Teaching post-intermediate macroeconomics with a dynamic 3-equation model

Abstract

The 3-equation model by Carlin and Soskice (2014) introduces the current consensus in modern monetary macroeconomics to undergraduates through a static framework in which adjustment occurs via the monetary policy rule of an inflation-targeting central bank. In this article, the authors present a dynamic extension of this model and an Excel-based simulation tool for upper-level undergraduate and master’s-level macroeconomics courses. This dynamic framework allows instructors and students to tackle conceptual issues (e.g., understanding a world with output growth and steady inflation) and contemporary applications (e.g., hysteresis and secular stagnation) that are difficult to interpret in static models. Depending on the goals of the course, instructors can either cover the full presentation of the model or instead use the simulation tool to compare scenarios.

Keywords:

• Dynamic macroeconomics
• economic education
• post-intermediate macroeconomics
• 3-equation model

JEL CODES:

• A22
• A23
• C61
• E52
• E58

Notes

1 A widening gap between intermediate- and graduate-level macroeconomics (Colander 2005) has meant that suitable teaching materials for advanced undergraduate and master’s-level macroeconomics courses are limited. In turn, this scarcity of teaching materials has important implications for the role of macro in undergraduate- and master’s-level curricula—for example, making it difficult for students to move beyond intermediate macroeconomics to do independent research in macroeconomics or write macro-oriented theses.

2 Long-term unemployment following extended growth slowdowns can, for example, lead to skills deterioration, which has long-term consequences (Layard and Nickell 1986). Hysteresis has long been emphasized in Keynesian theory (e.g., Arestis and Skott 1993; Cross 1993; Setterfield 1993; Dutt 1997), and has also received renewed widespread attention in the years after the GFC. DeLong and Summers (2012), for instance, show that, in the context of demand constraints and 0 nominal interest rates in the post-2008 U.S. economy, hysteresis implies an important role for fiscal policy in macroeconomic stabilization. See also Carlin and Soskice (2014, 564–68).

3 Three useful mathematical references on two-equation systems of first-order linear difference equations include chapter 9 of Gandolfo (2010), chapter 5 of Shone (2002), and chapter 28 of Pemberton and Rau (2016). For solving a single first-order linear difference equation, students can refer to chapter 3 of Gandolfo (2010), chapter 3 of Shone (2002), or chapter 23 of Pemberton and Rau (2016).

4 Because $\frac{y_t-y_{t-1}}{y_{t-1}}=b_n-1$ demand-side growth is positive or negative when $b_n\gtrless 1$ Given that values close to 0 imply unrealistically high rates of demand contraction, we also assume $b_n > 0$ to avoid complicating the model. For instance, at the height of the 2020 COVID-19 lockdowns in the United States, when the contraction in consumption exceeded previous records, consumption demand fell by approximately 30 percent (Chetty et al. 2020). With constant investment and government spending, this decline implies a $b_n$ of approximately 0.7.

5 While the primary focus in this article is on interest-rate-based monetary policy, this IS curve also captures the roles of both fiscal and nontraditional monetary policy, both of which are increasingly central parts of the post-2008 policy framework. To consider, for instance, expansionary fiscal policy following economic contractions, one can explore offsetting increases in the demand-side growth rate in the simulations below.

6 If, instead, the real interest rate only affects the level of demand, then—with positive (negative) demand growth—stable inflation could be achieved only by continually decreasing (increasing) the real interest rate. We thank an anonymous reviewer for this point.

7 More specifically, $\alpha$ is the slope of the wage-setting curve, and defines how the real wage consistent with wage-setting behavior reacts to the output gap (see Carlin and Soskice 2014, 65). In turn, expected inflation affects prices via wage-setting behavior by serving, for example, as a benchmark for wage negotiations that minimizes bargaining costs.

8 Note that, while inflation converges to ${\pi }^s={\pi }^T-\frac{\alpha y_e}{\chi }$ for $b < 1$, ${\pi }^s$is negative and large in absolute value for reasonable parameter values. Thus, with negative growth, inflation most likely converges to a level of deflation that is large in absolute value. The impact of different parameter choices for this stationary solution can be explored using the simulation tool.

9 This system can be solved using similar steps as with demand-side growth in equations (3) and (6), laid out in appendix A. We leave solving the system in equations (9) and (10) as an exercise.

10 The loss function is $L={\left(y_t-y_e\right)}^2+\beta {\left({\pi }_t-{\pi }^T\right)}^2$ For details on the central bank’s optimization problem that yields the monetary rule, see Carlin and Soskice (2014, 93–96 and 111).

11 The central bank can make these interventions so long as the economy is not at the ELB. Exercise #1 considers inflationary dynamics when the economy is at the ELB.

12 If one does not shock inflation, the simulations indicate that, when the central bank lowers interest rates to increase demand growth (such that $b=d=2\%$ inflation remains constant at ${\pi }^T$ (rather than falling below and then converging back to ${\pi }^T$). This difference reflects that, in our presentation, the central bank acts in a forward-looking manner, anticipates the supply-side shock, and adjusts the interest rate in advance. We make this choice for mathematical convenience; an alternative modeling approach that includes a lag in the monetary rule would substantially increase the mathematical complexity of the model by implying a second-order system of difference equations. By simply shocking the level of inflation, however, one can capture the main implications of a supply shock for inflation within a first-order system of difference equations, and demonstrate that inflation adjusts back to the target over time.

13 Again, this result holds as long as the economy is away from the ELB.

14 C&S also show impulse response functions for the static 3-equation model with the monetary rule (see, for example, 100–101). Comparing them with the simulations in this article highlights differences between the static and dynamic settings. Note, in particular, that in the static C&S setting, inflation is stable and equal to ${\pi }^T$ only when output equals the level of equilibrium output ($y_e$). After the monetary authority intervenes to match $b_n=d$ in the dynamic framework, however, inflation is stable and equal to ${\pi }^T$ while output is also growing.

15 See Carlin and Soskice (2014, 104–7) for discussion of the deflationary trap in the static 3-equation model.

16 See note 2 for references on hysteresis.

17 Allowing $y_e$ to also grow for supply-side reasons (such that $y_{e,t}=b\theta y_{t-1}+d\left(1-\theta \right)y_{e,t-1}$) introduces a 3-dimensional system of difference equations.

18 The simulation tool is available on the authors’ Web sites and upon request. Each worksheet includes an instruction panel on the right-hand side, which lists the parameters that can be adjusted in the simulation; the restrictions on each parameter; and a column in which students can adjust these parameter values. The boxes on the left populate with these parameter choices, and the graph plots output and inflation over time. The instruction box also includes a reset button that returns each parameter to the values in figures 1–4.

19 Recall from the discussion of figure 3 that one needs to shock both supply-side growth and inflation in period 1 when analyzing the case with an inflation-targeting monetary policymaker. Specifically, with negative supply-side growth from period 1 ($d<1$), inflation (${\pi }_1$) rises. Conversely, when $d>1$ from period 1, it then noncontemporaneously falls. The need to adjust both $d_1$ and ${\pi }_1$ is indicated in the instructions for the simulation tool. See also endnote 12 for more details.

20 See, for example, Summers (2015); Skott (2016); Cynamon and Fazzari (2017) on demand-side secular stagnation, and Gordon (2015) on supply-side secular stagnation.

21 Demand-side secular stagnation also points to the role of fiscal policy in stabilization. See DeLong and Summers (2012) for an analysis of the post-2008 U.S. economy, which shows that—due both to hysteresis and state-dependent multipliers that are larger in depressed economies—the stabilization potential of fiscal policy is particularly dramatic when the economy is demand-constrained and at the ELB. Recall from the IS curve that fiscal policy affects demand-side growth via the parameter $b_f$. Also see Setterfield (2007) for an example of a paper highlighting the role of fiscal policy in a similar 3-equation framework.

1 If $b>d$, then $\pi \to \infty$ as $t\to \infty$. Conversely, if $b<d$, then $\pi \to -\infty$ as $t\to \infty$. Note also that the only extraordinary, highly unlikely case in which inflation converges to the policymaker’s target is when demand- and supply-side growth rates are equal and their initial levels coincide ($b=d$ and $y_0=y_{e,0}$).