Two Illustrations of the Quantity Theory of Money

Questions & answers

Q1. What two “laws” of the quantity theory does the paper illustrate, and what theoretical model grounds them?

The paper illustrates “two central implications of the quantity theory of money: that a given change in the rate of change in the quantity of money induces (i) an equal change in the rate of price inflation; and (ii) an equal change in nominal rates of interest” (Introduction). Lucas grounds both as necessary characteristics of the stationary solution to Miguel Sidrauski’s monetary version of the Solow-Swan growth model – that is, as statements about long-run average behavior of an idealized model economy rather than short-run dynamics, and notes this differs from, though is not inconsistent with, Milton Friedman’s emphasis on the stability of the money demand function, “a property which is neither necessary nor sufficient for the quantity theory to obtain in the sense used here” (Section I, footnote 1).

Q2. What qualifications does Lucas attach to these laws before testing them empirically?

Two: first, Sidrauski’s baseline model omits the Mundell-Tobin effect, whereby higher inflation, by lowering the real yield on money, could shift saving toward real capital and so dampen the nominal-interest response to money growth below one-for-one – though Lucas notes taxation of interest income works in the opposite direction and that “in general, one does not want to view this effect as ruled out on prior, logical grounds.” Second, “theory at this level gives no guidance as to the measurement of the quantity of money,” so the choice to define money as M1 in the empirical work that follows is acknowledged as arbitrary, “though it is likely that very similar results would have been obtained under a variety of other choices” (Section I).

Q3. What is the cross-country benchmark that motivates Lucas’s time-series filtering approach?

Robert Vogel’s study of sixteen Latin American economies, 1950-69, plotting annual money-growth/inflation pairs, produces a scatter whose grand mean lies almost exactly on a theoretical 45-degree line – “it is hard to imagine a nonvacuous economic prediction obtaining stronger confirmation than that shown in Figure 1” (Section I). Because that cross-country comparison of long-run averages is the theoretically ideal test but data limitations (interest rates in high-inflation economies are rarely market-determined) prevent extending it to the Fisherian interest-rate law, Lucas’s paper develops an atheoretical filtering method meant to extract an analogous long-run signal from continuous time-series data for a single economy.

Q4. What does Lucas’s exponential-smoothing filter do, and how is it meant to isolate the quantity-theoretic signal?

The filter is a two-sided, one-parameter family of exponentially weighted moving averages (equation 1, parameter beta in (0,1)) applied identically to money growth, inflation, and the interest rate, constructed so that as beta approaches one the filtered series converge to their long-run sample-average values (Section II). Lucas motivates it with a signal-extraction interpretation: imagining the observed series as a slowly changing “signal” (dominated by quantity-theoretic monetary forces) plus higher-frequency “noise” (other real, business-cycle forces), the filter’s weight on distant observations rises as the ratio of noise variance to signal variance rises – the same statistical logic John Muth used for the permanent-income hypothesis (Section II, footnote 8). Lucas is explicit that “this purely statistical rationale…has no basis in economic theory” on its own, but offers it as a plausible approximation to an economy whose monetary-policy regime changes only rarely and slowly, with business-cycle activity superimposed at higher frequencies.

Q5. What do the resulting scatter plots show as the filter’s smoothing parameter increases?

At beta = 0.5 “it is evident that a filter…does not quite extract the quantity-theoretic signal,” but “a beta-value of 0.9 reveals a clear 45-degree line, as predicted by the quantity theory, and produces a picture about as clear as Vogel’s cross-country estimates,” with beta = 0.95 “clearer still” (Section III, Figures 2-11). This pattern holds separately for the inflation-versus-money-growth scatter and the interest-rate-versus-money-growth scatter, so that at high smoothing both quantity-theoretic laws – the inflation law and the Fisherian nominal-rate law – are visually confirmed in the postwar U.S. time series.

Q6. Why does Lucas also plot smoothed unemployment against smoothed money growth, and what point is he making?

To demonstrate that filtering by itself can create the appearance of order with no economic meaning: “subjecting any two series to moving-average filtering of the type used here will cause a ‘pattern’ of some kind to emerge,” and the unemployment-versus-money-growth plot shows exactly that – “order of a sort emerging from confusion but it is an order that makes no sense economically” (Section III, Figures 12-13). Lucas’s point is that the quantity-theoretic plots are convincing not because smoothing alone produces a 45-degree line, but because that specific line “is an implication of a coherent economic theory,” which the unemployment comparison is not.

Q7. How does Lucas relate his “quantity-theoretic” filtered component to the anticipated/unanticipated money decomposition used by Sargent and Barro?

He states his opinion that “all of what I have called X0t(0.9) should be identified as anticipated in the Sargent-Barro sense, and in addition, that much of my X0t - X0t(0.9) should also be thought of as anticipated” – i.e., even some of the higher-frequency residual beyond the heavily smoothed trend is plausibly anticipated by agents, even though it is filtered out of his quantity-theoretic signal (Section IV). This is exactly what he means by calling his approach a “minimal” use of the quantity theory: “by using weaker theory, one is more confident that his filter has not incorrectly labeled noise as signal,” but “there is no doubt that the methods used in this paper have not fully extracted from the series all that the quantity theory can account for” (Section IV) – the method deliberately trades completeness for robustness.

Q8. What is Lucas’s own final assessment of what this evidence says about the inflation and interest rates of the 1970s?

“Both the inflation and the high interest rates of the 1970’s are well accounted for by the quantity theory or, to put the same point backwards, any nonmonetary explanation of these trends would lead to large, unexplained deviations from the relationships depicted so clearly in Figures 8-11” (Section III). Lucas also cautions that this kind of confirmation is contingent on the data: the method yields clear results “only if a good enough ’experiment’ has been run by ’nature’ over the sample period used” – i.e., only because U.S. monetary growth varied enough over 1953-77 to trace out the relationship, just as Vogel’s cross-country result depended on sufficient variation in money growth across the sixteen economies he studied (Section III).