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

This 1980 Journal of Econometrics paper by William A. Barnett lays out the theoretical foundation for treating monetary aggregates such as M2 as genuine economic quantities and derives the Divisia index as the theoretically correct way to measure them, arguing that the simple-sum index conventionally used by central banks “is severely defective.” Barnett shows that a monetary aggregate is meaningful as an economic variable only if the representative consumer’s utility function is blockwise weakly separable in the relevant monetary assets – utility can be written as a function of “other goods” and a subutility function over the assets to be aggregated – and that, given this separability plus linear homogeneity of the subutility function, Green’s (1964) theorem permits two-stage (or n-stage) budgeting in which total monetary expenditure is allocated first across blocks, such as transaction balances versus a passbook-savings aggregate, and then within each block across its component assets. Extending his own 1978 user-cost formula to incorporate taxation, Barnett prices each asset’s foregone return relative to a benchmark bond yield and estimates nested CES demand systems by full-information maximum likelihood on quarterly U.S. data, 1970Q1-1978Q1: across three types of passbook-savings institutions (commercial bank, savings and loan, mutual savings bank) the estimated elasticity of substitution is high, sigma = 2.66 (beta-hat = 0.62), so high that a near-linear approximation “may be a reasonable approximation” at that level, though unequal estimated weights (alpha-hat = 0.55, 0.26, 0.20) still statistically reject simple summation; but between transaction balances and the passbook aggregate substitutability is much lower, sigma = 0.28 (beta-hat = -2.53), which Barnett calls “too low to justify a linear approximation,” so simple-sum M2 is inappropriate at that level of aggregation. Barnett then shows that the Diewert-superlative Tornquist-Theil Divisia index and the Fisher Ideal index – both exact or near-exact for a broad class of well-behaved aggregator functions – agree to three decimal places when computed for M3 over 1968Q1-1978Q1, while simple-sum M3 velocity over the same period declines secularly (from 1.0 in 1968Q1 to about 0.906 in 1978Q1) even as Divisia M3 velocity stays comparatively stable (roughly 1.00-1.07); Barnett interprets this as evidence that simple-sum aggregation misreads 1970s substitution toward higher-yielding, less-regulated monetary assets as a decline in the “quantity of money,” when in fact the Divisia-measured aggregate is roughly unchanged.

The User Cost of Money

This 1978 Economics Letters paper by William A. Barnett asks what the correct price of holding a monetary asset actually is – the “user cost” or equivalent rental price of the liquidity services a monetary asset provides – and derives that price rigorously from an explicit consumer optimization model rather than from informal reasoning about stocks and flows. Barnett sets up a discrete-time Fisherine intertemporal consumption-allocation model in which a representative consumer, over a planning horizon of T+1 periods, chooses paths of goods consumption, holdings of n monetary assets, and bond holdings subject to a period-by-period budget constraint; monetary assets are assumed blockwise weakly separable from the other arguments of utility, and labor supply is exogenous. Solving the period constraints backward from the terminal bond holding yields a single intertemporal wealth constraint whose left-hand side prices monetary-asset holdings each period at their user cost. For the current period this user cost reduces to p_it = p*_t (R_t - r_it)/(1+R_t), where p*_t is the aggregate price index, R_t is the yield on the benchmark bond, and r_it is the nominal own-yield on monetary asset i – the same formula Donovan (1977) had proposed on informal grounds. Barnett’s contribution is to show this formula is in fact correct and unique within an explicit optimizing model, without needing to assume any particular functional relationship between asset stocks and the service flows they yield. The formula measures the opportunity cost of holding asset i rather than the benchmark bond, deflated by the gross benchmark rate; it does not depend directly on the inflation rate, though nominal yields can be expected to embed expected inflation, and the paper notes (in a footnote) that the user cost of monetary assets relative to durables rises as expected inflation rises. As a pure theory paper, it reports no empirical estimates; its results are the derivation itself and the scope conditions – discrete time (approximating a continuous-time model over longer intervals), constant within-period portfolio stocks, end-of-period interest payment, and exogenous labor supply – under which the formula holds. This user-cost formula subsequently became the foundation for Barnett’s (1980) Divisia monetary aggregation.

William A. Barnett | Macro Paper Warehouse

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

The User Cost of Money