMC1(y1) = 2
SMC2(y2) = 1.
Thus, no matter how the firm allocates its output, the marginal cost in the first plant is higher than the marginal cost in the second plant. Thus the cost-minimizing allocation assigns all the output to the second plant:
y*1 = 0 and y*2 = y.