On the Mitra–Wan forest management problem in continuous time

Abstract

International audienceThe paper provides a continuous-time version of the discrete-time Mitra–Wan model of optimal forest management, where trees are harvested to maximize the utility of timber flow over an infinite time horizon. The available trees and the other parameters of the problem vary continuously with respect to both time and age of the trees, so that the system is ruled by a partial differential equation. The behavior of optimal or maximal couples is classified in the cases of linear, concave or strictly concave utility, and positive or null discount rate. All sets of data share the common feature that optimal controls need to be more general than functions, i.e. positive measures. Formulas are provided for golden-rule configurations (uniform density functions with cutting at the ages that solve a Faustmann problem) and for Faustmann policies, and their optimality/maximality is discussed. The results do not always confirm the corresponding ones in discrete time

A Ricardian Model of Forestry

“…Various general remarks are here due. First, note that, except for the properties of the productivity function, 6 the basic Mitra-Wan model in continuous-time in Fabbri et al (2015) is recovered for b i = 0, l i (s) = 0, and for N = 1. 7 8 However, adding di↵erent lands in a single final good model (as the one in Fabbri et al (2015)) would not change much the structure of the stationary states.…”

“…First, note that, except for the properties of the productivity function, 6 the basic Mitra-Wan model in continuous-time in Fabbri et al (2015) is recovered for b i = 0, l i (s) = 0, and for N = 1. 7 8 However, adding di↵erent lands in a single final good model (as the one in Fabbri et al (2015)) would not change much the structure of the stationary states. Since candidates for stationary equilibria are the strategy-trajectory components of the modified golden rules and, as for the RamseyCass-Koopmans aggregate model, generically only a unique strategy-trajectory couple can belong to a modified golden rule of the model, the quantity of timber produced in a stationary equilibrium turns out to be independent from the demand of timber (see Mitra & Wan, 1985;Fabbri et al, 2015).…”

“…We recall that optimal forest management has developed since the late 1970s following Samuelson's review paper (Samuelson, 1976), and mainly focuses on the dynamics of forest rotation on a fully occupied plot of land, where new trees are immediately replanted after old trees are cut (see, e.g. Mitra & Wan, 1985, 1986Salo & Tahvonen, 2002, 2003Khan & Piazza, 2012;Fabbri et al, 2015). Immediate replanting and continuous full occupation of the land are justified by the assumptions that cutting and replanting costs are null and that the productive life of trees is finite.…”

“…According to Salant (2013), this shift is long overdue in literature both for the study of the equilibrium price system -still largely unknown -and as a first step towards the analysis of models in which externalities or other distortions are present 3 . In building our Ricardian model, we borrow the continuous-time production structure of Fabbri et al (2015), instead of using the original discrete-time Mitra-Wan formulation. The continuous-time model is mathematically more challenging than the discrete-time counterpart for two reasons: a) the evolution of state variables is governed by a partial di↵erential equation, as they represent a continuum of (vintage) capital goods: trees of di↵erent ages; b) since cost minimization and competitive arbitrage imply that cutting trees is profitable only at a finite set of ages, distributed controls concentrating on single points need be allowed.…”

“…Hence, in order for the model to be endowed with meaningful price-supported stationary states, intensity levels of the production processes need to be chosen in a very large space (a space of measures). However, once the technical di culties are overcome (see Fabbri et al, 2015), the clear distinction between stock and flow variables, which is lacking in discrete time (see Foley, 1975), turns into an advantage in the interpretation of the price system: the theory of long run production prices and the role of the Hotelling rule become transparent. The paper is organized as follows.…”