The natural interest rate is of great relevance to central banks, but it is difficult to measure. We show that in a standard microfounded monetary model, the natural interest rate co-moves with a transformation of the money demand that can be computed from actual data. The co-movement is of a considerable magnitude and independent of monetary policy. An optimizing central bank that does not observe the natural interest rate can take advantage of this co-movement by incorporating the transformed money demand, in addition to the observed output gap and inflation, into a simple but optimal interest rate rule. Combining the transformed money demand and the observed output gap provides the best information about the natural interest rate.

The natural interest rate, the real rate of return in the natural economy with flexible prices instead of sticky prices, is of great relevance to modern central banks. For example, interest rate rules like the one proposed by Taylor (1993) suggest that a central bank sets its policy rate equal to the natural interest rate plus the inflation rate when the inflation rate is at its target and the output gap is closed. More generally, monetary policy is considered contractionary if the actual real rate exceeds the natural interest rate, while it is considered expansionary if the actual real rate is below the natural interest rate.

The caveat to these straightforward principles is that the natural interest rate is fairly difficult to measure. Economic theory suggests that at business cycle frequency, the natural interest rate varies over time with shocks to technology, preferences, and absorption. A number of empirical studies show, however, that the uncertainty surrounding estimates of the natural interest rate is substantial, and this uncertainty undermines the usefulness of the natural interest rate as a practical guide to monetary policymakers.

We show that money demand can play a role in measuring the natural interest rate and therefore, in the optimal monetary policy rule. To illustrate this, we revisit a basic microfounded monetary model with a money-in-utility (MIU) specification of money demand. In this model, we derive a transformation of the money demand that can be computed from actual data and that we call the money gap. The money gap co-moves with the natural interest rate because it reflects, among other things, shocks to the marginal utility of consumption. These shocks also alter the time path of natural consumption and, therefore, the natural interest rate. The comovement between the money gap and the natural interest rate is independent of the specific model used, and we extend the basic model to a quantitative model with a dynamic money demand function to show this.

We find that the correlation between the money gap and the natural interest rate is of considerable magnitude, is independent of monetary policy, and is thus, immune to the Lucas (1976) critique. These are welcome characteristics from the perspective of a central bank that uses the money gap to measure the natural interest rate more accurately. Based on the quantitative model calibrated for the euro area, our estimate of the correlation between the money gap and the natural interest rate is between 0.3 and 0.45. Results for the U.S., indicate a somewhat lower but still notable correlation. The correlation is immune to changes in the monetary policy regime because, by construction, the money gap omits the variation in money demand that results from the opportunity costs and the transaction volume of money demand. This insulates the money gap against the direct and indirect effects of changes in the policy regime, just like the natural interest rate.

A central bank can take advantage of the co-movement between the money gap and the natural interest rate. To demonstrate the appropriate role of the money gap for monetary policy, we suppose that the optimizing central bank selects the coefficients of a typical interest rate rule that is modified to account for the lack of information on the side of central banks. With full information, the interest rate rule incorporates the natural interest rate, the output gap between the actual and the natural level of output, and the inflation rate. However, two of these variables, the natural interest rate and the output gap, are not readily observable in the real world. We operationalize this lack of information in the modified interest rate rule by replacing the natural interest rate with the money gap, which can be computed from the actual data. Furthermore, we constrain the central bank to only observe the output gap conflated with noise.

Our analytical solution of the policy problem implies that the optimizing central bank selects a positive coefficient of the money gap in the modified interest rate rule. The principles underlying the interest rate rule suggest that the central bank, provided it observes the natural interest rate directly, varies the policy rate one-for-one with the natural interest rate to counter the impact of the natural interest rate on aggregate demand. Selecting a positive coefficient of the money gap helps the central bank to approximate this ideal policy if it cannot observe the natural interest rate directly. Furthermore, we find that the optimizing central bank combines the money gap with the observed output gap in a way that provides the best information about the natural interest rate. The central bank considers both the money gap and the observed output gap at the same time because their combination improves upon the information that each indicator provides individually.

These results are fairly robust. We use the basic model to establish them analytically and to develop the core findings. Then, we use the quantitative model that features a dynamic money demand function and that can only be solved numerically to extend our analysis to more sophisticated policy rules. The analytical results hold up well and provide a suitable guideline for the numerical results. We also show that, in the quantitative model, lags of the money gap (rather than leads) co-move tightly with the contemporaneous natural interest rate. Consequently, in the optimal policy rule, the lagged money gap performs better as an indicator of the natural interest rate. We trace the pronounced lead-lag structure between the money gap and the natural interest rate back to the habit formation in consumption, which alters the time profile of the natural interest rate. Furthermore, while in the real world the information content of the money gap may change over time, our numerical results suggest that, as a rule, the central bank will be better off taking the money gap into account instead of not appropriately responding to the movements in the natural interest rate that the money gap indicates.

It is worth mentioning that our approach considers only a narrow (non-interest bearing) concept of money. However, while there are multiple concepts of money, broadening the model along this dimension would only add to the number of monetary indicator variables that the central bank would combine optimally according to their informational content. Here we focus on only one of possibly many money gaps to portray a link between money demand and the natural interest rate that the literature has not explored so far.

Our study is related to two branches in the literature. The first branch emphasizes that the natural interest rate is of great relevance to central banks and attempts to measure it using the information contained in the money demand. Our study differs from this branch in that we explore a new link between the money demand and the natural interest rate that is complementary to the link that has been explored so far. Furthermore, in contrast to the literature, we derive the consequences of the new link for the optimal monetary policy rule.

Andres, Lopez-Salido, and Nelson (2009) build upon Andres, Lopez-Salido, and Valles (2006) and belong to the first branch of the literature. They show that the money demand contains information about the natural interest rate if the money demand serves as a summary index of unobserved yields and, therefore, is forward looking.2 Arestis, Chortareas, and Tsoukalas (2010) measure the natural rate of output, which is related to the natural interest rate, also using the information contained in the forward-looking money demand and obtain an estimate of the natural output that is fairly precise. In contrast, Laubach and Williams (2003), Mesonnier and Renne (2007), and Edge, Kiley, and Laforte (2008) measure the natural interest rate without using the information contained in the money demand and find estimates of the natural interest rate that are fairly imprecise and subject to substantial measurement error.

The second branch of the related literature studies the indicator role of money demand when monetary policymakers lack information about the state of the economy (Dotsey and Hornstein (2003), Coenen, Levin, and Wieland (2005), Lippi and Neri (2007), Beck and Wieland (2007), Beck and Wieland (2008), Scharnagl, Gerberding, and Seitz (2010), Unsal, Portillo, and Berg (2010)).3 A common feature of this literature is that it treats the money demand residual as mutually independent of all other structural shocks in the economy. In the typical microfounded model that we use, however, the money demand residual contains one component that also drives the natural interest rate and, therefore, is not mutually independent of all other shocks. A number of authors have mentioned this characteristic of the money demand residual (see, e.g., McCallum and Nelson (1999), Neiss and Nelson (2001), Nelson (2002), Woodford (2003), Favara and Giordani (2009), and Sargent and Surico (2011)), but the implications for the indicator ro e of money demand have remained largely unexplored.

In the next section, we briefly recap the MIU specification of money demand and isolate the link between the money demand and the natural interest rate that we explore. Section III describes our transformation of money demand, which we call the money gap, and derives the correlation between the money gap and the natural interest rate. In Section IV, we demonstrate the usefulness of the money gap for the optimal monetary policy rule in the basic model. Section V contains the quantitative analysis of the link between the money gap and the natural interest rate and of the consequences for the optimal policy rule in the quantitative model with a dynamic money demand function, and Section VII concludes.

IMF.World Bank.Author/Editor: Berger, Helge; Weber, Henning

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