1 A general continuous global approach to: - Optimal forest management with respect to the global warming problem and global economics - One section of the lectures by Peter Lohmander at UPV, Polytechnical University of Valencia, Spain, February 2010 Peter Lohmander Professor of forest management and economic optimization SLU, Faculty of Forest Sciences Dept. of forest economics Umeå, Sweden

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8 Second order minimum conditions:

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12 Observations of the first and second order conditions:

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17 So, if and or then

18 Then, the solution to represents a (locally) unique minimum.

19 A numerically specified example:

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21 Comparative statics analysis:

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27 Explicit solution of the example for alternative values of K

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35 Dynamic approach analysis

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42 k is selected in way such that the two equations become identical. This way, the equations only determine the ratio B/A, not the values of A and B. This is necessary since we must have some freedom to determine A and B such that they fit the initial conditions. With two roots (that usually are different), we (usually) get two different ratios B/A. This makes it possible to fit the parameters to the (two dimensional) initial conditions (x(0),y(0)).

43 One way to determine the value(s) of k is to use this equation:

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45 Another way to get to the same equation, is to make sure that the two equations give the same value to the ratio B/A.

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47 Lets us solve the equation!

48 No cyclical solutions! Observe that the expression within the square root sign is positive. As a consequence, only real roots, k, exist. For this reason, cyclical solutions to the differential equation system can be ruled out.

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51 We may observe that As a consequence, both roots to to the equation are strictly negative. Therefore, divegence from the equilibrium solution is ruled out.

52 With only strictly negative roots, we have a guaranteed convergence to the equilibrium. However, this does not have to be monotone. With two different roots (k1 and k2) and with parameters A1 and A2 with different signs (and/or parameters B1 and B2 with different signs), the sign(s) of the deviation(s) from the equilibrium value(s) may change over time.

53 Derivation of the roots in the example:

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56 We may determine the path completely using the initial conditions

57 We also use the earlier derived results:

58 Using the derived roots, we get:

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60 Let us use the initial conditions and determine the parameters!

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64 Using the figures from the example, we get:

65 The solutions to the numerically specified example X(t) = ·EXP( ·t) ·EXP( ·t)

66 Y(t) = ·EXP( ·t) ·EXP( ·t)

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68 The cost function from different perspectives: (based on the numerically specified example)

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71 Numerical solution of the example problem using direct minimization: Costmin Valencia Lohmander

72 K = 600 model: min = C; k = 600; C = cu + cf + ca + cw; cu = 5*u+1/40*u^2; cf = 10*f + 1/600*f^2; ca = 1/40*(k-u-w)^2; cw = 14*w+1/200*w^2; f = k-u; a = anet = a - 31*equ+15*eqw= *k; 5*equ+6*eqw = *k; end

73 Variable Value Reduced Cost C K CU CF CA CW U F W A ANET EQW EQU

74 K = 800 model: min = C; k = 800; C = cu + cf + ca + cw; cu = 5*u+1/40*u^2; cf = 10*f + 1/600*f^2; ca = 1/40*(k-u-w)^2; cw = 14*w+1/200*w^2; f = k-u; a = anet = a - 31*equ+15*eqw= *k; 5*equ+6*eqw = *k; end

75 Variable Value Reduced Cost C K CU CF CA CW U F W A ANET EQW EQU

76 K = 1000 model: min = C; k = 1000; C = cu + cf + ca + cw; cu = 5*u+1/40*u^2; cf = 10*f + 1/600*f^2; ca = 1/40*(k-u-w)^2; cw = 14*w+1/200*w^2; f = k-u; a = anet = a - 31*equ+15*eqw= *k; 5*equ+6*eqw = *k; end

77 Variable Value Reduced Cost C K CU CF CA CW U F W A ANET EQW EQU

78 Numerical approximation of the dynamics: ! dynsim; ! Peter Lohmander Valencia ; model: sets: time/1..100/:x,y,dx,dy; endsets cxx = 31/300; cxy = 15/300; cyx = 15/300; cyy = 18/300; x(1) = -100; y(1) = -80;

time(t): dx(t)= -( cxx*x(t) + cxy*(y(t)) time(t): dy(t)= -( cyx*x(t) + cyy*(y(t)) time(t)| t#GT#1: x(t)= x(t-1) + dx(t-1) time(t)| t#GT#1: y(t)= y(t-1) + end

80 Variable Value CXX CXY E-01 CYX E-01 CYY E-01 X( 1) X( 2) X( 3) X( 4) X( 5) X( 6)

81 X( 19) X( 20) X( 21) X( 22) X( 23) X( 24) X( 25) X( 26) X( 27) X( 28)

82 X( 40) X( 41) X( 42) X( 43) X( 44) X( 45)

83 Y( 1) Y( 2) Y( 3) Y( 4) Y( 5) Y( 6) Y( 7) Y( 8)

84 Y( 94) Y( 95) Y( 96) Y( 97) Y( 98) Y( 99) Y( 100)

85 References Lohmander, P., Adaptive Optimization of Forest Management in a Stochastic World, in Weintraub A. et al (Editors), Handbook of Operations Research in Natural Resources, Springer, Springer Science, International Series in Operations Research and Management Science, New York, USA, pp , #reader-linkhttp:// #reader-link Lohmander, P,. Energy Forum, Stockholm, 6-7 February 2008, Conference program with links to report and software by Peter Lohmander: Lohmander, P., Ekonomiskt rationell utveckling för skogs- och energisektorn i Sverige, Nordisk Papper och Massa, Nr 3, 2008 Lohmander, P., Mohammadi, S., Optimal Continuous Cover Forest Management in an Uneven-Aged Forest in the North of Iran, Journal of Applied Sciences 8(11), Lohmander, P., Guidelines for Economically Rational and Coordinated Dynamic Development of the Forest and Bio Energy Sectors with CO2 constraints, Proceedings from the 16th European Biomass Conference and Exhibition, Valencia, Spain, June, 2008 (In the version in the link, below, an earlier misprint has been corrected. )

86 Lohmander, P., Economically Optimal Joint Strategy for Sustainable Bioenergy and Forest Sectors with CO2 Constraints, European Biomass Forum, Exploring Future Markets, Financing and Technology for Power Generation, CD, Marcus Evans Ltd, Amsterdam, 16th-17th June, Lohmander, P., Ekonomiskt rationell utveckling för skogs- och energisektorn, Nordisk Energi, Nr. 4, 2008 Lohmander, P., Optimal resource control model & General continuous time optimal control model of a forest resource, comparative dynamics and CO2 consideration effects, SLU Seminar in Forest Economics, Umea, Sweden, Lohmander, P., Tools for optimal coordination of CCS, power industry capacity expansion and bio energy raw material production and harvesting, 2nd Annual EMISSIONS REDUCTION FORUM: - Establishing Effective CO2, NOx, SOx Mitigation Strategies for the Power Industry, CD, Marcus Evans Ltd, Madrid, Spain, 29th & 30th September Lohmander, P., Optimal CCS, Carbon Capture and Storage, Under Risk, International Seminars in Life Sciences, Universidad Politécnica de Valencia, Thursday

87 Lohmander, P., Economic forest production with consideration of the forest and energy industries, E.ON International Bioenergy Conference, Malmo, Sweden, Lohmander, P., Optimal dynamic control of the forest resource with changing energy demand functions and valuation of CO2 storage, UE2008.fr, The European Forest-based Sector: Bio-Responses to Address New Climate and Energy Challenges? Nancy, France, November 6-8, (See also later versions 2009) Lohmander, P., Optimal dynamic control of the forest resource with changing energy demand functions and valuation of CO2 storage, The European Forest-based Sector: Bio-Responses to Address New Climate and Energy Challenges, Nancy, France, November 6-8, 2008, Proceedings: (forthcoming) in French Forest Review (2009) Abstract: Page 65 of: Presentation as pdf: ecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdf Conference: ecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdfhttp:// ECOFOR, (in French) Summary of results by Peter Lohmander (on page 8) in “Evaluation du developpement de la bioenergie”, in Bulletin d’information sur les forets europeennes, l’energie et climat, Volume 157, Numero 1, Lundi 10 novembre IISD, Summary of results by Peter Lohmander (on page 6) in “Evaluation of Bioenergy Development”, in European Forests, Energy and Climate Bulletin, Published by the International Institute for Sustainable Development (IISD) Vol. 157, No. 1, Monday, 10 November,

88 Lohmander, P., Integrated Regional Study Stage 1., Presentation at the E.ON - Holmen - Sveaskog - SLU Research Meeting, Norrköping, Sweden, – , Lohmander, P., Öka avverkningen och hjälp Sverige ur krisen, VI SKOGSÄGARE, Debatt, Nr. 1, Lohmander, P., Economic Forest Production with Consideration of the Forest and Energy Industries (SLU ), Lohmander, P., Rational and sustainable international policy for the forest sector with consideration of energy, global warming, risk, and regional development, SLU, Umea, , Lohmander, P., Strategic options for the forest sector in Russia with focus on economic optimization, energy and sustainability (Full paper in English with short translation to Russian), ICFFI News, Vol. 1, Number 10, March

89 International seminar, ECONOMICS OF FORESTRY AND FOREST SECTOR: ACTUAL PROBLEMS AND TRENDS, St Petersburg, Russia, March 2009, Lohmander, P., Satsa på biobränsle, Skogsvärden, Nr 1, Lohmander, P., Stor potential för svensk skogsenergi, Nordisk Energi, Nr. 2, Lohmander, P., Strategiska möjligheter för skogssektorn i Ryssland Nordisk Papper och Massa, Nr 2,

90 Lohmander, P., Economic forest production with consideration of the forest- and energy industries, Project meeting presentation, Stockholm, Sweden, , Lohmander, P., Derivation of the Economically Optimal Joint Strategy for Development of the Bioenergy and Forest Products Industries, European Biomass and Bioenergy Forum, MarcusEvans, London, UK, 8-9 June, 2009, & ttp:// ttp:// Lohmander, P., Rational and sustainable international policy for the forest sector - with consideration of energy, global warming, risk, and regional development, Preliminary plan, ,

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93 A general continuous global approach to: - Optimal forest management with respect to the global warming problem and global economics - One section of the lectures by Peter Lohmander at UPV, Polytechnical University of Valencia, Spain, February 2010 Peter Lohmander Professor of forest management and economic optimization SLU, Faculty of Forest Sciences Dept. of forest economics Umeå