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.

VARMA representation of DSGE models

This 2016 Economics Letters paper by Stephen D. Morris asks how concise the VARMA (vector autoregressive moving-average) representation of a DSGE model can be made, and whether the model’s structural parameters can be locally identified from that representation. Starting from the general “ABCD” state-space form of a DSGE model (Fernandez-Villaverde et al. 2007) – states X_t (m x 1), observables Y_t (n x 1), transition matrix A, and shock-loading matrices B, C, D – Morris shows that a prior general result (Ravenna 2007) implying every such model has a VARMA(n+m, n+m-1) representation is far larger than necessary in practice: for the Smets-Wouters (2007) model, with n=7 observables and m=12 states, that bound is a “prohibitively large” VARMA(19,18). Under Assumption 1 – observables no more numerous than states (n <= m) and an invertible shock-loading matrix D, i.e., the model is not stochastically singular – Morris derives a condition (Proposition 1) on the rank of an observability-type matrix Psi(kappa) that pins down a much lower-order VARMA(kappa+2, kappa+1) representation, where kappa is the minimum number of lags of the observables needed before the unobserved states become indirectly recoverable through them; corollaries collapse this further to VARMA(2,1), VARMA(1,1), or even VAR(1) representation under additional structure (n=m and C_X invertible; then A_Y=0 and C_Y=0; then an invertible map from X_t to Y_t). Applied to the Smets-Wouters (2007) model, checking that Psi(1) has full column rank 7 at reasonable parameter values shows the model actually has a VARMA(3,2) representation – far more concise than the VARMA(19,18) implied by Ravenna’s bound. Morris then shows that the largest subset of DSGE structural parameters that is locally identifiable from the entire model likelihood is also locally identifiable from the identifiable VARMA parameter subset alone – the nonzero AR coefficients phi together with vech of the innovation covariance matrix Omega – meaning no information outside this reduced parameter set aids local identification of the structural parameters; for Smets-Wouters, the paper reports the Jacobian of this map is 161x36 with full column rank 36 “for a range of reasonable parameterizations,” implying 36 of the model’s 41 structural parameters are locally identified this way and yielding 125 over-identifying restrictions useful for GMM-based testing. The paper is pure theory: it presents no data and no empirical estimation, its main results are stated as holding under Assumption 1 and as local (not global) identification results valid at generic rather than all parameter points, and it does not claim its VARMA(kappa+2, kappa+1) representation is the uniquely minimal one in every case.

Economics Letters | Macro Paper Warehouse

The User Cost of Money

VARMA representation of DSGE models