Hostname: page-component-7f64f4797f-p4db6 Total loading time: 0 Render date: 2025-11-04T00:19:10.968Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON THE OPTIMAL CONTROL OF STOCKS ACCUMULATING WITH A DELAY

Published online by Cambridge University Press:  13 July 2010

Ralph Winkler
Ralph Winkler*
Affiliation:
University of Bern
*
Address correspondence to: Ralph Winkler, Department of Economics and Oeschger Centre for Climate Change Research, University of Bern, Schanzeneckstrasse 1, CH-3001 Bern, Switzerland; e-mail: mail@ralph-winkler.de.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a generic optimal control problem with a stock that accumulates with constant delay. We show that the optimal system dynamics reduces to a system of ordinary differential equations, implying monotonic optimal paths, if the objective function is additively separable in the stock and the control. This is, however, not true for general objective functions, which may exhibit nonmonotonic and oscillatory optimal paths. The reason is that the impact of the stock on the objective depends on the current level of the control, whereas the control influences the dynamics of the stock with a delay.

Keywords

Additively Separable ObjectiveDelayed Optimal ControlDifferential–Difference EquationsTime to Build

Information

Type
Notes
Information
Macroeconomic Dynamics , Volume 15 , Issue 4 , September 2011 , pp. 565 - 578
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Asea, P.K. and Zak, P.J. (1999) Time-to-build and cycles. shape Journal of Economic Dynamics and Control 23, 1155–1175.Google Scholar
Bambi, M. (2008) Endogenous growth and time-to-build: The AK case. shape Journal of Economic Dynamics and Control 32, 1015–1040.CrossRefGoogle Scholar
Bellman, R. and Cooke, K.L. (1963) shape Differential–Difference Equations. New York: Academic Press.Google Scholar
Boucekkine, R., de la Croix, D., and Licandro, O. (2002) Vintage human capital, demographic trends, and endogenous growth. shape Journal of Economic Theory 104, 340–375.Google Scholar
Boucekkine, R., de la Croix, D., and Licandro, O. (2004) Modelling vintage structures with DDEs: Principles and applications. shape Mathematical Population Studies 11, 151–179.Google Scholar
Boucekkine, R., Germain, M., and Licandro, O. (1997) Replacement echoes in the vintage capital growth model. shape Journal of Economic Theory 74, 333–348.CrossRefGoogle Scholar
Boucekkine, R., Germain, M., Licandro, O., and Magnus, A. (1998) Creative destruction, investment volatility, and the average age of capital. shape Journal of Economic Growth 3, 361–384.CrossRefGoogle Scholar
Boucekkine, R., Germain, M., Licandro, O., and Magnus, A. (2001) Numerical solution by iterative methods of a class of vintage capital models. shape Journal of Economic Dynamics and Control 25, 655–669.Google Scholar
Boucekkine, R., Licandro, O., Puch, L.A., and del Rio, F. (2005) Vintage capital and the dynamics of the AK model. shape Journal of Economic Theory 120, 39–72.Google Scholar
Brandt-Pollmann, U., Winkler, R., Moslener, U., and Schlöder, J. (2008) Numerical solution of optimal control problems with constant control delays. shape Computational Economics 31, 181–206.Google Scholar
d'Albis, H. and Augeraud-Véron, E. (2009) Competitive growth in a life-cycle model: Existence and uniqueness. shape International Economic Review 50, 459–84.CrossRefGoogle Scholar
de la Croix, D. and Licandro, O. (1999) Life expectancy and endogenous growth. shape Economics Letters 65, 255–263.Google Scholar
Diehl, M., Leineweber, D.B., and Schäfer, A.A.S. (2001) Muscod-II users' manual. Preprint 2001-25, Interdisciplinary Center for Scientific Computing, University of Heidelberg.Google Scholar
El-Hodiri, M.A., Loehman, E., and Whinston, A. (1972) An optimal growth model with time lags. shape Econometrica 40, 1137–1146.Google Scholar
Falk, I. and Mendelsohn, R. (1993) The economics of controlling stock pollutants: An efficient strategy for greenhouse gases. shape Journal of Environmental Economics and Management 25, 76–88.Google Scholar
Gandolfo, G. (1996) shape Economic Dynamics, 3rd completely revised and enlarged edition. Berlin: Springer.Google Scholar
Goulder, L.H. and Mathai, K. (2000) Optimal CO2 abatement in the presence of induced technological change. shape Journal of Environmental Economics and Management 39, 1–38.Google Scholar
Hayek, F.A. (1941) shape The Pure Theory of Capital. London: Routledge & Kegan Paul.Google Scholar
Jevons, W.S. ([1871]1911) shape The Theory of Political Economy, 4th ed. First published in 1871, second, enlarged edition in 1879. London: Macmillan.Google Scholar
Kalecki, M. (1935) A macroeconomic theory of business cycles. shape Econometrica 3, 327–344.Google Scholar
Kamien, M.I. and Schwartz, N.L. (1992) shape Dynamic Optimization. The Calculus of Variations and Optimal Control in Economics and Management, 2nd ed.Amsterdam: Elsevier.Google Scholar
Kolmanovskii, V.B. and Myshkis, A.D. (1999) shape Introduction to the Theory and Applications of Functional Differential Equations. Dordrecht, The Netherlands: Kluwer.Google Scholar
Kydland, F.E. and Prescott, E.C. (1982) Time to build and aggregate fluctuations. shape Econometrica 50, 1345–1370.CrossRefGoogle Scholar
Leineweber, D.B., Bauer, I., Bock, H.G., and Schlöder, J.P. (2003) An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization: I. Theoretical aspects. shape Computers and Chemical Engineering 27, 157–166.Google Scholar
Moslener, U. and Requate, T. (2007) Optimal abatement in dynamic multi-pollutant problems when pollutants can be complements or substitutes. shape Journal of Economic Dynamics and Control 31, 2293–2316.Google Scholar
Moslener, U. and Requate, T. (2008) The dynamics of optimal abatement strategies for multiple pollutants—An illustration in the greenhouse. shape Ecological Economics 68, 1521–1534.Google Scholar
Rustichini, A. (1989) Hopf bifurcation for functional differential equations of mixed type. shape Journal of Dynamics and Differential Equations 1, 145–177.Google Scholar
von Böhm-Bawerk, E. ([1889]1921) shape Kapital und Kapitalzins. Positive Theorie des Kapitals [Capital and Interest. The Positive Theory of Capital], 4th ed.London: Macmillan.Google Scholar
Winkler, R. (2008) Optimal Control of Pollutants with Delayed Stock Accumulation. Working paper 08/91, CER-ETH—Center of Economic Research at ETH Zurich.Google Scholar
Winkler, R. (2009) Time-to-Build in the AK Growth Model: Short-Run Dynamics and Response to Exogenous Shocks. Mimeo, CER-ETH—Center of Economic Research at ETH Zurich.Google Scholar