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Iván Carriño (Fundación Libertad y Progreso) pointed me out to an interesting exchange about the Keynesian multiplier. Steven Landsburg writes a post (follow-up here) using Rothbard’s critique (in Man, Economy, and State) of the Keynesian multiplier. Daniel Kuehn criticizes Landsburg’s (and therefore Rothbard’s) example (here, here and here). Paul Krugman also criticizes Landsburg saying that he used a “stupid argument.”

What’s going on? Is Rotbhard’s and Landsburg’s example so ill-founded as Kuehn argues? I’ll briefly summarize Rothbard’s and Landsburg’s example, then Kuehn’s reply and finally what I think has been left out.

The Landsburg (Rothbard) Multiplier

Here I closely follow Landsburg’s post.

This is how the textbook version of the Keynesian multiplier goes. We start with the well known accounting identity:

  • Y = C + I + G.

Y is the value of final goods and services produced, C is household consumption, I is investment (firms), and G is government spending. If we notice that people spend 80% of their income, then we can obtain the Keynesian multiplier as follows:

  • (1) Y = C + I + G
  • (2) Y = 0.80Y + I + G
  • (3) 0.20Y = I + G
  • (4) Y = 5*(I + G)

The number 5 is the Keynesian multiplier and 0.80 is the marginal propensity to consumption (MPC) [Note that 5 = 1/(1-MPC).] Therefore, for every USD increase in G, national incomes increases five times. I’d add that given that investment depends on entrepreneurs who are subject to suffer from animal spirits, government spending is a more reliable (and stable) variable to control Y (which is the aggregate demand).

Then, Landsburg argues that if this reasoning is correct, it should not give us ridiculous results in a different scenario. Assume then, a different accounting identity: Y = L + E. Where L is Landsburg’s income and E is everyone’s else income. Given that Landsburg is only one person among many, the share of E on total income is very high, say 99.999999%. Then, if we repeat the above algebraic steps we get Landsburg’s multiplier:

  • Y = 100.000.000L.

To increase Landsburg’s income is a much more efficient way of increasing national income than the government spending (fiscal policy.) Just like the textbook policy implications of the Keynesian multiplier is to increase government spending, the policy implications of Landsburg’s example is to send him money to spend it on whatever he wants.

If, as it’s clear from this result, there’s something wrong with Landsburg’s multiplier, then the Keynesian multiplier can’t be right. There has to be something else going on. Even if there is such thing as a Keynesian multiplier bigger than one, it can’t be for the reasons described in the typical economic textbook.

Yes, this may be considered a caricature of what the Keynesian model is saying. But that’s precisely the point. The fact that the textbook examples of the fiscal multiplier don’t look ridiculous doesn’t mean they are not.

Daniel Kuehn’s Critique

Daniel Kuehn argues that Landsburg’s example is misleading because he doesn’t separate between income and expenditure. This is a reference to the Keynesian cross, where the expenditure function is a linear function of the form:

  • E = MPS*Y + (G + I),

where the constant  is (G + I) and MPS is the slope (Y is in the horizontal axis and E in vertical axis.) If Y = 0, then E = G + I (autonomous consumption.) Because Landsburg drops the expenditure, Kuehn argues, the multiplier calculation is faulty.

I’m not sure that’s the case. National income (GDP) can be measured either from the so called expenditure approach or the income approach. The expenditure approach is the Y = C + I + G that we had above. Income (Y) is spent in C, I or G and can’t be spend on anything else.

The income approach is, rather than looking at how income is spent, looking at all the income sources. If you look a the income sources or all spending options, then both approaches should give the same results. To look at income or expenditure is to look at two sides of the same coin. Then, by changing “expenditure (C, I and G)” for “Landsburg’s and everyone’s else income (L + E),” Landsburg is just changing the point of view in the same equation. The following identity shows that income equals (1) all expenditure, (2) all sources of income and (3) everyone’s income (regardless of the source).

  • Y = C + I + G = wL + rK = L + E

I don’t see, then, the problem that Kuehn mentions. Because  C + I + G = L + E, the expenditure that Kuehn mentions cannot be present in one but missing on the other. It is not that the expenditure is not present in Landsburg’s example, it’s just that the “constant component” (G + I) and the “variable component” (C) are bundled together, rather than clearly separated.

The Opportunity Cost (or the Crowding Out)

At the end of his post Kuehn mentions something very important.”[T]hat 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.” (italics are mine)

Recall equation (4): Y = 5*(I + G)

Two cases to consider. First, that any effect on Y due to an increase in G is compensated by a decrease in I. Therefore, if all the increase in G is financed with debt that crowds out investment (and there’s no effect on C) the net effect on Y is zero. This is the easiest case.

Second, an increase in G that is financed through taxation (assume no effect on I) should reduce consumption. Therefore, the multiplier effect that is gained through G is lost through C’s multiplier. The later, however, can be different from the Keynesian multiplier. For simplicity assume I = 0, if C = 0.80Y, then G = 0.20Y; therefore:

  • Y = C + 0.20Y
  • Y = 1.25C

At first sight, this suggest that what is lost from C is less that what is gained from G. If G increases by 1USD, the net effect on Y is 5-1.25 = 3.75/USD rather than 5. Shouldn’t they be the same? Why one less USD spent by the household has a smaller multiplier than the same USD spent by the government? The increase in G as discussed in this context comes from a redistribution of income in favor of G, not from an increase in productivity that needs to be allocated. The value of the multiplier should be attached to the USD that is being redistributed, not to who spends it (this is what Rothbard and Landsburg are showing with their examples.)

The problem is that the multiplier values depend on the weight of the expenditure values (C, I and G) on total income (Y), not on the marginal effect of 1 USD. The value of 5 in the Keynesian multiplier depends on C being 80% of total income, not on how much wealth the next USD is going to spend. The share and the marginal propensities are equal on the particular (but unreal) case where they don’t change when disposable income changes. That marginal propensities change is what comment #37 by edarniw in this post is showing:

This is maybe the only legitimate criticism I’ve seen so far (most of the stuff by Kuehn goes off on some weird tangent). That is: in the first case you can argue that the second equation is a “behavioral” equation and in the second case the second equation is an “empirical regularity”. But I don’t think that it being an “empirical regularity” nullifies its case.

You can still demonstrate the absurdity with an example where there’s a clear “behavioral” assumption.

Example:

W = A + B

W=water in a cup

A=water poured in first 30 seconds

B=water poured after the first 30 seconds

Suppose we regularly observe that Jon, who’s pouring the water, fills about 80% of the final amount in the first 30 secs. As a result, we think that:

A=.8*W

Which implies W=5*B

Thus, we might (mistakenly) tell Jon that if he poured an additional ounce of water after the first 30 secs, total water in the cup would rise by 5 ounces!

The abuse of mathematics in the discipline has produced many problems. One of them is to confuse the direction of causality. It is not because government spends money that we have income, it is because we produce income that there can be government spending. That’s the story behind the water example. To argue for an increase in income due to an increase in government spending is to confuse cause an effect.