# The True Multiplier

I ended “The Keynesian Multiplier: Phony Math” on this note:

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.

Before I get to my estimate of the true multiplier — the one that does lasting damage to the economy — I must say more about the multiplier of Keynesian mythology.

As shown in “The Keynesian Multiplier: Phony Math,” the Keynesian multiplier rests on a mathematical illusion. It is nevertheless possible that an exogenous increase in spending really does yield a short-term, temporary increase in GDP.

How would it work? The following example goes beyond the bare theory of the Keynesian multiplier, and addresses several practical and theoretical reservations about it (some which I discuss in “A Keynesian Fantasy Land” and “The Keynesian Fallacy and Regime Uncertainty“):

1. Annualized real GDP, denoted as Y, drops from \$16.5 trillion a year to \$14 trillion a year because of the unemployment of resources. (How that happens is a different subject.)
2. Government spending (G) is temporarily and quickly increased by an annual rate of \$500 billion; that is, ∆G = \$0.5 trillion. The idea is to restore Y to \$16 trillion, given a multiplier of 5 (In standard multiplier math: ∆Y = (k)(∆G), where k = 1/(1 – MPC); k = 5, where MPC = 0.8.)
3. The ∆G is financed in a way that doesn’t reduce private-sector spending. (This is almost impossible, given Ricardian equivalence — the tendency of private actors to take into account the long-term, crowding-out effects of government spending as they make their own spending decisions. The closest approximation to neutrality can be attained by financing additional G through money creation, rather than additional taxes or borrowing that crowds out the financing of private-sector consumption and investment spending.)
4. To have the greatest leverage, ∆G must be directed so that it employs only those resources that are idle, which then acquire purchasing power that they didn’t have before. (This, too, is almost impossible, given the clumsiness of government.)
5. A fraction of the new purchasing power flows, through consumption spending (C), to the employment of other idle resources. That fraction is called the marginal propensity to consume (MPC), which is the rate at which the owners of idle resources spend additional income on so-called consumption goods. (As many economists have pointed out, the effect could also occur as a result of investment spending. A dollar spent is a dollar spent, and investment  spending has the advantage of laying the groundwork for economic growth, unlike consumption spending.)
6. A remainder goes to saving (S) and is therefore available for investment (I) in future production capacity. But S and I are ignored in the multiplier equation: One story goes like this: S doesn’t elicit I because savers hoard cash and investment is discouraged by the bleak economic outlook. This is probably closer to the mark: The multiplier would be infinite (and therefore embarrassingly inexplicable) if S generated an equivalent amount of I, because the marginal propensity to spend (MPS) would be equal to 1, and the multiplier equation would look like this: k = 1/(1 – MPS) = ∞, where MPS = 1.
7. In any event, the initial increment of C (∆C) brings forth a new “round” of production, which yields another increment of C, and so on, ad infinitum. If MPC = 0.8, then assuming away “leakage” to taxes and imports, the multiplier = k = 1/(1 – MPC), or k = 5 in this example.  (The multiplier rises with MPC and reaches infinity if MPC = 1. This suggests that a very high MPC is economically beneficial, even though a very high MPC implies a very low rate of saving and therefore a very low rate of growth-producing investment.)
8. Given k = 5,  ∆G = \$0.5T would cause an eventual increase in real output of \$2.5 trillion (assuming no “leakage” or offsetting reductions in private consumption and investment); that is, ∆Y = [k][∆G]= \$2.5 trillion. However, because G and Y usually refer to annual rates, this result is mathematically incoherent; ∆G = \$0.5 trillion does not restore Y to \$16.5 trillion.
9. In any event, the increase in Y isn’t permanent; the multiplier effect disappears after the “rounds” resulting from ∆G have played out. If the theoretical multiplier is 5, and if transactional velocity is 4 (i.e., 4 “rounds” of spending in a year), more than half of the multiplier effect would be felt within a year from each injection of spending, and about two-thirds would be felt within two years of each injection. It seems unlikely, however, that the multiplier effect would be felt for much longer, because of changing conditions (e.g., an exogenous boost in private investment, private reemployment of resources, discouraged workers leaving the labor force, shifts in expectations about inflation and returns on investment).
10. All of this ignores that fact that the likely cause of the drop in Y is not insufficient “aggregate demand,” but a “credit crunch” (Michael D. Bordo and Joseph G. Haubrich in “Credit Crises, Money, and Contractions: A Historical View,” Federal Reserve Bank of Cleveland, Working Paper 09-08, September 2009), “Aggregate demand” doesn’t exist, except as an after-the-fact measurement of the money value of goods and services comprised in Y. “Aggregate demand,” in other words, is merely the sum of millions of individual transactions, the rate and total money value of which decline for specific reasons, “credit crunch” being chief among them. Given that, an exogenous increase in G is likely to yield a real increase in Y only if the increase in G leads to an increase in the money supply (as it is bound to do when the Fed, in effect, prints money to finance it). But because of cash hoarding and a bleak investment outlook, the increase in the money supply is unlikely to generate much additional economic activity.

To top it off, a somewhat more realistic version of multiplier math — as opposed to the version addressed in “The Keynesian Multiplier: Phony Math” — yields a maximum value of k = 1:

How did I do that? In step 3, I made C a function of P (private-sector GDP) instead of Y (usually taken as the independent variable). Why? For the years 1929-2012 (excluding 1941-46, when the massive war effort and its aftermath drastically reduced C), C was more closely linked to P than to Y. (The true consumption function turns out to be C = – \$148 trillion in 2012 \$ + 0.838P.)

Math trickery aside, there is evidence that the Keynesian multiplier is less than 1. Robert J. Barro of Harvard University opens an article in The Wall Street Journal with the statement that “economists have not come up with explanations … for multipliers above one.”[1]

Barro continues:

A much more plausible starting point is a multiplier of zero. In this case, the GDP is given, and a rise in government purchases requires an equal fall in the total of other parts of GDP — consumption, investment and net exports….

What do the data show about multipliers? Because it is not easy to separate movements in government purchases from overall business fluctuations, the best evidence comes from large changes in military purchases that are driven by shifts in war and peace. A particularly good experiment is the massive expansion of U.S. defense expenditures during World War II. The usual Keynesian view is that the World War II fiscal expansion provided the stimulus that finally got us out of the Great Depression. Thus, I think that most macroeconomists would regard this case as a fair one for seeing whether a large multiplier ever exists.

I have estimated that World War II raised U.S. defense expenditures by \$540 billion (1996 dollars) per year at the peak in 1943-44, amounting to 44% of real GDP. I also estimated that the war raised real GDP by \$430 billion per year in 1943-44. Thus, the multiplier was 0.8 (430/540). The other way to put this is that the war lowered components of GDP aside from military purchases. The main declines were in private investment, nonmilitary parts of government purchases, and net exports — personal consumer expenditure changed little. Wartime production siphoned off resources from other economic uses — there was a dampener, rather than a multiplier….

There are reasons to believe that the war-based multiplier of 0.8 substantially overstates the multiplier that applies to peacetime government purchases. For one thing, people would expect the added wartime outlays to be partly temporary (so that consumer demand would not fall a lot). Second, the use of the military draft in wartime has a direct, coercive effect on total employment. Finally, the U.S. economy was already growing rapidly after 1933 (aside from the 1938 recession), and it is probably unfair to ascribe all of the rapid GDP growth from 1941 to 1945 to the added military outlays. (“Government Spending Is No Free Lunch,” The Wall Street Journal (online.WSJ.com), January 22, 2009)

This is from Valerie A. Ramsey of  the University of California-San Diego and the National Bureau of Economic Research:

…[I]t appears that a rise in government spending does not stimulate private spending; most estimates suggest that it significantly lowers private spending. These results imply that the government spending multiplier is below unity. Adjusting the implied multiplier for increases in tax rates has only a small effect. The results imply a multiplier on total GDP of around 0.5. (“Government Spending and Private Activity,” January 2012)

The price of the demonstrably small Keynesian multiplier is a “temporary” increase in G — an increase that is likely to be permanent and therefore harmful to economic growth.

If that is the case, whence the continuing clamor for “temporary” increases in government spending? It’s because the Keynesian multiplier isn’t just a phony theory; it’s a “religious” tenet shared by economists, pundits, and policy-makers who are true believers in big government. True believers aren’t swayed by such considerations as slower growth and the loss of freedom that accompanies government interventions in private affairs. True believers — Paul Krugman, Brad DeLong, Joseph Stiglitz, and their ilk — always claim that government should spend more, not just in recessionary times. Their preachings bolster the pro-government-spending biases of most pundits and a large fraction of politicians.

As for “temporary” increases in government spending, they usually are as temporary as the infamous “temporary government buildings” in Washington, D.C. Consider this:

Sources: See footnote 3.

Look at defense spending, which actually fluctuates noticeably. Then look at the two lines for government spending, especially the top line. What do you see? An almost unbroken rise in total government spending (including transfer payments). There’s no give-back of any significance. When government officials latch onto your money, they find a way to keep it. If a reduction in defense spending seems to be in order, the money is shifted to other government programs. If there’s a cut in government programs that don’t provide “social insurance,” the money is shifted to government programs that do provide “social insurance” (mainly Social Security, Medicare, and Medicaid). In other words, our rulers consider our money to be their money, and they find ways to keep it, and to grab more and more of it. That’s why the private sector’s share of GDP — the gap between GDP and top-line government spending — shrank almost steadily from 1947 (the first year of full demobilization after World War II) through 2012.

Let’s now talk about the true multiplier on government spending, which I denote as K. Before I spring some equations on you, I want to take a brief tour of the economy’s performance since World War II. Consider the long, upward trend in government spending (G)[2] as a fraction of GDP:

Sources: See footnote 3.

The decline of G/GDP in the 1990s can be attributed to the “peace dividend” — the accelerated reduction of defense spending following the end of the Cold War and the Gulf War — and to the overrated “Clinton boom.” (This so-called boom featured a real growth rate of 3.4 percent from 1993 to 2001, which is unimpressive by historical standards. For example, the overall rate of growth from the first quarter of 1947 to the first quarter of 1993 — recessions and all — was 3.4 percent.)

In any event, the cumulative effect of rising G/GDP on the rate of growth is evident here:

Derived from a spreadsheet published by the Bureau of Economic Analysis, Current Dollar and “Real” Gross Domestic Product.

This graph tells the same story in a different way:

Derived from the same source as the preceding graph. My definition of a recession is given here.

Putting it all together, for the period 1947-2012 I estimated[3] the year-over-year percentage change in GDP (denoted as Y%) as a function of G/GDP (denoted as G/Y):

Y% = 0.09 – 0.17(G/Y)

Solving for Y% = 0 yields G/Y = 0.53; that is, Y% will drop to zero if G/Y rises to 0.53 (or thereabouts)[4]. At the present level of G/Y (about 0.4), Y% will hover just above 2 percent, as it has done in recent years. (See the graph immediately above.)

If G/Y had remained at 0.234, its value in 1947:

• Real growth would have been about 5 percent a year, instead of 3.2 percent (the actual value for 1947-2012).
• The total value of Y for 1947-2012 would have been higher by \$500 trillion (98 percent).
• The total value of G would have been lower by \$61 trillion (34 percent).

The last two points, taken together, imply a cumulative government-spending multiplier (K) for 1947-2012 of about -8. That is, aggregate output in 1947-2012 declined by 8 dollars for every dollar of government spending above the amount represented by G/Y = 0.234.

But -8 is only an average value for 1947-2012. It gets worse. The reduction in Y is cumulative; that is, every extra dollar of G reduces the amount of Y that is available for growth-producing investment, which leads to a further reduction in Y, which leads to a further reduction in growth-producing investment, and on and on. (Think of the phenomenon as negative compounding; take a dollar from your savings account today, and the value of the savings account years from now will be lower than it would have been by a multiple of that dollar: [1 + interest rate] raised to nth power, where n = number of years.) Because of this cumulative effect, the effective value of K in 2012 was about -14.

For this, the U.S. government should “stimulate” the economy with a burst of “temporary” spending? Think again.

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Footnotes:

[1] Theoretical estimates of the Keynesian multiplier [k = 1/(1 – b)] are always greater than 1. How much greater depends only on the value assigned to b, the marginal propensity to consume. The story about “rounds” of additional consumption spending, the sum of which asymptotically approach the value of the multiplier, is just that — a story, a rationalization of phony multiplier math:

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.

Why is 1 the true upper limit of the Keynesian multiplier? I offer the following explanation by way of a thought experiment.

Consider a simple economy with two producers who trade with each other. One (the baker) makes bread; the other (the dairyman) makes butter. With fixed capital, which is kept in repair but never improved or expanded, the output and consumption of bread and butter will change for any or all of four reasons: (1) changes in the tastes and preferences of the producers (as consumers); (2) changes in the availability of resources because of ambient conditions (e.g., a flood that disruptions production, weather conditions that affect the yield of wheat, the health of the baker and the dairyman); (3) innovations that lead to an increase the quality or quantity of output, without requiring additional inputs; and (4) a disruption of credit.

None of these conditions can be remedied at zero cost, that is, simply by printing money in the hope of re-employing the unemployed resources. The fourth condition warrants elaboration. Suppose that the baker had been relying on advances of butter from the dairyman. Then the dairyman cuts off those advances — perhaps because his churn is being repaired., or because he has less tolerance for risk. The baker’s resulting loss of energy causes him to produce less bread, which reduces the rate at which the baker and dairyman exchange bread and butter, which exacerbates the baker’s loss of energy and initiates a decline in the dairyman’s energy, and so on. (You should recognize this as an analog of the process by which an economy is thought to fall into recession or depression. The pervasiveness of the “credit crunch” as a cause of or major factor in recessions and depressions is documented by Michael D. Bordo and Joseph G. Haubrich in “Credit Crises, Money, and Contractions: A Historical View,” Federal Reserve Bank of Cleveland, Working Paper 09-08, September 2009.) No amount of “stimulus” will cause the dairyman to restore credit to the baker, unless and until the dairyman is convinced of the baker’s creditworthiness. (We’re in a barter economy, so throwing money at the baker won’t make him any more credit-worthy or lead to a rise in real output; it will just spread the pain. Nor does throwing money around do much to help in a money-based economy; it doesn’t (a) fix the underlying problem (e.g., a badly managed business in a highly competitive industry) or (b) yield a one-for-one increase in output, inasmuch as the thrown money is inevitably misdirected and must therefore cause some degree of price inflation instead of spurring real output.)

In the absence of any of the kinds of change discussed above, the simple baker-dairyman economy can be characterized as a static circular flow: the same inputs yield the same kinds of outputs, which are consumed at the same rate, year after year. Specifically, the baker keeps some of his output and trades some of it for butter; the dairyman keeps some of his output and trades some of it for bread. The two producers, acting as consumers, use all of the bread and butter that they produce. As a result, they produce the same amount of bread and butter in the next year, which they consume. And so on.

A heretofore unemployed third party enters: a maker of jam, who also wants to consume bread and butter. Where did he come from? Perhaps he’s a son of the baker or the dairyman who has just become old enough to strike out on his own. In any event, his production of jam doesn’t reduce the output of bread and butter; he is drawing on heretofore unused resources that the baker and dairyman couldn’t employ because they are fully engaged in their respective activities.

The producers of bread, butter, and jam decide to engage in three-way trade, so that every one of them becomes a consumer of all three items. The producers of bread and butter willingly consume less bread and butter than before in order to have jam. The producer of jam, of course, trades some of his output for some bread and butter.  As a result of these voluntary trades, all three producers are better off than they were before. (If they weren’t, they wouldn’t trade with each other.)

Because the employment of previously unemployed resources makes everyone better off, those previously unemployed resources will continue to be employed, barring changes of the kind discussed above. That is, the circular flow will continue, but at a higher level of output and consumption. But it won’t expand beyond the level attained when the jam-maker entered the scene and entered into three-way trade with the baker and dairyman.

In sum, the initial “burst” of new employment and output, represented by the jam-maker’s production, doesn’t cause a “ripple effect” that leads to further expansion of output and consumption. Any further expansion would have to be caused by (and limited to) the employment of yet another heretofore unemployed resource; that is, yet another shot of “stimulus.”

Conclusion: The maximum multiplier on the employment of a previously unemployed resource is 1. But it can be (and probably will be) less than 1 if a fourth party (government) tries to entice additional output by issuing directives (spending money) without perfect knowledge of current economic variables (including, but not limited to, knowledge of tastes and preferences, states of health, and producers’ and consumers’ plans and expectations). A good example of government failure: directing (issuing money to) the fully employed baker and dairyman to produce more bread and butter,  while failing to direct (issue money to) the jam-maker to make enough jam to fully employ himself.

[2] G, as I use it here, stands for all government spending, including so-called transfer payments. A transfer payment (usually “social insurance“) is just another way of moving claims on resources from those who earned such claims to those who didn’t earn them. In this respect, transfer payments are no different than other government programs, which involve the coercion of taxpayers to compensate government employees and contractors who do unproductive and counterproductive work (e.g., writing and enforcing regulations). Transfer payments, like other government programs, require administration by vast, costly bureaucracies. In sum, there is no substantive difference between transfer payments and other kinds of government spending; transfer payments are therefore properly considered government spending, arbitrary national-income accounting flim-flam to the contrary notwithstanding.

[3] This equation is based on estimates of GDP and government spending  for 1946-2012. The equation, the intercept, and the coefficient on G/GDP are significant at the 0.05 level and better. The standard error of the estimate is 0.024 percentage points; there is a 95-percent probability that the zero growth point lies between G/Y = 0.41 and 0.75. To derive the equation, I obtained current-dollar estimates of GDP and G from the Bureau of Economic Analysis, GDP and the National Income and Product Accounts (NIPA) Historical Tables, 1.1.5 Gross Domestic Product, line 1, and 3.1 Government Current Receipts and Expenditures, lines 33 and 34. I deflated the current-dollar values using the GDP deflator published at MeasuringWorth.com.

[4] This is is remarkably close to the zero-growth estimate of 0.55 that I derived in “Estimating the Rahn Curve: Or How Government Inhibits Economic Growth.” The equation given in that post is based on a longer view of the effects of government spending on growth. In retrospect, that longer view gives too much weight to the bygone era of low government spending, an era that ended with the Great Depression. I stand by the equation in this post as indicative of reality in the post-World War II era of rampant growth in government spending.