The Canonical New Keynesian Monetary Policy Model: A ...

Paper presented at the 2018 International System Dynamics Conference, Reykjavik

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The Canonical New Keynesian Monetary Policy Model: A System Dynamics Translation

I. David Wheat1 and Marianna Oleskevych2

Abstract

Our purpose in this paper is to make accessible to the system dynamics community a simplified version of the workhorse monetary policy model of many central banks: the New Keynesian Monetary Policy Model (NKMP). Specifically, we translate into system dynamics (SD) modeling language the well-known 3-equation, canonical version of the NKMP. Of course, even the canonical version in daily use actually contains 30 or 40 equations, but being able to summarize a useful mid-size model in such a compact way has great appeal to policy makers. Our contribution is to add stock-and-flow structure and emphasize the significance of delays inherent in stock-adjustment processes in real-world systems represented by such models, while retaining simplicity and improving transparency. A prerequisite for SD-based monetary policy models to be taken seriously by central bank economists is a good-faith effort to communicate with them on their terms, and a SD translation of models (theories) currently used by those economists is a good place to start. Our long run goal is to demonstrate the value added to their theories by an SD approach and reach a position to be able to suggest extensions and modifications to those theories. Key words: New Keynesian, monetary policy, value added, Taylor Rule, central bank, simulation Introduction System dynamics modelers engage in two types of policy analysis, both of which are important. One type might be called 'policy design' for the purpose of modifying the structure of existing systems to alleviate problematic behavior. This type of modeling is described in Wheat (2015). The other type of policy analysis is a necessary prerequisite for policy design, and that is the modeling of existing policies for the purpose of discovering whether they are part of the dynamic problem or potentially part of the solution. J. W. Forrester's earliest work; e.g., Industrial Dynamics (Forrester, 1961) and Urban Dynamics (Forrester 1969) laid the foundation for both types of policy analysis and did so in both the private and public sectors. This paper originated with a growing awareness that the way monetary policy makers think about their problems (their mental models of those problems) has become increasingly institutionalized inside central banks in the form of simple, small quantitative models (computer models for their laptops, or even iPads). These models are not used to make forecasts. Instead, they are used to promote critical thinking and provoke internal debate about systemic relationships in the part of the macroeconomy of greatest concern to them. In the mid-20th century, the static IS-LM model served that purpose. Today, it is the New Keynesian Monetary Policy Model (NKMP) expounded by Clardia, Gali, and Gertler (1999) and extended by countless others over the past two decades. The NKMP

1 University of Bergen, Norway; National University of Kyiv-Mohyla Academy, Ukraine; ISM University of Management and Economics, Lithuania. 2 Ivan Franko National University of Lviv, Ukraine.


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model is best known in its simple 3-equation canonical form, but it actually used in slightly more complex versions adapted to the structure of specific nations' economies and monetary policy requirements. In this paper, we draw upon the NKMP version extended by De Grauwe (2012) to incorporate 'bounded rationality' expectations (Simon, 1997) as an alternative to the 'rational expectations' that is more common in the NKMP literature. Our contribution is to add stock-and-flow structure to the NKMP and highlight the significance of delays inherent in the stock-adjustment processes that are largely ignored in the NK literature. We strive, however, to retain the simplicity of the NKMP while improving its transparency. Our goal is more than enriching the scholarly and pedagogical literature, as important as those tasks are. We are working with economists at the National Bank of Ukraine (NBU) who have developed their own version of the NK monetary policy model, and the modeling work discussed in this paper is our first step in developing a robust SD version of the NBU model. Model Structure In DeGrauwe's 'fundamentalist' version of the New Keynesian monetary model (2012, pp. 3-11), inflation in the current time period depends on its value in the previous period, the central bank's target value in the next period, and the output gap in the current period. With the output gap measured in terms of its percentage deviation from the long-term trend of real GDP, the inflation equation is expressed as inflation t = a1*inflation t-1 + (1-a1)*inflation target t+1 + b1*output gap t (NK1) The parameter a1 (0<a1<1) reflects the credibility of the central bank's monetary policy makers. If the central bank commits to an inflation target, is capable of reaching that target, and has demonstrated its capability well enough for its commitment to be credible, a1 will be a small fraction; ideally zero from the perspective of policy makers. Otherwise, current inflation is more likely to be closer to its previous value; if a1 = 1, the central bank target has no credibility. The parameter b1 reflects the unitless output gap effect on inflation. Key equations in the canonical NK monetary policy model typically add a stochastic exogenous shock term, which we have omitted from (NK1). Nevertheless, the stochastic shock terms in our model are similar to those used when testing NK models. Equation (NK1) appears as 'indicated inflation' that adjusts over time in our SD version of the NK model. See Figure 1. If a1=0, indicated inflation equals the central bank's inflation target plus the effect of the output gap. If a1=1, the central bank's target has no effect on inflationary expectations, and indicated inflation

Figure 1. Inflation as a Function of Previous Inflation, Inflation Target, and Output Gap


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equals current inflation plus the output gap effect. In the default settings for our model, a1=0.5, as in DeGrauwe (2012, p. 36) After indicated inflation is determined, inflation adjusts towards that value over time. An adjustment time constant of 1/3 year was selected in order to approximate the behavior observed empirically; i.e., that the full adjustment of the gap between inflation and its indicated value would occur in about one year after a change in the output gap.3 The one-year period for a full adjustment of inflation is suggested by empirical observations at the Bank of England:

The empirical evidence is that on average it takes up to about one year in this and other industrial economies for the response to a monetary policy change to have its peak effect on demand and production, and that it takes up to a further year for these activity changes to have their fullest impact on the inflation rate. (Bank of England, 1999)

Here are the equations and parameters in Figure 1: Inflation(t) = Inflation(t - dt) + (∆ inflation) * dt per year (SD1) ∆ inflation = (indicated inflation - Inflation) / inflation adj time per year/yr (SD2) indicated inflation = a1*Inflation + (1-a1)*inflation target + b1*output gap per year (SD3) output gap = 100*(GDP - Potential GDP*normal capacity utilization) per year (SD4) / Potential GDP*normal capacity utilization inflation adj time = 0.33 years (SD5) a1 = 0.5 unitless (SD6) b1 = 0.5 unitless (SD7) inflation target = 2 per year (SD8) Before considering the NK equation for aggregate demand, we want to clarify the reasoning behind the output gap equation (SD4). Figure 2 displays the inputs to that equation. In the literature, potential GDP is determined on the supply side of a macroeconomics model, and it is usually assumed to be growing as a consequence of growth in accessible material factors of production (labor and capital) and productivity trends. For a given year, potential GDP can be estimated based on an assumed growth trend, and that estimate can be compared with measured aggregate demand for that same year. The percentage difference between measured aggregate demand and estimated potential GDP 3 Admittedly, this is an ad hoc parameter value selection. It is not merely to compensate for gap adjustments formulated as exponentially declining processes; where a passage of time corresponding to the sum of three adjustment time constants is required to close about 95 per cent of the initial gap (Sterman, 2000, pp. 279-280). This is a more complex process because indicated inflation is not a constant value; it changes with the output gap and with prior inflation. Yet the seemingly separate impacts on indicated inflation are actually interrelated and have offsetting effects, and the choice of the 'one-third' time constant generates an overall adjustment time for inflation that approximates what is observed empirically after an output gap emerges. The same reasoning applies to the choice of the adjustment time constant for aggregate demand (AD) in Figure 3.


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is considered the output gap. That is the same calculation we have done, but we have assumed potential GDP to be constant over the short run over which monetary policy seeks attainment of its goals (2 to 3 years). This is a standard assumption in short-run monetary policy analysis, and reflects the fact that time is needed to expand productive capacity.4 That is what is displayed in Figure 2.

Figure 2. Output Gap as a Function of Aggregate Demand and Capacity Utilization

Although potential GDP will be constant, aggregate demand will be changing in our model (as can be seen below in Figure 3). If AD is growing, the short-run response to the need for increased output is assumed to be met by increases in capacity utilization--overtime for workers and more intensive use of capital facilities. And the converse is true if AD is declining. Normal capacity utilization is assumed to be 80 per cent of potential GDP, and if capacity utilization rises to, say 82 per cent, that translates into a positive output gap that produces inflationary pressures due to more intensive utilization of resources. Listed below are the new equations displayed in Figure 2, with the output gap equation repeated for the reader's convenience. (HUA is the abbreviation for Ukraine's currency, the hryvnia). output gap = 100*(GDP - (Potential GDP*normal capacity utilization)) per year (SD4) / (Potential GDP*normal capacity utilization) GDP = Potential GDP*capacity utilization HUA/year (SD9) capacity utilization = SMTH1(desired capacity utilization, CU adj time) unitless (SD10) desired capacity utilization = desired GDP/Potential GDP unitless (SD11) desired GDP = SMTH3(AD, time to avg AD) HUA/year (SD12)

4 Our model actually provides for growth in potential GDP, in response to changes in desired GDP. As expected, little growth occurs within our five-year simulation period and potential GDP has negligible effect on the output gap. To avoid unnecessary details, therefore, the potential GDP growth structure has not been displayed in any of the diagrams.


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normal capacity utilization = .80 unitless (SD13) CU adj time = 1/12 years (SD14) Potential GDP = 2500 billion HUA/year (SD15) time to avg AD = 1/12 years (SD16) initial AD = 2000 billion HUA/year (SD17) Now we examine the second key formulation in the New Keynesian monetary policy model: the aggregate demand equation as presented in De Grauwe's fundamentalist version (De Grauwe 2012, pp. 3-9) with variable names easily compared to those in our SD version: output gap t = a2*output gap t-1 + (1-a2)* target output gap t+1 + b2*(nominal IR t - expected inflation t+1) (NK2) Similar to a1 in equation (NK1), the parameter a2 (0<a2<1) in equation (NK2) reflects the credibility of the central bank's monetary policy makers, with the maximum credibility reflected in a value of zero. The second term in the equation is the expected real interest rate, and it reflects a hypothesis that can be expressed this way: when inflation is expected to be higher, the expected real interest falls, and the output gap increases. The parameter b2 is an elasticity concept that indicates the strength of the effect on the output gap when the real interest rate changes. Although equation (NK2) is often called the 'aggregate demand' equation in the NK literature (e.g., De Grauwe, 2012, p. 30), it refers to the output gap instead of aggregate demand. However, given the assumption that potential GDP is constant in the short-run monetary policy period, a change in the output gap can only result from a change in aggregate demand and/or a change in capacity utilization, as discussed previously.5 Equation (NK2) can be revised in a way that is compatible with these conditions yet maintains its essential proposition. First, however, we elaborate on the rationale for the alternative formulation. No economic agents (except the central bank) have an output gap decision rule; they may have rules for spending, capacity utilization, production, or potential production, but not for the output gap per se. A private sector decision rule in the model therefore should not rely on closing an output gap. Instead, the focus should be on the AD gap, as noted above. Another justification for revising the presentation of (NK2) is the ambiguity in the macroeconomics literature concerning GDP, output, AD, spending, sales, and income. These terms tend to be used interchangeably when, in fact, important distinctions should be noted. GDP (= output) and AD (= spending = sales) are numerically equal only when a model is in equilibrium (specifically, when there is no net change in inventories). In our model, we distinguish between AD and GDP. Nevertheless, we accept the simplifying assumption that income accrues (even if not simultaneously paid) to the factors of production when output occurs, so that indicated income has an accrual equality relationship with GDP in this simple model; in other words, income = GDP. 5 Even if potential GDP were adjusting, it would be in response to changes in AD, and it would be adjusting much more slowly than observed changes in AD. Even in that case, most of the shock to the output gap would be due to changes in AD and capacity utilization.


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The first step is to define the AD gap in terms of our SD model. AD gap = indicated AD - AD where indicated AD = income * (real interest rate/reference real interest rate)^b2

and b2 has the same elasticity interpretation as in equation (NK2) because we are using the nonlinear formulation instead of the linearized logarithmic form used in NK models. In equilibrium, the interest rate term equals 1 regardless of the value of b2; indicated AD

equals income; and AD equals indicated AD. Only then can the AD gap close to zero. Here are the new equations and parameter values for the aggregate demand formulation displayed in Figure 3, plus some not displayed: AD(t) = AD(t - dt) + (∆ AD) * dt HUA/year (SD18) ∆ AD = (indicated AD - AD) / AD adj time constant HUA/year/year (SD19) indicated AD = income * (real interest rate/reference real interest rate)^b2 HUA/year (SD20) real interest rate = Nominal Interest Rate - Inflation per year (SD21) income = GDP HUA/year (SD22) initial Nominal Interest Rate = reference nominal interest rate per year (SD23) reference nominal interest rate = reference real interest rate + reference inflation per year (SD24) reference real interest rate = 2 per year (SD25) reference inflation = inflation target = 2 per year (SD26) AD adj time constant = 1/3 years (SD27) b2 = interest rate elasticity of AD = -.02 unitless (SD28)

Figure 3. Aggregate Demand as a Function of Income and Interest Rates

Before discussing the third key equation in the NK monetary policy model, let us pause for a thought experiment. In Figure 3, assume there is an exogenous shock that increases AD. That would increase desired GDP, capacity utilization, actual GDP, income, and indicated AD, thus closing a reinforcing feedback loop that would push AD even higher.


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Meanwhile, the rise in GDP without an offsetting rise in Potential GDP would create a positive output gap and eventually boost inflation. Rising inflation would lower the real interest rate and raise indicated AD even more--another reinforcing feedback process. Moreover, both loops reinforce each other. From the perspective of an inflation fighter, this is a vicious circle that needs to be arrested by a counteracting policy intervention. In this situation, raising the nominal interest rate more than the increase in inflation would cause the real interest rate to rise instead of fall, thus counteracting the influence of the two reinforcing loops. The possibility of using the nominal interest rate as an instrument of monetary policy leads us to consider the final key equation in the NK model: the monetary policy interest rate rule. Again following De Grauwe (2012, p. 4), we see that the NK interest rate equation is a version of the Taylor Rule (Taylor 1993), which provides guidance to monetary policy makers seeking to eliminate inflation gaps and output gaps by raising or lowering nominal interest rates. nominal interest rate t = c1*inflation gap + c2*output gap t + c3*nominal interest rate t-1 (NK3)

where inflation gap = inflation t - inflation target

The parameters c1, c2, and c3 reflect the strength of the influence of each term in the equation. The value selected for c1 is particularly important. Unless inflationary expectations are well attached to the central bank's inflation target, c1 must be greater than 1.0 in order to have the desired effect of offsetting inflation's impact on the real interest rate, as noted in the 'thought experiment' above. A value equal to 1.0 could stabilize inflation but not at its target level. And a value less than 1.0 could not arrest a vicious inflationary circle. This 'Taylor Principle' will be illustrated when model behavior is examined below. Usually, parameter c2 is set to a value lower than c1, reflecting a higher priority assigned to inflation gaps compared to output gaps. A zero value would reflect a pure inflation targeting policy with no regard for output effects, whereas in Taylor's original model (1993, p. 202) designed to explain the historic pattern of the U.S. Fed Funds Rate, Taylor found that a value of 0.5 (coupled with the inflation gap parameter) was consistent with the data pattern. That finding is consistent with the Fed's dual mandate to stabilize both prices and output. The basics of the Taylor Rule are easily grasped: raise nominal interest rates when inflation exceeds target inflation or output exceeds potential output, or both, and conversely. In equilibrium, when both gaps are zero, the nominal rate should stabilize at the desired rate. This suggests c3 in De Grauwe's equation (NK3) should be assigned a value of 1.0. In Taylor's original paper (1993, p. 202), the third term was the sum of a variable (actual inflation over the past year) and a parameter (an assumed value for the long-term real interest rate) with the c3 coefficient equal to 1.0. With the increasing emphasis on the need to anchor inflationary expectations, a more conservative approach is for the third term to be a reference nominal interest rate (e.g., equation SD24) that incorporates the central bank's inflation target instead of actual inflation and also sets c3 = 1.0. That is the approach taken in this paper.


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Figure 4 displays the completed SD model, with the addition of the structure needed for the interest rate policy rule. Below the diagram is the list of new equations and parameter values added in Figure 4.

Figure 4. Nominal Interest Rate as a Function of the Target Rate set by Monetary Policy

Nominal Interest Rate(t) = Nominal Interest Rate(t - dt) + (∆_nominal IR)*dt per year (SD29) ∆ nominal IR = (target nominal interest rate - Nominal Interest Rate) per year/year (SD30) / nominal IR adj time constant target nominal interest rate = reference nominal interest rate per year (SD31) + c1*inflation gap perceived + c2*output gap perceived reference nominal interest rate = reference real interest rate + inflation target per year (SD32) inflation gap perceived = DELAY(inflation gap, data delay ) per year (SD33) output gap perceived = DELAY(output gap, data delay) per year (SD34) inflation gap = Inflation - inflation target per year (SD35) c1 = inflation gap effect on target rate = 1.5 unitless (SD36) c2 = output gap effect on target rate = .5 unitless (SD37) nominal IR adj time constant = 1/50 year (SD38) data delay = 1/12 year (SD39) output gap = 100*(GDP - Potential GDP*normal capacity utilization) per year (SD4) / Potential GDP*normal capacity utilization reference nominal interest rate = reference real interest rate + reference inflation per year (SD24) reference real interest rate = 2 per year (SD25) reference inflation = inflation target = 2 per year (SD26)


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A Guide to Testing the Model's Behavior Our system dynamics version of the New Keynesian Monetary Policy Model is available for inspection and testing and for conducting simulation experiments. A copy has been posted as 'additional materials' accompanying this paper in the SD Conference submission portal. Users will need Stella Architect to access both the model and the interface; even the 30-day trial version should provide full access. The free Stella Player (available from isee systems, inc.) provides access to the model layer only. Others interested in the model but having no Stella software to run it are encouraged to play with the model online, available at https://exchange.iseesystems.com/public/david-wheat/sd-nk/index.html#page1. A modern browser is all that is needed to conduct simulation experiments at the interface layer, and examine the structure of the model with Stella's story-telling feature. In this section of the paper, we illustrate the method of conducting simulation experiments. After opening the model, click on the Model Behavior button to view a screen (Figure 5) that enables you to Run the model and observe the behavior of several key variables, including inflation, the policy interest rate (policy rate), and the output gap. Other graph pages will be added from time to time.

Figure 5. Model Behavior Screen

In the middle of the screen are the basic parameters of the Monetary Policy Rule ('Taylor Rule'). You may set the inflation target, and the coefficients that determine the impact of inflation gaps and output gaps on the desired policy rate. On the left are default parameter values that can also be modified. The model is initialized in equilibrium to facilitate understanding the impact of shock tests and changes in parameter value assumptions. At top left, there are two buttons that generate exogenous shocks to aggregate demand. The 'temporary' button triggers a 5 per


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cent increase in AD at the beginning of the second quarter (time = 0.25) and the shock terminates at the end of that quarter (time = 0.50). Run the model with the temporary shock activated and observe the resulting patterns of inflation and the policy rate. With the default settings, inflation approaches its 2 per cent inflation target after about two years, which is consistent with the empirical findings at the Bank of England noted in the Model Structure section of the paper. See panel (a) in Figure 6. Moving the sliders that control the parameters in the policy rate equation illustrates the different time paths of the key variables. Very aggressive settings usually enable reaching the targets sooner, but at the risk of generating instability in prices, output, and interest rates. Timid settings fail to bring inflation back under control within political acceptable time horizons. In the thought experiment about adjusting the nominal interest rate, the 'Taylor Principle' was mentioned in regard to raising the policy rate more than the increase in inflation, in order to offset the expected decline in the real interest rate. In panel (b) of Figure 6, we see the results of failing to follow that principle in Runs 2 and 3 after adhering to it in Run 1. When the weight on the inflation gap was set to 1.0 in Run 2, inflation stabilized but at a level above the target. When the weight was reduced to 0.5 in Run 3, inflation soared, due to the two powerful reinforcing loops described in the thought experiment.6

(a) Results with Default Settings (b) Failure to Follow 'Taylor Principle' (see text)

Figure 6. Sample Simulation Results The 'permanent' button also shocks AD, but does so with a continuous stochastic impulse with a normal distribution that has a zero mean and a standard deviation equal to 2 per cent of AD. This 'white noise' disturbance generates cyclical behavior in the model, which indicates that the observed behavior pattern is due to the endogenous structure of the model and not due to cyclical input to the model. Readers are encouraged to 'play' with the model and to give us suggestions for improving the basic structure and, of course, fixing any mistakes that are found.

6 One additional non-default setting was used in the simulation experiment that produced the behavior displayed in panel (b) of Figure 6: the credibility of the central bank was zero in the inflation equation SD3 (the a1 parameter was set to 1.0). Note that this parameter change also resulted in higher inflation in panel (b) than in panel (a) when the 'a1' parameter was set to 0.50.


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Next Steps This paper has two purposes. The current purpose is to make available to the SD community of economic modelers an SD-based canonical version of a monetary policy model that is well known outside our community and, in fact, is used in one way or another inside policy analysis departments at many central banks throughout the world. The New Keynesian Monetary Policy Model--whatever one thinks about the inflation theory at its core--has become the 'workhorse' model for structuring policy makers' thinking about inflationary (and deflationary) conditions and the advantages and limitations of relying on the short-term interest rate to carry out monetary policy. While all models are a work-in-progress, this one is developed well enough that interested modelers could extend it and improve in a number of ways. For example, the formulation of expectations could be improved along the lines being developed by De Grauwe (2012). De Grauwe not only recognizes that 'bounded rationality' (Simon, 1997) is a more relevant organizing construct than 'rational expectations', he is actively engaged in a research agenda that promises a theoretically coherent quantitative representation of bounded rationality that should be of interest to SD modelers. The next steps in our own work will be to add structure to this canonical model so that it more accurately represents the model used by economists at the National Bank of Ukraine (NBU). The highest priority is the addition of foreign exchange interventions to the model because inflationary pressures in Ukraine's economy are both imported and home-grown, and both the policy interest rate and the exchange rate are important monetary policy tools at NBU. And, of course, as the model's structure more closely resembles the NK version used at NBU, we will give more attention to calibrating the model. But it is our plan to go beyond just replicating an NK model. While having a sound SD model that faithfully represents an NK theory serves scholarly and pedagogical purposes, our hope is that is that we can extend our model and demonstrate to policy makers the value that can be added by robust system dynamics models and that we can make those models accessible to economists outside the SD community. References Bank of England (1999). The Transmission Mechanism of Monetary Policy, p. 167.

https://www.bankofengland.co.uk/-/media/boe/files/quarterly-bulletin/1999/the-transmission-mechanism-of-monetary-policy

Clardia, R., Gali, J. and Gertler, M. (1999). The Science of Monetary Policy: A New Keynesian Perspective. Journal of Economic Literature XXXVII: 1661-1707.

De Grauwe, P.D. (2012). Lectures on Behavioral Economics. Oxford: Princeton University Press.


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Forrester, J. W. (1961) Industrial Dynamics. Cambridge: MIT Press. Forrester, J.W. (1969). Urban Dynamics. Waltham, MA: Pegasus Communications.

Sterman, J.D. (2000). Business Dynamics: Systems Thinking and Modeling for a Complex World. Boston: McGraw-Hill.

Simon, H.A. (1997). Models of Bounded Rationality: Empirically Grounded Economic Reason, Volume 3. Cambridge USA: MIT Press.

Taylor, J.B. (1993). Discretion versus Policy Rules in Practice. Carnegie-Rochester Conference Series on Public Policy, 39, pp. 195-214.

Wheat, I.D. (2015). Model-based Policy Design that Takes Implementation Seriously, in E.W. Johnston (ed.) Governance in the Information Era: New York: Routledge.

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