A Restatement of the Quantity Theory of Money

Allais seeks a single, operational formulation of the demand for money that applies without modification across ordinary peacetime conditions and full-blown hyperinflations, across different countries, and across different historical periods — rather than the ad hoc, situation-specific specifications common in the quantity-theory literature. His answer is the “hereditary, relativistic, and logistic” (H.R.L.) formulation, derived analytically from eight a priori postulates about the psychology of monetary decision-making, which he then fits to real data with only two free constants per series and three constants held fixed across all series and countries.

The hereditary postulate states that the “psychological rate of expansion” z is not the current rate of growth of nominal total outlay but an exponentially-weighted average of the entire past history of that growth rate, computed on a psychological time scale (Eq. 2.9). More recent growth episodes receive more weight, but no past growth ever drops out of the average entirely. This is what allows the model to explain why the velocity of circulation is not constant, as the classical quantity theory typically assumes, but instead a determinate function of accumulated past experience — money holders’ behavior today is conditioned by the whole trajectory of inflation or growth they have lived through, not simply its current level.

The relativistic postulate posits a psychological time scale t’, distinct from calendar time, on which the instantaneous coefficient of forgetfulness is a constant (Eq. 2.2): agents’ rate of “forgetting” past monetary experience, measured in calendar time, can itself change (for example, accelerating during hyperinflation), but is constant when re-expressed in units of this psychological clock. This is the device that lets a single hereditary-weighting structure describe both slow, decades-long adjustment in stable economies and the compressed, rapid adjustment observed within a few years of hyperinflation.

The first four postulates (relativistic, hereditary, invariance, and constant velocity in psychological time) fix the functional form of the demand function, while the remaining three (asymptotic behavior as hyperinflation ends, conjunctural symmetry between expansionary and recessionary episodes, 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. The postulates together yield the logistic function psi(Z) = (1+b)/(1+be^(alphaZ)), which with alpha=1 and b=1 becomes the “common H.R.L. law” psi(Z) = 2/(1+e^Z), governed by the dynamic equation dZ/dt = x - 0.002(1+e^Z)Z, where x(t) is the instantaneous growth rate of nominal total outlay (Eqs. 2.51-2.52). X_0=0.004/month is identified with a roughly 5% per annum equilibrium real interest rate estimated for the United States over 1880-1956 (Eqs. 2.49-2.50) and is therefore, by Allais’s own account, not derived independently of a US-specific calibration.

Allais fits the common H.R.L. law by nonlinear least squares to fifteen series from nine countries and monetary episodes — France (1898-1913, 1919-1938, 1947-1962), Great Britain (1925-1940, 1952-1962), the United States (1895-1915, 1918-1941, 1946-1958), and seven hyperinflations (Germany, Austria, Greece, Hungary I and II, Poland, the U.S.S.R.) drawn from Friedman’s, Allais’s own, and Cagan’s data — using only two free integration constants per series (an initial scale of desired balances and an initial value of Z), with no country-specific universal constants. The fit is extremely close: eleven of the fifteen correlation coefficients between log observed and log calculated money exceed 0.99 (eight exceed 0.995, two exceed 0.999), the mean unexplained variance across all fifteen series is 1.9%, and pooling all 389 paired observations of money and income yields a correlation of 0.9984 in levels and 0.9930 in logs (Table 2, Chart 8). Allais writes that “the results that have been obtained are so extraordinarily close that they seem too good to be true” (p. 1149).

The same three universal constants (alpha=1, b=1, X_0=0.004) are applied without modification to series ranging from near-stable growth (US national income varying only from 1.08 to 2.84 over 1918-1941) to extreme hyperinflation — the German price index rising from 15 to about 1.09x10^9 between December 1919 and October 1923, and the second Hungarian hyperinflation seeing the price level rise from 105 to roughly 4x10^29 between July 1945 and July 1946. Only the two per-series integration constants are allowed to differ. The France 1820-1870 series that Allais elsewhere calls the sole “significant exception,” attributing the discrepancy to money-supply and national-income figures he describes as “highly questionable,” belongs to his earlier (1954) study recapped in Section I of this paper — it is not one of the fifteen series fitted in this paper’s own Table 2, and no comparable exception is reported among those fifteen.

No: Allais emphasizes that the calculated M depends only on observed national income R (or the price level P for hyperinflations), not on the observed money stock M itself* — the two free integration constants (an initial scale of desired balances and an initial Z) fix initial conditions only, after which M* evolves as a function of income growth alone. The close match between M* and observed M is therefore, in Allais’s account, a genuine out-of-sample test of the demand relationship rather than a fit imposed by construction.

Allais shows that when the coefficient of forgetfulness is held constant, his formulation is mathematically equivalent to Cagan’s (1954) adaptive-expectations model of hyperinflation demand for money, since both reduce to the same differential equation with exponentially declining weights on past growth rates (Section IV.C). The key difference Allais draws is interpretive rather than mathematical: Cagan’s variable represents a forward-looking expectation of future inflation, while Allais’s z is explicitly a backward-looking summary of remembered past experience. The 1966 H.R.L. law generalizes both Cagan’s and Allais’s own earlier 1954 formulation by allowing the coefficient of forgetfulness to be a function of Z itself rather than a fixed constant.

Allais concludes the quantity theory of money is “fundamentally correct, provided that it is redefined on the basis of the hereditary and relativistic formulation” (Section IV.B): a proportional relationship between the price level and M/Q does hold at any given moment (Eq. 4.5), but the coefficient of proportionality is not constant, instead varying with the history of past outlay growth summarized by Z — so both the classical quantity-theory view and views emphasizing variable velocity are, in his account, partially correct. He is explicit that the H.R.L. law is a theory of money demand alone and “but one element of a much greater whole” (p. 1154): a complete account of monetary dynamics also requires a theory of how the inflationary or deflationary gap between actual money M and desired money M_D feeds back into total outlay, which he flags as needing further development. He also notes the X_0=0.004 constant is validated specifically against 1880-1956 US data under an assumed ~5% equilibrium real rate, so different periods or countries with a different equilibrium real rate would require a different X_0, and that allowing the constants to vary by series would further reduce the residual error — the universal constants are advanced on a priori theoretical grounds and represent “an order of magnitude” rather than a best possible fit.