Question

Marginal Cost—Booz, Allen & Hamilton

Booz, Allen & Hamilton Inc. is a large management consulting firm.* One service it provides to client companies is profitability studies showing ways in which the client can increase profit levels. The client company requesting the analysis presented in this case is a large producer of staple food. The company buys from farmers and then processes the food in its mills, resulting in a finished product. The company sells some food at retail under its own brands and some in bulk to other companies who use the product in the manufacture of convenience foods.

The client company has been reasonably profitable in recent years, but the management retained Booz, Allen & Hamilton to see whether its consultants could suggest ways of increasing company profits. The management of the company had long operated with the philosophy of trying to process and sell as much of its product as possible, since, they felt, this would lower the average processing cost per unit sold. The consultants found, however, that the client's fixed mill costs were quite low, and that, in fact, processing extra units made the cost per unit start to increase. (There are several reasons for this: the company must run on three shifts, machines break down more often, and so on.)

In this application, we shall discuss the marginal cost of two of the company's products. The marginal cost (approximate cost of producing an extra unit) of production for product A was found by the consultants to be approximated by the linear function

y = .133x + 10.09,
where x is the number of units produced (in millions) and y is the marginal cost. (Here the marginal cost is not a constant, as it was in the examples in the text.)

For example, at a level of production of 3.1 million units, an additional unit of product A would cost about

y = .133(3.1) + 10.09$10.50.
At a level of production of 5.7 million units, an extra unit costs $10.85. Figure 1 shows a graph of the marginal cost function from x = 3.1 to x = 5.7, the domain over which the function above was found to apply.

[ Chart 1 ]
The selling price for product A is $10.73 per unit, so that as shown in Figure 1, the company was losing money on many units of the product that it sold. Since the selling price could not be raised if the company were to remain competitive, the consultants recommended the production of product A be cut.

For product B, the Booz, Allen & Hamilton consultants found a marginal cost function given by

y = .0667x + 10.29,
with x and y as defined above. Verify that at a production level of 3.1 million units, the marginal cost is about $10.50, while at a production level of 5.7 million units, the marginal cost is about $10.67. Since the selling price of this product is $9.65, the consultants again recommended a cutback in production.

The consultants ran similar cost analyses of other products made by the company and then issued their recommendation: the company should reduce total production by 2.1 million units. The analysts predicted that this would raise profits for the products under discussion for $8.3 million annually to $9.6 million—which is very close to what actually happened when the client took the advice.

Exercises

1.   At what level of production, x, was the marginal cost of a unit of Product A equal to the selling price?
2.   Graph the marginal cost function for product B from x = 3.1 million units to x = 5.7 million units.
3.   Find the number of units for which the marginal cost equals the selling price for product B.
4.   For product C, the marginal cost of production is
y = .133x + 9.46.
(a)   Find the marginal costs at production levels of 3.1 million units and 5.7 million units.
(b)   Graph the marginal cost function.
(c)   For a selling price of $9.57, find the production level for which the cost equals the selling price.

Answer 1