When economists defend the use of mathematical modeling, they often argue - as Ryan does - that mathematical modeling is good because it makes you lay our your assumptions clearly. If you lay out your assumptions clearly, you can think about how plausible they are (or aren't). But if you hide your assumptions behind a fog of imprecise English words, you can't pin down the assumptions and therefore you can't evaluate their plausibility.
True enough. But here's another thing I've noticed. Many economists insist that the realism of their assumptions is not important - the only important thing is that at the end of the day, the model fits the data of whatever phenomenon it's supposed to be modeling. This is called an "as if" model. For example, maybe individuals don't have rational expectations, but if the economy behaves as if they do, then it's OK to use a rational expectations model.
So I realized that there's a fundamental tradeoff here. The more you insist on fitting the micro data (plausibility), the less you will be able to fit the macro data ("as if" validity). I tried to write about this earlier, but I think this is a cleaner way of putting it: There is a tradeoff between macro validity and micro validity.
How severe is the tradeoff? It depends. For example, in physical chemistry, there's barely any tradeoff at all. If you use more precise quantum mechanics to model a molecule (micro validity), it will only improve your modeling of chemical reactions involving that molecule (macro validity). That's because, as a positivist might say, quantum mechanics really is the thing that is making the chemical reactions happen.
In econ, the tradeoff is often far more severe. For example, Smets-Wouters type macro models fit some aggregate time-series really well, but they rely on a bunch of pretty dodgy assumptions to do it. Another example is the micro/macro conflict over the Frisch elasticity of labor supply.
Why is the macro/micro validity tradeoff often so severe in econ? I think this happens when an entire theoretical framework is weak - i.e., when there are basic background assumptions that people don't question or tinker with, that are messing up the models.
For example, suppose our basic model of markets is that prices and quantities are set based purely on norms. People charge - and pay - what their conscience tells them they ought to, and people consume - and produce - the amount of stuff that people think they ought to, in the moral sense.
Now suppose we want to explain the price and quantity consumed of strawberries. Microeconomists measure people's norms about how much strawberries ought to cost, and how many strawberries people ought to eat. They do surveys, they do experiments, they look for quasi-experimental shifts that might be expected to create shifts in these norms. They get estimates for price and quantity norms. But they can't match the actual prices and quantities of strawberries. Not only that, they can't match other macro facts, like the covariance of strawberry prices with weather in strawberry-growing regions. (A few microeconomists even whisper about discarding the idea of norm-driven prices, but these heretics are harshly ridiculed on blogs and around the drink table at AEA meetings.)
So the macroeconomists take a crack at it. They make up a class of highly mathematical models that involve a lot of complicated odd-sounding mechanisms for the creation of strawberry-related norms. These assumptions don't look plausible at all, and in fact we know that some of them aren't realistic - for example, the macro people assume that weather creates new norms that then spread from person to person, which is something people have never actually observed happening. But anyway, after making these wacky, tortured models, the macro people manage to fit the facts - their models fit the observed patterns of strawberry prices and strawberry consumption, and other facts like the dependence on weather.
Now you get to choose. You can accept the macro models, with all of their weird assumptions, and say "The economy works as if norms spread from the weather", etc. etc. Or you can believe the micro evidence, and argue that the macro people are using implausible assumptions, and frame the facts as "puzzles" - the "strawberry weather premium puzzle" and so on. You have a tradeoff between valuing macro validity and valuing micro validity.
But the real reason you have this tradeoff is because you have big huge unchallenged assumptions in the background governing your entire model-making process. By focusing on norms you ignore production costs, consumption utility, etc. You can tinker with the silly curve-fitting assumptions in the macro model all you like, but it won't do you any good, because you're using the wrong kind of model in the first place.
So when we see this kind of tradeoff popping up a lot, I think it's a sign that there are some big deep problems with the modeling framework.
What kind of big deep problems might there be in business cycle models? Well, people or firms might not have rational expectations. They might not act as price-takers. They might not be very forward-looking. Norms might actually matter a lot. Their preferences might be something very weird that no one has thought of yet. Or several of the above might be true.
But anyway, until we figure out what the heck is, as a positivist might say, really going on in economies, we're going to have to choose between having plausible assumptions and having models that work "as if" they're true.