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

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 !

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.

• Thanks for your reply. I disagree with you, sorry. If $\frac{dQ}{dq_i} = 0$, then the total output does not depend on any of the individual productions : if $q_i$ changes, $Q$ does not change. How can that be ? George Stigler said this was 1, not 0. – Cecil Faux Oct 29 '17 at 10:31
• And the idea of a continuum of firms does not bear any relation to reality : not only $\aleph_0$ but $\aleph_1$ firms ! – Cecil Faux Oct 29 '17 at 10:38
• Does the value of an integral change when you change the value of the integrand at a single point? As for relation to reality, the idea of a continuum of firms is meant to model a situation where there are very many other firms. It is an approximation, in the same way pretty much any model is. It may well be a bad approximation, but that is a different criticism from it being contradictory. – Theoretical Economist Oct 29 '17 at 10:46
• @CecilFaux could you point me to where Stigler says $\mathrm d Q / \mathrm d q_i = 1$? Also, if you don’t like assuming a continuum of firms, see Michael’s suggestion above of looking at (Nash) equilibria in a Cournot competition model when there are a large (but finite) number of firms. – Theoretical Economist Oct 29 '17 at 14:16
• I see. Having looked at Stigler (1957), that is because Stigler starts with a model where sellers have some market power. (Notice that he assumes a finite number of firms.) As the number of firms increases, this market power vanishes. – Theoretical Economist Oct 29 '17 at 17:12

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.

• Ok for the dependance of prices and individual firm production. I have yet to find a proper (mathematical) definition of a price-taking firm (that will be my second question on this site). I would be interested to know hypotheses that guarantee the two behaviors you speak do not differ much. Would you have a reference by any chance ? (Besides the continuum of firms which seems much too far-fetched.) – Cecil Faux Oct 28 '17 at 14:14
• I am reading Microenomic Theory (by Mas-Collel &...) and Keen at the same time. Mas-Collel is clear and very well written (I am a maths professor) and quite possibly there are calculations mistakes in Keen, which is painful to read because of the lack of formulas. But Keen's chapter I demonstrates quite clearly that it is impossible to prove that the market demand is decreasing and makes fun of some of the attemps (such as Gorman's) to add ad hoc but absurd hypotheses. Maybe you differ on that but then again thanks for the references. – Cecil Faux Oct 28 '17 at 14:23
• @CecilFaux Just look up Cournot competition for an oligopoly model in which the price taking assumption is justified "in the limit". For demand theory, you can take a look at the chapter by Shafer and Sonnenschein in the Handbook of Mathematical Economics referenced by Keen. It is clear that Keen doesn't even understand the difference between demand and excess demand. Werner Hildenbrand has a book on market demand that gives reasonable sufficient conditions for falling market demand. – Michael Greinecker Oct 28 '17 at 14:29
• Very interesting. You are right about Keen mixing demand and excess demand. Hildenbrand is referenced in Mas-Collel (chapter 4), I will also read him here, I guess. Not at all sure I will be convinced (from a mathematical point of view) but thanks for the food for thought. – Cecil Faux Oct 29 '17 at 10:46
• @MichaelGreinecker A department at my university just announced a conference, with Steve Keen as their keynote speaker... – Giskard Nov 14 '19 at 5:55