Perfect Competition Equals Monopoly

If a first year economics student learns anything, it is that perfect competition is good and monopoly is bad. The logic on which that mantra is based appears convincing, but as Size Does Matter argues, it is actually seriously flawed. These pages provide the maths behind the verbal arguments in Chapter 4.

An essential aspect of the 'proof' of the superiority of a perfectly competitive industry (when compared to a monopoly, or indeed any other market structure) is that the marginal cost function of the monopoly is identically equal to the supply curve of the competitive industry, as shown in Figure 1:

Given this equality--and given the models' assumptions that monopolies set price by equating marginal cost to marginal revenue, while a competitive industry sets marginal cost equal to price--then the competitive industry will produce a higher output at a lower price.

 However, if the two curves differ--if the monopolist's marginal cost curve is not identically equal to the supply curve of the competitive industry (which is in turn the sum of the marginal cost curves of all the firms in the industry) then it is quite possible that the monopolist could produce a higher output at a lower price, as shown in Figure 2 below.

Since marginal cost is derived from marginal product, the equivalence of monopoly and perfectly competitive aggregate marginal cost curves requires that the marginal products are also identical (assuming, as economists do, that both a monopoly and the competitive industry face a competitive market for their inputs). This in turn means that the total product curves--the integral of the marginal product curves--can differ only by a constant. If we consider labour as the variable input, where there is zero output for zero variable input, this constant can be set to zero. This situation is shown in Figure 3, where I have for the moment assumed--as do economists--that this condition can be true of total product curves drawn as economists like to draw them, with initially rising and then diminishing marginal productivity:

The condition can actually be stated much more accurately and succinctly in an equation which states the equivalence of the total product curves of the competitive industry and the monopoly:

where f is the output function for the competitive firms, and g the output function for the monopoly.

In words, this equation says that (assuming, without loss of generality, that all competitive firms are the same size) that the output of n competitive firms, each employing x workers, must be identical to the output of a single monopoly employing nx workers, for all values of n and x in the relevant range. If this condition does not apply, then the marginal cost curve of the monopolist will differ from the supply curve for the competitive industry, and the question of which industry provides the higher output and lower cost can't be answered.

Economists quite merrily draw the marginal cost curves for both monopolies and competitive industries as upward sloping when they make this argument in favour of competitive firms and against monopolies--just consult any introductory microeconomics text. But this equation imposes strict limits on the shape of the marginal cost curves: they must be horizontal straight lines! This is explained verbally in the extension to the book at More/Size, but it is much more easily shown using some simple calculus.

We start with the condition that

and differentiate this with respect to n:

The chain rule for differentiation tells us that

putting this into the previous equation yields:

The initial equation gives us a second way of describing f(x):

We can now equate these two expressions for f(x):

Rearranging this yields:

If we make the substitution that , this becomes

The left hand side is equivalent to  (since ln(x) is defined as ). Substituting this yields

Integrating both sides gives:

Taking exponentials, we find that:

In other words, the production function for the monopolist is a straight line--marginal returns are constant, with each additional worker hired adding C units of output.

We can now use the initial expression to work out what f() is:

So f() is also a straight line function with a slope of C. With both production functions f() and g() being straight lines, the marginal product in both cases is a constant C. The marginal cost 'curves' are both therefore the same horizontal straight line.

This doesn't present insurmountable problems for the model of monopoly--since a downward-sloping demand curve will intersect with the horizontal marginal cost curve to yield a determinate price and output. But it makes the model of 'perfect competition' untenable, since in this model the marginal revenue/price/demand curve faced by the individual firm is also supposed to be a horizontal straight line. If the marginal cost curve and the marginal revenue curve are both horizontal straight lines, then either they intersect everywhere, or they intersect nowhere. Either way, the model of perfect competition becomes indeterminate.

It therefore appears that neoclassical economics has to abandon its practice of promoting perfect competition over monopoly, since the conditions which are required to make a definitive comparison contradict the model of perfect competition.

This is a classic case of the 'you can't have your cake and eat it too' dilemma which recurs so frequently with economic theory. If economists want to be able to say that perfect competition is 'good' and monopoly 'bad', one of the conditions required to be definitive about this--that marginal cost curves are identical--means that their model collapses. If they accept the mathematical reality that they can't insist that marginal cost curves are identical, then they are no longer able to say on theoretical grounds alone whether perfect competition or monopoly is the preferred situation.

This is, however, just the tip of the iceberg of the problems for the neoclassical model of the firm. It is also possible to show that the a key assumption of the model--that the output of a single firm does not affect the market price--is logically inconsistent with the concept of a downward sloping demand curve, and that key feature of the competitive model--that price equals marginal cost--is not possible if firms are truly profit maximisers. These issues are covered here.