Ricardo Reis and I addressed this question in a paper a few years ago, called What Measure of Inflation Should a Central Bank Target? (published version). The abstract:
This paper assumes that a central bank commits itself to maintaining an inflation target and then asks what measure of the inflation rate the central bank should use if it wants to maximize economic stability. The paper first formalizes this problem and examines its microeconomic foundations. It then shows how the weight of a sector in the stability price index depends on the sector's characteristics, including size, cyclical sensitivity, sluggishness of price adjustment, and magnitude of sectoral shocks. When a numerical illustration of the problem is calibrated to U.S. data, one tentative conclusion is that a central bank that wants to achieve maximum stability of economic activity should use a price index that gives substantial weight to the level of nominal wages.With this conclusion in mind, let's look at growth in nominal compensation per hour:
As judged by this series, inflationary pressures look reasonably well contained at the moment.
But notice what happened in the late 1990s. Reis and I commented on this episode in our conclusion:
Consider how a monetary policymaker in 1998 would have reacted to these data. Under conventional inflation targeting, inflation would have seemed very much in control, as the CPI inflation rate of 1.5 percent was the lowest in many years. By contrast, a policymaker trying to target a stability price index would have observed accelerating wage inflation. He would have reacted by slowing money growth and raising interest rates (a policy move that in fact occurred two years later). Would such attention to a stability price index have restrained the exuberance of the 1990s boom and avoided the recession that began the next decade? There is no way to know for sure, but the hypothesis is intriguing.
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