We have now discussed the data which the firm needs for its decision-making—the demand for its products and the cost of supplying them. But, even with this information, in order to determine what decisions are optimal it is still necessary to find out the businessman's aims. The decision which best serves one set of goals will not usually be appropriate for some other set of aims.

1. Alternative Objectives of the Firm

There is no simple method for determining the goals of the firm (or of its executives). One thing, however, is clear. Very often the last person to ask about any individual's motivation is the person himself (as the psychoanalysts have so clearly shown). In fact, it is common experience when interviewing executives to find that they will agree to every plausible goal about which they are asked. They say they want to maximize sales and also to maximize profits; that they wish, in the bargain, to minimize costs; and so on. Unfortunately, it is normally impossible to serve all of such a multiplicity of goals at once.

For example, suppose an advertising outlay of half a million dollars minimizes unit costs, an outlay of 1.2 million maximizes total profits, whereas an outlay of 1.8 million maximizes the firm's sales volume. We cannot have all three decisions at once. The firm must settle on one of the three objectives or some compromise among them.

Of course, the businessman is not the only one who suffers from the desire to pursue a number of incompatible objectives. It is all too easy to try to embrace at one time all of the attractive-sounding goals one can muster and difficult to reject any one of them. Even the most learned have suffered from this difficulty. It is precisely on these grounds that one great economist was led to remark that the much-discussed objective of the greatest good for the greatest number contains one "greatest" too many.

It is most frequently assumed in economic analysis that the firm is trying to maximize its total profits. However, there is no reason to believe that all businessmen pursue the same objectives. For example, a small firm which is run by its owner may seek to maximize the proprietor's free time subject to the constraint that his earnings exceed some minimum level, and, indeed, there have been cases of overworked businessmen who, on medical advice, have turned down profitable business opportunities.

It has also been suggested, on the basis of some observation, that firms often seek to maximize the money value of their sales (their total revenue) subject to a constraint that their profits do not fall short of some minimum level which is just on the borderline of acceptability. That is, so long as profits are at a satisfactory level, management will devote the bulk of its energy and resources to the expansion of sales. Such a goal may, perhaps, be explained by the businessman's desire to maintain his competitive position, which is partly dependent on the sheer size of his enterprise, or it may be a matter of the interests of management (as distinguished from shareholders), since management's salaries may be related more closely to the size of the firm's operations than to its profits, or it may simply be a matter of prestige.

In any event, though they may help him to formulate his own aims and sometimes be able to show him that more ambitious goals are possible and relevant, it is not the job of the operations researcher or the economist to tell the businessman what his goals should be. Management's aims must be taken to be whatever they are, and the job of the analyst is to find the conclusions which follow from these objectives—that is, to describe what businessmen do to achieve these goals, and perhaps to prescribe methods for pursuing them more efficiently.

The major point, both in economic analysis and in operations-research investigation of business problems, is that the nature of the firm's objectives cannot be assumed in advance. It is important to determine the nature of the firm's objectives before proceeding to the formal model-building and the computations based on it. As is obviously to be expected, many of the conclusions of the analysis will vary with the choice of objective function. However, as some of the later discussion in this chapter will show, a change in objectives can, sometimes surprisingly, leave some significant relationships invariant. Where this is true, it is very convenient to find it out in advance before embarking on the investigation of a specific problem. For if there are some problems for which the optimum decision will be the same, no matter which of a number of objectives the firm happens to adopt, it is legitimate to avoid altogether the difficult job of determining company goals before undertaking an analysis.

2. The Profit-Maximizing Firm

Let us first examine some of the conventional theory of the profit-maximizing firm. In the chapter on the differential calculus, the basic marginal condition for profit maximization was derived as an illustration. Let us now rederive this marginal-cost-equals-marginal-revenue condition with the aid of a verbal and a geometric argument.

The proposition is that no firm can be earning maximum profits unless its marginal cost and its marginal revenue are (at least approximately) equal, i.e., unless an additional unit of output will bring in as much money as it costs to produce, so that its marginal profitability is zero.

It is easy to show why this must be so. Suppose a firm is producing 200,000 units of some item, x, and that at that output level, the marginal revenue from x production is $1.10 whereas its marginal cost is only 96 cents. Additional units of x will, therefore, each bring the firm some 14 cents = $1.10 — 0.96 more than they cost, and so the firm cannot be maximizing its profits by sticking to its 200,000 production level. Similarly, if the marginal cost of x exceeds its marginal revenue, the firm cannot be maximizing its profits, for it is neglecting to take advantage of its opportunity to save money—by reducing its output it would reduce its income, but it would reduce its costs by an even greater amount.

We can also derive the marginal-cost-equals-marginal-revenue proposition with the aid of Figure 1. At any output, OQ, total revenue is represented by the area OQPR under the marginal revenue curve (see Rule 9 of Chapter 3). Similarly, total cost is represented by the area OQKC immediately below the marginal cost curve. Total profit, which is the difference between total revenue and total cost is, therefore, represented by the difference between the two areas—that is, total profits are given by the lightly shaded area TKP minus the small, heavily shaded area, RTC. Now, it is clear that from point Q a move to the right will increase the size of the profit area TKP. In fact, only at output OQ_mwill this area have reached its maximum size—profits will encompass the entire area TKMP.

But at output OQ_mmarginal cost equals marginal revenue—indeed, it is the crossing of the marginal cost and marginal revenue curves at that point which prevents further moves to the right (further output increases) from adding still more to the total profit area. Thus, we have once again established that at the point of maximum profits, marginal costs and marginal revenues must be equal.

Before leaving the discussion of this proposition, it is well to distinguish explicitly between it and its invalid converse. It is not generally true that any output level at which marginal cost and marginal revenue happen to be equal (i.e., where marginal profit is zero) will be a profit-maximizing level. There may be several levels of production at which marginal cost and marginal revenue are equal, and some of these output quantities may be far from advantageous for the firm. In Figure 1 this condition is satisfied at output OQ_tas well as at OQ_m. But at OQ_tthe firm obtains only the net loss (negative profit) represented by heavily shaded area RTC. A move in either direction from point Q_twill help the firm either by reducing its costs more than it cuts its revenues (a move to the left) or by adding to its revenues more than to its costs. Output OQ_tis thus a point of minimum profits even though it meets the marginal profit-maximization condition, "marginal revenue equals marginal cost."

This peculiar result is explained by recalling that the condition, "marginal profitability equals zero," implies only that neither a small increase nor a small decrease in quantity will add to profits. In other words, it means that we are at an output at which the total profit curve (not shown) is level—going neither uphill nor downhill. But while the top of a hill (the maximum profit output) is such a level spot, plateaus and valleys (minimum profit outputs) also have the same characteristic—they are level. That is, they are points of zero marginal profit, where marginal cost equals marginal revenue.

We conclude that while at a profit-maximizing output marginal cost must equal marginal revenue, the converse is not correct—it is not true that at an output at which marginal cost equals marginal revenue the firm can be sure of maximizing its profits.

3. Application: Pricing and Cost Changes

The preceding theorem permits us to make a number of predictions about the behavior of the profit-maximizing firm and to set up some normative "operations research" rules for its operation. We can determine not only the optimal output, but also the profit-maximizing price with the aid of the demand curve for the product of the firm. For, given the optimal output, we can find out from the demand curve what price will permit the company to sell this quantity, and that is necessarily the optimal price. In Figure 1, where the optimal output is OQ_mwe see that the corresponding price is Q_mP_mwhere point P_mis the point on the demand curve above Q_m(note that P_mis not the point of intersection of the marginal cost and the marginal revenue curves).

It was shown in the last section of Chapter 4 how our theorem can also enable us to predict the effect of a change in tax rates or some other change in cost on the firm's output and pricing. We need merely determine how this change shifts the marginal cost curve to find the new profit-maximizing price-output combination by finding the new point of intersection of the marginal cost and marginal revenue curves. Let us recall one particular result for use later in this chapter—the theorem about the effects of a change in fixed costs. It will be remembered that a change in fixed costs never has any effect on the firm's marginal cost curve because marginal fixed cost is always zero (by definition, an additional unit of output adds nothing to fixed costs). Hence, if the profit-maximizing firm's rents, its total assessed taxes, or some other fixed cost increases, there will be no change in the output-price level at which its marginal cost equals its marginal revenue. In other words, the profit-maximizing firm will make no price or output changes in response to any increase or decrease in its fixed costs! This rather unexpected result is certainly not in accord with common business practice and requires some further comment which will be supplied presently.

4. Extension: Multiple Products and Inputs

The firm's output decisions- are normally more complicated, even in principle, than the preceding decisions suggest. Almost all companies produce a variety of products and these various commodities typically compete for the firm's investment funds and its productive capacity. At any given time there are limits to what the company can produce, and often, if it decides to increase its production of product x, this must be done at the expense of product y. In other words, such a company cannot simply expand the output of x to its optimum level without taking into account the effects of this decision on the output of y.

For a profit-maximizing decision which takes both commodities into account we have a marginal rule which is a special case of Rule 2 of Chapter 3:

Any limited input (including investment funds) should be allocated between the two outputs x and уin such a way that the marginal profit yield of the input, i, in the production of x equals the marginal profit yield of the input in the production of y.

If the condition is violated the firm cannot be maximizing its profits, because the firm can add to its earnings simply by shifting some of г out of the product where it obtains the lower return and into the manufacture of the other.

Stated another way, this last theorem asserts that if the firm is maximizing its profits, a reduction in its output of x by an amount which is worth, say, $5, should release just exactly enough productive capacity, C, to permit the output of уto be increased $5 worth. For this means that the marginal return of the released capacity is exactly the same in the production of either x or y, which is what the previous version of this rule asserted.³

Still another version of this result is worth describing: Suppose the price of each product is fixed and independent of output levels. Then we require that the marginal cost of each output be proportionate to its price, i.e., that

where P_xand MC_Xare, respectively, the price and the marginal cost of x, etc.

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