The stochastic Mitra–Wan forestry model: risk neutral and risk averse cases

We extend the classic Mitra and Wan forestry model by assuming that prices follow a geometric Brownian motion. We move one step further in the model with stochastic prices and include risk aversion in the objective function. We prove that, as in the deterministic case, the optimal program is periodic both in the risk neutral and risk averse frameworks, when the benefit function is linear. We find the optimal rotation ages in both stochastic cases and show that they may differ significantly from the deterministic rotation age. In addition, we show how the drift of the price process affects the optimal rotation age and how the degree of risk aversion shortens it. We illustrate our findings for an example of a biomass function and for different values of the model’s parameters.

[1]  M. Insley,et al.  On Solving the Multirotational Timber Harvesting Problem with Stochastic Prices: A Linear Complementarity Formulation , 2005 .

[2]  M. Insley,et al.  Regime switching in stochastic models of commodity prices: An application to an optimal tree harvesting problem , 2012 .

[3]  Markku Kallio,et al.  OPTIMAL HARVESTING OF FOREST AGE CLASSES UNDER PRICE UNCERTAINTY AND RISK AVERSION , 2006 .

[4]  M. Clutter,et al.  Modeling and forecasting pine sawtimber stumpage prices in the US South by various time series models , 2010 .

[5]  Hanne K. Sjølie,et al.  A review of recent developments and applications of partial equilibrium models of the forest sector , 2013 .

[6]  A preference foundation for constant loss aversion , 2012 .

[7]  K. Löfgren,et al.  IMPACT OF RISK AVERSION ON THE OPTIMAL ROTATION WITH STOCHASTIC PRICE , 2008 .

[8]  Michel Denuit,et al.  Risk measurement with equivalent utility principles , 2006 .

[9]  Santanu Roy,et al.  Sustained positive consumption in a model of stochastic growth: The role of risk aversion , 2012, J. Econ. Theory.

[10]  M. Ali Khan,et al.  On the Mitra-Wan forestry model: A unified analysis , 2012, J. Econ. Theory.

[11]  Laura Schechter,et al.  Risk aversion and expected-utility theory: A calibration exercise , 2006 .

[12]  Erkki Koskela,et al.  Optimal Harvesting Under Resource Stock and Price Uncertainty , 2005, SSRN Electronic Journal.

[13]  Thomas A. Thomson,et al.  Optimal forest rotation when stumpage prices follow a diffusion process , 1992 .

[14]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[15]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[16]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[17]  Thomas Knoke,et al.  May risk aversion lead to near-natural forestry? A simulation study , 2011 .

[18]  Alexander Shapiro,et al.  Conditional Risk Mappings , 2005, Math. Oper. Res..

[19]  Bernardo K. Pagnoncelli,et al.  The optimal harvesting problem under price uncertainty , 2014, Ann. Oper. Res..

[20]  Freddy Delbaen,et al.  A von Neumann–Morgenstern Representation Result without Weak Continuity Assumption , 2011 .

[21]  T. Knoke,et al.  Mixed species plantations in Southern Chile and the risk of timber price fluctuation , 2010, European Journal of Forest Research.

[22]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[23]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[24]  Tapan Mitra,et al.  On the faustmann solution to the forest management problem , 1986 .

[25]  J. Buongiorno,et al.  Space-Time Modeling of Timber Prices , 2006 .

[26]  W. Reed,et al.  The tree-cutting problem in a stochastic environment: The case of age-dependent growth , 1989 .

[27]  Frank Riedel,et al.  Dynamic Coherent Risk Measures , 2003 .

[28]  Alexander Shapiro,et al.  Solving multistage asset investment problems by the sample average approximation method , 2006, Math. Program..

[29]  Andrew J. Plantinga,et al.  The Optimal Timber Rotation: An Option Value Approach , 1998 .

[30]  Stanislav Uryasev,et al.  Conditional Value-at-Risk for General Loss Distributions , 2002 .

[31]  Erkki Koskela,et al.  Does Risk Aversion Accelerate Optimal Forest Rotation Under Uncertainty? , 2004 .

[32]  Seppo Salo,et al.  On Equilibrium Cycles and Normal Forests in Optimal Harvesting of Tree Vintages , 2002 .

[33]  Peichen Gong,et al.  Risk-aversion and the short-run supply of timber , 2001 .

[34]  M. Penttinen,et al.  Impact of stochastic price and growth processes on optimal rotation age , 2006, European Journal of Forest Research.

[35]  Seppo Salo,et al.  On the economics of forest vintages , 2003 .

[36]  Isao Shoji,et al.  Searching for an optimal rotation age for forest stand management under stochastic log prices , 1998, Eur. J. Oper. Res..

[37]  A. Guttormsen,et al.  Real options in the forest: what if prices are mean-reverting? , 2002 .

[38]  T. Knoke,et al.  Mixed forests reconsidered: A forest economics contribution on an ecological concept , 2005 .

[39]  Kazuo Murota,et al.  Discrete convex analysis , 1998, Math. Program..

[40]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[41]  Erkki Koskela,et al.  Taxation and Rotation Age Under Stochastic Forest Stand Value , 2004, SSRN Electronic Journal.

[42]  M. Insley.  A Real Options Approach to the Valuation of a Forestry Investment , 2002 .

[43]  Shashi Kant,et al.  Valuation of Timber Harvesting Options Using a Contingent Claims Approach , 2009, Land Economics.

[44]  Tapan Mitra,et al.  Some Theoretical Results on the Economics of Forestry , 1985 .

[45]  M. Moretto,et al.  Land conversion pace under uncertainty and irreversibility: too fast or too slow? , 2012 .