Dietz Vollrath explains the "mathiness" debate (and also Euler's theorem in a part of the post I left out). Glad he's interpreting Romer -- it's very helpful:
What Assumptions Matter for Growth Theory?: The whole “mathiness” debate that Paul Romer started tumbled onwards this week... I was able to get a little clarity in this whole “price-taking” versus “market power” part of the debate. I’ll circle back to the actual “mathiness” issue at the end of the post.
There are really two questions we are dealing with here. First, do inputs to production earn their marginal product? Second, do the owners of non-rival ideas have market power or not? We can answer the first without having to answer the second.
Just to refresh, a production function tells us that output is determined by some combination of non-rival inputs and rival inputs. Non-rival inputs are things like ideas that can be used by many firms or people at once without limiting the use by others. Think of blueprints. Rival inputs are things that can only be used by one person or firm at a time. Think of nails. The income earned by both rival and non-rival inputs has to add up to total output.
Okay, given all that setup, here are three statements that could be true.
- Output is constant returns to scale in rival inputs
- Non-rival inputs receive some portion of output
- Rival inputs receive output equal to their marginal product
Romer’s argument is that (1) and (2) are true. (1) he asserts through replication arguments, like my example of replicating Earth. (2) he takes as an empirical fact. Therefore, (3) cannot be true. If the owners of non-rival inputs are compensated in any way, then it is necessarily true that rival inputs earn less than their marginal product. Notice that I don’t need to say anything about how the non-rival inputs are compensated here. But if they earn anything, then from Romer’s assumptions the rival inputs cannot be earning their marginal product.
Different authors have made different choices than Romer. McGrattan and Prescott abandoned (1) in favor of (2) and (3). Boldrin and Levine dropped (2) and accepted (1) and (3). Romer’s issue with these papers is that (1) and (2) are clearly true, so writing down a model that abandons one of these assumptions gives you a model that makes no sense in describing growth. ...
The “mathiness” comes from authors trying to elide the fact that they are abandoning (1) or (2). ...
[There's a lot more in the full post. Also, Romer comments on Vollrath here.]