Suppose you had an economy where half the agents are "Hand-To-Mouth" and have a Marginal Propensity to Consume of one (Ct=Yt), and the other half are "Autonomous" and have a Marginal Propensity to Consume of zero (Ct=At where At is exogenous with respect to their current income). If the two types of agents initially have the same income, the Marginal Propensity to Consume for the economy as a whole would be 0.5 and so the Old Keynesian Multiplier would be 2.
For simplicity, ignore all other forms of expenditure, and suppose that aggregate income is determined by desired aggregate consumption expenditure (It's an economy with excess supply of goods, where output is demand-determined).
Suppose something causes the Autonomous agents to increase their consumption by 1%. The first round effect is to increase aggregate income by 0.5% (because they are only half the economy). In the second round the Hand-To-Mouth agents increase their consumption by 0.5%, which increases aggregate income by an additional 0.25%. And so on, until the increase in aggregate income converges to 1% in the limit of the multiplier process.
[Update: to paraphrase Kalecki(?): Hand-To-Mouth agents spend what they earn, so Autonomous agents earn what they spend.]
Even though half the agents are Hand-To-Mouth, this economy gives exactly the same results as one where all the agents are Autonomous. The Hand-To-Mouth agents end up mimicking the Autonomous agents.
I don't need the assumptions that it is exactly half of each type of agent, and that they start out with exactly the same level of income. Because the smaller the percentage of agents that are Autonomous, the bigger the multiplier effect of a change in their consumption.
But I do need the assumption that some agents are Autonomous, because at the limit, where all are Hand-To-Mouth, equilibrium income is indeterminate in this economy. But I can still get a determinate answer for the multiplier if I look at what happens in the limit, as the number of rounds in the multiplier process approaches infinity, and as the percentage of Hand-To-Mouth agents approaches zero.
Now suppose that the Autonomous agents are infinitely-lived consumption-smoothers, like the agents in a standard simple New Keynesian model. Their consumption function is C(t)/Et[C(t+1]=some negative function of the real interest rate r(t), which is assumed to be set by the central bank. Standard Euler equation stuff.
You can see why "Forward Guidance" matters a lot in New Keynesian models. Anything that causes expected future consumption to increase by 1% causes current consumption to increase by the same 1%, regardless of how far in the future agents expect that thing to happen. And that thing that happens could be an announcement by the central bank that it will cut the real interest rate 100 years in the future. The effect propagates back down the chain all the way to the present, without petering away.
But you can also see that equilibrium income is indeterminate in this model. Because if I have one equilibrium, then assume that consumption is 1% higher than that equilibrium for the current and all future periods, for both types of agent, I have a second equilibrium. Which means that I can only talk about the multiplier effects of some exogenous change if I imagine the agents in my model thinking through the first round, then second round, then third round, etc. effects of the multiplier process. Just like my professors did on the blackboard when I was but a lad and they were teaching me Old Keynesian macroeconomics. (But it's got bugger all to do with inventories (think haircuts); it's all about updating expectations either in real time or virtual time.) You take the limit of the multiplier process, and refuse to answer what happens at the limit as opposed to in the limit.
You should be able to see where this post is going. The Hand-To-Mouth agents don't care about the future; they only care about their current income. But in any period they mimic what the Autonomous agents are doing. And when the Autonomous agents hear the central bank's announcement about what it will do a century in the future, they figure out how they will respond, and how the Hand-To-Mouth will respond to their responses, and to their own responses, and so on. And they will see that their figurings are converging on an answer where the effect of the central bank's announcement is exactly the same whether it is an announcement about what the bank will do next month or next century. Their permanent income and current income, and permanent consumption and current consumption, all rise by the same percentage.
Take a 2-period example to see how it works. The Autonomous agents have a 0.5 Marginal Propensity to Consume out of current income. So if they get news from the central bank that leads them to expect that future income will be 1% higher than they had previously expected, their first round effect is to increase their consumption by 0.5% in both the current and future periods, which means (since they are half the economy) that aggregate income will increase by 0.25% in both the current and future periods. In the second round of their figuring, they figure the Hand-To-Mouth agents will respond by increasing their consumption by 0.25% in both periods, which means aggregate income increases by an additional 0.125% in both periods. But then the Autonomous agents decide to increase their consumption by an additional 0.125% in both periods. And so on. And (0.25+0.125+0.125)+(0.0625+0.03125+0.3125)+......etc. = 0.5+0.25+.....etc. = one. So in the limit of their figurings, their expectation of a 1% increase in future income becomes a self-fulfilling rational expectation, and also causes current income to increase by 1% too.
The Hand-To-Mouth agents create a Cross-Section Multiplier; the Autonomous agents create a Time-Series Multiplier. The two Multipliers multiply each other.
I think that's the intuition behind the new(ish) paper by Emmanuel Farhi and Ivan Werning pdf (HT Christian Odendahl). But (as usual) I'm not 100% sure about that, because I'm not very good at math, and so I had to try to reverse-engineer their model. What's new, interesting, and important is that the future can matter just as much even though most people don't care about the future. And I think that's because there are two multipliers multiplying each other.
[Update: I changed the title of the post, to better reflect the content. On reflection: the Cross-Section Multiplier is Old Keynesian; the Time-Series Multiplier is New Keynesian. They work together.]