Elsevier

Automatica

Volume 47, Issue 9, September 2011, Pages 1868-1877
Automatica

Numerical solution of a conspicuous consumption model with constant control delay

https://doi.org/10.1016/j.automatica.2011.06.004Get rights and content
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Abstract

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account.

This non-standard optimal control problem is difficult to solve analytically, and solutions depend on the variable model parameters. Therefore, we use a numerical result-driven approach. We propose a structure-exploiting direct method for optimal control to solve this challenging optimization problem. In particular, we discretize the uncertainties in the model formulation by using scenario trees and target the control delays by introduction of slack control functions.

Numerical results illustrate the validity of our approach and show the impact of uncertainties and delay effects on optimal economic strategies. During the recession, delayed optimal prices are higher than the non-delayed ones. In the normal economic period, however, this effect is reversed and optimal prices with a delayed impact are smaller compared to the non-delayed case.

Keywords

Optimal control
Optimization under uncertainties
Control delays
Economic system
Recession

Cited by (0)

Tony Huschto was born in Bad Muskau, Germany, in 1985. He studied mathematics at the Brandenburg University of Technology in Cottbus, Germany, and at the Högskolan Halmstad, Sweden. There he obtained his diploma in 2009 and his master’s degree in financial mathematics in 2008. Currently he is a Ph.D. student at the Interdisciplinary Center for Scientific Computing of the University of Heidelberg. His research interests are in stochastic differential equations and optimal control, as well as their applications to economics.

Gustav Feichtinger was born in Wiener Neustadt, Austria, in 1940. He graduated from the University of Vienna in 1963 (Ph.D. in Mathematics). He was assistant and associate professor between 1965 and 1972 at the University of Bonn, Germany, where he got his Habilitation in Statistics and Operations Research. From 1972 to 2008 he was full professor for OR at the Vienna University of Technology. He is currently professor emeritus at the Institute for Mathematical Methods in Economics at the Vienna University of Technology. His research interests include optimal control, differential games, nonlinear dynamical systems, the economics of ‘deviant’ behavior, and population dynamics.

Richard F. Hartl holds a chair of Production and Operations Management at the University of Vienna. He studied mathematics at the University of Technology in Vienna, where he also obtained his Ph.D. He was Post-Doctoral fellow at the University of Toronto and Assistant and Associate Professor at the University of Technology in Vienna. Prior to his present appointment he was professor at the OvG University in Magdeburg, Germany. He received several recognitions for his work including Senior Extramural Fellow of the Center for Economic Research (CentER), University of Tilburg. His research interest involves application of OR methods in Production and Operations Management as well as application of optimal control in economics and management.

Peter M. Kort was born in Zierikzee, the Netherlands in 1961. He received a M.Sc. degree in 1984 from the Erasmus University Rotterdam and a Ph.D. from Tilburg University in 1988. In 1991 he became research fellow of the Royal Netherlands Academy of Arts and Sciences. Currently he is full professor in Dynamic Optimization in Economics and Operations Research at Tilburg University and Guest Professor at the University of Antwerp. His main research areas are applied optimal control, dynamics of the firm, real options theory, and strategic industrial organization, in which he published about 85 papers in refereed international journals.

Sebastian Sager studied mathematics in Montpellier and Heidelberg. He obtained his Ph.D. in 2006 from the Universität Heidelberg. After a year as postdoctoral fellow in Madrid, he got a position as a Junior Research Group leader at the Interdisciplinary Center for Scientific Computing in Heidelberg. He received the Dissertation Prize of the German Operations Research Society in 2006 and the Klaus Tschira Award for Achievements in Public Understanding of Science in 2007. His main interests are in mixed-integer nonlinear optimization and optimal control.

Andrea Seidl was born in Vienna, Austria, in 1982. She received a masters degree in “economics and computer science” in 2005 and a Ph.D. degree in 2009 both from the Vienna University of Technology. Currently she works as a postdoctoral research assistant at the Institute for Mathematical Methods in Economics at the Vienna University of Technology. Her research interests include optimal control and its applications, in particular multi-stage models.

This research was supported by the Heidelberg Graduate School Mathematical and Computational Methods for the Sciences and the Austrian Science Fund (FWF) under Grant P21410-G16, which is gratefully acknowledged. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Qing Zhang under the direction of Editor Berç Rüstem.