Classical turnpike theory and the economics of forestry☆
Research highlights
► We develop two classical turnpike results for the Mitra–Wan forestry model. ► The transition production set does not satisfy assumptions of inaction and free-disposal. ► We consider approximately optimal, large but finite-horizon programs. ► We work with a novel non-interiority assumption on concave benefit functions. ► We list three open questions.
Introduction
The results presented in this paper can perhaps be best introduced by asking what is “classical turnpike theory”? and what has it to do with the “economics of forestry”? We begin with the first question.
The origin of the subject is easily dated to Paul Samuelson’s (1949) Rand Memorandum, and its essential motivation is to show that the solutions of an intertemporal allocation problem over a large but finite time horizon, with given initial and terminal stocks, stay approximately close, most of the time, to maximally sustainable programs free from any considerations pertaining to a terminal horizon. However, despite the commitment in Ramsey (1928), it is only by 1965 that an infinite-horizon variational problem is accepted as a legitimate vehicle to address issues of intertemporal allocation. In an undiscounted setting, Gale (1967) focussed on the problem of the existence of an optimal program, and resolved it by conditions ensuring the asymptotic convergence of good programs, a methodological procedure that he referred to as a “round-about method.”1 Even though this asymptotic property of good programs is mentioned in passing as a “turn-pike” theorem, Gale reverted to the classical conception in his 1970 expository paper. Along with monotonicity properties, he situates classical turnpike theory within the broader rubric of qualitative properties of optimal programs.2 In this paper, we revert to the Samuelsonian classical conception pertaining to large but finite optimal programs, and investigate what McKenzie (1976) was subsequently to term the middle turnpike.3
The resurgence of interest in the economics of forestry can also be dated to Samuelson (1976a) and to his sighting of Faustman’s (1849) analysis. However, it remained for the remarkable articles of Mitra and Wan, 1985, Mitra and Wan, 1986, and following them, Salo and Tahvonen, 2002, Salo and Tahvonen, 2003, to take his market analysis and recast it into an optimal Ramseyian planning, framework. Once this link is forged, and the conceptual markers of the theory of intertemporal resource allocation are identified in the forestry model – the golden-rule stocks, the golden-rule prices and corresponding maximal sustainable timber yields as Ramsey’s bliss point – we are naturally lead to ask whether forest management programs for large but finite time horizons follow a turnpike. To go to the epigraph from Gale, there appears to be no reason why they should, and this scepticism leads directly into our second question concerning the relevance of classical turnpike theory to the Mitra–Wan (MW) forestry model. Here the usual justifications of turnpike theory as resolving qualitative puzzles and computational difficulties take on an added force, and turnpike theory attains a normative significance perhaps even greater than that in capital theory in the abstract and in the large. And to be sure, the resulting theory includes, as a special case, the situation of a profit-maximizing forest manager, with the infinite horizon as the formalization of the fact that there is no specified time-period for the end of his or her firm, with such an application involving a demand function for timber being very much facilitated under the weakened assumptions on benefit (felicity) functions that we pursue in this paper.4
Unlike the aggregative Ramsey–Cass–Koopmans optimal growth model of modern macroeconomics, the issue in the economics of forestry is the lack of knowledge of the optimal policy correspondence and of transition dynamics. What should the planner, or the forest manager, do today in terms of optimal policy over a fixed time-horizon rather than take satisfaction that any arbitrary initial configuration converges to the maximally sustainable forest configuration in the very long run? In any case, at least since Faustman, periodicity seems to be the rule, and the charting of an optimal policy correspondence with a non-linear benefit (felicity) function remains, even until now, totally uncharted.5 And so the question arises whether staying with the maximally sustainable configuration is “good enough,” and thereby leads only to an increased reliance on turnpike results.6 Furthermore, a forest, unlike a given stock of capital, is much more than a durable input for the production of desired commodities – it is desirable in itself, and if not a “way of life” of so-called endogenous and native communities, a stock imbricated by externality considerations and entrusted by one generation to another.7 As such, a forest configuration also enters as an argument in the benefit (felicity) function, and thereby further complicates the difficulties of determining what the planner has to do “tomorrow and the day after” rather than the long-run. But more to the point, such results allay fears and furnish a reassurance that a planner’s departure from an initial forest configuration to one yielding maximal sustained timber yields, even when he has to return to future generations the forest in the same state that he was given it, is not betraying this trust. However such a theory can only be constructed on the shoulders of a theory for the simpler case considered here when the felicities depend only on timber yields.8
With the relevance of turnpike theory to the economics of forestry established and out of the way, one can turn to the more immediately antecedent literature and delineate the precise contribution of this paper. Given the extensive work on turnpike theory associated with the names of Samuelson, McKenzie, Gale and their followers, why can one not simply appeal to the standard results? What is the need for additional modifications? In particular, why are the results presented in this paper not straightforward applications of the recent extension of the theory sketched in Khan and Zaslavski (2010a)? The answers to these questions require a technically more focussed discussion. We turn to this.
Even though the Robinson–Solow–Srinivasan (RSS) model and the MW model are different models with entirely different interpretive registers, the subtle analytical connections between them are undeniable, and the recent RSS revisitation of turnpike theory in Khan and Zaslavski (2010b) worth noting. It involves at least four disparate elements: (i) the irrelevance, in principle, of necessary first order Euler–Lagrange conditions, and indeed of differentiability of the felicity function at the golden-rule stock, (ii) the identification of asymptotic convergence of good programs as a sufficient condition for classical turnpike theory, and therefore for the asymptotic convergence of optimal programs, (iii) the derivation of asymptotic stability of optimal programs from the classical turnpike result, which is to say, the derivation of results on the early and late turnpikes as a consequence of a result on the middle turnpike, (iv) a focus on approximately optimal large but finite programs. Only points (i) and (ii) need further supplementation in the context of results that we report here, and we take them in turn. As regards (i), it is now well-understood that the golden-rule stock in the RSS model, is not in the interior of the transition set, and even for the case with a single type of machine when it is in the interior, the reduced form utility function is not differentiable at it even with a linear felicity function. In part, this is precisely what gives the RSS model its continuing interest. The same occurs in the MW model. As regards (ii), it allows us to move away from the dichotomy of linear and strictly concave felicity functions to a more productive sufficiency condition, something essential for the RSS model where even strictly concave felicity functions do not lead to strictly concave reduced-form utility functions as is required by the theory. In particular, such a condition allows a turnpike theorem when the felicity function is linear and the marginal rate of transformation ξσ ≠ 1. It is this issue that finds its most satisfactory culmination in the MW model.
In recent work, Khan and Piazza (2009) furnish for the MW forestry model a non-interiority condition that is necessary and sufficient for asymptotic convergence of good programs when the benefit (felicity) function is assumed only to be concave and not necessarily differentiable.9 And so the natural question arises as to whether one can construct a robust turnpike theory of the classical type for the economics of forestry, as is done in Khan and Zaslavski (2010a) for the (RSS) choice of technique problem in development planning. Theorems A and B presented in Section 3 below answer this question in the affirmative, and constitute the principal results of this work. They also yield as direct corollaries results on the asymptotic convergence of optimal programs (Theorem 2.4), and results on the “bunching” or “approximate bunching” of optimal forest configurations. Unlike the RSS model, there is no natural ordering on the transition production set in the MW model, and this necessitates novel and different arguments, and as it happens, more constructive ones than those presented in Khan and Zaslavski (2010a). Sections 4 and 5 present the statements of several results that are both needed in the proofs of the results of Section 3, and are interesting in their own right. We take care to comment on aspects in which they depart from the corresponding arguments in Khan and Zaslavski (2010a); the formalities of the proofs themselves are confined to the Appendix A. Section 6 concludes the paper with a delineation of three directions and the open questions associated with each. In the next section, by way of introducing the reader to the notation and the terminology, we present the basic analytics of the model. This material is by now well-understood, but for the sake of completeness, we present results that being phrased in terms of our non-interiority condition, generalize corresponding results in Khan and Piazza (2010).
Section snippets
The Mitra–Wan tree farm and the non-interiority condition
We begin by introducing some notation. Let be the set of non-negative integers and () the set of real (non-negative) numbers. We shall work in the n − 1-dimensional simplex . For any we denote the inner product by and the supreme norm of x by ||x||∞.
In addition to its original formulation (Mitra and Wan, 1985, Mitra and Wan, 1986), an outline of the Mitra–Wan forestry model is also available in Mitra, 2005, Mitra, 2006. Here we depart from the original
Principal results
We now present the principal results of this work. As emphasized in the introduction, both results are classical in that they pertain to the following situation: if the initial and terminal configurations of a forest are given, then with enough time at his or her disposal, the planner knows that every optimal (or approximately optimal) program ought to stay arbitrarily near the golden-rule configuration, during an arbitrarily large fraction of the total time. However, the less of an error that
Preliminary substantive results
In this section and the next we begin developing the technical arguments needed to prove the principal results of this work. The five propositions presented here develop intuition into the basic dynamics underlying the MW model, and even though the proofs are notationally somewhat complex, the essential ideas are simple.16 We
Four substantive lemmas
With these preliminary results out of the way, we can turn to the deeper substance of the argumentation. As in Khan and Zaslavski (2010a), it revolves around the four footholds presented below as Lemmas Corollary 5.1, Lemma 5.3, Lemma 5.4: the visiting lemma, the stability lemma, the value-loss lemma and the aggregate value-loss lemma. However, before we take each in turn, it is worth elaborating on what was already emphasized in the introduction: that even though these results are inspired by
Concluding remarks
We now conclude the non-technical part of this work by delineating three directions in which the results demand extension and further investigation.
The first of these is the rather immediate question as to how much of the theory can be salvaged when the non-interiority Condition 2.1 does not hold? Since this condition is necessary and sufficient for asymptotic convergence of good programs, and it is easy to provide examples of periodic optimal programs when it does not hold, perhaps the obvious
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Cited by (4)
On a forest as a commodity and on commodification in the discipline of forestry
2016, Forest Policy and EconomicsStochastic Mitra–Wan forestry models analyzed as a mean field optimal control problem
2023, Mathematical Methods of Operations ResearchThe mitra-wan forestry model: A discrete-time optimal control problem
2015, Natural Resource ModelingMultiple forest stocks and harvesting decisions: The enhanced green golden rule
2013, Post-Faustmann Forest Resource Economics
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This work was initiated during Piazza’s visit to Johns Hopkins in May 2008, continued during her visit to the University of Illinois at Urbana-Champaign in April–May, 2009, and completed when Khan held the position of Visiting Professor at the University of Queensland, June–July 2009, and at the Nanyang Technological University, August, 2009. The authors thank Joseph Harrington, Flavio Menezes, Euston Quah, Anne Villamil and Nicholas Yannelis, and each department, for their hospitality. They also thank Tapan Mitra for correspondence and conversation, and two anonymous referees of this journal for their suggestions. Adriana Piazza gratefully acknowledges the financial support of FONDECYT under Grant # 3080059 and Grant # 11090254 and that of Programa Basal PFB 03, CMM, U. de Chile.