Theory of the Firm

As a first step toward a better understanding of supply, we need to think about the operations of suppliers. In economics, this is often called the "Theory of the Firm."

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).

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

• Proprietorships
• partnerships
• corporations

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

Objectives

The operations of the firm will, of course, depend on its objectives. We will follow the neoclassical tradition by assuming that firms aim at maximizing their profits.

There are two reasons for this. 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

Profit is defined as revenue minus cost.

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 proprietor 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.

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."

Production Function

Firms earn profits by selling goods and services, which in turn are outputs. Production is the transformation of inputs into outputs. Inputs are the factors of production -- land, labor, and capital -- plus raw materials and business services.

The transformation of inputs into outputs is determined by the technology in use. Limited quantities of inputs will yield only limited quantities of outputs. The relationship between the quantities of inputs and the maximum quantities of outputs produced is called the "production function."

But how do these outputs change when the input quantities vary?

Short and Long Run

A key distinction here is between the short and long run.

Some inputs can be varied flexibly in a relatively short period of time. We conventionally think of labor and raw materials as "variable inputs" in this sense. Other inputs require a commitment over a longer period of time. Capital goods are thought of as "fixed inputs" in this sense.

Thus, we distinguish between the short run and the long run as follows:

In the perspective of the short run, the number and equipment of firms operating in each industry is fixed.

In the perspective of the long run, all inputs are variable and firms can come into existence or cease to exist, so the number of firms is also variable.

Diminishing Returns

Another key concept here is diminishing returns.

The idea of diminishing returns comes to us from Thomas Malthus. Malthus is best known for his pessimistic idea that population growth would force incomes down to the subsistence level. What we are interested in here is not his conclusion, but the reasoning that took him there.

Malthus argued that land is a fixed input, but the growth of population makes labor a variable input. Malthus proposed a general law of economics: when a fixed input is combined in production with a variable input, using a given technology, increases in the quantity of the variable input will eventually depress the productivity of the variable input. (Malthus argued that decreasing productivity of labor would depress incomes).

There is plenty of evidence, both observational and statistical, that the Law of Diminishing Returns is valid. (However, the technology has not been fixed, so the conclusion -- subsistence incomes -- does not follow).

The John Bates Clark Model

Decades later, economist John Bates Clark borrowed that idea for his model of the firm, which is still the basic one in neoclassical economics.

Clark focused on the short run, so that technology is given and the capital equipment of the firm is a fixed input. He considered labor to be the variable input.

His next step -- following the example of the theory of demand, which was then a hot new development -- was to distinguish between the average and marginal productivity of labor. As before, it is especially important to focus on the marginal change.

Marginal Productivity

Productivity, by definition, is a ratio of output to labor input. We often refer to average productivity:

Average productivity is an important concept, especially in macroeconomics. In microeconomics, however, we will focus more on the marginal productivity:

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, the marginal productivity of the variable input will eventually decline.

Marginal Productivity Example

Here is an example of a hypothetical firm with the quantities of labor it might employ, and the output and average and marginal productivities.

Table 1

Labor	Output	Average Productivity	Marginal Productivity
0	0	0
			9.45
100	945	9.45
			8.35
200	1780	8.90
			7.25
300	2505	8.35
			6.15
400	3120	7.80
			5.05
500	3625	7.25
			3.95
600	4020	6.70
			2.85
700	4305	6.15
			1.75
800	4480	5.60
			0.65
900	4545	5.05
			-0.45
1000	4500	4.50

Output Diagram

Here is a picture of the relationship between the variable input and the output -- the short run production function -- in the example. Notice how the slope gets flatter: as the variable input increases, output increases at a decreasing rate.

Average and Marginal Productivity Diagram

Here are the average and marginal productivities in the same example. Notice how the marginal productivity declines faster than the average productivity, pulling the average productivity down after it.

Maximization of Profits

In his approach to the theory of the firm, John Bates Clark was following the example of consumer demand theory, which we have already studied and which had already been worked out. Thus, Clark borrowed the equimarginal principle from the literature on consumer demand. In demand theory, the consumer is supposed (in effect) to ask: "I want to maximize my utility. How many burgers should I buy in order to do that?" The answer is found in the equimarginal principle, MP/P = same for all goods or, in other words, MB=price. Clark has the proprietor of the firm ask, in effect, "I want to maximize my profits. How much labor should I hire, in order to do that?" Again, the answer will be in the equimarginal principle.

But, of course, there is a bit more to it than just that.

• The price of output is a given constant.
• The wage (the price of labor per labor hour) is a given constant.

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 answer to the second question 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.

Adding one more unit to the labor input, we have

VMP=wage

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.

Applying the Principle

We want to apply the equimarginal principle in the form MB=price. We can immediately identify the wage as the price of labor in the John Bates Clark model. But what is the marginal benefit of hiring labor? In order to benefit, the firm must direct the labor to produce goods and services, but must also sell those goods and services. Thus the marginal benefit, in this application, will be the product of the selling price of the output times the marginal product of the labor.

The way to maximize profits then is to hire enough labor so that

p*MP=wage

where p is the price of output and MP the marginal productivity of labor.

Profit Maximization Example

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:

Table 1

Labor	Marginal Productivity	p*MP	Wage	Accounting Profit
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

Visualizing Profit Maximization

What we see in the table is that the transition from 400 to 500 units of labor gives p*MP=505, very nearly p*MP=wage. And that is the highest profit. So the profit-maximizing labor force is about 500 units -- probably just slightly more than 500, but 500 as nearly as the example can tell.

Here is a picture of the profit-maximizing hiring in this example:

Profit Maximization

Notice the shaded area between the p*MP curve and the price (wage) line, somewhat like the consumers' surplus in demand theory. It is the accounting profit -- that is, the sum of economic profit and the return to capital -- while the rectangular area below the wage line and left of the labor=500 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.

Allocation of Labor Between Two Fields

Here are the production functions for the two fields.

Here are the two Production Functions

Figure 1. Production Functions for Two Fields

Allocation of Labor Between the Two Fields
and Total Output

We see here the labor on the less fertile Field 1, labor left over for Field 2, and the total output of the two fields. As we can see, it is best to allocate some labor to Field 1, even though it is less fertile overall.

Optimum Allocation

The Marginal Productivity of Labor on the two fields is the key to an answer.

In this diagram, labor on the north field is measured from left to right, and labor on the south field from right to left.

Marginal Productivity and the Equimarginal Principle

This is a quite general principle, which we may state as follows.

Rule:

When the same product or service is being produced in two or more units of production, in order to get the maximum total output, resources should be allocated among the units of production in such a way that the marginal productivity of each resource is the same in all units of production.

Production and Cost

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; but we will want to reinterpret the model in terms of cost -- since the cost structure of the firm is important in itself, and important for an understanding of supply.