The impossibility of perfect competition

Perfect competition is the one of the very first concepts to which a new student of economics is introduced. It therefore plays a vital role in economic pedagogy. Though many advances in economic theory are based on attempting to move away from reliance on the concept of perfect competition, it remains a key reference point for economic theory. It is therefore very disturbing that its foundation is a set of basic mathematical errors.

The alleged unique attribute of a perfectly competitive industry is that the market price equals the marginal cost of production, as a consequence of the competitive profit maximising behaviour of myriad non-collusive small firms. Individual self-interest and social welfare are reconciled, because the profit-maximising behaviour of individual firms leads to the socially optimum outcome: that the marginal benefit of output to society equals the marginal cost of production.

If the model is accurate, then it should be provable that any individual firm which departs from this level of output will suffer a reduction it its profit. However, careful mathematics shows that any firm which reduces its output level below that at which P=MC will increase its profit (thanks to John Legge of La Trobe University for developing the argument below). Further analysis shows that all other firms in the industry will benefit from the reduction in production, and that--under the usual assumptions of rising marginal cost and downward sloping market demand--the equilibrium situation for 'perfectly competitive' firms is exactly the same as for 'imperfect competition' and monopoly: equilibrium occurs where MC=MR<P.

We start from

where total revenue can be disaggregated further into price times quantity:

Maximum profit occurs where the first differential of profit is zero:

where the theory assumes that  so that

This equals zero--so that p is a maximum--where

Now consider a firm producing at this level, which changes its output by dq. Its new profit situation will then be

Expanding the functions, we have

Expanding this yields

Rearranging yields

The first bracketed term is p(q), while terms in P and  can be cancelled since they are equal at this output level:

Subtracting the two profit terms yields

So the change in profit for a change in output of  dq will be:

The first term in this expression is negative (neoclassical economists might insist that it is zero, but that raises another pitfall which is discussed below); the second term is positive, since q will always be larger than dq; the sign of the third term dq depends on whether the firm increases or decreases its output.

If dq is positive---if the firm increases its output---then it will reduce its profit; so far, so good. But if dq is negative---if the firm reduces its output---then it will increase its profit. Therefore the firm, if it is a profit maximiser, will reduce its output below the level at which price equals marginal cost. Therefore price equals marginal cost is not a profit maximising equilibrium.

Now let's consider the probable neoclassical rejoinder, that we have wrongly assumed that is negative, when the theory assumes that changes in output by a single firm can't affect the market price. The mathematically competent reader will already have spotted the conundrum this leads to, but it's worth spelling out in full.

 What if price doesn't change due to the change in output by one firm?

Neoclassical economists might object to the expansion for price, on the assumption that the actions of one firm can't affect the market price. What happens if we allow this?

where we assume that  so that we get instead

Making the substitution that , we get:

Since we have started from the position at which price equals marginal cost, the final term is zero. This leaves us with

So if we allow this objection, we get the contradiction that dq more or less output than the profit maximising level generates the same level of profit as the profit maximising level. But we began by assuming that there was a single profit maximising level of output---we have another logical contradiction. Hence this objection does not save the theory (and below we show it's fallacious in the section on conjectural variation), and cannot be allowed on logical grounds.

How do other firms fare?

Now let's consider the other firms in the industry. We've already shown that a single firm which reduces its output will increase its profit. The sole difference between this firm and the rest of the industry is that the other firms don't change their output level. However, they all benefit from the increased price. Thus the situation of the other firms in the industry is

since  is unambiguously positive in the case in which the single firm has reduced its output so that dq is negative (we have a negative times a positive times another negative: two negatives multiplied together give a positive)

So any other firms producing where P=MC unambiguously benefit from the fall in output. P=MC therefore cannot be an equilibrium for a competitive industry.

Clearly, with these results, there must be an error in the assumptions economists use to develop the model of 'perfect' competition. A popular procedure called 'conjectural variation' reveals the contradiction (this proof was developed by Geoff Fishburn of UNSW).

Where's the error? Conjectural Variation

The model of perfect competition assumes that firms are atomistic and non-collusive: they do not respond to the actions of other firms, but merely to changes in demand and their own conditions of supply. If we therefore define total output Q as

where qi is the output of the ith firm and QR the output of the rest of the industry, then this assumption means that

The 'conjectural variation'--how much all other firms alter their output in response to a variation by one firm--is thus assumed to be zero.

The marginal revenue for the ith firm is

The model of perfect competition is allegedly distinguished from other market structures by the result that

This means that the final term above must be zero. Economists normally simply assume that , but the concept of conjectural variation lets us explore what this assumption requires in more detail

This term can only equal zero if , since both qi and  are non-zero.

Therefore  or

But this contradicts the initial assumption that .

The reconciliation is obvious: the concept of a horizontal demand curve for the individual firm is incompatible with the concept of a downward sloping demand curve for the entire industry. The only way they can be made compatible is if all other firms in the industry respond to a reduction in output by one firm by increasing their output to the same extent, thus holding total industry output constant--and hence keeping price constant. But as our analysis above shows, profit maximizing firms would have no incentive to do this: if a single firm reduced its output, it would increase its profit and so would all other firms, whereas an increase in the output of any single firm reduces its profit. The dynamics of profit maximizing behaviour would therefore drive output down from where price equals marginal cost. The question then arises as to where will the reduction in output stop?

What is the equilibrium?

Let's consider the stability of the point at which marginal revenue equals marginal cost for a competitive industry. This is supposed to be the equilibrium for all other forms of markets, but not for a perfectly competitive industry. But what if a competitive industry is actually producing where marginal revenue equals marginal cost, and marginal revenue is less than price?

We have the general proposition that  and

where   so that

So our profit maximisation substitution now is that

We consider a firm that changes its output by dq, where this can be either positive or negative. We get

where  for profit maximisation under the usual general neoclassical assumptions:

Since  and

So MR=MC is an equilibrium--in that any firm which changes its output level from this point will reduce its profit level--while MC=P is not an equilibrium. The 'perfect competition' outcome of P=MC can only apply if firms deliberately produce more than the profit maximising level. Since this notion goes against the entire neoclassical agenda, the only logical deduction is that the model of perfect competition is untenable.

But that's not the end of the problems for the neoclassical model of the firm. Not only is the theory's ideal of price equal to marginal cost unachievable, it isn't even true that equating marginal revenue to marginal cost maximises profits.

This issue was raised briefly in Chapter 3 of Debunking Economics; this is an appropriate point to go into more detail

The dynamics of profit maximisation

The neoclassical analysis of profit maximisation begins with the self-evident proposition that profit is the difference between revenue and cost:

It then states that both total revenue and total cost are functions of the quantity produced, where TR is price times quantity, and price assumed to be either constant (for a competitive firm--something we have debunked above) or falling as output rises, while total cost per unit of output is assumed to be rising as quantity increases (in the relevant range; it is normally shown as falling due to economies of scale early on).

If this is conceded, then it is easily shown that equating marginal revenue to marginal cost maximises profits:

which equals zero where MR=MC; hence profit is at a maximum.

However, it is not true that revenue and cost are solely functions of quantity; they are also functions of time, of location, and many other factors. Considering just the first two, a more general statement of the profit function is:

If we consider time alone, then we have

Now the term of interest is not  but dp. We are also interested in making dp as big as possible (so that the change in profit is as big as possible). We can now expand:

This expression can be expanded to

From any given initial level of profit, it should be the firm's dynamic objective to maximise this amount with respect to time. It should therefore want to make the differential with respect to time as large as possible:

The neoclassical method of maximising profit would only maximise the rate of growth of profit if, for all time, . But this is only possible if there is some quantity which is the appropiate quantity to produce for all time--and there is no such quantity. Also, neoclassical economics could give no advice about how to maximize the rate of growth of profit, since as a static discipline it has no advice on how to make profit (or anything else) grow over time.

Contrary to neoclassical theory, since in the real world the rate of change of quantity with respect to time is generally positive, then to maximise the rate of growth of profits, MR-MC should be greater than zero. The neoclassical profit maximisation rule is only valid in a static economy in which . Its advice is irrelevant in a dynamic setting--in fact, counterproductive, since it will lead to a lower rate of growth of profit.

This is a simple but profound instance of the way in which static reasoning—which ignores time—can give incorrect guidance when applied to a dynamic topic like a firm’s endeavours to earn a profit.

This embarrassingly simple mathematics supports a contention made 75 years ago by Sraffa—that economics should abandon the Marshallian analysis of the firm.