Take just one example: the Calvo pricing model. In that model, the Calvo fairy visits each firm at random, taps it with her wand, and lets it change its price. The probability of her visiting in any period is 1/n, so she visits each firm on average every n periods. Firms know this and set prices rationally when she visits.
Make one very small change to that model. Assume she is not random. Assume she visits a fraction 1/n firms each period, and visits each firm once every n periods. Firms know this and set prices rationally when she visits. That's a different model. I like it better than the first model; but it's a nightmare to solve.
Both those models are equally microfounded. Or equally not microfounded, because the fairy herself is, well, just a fairy, and not a real person. She's an ad hoc fairy, who is just a metaphor for our ignorance about why the price of money (the reciprocal of the price level) doesn't behave like the prices of other financial assets, the ones that are traded on centralised exchanges against money. The price of money won't be determined like the prices of those other financial assets, because money is the medium of exchange and the medium of account. So money can't be traded on one centralised exchange with a price of its own. It wouldn't be money if it were. But I digress.
I don't really like either model. But given a choice between only those two models, I prefer the second model to the first. I think it fits the microdata better, and I think it fits the macrodata better too. You wouldn't think it would matter much whether she's a random fairy or a non-random fairy, but it does. The non-random fairy generates inflation-inertia and the random fairy doesn't. You get a sticky inflation rate, and not just a sticky price level, with the non-random fairy.
Trouble is, I can solve the first model, but I can't solve the second model. So if I had to build a formal microfounded macromodel, and solve it, I would be forced to choose the first model and reject the second model.
There's a trade-off between the microfoundations we like and the microfoundations we can solve.
What can I do? I have three options:
1. I can assume microfoundations I don't like (with the random fairy) and solve the macromodel.
2. I can assume microfoundations I like, or, at least, like better than the first (with the non-random fairy), and wave my hands and talk about what I think the macromodel would say if I could solve it.
3. I can write down an equation for an ad hoc Phillips Curve with inflation inertia, wave my hands and say that I think it is roughly what would happen with the non-random fairy, add it to the macromodel, [Update: wave my hands and say I think it is consistent with the agents' behaviour that underpins the other equations in the model], and then solve it.
I know, from previous experience, that sometimes I get things wrong when I wave my hands. Sometimes, when I do the math, the results turns out differently than I thought they would.
This isn't an easy choice.
Maybe that's an argument for a fourth option:
4. Same as option 2, except you also do a few computer simulations ("agent based modelling"?) as a check to see if your hand-waving intuition is roughly right.
Hmmm. When I started to write this post, I didn't think it would end up as an argument for that conclusion. I am prejudiced against that conclusion, because computer simulations are something I have never done and wouldn't like doing and would be no good at doing. Oh well. Someone else can do it. But I wouldn't trust any of their results unless they confirm my hand-waving intuition.
[Update: though maybe there's still a trade-off, between microfoundations we like, and microfoundations we can program the computer to solve?]
This post is a sort of follow-up to Noah Smith's good post.