# Who controls monetary policy

### Optimal-Control Monetary Policy in the FRB/US Model

Flint Brayton, Thomas Laubach. and David Reifschneider

**Introduction**

The question of how best to conduct monetary policy has been studied by economists for a long time. Over the past 25 years or so, attention has focused on systematic approaches to setting the short-term interest rate in a manner that effectively balances policymaker objectives. One area of research on this topic has studied simple feedback rules; another has examined optimal-control (OC) techniques. Interest rate projections based on both of these methods have for some time been regularly reported to the Federal Open Market Committee (FOMC) of the Federal Reserve to help inform its policy decisions. 1 While the economic consequences of applying these methods have been evaluated within many different models, these models are by necessity an abstraction of a much more complex economic reality, and hence the actual strategies followed by the Federal Reserve and other central banks necessarily retain an important judgmental component.

In a simple feedback rule, the central bank's policy interest rate responds to movements in a small number of macroeconomic factors, such as the current amount of labor market slack and the deviation of the rate of inflation from its target. The precise definitions of these factors and the magnitudes of their response coefficients can be chosen to provide the best obtainable outcome with respect to policymaker objectives for a specific macroeconomic model, or they can be chosen to provide good outcomes across a range of plausible models. 2 Under an OC policy, the current and expected future path of the policy interest rate is instead typically calculated with a procedure that minimizes an intertemporal policymaker loss function subject to a specific model of the dynamics of the macroeconomy and a baseline outlook summarizing the exogenous forces affecting the economy going forward. Depending on the size of the model, an OC interest rate path may be a function of many more macroeconomic variables than is the case in a simple feedback rule. OC policies can be calculated under two alternative assumptions. Under the first, policymakers commit to a plan and its intended effects on real activity and inflation, thereby constraining their future policy actions. Under the second, policymakers do not view themselves as constrained by past policy plans and so optimize on a period-by-period basis. The OC policies presented in this note are of the commitment type.

Both feedback rules and OC policies provide useful benchmarks for the deliberations of monetary policymakers. Simple feedback rules are relatively easy to understand and communicate to the public, and potentially robust to uncertainty about the structure of the macroeconomy. While OC policies are more complex and hence likely to be less robust, they nonetheless lead to expected outcomes that are the best obtainable under the assumptions with which they are constructed. Moreover, unusual situations may arise that are not well suited to the design of simple feedback rules (which are generally intended to moderate typical cyclical disturbances), but instead favor the fine-tuning specificity of OC policies. Arguably, this has been the case in recent years, with persistently weak real activity, inflation generally running below the Federal Reserve's long-run goal, and the federal funds rate at its effective lower bound since late 2008. In these atypical circumstances, the OC approach under commitment is able to illustrate the potential benefits of signaling that the central bank intends to pursue an accommodative policy for a longer time into the future than would ordinarily be expected, thereby supporting real activity and inflation today through expectations of lower real interest rates and better economic conditions in the future (English et al. 2013). 3

These benefits notwithstanding, another theme of this note is that, while useful, OC policies need to be treated with caution. In practical terms, implementation of the OC policy approach requires a specific policymaker loss function, a specific macroeconomic model that relates the goal variables of policy to short-term interest rates, and, as the approach is implemented in staff material prepared for the FOMC, an initial projection for real activity, inflation, and other factors conditioned on some path for short-term interest rates. The sensitivity of OC policy prescriptions to variations in all these assumptions is a limitation of this approach. 4

This note presents interest rate paths constructed using the Federal Reserve Board's FRB/US model in conjunction with baseline forecasts of real activity, inflation and interest rates that are consistent with the FOMC's Summary of Economic Projections (SEP) as published at different points in time. In all cases, the OC policy path satisfies the effective lower bound on nominal interest rates. 5 We illustrate the dependence of estimates of OC policies on the three key ingredients listed above by considering results generated using different specifications of the loss function, different characterizations of the structure of the economy as embedded in alternative versions of the FRB/US model, and initial projections made at different points in time.

**Optimal-Control Policy: A Baseline Case**

The description of the interactions over time among the policy instrument--in the FOMC's case, the federal funds rate--and the goal variables of policy is at the heart of OC policy computations. We therefore start by highlighting a few key features of the model we use for this exercise, the FRB/US model. While the precise paths that we present are clearly specific to our choice of model, the main qualitative insights are valid in many models that are currently used to analyze the effects of monetary policy on real activity and inflation. More detailed information about the FRB/US model is available elsewhere. 6

Monetary policy most directly affects real activity and inflation in FRB/US through its effects on longer-term real interest rates. Several components of aggregate demand, such as consumer purchases of motor vehicles and other durable goods, business fixed investment, and residential construction, are modelled as

depending on various real interest rates with maturities that range between 5 and 30 years. Therefore, expectations of future short-term interest rates many years into the future, together with long-horizon expectations of inflation, play an important role in the transmission of monetary policy in FRB/US. Other channels of monetary transmission, such as effects on the value of households' equity holdings and the foreign exchange value of the dollar, also depend on expectations of short-term interest rates far into the future. Finally, consumption and investment depend on expectations of future household income and business sales. Monetary policy thus can have substantial effects on current real activity and inflation to the extent it can affect expectations of the future course of interest rates, inflation, and real activity. A crucial assumption of the simulation results presented in this note is that financial market participants and wage-and-price setters fully understand the central bank's monetary strategy and its implications for the future evolution of the economy, including the path of the federal funds rate. 7 In addition, and also crucially, these agents view the OC strategy as a credible commitment to which policymakers will adhere in the future. 8

Turning to the objective function of monetary policymakers, let *t* = 0 denote the quarter in which policymakers choose an optimal path for the federal funds rate *i _{t}* . defined as the path which minimizes the expected value of a specified loss function at quarter 0, denoted

*L*.

_{t}In this expression, *π _{t}* denotes the inflation rate in quarter

*t*(defined as the four-quarter percent change of the PCE deflator),

*u*the unemployment rate,

_{t}*u **its longer-run normal value, and

*Δi*the first difference of the federal funds rate. Because all three arguments in this loss function are squared, losses equal zero only when inflation is at 2 percent, the unemployment rate is at

_{t}*u **. and the nominal federal funds rate is constant; otherwise, losses are positive. Because policymakers have only one instrument to hit three targets, losses are almost inevitable although they can be minimized by selecting the appropriate interest rate path.

This loss function embeds a number assumptions. The objectives of policy are to stabilize inflation around 2 percent and the unemployment rate around *u **. For the baseline case, we assume equal weights on both arguments. This formulation may be seen as consistent with the Federal Reserve's mandate to promote maximum employment and price stability--where the latter has been judged consistent with 2 percent inflation by the FOMC--while the term penalizing changes in the federal funds rate reflects a desire to avoid abrupt changes in the policy instrument. 9 The objective function is symmetric around the target values, in that a given deviation of one variable to one side of its target value is as costly as the same-sized miss to the other side. In addition, future losses are discounted at a quarterly rate of 1 percent; thus, a given loss projected to occur 40 quarters into the future (such as a one-percentage-point departure of the unemployment rate from its longer-run normal value) is only two-thirds as costly as the same-sized loss incurred in the current period.As all the baselines used in our analysis assume that the economy converges over time to a long-run equilibrium in which the inflation rate is 2 percent, the unemployment rate settles at *u **. and the federal funds rate remains constant at its longer-run value, beyond a certain point (called *T* in the above summation) expected losses are zero. Thus, we evaluate the loss function for only 20 years into the future, sufficient time for the economy to have essentially settled back into its long-run equilibrium. 10

The remaining ingredient needed to compute an OC interest rate path is an initial projection for the economy that summarizes, among other things, the exogenous forces expected to affect the economy over the optimization horizon. For illustrative purposes, we begin by constructing a model forecast that starts in the third quarter of 2012 and is consistent with the SEP projections of real activity, inflation, and the federal funds rate made by FOMC participants at the time of its September 2012 meeting. 11 This time period is of interest in part because it corresponds with that used to generate the OC policy path reported in a speech given by then Vice-Chair Yellen in November of that year. 12 For the unemployment rate, PCE price inflation, core PCE price inflation, and real GDP growth, the initial projection matches the midpoint of the central tendency reported in the September 2012 SEP for the years 2012 through 2015; thereafter, these variables converge to the midpoints of the central tendency of their longer-run values. Similarly, the initial projection for the federal funds rate matches the median projection from the September 2012 SEP at year-end 2012 to 2015 (adjusted to a quarterly average basis), and thereafter converges to the median projection of its longer-run normal value. 13

Figure 1 presents two sets of paths for the federal funds rate, the real yield on 10-year Treasury notes, the unemployment rate, and 4-quarter PCE inflation--one consistent with the September 2012 SEP and one based on OC policy. In both cases, the real 10-year Treasury yield at each point in time is calculated to be consistent with the relevant (perfectly-anticipated) paths of the federal funds rate and inflation over the next 10 years, plus an assumption about the evolution of the term premium beyond late 2012; because spending decisions in FRB/US depend importantly on longer-term real interest rates, the real 10-year Treasury yield is a useful indicator of the transmission of monetary policy to real activity and inflation. In the OC policy simulation we assume that the loss function weights on the three terms are all equal to 1.

Figure 1: Predicted Outomes Under OC Policy Starting in Late 2012

Source: www.federalreserve.gov

Category: Bank

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