Abstract
This paper generalizes the classical discounted utility model introduced in Samuelson (Rev. Econ. Stud. 4:155–161, 1937) by replacing a constant discount rate with a function. The existence of recursive utilities and their constructions are based on Matkowski’s extension of the Banach Contraction Principle. The derived utilities go beyond the class of recursive utilities studied in the literature and enable a discussion on subtle issues concerning time preferences in the theory of finance, economics or psychology. Moreover, our main results are applied to the theory of optimal economic growth related with resource extraction models with unbounded utility function of consumption.
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References
Alvarez, F., & Stokey, N. (1998). Dynamic programming with homogeneous functions. Journal of Economic Theory, 82, 167–189.
Becker, R. A., & Boyd, J. H. III (1997). Capital theory, equilibrium analysis and recursive utility. New York: Blackwell.
Benzion, U., Rapoport, A., & Yagil, J. (1989). Discount rates inferred from decisions: an experimental study. Management Science 35, 270–284.
Berge, C. (1963). Topological spaces. New York: MacMillan.
Bhattacharya, R., & Majumdar, M. (2007). Random dynamical systems: theory and applications. Cambridge: Cambridge University Press.
Blackwell, D. (1965). Discounted dynamic programming. Annals of Mathematical Statistics, 36, 226–235.
Boyd, J. H. III (1990). Recursive utility and the Ramsey problem. Journal of Economic Theory, 50, 326–345.
Boyd, J. H. III (2006). Discrete-time recursive utility. In R. A. Dana, C. Le Van, T. Mitra, & K. Nishimura (Eds.), Handbook of optimal growth 1, discrete time (pp. 251–272). New York: Springer.
Denardo, E. V. (1967). Contraction mappings in the theory underlying dynamic programming. SIAM Review, 9, 165–177.
Dugundji, J., & Granas, A. (2003). Fixed point theory. New York: Springer.
Epstein, L., & Hynes, J. A. (1983). The rate of time preference and dynamic economic analysis. Journal of Political Economy, 91, 611–635.
Feinberg, E. A. (2002). Total reward criteria. In E. A. Feinberg & A. Shwartz (Eds.), Handbook of Markov decision processes (pp. 173–207). Boston: Kluwer.
Frederick, S., Loewenstein, G., & O’Donoghue, T. (2002). Time discounting and time preference: a critical review. Journal of Economic Literature, 11, 351–401.
Green, L., Myerson, J., & McFadden, E. (1997). Rate of temporal discounting decreases with amount of reward. Memory & Cognition, 25, 715–723.
Hernández-Lerma, O., & Lasserre, J. B. (1996). Discrete-time Markov control processes: basic optimality criteria. New York: Springer.
Hernández-Lerma, O., & Lasserre, J. B. (1999). Further topics on discrete-time Markov control processes. New York: Springer.
Hinderer, K. (1970). Foundations of non-stationary dynamic programming with discrete time parameter. In Lecture notes in oper. res., Vol. 33. New York: Springer.
Jaśkiewicz, A., Matkowski, J., & Nowak, A. S. (2011). Persistently optimal policies in stochastic dynamic programming with generalized discounting. Working Paper. Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Poland.
Kirby, K. N. (1997). Bidding on the future: evidence against normative discounting of delayed rewards. Journal of Experimental Psychology. General, 126, 54–70.
Koopmans, T. C. (1960). Stationary ordinal utility and impatience. Econometrica, 28, 287–309.
Kreps, D. M., & Porteus, E. L. (1978). Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46, 185–200.
Le Van, C., & Morhaim, L. (2002). Optimal growth models with bounded or unbounded returns: a unifying approach. Journal of Economic Theory 105, 158–187.
Le Van, C., & Vailakis, Y. (2005). Recursive utility and optimal growth with bounded or unbounded returns. Journal of Economic Theory 123, 187–209.
Lucas, R. E., & Stokey, N. (1984). Optimal growth with many consumers. Journal of Economic Theory, 32, 139–171.
Marinacci, M., & Montrucchio, L. (2010). Unique solutions for stochastic recursive utilities. Journal of Economic Theory, 145, 1776–1804.
Matkowski, J. (1975). Integral solutions of functional equations. Dissertationes Mathematicae, 127, 1–68.
Ramsey, F. P. (1928). A mathematical theory of saving. Econometrics Journal, 38, 543–599.
Rincón-Zapatero, J. P., & Rodriguez-Palmero, C. (2003). Existence and uniqueness of solutions to the Bellman equation in the unbounded case. Econometrica, 71, 1519–1555.
Rincón-Zapatero, J. P., & Rodriguez-Palmero, C. (2009). Corrigendum to “Existence and uniqueness of solutions to the Bellman equation in the unbounded case. Econometrica, 71, 1519–1555 (2003)”. Econometrica, 77, 317–318.
Samuelson, P. (1937). A note on measurement of utility. Review of Economic Studies 4, 155–161.
Schäl, M. (1975). Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 32, 179–196.
Stokey, N. L., Lucas, R. E., & Prescott, E. (1989). Recursive methods in economic dynamics. Cambridge: Harvard University Press.
Strauch, R. (1966). Negative dynamic programming. Annals of Mathematical Statistics, 37, 871–890.
Thaler, R. H. (1981). Some empirical evidence on dynamic inconsistency. Economics Letters, 8, 201–207.
Uzawa, H. (1968). Time preference, the consumption function, and optimum asset holding. In J. N. Wolfe (Ed.), Value, capital and growth: papers in honor of Sir John Hicks (pp. 485–504). Edinburgh: Edinburgh University Press.
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Jaśkiewicz, A., Matkowski, J. & Nowak, A.S. On variable discounting in dynamic programming: applications to resource extraction and other economic models. Ann Oper Res 220, 263–278 (2014). https://doi.org/10.1007/s10479-011-0931-2
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DOI: https://doi.org/10.1007/s10479-011-0931-2