Supplement 5:  Cost minimization for a multiplant firm
The situation in which a multiplant firm is considering the cost minimizing way to allocate a fixed total output across its plants represents a good opportunity to explore the concept of marginal decision making and the distinction between marginal cost and average cost.
Consider a firm that produces its output using two plants.  The plants use somewhat different technologies and so have different cost functions.  Suppose that the firm is currently producing in such a way that the cost per unit of output in plant 1 is $50 and the cost per unit of output in plant 2 is $58.  Would it make sense for the firm to shift some of its production from plant 2 to plant 1?
To get students to see that it does not necessarily follow that plant 1 should produce more and plant 2 less, it may be helpful to consider a firm that wishes to produce 32 units of output.  Let q_1 and q_2 stand for the output at the different plants.  Plant 1 has a total cost function given by the equation
											(S5.1)
Plant 2 has a total cost function given by the equation
										(S5.2)
Figure S5.1 below shows the set up of this problem on an Excel worksheet. 
Figure S5.1:  Tabular set of multiplant cost minimization problem

Originally, students are asked to find how many of the 32 units of output should be produced in each plant and to keep track of the marginal and average cost in each plant as the output in each plant changes.  Formulas for marginal cost are computed by hand using calculus and then entered into cells F1 and F2.  Average cost cells are computed by taking C/q.  Q is simply the sum of q_1 and q_2.  The optimal distribution of output can be found using the Tools/Solver command.  In Figure S5.1, the Solver would be instructed to minimize cell D3 by changing the range of cells B1 through B2 (denoted in Excel as B1:B2) subject to the constraint that cell B3 equals 32.  
Notice that the cost minimizing results are characterized in this case by AC_1 < AC_2.  Students are encouraged to ensure that they are at the cost minimizing combination by attempting to respond to the AC inequality and producing more in plant 1 and less in plant 2.  Substituting q_1 = 9 and q_2 = 23, however, increases total cost to $1,800.  Finally, to see that the equality of marginal cost values is no accident, students are asked to repeat the process for 40 units of output and then 24.  Hopefully, such an exercise helps to drive home the point that the optimal allocation of output occurs where the marginal cost in each plant is driven into equality.  Whether or not the average cost in each plant is the same is largely irrelevant.  The basic approach in this problem can also be used to explore a situation where a firm is trying to maximize the output from a fixed amount of labor (marginal products will be equalized) or maximize the revenue from a certain output (marginal revenue from each customer class will be equalized).  The optimal price discrimination application can then be followed up by a situation in which the firm must follow a single price strategy.  This involves using the Tools/Solver command and maximizing the revenue cell subject to the constraint that the two price cells are equal.

Reference

Whigham, David.  1998.  Quantitative Business Methods Using Excel.  New York:  Oxford University Press.