1. 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 901 83 Umeå, Sweden http://www.Lohmander.com

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8. 8 Second order minimum conditions:

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12. 12 Observations of the first and second order conditions:

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17. 17 So, if and or then

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

19. 19 A numerically specified example:

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21. 21 Comparative statics analysis:

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27. 27 Explicit solution of the example for alternative values of K

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35. 35 Dynamic approach analysis

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42. 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. 43 One way to determine the value(s) of k is to use this equation:

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45. 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. 47 Lets us solve the equation!

48. 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. 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. 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. 53 Derivation of the roots in the example:

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56. 56 We may determine the path completely using the initial conditions

57. 57 We also use the earlier derived results:

58. 58 Using the derived roots, we get:

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

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64. 64 Using the figures from the example, we get:

65. 65 The solutions to the numerically specified example X(t) = -106.63·EXP(- 0.13612·t) + 6.61·EXP(- 0.02718·t)

66. 66 Y(t) = - 69.95·EXP(- 0.13612·t) - 10.07·EXP(- 0.02718·t)

67. 67

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

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71. 71 Numerical solution of the example problem using direct minimization: Costmin Valencia Lohmander 2010-02-22

72. 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 = k-w; @free(anet); anet = a - u; @free(eqw); 31*equ+15*eqw=1500 + 16*k; 5*equ+6*eqw = -1400+5*k; end

73. 73 Variable Value Reduced Cost C 8975.806 0.000000 K 600.0000 0.000000 CU 4995.578 0.000000 CF 2516.909 0.000000 CA 1463.319 0.000000 CW 0.000000 0.000000 U 358.0645 0.000000 F 241.9355 0.000000 W 0.000000 1.903226 A 600.0000 0.000000 ANET 241.9355 0.000000 EQW -53.15315 0.000000 EQU 383.7838 0.000000

74. 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 = k-w; @free(anet); anet = a - u; @free(eqw); 31*equ+15*eqw=1500 + 16*k; 5*equ+6*eqw = -1400+5*k; end

75. 75 Variable Value Reduced Cost C 13952.25 0.000000 K 800.0000 0.000000 CU 6552.228 0.000000 CF 4022.401 0.000000 CA 2196.271 0.000000 CW 1181.353 0.000000 U 421.6216 0.000000 F 378.3784 0.000000 W 81.98198 0.000000 A 718.0180 0.000000 ANET 296.3964 0.000000 EQW 81.98198 0.000000 EQU 421.6216 0.000000

76. 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 = k-w; @free(anet); anet = a - u; @free(eqw); 31*equ+15*eqw=1500 + 16*k; 5*equ+6*eqw = -1400+5*k; end

77. 77 Variable Value Reduced Cost C 19357.66 0.000000 K 1000.000 0.000000 CU 7574.872 0.000000 CF 5892.379 0.000000 CA 2615.068 0.000000 CW 3275.339 0.000000 U 459.4595 0.000000 F 540.5405 0.000000 W 217.1171 0.000000 A 782.8829 0.000000 ANET 323.4234 0.000000 EQW 217.1171 0.000000 EQU 459.4595 0.000000

78. 78 Numerical approximation of the dynamics: ! dynsim; ! Peter Lohmander Valencia 20100222; 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;

79. 79 @FOR( time(t): dx(t)= -( cxx*x(t) + cxy*(y(t)) )); @FOR( time(t): dy(t)= -( cyx*x(t) + cyy*(y(t)) )); @FOR( time(t)| t#GT#1: x(t)= x(t-1) + dx(t-1) ); @FOR( time(t)| t#GT#1: y(t)= y(t-1) + dy(t-1) ); @for(time(t): @free(x(t))); @for(time(t): @free(y(t))); @for(time(t): @free(dx(t))); @for(time(t): @free(dy(t))); end

80. 80 Variable Value CXX 0.1033333 CXY 0.5000000E-01 CYX 0.5000000E-01 CYY 0.6000000E-01 X( 1) -100.0000 X( 2) -85.66667 X( 3) -73.30444 X( 4) -62.64442 X( 5) -53.45430 X( 6) -45.53346

81. 81 X( 19) -3.638081 X( 20) -2.705807 X( 21) -1.912344 X( 22) -1.238468 X( 23) -0.6675825 X( 24) -0.1853596 X( 25) 0.2205703 X( 26) 0.5608842 X( 27) 0.8447973 X( 28) 1.080263

82. 82 X( 40) 1.894182 X( 41) 1.881254 X( 42) 1.863429 X( 43) 1.841555 X( 44) 1.816360 X( 45) 1.788465

83. 83 Y( 1) -80.00000 Y( 2) -70.20000 Y( 3) -61.70467 Y( 4) -54.33716 Y( 5) -47.94471 Y( 6) -42.39532 Y( 7) -37.57492 Y( 8) -33.38500

84. 84 Y( 94) -0.7735111 Y( 95) -0.7524823 Y( 96) -0.7320263 Y( 97) -0.7121272 Y( 98) -0.6927697 Y( 99) -0.6739392 Y( 100) -0.6556210

85. 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 525-544, 2007 http://www.amazon.ca/gp/reader/0387718141/ref=sib_dp_pt/701-0734992- 1741115#reader-linkhttp://www.amazon.ca/gp/reader/0387718141/ref=sib_dp_pt/701-0734992- 1741115#reader-link Lohmander, P,. Energy Forum, Stockholm, 6-7 February 2008, Conference program with links to report and software by Peter Lohmander: http://www.energyforum.com/events/conferences/2008/c802/program.php http://www.lohmander.com/EF2008/EF2008Lohmander.htm http://www.energyforum.com/events/conferences/2008/c802/program.php http://www.lohmander.com/EF2008/EF2008Lohmander.htm 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), 2008 http://ansijournals.com/jas/2008/1995-2007.pdf http://www.Lohmander.com/LoMoOCC.pdf http://ansijournals.com/jas/2008/1995-2007.pdf http://www.Lohmander.com/LoMoOCC.pdf 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, 02-06 June, 2008 (In the version in the link, below, an earlier misprint has been corrected. ) http://www.Lohmander.com/Valencia2008.pdfhttp://www.Lohmander.com/Valencia2008.pdf

86. 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, 2008 http://www.Lohmander.com/Amsterdam2008.ppt http://www.Lohmander.com/Amsterdam2008.ppt 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, 2008-09-18 http://www.lohmander.com/CM/CMLohmander.ppthttp://www.lohmander.com/CM/CMLohmander.ppt 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 2008 http://www.lohmander.com/Madrid08/Madrid_2008_Lohmander.ppt http://www.lohmander.com/Madrid08/Madrid_2008_Lohmander.ppt Lohmander, P., Optimal CCS, Carbon Capture and Storage, Under Risk, International Seminars in Life Sciences, Universidad Politécnica de Valencia, Thursday 2008-10-16 http://www.lohmander.com/OptCCS/OptCCS.ppt http://www.lohmander.com/OptCCS/OptCCS.ppt

87. 87 Lohmander, P., Economic forest production with consideration of the forest and energy industries, E.ON International Bioenergy Conference, Malmo, Sweden, 2008-10-30 http://www.lohmander.com/eon081030/eon081030.ppthttp://www.lohmander.com/eon081030/eon081030.ppt 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, 2008 http://www.lohmander.com/Nancy08/Nancy08.ppt (See also later versions 2009)http://www.lohmander.com/Nancy08/Nancy08.ppt 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: http://www.gip-ecofor.org/docs/34/rsums_confnancy2008__20081105.pdf Presentation as pdf: http://www.gip- ecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdf Conference: http://www.gip-ecofor.org/docs/34/nancy2008englishprogramme20081106.pdfhttp://www.gip-ecofor.org/docs/34/rsums_confnancy2008__20081105.pdfhttp://www.gip- ecofor.org/docs/nancy2008/ppt_des_presentations_orales/lohmander_session_3.1.pdfhttp://www.gip-ecofor.org/docs/34/nancy2008englishprogramme20081106.pdf 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 2008 http://www.gip-ecofor.org/docs/34/nancy2008synthseiisd.pdfhttp://www.gip-ecofor.org/docs/34/nancy2008synthseiisd.pdf 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) http://www.iisd.org/, Vol. 157, No. 1, Monday, 10 November, 2008 http://www.iisd.ca/download/pdf/sd/ymbvol157num1e.pdf http://www.iisd.ca/download/pdf/sd/ymbvol157num1e.pdf

88. 88 Lohmander, P., Integrated Regional Study Stage 1., Presentation at the E.ON - Holmen - Sveaskog - SLU Research Meeting, Norrköping, Sweden, 2008-12-10 – 2008-12-11, http://www.lohmander.com/NorrDec08/NorrDec08.ppt, http://www.lohmander.com/NorrDec08/NorrDec08.pdf, http://www.lohmander.com/NorrDec08/NorrDec08RawData.xls http://www.lohmander.com/NorrDec08/NorrDec08.ppt http://www.lohmander.com/NorrDec08/NorrDec08.pdf http://www.lohmander.com/NorrDec08/NorrDec08RawData.xls Lohmander, P., Öka avverkningen och hjälp Sverige ur krisen, VI SKOGSÄGARE, Debatt, Nr. 1, 2009 http://www.lohmander.com/PLdebattVIS2009nr1.pdfhttp://www.lohmander.com/PLdebattVIS2009nr1.pdf Lohmander, P., Economic Forest Production with Consideration of the Forest and Energy Industries (SLU 2009-01-29), http://www.lohmander.com/SLU09/SLU09.pdf http://www.lohmander.com/SLU09/SLU09.ppthttp://www.lohmander.com/SLU09/SLU09.pdf http://www.lohmander.com/SLU09/SLU09.ppt Lohmander, P., Rational and sustainable international policy for the forest sector with consideration of energy, global warming, risk, and regional development, SLU, Umea, 2009-02-18, http://www.lohmander.com/IntPres090218.ppthttp://www.lohmander.com/IntPres090218.ppt 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 2009 http://www.Lohmander.com/RuMa09/RuMa09.htm http://www.Lohmander.com/RuMa09/RuMa09.htm

89. 89 International seminar, ECONOMICS OF FORESTRY AND FOREST SECTOR: ACTUAL PROBLEMS AND TRENDS, St Petersburg, Russia, March 2009, http://www.lohmander.com/RuMa09/ProgramRuMa09.pdfhttp://www.lohmander.com/RuMa09/ProgramRuMa09.pdf Lohmander, P., Satsa på biobränsle, Skogsvärden, Nr 1, 2009 http://www.Lohmander.com/PL_SV_1_09.jpg http://www.Lohmander.com/PL_SV_1_09.jpg Lohmander, P., Stor potential för svensk skogsenergi, Nordisk Energi, Nr. 2, 2009 http://www.Lohmander.com/Information/ne1.jpg http://www.Lohmander.com/Information/ne2.jpg http://www.Lohmander.com/Information/ne3.jpg http://www.Lohmander.com/PL_SvSE_090205.pdf http://www.Lohmander.com/PL_SvSE_090205.doc http://www.Lohmander.com/Information/ne1.jpg http://www.Lohmander.com/Information/ne2.jpg http://www.Lohmander.com/Information/ne3.jpg http://www.Lohmander.com/PL_SvSE_090205.pdf http://www.Lohmander.com/PL_SvSE_090205.doc Lohmander, P., Strategiska möjligheter för skogssektorn i Ryssland Nordisk Papper och Massa, Nr 2, 2009 http://www.Lohmander.com/PL_NPM_2_2009.pdf http://www.Lohmander.com/PL_RuSwe_09.pdf http://www.Lohmander.com/PL_RuSwe_09.doc http://www.Lohmander.com/PL_NPM_2_2009.pdf http://www.Lohmander.com/PL_RuSwe_09.pdf http://www.Lohmander.com/PL_RuSwe_09.doc

90. 90 Lohmander, P., Economic forest production with consideration of the forest- and energy industries, Project meeting presentation, Stockholm, Sweden, 2009-05-11, http://www.lohmander.com/EON_090511.ppthttp://www.lohmander.com/EON_090511.ppt 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, http://www.lohmander.com/London09/London_Lohmander_09.ppt & ttp://www.lohmander.com/London09.pdf http://www.lohmander.com/London09/London_Lohmander_09.ppt ttp://www.lohmander.com/London09.pdf Lohmander, P., Rational and sustainable international policy for the forest sector - with consideration of energy, global warming, risk, and regional development, Preliminary plan, 2009-08-05, http://www.lohmander.com/ip090805.pdf http://www.lohmander.com/ip090805.pdf

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93. 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 901 83 Umeå