Economic Monetary Aggregates: An Application of Index Number and Aggregation Theory

Barnett argues that simple-sum aggregates like M2 – which add up the dollar balances of different monetary assets with equal weight – are “badly designed as an economic monetary quantity index” and that “the use of simple sum monetary quantity aggregates as economic indices of the quantity of monetary services should be discontinued” (Section 8, p. 34). His diagnosis is that simple summation implicitly assumes every component asset is a perfect substitute for every other in a fixed one-to-one ratio; it therefore cannot distinguish a genuine change in the quantity of monetary services from a reallocation among assets driven by relative yield changes.

A monetary aggregate is meaningful only if the representative consumer’s utility function is blockwise weakly separable in the assets being aggregated – formally, overall utility over the monetary components and other goods can be written as u(f(monetary components), other goods) for some subutility function f (Section 2, pp. 12-13). Barnett states the logic directly: “If the concept of money has meaning, then it follows that an aggregate of monetary assets must exist which is treated by the economy as if it were a single good, which we thereby can call ‘money.’” Without this separability, he argues, any aggregate “is inherently arbitrary and spurious and does not define an economic variable” (p. 13).

The simple-sum index is the degenerate limiting special case of the framework in which components are perfect substitutes in identical ratios – indifference curves that are straight lines at 45 degrees – with a Leontief (minimum-cost) dual price index. This requires what Barnett calls the “Linearity Conditions”: infinite elasticities of substitution between every pair of components within the aggregate (Sections 6.2 and 8, pp. 28, 34). Because real monetary assets pay different yields and are held for different purposes, this is a strong and, Barnett argues, generally false assumption.

Given blockwise weak separability and linear homogeneity of the subutility functions, Green’s (1964) Theorem 4 lets the consumer’s decision decentralize into nested two-stage (or n-stage) budgeting: a first stage allocates total monetary expenditure between transaction balances and a passbook-savings aggregate, and a second stage allocates passbook expenditure across the three institution types offering passbook accounts (Sections 3.4 and 4, pp. 21-24). This nested structure is what lets Barnett estimate separate CES demand systems at each level rather than one large system over every asset at once.

Extending his own 1978 formula to incorporate taxation, Barnett defines the current-period user cost of asset i as pi_it = p_t(R_t - r_it)(1 - tau_t) / [1 + R_t(1 - tau_t)]*, where R_t is the benchmark bond yield, r_it is the asset’s own nominal yield, tau_t is the marginal tax rate, and p*_t is the aggregate price index (Eq. 3.5, Section 3.2, pp. 18-20). The real user cost, pi*_it = (R_t - r_it)(1 - tau_t)/[1 + R_t(1 - tau_t)], is independent of p*_t. An asset becomes a “free good” (zero user cost) when its own yield equals the benchmark rate, and user costs of monetary assets rise relative to durables as expected inflation rises (pp. 19-20, fn. 18). This user cost, not the dollar balance alone, is the expenditure-share weight later used in the Divisia index.

FIML estimates of the CES system over passbook accounts at three institution types (Table 1) give beta-hat = 0.62, implying an elasticity of substitution of sigma = 1/(1-beta) = 2.66 – high enough that a near-linear (Laspeyres-type) quantity index “may be a reasonable approximation to the theoretical quantity index” at this level. Even so, simple summation is still statistically rejected because the estimated intensity weights, alpha-hat = (0.55, 0.26, 0.20), are unequal (tail area for equal weights < 0.00001); commercial-bank passbook accounts contribute the most to the aggregate (Section 6.3, Table 1, pp. 28-30).

Between transaction balances and the passbook-savings aggregate, substitutability is far lower: beta-hat = -2.53, giving an elasticity of substitution of only sigma = 1/(1-(-2.53)) = 0.28, which Barnett calls “far lower than between passbook accounts at different institution types” and “too low to justify a linear approximation.” The corresponding weights are (alpha-hat_1, alpha-hat_2) = (0.77, 0.23) with an estimated first-order autocorrelation coefficient rho-hat = 0.96 (Section 7.2, Table 2, pp. 32-33). Because simple summation requires near-infinite substitutability, this result implies the conventional simple-sum M2 index is inappropriate at the transaction-passbook level even though a near-linear approximation might be tolerable within the passbook block alone.

The Törnquist-Theil Divisia index defines the aggregate’s growth rate as a share-weighted average of the growth rates of its components, log Q_t - log Q_{t-1} = sum_i sit (log m_it - log m{i,t-1}), where s_it is the average of current and lagged expenditure shares computed from user costs (Eq. 10.1, p. 38).** Following Hulten (1973), the continuous-time Divisia index is exact for any consistent aggregator function; Diewert (1976) shows both the Törnquist-Theil Divisia and the Fisher Ideal index are “superlative,” meaning they are exact for aggregator functions that can second-order-approximate any linearly homogeneous function. Computed for the M3 aggregate over 1968Q1-1978Q1, the two superlative indices’ implied velocities are identical to three decimal places, so “the choice between those two indices is of no importance” (Section 10.2, Table 3, p. 39); Barnett favors the Törnquist-Theil Divisia specifically because it admits a natural interpretation as a weighted average of component growth rates and because it is the natural discrete-time approximation to the continuous-time Divisia line integral (pp. 38-39).

Simple-sum M3 velocity declines secularly from 1.0 in 1968Q1 to about 0.906 in 1978Q1, a range of 0.201, while Divisia (and Fisher Ideal) M3 velocity is comparatively stable, ranging roughly 1.00-1.07 – a range of only 0.089, less than half as wide – and extending the aggregate to M3+ (adding repurchase agreements, dealer commercial paper, bankers’ acceptances, and short-dated negotiable Treasury securities) further stabilizes Divisia velocity while simple-sum velocity keeps trending the “wrong” way (Section 10.2, Fig. 1, Table 3, pp. 39-43). Barnett’s proposed mechanism: during the 1970s, as market rates on unregulated assets rose relative to Regulation Q-constrained deposits, consumers substituted toward higher-yielding, less-regulated assets. Simple summation reads this substitution as a decline in the “quantity of money” (falling velocity), but because the Divisia index weights each component by its user-cost expenditure share, it registers the same substitution as an internal reallocation that leaves the true monetary aggregate roughly unchanged – so its velocity stays stable (pp. 12, 41-43).