Gene Callahan on the Landsburg-Rothbard-Hazlitt Cross

First, Bob Murphy thinks that we should judge the quality of an idea on the basis of the two equations that make it up and not in the context of all the economics that's taught around it. I think that's stupid and am not going to spend more than these two sentences on it.

Now, on to Gene. He's apparently posted on this critique before (HT noiselull).

It's an argument that was developed in Landsburg's comment section too. I didn't think it made much sense at the time Gene's is a little clearer but I still think there are some problems. First, note that this is different from my criticism. My criticism, in a nutshell, was:

1. Landsburg dropped the expenditure side of the income-expenditure identity so he basically had a one-to-one mapping of income on income, then  

2. Without the expenditure side and with a somewhat confusing conflation of MPC and income share he had an implicit MPC (implicit based on how he was doing the math) of 0.99999999, but probably more accurately (because he left out his own consumption) an implicit MPC of 1.  

3. This was all a little obscured because expenditure was dropped, but once you reconstructed the expenditure side it was a "valid" multiplier result.  

4. "Valid" is in quotation marks because it's only "valid" if you completely ignore... um... economics.

Now, REALLY on to Gene. This is his argument, reproduced in its entirety, which is a little different:

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"Murray Rothbard never cared if an argument he offered was sound, but only about whether it seemed to make his opponent look stupid. Consider, for instance, his "reductio" of the Keynesian multiplier:

Social Income = Income of (insert name of any person, say the reader) + Income of everyone else.

Let us use symbols:

Social income = Y

Income of the Reader = R

Income of everyone else = V

We find that V is a completely stable function of Y. Plot the two on coordinates, and we find historical one-to-one correspondence between them. It is a tremendously stable function, far more stable than the “consumption function.” On the other hand, plot R against Y. Here we find, instead of perfect correlation, only the remotest of connections between the fluctuating income of the reader of these lines and the social income. Therefore, this reader’s income is the active, volatile, uncertain element in the social income, while everyone else’s income is passive, stable, determined by the social income.

Let us say the equation arrived at is: V = .99999 Y

Then, Y = .99999 Y + R
.00001 Y = R

Y = 100,000 R

This is the reader’s own personal multiplier, a far more powerful one than the investment multiplier. To increase social in- come and thereby cure depression and unemployment, it is only necessary for the government to print a certain number of dollars and give them to the reader of these lines. The reader’s spending will prime the pump of a 100,000-fold increase in the national income.

-- Man, Economy, and State, p. 867-868

Of course, what has been ignored is the relationship of R to V. In the simplest version of the Keynesian multiplier story, an essential element is that C remains a constant percentage of Y (ignoring autonomous consumption) regardless of what happens to G or I. Therefore, an increase in G or I must lead to enough of an increase in C to keep that percentage where it was. Now, that may be an invalid assumption, but Keynesians have a story as to why we'd expect something like that to happen, and then they actually go and do empirical work to discover to what extent it really does happen.

But if we look at the relationship of R to V, we would expect to find that they vary inversely: If my income was .00001 Y and it doubled the next year, the income of everyone else simply dropped to .99998 Y. Rothbard has played a trick here: because V is such a huge percentage of Y, he can claim it is "a completely stable function of Y." But the correct thing to look at is: When R changes, does V respond so as to maintain its ratio to Y? While the Keynesian story may be wrong, at least it represents a theory as to why the ratio might be roughly constant. Rothbard, on the other hand, has absolutely no theory for positing that V and R are behave like they do in his model, and thus his example is hardly "in keeping with the Keynesian method."

The correct answer for the multiplier in Rothbard's example is roughly 1 rather than 100,000. Rather a bad error. But who cares? For readers who did not bother to analyze the example carefully, it sure made the Keynesians look dumb!"

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So one thing I don't like about this is that it seems to accept the Landsburg-Rothbard-Hazlitt conflation of income share and MPC.

Again, that conflation doesn't matter if we are dealing with an expenditure equation, but if we're dealing with an income equation it messes things up because on the income side share of income is not necessarily equal to MPC. I may have 0.00000001 share of total income but my MPC is a lot higher than that. Everyone else has 0.99999999 share of total income but their MPC is almost certainly much lower than that. This is all clear if you have an expenditure equation because then we remember our economics and what we learned about supply and that we need to save some so that we can invest. But if you are just tinkering with an income side you get stupid results like MPC = 1 or near 1.

So Gene has a point - if you're going to use income share at least use it right. Gene shows more clearly than what I was picking up in the comment section how Landsburg-Rothbard-Hazlitt do not use it right. But I think that still misses the bigger point of what dropping the expenditure side of the Keynesian cross does to the logic of the whole theory.