A Restatement of the Quantity Theory of Money

This 1966 American Economic Review paper by Maurice Allais asks whether the quantity theory of money can be given a single operational formulation that holds uniformly across normal times and hyperinflations, across countries, and across historical periods, and proposes the “hereditary, relativistic, and logistic” (H.R.L.) formulation of money demand as that reformulation. Allais derives the H.R.L. law from eight a priori postulates about the psychological basis of monetary behavior: the “relativistic” postulate posits a psychological time scale on which the coefficient of forgetfulness is constant; the “hereditary” postulate makes the psychological rate of expansion of total outlay, z, an exponentially-weighted average of all past rates of growth of total outlay (so current money demand depends on the entire history of outlay growth, more recent history weighted more heavily); an invariance postulate holds that the function relating relative desired money balances to psychological time is the same across all times and countries; a constant-velocity-in-psychological-time postulate; and a “logistic” postulate that the relative change in desired balances is proportional to the relative gap between desired balances and their maximum, yielding a logistic function of the dimensionless variable Z = z/X_0. Three further postulates (asymptotic behavior at the end of hyperinflation, conjunctural symmetry between expansion and recession, and temporal symmetry between past and future) pin down the model’s three universal constants at alpha=1, b=1, and X_0=0.004 per month (derived from a roughly 5% per annum equilibrium real interest rate for the United States, 1880-1956), giving the fully specified common law psi(Z) = 2/(1+e^Z) with dZ/dt = x - 0.002(1+e^Z)Z, where x is the instantaneous rate of growth of nominal total outlay (or national income). Allais then fits this single law, with only two free series-specific integration constants (an initial scale of desired balances and an initial value of Z) and no country- or period-specific universal constants, by nonlinear least squares to fifteen time series from nine countries and monetary regimes: annual/quarterly data for France (1898-1913, 1919-1938, 1947-1962), Great Britain (1925-1940, 1952-1962), and the United States (1895-1915, 1918-1941, 1946-1958), plus seven hyperinflation episodes (Germany, Austria, Greece, Hungary I and II, Poland, and the U.S.S.R. in the 1920s-1940s) using Cagan’s hyperinflation data. The calculated M* depends only on observed national income or the price level, not on observed money M, so the fit is not circular. The fit is extremely close: of the fifteen correlation coefficients between observed and calculated log money, eleven exceed 0.99, eight are at least 0.995, and two exceed 0.999, with a mean unexplained variance of 1.9% across all series, and pooling all 389 paired observations gives a correlation of 0.9984 (0.9930 in logs); Allais remarks that these results “seem too good to be true.” The same three universal constants fit series ranging from near-stable income growth in the interwar United States to the German hyperinflation (price index rising from 15 to about 1.09 billion) and the second Hungarian hyperinflation (price level rising by a factor of roughly 4x10^29); the only “significant exception” Allais reports anywhere in the paper belongs to his earlier (1954) study recapped in Section I — the France 1820-1870 series, built on money and income figures he calls “highly questionable” — and is not one of the fifteen series in this paper’s own Table 2 (whose own French series run 1898-1913, 1919-1938, and 1947-1962), among which no comparable exception is flagged. Allais interprets the results as showing that the quantity theory is “fundamentally correct” once redefined on this hereditary and relativistic basis: a proportional relationship between the price level and M/Q holds instantaneously, but its coefficient of proportionality is not a constant, instead varying with the history of past outlay growth summarized by Z, so that both the traditional quantity-theory view and the anti-quantity-theory view are partially right. He is careful to note that the H.R.L. law is a theory of money demand only, that a full account of monetary dynamics also requires a theory of how the gap between actual money M and desired money M_D (a gap Allais argues is not fully captured by observed M* alone) feeds back into total outlay, and that X_0=0.004 is calibrated to a roughly 5% annual real rate specific to the US sample and may differ in other places or periods; allowing the constants to vary by series would improve the fit further, but the universal constants are advanced on a priori theoretical grounds rather than as the best attainable fit.

Maurice Allais | Macro Paper Warehouse

A Restatement of the Quantity Theory of Money