doi: 10.3934/dcdsb.2021224
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An optimal control problem of monetary policy

University of Urbino, Via Saffi, 42, Urbino, 61029, Italy

*Corresponding author: Andrea Bacchiocchi

FINANCIAL SUPPORT FROM THE RESEARCH PROJECT ON "MODELS OF BEHAVIORAL ECONOMICS FOR SUSTAINABLE DEVELOPMENT" FINANCED BY DESP-UNIVERSITY OF URBINO IS GRATEFULLY ACKNOWLEDGED.

Received  December 2020 Revised  June 2021 Early access September 2021

Fund Project: We thank Gian Italo Bischi and Fabio Lamantia for helpful suggestions and comments. The usual disclaimer applies

This paper analyses an optimal monetary policy under a non-linear Phillips curve and linear GDP dynamics. A central bank controls the inflation and the GDP trends through the adjustment of the interest rate to prevent shocks and deviations from the long-run optimal targets. The optimal control path for the monetary instrument, the interest rate, is the result of a dynamic minimization problem in a continuous-time fashion. The model allows considering various economic dynamics ranging from hyperinflation to disinflation, sustained growth and recession. The outcomes provide useful monetary policy insights and reveal the dilemma between objectives faced by the monetary authority in trade-off scenarios.

Citation: Andrea Bacchiocchi, Germana Giombini. An optimal control problem of monetary policy. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021224
References:
[1]

C. AltavillaL. BurlonM. Giannetti and S. Holton, Is there a zero lower bound?, The Effects of Negative Policy Rates on Banks and Firms ECB Working Paper Series, (2020).   Google Scholar

[2] R. J. Barro, Determinants of economic growth, MIT Press, Cambridge, Mass, 1997.   Google Scholar
[3]

G. I. Bischi and R. Marimon, Global stability of inflation target policies with adaptive agents, Macroeconomic Dynamics, 5 (2001), 148-179.   Google Scholar

[4]

C. Briault, The costs of inflation, Bank of England Quarterly Bulletin, February, (1995), 33-45.   Google Scholar

[5]

M. K. Brunnermeier and Y. Koby, The reversal interest rate: An effective lower bound on monetary policy, Working Paper. Princeton University, 2016. Google Scholar

[6]

M. Bruno and W. Easterly, Inflation and growth: In search of a stable relationship, Proceedings, Federal Reserve Bank of St. Louis, 78 (1996), 139-146.   Google Scholar

[7]

G. Chow, Control methods for macroeconomic policy analysis, The American Economic Review, 66 (1976), 340-345.   Google Scholar

[8]

P. B. ClarkD. Laxton and D. Rose, Asymmetry in the U.S. output-inflation nexus: Issues and evidence, IMF Staff Paper International Monetary Fund, Washington, 43 (1996).   Google Scholar

[9]

G. Debelle and D. Laxton, Is the Phillips curve really a curve? Some evidence for Canada, the United Kingdom, and the United States, IMF Staff Papers, International Monetary Fund, Washington, 44 (1997).   Google Scholar

[10] G. W. Evans and S. Honkapohja, Learning and Expectations in Macroeconomics, Princeton University Press, Princeton, NJ., 2001.   Google Scholar
[11]

G. Ferrero, Monetary policy, learning and the speed of convergence, J. Econom. Dynam. Control, 31 (2007), 3006-3041.  doi: 10.1016/j.jedc.2006.10.003.  Google Scholar

[12]

A. J. Filardo, New evidence on the output cost of fighting inflation, Economic Review, Third Quarter, (1998), 33-61.   Google Scholar

[13]

M. Friedman, The role of monetary policy, American Economic Review, 58 (1968), 267-291.   Google Scholar

[14] J. Galí, Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, Princeton, Princeton University Press, 2008.   Google Scholar
[15]

R. J. Gordon, Modern theories of inflation in Macroeconomics: Theory and policy, 2$^nd$ edition, Chap. 22.4, Boston, Massachusetts. McGraw-Hill, 1988. Google Scholar

[16]

J. M. Keynes, The General Theory of Employment, Interest, and Money. Reprinted from the 1936 Original With A New Introduction by Paul Krugman and An Afterword by Robert Skidelsky. Palgrave Macmillan, Cham, 2018. doi: 10.1007/978-3-319-70344-2.  Google Scholar

[17]

J. Kodera and V. Q. Tran, Monetary policy as an optimal control problem, European Financial and Accounting Journal, 8 (2013), 18-38.   Google Scholar

[18]

A. MussoL. Stracca and D. J. van Dijk, Instability and nonlinearity in the euro area Phillips curve, International Journal of Central Banking, (2009), 181-212.   Google Scholar

[19]

A. R. Nobay and D. A. Peel, Optimal monetary policy with a nonlinear Phillips curve, Economics Letters, 67 (2000), 159-164.   Google Scholar

[20]

A. Orphanides and J. C. Williams, Learning, expectations formation, and the pitfalls of optimal control monetary policy, J. Monetary Economics, Supplement, 55 (2008), S80-S96. doi: 10.1016/j.jedc.2005.06.009.  Google Scholar

[21]

E. S. Phelps, Phillips curves, expectations of inflation and optimal unemployment over time, Economica, 34 (1967), 254-81.   Google Scholar

[22]

A. W. Phillips, The relation between unemployment and the rate of change of money wage rates in the united kingdom, 1861-1957, Economica, New Series, 25 (1958), 283-299.   Google Scholar

[23]

W. Semmler and W. Zhang, Nonlinear Phillips curves, endogenous NAIRU and monetary policy, Contributions to Economic Analysis, 277 (2006), 483-515.   Google Scholar

[24]

J. Stiglitz and E. Joseph, Reflections on the natural rate hypothesis, J. Economic Perspectives, 11 (1997), 3-10.   Google Scholar

[25]

E. O. Svensson Lars, Inflation forecast targeting: Implementing and monitoring inflation targets, European Economic Review, 41 (1997), 1111-1146.   Google Scholar

[26]

G. Tabellini, Money, debt and deficits in a dynamic game, J. Economic Dynam. Control, 10 (1986), 427-442.  doi: 10.1016/S0165-1889(86)80001-X.  Google Scholar

[27]

J. B. Taylor, Discretion versus policy rules in practice, Carnegie-Rochester Conference Series on Public Policy, 39 (1993), 195214. Google Scholar

[28]

J. Tinbergen, Economic policies. Principles and design, Amsterdam: North Holland, 1956. Google Scholar

[29]

F. TramontanaL. Gardini and P. Ferri, The dynamics of the NAIRU model with two switching regimes, J. Econom. Dynam. Control, 34 (2010), 681-695.  doi: 10.1016/j.jedc.2009.10.014.  Google Scholar

[30]

D. Turner, Speed limit and asymmetric effects from the output gap in the seven major countries, OECD Economic Studies, 24 (1995), 57-88.   Google Scholar

show all references

References:
[1]

C. AltavillaL. BurlonM. Giannetti and S. Holton, Is there a zero lower bound?, The Effects of Negative Policy Rates on Banks and Firms ECB Working Paper Series, (2020).   Google Scholar

[2] R. J. Barro, Determinants of economic growth, MIT Press, Cambridge, Mass, 1997.   Google Scholar
[3]

G. I. Bischi and R. Marimon, Global stability of inflation target policies with adaptive agents, Macroeconomic Dynamics, 5 (2001), 148-179.   Google Scholar

[4]

C. Briault, The costs of inflation, Bank of England Quarterly Bulletin, February, (1995), 33-45.   Google Scholar

[5]

M. K. Brunnermeier and Y. Koby, The reversal interest rate: An effective lower bound on monetary policy, Working Paper. Princeton University, 2016. Google Scholar

[6]

M. Bruno and W. Easterly, Inflation and growth: In search of a stable relationship, Proceedings, Federal Reserve Bank of St. Louis, 78 (1996), 139-146.   Google Scholar

[7]

G. Chow, Control methods for macroeconomic policy analysis, The American Economic Review, 66 (1976), 340-345.   Google Scholar

[8]

P. B. ClarkD. Laxton and D. Rose, Asymmetry in the U.S. output-inflation nexus: Issues and evidence, IMF Staff Paper International Monetary Fund, Washington, 43 (1996).   Google Scholar

[9]

G. Debelle and D. Laxton, Is the Phillips curve really a curve? Some evidence for Canada, the United Kingdom, and the United States, IMF Staff Papers, International Monetary Fund, Washington, 44 (1997).   Google Scholar

[10] G. W. Evans and S. Honkapohja, Learning and Expectations in Macroeconomics, Princeton University Press, Princeton, NJ., 2001.   Google Scholar
[11]

G. Ferrero, Monetary policy, learning and the speed of convergence, J. Econom. Dynam. Control, 31 (2007), 3006-3041.  doi: 10.1016/j.jedc.2006.10.003.  Google Scholar

[12]

A. J. Filardo, New evidence on the output cost of fighting inflation, Economic Review, Third Quarter, (1998), 33-61.   Google Scholar

[13]

M. Friedman, The role of monetary policy, American Economic Review, 58 (1968), 267-291.   Google Scholar

[14] J. Galí, Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework, Princeton, Princeton University Press, 2008.   Google Scholar
[15]

R. J. Gordon, Modern theories of inflation in Macroeconomics: Theory and policy, 2$^nd$ edition, Chap. 22.4, Boston, Massachusetts. McGraw-Hill, 1988. Google Scholar

[16]

J. M. Keynes, The General Theory of Employment, Interest, and Money. Reprinted from the 1936 Original With A New Introduction by Paul Krugman and An Afterword by Robert Skidelsky. Palgrave Macmillan, Cham, 2018. doi: 10.1007/978-3-319-70344-2.  Google Scholar

[17]

J. Kodera and V. Q. Tran, Monetary policy as an optimal control problem, European Financial and Accounting Journal, 8 (2013), 18-38.   Google Scholar

[18]

A. MussoL. Stracca and D. J. van Dijk, Instability and nonlinearity in the euro area Phillips curve, International Journal of Central Banking, (2009), 181-212.   Google Scholar

[19]

A. R. Nobay and D. A. Peel, Optimal monetary policy with a nonlinear Phillips curve, Economics Letters, 67 (2000), 159-164.   Google Scholar

[20]

A. Orphanides and J. C. Williams, Learning, expectations formation, and the pitfalls of optimal control monetary policy, J. Monetary Economics, Supplement, 55 (2008), S80-S96. doi: 10.1016/j.jedc.2005.06.009.  Google Scholar

[21]

E. S. Phelps, Phillips curves, expectations of inflation and optimal unemployment over time, Economica, 34 (1967), 254-81.   Google Scholar

[22]

A. W. Phillips, The relation between unemployment and the rate of change of money wage rates in the united kingdom, 1861-1957, Economica, New Series, 25 (1958), 283-299.   Google Scholar

[23]

W. Semmler and W. Zhang, Nonlinear Phillips curves, endogenous NAIRU and monetary policy, Contributions to Economic Analysis, 277 (2006), 483-515.   Google Scholar

[24]

J. Stiglitz and E. Joseph, Reflections on the natural rate hypothesis, J. Economic Perspectives, 11 (1997), 3-10.   Google Scholar

[25]

E. O. Svensson Lars, Inflation forecast targeting: Implementing and monitoring inflation targets, European Economic Review, 41 (1997), 1111-1146.   Google Scholar

[26]

G. Tabellini, Money, debt and deficits in a dynamic game, J. Economic Dynam. Control, 10 (1986), 427-442.  doi: 10.1016/S0165-1889(86)80001-X.  Google Scholar

[27]

J. B. Taylor, Discretion versus policy rules in practice, Carnegie-Rochester Conference Series on Public Policy, 39 (1993), 195214. Google Scholar

[28]

J. Tinbergen, Economic policies. Principles and design, Amsterdam: North Holland, 1956. Google Scholar

[29]

F. TramontanaL. Gardini and P. Ferri, The dynamics of the NAIRU model with two switching regimes, J. Econom. Dynam. Control, 34 (2010), 681-695.  doi: 10.1016/j.jedc.2009.10.014.  Google Scholar

[30]

D. Turner, Speed limit and asymmetric effects from the output gap in the seven major countries, OECD Economic Studies, 24 (1995), 57-88.   Google Scholar

Figure 1.  $ \gamma = 0.1 $
Figure 2.  $ \omega = 0.5 $, $ i_{min} = -2 $ and $ \bar{i} = 2 $
Figure 3.  $ \pi_0 = 2, \ y_0 = 2, \ T = 5, \ \alpha = 0.5, \ \beta = 0.5, \ \gamma = 0.1, \ \eta = 0.05, \ \omega = 0.8, \ \phi = 0.5 $
Figure 4.  $ \pi_0 = -1, \ y_0 = -2.5, \ T = 5, \ \alpha = 0.5, \ \beta = 0.5, \ \gamma = 0.1, \ \eta = 0.1, \ \omega = 0.3, \ \phi = 0.5 $
Figure 5.  $ \pi_0 = -2, \ y_0 = 2, \ T = 5, \ \alpha = 0.5, \ \beta = 0.5, \ \gamma = 0.1, \ \eta = 0.05, \ \omega = 0.8, \ \phi = 0.5 $
Figure 6.  $ \pi_0 = 2, \ y_0 = -2, \ T = 5, \ \alpha = 0.5, \ \beta = 0.5, \ \gamma = 0.1, \ \eta = 0.1, \ \omega = 1.2, \ \phi = 0.6 $
Figure 7.  $ \pi_0 = 2, \ y_0 = -2, \ T = 5, \ \alpha = 0.9, \ \beta = 0.1, \ \gamma = 0.1, \ \eta = 0.1, \ \omega = 1.2, \ \phi = 0.6 $
Figure 8.  $ \pi_0 = 2, \ y_0 = -2, \ T = 5, \ \alpha = 0.2, \ \beta = 0.8, \ \gamma = 0.1, \ \eta = 0.1, \ \omega = 1.2, \ \phi = 0.6 $
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