This appendix provides the derivation of the economic capital stock of money, as the discounted expected value of the monetary service flow. While general equilibrium theory deals with demand and supply for flows, wealth effects depend on capital stock.
In appendix A, when we were concentrating on the demand and supply for flows, we needed only to work with conditional current period models under the assumption of intertemporal separability. But since the discounted present value of the future flows depends on the entire intertemporal decision, we now return to the consumer’s full intertemporal decision.
The relevant utility function is equation (A.2) in appendix A subject to the T + 1 budget constraints in equation (A.3) in appendix A. To be able to deal with the full intertemporal decision, we now need the Fisherine wealth constraint acquired by combining all of the T + 1 flow of funds constraints, (A.3), in appendix A The procedure resembles the Bellman method for solving dynamic programming problems, since it starts at the end of the decision and works backward. In particular, first solve (A.3) in appendix A for the benchmark asset quantity, AS,and write the resulting equation for each s between t and t + T. Then back-substitute for As, starting from At+T, and working down to At, always substituting the lower subscripted equation into the next higher one. The result is the following discounted wealth allocation constraint, permitting the full intertemporal allocation decision to be to maximize (A.2) in appendix A subject to this one wealth constraint:
From that factorization of the intertemporal constraint, we see immediately from (B.1) that the forward user cost of the services of the monetary asset, mit, in period s is
It follows from (B.2) that the current-period nominal user cost of monetary asset, mit, is
while the corresponding real user cost of monetary asset, mit, is πrit=πitp*t Note that (B.4) is in accordance with appendix A’s equation (A.5), which we now have formally derived.
The economic stock of money (ESM), as defined by Barnett (1991) under perfect foresight, follows immediately from the manner in which monetary assets are found to enter the derived wealth constraint, (B.1). As a result the formula for the economic stock of money under perfect foresight is
The economic stock of money is thereby found to be the discounted present value of expenditure flow on the services of all monetary assets, with each asset priced at its user cost. Let Mis be the nominal balance of monetary asset i in period s, so that Mis = p*smis. Using the definition in equation (B.3), Vt becomes
A mathematically equivalent alternative form of (B.6) can be derived from quantity and user-cost flow aggregates, discounted to present value. Dual to any exact quantity flow aggregate, a unique price aggregate exists. The price aggregate equals the minimum cost of consuming one unit of the quantity aggregate. Let Πt = Π(πt) be the nominal user cost aggregate that is dual to the exact, real monetary quantity aggregate, Mt = M(mt). By Fisher’s factor reversal, the product of the quantity and user cost price aggregate must equal expenditure on the components, so that
where (TE)S is total nominal expenditure on the monetary services of all monetary components. Alternatively, instead of using real quantities and nominal user costs, we can use nominal quantities and real user costs to acquire
where πris = πis / p*s = (Rs - ris) / (1 + Rs) is the real user cost of monetary asset i in period s, while Ms = (M1s,. . .,Mns)’ is the vector of nominal balances and nrs ={πr1s,. . .,πrns)’ = πsp*s is the vector of real user costs. Since M is the aggregator function, M(MS) is aggregate nominal balances and is a scalar.
Therefore Vt can be rewritten as follows:
Note that equation (B.9) provides a connection between the Divisia aggregate flow index, M(ms), and the discounted money stock, Vt. Also observe that the formula contains a time-varying discount rate.
The theory reviewed above assumes perfect foresight. It has been shown by Barnett (1995) and Barnett, Liu, and Jensen (1997) that all of the results on user costs and on Divisia aggregation, including (B.2) and (B.4), along with the Divisia indexes, (A.79) and (A.80) in appendix A, carry over to risk neutrality, so long as all random interest rates and prices are replaced by their expectations. Under risk aversion, a “beta”-type correction for risk aversion is shown in appendix D to appear in those formulas. The further extension to intertemporal nonseparability also is contained in that appendix. The derivations under risk aversion do not use the discounted Fisherine wealth constraint (B.1) but rather are produced from the Euler equations, which solve the stochastic dynamic programming problem of maximizing expected intertemporally separable utility, subject to the sequence of random flow-of-funds constraints (A.3) in appendix A.1
We introduce the expectations operator, Et, to designate expectations conditional on all information available at current period t. In accordance with consumption-based capital asset-pricing theory (e.g., see Blanchard and Fischer (1989, p. 292), the general formula for the economic capital stock of money under risk becomes
where ?s is the subjectively discounted intertemporal rate of substitution between consumption of goods in current period t and in future period s, and is a generalization of the subjective discounting in equation (A.7) in appendix A. In general, ?s is random and can be correlated with current and future values of (TE)S. Assuming maximization of expected intertemporal utility subject to the sequence of flow of funds equations (A.3) in appendix A, Barnett (1995) and Barnett, Liu, and Jensen (1997) have derived the relevant Euler equations for ?s under intertemporal separability. If we further assume risk neutrality, as in Blanchard and Fischer (1989, p. 294), it follows that which becomes (B.9) under perfect foresight. Results relevant to extending to risk aversion are provided in appendix D to this book.
Rotemberg (1991) and Rotemberg, Driscoll, and Poterba (1995) introduced the currency equivalent index (CE index),
as a flow index under assumptions stronger than needed to derive the Divisia monetary flow index. But Barnett (1991) proved that the CE index can be interpreted to be a stock index, rather than a flow index, and is therefore called the Rotemberg money stock in that paper. In particular, he showed that the CE index is a special case of the ESM, (B.11), under the assumption of martingale expectations, in addition to the assumption of risk neutrality.
Following Barnett’s proof, assume that Mit, rit, and Rt follow martingale processes. Then we can see from (B.6) that under risk neutrality equation (B.11) can be rewritten as
This shows that the CE index is a special case of the economic stock of money, when the conditional expectation of the future value of each variable is equal to its current value.
From equation (B.13) under martingale expectations, we furthermore can show that the CE index is proportional to the Divisia current-period monetary flow aggregate, as follows:
We define the simple-sum aggregate, VtSSI, by
As a flow index, this index requires that all monetary components be perfect substitutes, so that linear aggregation is possible, and furthermore that the coefficients in the resulting linear quantity aggregator function be equal for all components. But we also can acquire that index as a stock index of a joint product under the assumption of martingale expectations, since it then follows that
where IYt = Σni=1 [rit /Rt]Mit is the discounted investment yield part of the simple-sum aggregate and VtCE is the discounted monetary service flow part.
Hence the simple-sum monetary aggregate, treated as a stock index, is the stock of a joint product, consisting of the discounted monetary service flow and the discounted investment yield flow. For the SSI to be valid as an economic money capital stock measure, all monetary assets must yield no interest. Clearly, investment yield cannot be viewed as a monetary service, or the entire capital stock of the country would have to be counted as part of the money stock, so the investment yield part, IYt, must be removed. The simple-sum monetary aggregates confound together the discounted monetary service flow and the nonmonetary investment-motivated yield. The simple-sum aggregates will overestimate the actual monetary stock by the amount of the discounted nonmonetary services. Furthermore the magnitude of the simple-sum aggregates’ upward bias will increase, as more interest-bearing monetary assets are introduced into the monetary aggregates.
Under martingale expectations and the assumption that all monetary assets yield zero interest, it follows that
In short, the ESM is the general formula for measuring money stock and fully nests the CE and the SSI as special cases. As financial innovation and deregulation of financial intermediation have progressed, the assumption that all monetary assets yield zero interest rates has become increasingly unrealistic.
The previous section showed that the economic monetary stock provides a general capital stock formula nesting the currency equivalent (CE) index and the simple-sum index as special cases. Each of these results requires martingale expectations, and the simple-sum result further requires that every monetary asset pay a nominal return of zero. Empirical results using those assumptions and that theory are available in Barnett and Zhou (1994a). More recently Barnett, Chae, and Keating (2006) have shown how the ESM can be computed without making either of those restrictive assumptions, but retaining the assumption of risk neutrality. They propose and apply three methods of approximating Vt. The first two linearize the function in terms of expected future variables, and the third imposes a set of convenient covariance restrictions. Appendix D to this book provides extensions to the case of risk aversion.
1. Under risk, equation (B.1) is a state-contingent random constraint. Neither (B.l) nor its expectation is used in Bellman’s method for solving and is not useful in producing the extension to risk aversion in Barnett (1995) and Barnett, Liu, and Jensen (1997).