The User Cost of Money

Barnett sets out to derive, from an explicit consumer optimization model, the correct “user cost” or equivalent rental price to impute to the services of a monetary asset – because prior imputations of such a price “have not been based upon an explicit assumption structure or an explicit model” (Abstract). Without a model, there was no way to verify that a proposed user-cost formula was actually the price implied by rational intertemporal choice, or to know what assumptions it silently relied on.

The model is a discrete-time Fisherine intertemporal consumption-expenditure allocation problem in which a representative consumer, over a planning horizon of T+1 periods (t through t+T), chooses a path of goods consumption x_s, holdings of n monetary assets m_is, and bond holdings A_s to maximize a utility function u(m_t,…,m_{t+T}; x_t,…,x_{t+T}; A_{t+T}/p_{t+T}) subject to a period-by-period budget constraint* (Eq. 1). Each period’s budget constraint equates goods spending to labor income plus net proceeds from monetary assets and bonds carried over from the previous period, including their interest.

Stocks of monetary assets and bonds are held constant within each period, with portfolio adjustments occurring only at period boundaries, and interest on both bonds and monetary assets is paid at the end of each period – so interest earned during period t cannot be spent until period t+1. Labor supply is taken as exogenously given rather than chosen jointly with consumption, and monetary assets are assumed blockwise weakly separable from the other arguments of utility, meaning a subutility function over the monetary-asset vector alone is well defined. This separability assumption is what later permits treating money demand as a single-period, single-block problem in which every monetary asset’s price takes the user-cost form the paper derives (footnote 2).

The single wealth constraint (Eq. 2) is obtained by solving the period budget constraint (Eq. 1) for bond holdings A_s and back-substituting recursively from the terminal holding A_{t+T} down to A_t, producing one constraint that equates the discounted value of goods consumption plus the discounted user cost of monetary-asset holdings to discounted labor income, initial assets, and terminal bond value. The discount factor is ρ_s = 1 at s = t and ρ_s = the product of (1+R_u) for u running from t to s-1 for later periods – and Barnett flags explicitly that ρ_s is not simply the product of (1+R_u) up through period s, because R_s is paid at the start of period s+1 rather than during period s itself; getting this timing convention wrong would misstate the discounted user-cost formula.

Barnett shows the user cost of monetary asset i in the current period reduces to p_it = p_t (R_t - r_it) / (1 + R_t) (Eq. 3), where R_t is the benchmark bond yield, r_it is asset i’s own nominal yield, and p_t is the aggregate price index** – which the paper identifies as “the Donovan’s (1977) user cost formula.” Economically, the formula prices the opportunity cost of holding a unit of asset i instead of the benchmark bond: the forgone yield spread (R_t - r_it), deflated by the gross benchmark rate (1+R_t) and scaled up to nominal terms by the price level.

A non-interest-bearing asset (r_it = 0) has the maximal user cost p_t · R_t/(1+R_t), while an asset that pays the benchmark rate itself (r_it = R_t) has zero user cost* – in the model’s terms, such an asset provides no liquidity services at the margin, since holding it involves no yield sacrifice relative to the pure-return bond. The formula thus ranks monetary assets by how much of the benchmark return they forgo, which is exactly the intuition the user-cost concept is meant to capture, but here it falls directly out of the optimization rather than being assumed.

Barnett proves that Donovan’s (1977) formula, which had been derived “by general economic reasoning without the use of a model and under the assumption that the ‘services’ of financial assets are proportional to the stocks with a unitary proportionality constant,” is in fact the correct and unique user-cost price implied by an explicit Fisherine optimization model – and that this holds without needing any assumption about the functional relationship between asset stocks and the service flows they generate. That is the paper’s precise contribution: not a new formula, but a rigorous derivation showing an existing ad hoc formula was right, and showing it is right for a more general reason than originally supposed.

The current-period formula (3) does not enter expected inflation directly – it depends only on nominal yields R_t and r_it and the price level p_t – though the paper notes nominal interest rates “can be expected to incorporate expected inflation.”* In footnote 2, Barnett adds that because the user cost of non-monetary durables depends inversely on expected inflation, the user cost of monetary assets relative to durables rises as expected inflation rises – a relative-price implication rather than a claim that formula (3) itself contains an inflation term.

The model is explicitly discrete-time; the paper notes the time interval could be set as short as one day, and that the discrete model can be viewed as approximating an underlying continuous-time model when applied to empirical data collected at longer intervals. Labor supply is exogenous rather than jointly optimized, which is a simplification relative to a fully general model of consumption-leisure-money choice. And because this is a pure theory paper, it contains no data, no identification strategy, and no empirical estimates – its output is the derivation and the formula, not a quantitative application (that came later, in Barnett’s 1980 Divisia aggregation paper).