1 Introduction

The formulation of objectives for central banks is a classical issue in economics. The natural premise when society assigns an objective for its central bank is that it should aggrandize welfare for its citizens. The recent global financial crisis and sovereign debt crisis in Europe, which caused central banks to slash interest rates and unleash large purchases of government bonds to prevent deflationary spirals and another great depression, have stimulated a vibrant discussion about the role of central banks in the economy and what their objectives should be. The purpose of this paper is to contribute to this important discussion.

In the 1970s and 1980s, variable and high rates of price inflation led many countries to delegate the conduct of monetary policy to “instrument-independent” central banks. Drawing on learned experiences, many societies gave their central banks a clear mandate to pursue price stability and instrument independence to achieve it.¹ Advances in academic research, notably the seminal work of Rogoff (1985) and Walsh (1995), supported a strong focus on price stability as a means to enhance the independence and credibility of monetary policymakers. As discussed in further detail in Svensson (2010), an overwhelming majority of these central banks also adopted an explicit inflation target to further strengthen credibility and facilitate accountability. For example, the mandate of the European Central Bank is to maintain price-stability, without any explicit responsibility for economic activity.²

One exception to the reigning central banking practice is the U.S. Federal Reserve, which since 1977 has been assigned the so-called “dual mandate” which requires it to “promote maximum employment in a context of price stability”. Only as recently as January 2012, the Fed announced an explicit long-run inflation target but also made clear its intention to keep a balanced approach between mitigating deviations of both inflation and employment from target.

Our reading of the academic literature to date, perhaps most importantly the seminal work by Woodford (2003), is that a welfare-maximizing central bank should be assigned only a small weight to measures of economic activity in its objective.³ In the same model framework, Blanchard and Galí (2007) established that stabilizing inflation allows the central bank to simultaneously stabilize welfare-relevant measures of economic activity—a property known as the “divine coincidence”. Taken together, these findings suggest that the strong focus on inflation stabilization by prominent central banks like the ECB is sufficient for macroeconomic stabilization, and that the focus on resource utilization in the Fed’s mandate is redundant or even harmful.

But is a dual mandate really harmful, or could it in fact benefit societies? In this paper we revisit this issue. We argue that a high weight on standard measures of economic activity—e.g. the output gap—is beneficial for society as it serves as an overall proxy for other welfare relevant variables. To make this point, we start with an analysis of standard flexible inflation targeting mandates in the canonical New Keynesian sticky price and wage model of Erceg, Henderson and Levin (2000), EHL henceforth. This model is convenient for our purposes, because it allows us to derive some general analytical results for the output gap weight in a standard inflation-output gap loss function in a framework with both wage and price stickiness.

However, the analysis with the canonical model demonstrates that the merits of a dual mandate depend on the structure of the economy (distortions in goods and labor markets) as well as shocks creating a trade-off between inflation and output gap stabilization. We therefore complement this analysis with numerical simulations in an estimated medium-scale model. Specifically, we use the workhorse Smets and Wouters (2007) empirical model—SW henceforth—of the U.S. economy to examine how a simple objective for the central bank should be designed in order to approximate the welfare of households in the model economy as closely as possible. The SW model is very useful for our purposes, as it represents a prominent example of how the U.S. economy can be described by a system of dynamic equations consistent with optimizing behavior. As such, it should be less prone to the Lucas (1976) critique than other prominent studies on optimal monetary policy that are based on backward-looking models (see e.g. Rudebusch and Svensson, 1999, and Svensson, 1997).⁴ Moreover, many of the existing papers which use models based on optimizing behavior have often relied on simple calibrated models without capital formation.⁵ Even though policy recommendations are model consistent, their relevance may be questioned given the simplicity of these models and the fact that they have not been estimated.

By complementing our analytical results in the canonical EHL model with a normative analysis in an empirically realistic model, this paper provides both theoretically coherent and empirically relevant policy recommendations. We can, for instance, examine if the Federal Reserve’s strong focus on resource utilization improves households’ welfare relative to a simple mandate that focuses more heavily on inflation. Although the exact details of our results are based on U.S. data, we believe they are rather general and relevant for many other economies. Both VAR evidence (see e.g. Angeloni et al., 2003) and estimated New Keynesian models (see e.g. Christiano, Motto and Rostagno, 2010) suggest that the transmission of monetary policy, the structure of the economy, and shocks are very similar in the European economies.

The benchmark to which we compare the simple mandates is Ramsey policy. Even though it is optimal to implement the Ramsey policy directly, the overview of central banking mandates by Reis (2013) and Svensson (2010) shows that no advanced country has asked their central bank to implement such a policy for society. Instead, many central banks are mandated to pursue a simple objective that involves only a small number of economic variables.⁶ We believe there are several important reasons for society to assign the central bank a simple mandate. First, it would be for all practical purposes infeasible to describe the utility-based welfare criterion for an empirically plausible model, as it would include too many targets in terms of variances and covariances of different variables.⁷ Instead, a simple objective facilitates communication of policy actions with the public and makes the conduct of monetary policy more transparent. Second, a simple mandate also enhances accountability of the central bank, which is of key importance. Third and finally, prominent scholars like Svensson (2010) argue that a simple mandate is more robust to model and parameter uncertainty than a complicated state-contingent Ramsey policy.⁸

Our main findings are as follows. First, analytical results within the simple canonical sticky price-wage model of EHL shows that it is often optimal to place a high weight on resource utilization (output gap) in a simple mandate which omits nominal wage inflation and only includes price inflation and the output gap. In particular, this is the case if the sensitivity of the price and wage Phillips curves to the output gap both are of similar magnitude, an assumption which seems to hold empirically (see e.g. Christiano et al., 2010, and Smets and Wouters, 2007). This result is at odds with the conventional wisdom and is essentially driven by the fact that the output gap summarizes the welfare relevant frictions in the goods and labour markets.

Second, in line with the analytical results in the stylized model, our numerical analysis with the SW model shows that a positive weight on any of the typical resource utilization variables like the output gap, the level of output (as deviation from a linear trend), and the growth rate of output improves welfare significantly. And the results hold up when imposing realistic limitations on the extent to which monetary policy makers change policy interest rates. Moreover, among these measures, a suitably chosen weight on the model-consistent output gap delivers the lowest welfare loss. Specifically, we find that in a simple loss function—with the weight on annualized inflation normalized to unity—the optimized weight on the output gap is about 1. This value is considerably higher than the reference value of 0.048 derived in Woodford (2003) and the value of 0.25 assumed by Yellen (2012).⁹ The high weight on the output gap stems from several empirically relevant characteristics in the estimated model that reduce the importance of inflation relative to the output gap. These include a low elasticity of substitution between monopolistic goods, price indexation to lagged inflation by non-optimizing firms, and a low interest rate sensitivity of demand. In addition, the model features real rigidities which enables it to account jointly for the macroeconomic evidence of a low sensitivity of prices (wages) to marginal costs (labor wedge) and the microevidence of frequent price (wage) re-optimization (3-4 quarters). The moderate degree of price stickiness, in turn, makes inflation fluctuations less costly relative to output fluctuations. Our basic finding that the central bank should respond vigorously to resource utilization is consistent with the arguments in Reifschneider, Wascher and Wilcox (2013) and English, López-Salido and Tetlow (2013).

In the SW model, the chosen weight for the output gap has important implications for inflation volatility, as the model features a prominent inflation-output gap trade-off along the efficient frontier as defined in the seminal work of Taylor (1979) and Clarida, Galí and Gertler (1999). At first glance, this inflation-output gap trade-off may appear to be contradictory to Justiniano, Primiceri and Tambalotti (2013), who argue that there is no important trade-off between stabilizing inflation and the output gap. However, the different findings can be reconciled by recognizing that the key drivers behind the trade-off in the SW model—the price- and wage-markup shocks—are absent in the baseline model analyzed by Justiniano et al. (2013).¹⁰ While our reading of the literature is that considerable uncertainty remains about the role of these inefficient shocks as drivers of business cycle fluctuations, we want to stress that our results hold regardless. In particular, if inefficient shocks are irrelevant for business cycle fluctuations, then stabilizing inflation or output is approximately equivalent and attaching a high weight to output is still optimal. And as long as inefficient shocks do play some role—as in SW—then the high weight on output stabilization becomes imperative for the simple mandate to mimic Ramsey policy.

Our third important finding is that a large weight on the output gap remains optimal even when we assume that the gap is measured with significant errors in real time. When we calibrate the size and persistence of the measurement errors for the output gap following the work by Orphanides and Williams (2002), the optimal weight on the output gap drops only modestly from 1.04 to 0.96. In addition, even when the output gap is measured with significant errors in real time, it remains a better target variable than output as deviation from a linear trend or the GDP growth rate. So the common “measurement error” counterargument against targeting the output gap is without merit according to our analysis.

Fourth and finally, in line with Levin et al. (2005), we find that a loss function with nominal wage inflation and the hours gap provides a good, or even better approximation to the household true welfare function than a simple standard inflation-output gap based objective.¹¹ As is the case with the inflation-output gap based simple objective, the hours gap—defined as the difference between actual and potential hours worked per capita—should be assigned a large weight in such a loss function. The reason why targeting labor market variables provides a better approximation of the Ramsey policy is that the labor market in the SW model features large nominal wage frictions and markup shocks, and it is equally or potentially even more important to correct these frictions in factor markets than to correct the distortions in the product markets (sticky prices and price markup shocks).

This paper proceeds as follows. We start in Section 2 by describing how to compute the Ramsey policy and to evaluate the alternative monetary policies. In Section 3, we present the analytical results with the EHL model. In Section 4, we turn to the numerical analysis with the SW model. The robustness of the results in the SW model along some key dimensions is subsequently discussed in Section 5. Finally, Section 6 provides some concluding remarks and suggestions for further research.

2 The Utility-Based Welfare Criterion

We start by deriving and discussing our utility-based welfare criterion. Rotemberg and Woodford (1998) showed that—under the assumption that the steady state satisfies certain efficiency conditions—the objective function of households can be transformed into a (purely) quadratic function using the first-order properties of the constraints. With this quadratic objective function, optimization subject to linearized constraints would be sufficient to obtain accurate results from a normative perspective. Some assumptions about efficiency were unpalatable as exemplified by the presence of positive subsidies that would make the steady state of the market equilibrium equivalent to that of the social planner.¹² Therefore, many researchers—including Benigno and Woodford (2012)—extended the LQ transformation to a general setting without the presence of such subsidies. Benigno and Woodford (2012) demonstrated that the objective function of the households could be approximated by a (purely) quadratic form:

where X_t is a N × 1 vector with the model variables measured as their deviation from the steady state; therefore, $X_t^{\prime }{W^{H}X}_t$ is referred to as the quadratic approximation of the household utility function U(X_t). We discuss the second-order approximation of the utility function and the resource constraints around the distorted steady state in more detail in Appendix A.

Throghout the analysis, we assume that the central bank operates under commitment. We believe this is a good starting point for two reasons. First, the evidence in Debortoli, Maih and Nunes (2014) and Debortoli and Lakdawala (2016), suggests that the Federal Reserve operates with a high degree of commitment, at least before the zero lower bound became binding.¹³ Second, this assumption makes our analysis more comparable with the literature on simple interest rate rules, which also imply some form of central bank commitment.

We define Ramsey policy as a policy that maximizes (1) subject to the N − 1 constraints of the economy. While N is the number of variables, there are only N − 1 constraints provided by the SW model because the monetary policy rule is omitted. Unlike the efficient steady-state case of Rotemberg and Woodford (1998), second-order terms of the constraints do influence the construction of the W^H matrix in (1), and as detailed in Appendix C, we made assumptions on the functional forms for the various adjustment functions (for example, the capital utilization rate, the investment adjustment cost function, and the Kimball aggregators) that are consistent with the linearized behavioral equations in SW.

Since the constant term in (1) depends only on the deterministic steady state of the model, which is invariant across different policies considered in this paper, the optimal policy implemented by a Ramsey planner can be solved as

where the minimization is subject to the N − 1 constraints in the economy, which are omitted for brevity. Following Marcet and Marimon (2012), the Lagrange multipliers associated with the constraints become state variables. Accordingly ${\tilde{X}}_t^{\prime }\equiv [X_t^{\prime },{\varpi }_t^{\prime }]$ now includes the Lagrange multipliers ϖ_t as well. For expositional ease, we denote these laws of motion more compactly as ${\tilde{X}}_t^{*}\left(W^H\right)$.

Using (1) to evaluate welfare would require taking a stance on the initial conditions. Doing so is particularly challenging when Lagrange multipliers are part of the vector of state variables because these are not readily interpretable. We therefore adopt the unconditional expectations operator as a basis for welfare evaluation.¹⁴ The loss under Ramsey optimal policy is then defined by

Our choice of an unconditional expectation as the welfare measure is standard in the literature (see for instance Woodford, 2003). Furthermore, when the discount factor is close to unity—as is the case in our calibration—unconditional and conditional welfare are also quite similar.¹⁵

The Ramsey policy is a useful benchmark. Obviously, in theory a society could design a mandate equal to the Ramsey objective (1). But in practice societies do not; instead, central banks are typically subject to a mandate involving only a few variables. To capture this observation, we assume that a society provides the central bank with a loss function

where W^CB is a sparse matrix with only a few non-zero entries. The matrix W^CB summarizes the simple mandates and will be specified in detail in our analysis. Given a simple mandate, the optimal behavior of the central bank is

When the simple mandate does not coincide with the Ramsey policy, we have that W^CB ≠ W^H and therefore that ${\tilde{X}}_t^{*}\left(W^{CB}\right)\ne {\tilde{X}}_t^{*}\left(W^H\right)$.¹⁶ To compute the extent to which the simple mandate of the central bank approximates optimal policy, one can calculate its associated loss according to the formula:

The welfare performance of the simple mandate is then found by taking the difference between Loss^CB in eq. (6) and Loss^R in eq. (3). In our presentation of the results, we express this welfare difference in consumption equivalent variation (CEV) units as follows:

where $\overline{C}\left(\frac{\partial U}{\partial C}{|}_{s\mathrm{.}s\mathrm{.}}\right)$ can be interpreted as how much welfare increases when consumption in the steady state is increased by one percent. That is, CEV represents the percentage point increase in households’ consumption, in every period and state of the world, that makes them in expectation equally well-off under the simple mandate as they would be under Ramsey policy.¹⁷ Moreover, (7) makes it clear that our choice to neglect the policy-invariant constant in (1) when deriving the Ramsey policy in (2) is immaterial for the results in our paper since all alternative policies are evaluated as difference from the loss under Ramsey.

So far we have proceeded under the assumption that the law governing the behavior of the central bank specifies both the variables and the weights in the quadratic objective, i.e. W^CB in (4). But in practice, the mandates of central banks are only indicative and not entirely specific on the weights that should be attached to each of the target variables. A straightforward way to model this is to assume that society designs a law Ω that constrains the weights on some variables to be equal to zero, without imposing any restriction on the exact weight to be assigned to the remaining variables. When determining the simple mandate consistent with the law Ω, we assume the central bank is benevolent and selects a weighting matrix, W^CB*, which minimizes the expected loss of the society. Formally,

where the weighting matrix W^H is defined by (1).

To sum up, our methodology can examine the performance of simple mandates that central banks are typically assigned with. This statement is true whether the simple mandate specifies both the target variables and the exact weights, or whether the target variables are specified but the weights are loosely defined. In this latter case, our exercise can inform central banks of the optimal weights, and ultimately society about whether bounds on certain weights should be relaxed or not. Finally, it should be noted that our approach described above is similar in spirit to the extensive literature that has studied how simple interest rate rules should be designed to mimic optimal policy as closely as possible; see for example Juillard et al. (2006), Levin et al. (2005), Kim and Henderson (2005), and Schmitt-Grohé and Uribe (2007). The key difference with this literature is that we focus on simple mandates, and the variables that should be included in the simple mandate are not necessarily those that make simple rules mimic the Ramsey policy.

3 Analytical Results in a Canonical New Keynesian Model

This section considers the canonical sticky-price and sticky-wage model with fixed capital by Erceg, Henderson and Levin (2000) to build intuition for the analysis with the workhorse SW model with endogenous capital. The key insight is that stabilizing the output gap also helps stabilize additional welfare relevant variables. For this reason, it is desirable to attach a significant weight to the output gap in simple mandates that do not include all the welfare relevant targets. We show analytically that under certain conditions the weight on the output gap should be infinite. More generally, we derive an approximate expression for the weight on the output gap, which can be easily calculated using a few simple statistics.

The EHL model is characterized by the equations

Eqs. (9) and (10) are the New-Keynesian Phillips curves describing the evolution of price inflation $\left({\pi }_t^p\right)$ and wage inflation $\left({\pi }_t^{\omega }\right)$ as a function of the output gap $y_t^{gap}$, and the real wage gap $\left({\omega }_t^{gap}\right)$. The latter variable, defined in eq. (11), measures the deviation of the actual real wage ω_t from its frictionless counterpart $\left({\omega }_t^n\right)$. β denotes the discount factor, and the composite parameters κ_p and ϑ_p (κ_ω and ϑ_ω) are both inversely related to the probability of the firm (household) not being able to re-optimize its price (nominal wage), implying that their values fall when the degree of price (wage) stickiness increases.¹⁸

The quadratic approximation to the household utility around a non-distorted steady state is given by

where ${\lambda }_w^{opt}\equiv \frac{{∊}_{\omega }(1-\alpha )}{{∊}_p}\frac{{\vartheta }_p}{{\vartheta }_w}$ and ${\lambda }_y^{opt}\equiv ({\sigma }_c+\frac{{\sigma }_l+\alpha }{1-\alpha })\frac{{\vartheta }_p}{{∊}_p}$ denote the weights on wage inflation and output

gap relative to price inflation, whereas the parameters ∊_p and ∊_ω are the elasticities of substitution between goods and labour varieties, respectively, whereas σ_c denotes the intertemporal elasticity of substitution of consumption, σ_l the inverse of the Frisch elasticity of labour supply, and α is the (fixed) capital income share.

For our purposes, we consider that the central bank is assigned the following simple mandate,

which does not include one of the target variables in the social loss function $L_t^R$, namely wage inflation.¹⁹ Next, we study how to select the appropriate weight λ_y so that the actual central bank policy under this simple but suboptimal mandate is as close as possible to the optimal policy (i.e. minimize $L_t^{CB}-L_t^R$).

A critical feature of this economy is that it is not possible to simultaneously stabilize the output gap and the two inflation rates. For example, in response to changes in the natural real-wage—e.g. due to changes in productivity—perfectly stabilizing the output gap requires a change in the real wage, and thus a change in either prices or nominal wages (or both). As a result, as it can be seen from eqs. (9)-(11), it is not feasible to achieve simultaneously $y_t^{gap}=0,{\pi }_t^p=0$, and ${\pi }_t^w=0$.

Nevertheless, combining eqs. (9) and (10) gives that the composite inflation index ${\vartheta }_w{\pi }_t^p+{\vartheta }_p{\pi }_t^{\omega }$ evolves according to

This equation implies that perfectly stabilizing the output gap leads to perfect stabilization of the composite inflation index ${\vartheta }_w{\pi }_t^p+{\vartheta }_p{\pi }_t^{\omega }$, where a higher weight is attached to the inflation rate of the sector of the economy where nominal rigidities are more severe. Thus, stabilizing the output gap also mitigates the costs of nominal rigidities both in the goods and in the labor markets.

In what follows, we study under which circumstances such a policy is actually desirable. Because a complete analytical solution for the optimal simple mandate is infeasible, we present our results in two exercises. First, we solve the dynamic model in a case with equal slope of the price and wage Phillips curves. Second, we solve a static version of the model, with arbitrary slopes of the two Phillips curves.

4 Quantitative Analysis

We now turn to the quantitative analysis within the workhorse model of Smets and Wouters (2007), which is outlined in greater detail in Appendix C. The model includes monopolistic competition in the goods and labor market and nominal frictions in the form of sticky price and wage settings, while allowing for dynamic inflation indexation. It also features several real rigidities: habit formation in consumption, investment adjustment costs, variable capital utilization, and fixed costs in production. The model dynamics are driven by six structural shocks: the two inefficient shocks—a price-markup shock and a wage-markup shock—follow an ARMA(1,1) process, while the remaining four shocks (total factor productivity, risk premium, investment-specific technology, and government spending shocks) follow an AR(1) process. All the shocks are assumed to be uncorrelated, with the exception of a positive correlation between government spending and productivity shocks, i.e. $Corr(e_t^g,e_t^a)={\rho }_{ag}>0$.

Our main departure from the SW model concerns the behavior of the central bank. Rather than considering a Taylor-type interest rate rule and the associated monetary policy shock, we assume that the central bank sets its policies optimally, to best achieve its mandate—i.e. it minimizes its loss function subject to the structural constraints. The model parameters are fixed at the posterior mode of the SW original estimates.²⁴ An alternative approach would be to allow for both parameter and model uncertainty (see e.g. Walsh, 2005). However, we believe it is instructive to start out by performing our exercise in a specific model, under specific parameter values. Throughout the analysis, we discuss the sensitivity of our results to alternative parameterizations.

4.1 Benchmark Results

Table 1 reports our benchmark results. The benchmark simple mandate we consider reflects the standard practice of monetary policy, and is what Svensson (2010) refers to as “flexible inflation targeting.” Specifically, we use the framework in Woodford (2003) and assume that the simple mandate can be captured by the following period loss function

Table 1:

Benchmark Results for “Flexible Inflation Targeting” Mandate in eq. (21).

Note: CEV denotes the consumption equivalent variation (in percentage points) needed to make households indifferent between the Ramsey policy and the simple mandate under consideration according to eq. (7). The “Dual Mandate” refers to a weight of unity for the unemployment gap in the loss function (21), which translates into λ^a = 0.25 when applying a variant of Okun’s law. Finally, “Optimized Weight” refers to minimization of eq. (6) w.r.t. λ^a in eq. (21).

Table 1:

Benchmark Results for “Flexible Inflation Targeting” Mandate in eq. (21).

	xt: Output gap		xt: Output (dev from trend)		xt: Output growth (Ann.)
Simple Mandate	λa	CEV (%)	λa	CEV (%)	λa	CEV (%)
Woodford (2003)	0.048	0.471	0.048	0.554	0.048	0.611
Dual Mandate	0.250	0.140	0.250	0.276	0.250	0.404
Optimized Weight	1.042	0.044	0.542	0.244	2.943	0.302

Note: CEV denotes the consumption equivalent variation (in percentage points) needed to make households indifferent between the Ramsey policy and the simple mandate under consideration according to eq. (7). The “Dual Mandate” refers to a weight of unity for the unemployment gap in the loss function (21), which translates into λ^a = 0.25 when applying a variant of Okun’s law. Finally, “Optimized Weight” refers to minimization of eq. (6) w.r.t. λ^a in eq. (21).

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Table 1:

Benchmark Results for “Flexible Inflation Targeting” Mandate in eq. (21).

	xt: Output gap		xt: Output (dev from trend)		xt: Output growth (Ann.)
Simple Mandate	λa	CEV (%)	λa	CEV (%)	λa	CEV (%)
Woodford (2003)	0.048	0.471	0.048	0.554	0.048	0.611
Dual Mandate	0.250	0.140	0.250	0.276	0.250	0.404
Optimized Weight	1.042	0.044	0.542	0.244	2.943	0.302

Note: CEV denotes the consumption equivalent variation (in percentage points) needed to make households indifferent between the Ramsey policy and the simple mandate under consideration according to eq. (7). The “Dual Mandate” refers to a weight of unity for the unemployment gap in the loss function (21), which translates into λ^a = 0.25 when applying a variant of Okun’s law. Finally, “Optimized Weight” refers to minimization of eq. (6) w.r.t. λ^a in eq. (21).

$\begin{array}{cc}{L}_t^a=({\pi }_t^a-{\pi }^a{)}^2+{\lambda }^a{x}_t^2,& \left(21\right)\end{array}$

where ${\pi }_t^a$ denotes the annualized rate of quarterly inflation and x_t is a measure of economic activity with λ^a denoting its corresponding weight.

We consider three different measures of economic activity. Our first measure is the model-consistent output gap,

$\begin{array}{cc}{y}_t^{gap}=y_t=y_t^{pot},& \left(22\right)\end{array}$

i.e. the difference between actual and potential output where the latter is defined as the level of output that would prevail if prices and wages were fully flexible and inefficient markup shocks were excluded.²⁵ The second measure we consider is simply the level of output (as deviation from the deterministic labor-augmented trend, i.e. $y_t-{\overline{y}}_t$) Finally, we also consider annualized output growth in the spirit of the work on “speed-limit” policies by Walsh (2003).

The first two rows of Table 1 contain a comparison between two benchmark values of λ^a. In the first row of Table 1 we set λ^a = 0.048, corresponding to the welfare-maximizing weight on output-gap in Woodford (2003).²⁶ The second row of Table 1 examines instead the dual mandate. In a recent speech, Yellen (2012) describes the dual mandate through a simple loss function that assigns equal weights for annualized inflation and the unemployment gap (i.e. actual unemployment minus the NAIRU).²⁷ In addition, Yellen stipulates that the Federal Reserve converts the unemployment gap into an output gap according to a value of roughly 0.5—and such a value is based on the widely spread empirical specification of the Okun’s law $u_t-u_t^{pot}=(y_t-y_t^{pot})/2$. Accordingly, the unit weight on the unemployment gap converts into a weight of λ^a = 0.25 on the output gap.²⁸

As we can see from the second row in Table 1, increasing the weight on real activity from the value of Woodford (2003) to the value consistent with the dual mandate significantly reduces welfare losses, namely by a factor of three for our benchmark measure of economic activity (the output gap), and by about a factor of two for alternative measures (output level and output growth). In all cases, the welfare gains are large compared to similar studies in the monetary policy literature—e.g. larger than the threshold value of 0.05% used by Schmitt-Grohe and Uribe (2007).

The last row in Table 1 displays the results when the weight λ^a is optimized. The key finding is that the optimal value of λ^a is much higher than the values considered so far, for all measures of economic activity. For example, the optimized coefficient for the output gap is 1.042. Coincidentally, this is very similar to the unit weight on the unemployment gap as used in Yellen (2012). For the level of output (as deviation from trend), the optimized coefficient is lower (0.5) but still twice as high as implied by the dual mandate. In the case of output growth, the optimized coefficient is even higher (around 2.9), which essentially is a so-called speed-limit regime (see Walsh, 2003). Notably, our analysis shows that adopting a simple mandate with a high weight on any of the resource utilization measures improves welfare with respect to considering the model-based output gap but assigning to it a low weight—e.g. as in Woodford (2003).²⁹ This is since assigning a high weight to detrended output or output growth in the loss function helps reducing considerably the volatility of output gap, albeit not to the same extent as when targeting it directly.

To gauge the sensitivity of the CEV with respect to the weight assigned to resource utilization, Figure 1 plots the CEV as a function of λ^a for the three resource measures. Consistent with the results in Table 1, we see that there is quite some curvature of the CEV function for small values of λ^a for all three measures. Moreover, for the output gap we see that values of λ^a between 0.5 and 1.5 perform about equally well, whereas the mandate with detrended output has a higher curvature near the optimum. For output growth, the figure shows that any value above unity yields virtually the same CEV.

Consumption Equivalent Variation (percentage points) as Function of the Weight (λ^a) on Economic Activity.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the CEV (in %) for the simple mandate with inflation and: output gap (left panel), output level (middle panel), output growth (right panel) The coordinate with an ‘x’ mark shows the CEV for λ^a = 0.01, the ‘o’ mark shows the CEV for the optimized weight.

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Consumption Equivalent Variation (percentage points) as Function of the Weight (λ^a) on Economic Activity.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the CEV (in %) for the simple mandate with inflation and: output gap (left panel), output level (middle panel), output growth (right panel) The coordinate with an ‘x’ mark shows the CEV for λ^a = 0.01, the ‘o’ mark shows the CEV for the optimized weight.

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Consumption Equivalent Variation (percentage points) as Function of the Weight (λ^a) on Economic Activity.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the CEV (in %) for the simple mandate with inflation and: output gap (left panel), output level (middle panel), output growth (right panel) The coordinate with an ‘x’ mark shows the CEV for λ^a = 0.01, the ‘o’ mark shows the CEV for the optimized weight.

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To clarify the mechanism behind our results, we follow Taylor (1979), Erceg, Henderson and Levin (1998), and Clarida et al. (1999) and study the main trade-offs involved in stabilizing measures of inflation vs. measures of economic activity through variance frontiers. Figure 2 plots the variance of price or wage inflation (horizontal axis), together with measures of economic activity (vertical axis), while letting the weight λ^a vary from a small (0.01) to a large value (5.00). The slope of the resulting curve is referred to as the trade-off between the two variances. The upper panels refer to the benchmark loss function with price inflation and output gap. Panel A shows that there is a clear trade-off between stabilizing price inflation and the output gap. Indeed, a lower volatility of output gap is always associated with an increase in the volatility of price inflation. Instead, Panel B shows that there is not necessarily a trade-off between stabilizing output gap and wage inflation. For example, as long as λ^a < 0.1, reducing the volatility of output gap also reduces the volatility of wage inflation. In other words, and consistently with our theoretical results of Section 3, Figure 2 shows that increasing the weight λ^a up to a value of 0.1 helps stabilizing not only the output gap, but also wage inflation—i.e. a welfare relevant variable not explicitly targeted by the central bank in its loss function. In fact, the volatility of nominal wage inflation remains lower relative to a benchmark strict inflation targeting loss function (λ^a = 0.01) for values of λ^a up to 0.4 (not shown in Panel B). This explains why in this economy measures of economic activity should receive a relatively high weight in a central bank’s simple mandate that does not include all the welfare relevant targets.

Variance Frontier for Alternative Resource Utilization Measures.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the variance frontier for the simple mandate with inflation and: output gap (Panel A), output level (Panel C), output growth (Panel D). Panel B shows the variance combination of the output gap and the annualized nominal wage inflation when varying λ^a for the price inflation-output gap loss function (i.e. same loss function as in Panel A). The coordinate with an ‘×’ mark shows the volatility for λ^a = 0.01, the ‘o’ mark shows the volatility for the optimized weight, and the ‘+’ mark shows the volatility for λ^a = 5.

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Variance Frontier for Alternative Resource Utilization Measures.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the variance frontier for the simple mandate with inflation and: output gap (Panel A), output level (Panel C), output growth (Panel D). Panel B shows the variance combination of the output gap and the annualized nominal wage inflation when varying λ^a for the price inflation-output gap loss function (i.e. same loss function as in Panel A). The coordinate with an ‘×’ mark shows the volatility for λ^a = 0.01, the ‘o’ mark shows the volatility for the optimized weight, and the ‘+’ mark shows the volatility for λ^a = 5.

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Variance Frontier for Alternative Resource Utilization Measures.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the variance frontier for the simple mandate with inflation and: output gap (Panel A), output level (Panel C), output growth (Panel D). Panel B shows the variance combination of the output gap and the annualized nominal wage inflation when varying λ^a for the price inflation-output gap loss function (i.e. same loss function as in Panel A). The coordinate with an ‘×’ mark shows the volatility for λ^a = 0.01, the ‘o’ mark shows the volatility for the optimized weight, and the ‘+’ mark shows the volatility for λ^a = 5.

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The lower panels of Figure 2 plot variance frontiers when the measure of economic activity is given by the output level and output growth both in the loss function and the frontier itself. Panel D in the figure shows that the trade-off between stabilizing inflation and economic activity is most favorable when the resource utilization measure is output growth; the variance of annualized output growth can be reduced to nearly 1 percent without $\text{Var}\left({\pi }_t^a\right)$ increasing by much. Moreover, the flatness of the CEV witnessed in the right panel of Figure 1 for values of λ^a higher than optimal can be readily explained by the fact that panel D in Figure 2 shows that such values induce only small changes in the volatilities of inflation and output growth. For detrended output shown in panel C, the figure shows that the trade-off is most pronounced. Accordingly, values of λ^a higher than optimal translate into a higher curvature of the CEV function in Figure 1.

As noted in Section 2, a strength of the methodology used in this paper is that it can handle a non-efficient steady state. The results in Table 1 and Figure 1, however, are robust to allowing for subsidies to undo the steady-state distortions stemming from the presence of external habits, as well as firms’ and households’ monopoly power in price and wage setting. For detrended output and the output gap, the optimized weights are even larger when considering the efficient steady state; for example, λ^a equals 2.34 with an associated CEV of 0.0119 for the output gap when the steady state is efficient. For output growth, however, the optimized λ^a is notably lower (0.43). But given the flatness of the CEV function in Figure 1, it is not surprising that the exact weight for output growth can be somewhat sensitive to the specific assumptions. Even so, the optimized weight remains relatively large, reflecting the larger curvature for smaller values of λ^a. In principle, moving from a distorted to an efficient steady state could make a big difference when we consider a model with relatively large distortions in both goods and labor markets. However, in our model, the surge in steady state output when removing these distortions are to a large extent offset by removing external habit formation, so the efficient steady state level for output is only about 6 percent higher than our distorted steady state.³⁰

4.2 The Importance of Real Activity

The key message from Table 1 is that the rationale for including some measure of real activity in the central bank’s objective is much more important than previously recognized either in policy circles or in previous influential academic work (e.g. Woodford (2003) and Walsh (2005)). By perturbing some of the parameters and shocks in the model, we seek to nail down why the model suggests that a high weight on real economic volatility improves household welfare.

We begin the analysis by using the SW parameters in Table A.1 to recompute λ^a according to the analytic formula provided in Woodford (2003):

$\begin{array}{cc}{\lambda }^a\equiv \frac{16{\kappa }_x}{\left(\frac{{\varphi }_p}{{\varphi }_p-1}\right)},& \left(23\right)\end{array}$

where κ_x is the coefficient for the output gap in the linearized pricing schedule (i.e. in the New Keynesian Phillips curve), and $\frac{{\varphi }_p}{{\varphi }_p-1}$ is the elasticity of demand of intermediate goods. In the SW model, the NKPC is given by

$\begin{array}{cc}{\pi }_t-{\iota }_p{\pi }_{t-1}=\beta {\gamma }^{1-{\sigma }_c}({\text{E}}_t{\pi }_{t+1}-{\iota }_p{\pi }_t)+\frac{(1-\beta {\gamma }^{1-{\sigma }_c}{\xi }_p)(1-{\xi }_p)}{{\xi }_p(({\phi }_p-1){\epsilon }_p+1)}m{c}_t+{\varepsilon }_t^p\mathrm{.}& \left(24\right)\end{array}$

However, because the SW model features endogenous capital and sticky wages, there is no simple mapping between the output gap and real marginal costs within the fully fledged model. But by dropping capital and the assumption of nominal wage stickiness, we can derive a value of κ_x = 0.143 in the simplified SW model.³¹ From the estimated average markup ϕ_p, we then compute λ^a = 0.87. This value is considerably higher than Woodford’s (2003) value of 0.048 mainly for four reasons. First, the estimated gross markup in SW (1.61) implies a substantially lower substitution elasticity $(\frac{{\phi }_p}{{\phi }_p-1}=2.64)$ compared to Woodford’s value (7.88). If we replace Woodford’s value with the one estimated by SW, λ^a in eq. (23) rises to 0.30. Second, if we replace Woodford’s value of the intertemporal substitution elasticity (6.25) with the value estimated by SW (1.39), λ^a increases further to 0.59. Third, if we relax the assumption of firm-specific labor (the Yeoman-farmer model of Rotemberg and Woodford, 1997), we have that λ^a equals 0.80. The remaining small difference to the SW value (0.87) can largely be explained by the slightly higher degree of price stickiness in Woodford’s calibration.

Fourth and finally, real rigidities in the form of the Kimball aggregators for prices and wages play an important role as they enable the SW model to fit the macroevidence of a low sensitivity of price (wage) inflation to marginal costs (labor wedge) while at the same time be consistent with the microevidence suggesting frequent price (and wage) re-optimization every 3-4 quarters (see e.g. Klenow and Malin, 2010, and Nakamura and Steinsson, 2013). Had the estimated Smets and Wouters (2007) model not included this feature, the price stickiness parameter would have been considerably higher (about 0.9 as in Smets and Wouters, 2003), and the optimal weight on the output gap considerably lower (about 0.05 according to eq. 23). But again, such a high degree of price stickiness is at odds with the microevidence, and the very reason why the SW model features Kimball aggregators.

It is important to recognize that this analysis is only suggestive as it omits some of the key features—wage stickiness and endogenous capital—in the fully fledged model. The analysis in Section 3 demonstrated that the optimal λ^a depends on the relative degree of structural frictions in goods and labor markets. As a consequence, the obtained λ^a above will only partially reflect the true structure of the fully fledged SW model. Even so, the analysis suggests that a large part of the gap between Woodford’s (2003) value and our benchmark finding of λ^a = 1.042 in the output gap case stems from differences in household preferences (intertemporal substitution elasticity of consumption and Yeoman-Farmer assumption) and the estimated substitution elasticity between intermediate goods.

With these results in mind, we turn to exploring the mechanisms within the context of the fully fledged model. Our approach is to turn off or reduce some of the frictions and shocks featured in the model one at a time to isolate the drivers of the results. The findings are provided in Table 2. The first row restates the baseline results with the optimized weight. The second row presents the optimized weight on the real-activity term when dynamic indexation in price- and wage-setting is shut down, i.e. ι_p and ι_w are calibrated to zero. All the other parameters of the model are kept unchanged. As can be seen from the table, the calibration without indexation lowers the optimized weight for the output gap to roughly 0.3—about a third of the benchmark value. In the other columns where real activities are captured by the level and the growth rate of detrended output, the optimized weights are also found to be about a third of the benchmark values.

Table 2:

Perturbations of the Benchmark Model.

Note: “No Indexation” refers to setting ι_p = ι_w = 0; “No ${\varepsilon }_t^p\left({\varepsilon }_t^w\right)$ Shocks” refers to setting the variance of the price markup shock (wage markup shock) to zero; “Small ${\varepsilon }_t^w$ and ${\varepsilon }_t^p$ Shocks” means that the std. of these shocks are set to a 1/3 of their baseline values; and “No ${\varepsilon }_t^w$ and ${\varepsilon }_t^p$ Shocks” refers to setting the variance of both shocks to zero. “Large”means that the optimized value is equal or greater than 5.

Table 2:

Perturbations of the Benchmark Model.

	xt: Output gap		xt: Output (dev from trend)		xt: Output growth (Ann.)
Simple Mandate	λa	CEV (%)	λa	CEV (%)	λa	CEV (%)
Benchmark	1.042	0.044	0.542	0.244	2.943	0.302
No Indexation	0.318	0.042	0.179	0.220	0.817	0.285
No ${\varepsilon }_t^p$ Shocks	0.914	0.039	0.343	0.220	1.235	0.278
No ${\varepsilon }_t^w$ Shocks	2.094	0.020	0.355	0.213	1.267	0.226
Small ${\varepsilon }_t^p$ and ${\varepsilon }_t^w$ Shocks	1.268	0.024	0.112	0.167	0.157	0.180
No ${\varepsilon }_t^p$ and ${\varepsilon }_t^w$ Shocks	Large	0.016	0.161	0.150	0.025	0.134

Note: “No Indexation” refers to setting ι_p = ι_w = 0; “No ${\varepsilon }_t^p\left({\varepsilon }_t^w\right)$ Shocks” refers to setting the variance of the price markup shock (wage markup shock) to zero; “Small ${\varepsilon }_t^w$ and ${\varepsilon }_t^p$ Shocks” means that the std. of these shocks are set to a 1/3 of their baseline values; and “No ${\varepsilon }_t^w$ and ${\varepsilon }_t^p$ Shocks” refers to setting the variance of both shocks to zero. “Large”means that the optimized value is equal or greater than 5.

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Table 2:

Perturbations of the Benchmark Model.

	xt: Output gap		xt: Output (dev from trend)		xt: Output growth (Ann.)
Simple Mandate	λa	CEV (%)	λa	CEV (%)	λa	CEV (%)
Benchmark	1.042	0.044	0.542	0.244	2.943	0.302
No Indexation	0.318	0.042	0.179	0.220	0.817	0.285
No ${\varepsilon }_t^p$ Shocks	0.914	0.039	0.343	0.220	1.235	0.278
No ${\varepsilon }_t^w$ Shocks	2.094	0.020	0.355	0.213	1.267	0.226
Small ${\varepsilon }_t^p$ and ${\varepsilon }_t^w$ Shocks	1.268	0.024	0.112	0.167	0.157	0.180
No ${\varepsilon }_t^p$ and ${\varepsilon }_t^w$ Shocks	Large	0.016	0.161	0.150	0.025	0.134

Note: “No Indexation” refers to setting ι_p = ι_w = 0; “No ${\varepsilon }_t^p\left({\varepsilon }_t^w\right)$ Shocks” refers to setting the variance of the price markup shock (wage markup shock) to zero; “Small ${\varepsilon }_t^w$ and ${\varepsilon }_t^p$ Shocks” means that the std. of these shocks are set to a 1/3 of their baseline values; and “No ${\varepsilon }_t^w$ and ${\varepsilon }_t^p$ Shocks” refers to setting the variance of both shocks to zero. “Large”means that the optimized value is equal or greater than 5.

To understand why indexation makes the real-activity term much more important than in a model without indexation, it is instructive to consider the simple New Keynesian model with indexation and sticky prices only briefly discussed in Section 3.3. As shown by Woodford (2003), this model generates the following approximated welfare-based loss function,

$\begin{array}{cc}({\pi }_t-{\iota }_p{\pi }_{t-1}{)}^2+\lambda (y_t^{gap}{)}^2,& \left(25\right)\end{array}$

where ι_p is the indexation parameter in the pricing equation. Suppose further, for simplicity, that inflation dynamics in equilibrium can be represented by an AR(1) process π_t = ρπ_t–1+ε_t. In this simple setup, the welfare metric can be expressed as

$\begin{array}{cc}{\text{E}}_0\left[(\rho -{\iota }_p{)}^2({\pi }_{t-1}{)}^2+\lambda (y_t^{gap}{)}^2\right],& \left(26\right)\end{array}$

where the error term ε_t is absorbed into terms independent of policy. In more empirically relevant models like SW, inflation persistence (ρ) is in large part explained by the indexation parameters (ι_p and, in our sticky-wage framework, ι_w matter as well). Therefore, these two parameter values tend to be similar and the coefficient on the inflation term is accordingly smaller. Hence, in a loss function like ours (eq. 21) where the inflation coefficient is normalized to unity, the coefficient on real activity tends to become relatively larger—as evidenced in Table 1. Intuitively, in economies where prices have a component indexed to their lags, the distortions arising from inflation are not as severe. Consequently, there is less need to stabilize inflation.

Notably, even when we remove indexation to lagged inflation in price and wage settings, the optimal value of λ^a still suggests a very large role for targeting economic activity; in fact, the optimal value is still slightly higher than the value implied by the dual mandate.³² Moreover, one can observe from Figure 3 that dropping dynamic indexation is associated with a rather sharp deterioration in the CEV when λ^a is below 0.2. This finding suggests that a vigorous response to economic activity is indeed important even without indexation. This is a reassuring result. Even though SW showed that excluding indexation to lagged inflation in price and wage setting is associated with a deterioration in the empirical fit (i.e. reduction in marginal likelihood) of the model, there is only weak support of indexation in micro-data.

CEV (in percentage points) as Function of λ^a for Alternative Calibrations.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the CEV (in %) as a function of λ^a for three different calibrations. The solid line refers to the benchmark calibration. The dotted line refers to the calibration in which ι_p = ι_w = 0. The dashed line refers to the calibration in which $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$.

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CEV (in percentage points) as Function of λ^a for Alternative Calibrations.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the CEV (in %) as a function of λ^a for three different calibrations. The solid line refers to the benchmark calibration. The dotted line refers to the calibration in which ι_p = ι_w = 0. The dashed line refers to the calibration in which $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$.

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CEV (in percentage points) as Function of λ^a for Alternative Calibrations.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the CEV (in %) as a function of λ^a for three different calibrations. The solid line refers to the benchmark calibration. The dotted line refers to the calibration in which ι_p = ι_w = 0. The dashed line refers to the calibration in which $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$.

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Rows 3–6 in Table 2 examine the role of the inefficient markup shocks in the model. By comparing the CEV results in the third and fourth rows, we see that the wage markup shock contributes the most to the welfare costs of the simple mandate. But the key point is that even when one of these shocks is taken out of the model, the central bank should still respond vigorously to economic activity in order to maximize household welfare. Only when the standard deviations of both shocks are reduced or taken out completely (rows 5 and 6), λ^a falls for output and output growth. For the loss function with the model-consistent output gap, the weight λ^a is large when shocks are reduced (row 5), and is even larger when the standard deviations of both inefficient shocks are set to nil (row 6).

When both shocks are set to nil, any λ^a > 0.1 produces roughly the same CEV of about 0.016 although a λ^a ≥ 5 generates the lowest welfare loss relative to Ramsey as can be seen from Figure 3. This finding is supported by our analytical results in Section 3, which established that the weight on the output gap should be very high in a simple mandate like eq. (21) when the distortions in goods and labor markets are of similar magnitude. Even so, the flatness of the CEV as a function of λ^a in Figure 3 shows that there is only a weak trade-off between inflation and output gap stabilization in the absence of price- and wage-markup shocks. This suggests that the divine coincidence property holds approximately in this case, implying that the weight on the output gap is largely inconsequential. As an alternative to cutting the size of the markup shocks, we reduced the steady steady state gross markups from 1.61 (ϕ_p) and 1.5 (ϕ_ω), respectively, to 1.20 following the evidence in Christiano, Eichenbaum and Evans (2005). Also under this parametrization, we find that a large weight on economic activity is optimal. For instance, the optimized λ^a for the output gap equals 1.01.

In Figure 4, we depict variance frontiers when varying λ^a from 0.01 to 5 for alternative calibrations of the model. We also include the implied $\{\text{Var }\left({\pi }_t^a\right),\text{Var }\left(y_t^{gap}\right)\}$ combinations under Ramsey policy and the estimated SW policy rule with all shocks (marked by black ‘x’ and ‘+’ marks, respectively) and without the inefficient shocks (the blue ‘x’ and ‘+’ marks). As expected, we find that both the estimated SW rule and the Ramsey policy are outside the variance frontier associated with the simple mandate (solid black line), but the locus of $\{\text{Var }\left({\pi }_t^a\right),\text{Var }\left(y_t^{gap}\right)\}$ for the optimized λ^a is very close to the Ramsey policy. We interpret this finding as providing a strong indication that the simple mandate approximates the Ramsey policy well in terms of the actual paths of output gap and inflation, and not just in terms of CEV as seen from the results for the output gap in Table 1.^33,34 The locus of the estimated SW rule is a bit further away from the optimized simple mandate with λ^a = 1.042, and is associated with a higher (lower) variance of the output gap (price inflation).

Variance Frontiers for Alternative Calibrations.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the variance frontier for several calibrations: benchmark (solid line), $var\left({\varepsilon }_t^p\right)=0$ (dotted line), $var\left({\varepsilon }_t^w\right)=0$ (dashed-dotted line), and $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$ (dashed line). The ‘o’ mark shows the volatility for the optimized weight and benchmark calibration. The coordinates with an ‘x’ and the ‘+’ mark denote the Ramsey and SW policy rule, respectively. The box in the graph zooms in the case with $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$.

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Variance Frontiers for Alternative Calibrations.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the variance frontier for several calibrations: benchmark (solid line), $var\left({\varepsilon }_t^p\right)=0$ (dotted line), $var\left({\varepsilon }_t^w\right)=0$ (dashed-dotted line), and $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$ (dashed line). The ‘o’ mark shows the volatility for the optimized weight and benchmark calibration. The coordinates with an ‘x’ and the ‘+’ mark denote the Ramsey and SW policy rule, respectively. The box in the graph zooms in the case with $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$.

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Variance Frontiers for Alternative Calibrations.

Citation: IMF Working Papers 2017, 164; 10.5089/9781484309278.001.A001

Note: The figure plots the variance frontier for several calibrations: benchmark (solid line), $var\left({\varepsilon }_t^p\right)=0$ (dotted line), $var\left({\varepsilon }_t^w\right)=0$ (dashed-dotted line), and $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$ (dashed line). The ‘o’ mark shows the volatility for the optimized weight and benchmark calibration. The coordinates with an ‘x’ and the ‘+’ mark denote the Ramsey and SW policy rule, respectively. The box in the graph zooms in the case with $var\left({\varepsilon }_t^w\right)=var\left({\varepsilon }_t^p\right)=0$.

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Turning to the role of the markups shocks, we see that the trade-off between inflation and output gap volatility is reduced notably but still remains sizeable when we set the standard deviation of the wage markup shocks to nil (dash-dotted green line) following the baseline model of Justiniano et al. (2013). The reason that the central bank has to accept a higher degree of inflation volatility in order to reduce output gap volatility in this case is that we still have the price markup shock active in the model. The price markups, however, are less important for the trade-off as can be seen from the red-dotted line, which demonstrates a notable trade-off with only wage markup shocks in the model. Only when both the inefficient shocks are excluded, the trade-off is relatively small (dashed blue line in Figure 4, shown in more detail in the small inset box).

Since substantial uncertainty remains about the importance of markup shocks over the business cycle, it is important to consider the likely case where at least a small proportion of the observed variation in inflation and wages is in fact driven by inefficient price- and wage-markup shocks. The fifth row in Table 2 reports results in which the standard deviations of both the inefficient shocks have been set to a third of their baseline values. For the wage-markup shock, this alternative calibration can be motivated by the empirical work by Galí, Smets and Wouters (2011), who can distinguish between labor supply and wage markup shocks by including the unemployment rate as an observable when estimating a model similar to the SW model. For the price markup shock, our choice is more arbitrary and follows Justiniano et al. (2013) by assuming that almost 90 percent of the markup shock variances are in fact variations in the inflation target.³⁵

Even in this case, the table shows that the resulting λ^a is still high for the output gap. The reason is that if all shocks are efficient then a high λ^a is still optimal (recall Figure 3), and if some shocks are indeed inefficient then a high λ^a is required. Therefore, a high weight λ^a is a robust choice if there is uncertainty about the inefficiency of the shocks; a high weight is optimal but not economically important when all shocks are efficient, and optimal and economically meaningful as long as a small proportion of the shocks is indeed inefficient.

This analysis clarifies that our results and those of Justiniano et al. (2013) are not in contradiction. First, note that the analysis is not directly comparable because their no trade-off result refers to a shift from the estimated historical rule to Ramsey policy, while our focus is on different weights in the objective function. Second, and as just explained, as long as there is uncertainty about the inefficiency of the shocks and the presence of a trade-off, a high weight λ^a is a robust choice. In the next section, we show that this basic result holds up for some key modifications of the analysis.