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On two classical turnpike results for the Robinson–Solow–Srinivasan model

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Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 13))

Abstract

Turnpike theory, as originally conceived by Samuelson, pertains to optimal programs over a large but finite time horizon with given initial and terminal stocks. In this paper, we present two turnpike results in the context of a model proposed by Robinson, Solow and Srinivasan, and the subject of extensive recent analysis as the RSS model. Our results are classical except that they are phrased in terms of (i) approximately optimal programs, and (ii) golden-rule stocks rather than their parent facet, and they underscore the distinction between the original theory and the asymptotic stability of optimal infinite horizon programs. Our results, and the arguments used to prove them, go beyond the RSS model to contribute to the general theory.

Received: September 22, 2008

Revised: October 20, 2009

JEL classification: C62, D90, Q23

Mathematics Subject Classification (2000): 49J99, 54E52

The authors are grateful above all to Paul Samuelson: they were sustained by a letter from him during the rather long pre-publication travails of this paper. The authors are also grateful to Alex Ioffe: he waived anonymity as a referee, and to the extent that this version is easier to read than the previous one – negotiates better the “thicket of words” and the “jungle of mathematics” – it is due to him. Ali Khan acknowledges enlightening correspondence with Tapan Mitra, and thanks Adriana Piazza, Debraj Ray, and two wildly differing referees of TE for stimulating discussion regarding both substantive and methodological issues.

This paper is dedicated to the memory of David Cass: one of the authors learnt (in 1969–1970) the rudiments of capital theory and Pontryagin’s principle at his hands, and to his everlasting regret, was not afforded the chance of also working with him on his Yale Ph.D. dissertation.

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Notes

  1. 1.

    This quote joins the first two sentences of McKenzie’s introduction to its concluding sentence; see [21, pp. 1281, 1284]. For the indented quote above, see the fourth paragraph of the introduction to [21, p. 1282].

  2. 2.

    See [20, p. 844] and also the later Ely Lecture [23, § 5] titled “Turnpikes”.

  3. 3.

    In [23, § 5] McKenzie writes “Almost all the attention to asymptotic convergence has been concentrated on convergence to balanced paths, although it is not clear that optimal balanced paths will exist. This type of path is virtually impossible to believe in... I made this point in articles published in 1974 and 1976.” Also see a repetition of this rather basic point in [24, p. 390].

  4. 4.

    The phraseology is Tapan Mitra’s; the terms good and bad have precise technical meaning going back to Gale [8], as in the definitions below.

  5. 5.

    See [13, pp. 343, 347]. For overviews of Brock’s 1970 result and the average turnpike property, see [22, § 8, concluding paragraph] and [23].

  6. 6.

    The phrase “turnpike property of good programs” already serves as a counterpoint. Note that in his recent survey, Mitra uses the phrase “turnpike property” synonymously with asymptotic stability, [27, first paragraph]. Also see [26] and the reference to the “often called turnpike property” in [28, Introduction and Conclusion].

  7. 7.

    Such a trajectory will be of particular relevance to what follows in the sequel.

  8. 8.

    The question has to do with whether the von-Neumann growth model serves as a possible discontinuity in the evolution of the application of techniques of maximization and of the calculus of variations to capital theory; whether “maximization of an objective function had no part in his [von-Neumann’s] theory.” In his Nobel Lecture, [33], Samuelson motivates his work on the turnpike by recounting this episode, and McKenzie also refers to it in his Ely Lecture, [23]. Whether von-Neumann’s position is to be granted half or full credit is another interesting shade of difference between McKenzie and Samuelson.

  9. 9.

    For the latter, see [11], and under a title referring to a “turnpike of the third kind”, [16].

  10. 10.

    For a discussion of this model, as well as the antecedent references to the papers of these authors, see, for example, [11] and [14].

  11. 11.

    It is an interesting and enlightening exercise to base one reading of this current stage by coupling [21] and [35].

  12. 12.

    See [16, § 1] for references to this literature from the viewpoint of the RSS mode.

  13. 13.

    As [11] made clear, this resurgence is relative to [6, 34, 35]; also see [11] for their references to the earlier vintage of work by Robinson, Solow and Srinivasan.

  14. 14.

    As readers of Samuelson’s writings in financial economics are well aware, this is yet another way of lengthening the time horizon. For a rigorous replication of the Stiglitz analysis in the undiscounted case, but still with continuous time, see [12] and [41].

  15. 15.

    See Examples 2 and 3 in [11]. We also note here that Example 1 in this paper shows the existence of a two-period optimal cycle, thereby negating Stiglitz’ monotonicity result in the continuous case.

  16. 16.

    See [15] for the surprising differences between the linear and strictly concave theories. Note, however, that throughout this paper, we limit ourselves to an undiscounted setting, and do not comment on work, by now substantial, on the RSS model with a positive discount factor. It is also for this reason that in the discussion of the Ramseyian turnpike, there is no reference in the sequel to the theorems of Cass and Koopmans. We also draw the reader’s attention to work with non-concave felicities as in [42].

  17. 17.

    To quote McKenzie again, but now in the context of this more delimited turnpike theory, “The real ground for the result is the tendency for optimal paths to bunch together in the middle time, and this tendency is preserved even in models that are time-dependent;” see [21, p. 843]. In [19], McKenzie refers to the middle-turnpike for the von-Neumann model as the “Samuelson Turnpike”.

  18. 18.

    See [11, Theorem 3], and for a uniform asymptotic theorem, [16].

  19. 19.

    See, for example, the synthesis of value-loss methods and a novel theory of undiscounted dynamic programming presented in [13, 15]. And as mentioned in Footnote 14, we avoid any reference to the discounted setting.

  20. 20.

    Whether methods of non-smooth analysis, and in particular, the subdifferential of the felicity function w( ⋅) at the golden-rule stock \(\hat{x},\) can be substituted for the differential, remains a hope, though a distant one.

  21. 21.

    The one cautionary flag in the context of the earlier work on linear models relates to the fact that rather than the Ramseyian setting of the maximization of aggregate utility, the works relate to the von Neumann objective of the maximization of the utility of terminal capital stocks; see [18] and the references therein.

  22. 22.

    See [9] and [35] for an overview of “neoclassical growth theory,” with the latter also including the Ramsey aggregate capital accumulation model within its ambit; and [23] for the relevance of “turnpike theory” to the “new growth theory”.

  23. 23.

    For the reader innocent of economic theory, as we shall see below, the model is a rather standard one in discrete optimal control theory in infinite time.

  24. 24.

    Note that in restricting d to the open unit interval, we rule out the case of circulating capital. There is little doubt that our analysis can be extended to cover this case. See [7] for an analysis of the optimal policy function with circulating capital in a model that includes the two-sector RSS model.

  25. 25.

    This is a fundamental assumption that guarantees a unique golden-rule stock, and thereby a unique von-Neumann ray. For a detailed discussion in the context of the RSS model, the reader is referred to [11] and [15].

  26. 26.

    This paragraph is a response to a referee’s insistence that the choice of notation does not (i) contradict minimal aesthetical requirements, and (ii) present a serious obstacle to understanding of the material. We hope that these criteria are now being adhered to as a result of the explanation in this paragraph.

  27. 27.

    As a referee pointed out, “it is customary in modern literature to use different alphabets for vectors and numbers” and that “inner product is not a group operation.” On the other side, changing the notation now when it has been established in more than fifteen published papers on the RSS model would cause confusion in another branch of the literature.

  28. 28.

    See [11]. In this context, also see [21, Assumption II] for the general model.

  29. 29.

    As emphasized in their introduction and abstract, this is the point of departure of Khan–Mitra [11].

  30. 30.

    It bears emphasis that Stiglitz did not offer sufficient condition for the existence of an optimal program, and that these were furnished only recently by [12, 41]. But more to the point, his results applied only to the case of w( ⋅) being linear, and the problem of characterizing transition dynamics with concave functions remains open even in continuous-time models; see [6, 37]. In the discrete time setting, the problems has not be fully solved even for the case of one machine (n = 1), the two-sector RSS model, leave alone the multi-sectoral one being considered in this paper; see [11, 15].

  31. 31.

    To be able to prove the turnpike results reported here from the above suggested optimal control theory point of view has remained an open problem right from the inception of the subject.

  32. 32.

    See [24, p. 389] for 1964 as the original date of introduction of this reduced form model. For a comprehensive and complete diagrammatic analysis of the two-sector RSS model in the undiscounted case, see [14]. For the general case, Figs. 7 and 8 in [24], and Figs. 7.3–7.5 in [25].

  33. 33.

    We shall come back to this lemma in the discussion in § 4 below.

  34. 34.

    Note that the conditions furnished in Theorem 2.2 are only sufficient conditions; it is entirely conceivable that the convergence of good programs to the golden-rule stock, (the, so-called, turnpike property of good programs) holds with felicity functions that are only strictly concave only in the neighborhood of the golden-rule stock. See [8] for this assumption; it is highlighted in [24, p. 389].

  35. 35.

    As is well understood, the issue goes back to Ramsey’s original 1928 paper; see [25, p. 256] for a textbook treatment. In [4], what we call weakly optimal is termed weakly maximal.

  36. 36.

    As emphasized in the introduction, our primary concern in this paper is with optimal, and approximately optimal, large but finite programs.

  37. 37.

    See, for example, [24, pp. 387–388], and the reference to the unpublished dissertation of Takahashi for the first dimensionality results regarding the facet. For the use of the concept as a basic engine of analysis in the form of value-loss lines in the two-sector RSS model, see [14].

  38. 38.

    If we endow the parameter space \((a,d) \subset ({R}_{+} \times (0, 1)) \subset {R}_{+}^{2}\) with Lebesgue measure, the set {(a, d) : (1 ∕ a) = 2 − d} has zero measure.

  39. 39.

    See [11, Theorem 3 and Example 1], and, for the two-sector case, [13, 14, 15].

  40. 40.

    The terminology is McKenzie’s [23, § II and III respectively,] as already referred to in Footnote 17.

  41. 41.

    The parameter Γ is necessitated by a particular structural characteristic of the RSS model. The reader can refer to Fig. 3 to check out that in the two-sector RSS model, the maximal sustainable capital stock (1 ∕ ad) can never be attained by a finite program starting from an initial stock less than it. Note that in this special case, n = 1, and therefore a is a scalar.

  42. 42.

    Also see Inada [10] and his emphasis on uniformity considerations in turnpike theory. We leave for future work a detailed comparison of our results to those in these papers. It is clear that the RSS model is a particular case of the Leontieff model without joint production and many (but a finite number) techniques for the production of the consumption good.

  43. 43.

    An anonymous referee brought Theorem 5 in Chap. 4 of Arkin–Evstigneev [2] to our attention in the context of approximate optimality. This theorem establishes in a stochastic setting a version of the “middle turnpike theorem (in the spirit of Theorem A) for a family of uniformly good finite programs, and since optimal finite programs are uniformly good, the same is true for a family of approximately optimal programs.”

  44. 44.

    See [24, p. 389] for this point in the context of the general theory.

  45. 45.

    For a complete characterization of the policy correspondence in the two-sector RSS model, including the situation when the Standing Hypothesis does not necessarily hold, see [13, 14, 15].

  46. 46.

    The form that McKenzie chooses as final version suitable for a text, [25, Theorem 3] referred to above, obviously relies on [18, 19] in its evolution. However, since these latter references are phrased in terms of the maximization of a function of the terminal stock configuration, the Samuelson turnpike, so to speak, we do not discuss them in any detail.

  47. 47.

    See page 845 in [20] for the first quote, and pages 850, 851 for the second.

  48. 48.

    This is the principal result of [16, Theorem 3.3], which itself is a generalization of [11, Theorem 3].

  49. 49.

    It is of interest in the light of Theorem 2.3(ii) above, that McKenzie’s theorem 9.3 also furnishes an existence result that asserts the optimality of such a converging path, if it is maximal, (weakly maximal in the terminology used in this paper).

  50. 50.

    For a most fruitful exploitation of differentiable methods for this question in the context of the non-stationary aggregative one-good model, see [26, Theorem 3] and [28, Theorem 5.1].

  51. 51.

    An anonymous referee pointed out that methods of proof of Theorem 1 in [1] could possibly be used to provide a direct and unified proof of the corollaries. We leave this for future work.

  52. 52.

    In the succeeding line, y(t) stands for \(\bar{y}(t)\) and \(\tilde{y}(t)\) alternatively.

  53. 53.

    In the succeeding line, y(t) stands for \(\bar{y}(t)\) and \(\tilde{y}(t)\) alternatively.

  54. 54.

    Thus Radner’s emphasis on Karlin’s proof of the von-Neumann theorem is hardly incidental from this substantive point of view.

  55. 55.

    Note that it is only the assumption of strict convexity that does not hold in the RSS model; the von-Neumann ray is indeed unique by virtue of the fundamental assumption embodied in 2.1. Note also that in this statement, as well as loose treatment of the basic ideas in this section, we are using the golden-rule pair \((\hat{x},\hat{p})\) of the RSS model as the relevant analogue of the von-Neumann balanced growth path and the associated shadow prices. The assumption of exogenously given labor supply in each period obviously vitiates the constant returns to scale assumption in the von-Neumann model. It is of course the constant returns to scale assumption that necessitates the particular metric that Radner uses.

  56. 56.

    In addition to [31], see [10]. Also see the reference to Atsumi in [22] and [25, Lemma 4, p. 252]. A preliminary statement for the two-sector RSS model with strictly concave felicities, but one that ignores the complications of initial and terminal capital stocks, is available in [15, Footnote 4]. For the non-stationary case, see [21, Lemma 8.1].

  57. 57.

    In any case, as the two-sector RSS case brings out, the turnpike result does not hold in the case ξ = 1, and so it can hardly be proved by any method.

  58. 58.

    See Footnote 37 above and the accompanying text.

  59. 59.

    See [29] for the reference to Morishima (1961) and to [25] for that to McKenzie (1963). Malinvaud famous 1953 paper is also of obvious relevance in this connection.

  60. 60.

    See [23, Concluding paragraph of § 4]. It is of interest that both Samuelson and Koopmans avoid the use of Pontryagin’s principle and couch all their arguments in the classical phraseology of the calculus of variations.

  61. 61.

    See [36]; also [6, 37]. For a supplementation of Stiglitz’s arguments by existence theorems, see [12, 41].

  62. 62.

    To our knowledge, a result of this form is not available in the economic literature though it is well-known to students of optimal control and the variational calculus; see [40].

  63. 63.

    The distinction between the two kinds of trajectories was referred to in the introduction and illustrated in Fig. 2.

  64. 64.

    The fact that asymptotic stability (the late turnpike property) implies other turnpike properties is studied, in continuous time and in a general setting, in [40, Chaps. 3–5].

  65. 65.

    See Footnote 3 and the accompanying text.

  66. 66.

    The reader is invited to diagram the program in Fig. 3.

  67. 67.

    The reader is invited to diagram the trajectories (6.1) and (6.3) in Fig. 3.

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Authors and Affiliations

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

    M. Ali Khan

  2. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Alexander J. Zaslavski

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    Shigeo Kusuoka (Professor) (Professor)

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

    Toru Maruyama (Professor) (Professor)

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Khan, M.A., Zaslavski, A.J. (2010). On two classical turnpike results for the Robinson–Solow–Srinivasan model. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 13. Springer, Tokyo. https://doi.org/10.1007/978-4-431-99490-9_3

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