Skip to main content
SpringerLink
Log in

An overview of turnpike theory: towards the discounted deterministic case

  • Conference paper
Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 14))

Abstract

In the last 5 years, there has been extensive work on the existence and characterization of solutions of undiscounted optimal programs in simple discrete-time models of the ‘choice of technique’ in development planning and of lumber extraction in the economics of forestry. In this expository essay, we present a unified treatment of the characterization results in two of these models. Furthermore, with an eye towards extensions to the discounted setting, we present the general theory, both with or without “smoothness hypotheses” on the felicity function, and in continuous-time and discrete-time taking special care to distinguish asymptotic convergence of optimal programs from their classical turnpike properties. We show how the general results do not translate directly to the particular toy models examined here, and thereby suggest open problems for a more universal theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A preliminary version of this work was presented at the Workshop on Mathematical Economics held in Keio University, Tokyo during November 13–15, 2009. The authors are grateful to Professor Toru Maruyama for his invitation, and to him and Professors Alexander Ioffe, Alejandro Jofre, Boris Mordukhovich and Haru Takahashi for discussion and encouragement. Adriana Piazza gratefully acknowledges the financial support of Programa Basal PFB 03, Centro de Modelamiento Matemático (CMM), Universidad de Chile and that of FONDECYT under project 11090254. Ali Khan would like to acknowledge, in addition, the hospitality of the Departamento de Matemática, Universidad Técnica Federico Santa María, of the CMM and Facultad de Economía y Negocios at the Universidad de Chile during his visit in July 2010, when the final version of this work was completed.

  2. 2.

    For the precise references to these sentences in [15], see Footnote 34 below.

  3. 3.

    See [4, pp. 542–543], also his Fig. 10.14 with time on the horizontal axis and capital stocks on the vertical.

  4. 4.

    See [34] where McKenzie distinguishes three kinds of turnpikes: the early, late and middle turnpikes. In terms of this categorization, Bewley considers only the last two kinds of turnpikes. It is also worth pointing out that in [39], McKenzie uses a different classification scheme based on the Samuelson and Ramsey turnpike. It may be worth stating that in this essay, we are using the word as an adjective, as in “turnpike theory” or in “turnpike theorem”, rather than as a noun.

  5. 5.

    In this connection, see [25] for detailed references to the McKenzie’s oeuvre.

  6. 6.

    Also referred to as the “aggregate model”, or in the macroeconomic literature, as the Ramsey–Cass–Koopmans (RCK) model.

  7. 7.

    See [4, p. 544] where Bewley refers to his Theorems 10.79 and 10.80 in this connection.

  8. 8.

    See [4, p. 551] where the fact that the undiscounted “theory with multiple commodities” is not presented is footnoted. Also see a repetition of this statement on page 577.

  9. 9.

    These are the theorems referred to in Footnote 6. Subsequent to their presentation, he re-emphasizes the meaning of the phrase “true turnpike theorems”; see [4, p. 577, paragraph 4].

  10. 10.

    See, for example, Gale’s [14] justification for considering an infinite-horizon program, one that surely was in the mind of Ramsey, and goes back to [45].

  11. 11.

    In the context of Footnote 3, the word “turnpike” is being used as a noun.

  12. 12.

    For this qualitative–quantitative distinction, we follow Pietsch’s monumental history of Banach spaces; see [44].

  13. 13.

    We shall have occasion to return to this blanket claim in the conclusion to this essay, and qualify it; see Footnote 36 below and the text it footnotes.

  14. 14.

    See the bibliography of [6] and [3] for references to, and the reliance on, this classical literature even in the context of the particular subject-matter we discuss here.

  15. 15.

    This is reprinted without change in [12].

  16. 16.

    Fischer uses the abbreviation CSP to refer to the five volumes of Samuelson’s Collected Papers with the Roman numeral referring to the particular volume, and the Arabic numeral referring to the chapter in it.

  17. 17.

    See McKenzie [34] for a detailed discussion to Dosso; also Mckenzie’s 1963 papers [31, 32] and his 1987 Palgrave entry. The 1963 papers are masterly introductions to the pre-Ramsey literature, and devoted solely to what is being termed the “classical turnpike theorem” in this essay. Their introductions are useful complements to Fischer’s “fully worked out” phrase in the quotation below. Also see McKenzie’s description of the “Hicks’ pilgrimage” in [31, Footnote 1], and a terminological clarification of the term balanced growth in the subsequent footnote.

  18. 18.

    See the references in the papers cited in Footnote 16 above.

  19. 19.

    Thus we now have three different, albeit overlapping definitions of the term “turnpike theory.”

  20. 20.

    This stochastic literature is huge, and deserving of its own overview. Our basic point is that the deterministic results ought to be antecedent to it as a matter of logical priority. At any rate, the 2008 Palgrave entry of Brock–Dechert is again worth quoting: “infinite horizon stochastic multisector models are also basic in constructing econometrically tractable models to use in analysing data. Here, especially, is where stochastic versions of the turnpike theorem (explained below) are used. For example, it is used to justify use of laws of large numbers and central limit theorems in econometric time-series applications.”

  21. 21.

    See [6, Chap. 6] for a comparative results in continuous time.

  22. 22.

    Given the clarity of the phrase, one is led to inquire into the need for the turnpike terminology in the first place. It is almost as if it has been endowed with incantatory power, for approval or for dismissal of the literature! We make a serious attempt in this essay to avoid the turnpike terminology when we can satisfactorily do so.

  23. 23.

    See the comprehensive treatment of this theorem in Krantz–Parks [28].

  24. 24.

    After a consideration of examples in Chap. 3, titled “asymptotic stability and the turnpike property in some simple control problems,” the authors turn to the development of the general theory (in continuous-time) in in the subsequent four chapters. Their Sect. 6.7, and Chap. 7 is particularly relevant to the delineation of the discounted case that is being attempted in this essay.

  25. 25.

    This is the basic autonomous system considered in [6, Sect. 4.2] which should be referred to for the definitions and for simple examples. Note, however, that by following this text, we are not imposing non-negativity constraints of the state variables x.

  26. 26.

    See [6, Definition 1.2] and the discussion following it. Rockafellar [46] also considers optimal trajectories where the inequality is strict and refers to them as satisfying a “strong” overtaking property.

  27. 27.

    See [1, Footnote 2]. This is a rather prescient observation in the light of current interest in environmental economics. Perhaps such a more general model could be refereed to as the MRSS model: the multi-sectoral Ramsey–Samuelson–Solow model.

  28. 28.

    In the continuous-time case, the problem represented in (3) is referred to as an implicit programming problem in [6, Sect. 6.7].

  29. 29.

    The reader is referred to [1, Definition 2.4] for the precise definition. We also note that the symbol x is being used in two senses: the value of a stock at a particular time, as well as a function from the initial stock and the discount factor to ℓ ∞ n.

  30. 30.

    We leave it to the reader as an exercise to show that the continuous-time version of the model, as presented in [19, 54] and their references, is a particular case of the model spelt out in Sect. 2.

  31. 31.

    For the recent papers, see [47, 48] and [21, 22] and their references.

  32. 32.

    The continuous-time version of the model is difficult, and no longer an easily formulated particular version of the model spelt out in Sect. 2; see [42, 43] for the early formulations in the work of Kemp and Wan.

  33. 33.

    For differences in the model as presented in [42, 43] and here, see [22].

  34. 34.

    If footnotes could be labeled “pioneering,” surely this footnote (notationally modified) would be high on such a list. It is a fascinating exercise in the history of economic analysis to trace the work surveyed in Sect. 4 as an elaboration of this footnote.

  35. 35.

    In the context of this epigraph, the reader should note that the Kuhn–Tucker theorem is now referred to as the Karush–Kuhn–Tucker theorem, and what Gale refers to as the discount rate is the discount factor. The first sentence of the epigraph is taken from page 308, the second from pp. 314–315 and the third from p. 310.

  36. 36.

    We see Rockafellar’s recent survey [46] as being kindred in this motivation.

  37. 37.

    And so we end as we began: with the importance of having a clear and well-established terminology. See Footnote 12 above and the text it footnotes.

  38. 38.

    See, for example, Carlson et al. [6], Arkin–Evstegneev [3] and Zaslavski [58], and their references, to the work of both economists and mathematicians.

References

  1. Araujo, A., Scheinkman, J.A.: Comparative dynamics and the turnpike property. Econometrica 45, 601–620 (1977)

    Article  Google Scholar 

  2. Araujo, A., Scheinkman, J.A.: Notes on comparative dynamics. In: Green, J.R., Scheinkman, J.A. (eds.) General Equilibrium, Growth and Trade, pp. 217–226. Academic, New York (1979)

    Google Scholar 

  3. Arkin, V., Evstigneev, I.: Stochastic Models of Control and Economic Dynamics. Academic, New York (1987)

    Google Scholar 

  4. Bewley, T.F.: General Equilibrium, Overlapping Generations Models and Optimal Growth Theory. Harvard University Press, Cambridge (2007)

    Google Scholar 

  5. Brock, W.A.: On existence of weakly maximal programmes in a multi-sector economy. Rev. Econ. Stud. 37, 275–280 (1970)

    Article  Google Scholar 

  6. Carlson, D.A., Haurie, A.B., Leizarowitz, A.: Infinite Horizon Optimal Control: Deterministic and Stochastic Systems. Springer, Berlin (1991)

    Google Scholar 

  7. Cass, D.: Optimum growth in an aggregative model of capital accumulation. Rev. Econ. Stud. 32, 233–240 (1965)

    Article  Google Scholar 

  8. Cass, D.: Optimum growth in an aggregative model of capital accumulation: a turnpike theorem. Econometrica 34, 833–850 (1966)

    Article  Google Scholar 

  9. Dana, R.A., Le Van, C., Mitra, T., Nishimura, K. (eds.): Handbook of Optimal Growth, vol. 1. Springer, Berlin (2006)

    Google Scholar 

  10. Dechert, W. (ed.): Growth Theory, Nonlinear Dynamics, and Economic Modelling: Scientific Essays of William Allen Brock. Edward Elgar, Cheltenham (2001)

    Google Scholar 

  11. Dorfman, R., Solow, R.M., Samuelson, P.A.: Linear Programming and Economic Analysis. McGraw-Hill, New York (1958)

    Google Scholar 

  12. Durlauf, S., Blume, L. (eds.): The New Palgrave, vols. 1–7. Macmillan, New York (2008)

    Google Scholar 

  13. Eatwell, J., Milgate, M., Newman, P.K. (eds.): The New Palgrave, vols. 1–4. Macmillan, New York (1987)

    Google Scholar 

  14. Gale, D.: On optimal development in a multi-sector economy. Rev. Econ. Stud. 34, 1–18 (1967)

    Article  Google Scholar 

  15. Gale, D.: Nonlinear duality and qualitative properties of optimal growth. In: Abadie, J. (ed.) Integer and Nonlinear Programming, pp. 309–319. North Holland, Amsterdam (1970)

    Google Scholar 

  16. Halkin, H.: Necessary conditions for optimal control problems with infinite horizon. Econometrica 42, 267–273 (1974)

    Article  Google Scholar 

  17. Khan, M.A., Mitra, T.: On choice of technique in the Robinson–Solow–Srinivasan model. Int. J. Econ. Theory 1, 83–109 (2005)

    Article  Google Scholar 

  18. Khan, M.A., Mitra, T.: Discounted optimal growth in the two-sector RSS model: a geometric investigation. Adv. Math. Econ. 8, 349–381 (2006)

    Article  Google Scholar 

  19. Khan, M.A., Mitra, T.: Optimal growth in the two-sector RSS model: a continuous time analysis. In: Proceedings of the Seventh Portugese Conference on Automatic Control, Lisboa, Portugal. Electronic publication (2006)

    Google Scholar 

  20. Khan, M.A., Mitra, T.: Optimal growth in a two-sector RSS model without discounting: a geometric investigation. Jpn. Econ. Rev. 58, 191–225 (2007)

    Article  Google Scholar 

  21. Khan, M.A., Piazza, A.: On the Mitra-Wan Forestry Model: A Unified Analysis. Publicación Técnica CMM No. 226. Paper presented at Workshop in Economic Dynamics, National University of Singapore, July 30–August 1, 2009 (2009)

    Google Scholar 

  22. Khan, M.A., Piazza, A.: Classical turnpike theory and the economics of forestry. J. Behav. Econ. Organ. (2010, forthcoming)

    Google Scholar 

  23. Khan, M.A., Piazza, A.: On uniform convergence of undiscounted optimal programs in the Mitra-Wan forestry model: the strictly concave case. Int. J. Econ. Theory 6, 57–76 (2010)

    Article  Google Scholar 

  24. Khan, M.A., Zaslavski, A.J.: On a uniform turnpike of the third kind in the Robinson–Solow–Srinivasan model. J. Econ. 92, 137–166 (2006)

    Article  Google Scholar 

  25. Khan, M.A., Zaslavski, A.J.: On two classical turnpike results for the Robinson–Solow–Srinivasan (RSS) model. Adv. Math. Econ. 13, 47–97 (2010)

    Article  Google Scholar 

  26. Koopmans, T.C.: On the concept of optimal economic growth. In: Pontificae Academiae Scientarum Scripia Veria (ed.) Economic Approach to Development Planning, pp. 225–287. North Holland, Amsterdam (1965)

    Google Scholar 

  27. Koopmans, T.C.: Intertemporal distribution and “optimal” aggregate economic growth. In: Fellner, W., et al. (eds.) Ten Economic Studies in the Tradition of Irving Fisher, pp. 95–126. Wiley, New York (1967)

    Google Scholar 

  28. Krantz, S.G., Parks, H.R.: The Implicit Function Theorem: History, Theory and Applications. Birkhäuser, Basel (1998)

    Google Scholar 

  29. Levhari, D., Leviatan, N.: Stability in the saddlepoint sense. J. Econ. Theory 4, 88–93 (1972)

    Article  Google Scholar 

  30. Marimon, R.: Stochastic turnpike property and stationary equilibrium. J. Econ. Theory 47, 282–306 (1989)

    Article  Google Scholar 

  31. McKenzie, L.W.: The Dorfman–Samuelson–Solow turnpike theorem. Int. Econ. Rev. 4, 29–43 (1963)

    Article  Google Scholar 

  32. McKenzie, L.W.: The turnpike theorem of Morishima. Rev. Econ. Stud. 30, 169–176 (1963)

    Article  Google Scholar 

  33. McKenzie, L.W.: Accumulation programs of maximum utility and the von Neumann facet. In: Wolfe, J.N. (ed.) Value, Capital and Growth, pp. 353–383. Edinburgh University Press, Edinburgh (1968)

    Google Scholar 

  34. McKenzie, L.W.: Turnpike theory. Econometrica 43, 841–865 (1976)

    Article  Google Scholar 

  35. McKenzie, L.W.: A new route to the turnpike. In: Henn, R., Moeschlin, L.O. (eds.) Mathemtical Economics and Game Theory. Springer, Berlin (1977)

    Google Scholar 

  36. McKenzie, L.W.: Turnpike theory, discounted utility, and the von Neumann facet. J. Econ. Theory 27, 194–209 (1982)

    Article  Google Scholar 

  37. McKenzie, L.W.: A primal route to the turnpike and Lyapunov stability. J. Econ. Theory 30, 330–352 (1983)

    Article  Google Scholar 

  38. McKenzie, L.W.: Optimal economic growth, turnpike theorems and comparative dynamics. In: Arrow, K.J., Intrilligator, M. (eds.) Handbook of Mathematical Economics, vol. 3, pp. 1281–1355. North-Holland, New York (1986)

    Google Scholar 

  39. McKenzie, L.W.: Turnpikes. Am. Econ. Rev. (Papers and Proceedings) 88, 1–14 (1998)

    Google Scholar 

  40. McKenzie, L.W.: Classical General Equilibrium Theory. The MIT Press, Cambridge (2002)

    Google Scholar 

  41. Mitra, T.: Characterization of the turnpike property of optimal paths in the aggregative model of intertemporal allocation. Int. J. Econ. Theory 1, 247–275 (2005)

    Article  Google Scholar 

  42. Mitra, T., Wan, H.W., Jr.: Some theoretical results on the economics of forestry. Rev. Econ. Stud. LII, 263–282 (1985)

    Google Scholar 

  43. Mitra, T., Wan, H.W., Jr.: On the Faustmann solution to the forest management problem. J. Econ. Theory 40, 229–249 (1986)

    Article  Google Scholar 

  44. Pietsch, A.: History of Banach Spaces and Linear Operators. Birkhäuser, Berlin (2007)

    Google Scholar 

  45. Ramsey, F.: A mathematical theory of savings. Econ. J. 38, 543–559 (1928)

    Article  Google Scholar 

  46. Rockafellar, R.T.: Hamiltonian trajectories and saddle points in mathemtical economics. Contr. Cybern. 4, 1575–1588 (2009)

    Google Scholar 

  47. Salo, S., Tahvonen, O.: On equilibrium cycles and normal forests in optimal harvesting of tree vintages. J. Environ. Econ. Manage. 44, 1–22 (2002)

    Article  Google Scholar 

  48. Salo, S., Tahvonen, O.: On the economics of forest vintages. J. Econ. Dyn. Control 27, 1411–1435 (2003)

    Article  Google Scholar 

  49. Samuelson, P.A.: A catenary turnpike theorem involving consumption and the golden rule. Am. Econ. Rev. 55, 486–496 (1965). An erratum in 55, 864–866

    Google Scholar 

  50. Samuelson, P.A.: The general saddlepoint property of optimal control motions. J. Econ. Theory 5, 102–120 (1972)

    Article  Google Scholar 

  51. Samuelson, P.A., Solow, R.M.: A complete capital model involving heterogeneous capital goods. Q. J. Econ. 27, 537–562 (1956)

    Article  Google Scholar 

  52. Scheinkman, J.A.: On optimal steady states of n-sector growth models when utility is discounted. J. Econ. Theory 12, 11–30 (1976)

    Article  Google Scholar 

  53. Solow, R.M.: Modern capital theory. In: Brown, E.C., Solow, R.M. (eds.) Paul Samuelson and Modern Economic Theory. McGraw-Hill, New York (1983)

    Google Scholar 

  54. Stiglitz, J.E.: A note on technical choice under full employment in a socialist economy. Econ. J. 78, 603–609 (1968)

    Article  Google Scholar 

  55. von Neumann, J.: Über ein ökonomisches gleichungs-system und eine verallgemeinerung des Brouwerschen fixpunktsatzes. In: Menger, K. (ed.) Ergebnisse eines Mathematischen Kolloquiums, 8. Translated as: A model of general economic equilibrium. Rev. Econ. Stud. 13, 1945–1946, 1–9 (1935–1936)

    Google Scholar 

  56. Yano, M.: Note on the existence of an optimal capital accumulation path. J. Econ. Theory 27, 421–429 (1981)

    Article  Google Scholar 

  57. Zaslavski, A.J.: Optimal programs in the RSS model. Int. J. Econ. Theory 1, 151–165 (2005)

    Article  Google Scholar 

  58. Zaslavski, A.J.: Turnpike Properties in the Calculus of Variations and Optimal Control. Springer, New York (2005)

    Google Scholar 

  59. Zaslavski, A.J.: Turnpike results for discrete-time optimal control systems arising in economic dynamics. Nonlinear Anal. 67, 2024–2049 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Economics, The Johns Hopkins University, Baltimore, MD, 21218, USA

    M. Ali Khan

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Casilla 110-V, Valparaíso, Chile

    Adriana Piazza

Authors
  1. M. Ali Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adriana Piazza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Ali Khan .

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-0041, Japan

    Shigeo Kusuoka (Professor) (Professor)

  2. Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan

    Toru Maruyama (Professor) (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer

About this paper

Cite this paper

Khan, M.A., Piazza, A. (2011). An overview of turnpike theory: towards the discounted deterministic case. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 14. Springer, Tokyo. https://doi.org/10.1007/978-4-431-53883-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation