Old-Keynesian. The "Keynesian cross" is the most basic mechanism. (If you are worried that I'm making this up, see Greg Mankiw's Macroeconomics, p. 308 eighth edition, "Fiscal policy and the multiplier: Government Purchases.")
Consumption follows a "consumption function." If people get more income Y, they consume more C
C = a + m Y.
Output Y is determined by consumption C investment I and government spending G
Y = C+ I + G.
Put the two together and equilibrium output is
Y = a + mY + I + G
Y = (a + I + G)/(1-m).
So, if the marginal propensity to consume m=0.6, then each dollar of government spending G generates not just one dollar of output Y (first equation), but $2.5 dollars of additional output.
This model captures a satisfying story. More government spending, even if on completely useless projects, "puts money in people's pockets." Those people in turn go out and spend, providing more income for others, who go out and spend, and so on. We pull ourselves up by our bootstraps. Saving is the enemy, as it lowers the marginal propensity to consume and reduces this multiplier.
New-Keynesian. The heart of the New-Keynesian model is a completely different view of consumption. In its simplest version
Here consumption C, relative to trend, equals the sum of all future real interest rates i less inflation π i.e. all future real interest rates. The parameter σ measures how resistant people are to consuming less today and more tomorrow when offered a higher interest rate.
(This is just the integrated version of the standard first order condition, in discrete time
In this model too, totally wasted government spending can raise consumption and hence output, but by a radically different mechanism. Government spending raises inflation π . (How is not important here, that's in the Phillips curve.) Holding nominal interest rates i fixed, either at the zero bound or with Fed cooperation, more inflation π means lower real interest rates. It induces consumers to spend their money today rather than in the future, before that money loses value.
Now, lowering consumption growth is normally a bad thing. But new-Keynesian modelers assume that the economy reverts to trend, so lowering growth rates is good, and raises the level of consumption today with no ill effects tomorrow. (More in a previous post here)
This new-Keynesian model is an utterly and completely different mechanism and story. The heart of the New-Keynesian model is Milton Friedman's permanent income theory of consumption, against which old-Keynesians fought so long and hard! Actually, it's more radical than Friedman: The marginal propensity to consume is exactly and precisely zero in the new-Keynesian model. There is no income at all on the right hand side. Why? By holding expected future consumption constant, i.e. by assuming the economy reverts to trend and no more, there is no such thing as a permanent increase in consumption.
The old-Keynesian model is driven completely by an income effect with no substitution effect. Consumers don't think about today vs. the future at all. The new-Keynesian model based on the intertemporal substitution effect with no income effect at all.
Models and stories
Now, why is Grumpy grumpy?
Many Keynesian commentators have been arguing for much more stimulus. They like to write the nice story, how we put money in people's pockets, and then they go and spend, and that puts more money in other people's pockets, and so on.
But, alas, the old-Keynesian model of that story is wrong. It's just not economics. A 40 year quest for "microfoundations" came up with nothing. How many Nobel prizes have they given for demolishing the old-Keynesian model? At least Friedman, Lucas, Prescott, Kydland, Sargent and Sims. Since about 1980, if you send a paper with this model to any half respectable journal, they will reject it instantly.
But people love the story. Policy makers love the story. Most of Washington loves the story. Most of Washington policy analysis uses Keynesian models or Keynesian thinking. This is really curious. Our whole policy establishment uses a model that cannot be published in a peer-reviewed journal. Imagine if the climate scientists were telling us to spend a trillion dollars on carbon dioxide mitigation -- but they had not been able to publish any of their models in peer-reviewed journals for 35 years.
What to do? Part of the fashion is to say that all of academic economics is nuts and just abandoned the eternal verities of Keynes 35 years ago, even if nobody ever really did get the foundations right. But they know that such anti-intellectualism is not totally convincing, so it's also fashionable to use new-Keynesian models as holy water. Something like "well, I didn't read all the equations, but Woodford's book sprinkles all the right Lucas-Sargent-Prescott holy water on it and makes this all respectable again." Cognitive dissonance allows one to make these contradictory arguments simultaneously.
Except new-Keynesian economics does no such thing, as I think this example makes clear. If you want to use new-Keynesian models to defend stimulus, do it forthrightly: "The government should spend money, even if on totally wasted projects, because that will cause inflation, inflation will lower real interest rates, lower real interest rates will induce people to consume today rather than tomorrow, we believe tomorrow's consumption will revert to trend anyway, so this step will increase demand. We disclaim any income-based "multiplier," sorry, our new models have no such effect, and we'll stand up in public and tell any politician who uses this argument that it's wrong."
That, at least, would be honest. If not particularly effective!
You may disagree with all of this, but that reinforces another important lesson. In macroeconomics, the step of crafting a story from the equations, figuring out what our little quantitative parables mean for policy, and understanding and explaining the mechanisms, is really hard, even when the equations are very simple. And it's important. Nobody trusts black boxes. The Chicago-Minnesota equilibrium school never really got people to understand what was in the black box and trust the answers. The DSGE new Keynesian black box has some very unexpected stories in it, and is very very far from providing justification for old-Keynesian intuition.