Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-27T17:09:35.804Z Has data issue: false hasContentIssue false
  • English
  • Français

RENEWAL THEORY WITH EXPONENTIAL AND HYPERBOLIC DISCOUNTING

Published online by Cambridge University Press:  18 December 2007

J. A. M. van der Weide ,
Suyono  and
J. M. van Noortwijk
J. A. M. van der Weide
Affiliation:
Faculty of Electrical Engineering, Mathematics and Computer ScienceDelft University of TechnologyNL-2600 GA Delft, The Netherlands E-mail: j.m.vanderweide@tudelft.nl
Suyono
Affiliation:
Jurusan Matematika FMIPA Universitas Negeri JakartaJakarta Timur 13200, Indonesia E-mail: synjkt@yahoo.com
J. M. van Noortwijk
Affiliation:
HKV ConsultantsNL-8203 AC Lelystad, The Netherlands and Faculty of Electrical Engineering, Mathematics and Computer Science Delft University of Technology NL-2600 GA Delft, The Netherlands E-mail: j.m.van.noortwijk@hkv.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

To determine optimal investment and maintenance decisions, the total costs should be minimized over the whole life of a system or structure. In minimizing life-cycle costs, it is important to account for the time value of money by discounting and to consider the uncertainties involved. This article presents new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbolic, and no discounting). The main results include expressions for the first and second moment of the discounted costs over a bounded and unbounded time horizon as well as asymptotic expansions for nondiscounted costs.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 1 , January 2008 , pp. 53 - 74
Copyright
Copyright © Cambridge University Press 2008

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.)

References

1.Atherton, E. & French, S. (1998). Valuing the future: A MADA example involving nuclear waste storage. Journal of Multi-Criteria Decision Analysis 7(6): 304–321.3.0.CO;2-0>CrossRefGoogle Scholar
2.Brealey, R.A. & Myers, S.C. (2003). Principles of corporate finance, 7th ed.New York: McGraw-Hill/Irwin.Google Scholar
3.Dall'Aglio, G. (1964). Present value of a renewal process. The Annals of Mathematical Statistics 35(3): 1326–1331.CrossRefGoogle Scholar
4.den Iseger, P. (2006). Numerical transform inversion using Gaussian quadrature. Probability in the Engineering and Informational Sciences 20(2): 1–44.CrossRefGoogle Scholar
5.Feller, W. (1949). Fluctuation theory of recurrent events. Transactions of the American Mathematical Society 67(1): 98–119.CrossRefGoogle Scholar
6.Frederick, S., Loewenstein, G., & O'Donoghue, T. (2002). Time discounting and time preference: A critical review. Journal of Economic Literature 40(2): 351–401.CrossRefGoogle Scholar
7.Herrnstein, R.J. (1981). Self-control as response strength. In Bradshaw, C.M., Szabadi, E., & Lowe, C.F. (eds.), Quantification of steady-state operant behavior. Amsterdam: Elsevier/North-Holland.Google Scholar
8.Léveillé, G. & Garrido, J. (2001). Moments of compound renewal sums with discounted claims. Insurance: Mathematics and Economics 28(2): 217–231.Google Scholar
9.Loewenstein, G. & Prelec, D. (1992). Anomalies in intertemporal choice: Evidence and an interpretation. Quarterly Journal of Economics 107(2): 573–597.CrossRefGoogle Scholar
10.Mazur, J.E. (1987). An adjustment procedure for studying delayed reinforcement. In Commoms, M.L., Mazur, J.E., Nevin, J.A., & Rachlin, H. (eds.), Quantitative analysis of behavior: The effect of delay and intervening events on reinforcement value. Hillsdale, NJ: Erlbaum, pp. 55–73.Google Scholar
11.Morey, R.C. (1966). Some stochastic properties of a compound-renewal damage model. Operations Research 14(5): 902–908.CrossRefGoogle Scholar
12.Phelps, E.S. & Pollak, R.A. (1968). On second-best national saving and game-equilibrium growth. Review of Economic Studies 35(2): 185–199.CrossRefGoogle Scholar
13.Rackwitz, R. (2000). Optimization: The basis of code-making and reliability verification. Structural Safety 22(1): 27–60.CrossRefGoogle Scholar
14.Rackwitz, R. (2001). Optimizing systematically renewed structures. Reliability Engineering and System Safety 73(3): 269–279.CrossRefGoogle Scholar
15.Rackwitz, R., Lentz, A., & Faber, M. (2005). Socio-economically sustainable civil engineering infrastructures by optimization. Structural Safety 27(3): 187–229.CrossRefGoogle Scholar
16.Samuelson, P.A. (1937). A note on measurement of utility. Review of Economic Studies 4(2): 155–161.CrossRefGoogle Scholar
17.Smith, W.L. (1954). Asymptotic renewal theorems. Proceedings of the Royal Society of Edinburgh, Section A (Mathematical and Physical Sciences) 64: 9–48.Google Scholar
18.Smith, W.L. (1958). Renewal theory and its ramifications. Journal of the Royal Statistical Society, Series B 20(2): 243–302.Google Scholar
19.Smith, W.L. (1966). On necessary and sufficient conditions for the convergence of the renewal density. Transactions of the American Mathematical Society 104(1): 79–100.CrossRefGoogle Scholar
20.Tijms, H.C. (2003). A first course in stochastic models. New York: Wiley.CrossRefGoogle Scholar
21.van Noortwijk, J.M. (2003). Explicit formulas for the variance of discounted life-cycle cost. Reliability Engineering and System Safety 80(2): 185–195.CrossRefGoogle Scholar
22.Wagner, H.M. (1975). Principles of operations research, 2nd ed.Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
23.Washburn, A. (1992). Present values with renewals. Management Science 38(6): 846–850.CrossRefGoogle Scholar
24.Weitzman, M.L. (1998). Why the far-distant future should be discounted at its lowest possible rate. Journal of Environmental Economics and Management 36(3): 201–208.CrossRefGoogle Scholar
25.Weitzman, M.L. (2001). Gamma discounting. American Economic Review 91(1): 260–271.CrossRefGoogle Scholar
26.Wolff, R.W. (1989). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar