Mark Thoma explains the very basics of New Keynesian economics and I am very rude.
I came here clicking a link in a post where you indignantly deny that you are an old Keynesian. I ask what has been added by the very simple inter-temporal optimization ?
It seems to me that New Keynesian macro consists (as here) in writing models with optimizing agents which behave the way old Keynesian models behave. I ask why not cut out the middle man ?
The model as written has implications other than that there is something like an IS curve. As you note, the true expected value (that is rationally expected) of future GDP affects current GDP. Also the real interest rate affects the rate of growth of consumption.
The problem is that, to the extent new Keynessian models differ from old Keynesian models, the data are not kind to the new models. It is a fact that estimates of 1/theta made using aggregate data are tiny -- typically around 0.1. Such low intertemporal elasticities can be reconciled with the long term average rate of growth of consumption only by assuming Beta>1 which is crazy. It is also a fact that the scatter of growth of consumption on the econometricians forecast of r look a lot like scatters of independent variables. Also excess sensitivity is not in the model.
That which has been added is false and has been known to be false for decades.
Now, of course, there are more complicated new Keynesian models which also fit those facts. Like RBC new Keynesian economics can fit any summary statistic with just one arbitrary new variable for summary statistic.
The newness is all about intertemporal optimization without liquidity constraints. It clearly gives false implications. So the model is modified so that it acts just like an old Keynesian model. How is this a worthwhile activity ?
Notably, the micro foundations are not justified on the assumption that people really intertemporally optimize with rational expectations. The argument is that the economy could behave as if they did. If so, the case that it is better for macro models to have micro foundations is based on nothing at all. Anything might work. There is no reason to believe that the definitely false assumptions that new Keynesians like to make are better than any other definitely false assumptions.
What is wrong with Keynes or, for that matter, Hicks ? One might argue that it is unwise to treat a Phillips curve as an economic law holding in all times and places. This is, historically, the fact which convinced macro-economists to work on micro foundations. It was also argued by Keynes in spite of the disadvantage of writing before Phillips. What, of any value, have macroeconomists added to Keynes ? (note I am counting Hicks -- the move from LM to MP is a move from Hicks back to Keynes).
I close this long and very rude comment with the usual on methodology (for no particular reason).
Modern macro rests on the twin methodological pillars of "The Methodology of Positive Economics" and "Econometric Policy Evaluation: A Critique," but a better title for the second paper would be ""The Methodology of Positive Economics: A Critique," since it argues (correctly and as was well known at the time) that even if we don't care if a model is approximately true but only if it gives useful predictions, we must care if a model is approximately true. The model you present here definitely isn't.