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In recent monetary policy literature, optimal commitment policy and its variant from a timeless perspective have been studied with emphasis on welfare gains from policy commitment. These policies, however, involve a time-consistency problem called a stabilization bias in forward-looking models. We analyze Chari and Kehoe's [1990. Sustainable plans. Journal of Political Economy 98, 783–802] sustainable equilibrium and examine optimal sustainable policy, i.e. a policymaker's strategy in the best sustainable equilibrium. This paper shows that such a policy becomes consistent with the optimal commitment policy in sufficiently later periods. It also shows that whether the optimal sustainable policy can attain the Ramsey equilibrium outcome depends on the magnitude of shocks hitting the model economy. Moreover, the paper finds a sustainable policy that attains higher social welfare than discretionary policy does.

Since Kydland and Prescott's (1977) seminal work, the time-consistency problem of optimal commitment policy has been well known. Nevertheless, recent monetary policy literature has studied this optimal policy and its variant from a timeless perspective proposed by Woodford (1999), with emphasis on welfare gains from policy commitment.1 This recent literature has addressed the time-consistency problem called a stabilization bias, which arises from an inefficient trade-off in policymaking with private agents’ forward-looking behavior and differs from the well-known inflation bias studied in traditional literature starting from Kydland and Prescott (1977) and Barro and Gordon (1983a). In this paper we analyze Chari and Kehoe's (1990) sustainable equilibrium in a canonical model with the stabilization bias and examine a policymaker's strategy in the best sustainable equilibrium.2 This policy strategy is called optimal sustainable policy. The present paper shows first that discretionary policy induces the worst sustainable equilibrium. Using this result, it shows next that the entire set of sustainable equilibrium outcomes can be fully represented by private agents’ optimality condition and a set of inequalities called a sustainability constraint. 3 This constraint requires that in each period an equilibrium outcome in question attains at least as high a present value of social welfare as the worst sustainable equilibrium outcome induced by the discretionary policy. This paper then finds that the optimal sustainable policy is a policy strategy which specifies to continue a policy that yields the best sustainable equilibrium outcome, as long as this policy has been adopted in the past; otherwise, the strategy specifies to switch to the discretionary policy forever. To examine the optimal sustainable policy, this paper uses a Lagrange method of Marcet and Marimon (1998), who develop Kydland and Prescott's (1980) pioneering work. By maximizing a social welfare function within the entire set of sustainable equilibrium outcomes, this method generates a policy that attains the best sustainable equilibrium outcome in the presence of a commitment technology. Such a policy is referred to as optimal quasi-sustainable policy. The optimal sustainable policy now becomes a policy strategy that specifies to continue the optimal quasi-sustainable policy as long as it has been adopted in the past; otherwise, the strategy specifies to switch to the discretionary policy forever. This implies that the optimal sustainable policy is conducted by following the optimal quasi-sustainable policy and accomplishes the best sustainable equilibrium outcome in the absence of commitment technologies. 4 This is because private agents are policy takers and the optimal sustainable policy leads the policymaker to have no temptation to deviate from the optimal quasi-sustainable policy. By analyzing the optimal quasi-sustainable policy, this paper obtains three features of the optimal sustainable policy. First, the optimal quasi-sustainable policy is an intermediate one between the optimal commitment policy and the discretionary policy. Therefore, the best sustainable equilibrium outcome achieved by the optimal sustainable policy is also an intermediate one between the Ramsey equilibrium outcome and the worst sustainable equilibrium outcome attained, respectively, by the latter two policies. Second, the optimal quasi-sustainable policy becomes consistent with the optimal commitment policy in sufficiently later periods, regardless of values of the model parameters. This implies that even in the absence of commitment technologies, the optimal commitment policy can be credibly adopted after the policymaker keeps the optimal quasi-sustainable policy for a sufficiently long period.5 Last, if the sustainability constraint is never binding, the optimal quasi-sustainable policy is completely consistent with the optimal commitment policy and hence the time-consistency problem does not matter in that the optimal sustainable policy can achieve the Ramsey equilibrium outcome. The present paper shows that this holds if private agents are patient enough. This is because for a sufficiently high discount factor, sticking to the optimal commitment policy yields such a large present value of future social welfare that a policymaker's deviation from it never pays. Using this result, the paper examines whether the optimal sustainable policy can attain the Ramsey equilibrium outcome. It finds that the answer to this question depends on the magnitude of shocks hitting the model economy. If narrow bounds of the shocks are considered, the optimal sustainable policy can do so. For wide bounds of them, however, this is not the case because for a range of realistic calibrations of the model parameters, a certain lower bound on the discount factor for which the Ramsey equilibrium outcome is attainable is extremely close to one. Recent monetary policy literature assumes the Normal distributions for shocks and supports fairly wide bounds, suggesting that the optimal commitment policy may not be the desirable policy benchmark for actual policymakers, i.e. central banks, which do not possess commitment technologies perfectly. In addition to the analysis of the optimal sustainable policy, this paper finds a sustainable policy that achieves the best Markov equilibrium outcome and attains higher social welfare than the discretionary policy does, for any values of the model parameters.6 Among related literature, Ireland (1997) analyzes sustainable equilibrium in a model with the inflation bias. Albanesi et al. (2003) investigate Markov equilibrium in variants of Lucas and Stokey's (1983) cash-credit model and Christiano et al.'s (1997) limited participation model. These studies find that the inflation bias is unlikely to matter in their models. Loisel (2005) and Levine et al. (2007) are the two most related papers, which consider the stabilization bias. Loisel uses a similar model to that of the present paper and examines whether a particular trigger-like policy can generate the Ramsey equilibrium outcome. He assumes narrow bounds of shocks and thus the trigger-like policy is likely to yield the Ramsey equilibrium outcome in his model. Levine et al. use a medium-scale dynamic stochastic general equilibrium model along the lines of recent business cycle literature, e.g. Christiano et al. (2005), and investigate whether a monetary policymaker has a temptation to deviate from optimal commitment policy to discretionary policy. They find that the stabilization bias is unlikely to matter in their model. One reason for their result is that they do not include large shocks in simulation. Another reason, which is more crucial, is that their model contains a lot of frictions and a zero lower bound on nominal interest rates, and thereby welfare gains from policy commitment are so large that the policymaker's deviation from the optimal commitment policy to the discretionary policy does not pay. The results of Loisel and Levine et al. are in line with this paper's finding that whether the Ramsey equilibrium outcome is attainable without commitment technologies depends on the magnitude of shocks hitting the model economy. Their studies analyze only a particular sustainable equilibrium to address their questions. In stark contrast to these studies, this paper characterizes the entire set of sustainable equilibrium outcomes and examines the optimal sustainable policy in the canonical model with the stabilization bias. The remainder of this paper proceeds as follows. The next section describes a monetary policy design problem used in recent literature and reviews optimal commitment policy, its timeless-perspective variant, and discretionary policy. Section 3 examines optimal sustainable policy. Section 4 investigates whether the optimal sustainable policy can attain the Ramsey equilibrium outcome and then analyzes a sustainable policy that attains higher social welfare than the discretionary policy does. Finally, Section 5 concludes.

In this paper, we have examined sustainable equilibrium and optimal sustainable policy in a canonical model with the stabilization bias used in recent monetary policy literature. This paper has shown that the optimal sustainable policy becomes consistent with the optimal commitment policy in sufficiently later periods, regardless of values of the model parameters. It has also shown that in the presence of a sufficiently large shock, the optimal commitment policy may not be the desirable policy benchmark without commitment technologies, suggesting that the optimal sustainable policy is the proper benchmark for central banks, which do not possess such technologies perfectly. Moreover, the paper has found a sustainable policy that achieves the best Markov equilibrium outcome and attains higher social welfare than the discretionary policy does, for any values of the model parameters. One future research topic is a numerical investigation of the optimal (quasi-)sustainable policy. This is because the optimal sustainable policy is the desirable policy benchmark in the absence of commitment technologies and we have no idea about the exact responses of this policy to shocks. Thus, there is great interest in examining how such an optimal policy responds to the shocks, particularly when the sustainability constraint is binding, i.e. the optimal quasi-sustainable policy differs from the optimal commitment policy. In the existing literature there are two numerical methods for the optimal sustainable policy. One is suggested by Christiano and Fisher (2000), who use first-order conditions for optimal policy with inequality constraints. Another is proposed by Marcet and Marimon (1998), who use a Bellman equation called the “recursive saddle point functional equation”. Using these methods, we can analyze the optimal sustainable policy in more detail. Another topic is to examine the sustainability of a particular monetary policy rule. Since Taylor's (1993) pioneering work, policy rules have received much attention in monetary policy literature. Many recent studies assume that a monetary policymaker can credibly commit to a proposed policy rule. It is far from clear, however, exactly how or whether such credibility would arise. To address this question, this paper suggests examining whether the proposed policy rule leads to an outcome of a sustainable equilibrium. If the policy rule does so, the policymaker's commitment to it can be supported by a sustainable equilibrium in which the policymaker takes a policy strategy that specifies to continue the policy rule as long as it has been adopted in the past; otherwise, the strategy specifies to switch to a policy that induces the worst sustainable equilibrium outcome in the subsequent economy. Hence the commitment is credible. Otherwise, private agents know the policymaker's temptation to deviate from that policy rule, and thus such commitment cannot be credible. Therefore, the sustainability seems to be a requirement policy rules must meet. A companion paper by Kurozumi (2008) investigates the sustainability of Taylor style policy rules.