[Update 4 December: after long arguments with Adam P., which finally started to bear fruit in the comments of December 4th, I'm now clearer on what's essential in this model about the difference between "monetary exchange" and "barter exchange". It's this: In monetary exchange, there is no restriction on what the seller can do with the proceeds of a sale. He can spend the proceeds on anything, or save them. In barter exchange, there is a restriction on what the seller can do with the proceeds of a sale; he must spend all the proceeds on the buyer's goods. That is what is driving the results in this model. I think that is one important distinction between monetary and barter exchange. I have left the rest of the post as is.]
Keynesian macroeconomics in general, including New Keynesian macroeconomics in particular, makes absolutely no sense whatsoever in a barter economy. If people could trade goods and labour directly at zero transactions costs, without having to use monetary exchange, all Keynesian macroeconomics would be total rubbish. All Keynesian macroeconomics, either explicitly or implicitly, assumes monetary exchange. It's not just sticky prices that generate Keynesian results. It's sticky prices plus monetary exchange.
I used to think that the above paragraph was totally uncontroversial. I thought everybody understood this. I have learned they don't. So I'm going to do my best to explain why it's true.
I show that if we introduce barter into an otherwise nearly standard New Keynesian model, the solution immediately reverts to the perfectly competitive equilibrium, regardless of the degree of imperfect competition in the original model, regardless of any mistake made by the central bank in setting the nominal rate of interest, and regardless of sticky prices.