November  2021, 26(11): 5769-5786. doi: 10.3934/dcdsb.2021224

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 Published  November 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 and Continuous Dynamical Systems - B, 2021, 26 (11) : 5769-5786. 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). 

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

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

[4]

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

[5]

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

[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. 

[7]

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

[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). 

[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). 

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

[12]

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

[13]

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

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

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

[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.

[17]

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

[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. 

[19]

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

[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.

[21]

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

[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. 

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[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.

[30]

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

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). 

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

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

[4]

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

[5]

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

[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. 

[7]

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

[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). 

[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). 

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

[12]

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

[13]

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

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

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

[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.

[17]

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

[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. 

[19]

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

[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.

[21]

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

[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. 

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[28]

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

[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.

[30]

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

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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