The theory of the firm and industry equilibrium

Introduction

Economic theory consists of models designed to enhance our understanding of the behavior of economic agents and the interactions among them. This tutorial concentrates on the behavior of firms. Its level is elementary. It grew out of a second-year undergraduate course I taught at McMaster University in the winter semesters of 1997 and 1998. In 2025 I redesigned it to bring it in line with modern web standards. At that time, I also made some minor editorial changes.

One reason for being interested in understanding the behavior of firms is that we would like to compare the impact on human welfare of different ways of organizing production. Is the outcome in a competitive market likely to be desirable from the point of view of society? Is it likely to waste resources, or to use them effectively? What distribution of the benefits of production among the people in the economy will it generate? What are the answers to these questions for an industry in which there is only one firm, or one in which there are only a few firms?

The tutorial provides some extremely limited answers to these questions, using the analytical tools typically taught in introductory courses in economics—tools that were developed a century or more ago. Only the last section, on oligopoly, uses the more modern tool of game theory. For a modern approach to microeconomic theory that covers many topics besides (and, to my taste, more interesting than) the theory of the firm, see my book Models in microeconomic theory with Ariel Rubinstein (which is freely available in electronic form).

This tutorial covers the following topics.

1. Describing technology

Before performing any analysis, we need to have a way of describing the production possibilities of a firm. I describe several ways of doing so:

• output as a function of one input, holding other inputs fixed
• combinations of inputs that yield a given output
• output as a function of the scale at which all inputs are used.

2. The theory of a cost-minimizing firm

If a firm chooses its inputs to maximize its profit—a common assumption in economic models—then the combination of inputs that it uses minimizes the cost of producing its output.

3. The supply function of a profit-maximizing price-taking firm

If an industry contains many firms of more or less equal size, then we would expect the actions of a single firm to have little effect on the price, and a reasonable first approximation is to assume that each firm ignores the impact of its decisions on the price of output.

4. Competitive equilibrium in an exchange economy

A standard notion of equilibrium in a market containing a large number of economic agents is that of “competitive equilibrium”. I start by considering this notion in the context of an “exchange economy”, in which the problem is how to allocate goods between agents. I discuss also the criterion of “Pareto stability” (also known as “Pareto efficiency”).

5. Competitive equilibrium in an economy with production

For an economy with production, it is sometimes useful to distinguish two cases: the “short run”, in which the number of firms is given, and the “long run”, in which firms can enter and exit the industry.

6. Monopoly

I now turn to the opposite of a competitive industry: a monopoly (an industry in which there is a single firm). I show how the standard model leads to the conclusion that the outcome in such an industry is not Pareto efficient, and ask what policies might improve it.

7. Oligopoly

In many industries the number of firms is small. To understand such industries we need to model strategic behavior: we need a model of the outcome when a small number of decision-makers interact, each fully aware of the impact of its actions on the others. Game theory provides such a model.

Convention

I use the symbol “$” for a monetary unit because it appears to be widely understood as such, partly because of its use in several countries, including Canada, Mexico, Guyana, Fiji, Jamaica, Namibia, and Surinam.

Mathematical tools

The tutorial uses mathematical methods. Here are some mathematical facts you should know.

Graphs of functions

Linear function

The graph of a function y = mx + c (where m and c are constants) is a straight line with slope m; it intercepts the y axis at y = c.

Quadratic function

The graph of a function y = ax² + bx + c (where a, b, and c are constants) takes one of the following three forms:

• If a > 0 it is U-shaped, with a minimum at x = −b/2a and a minimum value of c − b²/4a.
• If a = 0 it is linear, with slope b and y-intercept c.
• If a < 0 its shape is an inverted U, with a maximum at x = −b/2a and a maximum value of c − b²/4a.

Rectangular hyperbola

The graph of a function y = a/x (where a is a constant) for x ≥ 0 is downward sloping, its slope increasing (i.e. becoming less negative) as x increases.

One-variable calculus

• For any number n ≠ 0 the derivative of f(x) = xⁿ is nxⁿ⁻¹.
• For any functions f and g, the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x).

Maximization and minimization

Let f be a (differentiable) function defined for x ≥ 0.

• If x* maximizes f(x) and x* > 0 then f'(x*) = 0 and f"(x*) ≤ 0.
• If x* minimizes f(x) and x* > 0 then f'(x*) = 0 and f"(x*) ≥ 0.