Short summary: It’s a really important paper. I think it’s too important to be true.
Gabaix’ irrationality fixes the pathologies of the standard model by making a stable model unstable, and hence locally determinate. Gabaix’ irrationality parameter M in [0,1] can thus substitute for the usual Taylor principle that interest rates move more than one for one with inflation.
Gabaix imagines — after three papers worth of careful math — that people pay less attention to future income when deciding on consumption than they should. Making today’s consumption less sensitive to future income, means expectations of future income are larger for any amount of today’s consumption. Thus, it makes model dynamics unstable.
But just a little irrationality won’t do. If you move a stable eigenvalue, say 0.8, by a bit, say 0.85, it’s still stable. You have to move it all the way past 1 before it does any good at all.
Thus, Gabaix puts irrationality right in the middle of monetary policy. If Gabaix is right, you simply cannot explain monetary policy in simple terms with money supply and money demand, or interest rate rises lower investment and inflation via a Phillips curve, as simple approximations that more complex models, perhaps involving some irrationality, improve on. Monetary policy is centrally about the Fed exploiting irrationality, full stop, and cannot be explained or understood at all without that feature.
More in the comments. There are too many equations and figures to mirror it here, so you have to get the pdf if you’re interested. This is for academics anyway.