"Rational" Expectations, the Optimal Monetary Instrument, and the Optimal Money Supply Rule

The paper asks whether, and how, the choice of monetary policy instrument (the money supply versus the interest rate) matters once the public’s expectations are modeled as rational rather than adaptive, working within a deliberately “ad hoc” macro model (Introduction, pp. 1-2). “By ad hoc we mean that the model is not derived from a consistent set of assumptions about individuals’ and firms’ objective functions and the information available to them. Despite this deplorable feature of the model, it closely resembles the macroeconometric models currently in use, which is our excuse for studying it” (p. 1). Following Poole [1970], the authors compare two policy strategies: pegging the interest rate period by period and letting the money supply be whatever satisfies money demand, versus setting the money supply period by period and letting the interest rate be whatever clears the system.

The model (Section 1, “The Ad Hoc Model,” pp. 2-4) consists of an aggregate supply schedule, an aggregate demand (“IS”) schedule, a portfolio-balance (“LM”) schedule, and an equation determining productive capacity, plus autoregressive processes for the exogenous variables. The aggregate supply equation relates output directly to productive capacity and to the gap between the current price level and the public’s own prior expectation of it, following the logic (attributed to Robert Lucas) that suppliers of labor and goods mistake a surprise rise in the aggregate price level for a rise in the relative price of the goods they sell, because they learn their own prices faster than they learn the aggregate price level – “the kind of aggregate supply schedule that Robert E. Lucas has used to explain the inverse correlation between observed inflation and unemployment depicted by the Phillips curve” (p. 3). The IS equation makes aggregate demand depend on the real interest rate (the nominal rate minus expected inflation) and on capacity/wealth; the LM equation makes real money demand depend positively on real income and negatively on the nominal interest rate; and the capacity equation determines next period’s productive capacity as a function of the real rate and exogenous variables.

The monetary authority is assumed to choose, each period, either the interest rate or the money supply to minimize a discounted quadratic loss function in output and the price level (Section 2, pp. 4-5). It must compare the minimum loss attainable under the best deterministic linear feedback rule for the interest rate against the minimum loss attainable under the best deterministic linear feedback rule for the money supply, and use whichever instrument delivers the lower loss.

No – with expectations governed by fixed distributed-lag schemes, “the usual exploitable tradeoffs between output and inflation” persist, and the choice between an interest-rate rule and a money-supply rule “depends, just as Poole asserted, on most of the parameters of the model including the covariance matrix of the disturbances” (Section 3, pp. 5-8; Introduction, p. 2). In other words, this version of the model is, in the authors’ words, “merely a variant of Poole’s model,” and it produces a unique period-by-period equilibrium when the interest rate is pegged, with instrument superiority remaining an empirical, model-specific question.

Because output in the aggregate supply schedule responds only to the price surprise (the gap between the realized price level and its previously expected value), and that surprise term is shown to depend only on the innovations in the exogenous processes, not on the deterministic parameters of whatever money-supply feedback rule is being followed (Section 4, pp. 8-11). The authors formally derive that productive capacity, and hence output, is an “exogenous process” whose distribution “does not depend on the parameters of the feedback rule for the money supply” – their result (a). Only unanticipated changes in money (via unanticipated changes in the price level) move output; any purely systematic, foreseeable component of the money-supply rule is fully anticipated and offset in expectations, leaving real activity unaffected.

Given the quadratic loss function, the optimal deterministic money-supply rule is the one that sets the expected value of next period’s price level equal to the loss function’s target price level – result (b) (Section 4, eqs. 21-25, pp. 9-11). Because the output distribution is invariant to the rule (Q5), the authority’s only remaining lever for reducing loss is steering the expected price level toward target; the paper solves explicitly for the feedback-rule coefficients that achieve this by working through the model’s expectational difference equation (with a terminal condition ruling out “speculative bubbles”).

The model cannot determine a unique equilibrium price level (or money supply) at all – this is result (c) (Section 5, pp. 11-13). Solving the interest-rate-rule version of the model yields a difference equation for the expected price path that, unlike the money-rule case, does not converge, so pinning down a solution would require arbitrarily fixing the expected price level at some future date as an external terminal condition – “a very much stronger terminal condition” than was needed under a money rule, and one the model gives no basis for choosing (p. 12).

Under an interest-rate rule, the public correctly expects the authority to accommodate whatever quantity of money is demanded at the pegged rate, so any given increase in the price level today is expected to be met, ceteris paribus, by an equal increase in the money supply – which means “one E_(t-1)p_t is as good as any other from the point of view of being rational,” leaving “nothing to anchor the expected price level” (Section 5, p. 13). The authors explicitly connect this to Wicksell’s classical static result that a pegged interest rate leaves the price level indeterminate in a full-employment, flexible-price economy, and note that the standard version of that result requires a vertical Phillips curve – yet “in our model…the Phillips curve is not vertical, but Wicksell’s indeterminacy still arises” (p. 13), showing the problem is not confined to the classical full-employment case.

The authors show that the money-supply rule optimal when there is no informational discrepancy remains optimal – and delivers the identical expected loss – even when the authority and the public have different information sets, as long as the authority does not try to exploit the gap (Section 6, pp. 13-16). They then ask whether the authority actually could do better by exploiting superior information, and conclude that “within our structure the answer seems to be that it can take advantage of a discrepancy, although necessarily in a limited and rather subtle way” – specifically only if the authority knows more than the public about the exogenous shock processes themselves, and only if it knows precisely how the public’s information differs from its own, something the authors call “farfetched” to assume and possibly “a very subtle and perhaps intractable econometric problem” to estimate in practice (pp. 15-16).

The authors are explicit that results (a)-(c) rest on an ad hoc structure and “should not be accepted as providing a suitable context within which to study macroeconomic policy” if taken literally, but they argue two specific ingredients of the model are not so easily dismissed: the rational-expectations hypothesis itself (whose main rival, fixed-weight autoregressive expectations, they view as “subject to so many objections”), and the Lucas aggregate-supply hypothesis, which “has proved difficult to dispose of empirically” (Section 7, “Concluding Remarks,” pp. 17-18). Because it is precisely these two ingredients – rational expectations combined with Lucas’s surprise-supply schedule – that the authors say “account for most of our results,” they judge the qualitative contrast between the rational-expectations and adaptive-expectations conclusions about systematic countercyclical policy to be “fairly robust to alterations of other features of the model,” such as the specific forms of the aggregate demand and portfolio-balance equations.

The paper is explicitly built as a rational-expectations reworking of Poole’s stochastic instrument-choice problem: with adaptive expectations the model collapses to “merely a variant of Poole’s model” with an empirically ambiguous answer, but substituting rational expectations for adaptive expectations overturns Poole’s framework, because changes to the parameters of the money-supply feedback rule now feed back into the parameters of the reduced-form equation for the price level itself – “This feature of the system is what renders Poole’s results inapplicable” (footnote 4, p. 18, citing Sargent and Wallace [1973] for an explicit illustration). The instrument-choice question is not merely reweighted under rational expectations; for the money-versus-output tradeoff it is resolved outright (output is invariant to the deterministic money rule), while for the interest-rate instrument it becomes a determinacy failure rather than a matter of degree.