Journal of Mathematical Economics 46 (2010) 1243–1246 Contents lists available at ScienceDirect Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco Short communication Delay equivalence in capital accumulation models  Jonathan P. Caulkins a , Richard F. Hartl b,∗ , Peter M. Kort c,d a Carnegie Mellon University, Qatar Campus, Heinz College’s School of Public Policy & Management, School of Information Systems & Management, 5000 Forbes Ave, Pittsburgh, PA 15213, USA b University of Vienna, School of Business, Economics, and Statistics, Bruennerstrasse 72, A-1210 Vienna, Austria c Tilburg University, Department of Econometrics & Operations Research and CentER, P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands d Department of Economics, University of Antwerp, Prinsstraat 13, 2000 Antwerp 1, Belgium article info Article history: Received 25 March 2010 Received in revised form 8 July 2010 Accepted 30 August 2010 Available online 16 September 2010 JEL classification: C61 D92 Keywords: Capital accumulation Delayed response Time-to-build Time-to-install/deliver Optimal control abstract We study delays in capital accumulation models. Our contribution is threefold. First, we identify a class of models that can be transformed into standard optimal control models without delay. Second, in the single state versions of these models the state trajectory is monotonic in the optimal solution. This is noteworthy given the common belief that adding delays facilitates oscillatory behavior of capital, output and investment. Third, we identify an equivalence result between time-to-install/deliver problems and time-to-build problems. © 2010 Elsevier B.V. All rights reserved. This paper studies capital accumulation models with delays. Capital accumulation models have been investigated exten- sively, but until recently usually without delays. That is striking inasmuch as there usually is a delay between the decision to launch a capital project and when that investment first bears fruit, whether the investment is in physical assets, such as building a factory, knowledge assets, such as inventing a new technology or product, or human capital, such as raising the educational level of one’s workforce. Here we in some sense defend the traditional emphasis on models without delays by showing that an important class of models with delays can be transformed into equivalent optimal control problems without delays. This result both extends the relevance of past work to some problems with delay and provides a strategy for analyzing certain types of models with delay. The equivalence holds regardless of the dimension of the state space. In the special case of one-dimensional models, the existence of an equivalent problem without delays implies that the optimal solution to the model with delays cannot involve oscillation. This is in agreement with, e.g., Benhabib and Rustichini (1991), who arrive at non-oscillatory behavior under exponential depreciation, which is also assumed by us. However, Benhabib and Rustichini (1991) also show that, as soon as we depart from the usually assumed exponential depreciation, then the optimal solution becomes oscillatory under  The authors like to thank an associate editor, two anonymous reviewers, Mauro Bambi, Raouf Boucekkine, Gustav Feichtinger, and Omar Licandro for their helpful comments. ∗ Corresponding author. Tel.: +43 1 4277 38091; fax: +43 1 4277 38094. E-mail address: richard.hartl@univie.ac.at (R.F. Hartl). 0304-4068/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2010.08.021