Theory of the Firm (copy)

In developing the supply and demand approach to economics, economists first worked out the basis of the demand curve. By treating the demand for a product or service as a rational decision by a (primarily) self-interested individual or family, economists were able to understand the relation of the demand for one product or service to the demands for other products and services and to many other forms of economic activity. It was natural to apply the same approach to supply. As a first step, we need to think about the decision-makers in supplying goods and services, and what a “rational decision” to supply goods and services would mean. In economics, this is often called the “Theory of the Firm.”

In the remainder of this chapter we will apply the concepts of marginal productivity and diminishing returns to the theory of the firm. First we will talk a bit about business firms and their role in a market economy, then we will return to the marginal productivity approach.


About Firms (copy)

A firm is a unit that does business on it’s own account. (Firm is from the Italian, “firma,” a signature, and the idea is that a firm can commit itself to a contract). Thus, the firm is the decision-maker in supplying goods and services.

There are three main kinds of firms in modern market economies:

A proprietorship (or proprietary business) is a business owned by an individual, the “proprietor.” Many “Mom and Pop stores” — and other “Mom and Pop” businesses — are proprietorships. Some proprietorships are too small even to employ one person full time. Craftsmen, such as plumbers and painters, may have “day jobs” and work as self-employed proprietors part time after hours. Computer programmers and others may also do that. At the other extreme, some proprietary businesses employ many hundreds of workers in a wide range of specializations. In a proprietorship, the proprietor is almost always the decision-maker for the business.
A partnership is a business jointly owned by two or more persons. In most partnerships, each partner is legal liable for debts and agreements made by any partner. Of course, this requires a great deal of trust, and thus partners generally know one another well enough to have that sort of trust. Family partnerships are very common for that very reason. (There are now a few “limited partnerships” in which some partners are protected from legal liability for the agreements made by others, beyond some limits). In many cases, one partner is designated as the managing partner and is the main decision-maker for the business.
A corporation has two characteristics that distinguish it from most proprietorships and partnerships:
  • Limited liability
  • Anonymous ownership
Limited liability means that the owner of shares in a corporation cannot lose more than a certain amount if the company fails. Usually the amount is the money paid to buy the shares. Anonymous ownership means that the owner of the shares can sell them without getting the permission of anyone other than the buyer. By contrast, in most partnerships, no one partner can sell out without getting the agreement of the other partners. In such a case the continuing partners will, of course, want to know about the new partner — he will not be an “anonymous owner.” In a typical corporation, the shareholders formally elect a board of directors, who in turn select the officers of the company. One of these officers, often called the “president,” will be the principle decision-maker for the firm, but he will be expected to make decisions in the interest of the shareholders.

While there are millions of proprietorships, typically very small, the biggest businesses are corporate and corporations are particularly important because of their size.


Objectives (copy)

As we recall, Malthus did not have firms in mind when he formulated the Law of Diminishing Returns. But this law has applications Malthus did not envision, and we will see how to apply the law to a business firm. In the Reasonable Dialog of economics in the nineteenth century, the development of these ideas was a bit indirect. In about the eighteen-seventies, economists were rethinking the theory of consumer demand. They applied a version of “diminishing returns” and the Equimarginal Principle to determine how a consumer would divide up her spending among different consumer goods. (We’ll get into that in another chapter). That worked pretty well, and so some other economists, especially the American economist John Bates Clark, tried using the same approach in the theory of the firm. These innovations were the beginning of Neoclassical Economics.

Following the Neoclassical approach, we will interpret “rational decisions to supply goods and services” to mean decisions that maximize — something! What does a supplier maximize? The operations of the firm will, of course, depend on its objectives. One objective that all three kinds of firms share is profits, and it seems that profits are the primary objective in most cases. We will follow the neoclassical tradition by assuming that firms aim at maximizing their profits.

There are two reasons for this assumption. First, despite the growing importance of nonprofit organizations and the frequent calls for corporate social responsibility, profits still seem to be the most important single objective of producers in our market economy. Thus it is the right place to start. Second, a good deal of the controversy in the reasonable dialog of economics has centered on the implications of profit motivation. Is it true, as Adam Smith held, that the “invisible hand” leads profit-seeking businessmen to promote the general good? To assess that question, we need to understand the implications of profit maximization.


Profit (copy)

Profit is defined as revenue minus cost, that is, as the price of output times the quantity sold (revenue) minus the cost of producing that quantity of output.

However, we need to be a little careful in interpreting that. Remember, economists understand cost as opportunity cost — the value of the opportunity given up. Thus, when we say that businesses maximize profit, it is important to include all costs — whether they are expressed in money terms or not.

For example, a cab-driver — the self-employed proprietor of an independent cab service — says: “I’m making a ‘profit,’ but I can’t take home enough to support my family, so I’m going to have to close down and get a job.” The proprietor is ignoring the opportunity cost of her own labor. When those opportunity costs are taken into account, we will find that he is not really making a profit after all.

Let’s say that the cab-driver makes $500 a week driving his cab, after all expenses (gasoline, maintenance, etc.) have been taken out. Suppose he can get wages (including tips!) of $800 driving for someone else, with hours no longer and about the same conditions otherwise. Then $800 is the opportunity cost of his labor, and after we deduct the opportunity cost from his $500 net as an independent cabbie, he is actually losing $300 per week.

This is one of the most important reasons for using the opportunity cost concept: it helps us to understand the circumstances that will lead people to get into and out of business.

Because accountants traditionally considered only money costs, the net of money revenue minus money cost is called “accounting profit.” (Actually, modern accountants are well aware of opportunity cost and use the concept for special purposes). The economist’s concept is sometimes called “economic profit.” If there will be some doubt as to which concept of profit we mean, we will sometimes use the terms “economic profit” or “accounting profit” to make it clear which is intended.


The Firm’s Decision (copy)

In the short run, then, there are only two things that are not given in the John Bates Clark model of the firm. They are the output produced and the labor (variable) input. And that is not actually two decisions, but just one, since labor input and output are linked by the “production function.” Either

  • the output is decided, and the labor input will have to be just enough to produce that output
  • the labor input is decided, and the output is whatever that quantity of labor can produce.

Thus, the firm’s objective is to choose the labor input and corresponding output that will maximize profit.

Let’s continue with the numerical example in the first part of the chapter. Suppose a firm is producing with the production function shown there, in the short run. Suppose also that the price of the output is $100 and the wage per labor-week is $500. Then let’s see how much labor the firm would use, and how much output it would produce, in order to maximize profits.

The relationship between labor input and profits will look something like this:

Figure 7(not reproduced here): Labor Input and Profits in the Numerical Example

In the figure, the green curve shows the profits rising and then falling and the labor input increases. Of course, the eventual fall-off of profits is a result of “diminishing returns,” and the problem the firm faces is to balance “diminishing returns” against the demand for the product. The objective is to get to the top of the profit hill. We can see that this means hiring something in the range of four to five hundred workers for the week. But just how many?

The way to approach this problem is to take a bug’s-eye view. Think of yourself as a bug climbing up that profit hill. How will you know when you are at the top?


The Marginal Approach (copy)

The bug’s-eye view is the marginal approach. However much labor is being employed at any given time, the really relevant question is, supposing one more unit of labor is hired, will profits be increased or decreased? If one unit of labor is eliminated, will profits increase or decrease? In other words, what does one additional labor unit add to profits? What would elimination of one labor unit subtract from profits?

We can break that question down. Profit is the difference of revenue minus cost. Ask, “What does one additional labor unit add to cost? What does one additional labor unit add to revenue?

The first question is relatively easy. What one additional labor unit will add to cost is the wage paid to recruit the one additional unit.

The second question is a little trickier. It’s easier to answer a related question: “What does one additional labor unit add to production?” By definition, that’s the marginal product — the marginal product of labor is defined as the additional output as a result of increasing the labor input by one unit. But we need a measurement that is comparable with revenues and profits, that is, a measurement in money terms. Since the price is given, the measurement we need is the Value of the Marginal Product:

Value of the Marginal Product
The Value of the Marginal Product is the product of the marginal product times the price of output. It is abbreviated VMP.

To review, we have made some progress toward answering the original question. Adding one more unit to the labor input, we have

increase in revenue = value of marginal product
increase in cost = wage

So the answer to “What will one additional labor unit add to profits?” is “the difference of the Value of the Marginal Product Minus the wage.” Conversely, the answer to “What will the elimination of one labor unit add to profits?” is “the wage minus the Value of Marginal Product of Labor.” And in either case the “addition to profits” may be a negative number: either building up the work force or cutting it down can drag down profits rather than increasing them.

So, again taking the bug’s-eye view, we ask “Is the Value of the Marginal Product greater than the wage, or less?” If greater, we increase the labor input, knowing that by doing so we increase profits by the difference, VMP-wage. If less, we cut the labor input, knowing that by doing so we increase profits by the difference, wage-VMP. And we continue doing this until the answer is “Neither.” Then we know there is no further scope to increase profits by changing the labor input — we have arrived at maximum profits.


Marginal Productivity (copy)

Productivity, by definition, is a ratio of output to labor input. In most statistical discussions of productivity, we refer to the average productivity of labor.

Average labor productivity is an important concept, especially in macroeconomics. In microeconomics, however, we will focus more on the marginal productivity. We can think of the

marginal productivity of labor as
the additional output as a result of adding one unit of labor, with all other inputs held steady and ceteris paribus.


Let’s have a numerical example to illustrate the application of the theory. Suppose that:

  • When 300 labor-daysper week are employed the firm produces 2505 units of output per week.
  • When 400 labor-days per week are employed the firm produces 3120 units of output per week.
  • It follows that the change in labor input, Labor, is 100.
  • It also follows that the change in output, Output, is 615.
  • Applying the formula above, we approximate the marginal productivity of labor by the quotient 615/100 = 6.15.
  • We can interpret this result as follows: over the range of 300 to 400 man-days of labor per week, each additional worker adds approximately 6.15 units to output.

Of course, if we had more information, we could get a closer approximation. For example, if we had the outputs for 310, 320, … 390 man-days of labor, we could see how MP varies within the range 300-400. But we can be sure that the values will be in the neighborhood of 6.15.

Now let’s think a little further about the Law of Diminishing Returns.


The Law of Diminishing Marginal Productivity (copy)

In his discussions of the Law of Diminishing Returns, Malthus did not distinguish between average and marginal productivity. However, in modern economics, we think of diminishing returns primarily in terms of marginal, not average, productivity. Thus, we would state the law this way:

Law of Diminishing Returns (Modern Statement):
When the technology of production and some of the inputs are held constant and the quantity of a variable input increases continually, the marginal productivity of the variable input will eventually decline.

The inputs that are held steady are called the “fixed inputs.” In these pages we are treating land and capital as fixed inputs. The inputs that are allowed to vary are called the “variable inputs.” In these pages we are treating labor as the variable input.

Another way to express the law of diminishing returns, is that, as the variable input increases, the output also increases, but at a decreasing rate. The marginal productivity of labor is the rate of increase in output as the labor input increases. To say that output increases at a decreasing rate when the variable input increases is another way to say that the marginal productivity declines.


Marginal Productivity Example (copy)

Let’s extend the numerical example in the page before last and see how marginal productivity varies over a wide range of labor inputs. Here is a hypothetical example of production with the inputs of land and labor held steady and varying quantities of labor, and the output and average and marginal productivities.


Labor Output Average
0 0 0
100 945 9.45
200 1780 8.90
300 2505 8.35
400 3120 7.80
500 3625 7.25
600 4020 6.70
700 4305 6.15
800 4480 5.60
900 4545 5.05
1000 4500 4.50



The relationship between average and marginal productivity in the diagram is important in itself, and we will see similar relationships in future chapters. So let’s look at it a little more closely. Average and marginal productivity will not always have the same slope. In general,

  • whenever average productivity is greater than marginal productivity, average productivity will slope downward.
  • whenever average productivity is less than marginal productivity, average productivity will slope upward.

The diagram does not show any values where average productivity is less, but a more complicated example might, and then we would see the second part of the relationship visualized.

To understand the relationship, think of it this way: as we add labor input, one unit after another, we add a bit more to output at each step. When the addition is greater than the average, it pulls the average up toward it. When the addition is less than the average, it pulls the average down toward it.

By taking the marginal approach — the bug’s-eye view — we have discovered the diagnostic rule for maximum profits. The way to maximize profits then is to hire enough labor so that


where p is the price of output and VMP = p*MP the marginal productivity of labor in money terms.

This is another instance of the Equimarginal Principle. The rule tells us that profits are not maximized until we have adjusted the labor input so that the marginal product in labor, in dollar terms, is equal to the wage. Since the wage is the amount that the additional (marginal) unit of labor adds to cost, we could think of the wage as the “marginal cost” of labor and express the rule as “value of marginal product of labor equal to marginal cost.”


Profit Maximization Example (copy)

In our numerical example, suppose that the price of output is $100 per unit and the wage is $500 per worker per period. Then the p*MP, wage, and profits will be something like this:

Labor Marginal
p*MP Wage Accounting
0 500 0
9.45 945
100 500 44500
8.35 835
200 500 78000
7.25 725
300 500 100500
6.15 615
400 500 112000
5.05 505
500 500 112500
3.95 395
600 500 102000
2.85 285
700 500 80500
1.75 175
800 500 48000
0.65 65
900 500 4500
-0.45 -55
1000 500 -50000


What we see in the table is that the transition from 400 to 500 units of labor gives p*MP=505, very nearly VMP=wage. And that is the highest profit. So the profit-maximizing labor force is about 500 units.

We can get a more exact answer by looking at a picture or tinkering with the program example a bit. Here is a picture of the profit-maximizing hiring in this example:

The picture suggests that the exact amount is a bit less than 500 units of labor. If you tinker with the program example enough, you will see that the exact profit maximizing labor input is 454.54545454545 … units of labor — a repeating decimal fraction.

Notice the shaded area between the VMP curve and the price (wage) line. n the picture, the area of the shaded triangle is the total amount of payments for profits, interest, and rent — in other words, everything the firm pays out for factors of production other than labor. The rectangular area below the wage line and left of the labor=454 line is shows the wage bill. Thus, the John Bates Clark model provides us with a visualization of the division of income between labor and property.


Profit Maximization (copy)

We can use the diagram also to understand why VMP=wage is the diagnostic that tells us the profit is at a maximum. Suppose the labor input is less than 500 — for example, suppose labor input is 200. Than an additional labor-day of labor will add about 7.8 units to output, and about $780 to the firm’s sales revenue, but only $500 to the firm’s costs, adding roughly $220 to profits. So it is profitable to increase the labor input from 200, or, by the same reasoning, from any labor input less than $500.

This difference between the VMP and the wage is the increase or decrease in profits from adding or subtracting one unit of labor. It is sometimes called the marginal profit and (as we observed in studying consumers’ marginal benefits) the absolute value of the marginal profits is a measure of unrealized potential profits. That’s why the businessman wants to adjust the labor input so that VMP-wage=0.

Let’s try one more example. Suppose the labor input is 800 labor-days per week. If the firm “downsizes” to 799 labor-days, it reduces its output by just about 1.2 units and its sales revenue by about $120, but it reduces its labor cost by $500, increasing profits by about $380. Thus a movement toward the VMP=wage again increases profits by realizing some unrealized potential profit.

The formula VMP=wage is a diagnostic for maximum profits because it tells us that there is no further potential to increase the profits by adjusting the labor input — marginal profit is zero.

The marginal productivity rule is the key to maximization of profits in the short run.


We have seen that the concept of marginal productivity and the law of diminishing marginal productivity play central parts in both the efficient allocation of resources in general and in profit maximization in the John Bates Clark model of the business firm.

The John Bates Clark model and the principle of diminishing marginal productivity provide a good start on a theory of the firm and of supply. In applying the marginal approach and the equimarginal principle to profit maximization, it extends our understanding of the principles of efficient resource allocation. Some key points in the discussion have been

  • the distinction between marginal productivity and average productivity
  • the “law of diminishing marginal productivity”
  • the rule for division of a resource between two units producing the same product: equal marginal productivities
  • the diagnostic formula VMP=wage, that tells us the input and output are adjusted to maximize profits in the business firm, in the short run
  • In the long run, there may be increasing, decreasing, or constant returns to scale. Increasing returns to scale will complicate things somewhat for the marginal productivity approach.

This has given us a start on the theory of the business firm. But we will want to reinterpret the model of the firm in terms of cost — since the cost structure of the firm is important in itself, and important for an understanding of supply.