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GENERALIZATIONS OF OPTIMAL GROWTH THEORY: STOCHASTIC MODELS, MATHEMATICS, AND METASYNTHESIS

Published online by Cambridge University Press:  10 August 2016

Stephen Spear  and
Warren Young
Stephen Spear*
Affiliation:
Tepper School of Business, Carnegie Mellon University
Warren Young
Affiliation:
Bar Ilan University
*
Address correspondence to: Stephen Spear, Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA; e-mail: [email protected].
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In previous papers [Spear and Young (2014, 2015)], we surveyed the origins, evolution, and dissemination of optimal growth, two-sector and turnpike models up to the early 1970s. Regarding subsequent developments in growth theory, a number of prominent observers, such as Fischer (1988), Stern (1991), and McCallum (1996), maintained that after significant progress in the 1950s and 1960s, economic growth theory “received relatively little attention for almost two decades” [Fischer (1988, p. 329)], and that “by the late 1960s early 1970s, research on the theory of growth more or less stopped” [Stern (1991, p. 259)]. Stern went on to say “the latter half of the 1980s saw a rekindling of growth theory, particularly in the work of Romer . . . and Lucas” (1991, p. 259), that is to say, in the form of “endogenous growth” models. McCallum, for his part, wrote (1996, p. 41), “After a long period of quiescence, growth economics has in the last decade (1986–1995) become an extremely active area of research.” Moreover, Brock and Mirman's (1972b) paper was the sole “extension” of Ramsey–Cass–Koopmans to a “stochastic environment” mentioned by McCallum (1996, 49).

Type
MD Survey
Information
Macroeconomic Dynamics , Volume 21 , Issue 2 , March 2017 , pp. 515 - 544
Copyright
Copyright © Cambridge University Press 2016 

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References

REFERENCES

Acemoglu, D. (2008) Introduction to Modern Economic Growth. Princeton, NJ: Princeton University Press.Google Scholar
Assad, A. (2011) Bellman, Richard E.. In Assad, A. and Gass, S. (eds.), Profiles in Operations Research. New York: Springer.Google Scholar
Bellman, R. (1951) On a General Class of Problems Involving Sequential Analysis. Rand research memorandum RM-647.Google Scholar
Bellman, R. (1952) On the theory of dynamic programming. Proceedings of the National Academy of Sciences 38, 716–719.Google Scholar
Bellman, R. (1954a) Some problems in the theory of dynamic programming. Econometrica 22, 37–48.CrossRefGoogle Scholar
Bellman, R. (1954b) The Theory of Dynamic Programming. Rand paper P-550.Google Scholar
Bellman, R. (1954c) The theory of dynamic programming. Bulletin of the American Mathematical Society 60, 503–515.CrossRefGoogle Scholar
Bellman, R. (1956) Dynamic Programming and Its Application to Variational Problems in Mathematical Economics. Rand paper P-796.Google Scholar
Bellman, R. (1957) Dynamic Programming. Princeton, NJ: Princeton University Press.Google ScholarPubMed
Bellman, R. (1958) Dynamic programming and its application to variational problems in mathematical economics. In Graves, L. (ed.), Proceedings of the Eighth Symposium in Applied Mathematics, American Mathematical Society, April 12–13, 1956, University of Chicago, pp. 115–138. New York: McGraw-Hill.Google Scholar
Bellman, R. (1963) Dynamic Programming and Mathematical Economics. Rand research memorandum R-3539-PR.Google Scholar
Bellman, R. (1984) Eye of the Hurricane: An Autobiography. Singapore: World Scientific.Google Scholar
Bellman, R. and Lee, E. (1984) History and development of dynamic programming. Control Systems Magazine (November), 24–28.Google Scholar
Blackwell, D. (1961) On the functional equation of dynamic programming. Journal of Mathematical Analysis and Applications 2, 273–276.Google Scholar
Blackwell, D. (1962) Discrete dynamic programming. Annals of Mathematical Statistics 33, 719–726.Google Scholar
Blackwell, D. (1964a) Memoryless strategies in finite state dynamic programming. Annals of Mathematical Statistics 35, 863–865.Google Scholar
Blackwell, D. (1964b) Probability bounds via dynamic programming. Proceedings of Symposia in Applied Mathematics 16, 277–280.Google Scholar
Blackwell, D. (1965) Discounted dynamic programming. Annals of Mathematical Statistics 36, 226–235.CrossRefGoogle Scholar
Bourguignon, F. (1974) A particular class of continuous time stochastic growth models. Journal of Economic Theory 9, 141–158.CrossRefGoogle Scholar
Brock, W. (1971) Sensitivity of optimal growth paths with respect to a change in target stocks. In Brockman, G. and Weber, W. (eds.), Contributions to the Von Neumann Growth Model, pp. 73–89. New York: Springer.Google Scholar
Brock, W. and MacGill, M. (1979) Dynamics under uncertainty. Econometrica 47, 843–868.CrossRefGoogle Scholar
Brock, W. and Mirman, L. (1970a) The Stochastic Modified Golden Rule in a One Sector Model of Economic Growth with Uncertain Technology. Cornell and Rochester mimeograph.Google Scholar
Brock, W. and Mirman, L. (1970b) The Stochastic Modified Golden Rule in a One Sector Model of Economic Growth with Uncertain Technology. Paper presented at the Econometric Society meeting, Detroit, December.Google Scholar
Brock, W. and Mirman, L. (1971) Optimal Economic Growth and Uncertainty: The Ramsey–Weizacker Case. Working paper 7, MSSB Workshop on the Theory of Markets and Uncertainty, Department of Economics, University of California at Berkeley.Google Scholar
Brock, W. and Mirman, L. (1972a) A one sector model of economic growth with uncertain technology: An example of steady state analysis in a stochastic control problem. In Balakrishnan, A. (ed.), Techniques of Optimization, pp. 407–419. New York: Academic Press Google Scholar
Brock, W. and Mirman, L. (1972b) Optimal economic growth and uncertainty: The discounted case. Journal of Economic Theory 4, 479–513.Google Scholar
Brock, W. and Mirman, L. (1973) Optimal economic growth and uncertainty: The no-discounting case. International Economic Review 14, 560–573.CrossRefGoogle Scholar
Cass, D. (1965) Optimum growth in an aggregative model of capital accumulation. Review of Economic Studies 37, 233–240.Google Scholar
Cutler, D. (1973) Comprehensive Identification of Past MSSB Projects and Participants. Memorandum, Center for Advanced Studies in the Behavioral Sciences, Stanford University, 12 July.Google Scholar
Dobell, A. (1970) Optimization in models of economic growth. In Hull, T. (ed.), Studies in Optimization 1 (Proceedings of the Toronto Symposium, 11–14 June 1968), pp. 1–27. New York: Society for Industrial and Applied Mathematics.Google Scholar
Doob, J. (1953) Stochastic Processes. New York: Wiley Google Scholar
Dvoretsky, A., Kiefer, J., and Wolfowitz, J. (1957) The inventory problem I and II. Econometrica 20, 187–222 and 450–466.Google Scholar
Fischer, S. (1988) Recent developments in macroeconomics. Economic Journal 98, 294–339.Google Scholar
Fischer, S. and Merton, R. (1984) Macroeconomics and finance: The role of the stock market. Carnegie Rochester Conference Series on Public Policy 21, 57–108.Google Scholar
Howard, R. (1960) Dynamic Programming and Markov Processes. New York: Wiley.Google Scholar
Ito, K. (1942) On stochastic processes. Japanese Journal of Mathematics 18, 261–301.Google Scholar
Ito, K. (1951) On stochastic differential equations. Memoirs of the American Mathematical Society 4, 1–51.Google Scholar
Ito, K. (1960) Lectures on Stochastic Processes. Bombay: Tata Institute of Research.Google Scholar
Ito, K. and McKean, H. (1964) Diffusion Processes and Their Sample Paths. New York: Springer.Google Scholar
Jeanjean, P. (1971) Optimal Growth with Stochastic Technology in a Closed Economy. Berkeley Center for Research in Management Science working paper.Google Scholar
Jeanjean, P. (1972) Optimal Growth with Stochastic Technology in a Multi-sector Economy. Ph.D. thesis, University of California, Berkeley.Google Scholar
Jeanjean, P. (1974a) Croissance optimale et incertitude: Un modéle a plusiers secteurs. Cahiers du Séminaire d'Économétrie 15, 47–99.Google Scholar
Jeanjean, P. (1974b) Optimal development programs under uncertainty: The undiscounted case. Journal of Economic Theory 7, 66–92.CrossRefGoogle Scholar
Karlin, S. (1955) The structure of dynamic programing models. Naval Research Logistics Quarterly 2, 285–294.Google Scholar
Koopmans, T. (1965) On the concept of optimal economic growth. In Study Week on the Econometric Approach to Development Planning, October 7–13, pp. 225–300. Rome: Pontifical Academy of Science.Google Scholar
Leland, H. (1974) Optimum growth in a stochastic environment. Review of Economic Studies 41, 75–86.Google Scholar
Levhari, D and Srinivasan, T. (1969) Optimal savings under uncertainty. Review of Economic Studies 36, 153–163.Google Scholar
Lucas, R. (1987) Models of Business Cycles. Oxford, UK: Blackwell.Google Scholar
Malinvaud, E. (1965) Croissances optimales dans un modele macroeconomique. In Study Week on the Econometric Approach to Development Planning, October 7–13, pp. 301–384. Rome: Pontifical Academy of Science.Google Scholar
McCallum, B. (1996) Neoclassical vs. endogenous growth analysis: an overview. Federal Reserve Bank of Richmond Economic Quarterly 82, 41–71.Google Scholar
McKenzie, L. (1968) Accumulation programs of maximum utility and the Von Neumann facet. In Wolf, J. (ed.), Value, Capital and Growth, pp. 353–383. Edinburgh: Edinburgh University Press.Google Scholar
McKenzie, L. (1999) [1998] The First Conferences on the Theory of Economic Growth. Working paper 459, Rochester Center for Economic Research, University of Rochester, January.Google Scholar
Merton, R. (1969) Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economics and Statistics 51, 247–257.Google Scholar
Merton, R. (1970) Analytical Optimal Control Theory as Applied to Stochastic and Non-stochastic Economies. Ph.D thesis, MIT.Google Scholar
Merton, R. (1971) Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373–413.CrossRefGoogle Scholar
Merton, R. (1973) An Asymptotic Theory of Growth under Uncertainty. Sloan School working paper 673-73, MIT.Google Scholar
Merton, R. (1975) An asymptotic theory of growth under uncertainty. Review of Economic Studies 42, 375–393.Google Scholar
Mirman, L. (1970a) The Steady State Behavior of a Class of One-Sector Growth Models with Uncertain Technology. Mimeograph, Department of Economics, University of Rochester.Google Scholar
Mirman, L. (1970b) Two Essays on Uncertainty and Economics. Ph.D. thesis, University of Rochester.Google Scholar
Mirman, L. (1971) Uncertainty and optimal consumption decisions. Econometrica 39, 179–185.CrossRefGoogle Scholar
Mirman, L. (1972) On the existence of steady state measures for one sector growth models with uncertain technology. International Economic Review 13, 271–286.Google Scholar
Mirman, L. (1973) The steady state behavior of a class of one-sector growth models with uncertain technology. Journal of Economic Theory 6, 219–242.Google Scholar
Mirman, L. and Zilcha, I. (1975) On optimal growth under uncertainty. Journal of Economic Theory 11, 329–339.CrossRefGoogle Scholar
Mirman, L. and Zilcha, I. (1976) Unbounded shadow prices for optimal stochastic growth models. International Economic Review 17, 121–132.Google Scholar
Mirman, L. and Zilcha, I. (1977) Characterizing optimal policies in a one sector model of economic growth under uncertainty. Journal of Economic Theory 14, 389–401.Google Scholar
Mirrlees, J. (1964a) Optimum economic policies under uncertainty. Paper presented at Rochester-SSRC Conference on Mathematical Models of Economic Growth.Google Scholar
Mirrlees, J. (1964b) Optimum Planning for a Dynamic Economy. Ph.D. thesis, University of Cambridge.Google Scholar
Mirrlees, J. (1964c) Structure of Optimum Policies in a Macroeconomic Model with Technical Change, Paper presented at the Econometric Society meeting, Zurich.Google Scholar
Mirrlees, J. (1965) Optimum Accumulation under Uncertainty, Paper presented at the First World Econometric Society meeting, Tokyo.Google Scholar
Mirrlees, J. (1966) Optimum accumulation under uncertainty. Econometrica 34 (5).Google Scholar
Mirrlees, J. (1967) Optimum growth when technology is changing. Review of Economic Studies 34, 95–124.Google Scholar
Mirrlees, J. (1974) Optimal accumulation under uncertainty: The case of stationary returns to investment. In Dreze, J. (ed.), Allocation under Uncertainty, pp. 36–50. London: Macmillan.Google Scholar
Newhouse, A. (1958) Review of Bellman's Dynamic Programming . American Mathematical Monthly 65, 788–789.Google Scholar
Olson, L. and Roy, S. (2006) Theory of stochastic optimal economic growth. In Dana, R., Van, C., Mitra, T., and Nishimura, K. (eds.), Handbook on Optimal Growth, pp. 297–335. New York: Springer.Google Scholar
Pesch, H. (2012) Carathéodory on the road to the maximum principle. In Optimization Stories, Documenta Mathematica, 317–329.Google Scholar
Pesch, H. and Plail, M. (2009) The maximum principle of optimal control. Control and Cybernetics 38, 973–995.Google Scholar
Pesch, H. and Plail, M. (2012) The cold war and the maximum principle of optimal control. In Optimization Stories, Documenta Mathematica, 331–343.Google Scholar
Phelps, E. (1960a) Capital Risk and Household Consumption Path: A Sequential Utility Analysis. Cowles Foundation discussion paper 101.Google Scholar
Phelps, E. (1960b) Optimal Inventory Policy for Serviceable and Reparable Stocks. Rand paper P-1996.Google Scholar
Phelps, E. (1961) The Accumulation of Risky Capital: A Discrete-Time Sequential Utility Analysis. Cowles Foundation discussion paper 10.Google Scholar
Phelps, E. (1962a) Optimal Decision Rules for the Procurement, Repair or Disposal of Spare Parts. Rand memorandum RM-2920-PR.Google Scholar
Phelps, E. (1962b) The accumulation of risky capital: A sequential utility analysis. Econometrica 30, 729–743.Google Scholar
Pontryagin, L., Bolyyanskii, V., Gamkriledze, R., and Mischenko, E. (1962) The Mathematical Theory of Optimal Processes. New York: Interscience.Google Scholar
Puterman, M. (1994) Markov Decision Processes: Discrete Stochastic Dynamic Programming. New York: Wiley.CrossRefGoogle Scholar
Radner, R. (1963) Notes on the Theory of Economic Planning. Technical report 9, Center for Research in Management Science, University of California at Berkeley.Google Scholar
Radner, R. (1964) Dynamic Programming of Economic Growth Technical report 17, Center for Research in Management Science, University of California at Berkeley.Google Scholar
Radner, R. (1966) Optimal growth in a linear-logarithmic economy. International Economic Review 7, 1–33.Google Scholar
Radner, R. (1967) Dynamic programming of economic growth. In Malinvaud, E. and Bacharach, M. (eds.), Activity Analysis in the Theory of Growth and Planning, Proceedings of a Conference Held by the International Economic Association, pp. 111–141. New York: St. Martin's Press.Google Scholar
Radner, R. (1971) Balanced stochastic growth at the maximum rate. In Bruckmann, G. and Weber, W. (eds.), Contributions to the Von Neumann Growth Model, pp. 39–62. New York: Springer Google Scholar
Radner, R. (1972) Optimal steady state behavior of an economy with stochastic production and resources. In Day, R. and Robinson, S. (eds.), Mathematical Topics in Economic Theory and Computation, pp. 99–112. Philadelphia: SIAM.Google Scholar
Radner, R. (1973) Optimal stationary consumption with stochastic production and resources. Journal of Economic Theory 6, 68–90.Google Scholar
Report of 1955 Ann Arbor Meeting, August 29–September 1, 1955, Econometrica 24 (April 1956), 198–210.Google Scholar
Report of the North American Regional Conference, Detroit, December 1970, Growth Models, Econometrica 39 (July 1971), 293–315, 344–350.Google Scholar
Samuelson, P. (1976) The periodic turnpike theorem. Nonlinear Analysis, Theory, Method and Applications 1, 3–13.Google Scholar
Sandmo, A. (1970) The effect of uncertainty on saving decisions. Review of Economic Studies 37, 353–360.Google Scholar
Spear, S. and Young, W. (2014) Optimum savings and optimal growth: The Cass-Malinvaud-Koopmans nexus. Macroeconomic Dynamics 18, 215–243.Google Scholar
Spear, S. and Young, W. (2015) Two-sector growth, optimal growth, and the turnpike: Amalgamation and metamorphosis. Macroeconomic Dynamics 19, 394–424.Google Scholar
Stern, N. (1991) Public policy and the economics of development. European Economic Review 35, 241–271.Google Scholar
Stiglitz, J. (1969) A Note on Behavior towards Risk with Many Commodities. Cowles Foundation discussion paper 262.Google Scholar
Tukey, J.W. (1942) Some notes on the separation of convex sets. Portugaliae Mathematica 3 (2), 95–102.Google Scholar
Uzawa, H. (1964) Optimal growth in a two-sector model of capital accumulation. Review of Economic Studies 31, 1–24.Google Scholar
Wald, A. (1950) Statistical Decision Functions. New York: Wiley Google Scholar
Wiener, N. (1923) Note on a paper of Banach. Fundamenta Mathematicae 4, 136–143.Google Scholar
Young, W. (2014) Real Business Cycle Models in Economics. New York: Routledge.Google Scholar