The point that I had hoped to make clear with my little proof was that Wren-Lewis does not constrain savings. I specifically set the change in savings equal to –1 so that the multiplier would be zero.

This is supposed to motivate you to tell me why I can’t do that.

Indeed, this is precisely what this graph from Krugman shows

See how Investment is just a horizontal line. And, we know that in the final analysis I = S.  Since, Investment is a horizontal line it cannot change. Hence savings cannot change.

S = –1 is not allowed.

However, the point that Scott was digging around was that this is not only a pretty significant assumption but it underlies the entire Old Keynesian framework.

Yet, neither Krugman nor Wren-Lewis come out and say this or suggest why Cochrane should accept this assumption or the Old Keynesian framework in general. This is important because Cochrane, indeed acknowledges, but then waves away the Old Keynesian model:

Yes, I’m aware that old Keynesian models do give a multiplier to tax financed spending. Also, some new Keynesian models such as Christiano, Eichenbaum and Rebelo (2009) predict huge government spending multipliers whether financed by taxes or by borrowing. However, tax-financed spending is usually thought to have a weaker (if any) effect, which is why the current policy debate is only about borrowing to spend.

Cochrane is saying: yeah you could get that if you were an old Keynesian but most people are suspect of that thinking nowadays.

I think the response to Cochrane and Lucas should go like this:

When the government raises taxes to fund additional spending then in theory the effect on aggregate demand depends crucially on what the money is spent on.

If the money is spent so that raises the marginal utility of consumption then that is going to tend to increase aggregate demand. Likewise if it is spent in a way that increases the marginal product of capital that will also tend to increase aggregate demand.

When Cochrane explicitly and Lucas implicitly thinks in terms of transferring purchasing power from one randomly chosen citizen to another there is no reason to expect that this will have any effect on either the marginal utility of consumption or the marginal product of capital.

Indeed, though Lucas trips himself up in the phrasing when he describes the government using money to purchase a bridge he is almost certainly describing a situation that will have an effect on the marginal utility of consumption and perhaps also the marginal product of capital.

Now, often I see people reasoning on the basis of whether or not the government will recruit idle resources to build the bridge or not. You can walk through the situation like that but I think its easier to see what’s going on by just assuming away that possibility and supposing that all of the resources recruited were not formerly idle.

In that case the government is pulling consumption away from taxpayers and pulling investment away from private investors in order to accumulate the resources necessary to build this bridge.

This mechanism should be highly intuitive to folks who think in a classical, seen-and-unseen, paradigm.

However, the analysis does not stop there. Neither the taxpayer nor the investor should just walk away accepting their fate. For the taxpayer his marginal utility of consumption has risen because he is now consuming less.

For the investor her return on investment has now risen because the marginal investment project has been taken from her. Both the taxpayer and the investor will attempt to borrow from the future to correct this imbalance.

You can think of the “stimulus” as actually occurring at this point. It is the effort of the consumer and the investor to correct for changes in the marginal utility of consumption and the marginal return on investment that expands aggregate demand.

If there are no resources available to fulfill this attempt then prices and interest rates will be driven up until the consumer and the investor are back into intertemporal balance. However, if resources are available then they will be recruited.

My hope is that this closes the intuition around why reasoning from random cash transfers or reasoning from a fully employed economy gives such different results from reasoning about a bridge purchase during a recession.

The key questions and differences are thoroughly microeconomic. What happens to the marginal utility of consumption? What happens to the marginal product of capital investment? What are consumers and investors able to do in response?