## Economics of Multiplant Operation An Example

An example can help clarify the relation between firm size and plant size. Consider Plainfield Electronics, a New Jersey-based company that manufactures industrial control panels. The firm's production is consolidated at a single Eastern-seaboard facility, but a multiplant alternative to centralized production is being considered. Estimated demand, marginal revenue, and single-plant production plus transportation cost curves for the firm are as follows:

Plainfield's total profit function is n = TR - TC

The profit-maximizing activity level with centralized production is the output level at which Mn = MR - MC = 0 and, therefore, MR = MC.

Setting marginal revenue equal to marginal cost and solving for the related output quantity gives

MR = MC \$940 - \$0.04Q = \$40Q + \$0.02Q \$0.06Q = \$900 Q = 15,000

and n = -\$0.03(15,000)2 + \$900(15,000) - \$250,000 = \$6,500,000

Therefore, profits are maximized at the Q = 15,000 output level under the assumption of centralized production. At that activity level, MC = MR = \$640, and Mn = 0.

To gain insight regarding the possible advantages of operating multiple smaller plants, the average cost function for a single plant must be examined. To simplify matters, assume that multiplant production is possible under the same cost conditions described previously. Also assume that there are no other multiplant economies or diseconomies of scale.

The activity level at which average cost is minimized is found by setting marginal cost equal to average cost and solving for Q:

= (\$250,000 + \$40Q + \$0.01Q2)/Q = \$250,000Q-i + \$40 + \$0.01Q

Q =B25,000,000

Average cost is minimized at an output level of 5,000. This output level is the minimum efficient plant scale. Because the average cost-minimizing output level of 5,000 is far less than the single-plant profit-maximizing activity level of 15,000 units, the profit-maximizing level of total output occurs at a point of rising average costs. Assuming centralized production, Plainfield would maximize profits at an activity level of Q = 15,000 rather than Q = 5,000 because market-demand conditions are such that, despite the higher costs experienced at Q = 15,000, the firm can profitably supply output up to that level.

Because centralized production maximized profits at an activity level well beyond that at which average cost is minimized, Plainfield has an opportunity to reduce costs and increase profits by adopting the multiplant alternative. Although the single-plant Q = 15,000 profit-maximizing activity level and the Q = 5,000 average cost-minimizing activity level might suggest that multiplant production at three facilities is optimal, this is incorrect. Profits were maximized at Q = 15,000 under the assumption that both marginal revenue and marginal cost equal \$640. However, with multiplant production and each plant operating at the Q = 5,000 activity level, marginal cost will be lowered and multiplant production will entail a new, higher profit-maximizing activity level. Notice that when Q = 5,000,

With multiple plants all operating at 5,000 units per year, MC = \$140. Therefore, it is profitable to expand production as long as the marginal revenue obtained exceeds this minimum MC = \$140. This assumes, of course, that each production facility is operating at the optimal activity level of Q = 5,000.

The optimal multiplant activity level for the firm, assuming optimal production levels of Q = 5,000 at multiple plants, can be calculated by equating MR to the multiplant MC = \$140:

MR = \$140 = MC \$940 - \$0.04Q = \$140 \$0.04Q = \$800 Q = 20,000

Given optimal multiplant production of 20,000 units and average cost-minimizing activity levels of 5,000 units for each plant, multiplant production at four facilities is suggested:

„ ,. , ,T , r„, , Optimal Multiplant Activity Level

Optimal Number of Plants = —,.-^—4—--±r.—-—

Optimal Production per Plant

\$540(20,000) - 4[\$250,000 + \$40(5,000) + \$0.01(5,0002)] \$8,000,000

Given these cost relations, multiplant production is preferable to the centralized production alternative because it results in maximum profits that are \$1.5 million larger. As shown in Figure 8.7, this follows from the firm's ability to concentrate production at the minimum point on the single-plant U-shaped average cost curve.

Finally, it is important to recognize that the optimal multiplant activity level of 20,000 units described in this example is based on the assumption that each production facility produces exactly 5,000 units of output and, therefore, MC = \$140. Marginal cost will only equal \$140 with production of Q = 5,000, or some round multiple thereof (e.g., Q = 10,000 from two plants, Q = 15,000 from three plants, and so on). The optimal multiplant activity-level calculation is more complicated when this assumption is not met. Plainfield could not produce Q = 21,000 at MC = \$140. For an output level in the 20,000 to 25,000 range, it is necessary to equate marginal revenue with the marginal cost of each plant at its optimal activity level. 