Firm with Two Plants

This Demonstration shows how to distribute the production of uniform goods between two or more (generally many) plants that have different total cost () functions. Put differently, it shows how to minimize total production costs of several plants subject to a given total production plan .
On the 3D plot, the quantity produced in the first plant is labeled and in the second plant . The constraint holds.

SNAPSHOTS

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DETAILS

The blue line in the 3D plot shows all optimal production combinations for different values of . Fixed costs in either plant (denoted as and , respectively) do not affect the distribution of production, which is logical because the sum of fixed costs exists even if no output is produced. At the same time, if the variable part of the costs for a plant are relatively lower, that plant starts to produce relatively more of the total output, all other things being equal.
This Demonstration deals with total cost, while the Demonstration "Monopoly with Two Plants" deals with marginal costs. As we may always get marginal costs from total costs, this Demonstration deals with a more general case that is solved as follows:
.
By differentiating with respect to and , this expression gives a system of two equations in two unknowns:
,
.
Here is, in fact, the marginal cost of the first plant and that of the second plant. The equation is the equality of marginal costs for each plant.
So, we have a solution:
,
.
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