The Barnett critique after three decades: A New Keynesian analysis

The paper asks whether Barnett’s (1980) critique of simple-sum monetary aggregation – that summing the nominal values of imperfectly substitutable liquid assets is theoretically inconsistent with aggregation theory – still holds inside a state-of-the-art dynamic, stochastic, New Keynesian macro model, and answers yes, “both qualitatively and quantitatively.” The authors extend a standard NK model to include currency and interest-bearing bank deposits as competing sources of liquidity services and show that “while a Divisia aggregate of monetary services tracks the true monetary aggregate almost perfectly, a simple-sum measure often behaves quite differently” (Abstract, p. 5). The paper is explicitly a revival and extension of Barnett’s critique into a modern DSGE setting, not a new empirical test.

The representative household’s utility depends on consumption and leisure net of a shopping-time cost, and shopping time falls as the household holds more of a CES-aggregated “effective money” bundle combining currency (N_t) and deposits (D_t). Formally, shopping time is h_t^s = (1/chi)(v_t P_t C_t / M_t^A)^chi (Eq. 2, p. 7), where the monetary aggregator is M_t^A = [v^(1/omega) N_t^((omega-1)/omega) + (1-v)^(1/omega) D_t^((omega-1)/omega)]^(omega/(omega-1)) (Eq. 3, p. 7). Here omega governs the elasticity of substitution between currency and deposits and v governs the steady-state currency share, so the household’s true liquidity demand already treats currency and deposits as imperfect, not perfect, substitutes – the assumption the simple-sum aggregate violates. A representative bank supplies deposits, holds a reserve ratio tau_t (an AR process), and sets the deposit rate through a zero-profit condition r_t^D = 1 + (r_t^L - 1)(1-tau_t) - x_t, where x_t is a financial-sector cost shock (Eq. 15, p. 8).

The paper compares (1) the model’s “true” theoretical aggregate M_t^A from the CES structure (Eq. 3), (2) a Divisia quantity index built from time-varying expenditure shares, and (3) a conventional simple-sum aggregate M_t^S = N_t + D_t (Eq. 24, p. 9) that adds the nominal values of currency and deposits with fixed weights of one. The Divisia quantity index growth rate is a share-weighted geometric average of the growth rates of currency and deposits, mu_t^Q = (mu_t^N)^((s_t^N+s_{t-1}^N)/2) times (mu_t^D)^((s_t^D+s_{t-1}^D)/2) (Eq. 30, p. 9), where the expenditure shares s_t^N and s_t^D are constructed from user costs and quantities (Eqs. 27-29, p. 9) and change endogenously with relative user costs. Unlike the simple-sum measure, the Divisia weights do not assume the two assets are perfect substitutes, and – critically – the Divisia index can be computed from observable quantities and prices without knowing the true functional form (3) or the parameters v and omega.

Across all six structural shocks, the Divisia quantity index and the true monetary aggregate produce impulse responses the authors describe as “indistinguishable” if plotted on the same graph (p. 11), and this result is robust across a wide range of assumed substitution elasticities, omega = 0.10, 0.50, 0.75, 1.5, 2.0, 5.0 (Fig. 3, p. 13). The theoretical basis is Diewert’s (1978) result that the Tornqvist-Theil Divisia index is superlative, i.e. a second-order accurate approximation to any linear-homogeneous aggregator function; the authors conclude that “specific knowledge of the true functional forms describing how households aggregate currency and deposits into a composite yielding liquidity services and specific knowledge of the parameters entering into those functional forms are not needed in constructing reliable monetary statistics” (p. 11) – i.e., the statistical agency does not need to know the household’s true preference parameters to construct a reliable monetary aggregate using Divisia methodology.

The simple-sum aggregate diverges most sharply from the true aggregate following reserves-demand shocks, deposit-cost shocks, and monetary-policy shocks, and the paper notes that following monetary policy shocks the simple-sum measure’s response can even differ in sign from the true aggregate’s. After a reserve-ratio shock, the simple-sum measure falls together with the true aggregate but with different timing and magnitude, because the Taylor-rule-driven adjustment of the monetary base does not offset the accompanying change in the money multiplier; after a deposit-cost shock (a “liquidity crunch”), the quantity index falls and the price index rises while the simple-sum measure diverges substantially. The authors summarize that “the more conventional simple-sum aggregate often behaves quite differently from its true and Divisia counterparts” (p. 11), including after preference and IS-type shocks (Fig. 2).

Yes: the Divisia price index tracks the model’s true opportunity-cost measure “almost as perfectly” as the Divisia quantity index tracks the true money stock, while a simple weighted-average opportunity cost measure diverges substantially after financial-sector and monetary-policy shocks (Figs. 4-6, pp. 13-15). The true opportunity cost follows Barnett’s (1978) user-cost formula, r_t - r_t^A = [v(r_t-1)^(1-omega) + (1-v)(r_t-r_t^D)^(1-omega)]^(1/(1-omega)) (Eq. 23, p. 9). The authors conclude that “for prices as well as quantities, ‘measurement matters’ in monetary economics” (p. 14).

Under the benchmark backward-looking Taylor rule (interest-smoothing 0.75, inflation-response 0.30, no output-gap term), a technology shock is accommodated efficiently – output moves with the efficient level and inflation is unaffected, consistent with the Kydland-Prescott (1982) view that technology-driven output movements should not be stabilized (p. 15) – while a preference (IS) shock generates inefficient overheating that the benchmark rule only partially offsets, and financial-sector shocks (reserve-ratio and deposit-cost) generate persistent output declines with essentially no change in inflation even under a more aggressive inflation-response coefficient (Figs. 7-8, pp. 15-16). The authors explain the financial-shock result mechanically: the policy-induced increase in the monetary base “does not suffice to offset the change in either the reserve ratio or the currency-deposit ratio – more generally, the change in the money multiplier – caused by these shocks” (p. 16), so an inflation-only rule cannot see or correct the underlying disturbance.

The paper finds that stabilizing output against financial-sector shocks requires adding an output-gap term to the Taylor rule, and that a backward-looking specification with a very large output-gap coefficient (rho_g = 100) can insulate output from all five non-technology shocks while still permitting an efficient output response to a technology shock (Figs. 9-10, pp. 17-18); the same large coefficient in a forward-looking rule instead produces indeterminacy.* The authors caution that this conclusion “may differ considerably from what is prescribed by a standard Taylor rule specification” (p. 17) and suggest that Divisia quantity and price indexes could serve as useful real-time indicators of financial conditions during crises, citing Barnett and Chauvet (2011) (p. 18).

The authors are explicit that all results come from a calibrated theoretical model with no empirical estimation, that the bank sector is “rudimentary” (the reserve-ratio and deposit-cost processes are exogenous AR processes rather than derived from optimal bank behavior, and footnote 5 notes additional parameter restrictions are needed just to keep the reserve ratio between 0 and 1), and that the model has no heterogeneity among depositors. They also flag that the optimal-policy results “lean quite heavily on the assumption that the monetary authority can identify successfully the various shocks and thereby measure accurately the output gap in real time” (p. 17, citing Orphanides 2003), and that while the Divisia-tracks-truth result is robust across all tested values of omega, the size of the simple-sum aggregate’s divergence from the truth itself depends on omega – shrinking as currency and deposits become closer substitutes (omega to infinity) and growing for small omega, with the benchmark omega=1.5 sitting in the middle of the empirically plausible range (Fig. 3).