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This paper characterizes the optimal monetary policy reaction function in the presence of a zero lower bound on the nominal interest rate. We analytically prove and numerically show that the function is highly non-linear, more expansionary, and more aggressive than the Taylor rule. We then test its empirical validity taking the case of Japan in the 1990s. Qualitatively, we find some evidence of non-linear monetary policy. Quantitatively, we find the actual monetary policy to be too contractionary during the first half of the decade, while the low interest policy during the latter half turns out to be fairly consistent with the simulated path

A zero lower bound on the nominal interest rate is becoming a serious concern. Many central banks, especially in industrialized countries, have been successful in reducing average inflation rates to a range of 0–3% in recent decades. In this kind of low inflation era, especially when a central bank is faced by a severe recession, a zero lower bound on the short-term nominal interest rate – a policy instrument for most of the central banks – could be a serious constraint for the implementation of monetary policy. In the extreme case, when the nominal interest rate actually binds at zero, a central bank will no longer be able to stimulate the economy via the nominal interest rate channel – a phenomenon also known as a liquidity trap. In such circumstances, standard monetary policy, by controlling the short-term nominal interest, will become totally ineffective and the economy will have to bear the cost of increased volatility. Blinder (2000), being keenly aware of this predicament, succinctly warns, ‘Don't go there.’ The recent trend of low inflation accompanied by the issues stemming from a zero lower bound is the basic reason why it is becoming a realistic and serious concern for many central banks. Reflecting the practical importance of a zero lower bound on the nominal interest rate, much research has been made regarding the conduct of monetary policy in the presence of such a constraint. The pioneering work is due to Fuhrer and Madigan (1997). They conducted an impulse response analysis taking into account of the zero lower bound and showed that stabilization policy is costly in the sense that it takes longer for the output and the inflation rate to return to a steady state than the case where there is no constraint. Orphanides and Wieland (1998), in their stochastic simulation study, showed that the probability of the economic state entering a liquidity trap will be lower when the inflation target is set higher and concluded that the social welfare loss can be reduced by setting a positive inflation target. Reifschneider and Williams (2000) provided an insightful study which shows that a variant of the Taylor rule is superior to a standard Taylor rule in stabilization capability by comparing the efficient policy frontier of stochastic simulation results. This implies that when the zero lower bound on the nominal interest rate is incorporated, the optimal policy is neither a linear function of state variables nor a standard Taylor rule. Orphanides and Wieland (2000) demonstrated that the optimal policy under the non-negativity constraint is a non-linear function of the inflation rate using a numerical method. Their numerical evidence suggests a central bank to adopt an ‘aggressive’ monetary policy as the nominal interest rate approaches the zero lower bound. Watanabe (2000) and Jung et al. (2001) investigated the optimal conditions for the termination of a ‘zero interest rate policy’ based on the forward-looking economy model following Woodford (1999). Using a simulation technique, they show that the optimal path of the nominal interest rate depends on historical policy conduct as well as a commitment for future policy conduct.1Hunt and Laxton (2003) investigated the role of an inflation target in the presence of the zero lower bound. Using the MULTIMOD simulation model, they show that targeting too low an inflation rate will induce a central bank to be susceptible to a deflationary spiral and suggest that the inflation rate should be targeted higher than 2% in the long run. Among multiple aspects of the zero bound problem, our focus is on the explicit form of the optimal monetary policy reaction function.2 The past studies are commonly aware of the risk of the zero lower bound (or the liquidity trap), which can seriously affect the stabilization function of the central bank. In fact, most of the past studies, such as Blinder (2000), Goodfriend (2000), Reifschneider and Williams (2000), Orphanides and Wieland (2000), and Hunt and Laxton (2003) point out the possibility that the optimal monetary policy is affected by the zero lower bound before the constraint actually binds. For example, Goodfriend (2000) claims that monetary policy must be ‘pre-emptive’ to prevent the constraint from binding. Unfortunately, however, the past studies have mainly relied on conjectures or numerical/simulation methods in showing this ‘pre-emptiveness’ feature of monetary policy. The main contribution of this paper is that we provide a mathematical foundation to this ‘pre-emptiveness’ feature of monetary policy in the presence of the zero lower bound. In particular, we investigate how the optimal monetary policy reaction function is affected when the zero lower bound of the nominal interest rate is explicitly incorporated to a Svensson (1997) and Ball (1999) type model.3 Based on the stylized framework of a central bank's dynamic optimization problem following Svensson (1997) and Ball (1999), we derive the analytical expression of the optimal monetary policy reaction function in the presence of the zero lower bound and prove that it is (i) highly non-linear, (ii) more expansionary, and (iii) more aggressive than the Taylor rule, which is otherwise the optimal policy in the absence of the zero bound. These features of the optimal policy reaction function are basically consistent with the policy implications shown by past studies, providing a solid analytical foundation to ‘pre-emptive’ monetary policy conduct in the presence of the zero lower bound. In the real world, perhaps the most notable episode where the conduct of monetary policy has been severely affected by the zero lower bound constraint was during the period of stagnation of the Japanese economy in the 1990s. There has been substantial debate regarding the Bank of Japan's (BOJ) monetary policy conduct during the 1990s. For instance, Ahearne et al. (2002) point out that the BOJ's monetary policy conduct was too slow in cutting the call rate during the early 1990s and conclude that this slow response has been one of the factors that caused Japan's prolonged stagnation during the entire decade.4Bernanke and Gertler (1999) also report similar simulation evidence based on the dynamic general equilibrium model incorporating a financial accelerator. On the other hand, McCallum (1999), Kamada and Muto (2000), and Yamaguchi (2002) show empirical evidence that the estimated Taylor rule can explain the BOJ's monetary policy conduct until the mid-1990s fairly well, while exhibiting a wide discrepancy between the actual call rate and the predicted call rate implied by the estimated Taylor rule during the latter half of the decade. This implies that the BOJ's low interest rate policy during the latter half of the 1990s was too expansionary, if we take the estimated Taylor rule for granted. In this paper, taking the case of Japanese monetary policy in the 1990s, we empirically test the validity of the qualitative and quantitative features implied in our model. Taking into account the left-censoring of the call rate due to the presence of the zero lower bound, we adopt the Tobit model in estimating the BOJ's policy reaction function during the 1990s. The Likelihood Ratio type test based on the Tobit estimation results (weakly) rejects the linearity of the BOJ's policy reaction function with respect to the inflation rate and the output gap, revealing some evidence for the concavity of the policy reaction function. This empirical evidence turns out to be qualitatively consistent with the ‘aggressive’ monetary policy conduct implied in our model. However, since the qualitative evidence from the Tobit estimation is unindicative about the desirable level of the call rate during the 1990s, we also conduct quantitative analysis based on the numerically approximated optimal policy reaction function. By comparing the actual path of the call rate and the simulated path of the call rate during the 1990s, it turns out that the BOJ's monetary policy conduct in the first half of the decade was too contractionary and too slow in responding to the declining output and disinflation, while the low interest rate policy by the BOJ in the latter half of the decade turns out to be fairly consistent with the simulated path implied by the optimal policy reaction function. The remainder of this paper is organized as follows. Section 2 describes the set up of the model and derives the analytical expression of the optimal policy reaction function. Section 3 discusses the numerical strategy in approximating the optimal policy reaction function and demonstrates the results. Section 4 addresses the empirical issues taking the case of Japan in the 1990s and tests the qualitative and quantitative implications of the optimal policy reaction function. Section 5 offers some concluding remarks and discusses the future extension of the model.

In this paper, we have studied the optimal policy reaction function where the zero lower bound of nominal interest rates might interfere with the conduct of monetary policy. The main contribution of this paper is that we have derived an analytical expression of the optimal monetary policy reaction function in the presence of the zero lower bound and proved the key properties that it is more expansionary and more aggressive than the Taylor rule. Although preceding research has pointed out or simulated these properties, to the best of our knowledge, none has derived an analytical expression of the optimal policy reaction function in the presence of the zero lower bound or formally proved the above properties. Further, in order to verify these analytical implications, we have numerically approximated the optimal policy reaction function using the method known as the collocation method. Conforming with the earlier numerical and simulation results demonstrated in Reifschneider and Williams (2000) and Orphanides and Wieland (2000), we have verified that, indeed, the reaction function is more expansionary and aggressive than the Taylor rule and concave (provided that the nominal interest rate is not binding at zero) in the inflation rate and the output gap. Based on this numerically approximated optimal reaction function, we have empirically tested the qualitative and quantitative implications taking the case of Japanese monetary policy conduct in the 1990s. According to our Tobit estimation results, we found some empirical evidence that BOJ's monetary policy conduct in the 1990s was qualitatively consistent – i.e., concave in inflation rate and output gap – with the optimal policy reaction function implied in the paper. Finally, in order to evaluate the BOJ's monetary policy conduct quantitatively, we have compared the actual path of the nominal interest rate with the simulated path implied by the numerically approximated policy reaction function. According to our quantitative analysis, we found the BOJ's monetary policy conduct in the first half of the 1990s to be too contractionary and too slow in responding to disinflation and declining output, while, in the latter half of the 1990s, we found the BOJ's low interest rate policy to be fairly consistent with the optimal policy reaction function implied in our model. One important remark should be made. The economy assumed in this paper is the simplest case, in the sense that state variables in the transition system are bivariate VAR(l) of the inflation rate and the output gap. In general, it is likely that the state variables in the ‘true’ transition system of the economy are not limited to the inflation rate and the output gap. Generally speaking, the transition system may be multivariate VAR(P) and not limited to the state variables of the inflation rate and the output gap. Yet, it is still notable that the optimal policy reaction function derived based on the simple VAR(l) transition system was able to capture the qualitative character of the BOJ's monetary policy conduct in the 1990s fairly well, despite the mixed evidence revealed by the quantitative analysis. It remains to be seen whether a more general specification of the transition system can characterize both the qualitative and quantitative features of Japanese monetary policy in the 1990s. Although this extension is interesting and important, this will be left for future research.