Response to Steve Keen

I am reading Prof. Steve Keen’s book Debunking Economics, which is rather good, so far (I’m about 100 pages into it). Towards the end of chapter 4, however, I have begun to question Prof. Keen’s maths. On p96-98, Prof. Keen develops a formula for calculating the optimal output of a company in a competitive market. I could not follow the logic. I contacted Prof. Keen and he pointed me to this article in which the logic which confused me is on p62.

Prof. Keen asserts that in an “atomic” market (i.e. where different companies do not react to each other’s changes in output volume),

$\displaystyle \frac{d q_j}{dQ} = 1$ for all $j$ $(1)$

where $\displaystyle q_j$ is the output of the $\displaystyle j^{th}$ company and $\displaystyle Q$ is the output of the market as a whole.

I argue that this results in a contradiction which is found by asking the value of $\displaystyle \sum_{\forall j} \frac{d q_j}{dQ}$ .

Given (1),

$\displaystyle \sum_{\forall j} \frac{d q_j}{dQ}=\sum_{\forall j}1=n$ $(2)$

where $\displaystyle n$ is the number of companies in the market. On the other hand, the total output of the market, $\displaystyle Q$ , is the sum of the outputs of each of the companies in the market. i.e.

$\displaystyle Q=\sum_{\forall j} q_j$

Taking derivatives of both sides of this equation with respect to $\displaystyle Q$ yields

$\displaystyle 1=\sum_{\forall j} \frac{d q_j}{dQ}$

which contradicts (2) in every case where $\displaystyle n\not=1$ , i.e. in any market with more than one company.

Indeed, I am uncertain as to whether it makes any sense to speak of $\displaystyle \frac{d q_j}{dQ}$ since $\displaystyle Q$ depends on all the $\displaystyle q_i$ , but the $\displaystyle q_i$ are quite explicitly independent of each other. Reverting to first principles,

$\displaystyle \frac{d q_j}{dQ}=\lim_{h \to 0} \frac{q_j(Q+h)-q_j(Q)}{h}$

$\displaystyle =\lim_{h \to 0} \frac{q_j(q_1+q_2+\ldots+q_n+h)-q_j(q_1+q_2+\ldots+q_n)}{h}$

I find it difficult to find any understanding of what the numerator in this limit expression means.

Prof. Keen suggests that I may be confusing partial derivatives with total derivatives. As can be seen, however, there are no partial derivatives involved. Having said that, the partial derivative of $\displaystyle q_j$ with respect to $\displaystyle Q$ is, indeed, equal to 1:

$\displaystyle Q=\sum_{\forall i}q_i=q_j+\sum_{i\not=j}q_i$

$\displaystyle \therefore q_j=Q-\sum_{i\not=j}q_i$

$\displaystyle \therefore \frac{\partial q_j}{\partial Q}=1$ $(3)$

However, this is by the by. Prof. Keen’s argument relies on the total derivative, not the partial derivative. If I am right, then this analysis breaks a key step between Prof. Keen’s equations (0.5) and (0.6).

Moving on

The argument above was where I started. But in preparing it, I have scrutinised Prof. Keen’s article more carefully and I now note another concern, this time on p63. In deriving equation (0.8) from (0.7), Prof. Keen implicitly uses another supposed consequence of “atomicity” which he first mentioned on p58, namely that:

$\displaystyle \frac{\partial q_j}{\partial q_i}=0$ $\displaystyle \forall$ $\displaystyle i\not=j$ $(4)$

This constitutes a system of partial differential equations. Added to this system is the boundary condition that $\displaystyle Q$ is the sum of the $\displaystyle q_i$ . The difficulty is that this boundary condition is inconsistent with the PDEs. This is shown very simply, following similar logic to the derivation of (3) above:

$\displaystyle Q=\sum_{\forall i}q_i=q_j+\sum_{i\not=j}q_i$

$\displaystyle \therefore q_j=Q-\sum_{i\not=j}q_i$

$\displaystyle \therefore \frac{\partial q_j}{\partial q_i}=-1$

This contradicts (4).

So what should the consequence of atomicity be? Simply that, all effects taken into account, a change in the output of one company has no effect in the output of any other company. i.e. a change in $\displaystyle q_i$ has no effect on $\displaystyle q_j$ for $i\not=j$ . This can be expressed in terms of a system of ordinary differential equations:

$\displaystyle \frac{d q_j}{d q_i}=0$ $\displaystyle \forall$ $\displaystyle i\not=j$ $(5)$

Critically, these equations are consistent with $\displaystyle Q$ being the sum of the $\displaystyle q_i$ :

$\displaystyle Q=\sum_{\forall i}q_i=q_j+\sum_{i\not=j}q_i$

$\displaystyle \therefore q_j=Q-\sum_{i\not=j}q_i$

$\displaystyle \therefore \frac{d q_j}{d q_i}=\frac{d Q}{d q_i}-1$ $(6)$

Taking (5), we now find no contradiction (whereas the argument foundered at this point when we were considering (4)). Further, we find we have simply derived another consequence of atomicity which Prof. Keen, himself, asserts on p58. Taking (5) and (6):

$\displaystyle 0=\frac{d Q}{d q_i}-1$

$\displaystyle \therefore \frac{d Q}{d q_i}=1$

This is part of Stigler’s 1957 observation, which Prof Keen quotes on p58.

We can conclude that, since all $\displaystyle q_i$ have a relationship with $\displaystyle Q$ , they necessarily also have a relationship with each other, but the net effects cancel out. This equates to saying that the partial derivatives are non-zero, but the total derivatives are zero.

So…

On the whole, this wrecks Prof. Keen’s argument. Unless I am badly misremembering undergrad maths. Any observations about where I have gone wrong would be appreciated.

2 thoughts on “Response to Steve Keen”

tom on April 1, 2013 at 10:32 am said:

PS, I have enabled LaTeX for comments. So you should be able to insert formulae in any response using the syntax described at http://en.support.wordpress.com/latex/.

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Russell Standish on April 1, 2013 at 9:24 pm said:

Dear Tom,

You are indeed correct that dq_i/dQ is not (necessarily) equal to 1. In fact, the model does not determine what dq_i/dQ should be. We acknowledge this error in a follow up paper http://arxiv.org/abs/1101.3409, which has been languishing in peer review for the last two years without even going in front of a reviewer (ie it has either been rejected on editorial grounds, or simply unacknowledged by the jounal editors at all). Makes you wonder about the peer review process.

It turns out that dq_i/dQ=1 is not needed for deriving the Keen equilibrium, just a symmetry assumption = , ie that the long time average of these quanties are the same for all firms. In that paper, we also consider the generalisation where that symmetry assumption is dropped.

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