Matt thinks that monetary frictions don't matter. Mike Woodford made the same mistake, and set a large fraction of macroeconomists off to work on New Keynesian models. Then, the financial crisis hit, and central banks started to engage in some unprecedented interventions, about which standard New Keynesian models had nothing to say.
In a basic Woodford model (see for example Mike's book, Interest and Prices), all monetary frictions are stripped away. In a Woodford world, the only friction comes about because of nominal price stickiness (and perhaps wage stickiness too), which leads to relative price distortions. What does a central bank do in a Woodford world? It sets the price of a bond which is a claim to "money" in the future, but this money is not actually held by anyone, in spite of the fact that all prices and wages are denominated in units of the stuff.
Why does central banking matter? It matters because a central bank can engage in intermediation activities that are not replicated in the private sector. A typical central bank has a monopoly on the issue of currency (in the US this is implicit), and on the large-value payments system. Thus, currency and bank reserves are liabilities that cannot be issued by private financial institutions. When the central bank issues its liabilities in order to buy assets, this in general matters. In particular, asset prices move. To understand how that process works, one has to model the frictions that make private intermediation useful, and the frictions that make assets of all kinds (including the ones conventionally called "money") useful in exchange. By "exchange," I mean exchange of all kinds, including retail exchange, and exchange among financial institutions.
Matt, for starters, you can read this blog post, this piece with Randy Wright, our chapter in the Handbook of Monetary Economics, and this forthcoming AER paper. The latter shows you why you need monetary frictions to understand the financial crisis and unconventional monetary policy.
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