In microeconomics : is this the contradiction in the atomicity of firms ?

Question

Let $p$ be the market demand. It is a function of the market production $Q$. Let $q_i$ be the production of firm $i$.

Reading Steve Keen (in Debunking Economics, chapter II) quoting George Stigler, I think the first wants to deduce the following contradiction in the atomicity of firms.

Using the chain rule, we get : $\frac{dp}{dq_i}= \frac{dp}{dQ} \frac{dQ}{dq_i}$.

The firm $i$ is price-taker, so the market price is the same whatever its production and so $\frac{dp}{dq_i}=0$.

The demand $p$ is a (strictly) decreasing function of $Q$ (supposing the law of demand is true). Thus $\frac{dp}{dQ} < 0$

The other firms than firm $i$ are not supposed to react to a change in production of firm $i$, so that $\frac{dQ}{dq_i}=1$.

We get : $0 < 0$. Is that the contradiction that Steve Keen means (or another way of expressing it) ?

Thank you so very much !

Answer 1

This does sound a lot like the “contradiction” that Keen tries to derive. The key to resolving it is to remember that firms are small relative to the market, so $$\frac{\mathrm dQ}{\mathrm dq_i} = 0.$$

One way to justify the above restriction is to assume there is a continuum of firms, so that each firm has zero measure, and $$Q = \int_{j \in I} q_j \, \mathrm dj,$$ where $I$ is the index set of firms.

Another way to justify the price-taking assumption (which means that price is equal to marginal cost) is to look at a Cournot competition model with a large number of firms, as Michael mentions in his answer to this question. Formally, suppose there are $n$ firms in the industry so that industry output is given by

$$ Q^s = \sum_{i=1}^n q_i, $$

where $q_i$ is the output of firm $i$. Market demand is given by the inverse demand curve

$$ p = a -bQ, $$ where $a,b > 0$. We normalise each firm's (constant) marginal cost to $0$, so that firm $i$'s profits are given by

$$ pq_i=(a-bQ^s)q_i = aq_i - bq_i \sum_{j=1}^n q_j.$$

The choice of $q_i$ that maximises the above expression solves

$$ a - b \sum_{j=1}^n q_j -b q_i = 0. $$

In other words,

$$ q_i^* (q_{-i}) = \frac{a - b\sum_{j \neq i}q_j}{2b} .$$

In a symmetric equilibrium, $q_i^* = q_j^* = q^*$, so the above best response function gives us

$$ q^* = \frac{a - (n-1)bq^*}{2b} \implies q^* = \frac{1}{n+1} \frac{a}{b}. $$

Hence, the equilibrium price is

$$ p^* = a - b \frac{n}{n+1}\frac{a}{b} = \frac{1}{n+1}a. $$

It is now easy to show that $p^* \to 0$ as $n \to \infty$, which is exactly the claim that the equilibrium price approaches marginal cost when the number of firms is large.

Answer 2

A price-taking firm takes prices as given, but that does not mean that the firm cannot influence prices; it just means that the firm ignores its own impact on prices.

Now the question is how sensible it is to assume that firms take prices as given. The usual view is that it is a reasonable assumption when the impact of a firm on prices is small enough that profit-maximizing behavior for given prices does not differ much from profit-maximization under actual prices. This is usually the case if there are many firms that are small relative to the market. There are various ways to make this precise. One can also work with models with a continuum of firms in which a single firm hs literally no impact on prices.

As a side remark, Steve Keen's book reveals a strange mixture of incompetence and dishonesty. Keen is ignorant of basic calculus. Keen, sometimes with coauthors, has come up with a nonsensical theory of the firm. Here is a note by Paulin Anglin pointing out several of the many problems. One of these problems is acknowledged here. Later, Keen (and Standish) wrote a survey of Keen's history of criticising the theory of the firm in which they did not acknowledge Anglin's criticism. I recommend reading books from people with higher scholarly standards than Keen.