Trend Inflation & Taylor Rule Indeterminacy

The study's central argument revolves around the substantial impact of even low levels of trend inflation on the dynamics of a basic new Keynesian DSGE model, particularly when monetary policy is governed by a contemporaneous Taylor rule. A key finding is that positive trend inflation significantly shrinks the determinacy region, the range of parameters for which the model yields a unique, stable equilibrium. This challenges the traditional Taylor principle, which posits that an inflation coefficient exceeding one is sufficient for equilibrium determinacy. The research demonstrates that neither the standard Taylor principle nor its generalized version (requiring a long-run nominal interest rate increase exceeding the inflation increase) guarantees local determinacy of equilibrium. This result holds true across various types of Taylor rules (contemporaneous, backward-looking, forward-looking, and hybrid), inertial policy rules, and different price indexation schemes. The implication is clear: the existing monetary policy literature, heavily reliant on Taylor rules, cannot afford to disregard average inflation in its theoretical and empirical analyses.

A crucial aspect of the analysis is the demonstration of how trend inflation affects the well-established Taylor principle for rational expectations equilibrium determinacy. This effect stems from the distortion of steady-state relative prices induced by trend inflation within the Calvo pricing framework. Ascari (2004) and Yun (2005) previously showed a highly nonlinear steady-state relationship between output and inflation. While the long-run Phillips curve exhibits a positive slope around the zero-inflation steady state, it inverts and becomes negatively sloped at even moderate positive trend inflation levels, reflecting the relative price distortions. This inversion means higher trend inflation is associated with lower steady-state output. The study underscores the profound implications of this feature, suggesting that many existing results in the literature, based on the zero-inflation steady state, may be misleading.

To address inconsistencies in the existing literature, the research extends the standard small-scale New Keynesian DSGE model to accommodate positive trend inflation. By incorporating a Taylor rule to represent the monetary authority's behavior, the study investigates the extent to which the model's properties change as trend inflation varies. Key findings demonstrate that even moderate trend inflation levels significantly modify three crucial aspects of the model: (i) the conditions for a determinate (unique) rational expectations equilibrium, (ii) the impulse response functions following a cost-push shock, and (iii) the unconditional variances of key variables like inflation and output. These findings challenge established understandings and highlight the need for a more nuanced approach to modelling monetary policy under conditions of non-zero inflation.

The study contrasts its findings with existing literature, noting that much of the extensive research on monetary policy rules has employed models approximated around the zero-inflation steady state (e.g., Clarida et al., 1999; Galí, 2003; Woodford, 2003; Taylor, 1999). It highlights the seminal work of Clarida et al. (2000), which, while estimating a Taylor rule on US data with an average inflation of approximately 4%, was based on a zero-trend inflation model. The authors show that incorporating positive trend inflation significantly alters the model's structural equations and determinacy region. The authors explicitly mention other relevant research, including Ascari (2004), Amano et al. (2007), Ascari and Ropele (2007), Cogley and Sbordone (2005), Karanassou et al. (2005), and Kiley (2007), differentiating their approach and contributions. The authors conclude that the zero-inflation steady state assumption is both empirically unrealistic and theoretically limiting, casting doubt on the validity of many existing results.

The paper's core argument centers on how trend inflation fundamentally alters the well-known Taylor principle for determining rational expectations equilibrium. This stems from the distortion of steady-state relative prices caused by trend inflation within a Calvo pricing framework. Research by Ascari (2004) and Yun (2005) highlighted the highly nonlinear relationship between output and inflation in the steady state. Crucially, the long-run Phillips curve, usually positively sloped around a zero-inflation steady state, inverts and becomes negatively sloped when even moderate positive trend inflation is present. This inversion reflects the relative price distortions created by trend inflation. Consequently, higher trend inflation correlates with lower output levels in the steady state. This nonlinearity has significant implications for the effectiveness and interpretation of monetary policy, particularly concerning the Taylor principle.

The standard Taylor principle, which suggests that an inflation coefficient greater than one in a Taylor rule is sufficient for equilibrium determinacy, is shown to be insufficient in the presence of trend inflation. The paper introduces a generalized Taylor principle, which requires the nominal interest rate to rise by more than the increase in inflation in the long run to ensure determinacy. However, even this generalized principle is profoundly influenced by the level of trend inflation. The impact of trend inflation on the relationship between inflation and output significantly alters the trade-off the central bank faces when adjusting policy parameters. The model shows that the trade-off between responding to inflation versus output in monetary policy changes dynamically with the level of trend inflation. This necessitates a re-evaluation of existing monetary policy strategies that do not fully account for this effect.

The research challenges the dominant approach in monetary policy literature that often relies on models approximated around a zero-inflation steady state. The authors argue that this simplification ignores the significant impact of trend inflation, which is demonstrably present in many economies. The paper underscores that the simple Taylor principle, widely used in empirical and theoretical monetary policy analysis, is insufficient to ensure equilibrium determinacy under positive trend inflation. The assumption of zero trend inflation in many existing models leads to potentially inaccurate results and policy recommendations. Therefore, the paper strongly suggests that the monetary policy literature must explicitly incorporate trend inflation in both its empirical and theoretical analyses to accurately capture the complexities of monetary policy effectiveness.

A core focus is on the conditions required for the determinacy of the rational expectations equilibrium (REE) within the New Keynesian DSGE model and how these conditions are fundamentally altered by trend inflation. The paper establishes that even moderate levels of trend inflation significantly shrink the determinacy region, the parameter space within which a unique, stable equilibrium exists. This finding is critical because it directly impacts the reliability of monetary policy prescriptions derived from models that do not explicitly account for trend inflation. The analysis rigorously examines the implications of this for the stability and predictability of the macroeconomic system under different monetary policy regimes. The research goes beyond simply stating the reduced determinacy region, delving into the specific mechanisms through which trend inflation undermines the stability of the equilibrium.

The research employs a standard contemporaneous Taylor rule to model the central bank's monetary policy actions. The study analyzes the interaction between the Taylor rule's coefficients (the responses to inflation and output gap) and the level of trend inflation to determine the conditions for REE determinacy. The analysis reveals that a simple inflation coefficient greater than one (the traditional Taylor principle) is insufficient to ensure determinacy when trend inflation is present. The paper demonstrates a crucial interaction between the responses to inflation and output in the Taylor rule, showing that the optimal policy response is dependent on the level of trend inflation. Under positive trend inflation, overreacting to output deviations can be destabilizing, leading to higher inflation in the long run.

The findings of the research have significant implications for monetary policy. The study indicates that under positive trend inflation, maintaining equilibrium determinacy requires a more nuanced approach to policymaking than what is suggested by models assuming zero trend inflation. Specifically, the central bank's response to inflation and output gap must be carefully calibrated to account for the changing dynamics induced by trend inflation. The presence of trend inflation worsens the trade-off between inflation and output stabilization. Policies that aim to aggressively reduce output deviations from steady state can inadvertently lead to even higher inflation levels due to the inverted long-run Phillips curve under positive trend inflation. This highlights the need for central banks to prioritize inflation stabilization, particularly in environments with significant trend inflation.

Beyond the generalized Taylor principle, the study identifies a second determinacy condition that is crucial for equilibrium stability. Even in the standard zero-inflation case, this condition places constraints on the central bank's ability to implement excessively pro-cyclical monetary policies. This second condition, however, becomes even more critical under positive trend inflation. The research demonstrates that the interplay between the two determinacy conditions is significantly altered by trend inflation, shrinking the viable policy space for central banks. Numerical simulations show that as trend inflation rises, the feasible monetary policies shift towards those that respond more aggressively to inflation and less to output deviations. For sufficiently high levels of trend inflation, even the classical Taylor rule parameters can lead to indeterminacy, highlighting the importance of considering trend inflation in practical monetary policy analysis.

The study rigorously examines the robustness of its findings by testing them across a wide range of Taylor rule specifications. This includes various types of Taylor rules such as contemporaneous, backward-looking, forward-looking, and hybrid rules. The consistency of the results across these different specifications strengthens the paper's conclusions. The incorporation of inertial Taylor rules, which reflect the gradual adjustment of interest rates often observed in practice, further supports the generalizability of the findings. This thorough exploration of different Taylor rule formulations demonstrates that the core findings regarding the impact of trend inflation on equilibrium determinacy are not artifacts of a particular model specification but rather represent a more fundamental characteristic of the New Keynesian DSGE model under trend inflation conditions.

The analysis investigates the influence of different price indexation schemes on the model's response to trend inflation. The study considers various indexation schemes, including indexation to trend inflation and indexation to past inflation rates, and varying degrees of overall indexation. By systematically varying these schemes, the robustness of the findings regarding the contraction of the determinacy region and the modified Taylor principle under trend inflation is assessed. This exploration of price indexation highlights how different price adjustment mechanisms interact with trend inflation to affect the overall dynamics of the model and the conditions for equilibrium determinacy. The results demonstrate that the core conclusions remain consistent regardless of the specific price indexation scheme implemented within the model.

The paper employs numerical simulations to support and extend its analytical results. These simulations are designed to test the sensitivity of the findings to changes in various parameters of the model. Different calibration values are used to assess the range of conditions under which the model's behavior is consistent with the main findings. The parameter values are explicitly discussed, sometimes noting deviation from common specifications and justifying those choices. The numerical simulations confirm the analytical insights, demonstrating the robustness of the results across different parameterizations. By exploring the model's behavior under various conditions, the researchers provide a comprehensive picture of how trend inflation impacts macroeconomic dynamics and the effectiveness of monetary policy.

The research utilizes impulse response functions (IRFs) to analyze the dynamic responses of key macroeconomic variables, such as output, inflation, and nominal interest rates, to a cost-push shock. This analysis explicitly examines how these responses change with varying levels of trend inflation. The IRFs illustrate the deterioration of the short-run output-inflation trade-off as trend inflation increases. Even with increasingly restrictive monetary policies implemented by the central bank, the flattening of the short-run New Keynesian Phillips Curve reduces the effectiveness of output adjustments in stabilizing inflation. The inclusion of price indexation to past inflation in some simulations allows an assessment of how this interacts with trend inflation's influence on the IRFs. These findings further support the paper's main argument that trend inflation significantly modifies the model's dynamics and the effectiveness of monetary policy.

The concept of the efficient policy frontier is introduced to illustrate the trade-off between inflation and output variability under different monetary policy settings. The efficient policy frontier, which links output and inflation variabilities (typical arguments in a central bank's loss function), is shown to be significantly affected by trend inflation. The analysis demonstrates that trend inflation shifts the efficient frontier northeast, resulting in higher variability for both inflation and output. This implies that achieving a given level of output stability requires accepting higher inflation variability, and vice versa, demonstrating a worsening trade-off. Price indexation, either to trend or past inflation, partially mitigates this negative effect, shifting the frontier southwest. This analysis shows that the optimal policy response under a Taylor rule is fundamentally altered in the presence of trend inflation, leading to a less favorable trade-off between inflation and output stabilization.

The conclusion summarizes the key findings, emphasizing the substantial impact of trend inflation on the properties of the New Keynesian DSGE model and its implications for monetary policy. The study highlights the limitations of much of the existing monetary policy literature, which predominantly relies on models approximated around a zero-inflation steady state. This simplification is criticized as both empirically unrealistic and theoretically restrictive. The researchers demonstrate that the model's behavior and, therefore, any conclusions drawn from it, are highly sensitive to even low and moderate levels of trend inflation, which are commonly observed in Western economies. The core finding that moderate levels of trend inflation significantly alter the determinacy region and impact the effectiveness of monetary policy is shown to be robust to variations in Taylor rule types, price indexation schemes, and parameter values. Therefore, the paper concludes that the literature on monetary policy rules cannot ignore trend inflation in both empirical and theoretical investigations. The research advocates for incorporating the model's long-run and short-run dynamics in future non-linear analyses.