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Numerical analysis of the disequilibrium monetary growth model: secular stagnation, slow convergence, and cyclical fluctuations

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Abstract

This study presents a monetary disequilibrium growth model and conducts numerical simulations to investigate how dynamic paths are affected by the initial conditions and the parameters of expectation formation. The main results are as follows. First, dynamic properties such as stable convergence and cyclical fluctuations depend on the type of expectation formation rather than on the initial regimes. Stable convergence takes an excessively long time when expectation formation is too rational and cyclical fluctuations appear when it is too adaptive. Second, when the economy converges to the steady state (i.e., the Walrasian equilibrium), persistent Keynesian unemployment is likely to appear along the dynamic path. Third, the dynamics of inflation expectation that contain the price dynamics in the feedback loop might play an important role in convergence to the steady state.

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Notes

  1. For theoretical and empirical contributions, see Smets and Wouters (2003) and Christiano et al. (2005).

  2. The idea of the ZLB is rooted in the study of Krugman (1998), who discusses the Japanese depression, with Summers (2014, 2015) spreading the idea. Eggertsson and Krugman (2012), Benigno and Fornaro (2018), and Eggertsson et al. (2019) also present New Keynesian models of the ZLB.

  3. For example, see Eggertsson et al. (2019) and Dupor et al. (2019). Schoder (2017, 2020) presents disequilibrium models that are extensions of the DSGE model.

  4. Backhouse and Boianovsky (2012) state that standard textbooks of macroeconomics distortedly explain the history of Keynesian economics as including disequilibrium analysis.

  5. Along the path on which KU continues, both nominal and real interest rates are positive and wages are lower than steady-state wages.

  6. This formulation is the same as that in Sargent (1987).

  7. As the households are identical, they are employees as well as asset holders.

  8. Furthermore, q satisfies the no-arbitrage condition in the asset market. That is, q is also determined to match demand to supply.

  9. This setting simplifies the households’ disposable income \(y_{di}\). The net of interest taxation is used in Sargent (1987), Chiarella and Flaschel (2000b) and Asada et al. (2011).

  10. These names follow Malinvaud (1977).

  11. As the system is discontinuous, the dynamics might stop on the boundaries of the regimes. This is called the pseudo equilibrium (Filippov 1988). In the numerical experiments, the pseudo equilibrium was not detected.

  12. Filippov (1988) specifies the solution to piecewise continuous differential equations, which is used not only in chemistry and electromagnetism but also in economics (Henry 1972; Ito 1979). In recent years, bifurcation analysis has also progressed in the field of mathematics (Kuznetsov et al. 2003; Guardia et al. 2011). The system used in \(\mathtt {DISODE45}\) is an improvement of the event-driven method proposed by Piiroinen and Kuznetsov (2008).

  13. The parameters used in Ogawa (2020) are estimated from quarterly data; hence, \(t = 2000\) corresponds to 500 years.

  14. We do not have to use a value of \(\beta\) less than unity. Flaschel et al. (2001) use \(\beta =0.6\), while Chiarella and Flaschel (2000a) use \(\beta =1\). Moreover, the value of Chiarella and Flaschel (1996) corresponds to \(\beta =1.1\). The size of \(\beta\) does not significantly affect the dynamics.

  15. Flaschel (1999) shows that the steady state can be characterized as KU if we assume that capacity utilization rate and employment rate are under unity in the steady state. By contrast, our model is based on the neoclassical model in which an economy is adjusted toward full capacity utilization and full employment in the long run.

  16. The white region shows that no convergence is observed.

  17. Nakayama and Oshima (1999) show empirical evidence that the proportion of rational expectation formation and that of adaptive expectation formation are roughly the same in Japan, which corresponds to \(\alpha = 0.5\) in our model.

  18. For instance, the expectation would be affected by the policy. The fiscal stimulus and the money market operation increases the aggregate goods demand so that the change of actural inflation affects the current inflation expectation. On the other hand, the change of growth rate of money supply (\(\mu\)) would directly change the expectation since the steady state value \(\pi _0\) changes.

  19. Note that \(m=M/(PK)\), \(b=B/(PK)\), and \(w=W/P\) retain high values and \(l^s=L^s/K\) is lower than its steady-state value.

  20. The price dynamics depend on \(y^d-y^s\), or the demand-supply gap divided by K.

  21. As the production factors are substitutable, the quantities of labor and capital do not directly restrict production.

  22. If the “natural” nominal interest rate \(r_0\) is below zero, permitting negative interest rate (repeal of the ZLB) would cure depression. However, the dual decision effect is not affected by this policy since the pessimistic expectation of sales plays a central role in depression prolonged by that effect.

  23. Their model lacks theoretical consistency when they introduce hybrid expectation. Therefore, we should carefully evaluate their conclusions.

  24. Ogawa (2020) varies the parameters \((\alpha , \beta )\) with a constant initial value and finds that the dynamics change from a monotone to a stable cycle around \(\alpha =0.94\) and a stable cycle to an unstable cycle around \(\alpha =0.98\). The mathematical analysis of the bifurcations of discontinuous dynamic systems, such as Kuznetsov et al. (2003) and Guardia et al. (2011), are restricted by low-dimensional cases.

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Authors and Affiliations

  1. Graduate School of Economics, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, Japan

    Shogo Ogawa

  2. Graduate School of Economics, Kyoto University, Kyoto, Japan

    Hiroaki Sasaki

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  1. Shogo Ogawa
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  2. Hiroaki Sasaki
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Correspondence to Shogo Ogawa.

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Appendix A: Cyclical dynamics

Appendix A: Cyclical dynamics

In this appendix, we see the cyclical dynamics under \(\alpha =1\). Figures 15 and 16 show an unstable cycle. The cyclical transitions of the regimes detected under \(\alpha =1\) are common, and every dynamic shows \(\text {WE}\rightarrow \text {RI}\rightarrow \text {CU}\rightarrow \text {KU}\rightarrow \text {WE}\rightarrow \cdots\). Furthermore, the qualitative dynamics of the variables are also common. This implies that some bifurcation might occur.Footnote 24 The fatal problem of this path is that KU occurs because of the relative increase in \(l^s\) rather than the decrease in \(y^d\). As Ogawa (2020) points out, this unexpected situation might arise from the setting of aggregate goods demand, which is unaffected by the income distribution. The dynamics affected by the income distribution are thus a future research issue.

Fig. 15
figure 15

Dynamics of the variables in an unstable cycle

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Fig. 16
figure 16

Dynamics of employment and production in an unstable cycle

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Ogawa, S., Sasaki, H. Numerical analysis of the disequilibrium monetary growth model: secular stagnation, slow convergence, and cyclical fluctuations. Evolut Inst Econ Rev 19, 369–394 (2022). https://doi.org/10.1007/s40844-021-00201-9

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