The Keynesian Multiplier: Phony Math

Much is wrong with the Keynesian multiplier. The arguments for it and explanations of it range from incoherent to inconsistent. In time, I’ll address those arguments and explanations. Today, however, I’ll focus on the phony math at the heart of the multiplier.

The bottom line: The Keynesian multiplier is a tautology that explains nothing. To see why, read on.

THE MULTIPLIER: AN IDEA THAT WON’T DIE

Why worry about the Keynesian multiplier, when old-style Keynesian economics has been supplanted — in the academy — by new Keynesianism, new classical macroeconomics, and the new neoclassical synthesis? Economist John Cochrane has the answer:

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. (“New vs. Old Keynesian Stimulus,” The Grumpy Economist, November 8, 2013)

People — in the academy, the media, and politics — love the story so much that Obama’s “stimulus package” was predicated on a multiplier of about 1.6.  That estimate was produced in January 2009 by Christina Romer, who was then chairwoman-designate of Obama’s Council of Economic Advisers. Another prominent economist, Alan Blinder (a member of the CEA under Clinton and former vice chairman of the Fed’s Board of Governors), cites a multiplier of 1.5.

Those of you who have at least a passing familiarity with the multiplier will ask what’s wrong with a multiplier of 1.5, or 1.6 — or even a multiplier of 5. That is, if government spends an extra $1 to employ previously unemployed resources, why won’t that $1 multiply and become $1.50, $1.60, or even $5 worth of additional output?

The answer, my friends, is found in the phony math by which the multiplier is derived, and in the phony story that was long ago concocted to explain the operation of the multiplier. The phony math and phony story led Keynes’s intellectual heirs and their followers to believe that the multiplier is greater than 1 — significantly greater, in the minds of true believers. And when people believe in something, it’s easy to find numbers to support the belief. This is especially true of macroeconomic aggregates, which reflect the influence of so many variables that it’s hard to pinpoint what causes what.

MULTIPLIER MATH

I’ll deal with the phony story in a future post. Today’s lesson is about the phony math. To show you why the math is phony, I’ll start with a derivation of the multiplier. That derivation begins with the accounting identity  Y = C + I + G, which means that  total output (Y) = consumption (C) + investment (I) + government spending (G). I could have used  a more complex identity that involves taxes, exports, and imports. But no matter; the bottom line remains the same, so I’ll keep it simple and use Y = C + I  + G.

Keep in mind that the aggregates that I’m writing about here — Y , C , I , G, and later S  — are supposed to represent real quantities of goods and services, not mere money.

Now for the derivation:

Derivation of investment-govt spending multiplier

So far, so good. Now, let’s say that b = 0.8. This means that income-earners, on average, will spend 80 percent of their additional income on consumption goods (C), while holding back (saving, S) 20 percent of their additional income. With b = 0.8, k = 1/(1 – 0.8) = 1/0.2 = 5.  That is, every $1 of additional spending — let us say additional government spending (∆G) rather than investment spending (∆I) — will yield ∆Y = $5. In short, ∆Y = k(∆G), as a theoretical maximum. (There are many reasons for the multiplier, even if it were real, to fall short of its theoretical maximum; see this post.)

How is it supposed to work? The initial stimulus (∆G) creates income (don’t ask how), a fraction of which (b) goes to C. That spending creates new income, a fraction of which goes to C. And so on. Thus the first round = ∆G, the second round = b(∆G), the third round = b(b)(∆G) , and so on. The sum of the “rounds” asymptotically approaches k(∆G). (What happens to S, the portion of income that isn’t spent? That’s part of the complicated phony story that I’ll examine in a future post.)

Note well, however, that the resulting ∆Y isn’t properly an increase in Y, which is an annual rate of output; rather, it’s the cumulative increase in total output over an indefinite number and duration of ever-smaller “rounds” of consumption spending.

The cumulative effect of a sustained increase in government spending might, after several years, yield a new Y — call it Y’ = Y + ∆Y. But it would do so only if ∆G persisted for several years. To put it another way, ∆Y persists only for as long as the effects of ∆G persist. The multiplier effect disappears after the “rounds” of spending that follow ∆G have played out.

The multiplier effect is therefore (at most) temporary; it vanishes after the withdrawal of the “stimulus” (∆G). (A permanent increase in G would adversely affect GDP in the longer term by diverting resources from more productive private uses and discouraging growth-producing investment spending.) The idea is that ∆Y should be temporary because a downturn will be followed by a recovery — weak or strong, later or sooner. (I can’t resist the observation that enthusiasts of big government relish the thought of any increase in G, hoping that it will become permanent.)

WHY MULTIPLIER MATH IS PHONY MATH

Now for my exposé of the phony math. I begin with Steve Landsburg, who borrows from the late Murray Rothbard:

. . . We start with an accounting identity, which nobody can deny:

Y = C + I + G. . . Since all output ends up somewhere, and since households, firms and government exhaust the possibilities, this equation must be true.

Next, we notice that people tend to spend, oh, say about 80 percent of their incomes. What they spend is equal to the value of what ends up in their households, which we’ve already called C. So we have

C = .8YNow we use a little algebra to combine our two equations and quickly derive a new equation:

Y = 5(I+G)That 5 is the famous Keynesian multiplier. In this case, it tells you that if you increase government spending by one dollar, then economy-wide output (and hence economy-wide income) will increase by a whopping five dollars. What a deal!

. . . [I]t was Murray Rothbard who observed 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 LThat 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.

The policy implications are unmistakable. It’s just Eco 101!! (“The Landsburg Multiplier: How to Make Everyone Rich,” The Big Questions blog, June 25, 2013)

Landsburg attributes the nonsensical result to the assumption that

equations describing behavior would remain valid after a policy change. Lucas made the simple but pointed observation that this assumption is almost never justified.

. . . None of this means that you can’t write down [a] sensible Keynesian model with a multiplier; it does mean that the Eco 101 version of the Keynesian cross is not an example of such. This in turn calls into question the wisdom of the occasional pundit [Paul Krugman] who repeatedly admonishes us to be guided in our policy choices by the lessons of Eco 101. (“Multiple Comments,” op. cit,, June 26, 2013)

It’s worse than that, as Landsburg almost acknowledges, when he observes (correctly) that Y = C + I + G is an accounting identity. That is to say, it isn’t a functional representation — a model — of the dynamics of economic exchange. Assigning a value to b (the marginal propensity to consume) — even if it’s an empirical value — doesn’t alter that fact that the derivation is nothing more than the manipulation of a non-functional relationship.

Consider the equation for converting temperature Celsius (C) to temperature Fahrenheit (F): F = 32 + 1.8C. It follows that an increase of 10 degrees C implies an increase of 18 degrees F. This could be expressed as ∆F/C = k* , where k* represents the “Celsius multiplier.” There is no mathematical difference between the derivation of the investment/government-spending multiplier (k) and the derivation of the Celsius multiplier (k*). And yet we know that the Celsius multiplier is nothing more than a tautology; it tells us nothing about how the temperature rises by 10 degrees C or 18 degrees F. It simply tells us that when the temperature rises by 10 degrees C, the equivalent rise in temperature F is 18 degrees. The rise of 10 degrees C doesn’t cause the rise of 18 degrees F.

Similarly, the Keynesian investment/government-spending multiplier simply tells us that if ∆Y = $5 trillion, and if b = 0.8, then it is a matter of mathematical necessity that ∆C = $4 trillion and ∆I + ∆G = $1 trillion. In other words, a rise in I + G of $1 trillion doesn’t cause a rise in Y of $5 trillion; rather, Y must rise by $5 trillion for C to rise by $4 trillion and I + G to rise by $1 trillion. If there’s a causal relationship between ∆G and ∆Y, the multiplier doesn’t portray it.

And that’s that.

ADDENDUM (12/18/13): PHONY MATH DOESN’T EVEN ADD UP

Well, that’s almost that. I couldn’t resist another jab at the multiplier.

Recall the story that’s supposed to explain how the multiplier works: The initial stimulus (∆G) creates income, a fraction of which (b) goes to C. That spending creates new income, a fraction of which goes to C. And so on. Thus the first round = ∆G, the second round = b(∆G), the third round = b(b)(∆G) , and so on. The sum of the “rounds” asymptotically approaches k(∆G). So, if b = 0.8, k = 5, and ∆G = $1 trillion, the resulting cumulative ∆Y = $5 trillion (in the limit). And it’s all in addition to the output that would have been generated in the absence of ∆G, as long as many conditions are met. Chief among them is the condition that the additional output in each round is generated by resources that had been unemployed.

In addition to the fact that the math behind the multiplier is phony, as explained above, it also yields contradictory results. If one can derive an investment/government-spending multiplier, one can also derive a “consumption multiplier”:

Derivation of consumption multiplier

Taking b = 0.8, as before, the resulting value of k-sub-c is 1.25. Suppose the initial round of spending is generated by C instead of G. (I won’t bother with a story to explain it; you can easily imagine one involving underemployed factories and unemployed persons.) If ∆C = $1 trillion, shouldn’t cumulative ∆Y = $5 trillion? After all, there’s no essential difference between spending $1 trillion on a government project and $1 trillion on factory output, as long as both bursts of spending result in the employment of underemployed and unemployed resources (among other things).

But with k-sub-c = 1.25, the initial $1 trillion burst of spending (in theory) results in additional output of only $1.25 trillion. Where’s the other $3.75 trillion? Nowhere. The $5 trillion is phony. What about the $1.25 trillion? It’s phony, too. The “consumption multiplier” of 1.25 is simply the inverse of b, where b = 0.8. In other words, Y must rise by $1.25 trillion if C is to rise by $1 trillion. More phony math.

If there is a multiplier on government spending, it’s bound to be negative. Stay tuned for more about the effect of government spending on economic output.

The sequel is here.