Forward-Looking Behavior and the Stability of a Conventional Monetary Policy Rule
Fuhrer, Jeffrey C., Moore, George R., Journal of Money, Credit & Banking
AT THE TURN OF THE CENTURY, Knut Wicksell introduced an analysis, "The Influence of the Rate of Interest on Prices," that has been influential among policymakers. In fact, a modem version of Wicksell's prescription dominates monetary policy discussions in the financial press today. Despite its simplicity, the modem version of Wicksell's model is not taught in school. We submit that one reason for this neglect is that this simple model of inflation cannot be captured in a two-dimensional phase diagram, the favored medium of exposition for dynamic economic models. It is a tribute to Wicksell's insight that, so far as we can tell, a three-dimensional model is the smallest that can accommodate it.
Writing during the gold standard and tangled in debates on bimetallism, Wicksell (1907) eagerly anticipated a banking system like the one we have today
if this most essential step on the way to a rational monetary system should be taken, if
the free coining of gold, like that of silver, should cease, and eventually the banknote
itself, or rather the entity in which the accounts of banks are kept, should become the
standard of value, then, and not till then, the problem of keeping the value of money
steady, the average level of money prices at a constant height, which evidently is to be
regarded as the fundamental problem of monetary science, would be solvable theoretically
and practically to any extent. And the means for solving it need not be sought in
some more or less fantastic scheme like a central issuing bank for all the world, as is
sometimes proposed, but simply in a proper regulation of general bank-rates, lowering
them when prices are getting low, and raising them when prices are getting high.
Nor would this system be at all artificial, because the point about which the rate of
interest would then oscillate, and to which it would constantly gravitate, would be precisely
. . . the marginal product of capital . . .
Modern monetary policy evidently does not attempt to control the price level, as Wicksell suggested, but rather the inflation rate. With the inflation rate at zero, it is highly unlikely that the Federal Open Market Committee would raise interest rates to reduce the consumer price index from a value of 150 to a value of 100.
Our contemporary version of Wicksell's simple policy prescription, translated into an inflationary environment, is as follows. In all the models discussed below, monetary policy is represented as a reaction function that raises the federal funds rate, f, when the inflation rate, [pi], is above its target,[pi].(1)
(1) f)(t) = [alpha][[pi](t) - [pi]].
The dot above f denotes its first derivative with respect to time, df/dt.
The next section describes five simple models. Each combines the reaction function in equation (1) with increasingly elaborate specifications of the inflation mechanism. The first model uses the simplest version that we can imagine--the one in which the rate of change of inflation depends on the short-term real rate of interest. This two-dimensional model is globally unstable. Unless it is resting at its long-run equilibrium, its solutions explode.(2) We then consider the ability of an alternative policy rule, which adjusts the level of the funds rate in response to inflation, to stabilize the two-dimensional model. We find that this rule can stabilize the two-dimensional model. However, the level rule corresponds less closely to Wicksell's proposed rule, and more importantly, requires knowledge of the equilibrium real interest rate. We focus on the policy rule of equation (1) for the remainder of the paper.
The following four models enrich the description of the inflation mechanism, either by making it depend on the long-term real rate of interest, or by using Calvo's (1983) specification of staggered contracts, or both. …