Romer argues that Euler's theorem (regarding homogenous functions) has the same intellectual status as the second law of thermodynamics writing "denies Euler’s theorem, which for economists is about the same as denying the second law of thermodynamics is for physicists."
I find this odd, because Euler's theorem is math while the second law of thermodynamics is a scientific hypothesis -- it doesn't just follow from standard assumptions it implies predictions which have been confirmed again and again and again. One exception would prove the second law of thermodynamics false (for example if someone actually made and demonstrated a perpetual motioni machine). I think Romer's post is an illustration of the fact that for economists, standard assumptions have the status of known facts.
In particular, I think Romer assumes, and assumes one must assume, that firms maximize profits. To be fair, I guess he really notes that Andolfatto assumes that firms maximize profits. This whole post might be a verbal quibble about leaving a shared assumption unstated.
update 2: Wow Paul Romer has commented on this post. He doesn't think that one must assume that firms maximize profits (my guess was just wrong -- sorry for casually typing it). Also he notes that Andolfatto discussed "a competitive equilibrium with price-taking behavior" and that equilibrium in this context implies profit maximization (as well as utility maximization and market clearing). In fact the assumption was stated by Romer. Again sorry about an incorrect quibble. end update 2 Romer considers two possibilities -- either there is market power or there is price taking. Price taking means that firms maximize profits taking prices as given and also that one firm can't affect market prices. The two meanings are identical if it assumed that firms are rational profit maximizers.
Romer writes about what must be true -- he claims that Andolfatto's claims are mathematically impossible. Now, I am sure that Andolfatto assumed rational profit maximization so I don't really have a defence of his post (which I haven't read (update: I have read it)).
I think I can focus by writing about what Romer wrote about something else which I haven't read
But the next level down in the hierarchy, followers like Boldrin and Levine are willing to just embarrass themselves:
It is widely believed that competitive equilibrium always results in prices equal to marginal cost. Hence the belief that competition is inconsistent with innovation. However widespread this belief may be, it is not correct. It is true only in the absence of capacity constraints, …(2008, p. 436)
They say they can ignore Euler’s Theorem, because in their bizarro version of a competitive equilibrium, prices for inputs do not equal marginal products.
But instead of presenting a competitive equilibrium of this type, they present a of an innovator who turns out to have market power. Their solution? The innovator has to be a price taker because they say so:
Making the initial single innovator behave competitively even in the very first period may be a source of misunderstanding. Since, by necessity, she has a monopoly in the initial period, why do we not take account of her incentive to restrict the initial supply…? (Boldrin and Levine, 2008, p. 438)
So in the analogy from physics, Boldrin and Levine say that it is possible to build a perpetual motion machine, but to their credit, at the last moment they back off and invoke the can-opener joke: “Well we haven’t actually built a perpetual motion machine, … so let’s just assume that we have one.”
What if Boldrin and Levine discussed a "bizarro version of a competitive equilibrium" but then don't present a competitive equilibrium of that type. The "bizarro" discussion would be an aside, perhaps a distraction, but not an error. It also isn't bizarre at all. If there is an absolute hard capacity constraint, then variable inputs do not always have a marginal product. The partial derivative of output with respect to the variable input does not exist when output hits capacity there is a kink in output as a function of that input (for other inputs given).
Similarly marginal cost doesn't exist at capacity -- it is the derivative of cost with respect to output and at capacity there is only a left derivative not a derivative -- if price is greater than marginal cost (as conventionally defined at that point as the left derivative) a price taking firm can't obtain greater profits by producing more -- that's what "capacity constraint" means.
I can sketch a hint of part of a growth model with price taking and capacity constraints. It is crazy, but I think it can be made consistent (hmm claiming consistency based on presenting a part of hmmm).
As a result of costly R&D new intermediate goods are sometimes invented. It is assumed that a continuum of agents of measure one has the same Eureka moment (at the same moment). Each has a limit to how much of the good they can produce so that if all produce to capacity 2 units of the good are produced in total. The good is sold to a perfectly competitive final goods sector where the marginal product at 2 units is 3. The cost of producing a unit of the good is 1 per unit. The inventors share a total profit of 2>0. Each inventor correctly assumes that the price will be 3 no matter what she does. Therefore each produces at maximum capacity. This is profit maximization with price taking where the profit maximizing point is a corner solution. No first order condition holds at a corner solution. Now it's crazy to assume that infinitely many people have the same idea at the same moment. But that's just because it is always crazy to assume perfectly perfect competition.
I suppose one might consider the concept of a capacity constraint to be bizarre, but there is nothing bizarro about the (super brief) discussion of the implications of capacity constraints
OK that was a long aside about a short aside. Now "Since, by necessity, she has a monopoly in the initial period, why do we not take account of her incentive to restrict the initial supply…?" I sure don't know the answer to that, but I can make up a story which doesn't seem bizarre to me which works except for the word "she". I think it is entirely possible that firms with market power act as price takers. This implies failing to respond to an economic incentive -- failing to maximize profits. In my story the firm has two divisions production and sales. The production division looks at the price for which the good has been sold, treats it as given and figures out how much to produce. The sales division sets the price in order to sell all that has been produced. There is no Walrasian auctioneer so people in sales must notice that they have to cut the price to sell all of the product -- they know the firm has market power. But, the salespeople have no incentive to tell the production people about this (they have poorly defined incentive contracts that they must sell a quota of the product for a price not lower than that charged by other sales people). The firm doesn't maximize profits, because of a failure of communication.
Is this really less plausible than the standard model in which a large number of people work together as one to maximize profits for the sake of shareholders they have never met ?
In the end (which I trust no reader has actually reached, because don't you have anything better to to with your time) this is a quibble. "Price taking" can be defined as both firms attempt to maximize profits assuming that their actions don't affect prices and their actions in fact don't affect prices. Certainly my attempt to defend fresh water economists by questioning the assumption of rational profit maximization is bizarre in the extreme. I guess the proviso "if we all assume rational profit maximization (as we all do) then ..." might be assumed to go without saying, and all I am saying is that Romer went with it without saying.