From Eq we have On the other hand as long

From Eq. (8), we have . On the other hand, as long as . Therefore, in this glutathione peroxidase case , the current monetary easing has a positive effect on output and the inflation rate.
Theoretical analysis In this section, we theoretically investigate the effects of an anticipated monetary policy by using Eq. (8). First, we note the effects of the current policy on the current economy. If , the current monetary easing has a positive effect on output and the inflation rate. What are the effects of an anticipated policy on the economy? The key parameter is the monetary policy response to inflation ϕ. We show that if , an anticipated monetary easing (m<0) has an inflationary effect and a positive effect on output: and for any i. This proof is straightforward because the entries of both A−1 and B are non-negative. Consider the intuition in the case of anticipated monetary easing. The expectation of future low real interest rates decreases the households’ incentives to save. In the case of , an anticipated monetary easing does not have the same effect of current policy. In this case, although the procedure to definitely confirm policy effects is complicated, we can at least show that ifan anticipated monetary policy has an inflationary effect as well as a negative effect on output; that is, and , . Condition (9) is satisfied by a frequently assumed parameterization such as . Thus, the effects of an anticipated monetary policy on inflation and output are opposite. In the case of anticipated monetary easing, the relatively high response of monetary policy rules to inflation raises the current real interest rate, and the degree of decrease in the future real interest rate becomes dampened. This motivates current savings, and as a result of high current savings, the future consumption increases. Thus, while a decrease in current output exerts deflationary pressure on the current inflation rate, an increase in future output creates inflationary pressure. Under condition (9), the latter exceeds the former and the result is current inflation. In the case of the rest, that is, , we show that as long as , although the sign of is not certain. Therefore, in the assumed parameter space (), the current and anticipated monetary easing always have inflationary effects. Summing up, we have the following proposition: It is noteworthy that the general restriction in the above proposition is equivalent to the Taylor principle and consistent with standard monetary policy (i.e., monetary easing leads to an increase in output and vice versa). Furthermore, since the condition of result (3) is sufficient to obtain , , we could find this in the case of .
Numerical example In this section, we quantitatively analyze the effects of an anticipated monetary policy on the economy. First, we assume the following: , , and . As for ϕ, we consider two cases, (i) ϕ=1.001 and (ii) ϕ=1.5. The parameterization of these cases yields Fig. 1, which depicts the value of . From the figure, we can confirm the proposition. A relatively high value of ϕ, typically assumed in explaining the new Keynesian model in a classroom, indicates that inflation rates and output move in opposite directions. Our analysis shows that theoretically an anticipated monetary policy can lead to rather unexpected results for output. Milani and Treadwell (2012), Gomes et al. (2013), and Verona et al. (2013) empirically investigate the effects of news shocks on the economy. While Milani and Treadwell (2012) use a new Keynesian model consisting of three equations (the IS curve, Taylor rule, and new Keynesian Philips curve), where each equation has a variable of own lag to incorporate inertia into the model, the other two construct a medium-scale new Keynesian model with interest rate smoothing. Although the terms for inertia make food pyramid difficult to obtain analytical solutions, empirical fits to actual data and the model are improved. Their estimation results indicate that the responses of news shocks to monetary policy on output and inflation is similar to case (i) in this section. As we show, the case occurs because of a relatively low response of the central bank to inflation. In the majority of empirical studies, including the papers mentioned above, that estimate DSGE models with a smoothing of interest rates, the term representing the smoothing (in our paper ϕ) is estimated at a lower value. Therefore, in an actual economy, news shocks to monetary policy easing might result in an increase in output and inflation.