I have not jumped into the recent stimulus debate. To be honest I've been pretty put off by the mudslinging, particularly from Cochrane's corner. But this was an interesting post from David Glasner on it. He makes the point that if you want to translate Lucas and Cochrane into a Keynesian model, we're talking about an MPC of zero, so that consumption is entirely determined by lifetime income.
This is all fine, but it's important not to reason an actual multiplier from a given MPC. Yes, we can get something that's been labeled "the multiplier", but what we actually care about (and what we estimate) is the change in national income in response to a change in government spending. That is generally smaller than the multiplier you get from a given MPC. Why? Because government spending can crowd out investment spending.
With crowding out, you don't even need an MPC of zero to get Cochrane's result.
Right now, of course, we're more likely to see crowding in than crowding out. I'm also not personally convinced by an MPC of zero.
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