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This paper examines optimal monetary policy with an explicit zero lower bound in a small open-economy model. The paper finds that the gains from commitment are increasing in the openness of the economy while the optimal rate of inflation is decreasing in the openness of the economy. These results imply that the main findings of Adam and Billi (2007) for a closed-economy model are also true for an open-economy model. Finally, the paper finds that the effectiveness of the exchange-rate channel as a stabilization tool in the low interest rate environment depends on whether the central bank can make a credible commitment. If the central bank cannot commit and makes monetary-policy decisions on a discretionary basis, the optimal path of the nominal exchange rate will exhibit an appreciation, rather than depreciation as suggested in the literature.

1. Introduction The recent experience of low interest rates in many countries has led to considerable interest in the design of monetary policy in the presence of a zero lower bound on nominal interest rates. Such a low interest rate environment presents a particular problem for monetary policy since the effectiveness of short-term nominal interest rates, a main instrument for most central banks, could be limited by the zero lower bound. Particularly, the central bank will not be able to offset severe contractionary disturbances by lowering the nominal interest rate by the desired amount. Thus, the central bank will not be able to stimulate the economy via the standard interest-rate channel. Academics and policymakers alike have made proposals regarding the conduct of monetary policy in the presence of such a constraint. One interesting approach that has been widely studied is for the central bank to resort to the exchange-rate channel and use exchange rates as a policy instrument, if the interest-rate channel becomes ineffective due to the zero lower bound. Prominent studies based on this approach include Orphanides and Wieland, 2000, Svensson, 2001, McCallum, 2002, Coenen and Wieland, 2003 and Coenen and Wieland, 2004. In a broader context, this is also related to a growing literature in which many authors have emphasized the importance of exchange rates as a distinct policy instrument in managing aggregate demand.1 This is indeed the strategy pursued by many real-world central banks,of which the most interesting are those in the Far East. Having experienced one of the lowest short-term nominal interest rates in the recent years, several Asian central banks have actively pursued several measures to influence their exchange rates. Even so, it appears that these central banks, especially those under floating exchange rates, have failed to achieve exchange rates that are deemed to be appropriate for their economic conditions. This raises a question whether such exchange-rate policy is desirable, especially if the attempt to achieve the target exchange rates leads to a conflict with other policy objectives. And in the case that such policy is indeed desirable, what is the implementation strategy that will allow the central banks to achieve their target exchange rates? This paper sets forth a framework to study optimal monetary policy in a small-open economy when the nominal interest rate is constrained by the zero lower bound. This paper relies on a full dynamic model with an explicit bound, under the standard assumptions of uncertainty and rational expectations, and derives optimal monetary policy under commitment as well as under discretion. Thus, this paper is closely related to Adam and Billi, 2006 and Adam and Billi, 2007 who studied optimal monetary policy in the presence of the zero lower bound but in a closed-economy model. It should be noted that extending the analysis of Adam and Billi into the open-economy setting is not straightforward. For one thing, with the standard assumptions of uncertainty and rational expectations, there exists no closed-form solution to the problem of optimal monetary policy with an explicit zero lower bound. To solve such problems, one has to resort to numerical analysis. In this paper, I rely on the collocation method, which is also used in Adam and Billi, 2006 and Adam and Billi, 2007.2 Like most algorithms in numerical analysis, a drawback of the collocation method is that the method may not work when applied to a model larger than the canonical closed-economy DSGE model. This is known as the curse of dimensionality, a situation in which numerical-analysis algorithms fail when applied to a large system of equations. As a contribution to the literature, this paper presents a strategy to cope with the curse of dimensionality when applying the collocation method to multi-sector models, such as an open-economy DSGE model. This paper derives three main results. First, the gains from commitment are increasing in the openness of the economy. This result implies that the main finding of Adam and Billi (2007) for a closed-economy model, that the welfare gains from policy commitment are more important once the zero lower bound is taken into account, is true for an open-economy model. Second, the optimal rate of inflation is decreasing in the openness of the economy. This result is consistent with the main finding of Billi (2007) for a closed-economy model that the optimal rate of inflation may not be as high as normally thought by policymakers. Finally, in response to a contractionary disturbance, the nominal exchange rate depreciates under the commitment equilibrium; however, under the discretionary equilibrium, the nominal exchange rate appreciates. Svensson (2004) finds, in a stylized, three-period model, that when the central bank pursues optimal policy under commitment, it is optimal to create a nominal depreciation to offset a decline in the natural interest rate. The result in this paper that when the central bank instead follows optimal policy under discretion, the optimal path of the nominal exchange rate exhibits a nominal appreciation, is however new. Thus, a contribution of this paper is that the effectiveness of the exchange rate as a stabilization tool under the low interest rate environment depends on whether the central bank can commit. If the central bank cannot commit and follows discretionary policy, the ability of the central bank to rely on the exchange-rate channel to stabilize the economy will be limited. The remainder of the paper proceeds in the following manner. Section 2 sets up the model used for the analysis. Section 3 discusses the curse of dimensionality when applying the collocation method to an open economy model and presents a method to derive the optimal path of the nominal exchange rate. Section 4 investigates the effect of the zero lower bound on the optimal target rate of inflation and the stabilization performance of monetary policy. Section 5 characterizes the optimal paths of the nominal exchange rate under commitment policy and discretionary policy. In Section 6, the importance of the exchange-rate channel in the low interest rate environment is examined. Section 7 concludes.

This paper derives optimal monetary policy with an explicit zero lower bound in a small open economy. The paper finds that the main findings of Adam and Billi, 2007 and Billi, 2007, which conduct similar studies in closed-economy models, are also true in the open-economy setting. In particular, the present paper finds that the gains from commitment are increasing in the openness of the economy while the optimal rate of inflation is decreasing in the openness of the economy. The present paper also lends support to Svensson’s (2004) result that under the commitment policy, it is optimal for the central bank to create a nominal depreciation in response to a decline in the natural interest rate. However, in the context of floating exchange rates, which becomes prevalent in both industrialized and emerging-market countries, the central bank cannot control exchange rates perfectly and thus may not necessarily be able to achieve a nominal depreciation at will. Particularly, this paper finds that if the central bank cannot commit and thus follows optimal monetary policy under discretion, the nominal exchange rate will instead appreciate in the presence of large contractionary disturbances. Therefore, the ability of the central bank to influence private agents’ expectations plays a key role on whether the central bank can rely on the exchange rate to stabilize the economy. The result that when the nominal interest rate is constrained by the zero lower bound, the optimal path of the exchange rate under the discretionary equilibrium exhibits a nominal appreciation is new in the literature. There are several earlier studies that propose using exchange rates as a policy instrument when the nominal interest rate is constrained by the zero lower bound. Orphanides and Wieland, 2000 and Coenen and Wieland, 2003, who also recommend that in the presence of the zero lower bound, the central bank should rely on the exchange-rate channel, suggest that the central bank can create a depreciation via a portfolio-balance effect. In particular, this can be done by expanding the monetary base on a large scale. This paper, on the other hand, argues that in order to create a depreciation successfully, the central bank also has to rely on the expectations channel. That is, the present paper implies that aggressive monetary-base expansion proposed by Orphanides and Wieland (2000) may be effective, but only if the central bank can make a credible promise to keep injecting liquidity into the economy going forward even when the deflationary pressure begins to subside. According to the analysis in this paper, by expanding the monetary base, but on a discretionary basis, and thereby only relying on the portfolio-balance effect, the central bank may not be able to generate a depreciation. It should be noted that this paper focuses on exchange-rate policy under the regime of floating exchange rates, which has become most prevalent among industrialized as well as emerging-market countries. This differs from McCallum (2002), who proposes using the exchange rate as the policy instrument when the nominal interest rate is constrained at zero. In particular, McCallum recommends setting the exchange rate in response to inflation and the output gap, in the same way as a Taylor-type interest rate rule. By committing to a rule with the exchange rate as the instrument, McCallum’s proposal requires that the central bank can control its exchange rate directly and thus can achieve any desired exchange rates in real time. One way to implement McCallum’s proposal is to adopt a crawling peg whereby the exchange rate is adjusted in every period to respond to inflation and the output gap. The present paper, on the other hand, examines scenarios in which countries operate under floating exchange rates. The policy recommendations in the present paper do not require countries to make a switch to fixed exchange rates. In subsequent research, I hope to consider several extensions to the work so far: First, the analysis in this paper is based on a calibrated model. The model parameters are calibrated such that we can capture the episode of low interest rates in several major economies, a phenomenon that has occurred only recently. When more data become available, the model can instead be estimated by a more sophisticated technique. Second, after we develop algorithms that can be more effective in dealing with the curse of dimensionality, we can incorporate additional imperfections, such as incomplete exchange rate pass-through or imperfect capital mobility, into the model. This will allow us to study the interactions between these imperfections and the effectiveness of the exchange-rate channel. Finally, under the commitment equilibrium, it is assumed that the central bank can make a credible promise that will constrain its action in the future. An important topic for future research is how to implement the commitment equilibrium. In other words, how can we design a mechanism that induces the central bank to deliver its own promise made from the past and thereby makes its promise credible among private agents.