# Further Reflections on the Keynesian Multiplier

In “Keynesian Multiplier: Fiction vs. Fact“, I piggyback on the insights of Murray Rothbard and Steven Landsburg to show that the fiscal multiplier is fool’s gold. In addition to showing this mathematically and empirically, I address the mechanics of the multiplier:

How is it supposed to work? The initial stimulus (∆G) [an exogenous — unfunded — increase in government spending] creates income (don’t ask how), a fraction of which (b) [the marginal propensity to consume] goes to C [consumption spending]. 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) [where k is the multiplier]….

Note well, however, that the resulting ∆Y [change in real, inflation-adjusted GDP] 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). The idea is that ∆Y should be temporary because a downturn will be followed by a recovery — weak or strong, later or sooner.

Further,

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 the clincher:

Taking b = 0.8, as before, the resulting value of kc 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 kc = 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.

The essential falsity of the multiplier can be found by consulting the equation of exchange:

In monetary economics, the equation of exchange is the relation:

MV = PQ

where, for a given period,

M is the total nominal amount of money supply in circulation on average in an economy.

V is the velocity of money, that is the average frequency with which a unit of money is spent.

P is the price level.

Q is an index of real expenditures (on newly produced goods and services).

Thus PQ is the level of nominal expenditures. This equation is a rearrangement of the definition of velocity: V = PQ/M. As such, without the introduction of any assumptions, it is a tautology. The quantity theory of money adds assumptions about the money supply, the price level, and the effect of interest rates on velocity to create a theory about the causes of inflation and the effects of monetary policy.

In earlier analysis before the wide availability of the national income and product accounts, the equation of exchange was more frequently expressed in transactions form:

MVT = PT

where,

VT is the transactions velocity of money, that is the average frequency across all transactions with which a unit of money is spent (including not just expenditures on newly produced goods and services, but also purchases of used goods, financial transactions involving money, etc.).

T is an index of the real value of aggregate transactions.

(Note the careful — but easily overlooked — qualification that quantities are for “a given period”, as I point out in the first block-quoted passage. One cannot simply add imaginary increases in real output over an unspecified span of time to an annual rate of output and obtain a new, annual rate of output.)

If the values for M, V, P, and Q are annual rates or averages, then MV = PQ = Y, the last of which I am using here to represent real GDP.

If the central government “prints” money and spends it on things (i.e., engages in deficit spending financed by the Federal Reserve’s open-market operations), then ∆G (the addition to the rate of G) = ∆M (the average annual increase in the money supply). What happens as a result of ∆M depends on the actual relationships between M and V, P, and Q. They are complex relationships, and they vary constantly with the state of economic activity and consumers’ and producers’ expectations. Even a die-hard Keynesian would admit as much.

If new economic activity (Y) is relatively insensitive to  ∆G, as it is for the many reasons detailed here, it is equally insensitive to ∆M. For one thing — one very important thing — ∆M may be absorbed almost entirely by an increase in VT without a concomitant increase in Q. That is to say, ∆G necessarily implies (in the short run) an increase in transactions velocity (VT) and it is most likely to be spent on resources that are already employed (i.e, either on things that were already being produced or by displacing private purchases of things that were already being produced).

The equality MV = PQ long predates Keynes’s General Theory, in which he introduces the multiplier, and so it was well known to Keynes. As it happens, the equality is at the heart of his multiplier:

The state of the economy, according to Keynes, is determined by four parameters: the money supply, the demand functions for consumption (or equivalently for saving) and for liquidity, and the schedule of the marginal efficiency of capital determined by ‘the existing quantity of equipment’ and ‘the state of long-term expectation’ (p 246).

Adjusting the money supply is the domain of monetary policy. The effect of a change in the quantity of money is considered at p. 298. The change is effected in the first place in money units. According to Keynes’s account on p. 295, wages will not change if there is any unemployment, with the result that the money supply will change to the same extent in wage units.

We can then analyse its effect from the diagram [reproduced below], in which we see that an increase in M̂ shifts r̂ to the left, pushing Î upwards and leading to an increase in total income (and employment) whose size depends on the gradients of all 3 demand functions. If we look at the change in income as a function of the upwards shift of the schedule of the marginal efficiency of capital (blue curve), we see that as the level of investment is increased by one unit, the income must adjust so that the level of saving (red curve) is one unit greater, and hence the increase in income must be 1/S'(Y) units, i.e. k units. This is the explanation of Keynes’s multiplier.

Here’s the diagram: If that is the explanation of Keynes’s multiplier, it is even more backward than the usual explanation that I shredded earlier. All it says is that if the real money supply (M̂) is increased (i.e., not translated into higher prices) due to an exogenous increase in government spending, the real interest rate (r̂) decreases. And if the decrease in the real interest rate leads to an increase in investment, Y must rise by enough to preserve the theoretical relationship between Y and saving (S) and investment (I).

In this case, Keynes depicts the multiplier as the effect of an increase in I resulting from an increase in M, which is really an increase in G (∆G) under the condition of less than full employment (whatever that is). The increase in I is made possible by the decrease in the real rate of interest. That’s odd, because the popular view of the multiplier is that it is the rise in real GDP that is directly attributable to a rise in government spending. Will the real multiplier please stand up?

Regardless, the relationship between the increase in I and the increase in Y is no less tautologous than it is in the usual explanation of the multiplier.Simply put, the increase in I is the increase that must result if Y increases, given an ex-post relationship between I and Y. There is no causality, except in the imagination of the proponent of increased government spending.

We are back where we started, with a mythical multiplier that explains nothing.