As misleading as comparing penetration by a rapist to penetration by photons but (mercifully) less offensive

UPDATE for the sake of traffic directed here by Krugman and DeLong: I've written a couple things at this point so the summary is:

1. This post (you're here. you don't need a link)
2. Henry Hazlitt is often promoted as a place for people to start with economics. I don't think that's wise.
3. The estimable Gene Callahan has an older post on this same argument from Rothbard, but I think his criticism doesn't go far enough - explanation here.
4. There is a legitimate problem with the Keynesian cross, but New Keynesianism addresses it.
5. One of my commenters makes my point much more succinctly - I reproduce that with an explanation of how it relates to my point here.

*****

He writes:

"Now, though I cannot seem to find a reference, I have a vague memory that it was Murray Rothbard who observed [<<< DPK: Well we're clearly off to a bad start] that the really neat thing about this argument is that you can do exactly the same thing with any accounting identity. Let’s start with this one:

Y = L + E

Here Y is economy-wide income, L is Landsburg’s income, and E is everyone else’s income. No disputing that one.

Next we observe that everyone else’s share of the income tends to be about 99.999999% of the total. In symbols, we have:

E = .99999999 Y

Combine these two equations, do your algebra, and voila:

Y = 100,000,000 L

That 100,000,000 there is the soon-to-be-famous “Landsburg multiplier”. Our equation proves that if you send Landsburg a dollar, you’ll generate $100,000,000 worth of income for everyone else."

No, no, no, no.

The Keynesian multiplier has two sides: income and expenditure. The economy's income today is the economy's expenditure yesterday. If you had a lot of income today but there was not a lot of expenditure today, then tomorrow you will have less income.

Landsburg drops the expenditure side of that equation. In effect, he swaps:

Y = C + I

for

Y = wL + rK

Except his income earners are himself and everybody else.

Wonder of wonders, when you set income equal to income you get a forty-five degree line.

So by botching the income/expenditure distinction Landsburg gives us a triviality. Even Say did better than that. Say at least kept income and expenditure separate. His mistake (initially) was assuming they were always equal. But on top of that triviality, Landsburg adds a genuine error.

The slope of the consumption component of the expenditure side of the Keynesian cross (the side that Landsburg is missing) is not strictly speaking determined by the consumption share of output, it's determined by the marginal propensity to consume. When we consider the economy as a whole, of course, we can think in terms of the consumption share of output (investment equals savings and all that). Normally we are talking about the economy as a whole, so it doesn't matter all that much that we make that distinction (when I TAed an intro macro class I always taught it as MPC, though).

Now as I highlighted above, we can't even really call Landsburg's 0.99999999 figure an MPC because it's not expenditure - it's income. But Landsburg is treating "L + E" like expenditure, so let's at least do it correctly for him. Either it's all consumed or part of it is not consumed. Landsburg seems to want E to consume all of it's income (hence the 0.99999999 coefficient) and since he is furnishing that dollar out of L to E presumably he is consuming all of his income, or 0.00000001 of total income. So actually the MPC for the economy is 1, and not 0.99999999, implying an infinite multiplier. I buy something from you for one dollar, you spend that whole dollar buying something from someone else for one dollar, etc. etc. and the dollars keep racking up. You can think of it as shifting the consumption curve (which is parallel to the 45-degree line) up such that there is no longer any stable equilibrium and consumption bounces off to eternity.

This brings me back to my initial point: Keynes made the multiplier go because he had income and expenditure. When you have income and income you get infinite solutions because you just have a tautology. When Landsburg mixed up MPC and income share (and implicitly assumed an MPC of 1 even though he mistakenly called it an MPC of 0.99999999, which gave him a very large equilibrium solution), he was essentially repeating the tautology under the guise of having performed the same exercise.

A very minor additional problem posed by Landsburg

So Landsburg confuses a couple things - he doesn't have an expenditure side so he has to smuggle it in later, and when he smuggles it in later he confuses MPC and income share which concealed the fact that he was only dealing in a single identity (if he had realized his implicit MPC was 1 and not 0.99999999 he might have picked up on what was going wrong).

Econ 101 students ought to know better because of course the Keynesian cross is pretty uniformly taught with both income and expenditure from the get-go!

But this does raise a ([very] minor) additional problem: if people consume a lot of their income then we can get similar outrageously high multipliers! Wouldn't the implication be that policymakers should encourage citizens to consume as much of their income as they can?

Of course not. Because every Econ 101 textbook I know of teaches aggregate supply before they teach aggregate demand. You can gussy that up with as much optimization as you want, but at the end of the day an Econ 101 student can still understand how that pins the multiplier down with the imperative of investment.

One more point

Take the consumption share of income, use that as an MPC for convenience sake, and get a multiplier. What is it - four? five? That's still outrageously high.

I've seen people actually cite this as proof that the Keynesian model is wrong.

Guess what folks: you have to factor in crowding out. As Keynes wrote in chapter 10 of the General Theory: "if the propensity to consume in various hypothetical circumstances is (together with certain other conditions) taken as given and we conceive the monetary or other public authority to take steps to stimulate or retard investment, the change in the amount of employment will be a function of the net change in the amount of investment".

Another way of putting this is that when the government spends money you can't just multiply that by the multiplier from the Keynesian cross! Or, put another way, you have to multiply government expenditure by the multiplier from the Keynesian cross, and subtract out the reduction in investment multiplied by the multiplier from the Keynesian cross (and of course also subtract out any reduction in consumption through taxation, etc., multiplied by a here-unspecified multiplier that will be different from "the multiplier"). In a sense it's misleading to call the empirical government spending multiplier "the multiplier" and to also call the Keynesian cross multiplier "the multiplier", but if you think about what the Keynesian cross is doing it's not all that hard to keep straight.