I am equally puzzled as the author here.
Thoughts ?
http://www.bondeconomics.com/2017/05/the-horrifying-mathematics-of.html
I am equally puzzled as the author here.
Thoughts ?
http://www.bondeconomics.com/2017/05/the-horrifying-mathematics-of.html
You and the author are equally puzzled simply because you have not paid attention in first year courses.
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.)
@ 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)
The dude gets it wrong right from the start.
The profit of firm i is given by f(x,u(i)), and not by the integral he has defined.
To expand: When considering the aggregate profits, the contribution of the i-th agent to the total profits is given by the integral in the post.
But this is not the same as the profit of firm i.
@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 ?
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.
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.
I am equally puzzled as the author here.
Thoughts ?
http://www.bondeconomics.com/2017/05/the-horrifying-mathematics-of.html
OP, do you measure theory, much?
The i-th agent's integral would imply that all agents contribute 0 to aggregate profits. No ?
For a normally distributed random variable X, P(X=x)=0 for any x. Does this imply the probability of x falling in the any range is 0?
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.
For a normally distributed random variable X, P(X=x)=0 for any x. Does this imply the probability of x falling in the any range is 0?
Exactly.
@ 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).
Eeeeeeeeekkkkkkkk!!!!!!!!!!!!
Christ this dude's a moron
i am amazed at these types, who discard scholarship by reflex and think the only things worth reading are by other third-rate bloggers. and they accuse *us* of being an echo chamber!
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."
do you need anything about the continuum [0,1], as opposed to the set of natural numbers N?
Here's one thing that might be a reason. Is there a uniform distribution on the natural numbers?
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.