I am equally puzzled as the author here.
Thoughts ?
http://www.bondeconomics.com/2017/05/the-horrifying-mathematics-of.html
The Horrifying Mathematics Of Infinitesimal Agents
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This critique is incoherent. It assumes that since the firm is atomistic, it essentially sells zero quantity, so that the firm is indifferent about what price to charge.
That this critique is false can be seen in the mathematics of a Dixit-Stiglitz monopolistic competition framework. In that framework, firms are atomistic but they price above marginal cost. (Of course, with free entry, prices are driven to equal average total cost, and there are no profits, but that need not be the case if entry is not free.)
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@ 9d31
I don't think he's made such assumption.
His point is that under this framework, any individual firm makes no profit solely based on the underlying maths. Regardless of quantity or price because that's what the integral dictates (being a set with a measure of 0 etc) -
This is a very bad critique generally. The model approximates reality, it doesn't mirror it.
@a832
Indeed, individual profit is merely f.
But then there's another paradox.
The i-th agent's integral would imply that all agents contribute 0 to aggregate profits. No ?Think about the integral as a limit.
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He's right that many papers in economics that purport to model a continuum of infinitesimal agents are mathematically incoherent. But he's not the first to notice this. There are older papers by Aumann and Judd on the subject, and a relevant paper by Yeneng Sun in the current issue of TE.
All posts in this thread above mine should be ignored.
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As everyone knows, the closed interval [0, 1] on the real line is a nondenumerable set. (A proved in the Theorem in Section 4 of Chapter 1 of Volume 1 of Elements of the Theory of Functions and Functional Analysis, by A.N. Kolmogorov and S.V. Fomin.) This means that we cannot express the set [0, 1] as the limit of a countable series of agents. In other words, we formally cannot view such a construct as being the limit of having "a lot" of firms.
So the blogger is not familiar with how decimal system works then.
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@ 9d31
I don't think he's made such assumption.
His point is that under this framework, any individual firm makes no profit solely based on the underlying maths. Regardless of quantity or price because that's what the integral dictates (being a set with a measure of 0 etc)He writes that "profits are equal to zero, no matter what decision the firm makes." I think this does imply that "the firm is indifferent about what price to charge," since all price-quantity pairs yield the same profits (by his incorrect mathematics).
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I also don't get why this is an issue--do you need anything about the continuum [0,1], as opposed to the set of natural numbers N? I always figured it was just much easier to talk about [0,1], especially when you're talking about shares of firms, but that nothing really relies on the "kind of infinity."
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I also don't get why this is an issue--do you need anything about the continuum [0,1], as opposed to the set of natural numbers N? I always figured it was just much easier to talk about [0,1], especially when you're talking about shares of firms, but that nothing really relies on the "kind of infinity."
The natural numbers isnt a compact set.