The effect of monetary policy inertia on the optimal monetary policy: a note
Abstract
This paper theoreticallyinvestigates the influence of monetary policy inertia on the optimal policy under both certainty equivalence and uncertainty. Moreover, it highlights the consequences of this effect. We utilize Williams’ (2013) simple static macroeconomic model, and modify it to match the goal of the present paper. We conclude that the effect of monetary policy inertia under certainty equivalence or uncertainty are identical. Its effect on the economy depends on the degree of the inertia. If the economy is close to the full inertia, then the economy has fewer fluctuations. Moreover, monetary policy inertia will be optimal if the inertia is full.
JEL Classification: E52, E58
Keywords: Monetary policy inertia, certainty equivalence, uncertainty, loss function
- Introduction
The latest financial crisis of 2007 and its consequences confirm that central banks work with different environments where perfect information is unavailable or uncertainty level is high. Hence, central banks are criticized by not acting aggressively enough to achieve their goals (see, for example, Krugman 2012). In general, uncertainty means lack of sureness about a state or an outcome.Economists introduced uncertainty in their models through the economy features, particularly by using either additive or multiplicative uncertainty[1] or both of them. The former indicates the ambiguity over the true state of the economy, and it is represented by the error term in any economic model. On the other hand, the latter denotes the opacity over the true value of the elasticity of the economic variables for a change in other economic variables of the model. Simply put, this type of uncertainty is presented by the parameters of the model.
When uncertainty is additive, the central bank can ignore the uncertainty and set policy as if everything is known with certainty. The conclusion was first suggested by Theil (1958) and Tinbergen (1952). In a seminal work Brainard (1967) focused on the multiplicative uncertainty. In this case, policymakers are unconfident of how an economic variable will react to a change in another economic variable. The high uncertainty of the estimated parameters leads to a high uncertainty about the path of the economy. Therefore, Brainard (1967) proposed that when policymakers are uncertain of the impact that their policy actions will have on the economy, it might be appropriate for them to adjust policy more carefully[2] and in smaller steps than they would if they had precise knowledge of the effects of their actions. This implies that multiplicative uncertainty leads to gradualism or inertia[3] in monetary policy.
Over the last decade, some economists such as: Sack (1998), Clardia et al. (1998), Goodhart (1999), Clardia et al. (2000), Sack (2000), Gerlach-Kristen (2004), Rudebusch (2006), and Sweidan (2011) contributed to study the inertia of monetary policy. Their results confirmeda considerable interest rate smoothing. In a seminal work ,Woodford (1999) showed in the context of a simple model of optimizing private sector behavior, that monetary inertial policy can be optimal. The justification of this conclusion is that small but persistent changes in short-term interest rates in response to economic shocks allow a larger effect of monetary policy on long rates and hence upon aggregate demand.Thus, this process is able to achieve the lowest expected value of the central bank objective function. At the same time, Woodford (1999) explained whypolicymakers prefer policies do not require the level of short-term interest rates to be too variable. He stated that at a low average level of nominal interest rate, such as zero nominal interest rate, policymakers cannot adopt interest-rate reductions in response to deflationary shocks. On the other hand, high nominal interest rates always imply distortions.
The present paper raises a question regarding the effect of monetary policy inertia on the optimal policy rate. The purpose of ourpaper is to theoretically investigate the influence of monetary policy inertia on the optimal monetary policy under both certainty equivalence and uncertainty. Besides, it seeks to highlight the consequences of this issue. This paper contributes theoretically to the literature by exploring this significant link between monetary policy inertia and optimal monetary policy. Our paper utilizes the model of Williams (2013). He investigated the implications of uncertainty for optimal monetary policy with an application to the current situation in the U.S. economy. The rest of the paper is organized as follows: Section 2 presents the model of the paper and how we modify it to match the goal of the current paper. Section 3 investigates the optimal monetary policy under certainty equivalence with and without monetary policy inertia. Section 4 explores optimal monetary policy under uncertainty with and without monetary policy inertia. Section 5 presents the influence of monetary policy inertia on the central bank loss function. Conclusions are made in the last section.
2. The Model
2.1 The Original Model
As stated above, we adopt and adjust Williams’ (2013) simple static macroeconomic model. The advantage of this model isthat it facilitates the analytical derivation of the optimal monetary policy, both under certainty and uncertainty. In addition, it provides obvious and decisive results. Williams’ (2013) model can be described as follows:
The central bank seeks to minimize the expected squared fluctuations in the inflation gap and the output gap. Thus, the central bank’s objective is to minimize the following quadratic loss function:
(1)
where stands for the inflation rate, denotes the central bank’s target rate of inflation, is the output gap, and denotes the expectation operator. The coefficient is the fixed weight (preference) that the central bank places on the deviation of actual output level from potential output.
The output gap in this economy is given by the following equation:
, (2)
where stands for the nominal interest rate, denotes the constant equilibrium nominal interest rate, is the error term and assumed to be normally distributedwith a zero mean and variance .
The inflation rate can be described by the following equation:
(3)
where stands for the error term and also assumed to be normally distributedwith a zero mean and variance.
2.2 Upgrading the Model
Let us assume that policymakers set the nominal interest rate in response to inflation and output gap. Hence, the Taylor rule has the following form:
, (4)
where is the nominal interest rate, : denotes the inflation rate, : stands for the output gap, : stands for white noise disturbances. and are the model coefficients. Equation (4) states that interest rate is set in response to the current values ofand. To outline the idea of monetary policy inertia, we borrow ideas from Gerlach-Kristen (2004), Rudebusch (2006), and Sweidan (2011). If monetary policy is inertial, adjusts its level gradually to reach the optimal level. Hence, we can re-write equation (4) as follows:
(5)
The parameter is assumed to lie between zero and unity[4]. If, full policy inertia exists,. On the contrary, if, monetary policy inertia is absent and equation (4) is valid. Rearranging equation (5) yields the following reaction function:
(6)
Technically, equation (6) has twoextrem options. First, if , then the interest rate is determined by Taylor rule and there is no room for monetary policy inertia. This means equation (4) is only valid. Second, if is close to one and the parameters and are statistically insignificant differ from zero, then the interest rate is determined by its inertia, and there is no rule for the traditional Taylor rule. The domine of this paper is in the second extreme case. We consider the inertia is exogenous and continuous. Hence, in order to add the idea of monetary policy inertia in the above-mentioned original model (Williams’ (2013) model), equation (6) will be reduced to the following form:
(7)
Equation (7) is the core idea of monetary policy inertia. Focusing on this portion of equation (6) simplifies the shape of the model. Equation (7) is a first order difference equation. Thus, we need to find a general solution of this difference equation[5] to preserve the original model in its static shape. Assume that in a certain previous period of time (say) the interest rate denoted by a variable[6]. We can then compute the following sequence:
Thus the solution after t periods of time will be.
Thus, as long as the model is static then the above solution can be written as follows:
(8)
3. Optimal Monetary Policy under Certainty Equivalence
3.1 Without monetary policy inertiacoefficient
Certainty equivalence means that the model coefficients and are known with certainty by the policymakers. Following Williams (2013), we assume that policymakers observe the occurrence of the shocks when the value of the policy instrument is chosen. Hence, there is no uncertainty at the time when the policy decision is made. Further, we assume that we work on the original model without the upgrading (without taking monetary policy inertia into consideration). Given all the above-mentioned information about the model[7], the optimal policy rate can be described as a linear reaction function in terms of the equilibrium interest rate and shocks:
(9)[8]
3.2 With monetary policy inertia coefficient
In this part of the paper, wetake monetary policy inertia coefficient into consideration in the original model. Minimizing (1) subject to (2), (3) and (8) yields the optimal policy which can be described as follows:
(10)
By comparing equations (9) and (10), it is obvious that monetary policy inertia’s parameter is part of all the coefficients that determine the optimal policy rate. This implies that monetary policy inertia controls the range by which the optimal policy reacts to any shock or equilibrium in the economy. In equation (10), if the economy has full inertia, then equations (10) and (9) are identical. This means that the original model assumes implicitly a full gradualism in the economy! However, if the economy is not conservative, the parameters of equation (10) will be large and this means the optimal policy reacts abruptly to any shock or equilibrium diversion in the economy. However, under high inertia level central banks are too slow to respond to new information and events. Thus, monetary policy looks like less effective, especially in the stabilization of short-run fluctuations in inflation or output gap.
Under certainty equivalence, we can compare the ability of optimal monetary policy to offset the influence of demand shocks. In the case of not including policy inertia coefficient in the model, the optimal policy can completely offset the aggregate demand shock, regardless of the central bank’s weight on output gap stabilization in its objective function.
(11)
On the contrary, in the case of including policy inertia coefficient in the model,the optimal policy cannot offset the aggregate demand shock[9] and as follows:
(12)
Equation (12) displays two different messages; first, if the economy has full inertia feature, then equations (11) and (12) are equal, which indicates that optimal policy reaction offsets completely demand shocks. This confirms the above-mentioned conclusion, which states that the original model presumes implicitly full gradualism! Second, if the economy is not conservative, then the policy reaction will offset more than the shock, and leads to deterioration in the aggregate output[10].
On the other hand, under certainty equivalence, we can link the capability of optimal monetary policy to affect inflation shocks. In the case of excluding policy inertia coefficientfrom the model, the optimal policy cannot offset the inflation shocks. Usually inflation or aggregate supply shocks create a short-run trade-off between output gap and inflation rate. But this rule can be violated, the optimal policy can completely offset the inflation shocks, if[11]. Nevertheless, in this case the optimal monetary policy creates a large output gap. We can present this idea as follows:
(13)
Equation (13) shows the above-mentioned idea. If, then. This means that the optimal monetary policy completely offset the inflation shock. However, if, then. This indicates that optimal monetary policy is unable to offset the inflation shock.
In the case of including policy inertia coefficient in the model, technically,we add another constraint to the original condition. Contrary to the previous finding, the optimal policy rate cannot offset the inflation shocks even if. We can present this idea as follows:
(14)
Equation (14) shows that even if the optimal monetary policy cannot offset the inflation shock. It shows another necessary condition which is to perform this mission. Overall, optimal monetary policy cancompletely offset the inflation shock if.
4. Optimal Monetary Policy under Uncertainty
In this part weassume that the policymaker is uncertain about key parameters of the model economy. Technically, this implies that policymaker is uncertain about the monetary transmission channel, and . We also assume that the parameter follows a Bayesian approach with a prior distribution that has mean and variance. Likewise, we presume that the parameter follows a Bayesian approach with a prior distribution that has mean and variance. In addition, the distributions of the two parameters are assumed to be independent of each other and independent ofthe shocks to the output gap and inflation equations. Recall thatin the previous section we assumed that the policymaker observe the occurrence of the shocks when the value of the policy instrument is chosen. Therefore, there is no uncertainty at the time the policy decision is made. This assumption is valid in this section too.
4.1 Without monetary policy inertiacoefficient
Given all the above-mentioned details about the model, the optimal monetary policy can be described as a linear reaction function in terms of the equilibrium interest rate and shocks as follows:
(15)[12]
Compare to the certainty equivalence case[13], equation (15) reveals that uncertainty parameters ( and ) mute the optimal policy reactions to the shocks in the economy. Their impacts are to reduce the reaction of optimal policy rate to the shocks in the economy. In this regard, Sack (2000)concluded that in the absence of parameter uncertainty, the optimal policy has more aggressive movements. However,the existence of parameter uncertainty limits the responsiveness of the interest rate. As a result, the optimal policy under parameter uncertainty can account for a considerable portion of the observed gradualism.
4.2 With monetary policy inertiacoefficient
We repeat what we did in section 3.2, but we take into consideration the distribution of the model’s parameters and as stated in section 4. Hence, we minimize (1) subject to (2), (3), (8), and the new uncertainty feature of the model’s parameter. It yields the optimal policy rate which can be described as follows:
(16)
Consistent with the above-mentioned results, if monetary policy inertia is full, then equations (15) and (16) are identical. But, if policy inertia is not full, then it works opposite to the effect of the uncertainty parameters. This suggests that lower inertia level magnifies the optimal policy reaction forto any shock or equilibrium diversion in the economy.
Even under uncertainty, we can compare the capability of optimal policy to offset the impact of demand shocks. In the case of not including policy inertia coefficient in the model, the optimal monetary policy can completely offset the aggregate demand shock, regardless the central bank’s weight on output gap stabilization in its objective function. Technically, we reach the exact result as in equation (11).But, in the case of including the inertia coefficient in the model,the optimal policy cannot offset the aggregate demand shock. Actually, it counterweigh more than the shock. The derived equation is exactly the same as equation (12).
Moreover,under uncertainty, we can investigate the proficiency of optimal monetary policy to affect inflation shocks. In the case of excluding policy inertia coefficientin the model, we get the same result precisely as equation (13). The result confirms that if, then the optimal policy rate cannot offset the inflation shocks. But, it can do this, if[14].But by including the inertia coefficientin the model, the result is different and identical to equation (14) of section 3.2. It displays that even if the optimal monetary policy cannot counterweight the inflation shock. It adds another necessary condition which is to perform this mission.
5. The optimal loss function
The goal of this part is to highlight the effect of monetary policy inertia on central bank’s loss function. Based on the analysis of the above two sections, it is apparent that the effect of monetary policy inertia on the optimal monetary policy is identical under certainty equivalence anduncertainty. Thus, we compute central bank loss function for only one case i.e. certainty equivalence then we make the comparison.
After substitution of the optimal policy rate without the inertia coefficient, the expected loss function is given by:
(17)
However, with monetary policy inertia coefficient, the expected loss function looks like:
(18)
Comparingequations (17) and (18) tells us that both of them are identical, if the optimal monetary rate has full inertia. For any value of policy inertia less than one, the expected loss function will be larger. The lower value of, the larger the expected loss function. This implies that monetary policy inertia is an optimum monetary policy if.