This is a rather long post that follows from my earlier one on Faustian bargains. Mainly for economists.
Although the debate about microfoundations normally talks about models, it is possible in some cases to also talk about individual relationships. This has the advantage of simplicity. I think the Phillips curve is a nice illustration of many of the key issues in the debate - even for those who do not believe in it! (I make the more general argument here.)
First, it illustrates for me why the debate is not, I repeat not, about whether the microfoundations approach is useful. Of course it is. Let’s think about the Phillips curve before microfoundations. This traditional Phillips curve had inflation at time t depending on expectations of inflation also at time t, and the output gap. This seemed at the time plausible: you could tell a story about lags in wage setting, so actors in the labour market had to guess the relevant real wage etc.
This is an example of the dangers of ad hoc theorising. The lags involved in wage decisions arise primarily because of contracts: wages are being fixed not just for the next period, but for a number of periods. Once you do the microfoundations properly, you find that inflation today depends on expected inflation next period: the New Keynesian Phillips curve (NKPC).
A detail? Unfortunately the ad hoc version implied that the output gap would move with expectation errors. So the dynamics of the business cycle became all about the inflation expectation error process. This was always implausible, even if you did not believe in rational expectations. With rational expectations, it appeared to deny the existence of Keynesian business cycles, and led to a whole generation of students being taught that anticipated monetary policy was ineffective.
So the traditional Phillips curve illustrates the dangers of ‘ad hoc’ theorising. But the relevant debate is not about whether microfounded relationships are useful, but whether they should be the only game in the academic town. So now let me give you various reasons why they should not, based on the Phillips curve example.
First, although various microfounded models suggest inflation should depend on expected inflation next period, the empirical evidence is more equivocal. A number of studies have found that inflation today depends on both expected inflation next period, but also on actual inflation last period (‘inflation inertia’). There are also empirical regularities (e.g. Phillips curve loops) that are difficult to rationalise with just a NKPC. As yet there is no generally accepted microfoundation for why this inflation inertia might occur, but given the key role this equation plays in policy it is important to look at the consequences of inflation inertia. That is what Steinsson did in a 2003 paper in JME, for example.  Now Steinsson’s introduction of ‘rule of thumb’ price setters was entirely ad hoc, with no microfoundation (traditional or behavioural) offered. Would those who decry theory by ‘hand waving’ as ad hoc think that article should not have been published in the JME?
At this point some might say that this example illustrates that there is no problem - papers with non-microfounded relationships do get published. Yet everyone knows that there is a large effective bias against publishing models that are not fully microfounded: Steinsson’s paper is the exception rather than the rule. (See my discussion of middlebrow models.) Indeed I started seriously thinking about these issues after I attended a conference where one respected participant attacked a couple of papers that did include inflation inertia, saying we must ‘respect microfoundations’. It was this that gave rise to my distinction between microfoundations purists and microfoundations pragmatists.
This example illustrates that we are often not confronted with a choice between two relationships: one microfounded and one not. Instead we have the ad hoc relationship, and we are still waiting for the microfounded version. Microfoundations takes time. There is a simple reason for this: microfoundations modelling is difficult, and it often relies on certain tricks to retain tractability. Those tricks may not be immediately obvious, and may take time to become ‘acceptable’. So in the early 1970s, before New Keynesian theory was developed, we only had the traditional (ad hoc) Phillips curve. In those circumstances, are we seriously suggesting that academic macroeconomics should have ignored this relationship because it was ad hoc? Surely we want to model what we can see, and not just what we can microfound.
Second, an aggregate relationship can be much more robust than any particular microfoundation. You get the result that inflation depends on expected inflation next period from different models of price rigidity: Calvo contracts of course (which can be said to mimic the impact of menu costs - see below), but also fixed contracts. Why tie things down to one particular price rigidity story, rather than going directly to an aggregate relationship which is compatible with more than one microfoundation?
The answer is often consistency. You need to demonstrate that your microfoundations are consistent across the model as a whole. Now this is clearly true for something like consumption and labour supply decisions. However models of nominal inertia are pretty separable from the rest of a model, or we employ tricks to ensure that they are. So the reasons for having to go through the Calvo set-up every time anyone wants to use a NKPC hardly seem compelling.
Third, just how great are these microfoundations anyway? The Calvo contract story has the expected probability of changing prices exogenous to the firm, and it is not obvious why this is a reasonable thing to assume. In practice microfoundation modellers wave their hands as much as anyone, talking about the impact of menu costs without formally modelling them. As I argue here, this is an ‘as if’ form of justification which is an important step removed from establishing model consistency based on only deep parameters.
If all this were not compelling enough, a belief that the microfoundations methodology is the only way of ‘doing macroeconomics properly’ has unfortunate side effects. It moves modellers away from the real world and into an imaginary world of yeoman farmers and the like. So some will react to the last paragraph by agreeing, saying that they were always dubious that Calvo contracts represented proper microfoundations, and that it is better therefore to ignore price stickiness until a more acceptable microfoundation can be found.
This is a bizarre reaction. The issue with price rigidity is that there are too many potential microfoundations, not too few. (See this well known paper by Alan Blinder for example.) The problem is in reality two fold. First, each reason is quite complex, so they are difficult to easily incorporate in a model. Tractability is the problem, not evidence. Second, even if you can do this for one particular motivation for price stickiness (e.g. Calvo contracts for menu costs), you are ignoring all the others - you ignore heterogeneity. Is this because you do not think these other reasons matter, or is it because the model is complicated enough as it is? The latter, obviously. Yes, of course it is important to do microfoundations modelling with heterogeneity - but this often results in a loss of clarity. If this heterogeneity in practice leads to a very similar aggregate outcome, why not go straight to the aggregate?
We are in danger of fooling ourselves here. We choose Calvo contracts to microfound price rigidity not because we believe this is how the majority of firms behave, but because it is tractable, and gives us an aggregate relationship (the New Keynesian Phillips curve) that is consistent with the data. Yet we then go on to argue that if any other aspect of the model is internally inconsistent in terms of its microeconomics with Calvo contracts, the model should be rejected. Finally, we then claim that anyone who uses a New Keynesian Phillips curve without doing this microfoundations check is not doing serious macroeconomics.
So we choose a microfoundation because it gives us the aggregate answer suggested by the data, and not because of evidence that this microfoundation is appropriate. We then insist that everything in that model has to be consistent with this microfoundation, and that our model has been built from only thinking about what individual agents do. Have we just replaced ad hoc with post hoc?
So, to summarise. The Phillips curve shows why microfoundations macro is really useful. However it also shows why microfoundations modelling should not be the only way of doing proper macroeconomics.
 Steinsson, J (2003), Optimal Monetary Policy in an economy with Inflation Persistence, Journal of Monetary Economics, 50, 1425-1456