Optimal Economic Control Strategies in Forest Resource Management by AFFENDI ANWAR A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Applied Economics Montana State University © Copyright by AFFENDI ANWAR (1974) Abstract: The investment decision is the most complex and difficult problem in forestry management because it involves large lapses of time and uncertainty of the future outcome resulting from the decision. This study attempts to analyze the forest production decision problem in general and management aspects related to investment, particularly those related to optimal thinning and rotation decisions. The decision model for this study is based on the application of modern control theory to the forest management problem. An appropriate line of approach to solve the control problem in forest production systems is stochastic dynamic programming which is capable of capturing the stochastic behavior of the system and incorporate risk factors of the decision explicitly. Another advantage of the method is thatit provides a powerful systematic search for obtaining numerical solutions to the problem, provided the required dynamic interrelationships of the system and transition probabilities can be estimated empirically. Based on this method a criterion function is formulated as maximization of the sum of expected discounted net returns to soil site as a modification of the Faustmann criterion in a statistical sense. To approach a specific problem in forest management, however, this study pursues another method to solve the control problem of a disease-infected lodgepole pine stand due to lack of available basic data. The control model for this specific problem is based on a modified simulation model which was previously developed by Myers et al. [1971]. By a modification of this model, the redirected computer simulation program resulted in an economic model which has a specific objective function consistent with the problem of managing a diseased timber stand. The criterion function of the model is generalized present value as an extension of the Faustmann criterion. By virtue of the certainty equivalence principle, the simulation model which is deterministic can be considered as a good approximation for solving stochastic problems in managing a forest production system due to the fact that the decision processes based on this model are carried out sequentially. The computer simulation model can serve as a management aid to help the forest manager in making decisions related to investment problems for any forest stand condition and economic factors which influence the system. Since the simulation results, in general, show a consistency with a priori logical reasoning, it is concluded that the approach can be expanded for application to other species and areas larger than the Rocky Mountain Region where this study was carried out.  OPTIMAL ECONOMIC CONTROL STRATEGIES IN FOREST RESOURCE MANAGEMENT by AFFENDI ANWAR A thesis submitted in partial fulfillment of the requirements for the degree , DOCTOR OF PHILOSOPHY i. ■ Applied Economics Approved: Chairman, Examining Committee Head, Major Department MONTANA STATE UNIVERSITY Bozeman, Montana October, 1974 Iii ■ ACKNOWLEDGMENTS I wish to acknowledge and express my thanks to many individuals for. their assistance and contribution to this study. First, I am indebted to my major professor. Dr. Oscar R. Burt, for his counsel and guidance . ■ during preparation of this thesis and throughout, the course, of my studies at Montana State University,. Bozeman, and during his one-year tenure at the University of California, Davis, where I had the oppor­ tunity to broaden my knowledge and experience. Mr, D. M. Cole at the Forest Science Laboratory and Mr. W. R. Driver at the Gallatin Forest District, Bozeman, Montana have supplied informa­ tion on the physicalr-biological aspects of Iodgepole pine stands for empirical analysis. Additional information on those aspects were obtained also from forest officers in Great Falls and Missoula, Montana. Without their valuable assistance the empirical part of this study would not have been possible. Thanks are due to Drs. Ward, McConnen, Stauber, Lund and Billis who served on the graduate committee and Dr. Verne House who provided extensive literature on forestry. During the period of my studies, I have been on.leave of absence from a position at Bdgor Agricultural University, Bogor, Indonesia. For the assistance and support from the institution, I am grateful. Additional financial help and moral support from my brother Kuswanda iv of Jakarta, Indonesia, during my studies is also gratefully acknow­ ledged. Appreciation is due to Dr. E. H. Ward and Mr. R. S. Sinaga who led my interest to studying Agricultural Economics. Also my thanks go to Mr. Bill Yaeger for his computer programming assistance and to Peggy Humphrey for typing the manuscript. . . . . . Special acknowledgment goes to the Agricultural Development Council, Inc., of New York for its financial support and the assis­ tance, of its Fellowship Secretary, Ms. Grace Tongue, and its Admin- •' . ■ .istrative Officer, Dr. Russell Stevenson, during my whole program of studies at both Montana State University and the University of California. My special appreciation is due to my wife. Pin, and my two children, Fifi and Reza, who endured their patience during the whole endeavor. V . TABLE OF.CONTENTS Page VITA.................. ii ACKNOWLEDGMENTS .................................. iii TABLE OF CONTENTS . . , . ........................ v LIST OF TABLES AND FIGURES........ vi ABSTRACT. .......................................... vii CHAPTER I: INTRODUCTION. . .................... I 1.1. Motivation of Study .................. ............ . I 1.2. Objectives and Plan of Analysis............ . .. . . . 4 CHAPTER II: .INVESTMENT MODEL FOR FORESTRY. . . . . . . . . . . 8 2.1. The Concept of Forest Production............. 8 2.2. Multiperiod Investment Decisions. . . . . . . . . . . . 12 2.3. An Investment Decision for Timber Stands. . . . . . . . 15 2.4. Consideration of Risk and Uncertainty................. 19 CHAPTER III: OPTIMAL CONTROL IN FOREST MANAGEMENT............ 23 3.1. Management Problem in Forestry................ 23 3.2. Formulation of General Control Problem. . . . . . . . . 25 3.3. Dynamic Programming Solution................... 29 CHAPTER IV: SIMULATION APPROACH TO CONTROL PROBLEM . . . . . . 47 4.1. The Setting...................... 47 4.2. The Impact of Disease Infestation . . . . . . . . . . . 48 4.3. Simulation Model. ........................ 52 4.4. Objective of the Control Process............ 57 CHAPTER V: APPLICATION TO LODGEPOLE PINE ........ . . . . . . . . 63 5.1. Data Description and Assumptions. . . . . . . . . . . . 63 5.2. Model Experimentation.................. 68 5.3. Simulation Results................................ .. . . 73 . CHAPTER VI: SUMMARY AND CONCLUSION . . . . . . . . . . . . . . . ■ 88 6.1. General ................ 88 6.2. Specific Problem.............. .. .... . . . . . . . 92 APPENDIX.......... gg BIBLIOGRAPHY............ 145 vi 'LIST OF TABLES AND FIGURES Table ■ Page 1 OPTIMAL DECISION RULES STARTING FROM BARE LAND UNDER AN INFINITE PLANNING HORIZON.. . . . . . . . . . . . . . . . . 74 ■ ■ '2 OPTIMAL DECISION RULES FOR CURRENT PRICE . . . . . . . . . 7.5 3 OPTIMAL DECISION RULES FOR THE HIGHEST PRICE . . . . . . . 76 , 4 OPTIMAL DECISION RULES FOR THE LOWEST PRICE. . . . . . . . 77 5 OPTIMAL DECISION RULES FOR ADVANCED TECHNOLOGY PRICE . . . 78 .6 OPTIMAL DECISION RULES FOR ADVANCED TECHNOLOGY PRICE WITH CURRENT ESTABLISHMENT COST. . . . . . . . . . . . 79 Figure Page 1 The Relation Between Stages and Chronological Time . . . . 34 2 Flow Chart of Main Program LPMIST2 ...................... 56 vii ABSTRACT The investment decision is the most complex and difficult problem, in forestry management because it involves large lapses of time and . uncertainty of the future outcome resulting from the decision. This study attempts to analyze the forest production decision problem in general and. management aspects related to investment, particularly those related to optimal thinning and rotation decisions. The decision model for this study is based on the application of modern control theory to the forest management problem. An appropriate line of approach to solve the control problem in forest production systems is stochastic . dynamic programming which is capable of capturing the.stochastic behavior of the system and incorporate risk factors of the decision explicitly. Another advantage of the method is that.it provides a powerful systematic search for obtaining numerical solutions to the problem, provided the required dynamic interrelationships of the sys­ tem and transition probabilities can be estimated empirically. Based on this method a criterion function is formulated as maximization of the sum of expected discounted net returns, to soil site as a modifica­ tion of the Faustmann criterion in a statistical sense. To approach a specific problem in. forest management, however, this study pursues another method to solve the control problem of a disease- infected lodgepole pine stand due to lack of available basic data. The control model for this specific problem is based on a modified simulai- tion model which was previously developed by Myers _et AL• [1971]. By a modification of this model, the redirected computer simulation program resulted in an economic model which has: a specific objective function consistent with the problem of managing a diseased timber stand. The criterion function of the model is generalized present value as an extension of the Faustmann criterion. By virtue of the certainty equivalence principle, the simulation model which.is deterministic can be considered as a good approximation for solving stochastic problems in managing a forest production system, due to the fact that the decision processes based on this model.are carried out sequentially. The computer simulation model.can serve as a management aid to help the forest manager in making decisions related to investment problems for any forest stand condition and economic factors which influence the system. Since the simulation results, in general, show.a consistency with a priori logical reasoningj it is con­ cluded that the approach can be expanded for application to other, species and areas larger than the Rocky Mountain Region where this study was carried out. CHAPTER I INTRODUCTION 1.1. Motivation of Study Among the most complex and difficult problems confronting a forest management authority are those involving investment decisions. Growing timber on forest land is an investment asset which has a changing value over time. Growing a timber stand can be considered as a long term investment decision where a long gestation period is needed between initial investment outlay and realization of all expected earnings. On the other hand, cutting timber at final harvest is an act of disinvest­ ment in which capital represented by the value of timber is released and the forest land is made available for a subsequent timber crop or other uses. Hence, there is opportunity cost attached to the land presently occupied by standing timber. Between the date of planting trees and final harvest, the forest manager may undertake a series of thinning actions and. other stand improvement activities such as clearing, fertilizing, pruning, disease control, etc. which involve additional investment decisions, in order to improve the future stand condition and hence to obtain a higher value of final harvest. The land committed to timber production, growing stock of timber, and cumulated stand improvement outlays constitute.the investment in a forest at any point in time. The long, gestation period . between investment outlays and receipt of revenues, combined with the 2 complex biological interrelationships of all management activities over the economic life of the forest, is what makes forest investment deci­ sions a difficult problem. Throughout the development of the principles of traditional forest resource management, various theoretical and empirical studies have been made to serve as guides for sound investment decisions in forestry. In an era of rapidly, changing technology, economic decisions grow in frequency and complexity, and wrong decisions increase in costs. With this additional impetus and advances in applied mathematics in recent years, quantitative methods for handling complex, economic problems have been introduced to the forestry field. Among others are those which are related to determination of forest management, including simultan­ eous optimization of rotation and thinning decisions (see Naslund, 1969 and Schrender, 1971) . However, both authors have mainly concen­ trated their studies on the discussion of a general analytical frame­ work for determining optimal rotations and thinnings without any empirical application of the results. Determination of optimal rotation and thinning decisions in forestry, by and large, is influenced by physical-biological aspects of the forest growth which constitute an "engineering" side of forest ■■ ■ . . .■ production systems. Hence the optimal decision will be specific to a. given forest environmental condition. . Therefore, if there are any 3 extraneous factors which influence this "engineering" aspect, the opti­ mal decision will be affected. Another factor which may influence the optimal decision is economic elements, including supply and demand of the commodity produced, capital, and input factors within the production system. Conceptually, optimal investment decisions in forest production systems can be approached from a more general equilibrium theory, but in practice, a model giving, the general equilibrium solution would often be extremely unwieldy. Hence, in order to formulate an operationally manageable model, we must often be willing to settle the problem for something less. Thus the investment problem in forestry for this study will be limited to partial equilibrium analysis where the price and cost structure of the commod­ ity, capital, and other input factors are determined outside the system of forest production. One specific aspect of the study will deal with the formulation of conceptual and empirical solutions to the forest investment problem so that it will help to give a better understanding of the overall problem. Since the major portion of forest investment decisions is dominated, by determination of the rotation and thinning policy, for clarity of further discussion, throughout the course of this study the term invest­ ment will refer to both rotation and thinning decisions interchangeably. Moreover, even though there are many intangible benefits that could be derived from the forest stand, this study will be limited to the measurable monetary benefits from wood production. 4 1.2. Objectives and Plan of Analysis There are two general objectives to be pursued in this study. The first and more theoretical objective is to formulate and to evaluate the optimal control strategy in art operational model suitable for analysis of decision making involved in the choice of a sound forest management policy in cases where complete basic data can be obtained— : for example, the basic data which could be obtained from a continuous forest inventory (C.F.I.). To obtain an optimal control strategy in forest production systems requires a multistage decision process; that is, a sequence of decisions that will optimize some objective function or criterion function. The appropriate criterion function for forest management policy is maximum discounted net returns with respect to soil site, but some method must be devised to account for the random occurrence of some measurable exogenous factors to the production sys­ tem, such as disease, insects, and other factors that may affect the forest growth during evolution of the system. Therefore, the ■decision criterion of this study will be maximization of expected soil value over the entire planning horizon in a statistical sense— a modi­ fication of the well-known Faustmann criterion (Faustmann,. 1849) . The strategy or conditional decision rule is applied sequentially as new information unfolds through time and the rule is chosen to maximize the above stated criterion. Mathematical analysis of this type of decision processes led to the development of dynamic, program- ming, a la Bellman [1957]. _!/ This type of model provides more than just a computational procedure for obtaining numerical results. It is also a logical framework and in some cases, analytical results can be. derived for ..specific, problems, such as in optimal control of natural resources (see Burt/ 1964, 1966 and Burt and Cummings, 1970). In fact, analytically, dynamic programming is one of the modern approaches for solving general control problems (see Intriligrator, 1971 and Robert, and Schulze, 1973). . The realm of modern control theory also includes the maximum principle of Pontryagin ej: al. [ 1962] as an extension Of the classical calculus of variations. This method will also be briefly reviewed in formulating the concept of optimal forest resource management. The second objective, which is more applied, is to analyze forest management decisions regarding ah optimal control strategy for dwarf mistletoe (Arceuthobium spicies). This disease has adverse effects oh forest growth and constitutes one of the exogenous factors which affects the production system, and hence will affect the optimal investment decision. However, due to lack of available basic data, l/ Sometimes dated versions of linear programming are also called "dynamic programming" which is intrinsically not dynamic because the optimization is made once and for all. 6 a somewhat different approach for.controlling diseased stands will be applied, as an approximate solution to more general control problems. The criterion function is still based.on a stochastic version of the Faustmann criterion, but with some modification. The physical-biological aspects of the production system for this study are obtained from a computer simulation program which has been developed by Myers et al. [1971]. This simulation model includes the dynamic physical-biological interrelationships between the level of disease infestation and growth of the forest which can be altered by thinning decisions at various stages of maturity of the forest. This . simulation study provides the necessary information which enables us to construct a dynamic decision model for controlling forest production In' order to build a decision model based on economic criteria, the simulation model required some modifications by inserting the computa­ tion of some economic elements into.the model. By varying some of the parameters of the system, results could then be obtained by calculating a performance measure of the system. The performance measure for this study is present value of the stream of net returns from wood produc­ tion over an infinite planning horizon. A search procedure on the parameters of the modified simulation model provides an approximation to the optimal decision rule. Information derived from this analysis could be of assistance to the forest management authority, which is 7 responsible for the execution of current forest management policy in the Rocky Mountain Region where the study was carried out. Chapter II of the study discusses the basic structure of. the forestry investment decision, and a simple investment decision model is also described. In Chapter III, the general optimal control prob­ lem in relation to forest investment decisions is dealt with. Then a more operational model of the investment problem with a dynamic program­ ming solution is presented. In Chapter IV the empirical model of the study using the simulation model as an approximate solution to the general control problem is presented. Finally, Chapter V contains an interpretation of the results of the empirical study and some conclu- . sions. CHAPTER II INVESTMENT MODEL FOR FORESTRY ■ 2,1. The Concept of Forest Production Forestry, the business of growing timber; has characteristics that distinctly give it special status among agricultural production activities. First, the long gestation period between initial input and the first harvested output is quite unique. Second, there is an extended period of autonomous growth in value associated with the initial, investment decision. Then eventually, there is a gradual decline in the productive capacity of trees to grow over an extended time due to . the biological" aging process. To produce a wood crop, one has to plant the trees and care for ' them through several years before there is any opportunity to receive J revenues. There is a long interval between the date when growing timber reaches a minimum size of marketability and. the date when trees ■ finish their growth. At any time between these two dates the timber may be judged ready for cutting on the basis of an economic criterion. The land has many alternative uses, but the trees on the land have virtually no use other than production of standing timber. Thus, an understanding of resources committed on a long-run basis is important . to any consideration of timber production.' Let us assume there exists an expected intertemporal production .function with many inputs and many outputs. The production function 9 associated with initial time t .is assumed to be twice differentiable: $tQ) = t̂Cq1, q2> • ••, qn) (I) where q̂ (j = I, 2..... n) are discrete dated inputs and outputs and n is the total number of dated inputs and outputs over some arbitrarily long but finite planning horizon. The condition q. < 0 signifies ahJ input and q. > 0 indicates an output. A given set of dated inputs will I result in a unique set of dated outputs which the firm values at v. (j =1, 2, ..., n). The v. are prices adjusted by the time discount J I . ’ structure. ^ Assuming profit maximizing behavior of the firm and perfect compe­ tition in the factor and product markets, an optimal production plan can be derived by means of the Lagrarigian function: . = I V q. - AS (<£). . (2) J = I j j The first order necessary conditions for a maximum of this Lagrangian will give a solution for output supply and input derived demand functions: A A q3'= qj Vl’ V2? '’*’ Vn̂ , J = I, 2, ..., n. (3) Given initial expectations, the demand function for inputs shows ■ how the firm would desire to allocate inputs over, time to maximize 10 profit. It also shows the stream of outputs which the firm expects as a result. The t-subscript on @ implies that a different production function exists at each time period t when a set of decisions are made. This production functipn is determined by all previous decisions and reflects the number of acres of standing timber of various ages at the point in time t, as well as any other fixed conditions resulting from all prev­ ious decisions of the firm. Moreover, because of some changes in production conditions, it is assumed that the firm makes periodic revisions of its plans based on new expectations. Thus each period the firm maximizes an appropriate Lagrangian of the above form. This maximization may in some periods call for the firm to invest in planting new trees expressed as derived demand for land.. When prices were sufficiently favorable in some previous period trees were planted. This decision process occurs recursively where in each period the forest manager uses updated expectations as timber becomes one period older. As long as reasonable economic stability is maintained, we may assume that there exists a wide span of periods where it is more profitable to keep the trees, rather than to cut them for timber, until it comes the time where additional net returns from presently standing timber are equal to opportunity cost of the land for other uses. One use of the land which may determine its opportunity cost is growing a new stand of trees.. 11 Each optimization problem faced by the firm at a particular time period t involves variables associated with time t and all future periods in the firm's planning horizon. The future variables must be considered jointly with the current variables in the optimization prob­ lem to determine optimum levels of the current variables., but these future variables are then ignored until the perpetually recurring -optimization problem is faced in the next time period, say t+1. At that time the entire process is repeated to get optimal levels of the variables associated with period t+1. In a sense, the variables in the optimization which are associated with future time periods are merely instruments or artificial variables used to derive optimal, levels of current variables. The underlying justification for this sequential dynamic optimiza­ tion approach is certainty equivalence decision theory of Simon [1955] .which was later expanded by Theil [1957] and Madansky [I960]. Although the theory only applies to quadratic criterion functions subject to linear constraints, it would appear to be a reasonable approximation to the general case of decision under uncertainty. The.above description of the concept of forest production is approached mainly from neo-classical theory of production related to . investment decisions. However, in order for the concept to be meaningful, in practice it must be operationally feasible. The following section 12 will discuss a more realistic model of investment decisions consistent with the above concept. 2.2. Multiperiod Investment Decisions Before we can go further to a formal investment decision model it is necessary to describe the procedure which must be employed to compare present and future receipts and outlays. Hence, a discussion of the discounting or present value concept is important to any invest­ ment . decision model. In real world situations, most decisions are periodic; hence, in facing economic problems which deal with time, the time span or planning horizon can be divided into several periods usually with equal intervals. Multiperiod investment decisions are characterized by a situation where factors of production invested during one time . period will affect, the level of output during subsequent periods. Therefore, there exists a functional relationship between input and output which has different dating. Further, we assume that there exists a market of capital where capital can be borrowed or lent at some given rate of interest. By using this interest rate, outlays or revenues incurred during different time periods can be made comparable by discounting (or compounding) them to one . common period'. Suppose P dollars were invested or borrowed for one time period at the rate r. . The value of r expresses the proportion of the amount of capital borrowed or lent which constitutes a cost, of capital to the 13 borrower or Income to the lender (investor) per period. Hence at the end of one period the investor would expect rP dollars in interest, and with return of the principal P , this would give the sum - P + rP = P (1+r) dollars. If the investment rate does not change over time, the investment would yield (1+r) = P(1+r) ̂dollars at the end of the subsequent period after the first, or in general, it would 2yield Sfc = Sfc_^(l+r) = P (1+r) after t time periods. Therefore, an amount of P dollars expected to be received after t time periods can always be exchanged, or equals P (1+r) ̂at the present, because the firm or individual who expects to receive the amount of. P dollars after , -tt time periods can borrow the amount of P(1+r) dollars at the present and he can repay with the amount P(1+r) t(l+r)t = P dollars at the later period. The expected future receipts = P (1+r) ̂from investment are called —tthe future or compounded value of P , while the amount P = (1+r) is called present value or. discounted value of Sfc after t periods of investment. Hence, the factor (1+r) by which the future value should be multiplied to get the present value P is called the discount factor. If we assume that within one time period we have more than one time compounding, say m times, then the future value Ŝ . becomes: t P d + ^ . (4) 14 Let us designate — = k, then r / . Sfc = P(1 + ̂ )rkt = P [(I + kJrt. (5) As compounding ,,within the time period becomes more and more fre­ quent , in the limit as nr*”, .hence: lim.S = Iim P[(l + i)k]rt = P[lim (I .+i)k ]rt.,. . (6) nrX” k-x»- k-x» K I kSince by definition Iim (I + y) = e, where e is the base of nat- k-x» k ural logarithm, then TtIim S. = Pe (7) m-x” ' or P = (lim S )e~rt. (8) mx” Discounting provides a method of transforming future incomes and outlays so that they are commensurate with the present. Through discounting, a future stream of net returns can be expressed as a single number which is called present value of the entire set. of future returns. For example, an entrepreneur expects to.receive . returns from an investment equal to , ..., R^ during time periods 0, I, 2, ..., T, respectively,and the. corresponding discount factors t tare g = I/(1+r) , t = 0, I, 2, ..., T. Present value of this.revenue stream is: 15 R = I R1-Bt , t = 0, I, 2, T (9) t=0 Z or for instantaneous discounting, T _ rt R = J R(t)e dt , 0 < t £ T. 0 While static theory assumes that the entrepreneur will maximize his profit in a single time period, in dynamic multiperiod cases, it is assumed that the entrepreneur will maximize present value of the difference between future incomes and outlays; or in short, he will maximize present value of the streams of future net returns from the investment. 2/ Having established the concept of multiperiod investment decisions now we are in a position to apply this concept to forestry.enterprises. The following discussion will deal with a very simple case of invest­ ment decisions concerning a growing stand of timber. 2.3. An Investment Decision for Timber Stands In the forestry enterprise, the investment decision is charac­ terized by a relatively large amount of initial outlays for timber 2/ If the decision maker is a public agency who is concerned with public investment projects, the same criteria still can be applied. A primary difference between private and public investment criteria lies in the rate of discount. The private rate reflects private opportunity cost of capital, while the social rate reflects the social opportunity cost of public projects as the value to society of the next best alternative use to.which the funds employed on the project could have been put. stand establishment, and receipts are dominated by the income from terminal harvest as contrasted to the .flow of annual costs of upkeep and returns from thinnings. In order to formulate a criterion function for the investment decision, let us use the following notation: T = length of the rotation cycle; B(T) = present value of all net returns; R(t) = net returns associated with.management of the timber stand at the. moment t, except for net sale value of final harvest; H(T) = net sale value of final harvest; r = interest rate; and t '3 = I/(1+r) is the discount factor for any period t. Present value of the stream of. net returns from the timber stand for a single rotation cycle can be expressed as T B(T) = f R(t)3t + H(T)3T , - (10) t=0 or for the continuous time discounting case: 3/ T B(T) = / R(.t)e~rtdt + H(T)e~rT (11) 0 However, formulation of the present value criterion in (10) and . (11) considers only one rotation cycle, which is not applicable to the .3/ In some cases, a continuous version of the present value cri­ terion is preferable, especially for analytical treatment of the model, 16 17 forestry enterprise based on a sustained yield principle. Based on this principle, the enterprise would evaluate the income stream not only from one rotation cycle, but from all future cycles within the planning horizon. Hence, the appropriate present value criterion for . investment decisions in forestry should be based on an expected income stream over an infinite planning horizon. Therefore, present value of net returns from growing stock of timber over an infinite chain of rota­ tions can be expressed as follows: T . B(T) = [/ R(t)e~rtdt + H(T)e"rT][l + e“rT + e~2rT + ... + e~lrT 0 I T + ...]--------R(t)e"rtdt + H(T)e_rT]. (12) I - e“ri 0 The expression in (12) is essentially the Faustmann criterion. 4/ Having established the criterion function, we can analytically deduce a decision rule for determining optimal length of the rotation cycle. For simplicity, we assume that other decisions such as the . thinning level are given; the forest entrepreneur is only concerned with determination of the optimal rotation. The object is to maximize 4/ Faustmann's solution to the problem, was later confirmed by G. A. D. Preinreich, who was apparently unaware of Faustmann's work in forestry nearly a.century earlier, in "The Economic Life of Industrial Equipment," Econometrics, VIII (January, 1940), pp. 12- 44. 18 present value of the infinite series of discounted future returns with respect to rotation length T. Therefore, the problem of investment becomes: T MaxB(T)=MaxI---— {/ R(t)e"rtdt + H(T)e^r1)]. (13) T T I - e 0 The necessary condition for the above problem can be obtained by differentiating (16) with respect to T and equating the results to . . zero, which gives the following: dB = dT ■ .{/ R(t)e rt + H(T).e rT} +-rT, (I - e rT)2 6 ' " (I - e“rT) {R(T)e rl + e“rT - rH(T)e-rT} = 0. Multiplying (14) by er^(l - e r"*") yields , (14) -rB(T) + R(T) + ^ - rH(T) = 0 (15) R(T) + df = r[H(T) + B (T)]- (16) A literal economic interpretation of (16) is that under an optimal rotation policy, we shall cut the timber when the current return R(T) for having the stand one more year plus the increase in the harvest dHvalue equals the interest on the sum of harvest value and soil rent. Soil rent is equal to B(T) which is present value of net returns 19 starting with bare timber land, following an optimal rotation policy ■ over an infinite planning horizon. The above analytical method, using elementary calculus for deter­ mining the optimal rotation in forestry, gives some insight as to the optimal decision rule to be followed by the forest entrepreneur. However as the problem becomes more and more realistic, and consequently more complicated, more sophisticated mathematical methods are required. The next chapter will examine some more comprehensive models for forest investment decisions. Before going into the analysis of more realistic models for investment, we will discuss the problem of uncertainty which is ah important difficulty faced by any decision maker in a real world situation. 2.4. Consideration of Risk and Uncertainty No discussion of economic decisions— such as determination of optimal investment in growing timber— can be complete without recog­ nizing and taking into account the effect of uncertainty of future events. The analytical model of investment in the last section was simplified by assuming perfect knowledge, of input-output relations and prices in the production system within the firm's planning horizon. . However a good economic decision model should be capable of providing managers with guides to take action when they face choices between alternative courses of action where the results are uncertain. Intro- duction of risk and uncertainty into the analytical model will 20 complicate the problem, but•from a practical decision viewpoint, it is justified by the additional realism obtained in approaching a real world problem. Although no one knows the exact outcome of future events, a real world problem of investment could be treated as if the probability distribution of the future outcomes were known rather than pretending to know the exact future outcomes. Knight [1921 ] distinguished risk and uncertainty as two different phenomena. Following his ideas, risk refers to the situation when parameters of probability distributions of future outcomes can be estimated so as to be actuarially insurable; or in other words, the variability of future outcomes can be empirically measured. Uncertainty, on the other hand, refers to a situation in which the parameters of the probability distribution of the outcomes cannot be empirically or quantitatively estimated. The importance of risk factors affecting forest management deci­ sions has long been recognized by forest economists, and most forest land managers are aware of the advantage in maintaining a flexible forest management program to reduce the difficulties of changing output of the forest in response to changes in the market. Markowitz made an important contribution to the theory of investment under risk when he developed his methods for determining an optimally diversified portfolio [1959 ]. He developed a logical framework to guide invest­ ment decisions by explicitly recognizing the risk factor. His model 21 provides a formal method for evaluating the advantage of diversifying investments as a means of reducing risk associated with the expected returns from a number of investments. Dowdle [1962] has applied the EV investment guide of Markowitz to diversify cost expenditures in forest investment activities. 5/ However this method'only applies to static investment decisions in forestry. A promising method applicable to dynamic investment in forestry which recognizes risk factors has been developed by Burt [1965].. He analyzes the problem of investment in the context of the general replacement model under risk. His discussion is related to a special case of Howard's model (Howard, I960), for which an analytical solu­ tion is derived. The replacement model is concerned primarily with the economic decision problem of asset life under conditions of chance failure or loss, i.e., possible loss of the asset due to exogenous influences outside the system. Though fiis model pertains to dynamic decisions of investment in forestry, the exogenous factors described 5/ E and V refer to expected income and variance of income, respectively The efficiency criterion of investment decisions can be considered as a combination of individual investments such that the variance, for the entire set of investments cannot be reduced without also reducing the mean. All possible combinations of such investments traces out what is commonly called the EV efficiency frontier. 22 in the analysis refer to a sudden extraneous shock, such as insect attack or fire which may cause a sudden destruction of the asset in a ■short period of time. This model is not applicable to our problem where the exogenous influence of dwarf mistletoe.on the standing timber system is chronic in nature. In the next chapter we present a stochastic dynamic programming solution to the control problem which can incorporate the risk factor into the investment decision model. CHAPTER III OPTIMAL CONTROL IN FOREST MANAGEMENT . 3.1. Management Problem in Forestry Sound management practices in forestry require knowledge about physical-biological aspects of the forest resource. If the primary objective in forest management is to produce timber, the basic physical resource with which a forest manager is concerned is land and the timber growing on it. The productivity of timber as a growing / . "machine" where the wood product is part of that "machine" depends on many factors. First, growth of timber depends on the site quality which reflects the capacity of a given area of land to grow a certain timber species. Another important factor to be considered in growing timber is stocking (tree density) which affects the rate, of growth of timber at a given.age and site quality. Genetic make-up of a tree species determines the capacity of timber to grow and withstand any exogenous influences which might affect its development. Those exogenous factors to be accounted for in growing timber are disease, insects, adverse climate, etc. For a certain tree species, there exists an optimum physical-biological combination among the elements comprising the forest environment which give an optimum growth of timber. Hence, a good knowledge of how to use those resources effectively is essential in forest management. 24 In order to obtain maximum benefit from the forest resource, full advantage of knowledge, skill, and art of silviculture and forest measurement should be taken. Those technical aspects of forestry which are important should be considered in making any decision to. solve management problems. However the concern of this study is investment decisions related to the economic problem of how to grow a. better timber stand which will result in maximum benefits over a given, time horizon. In order to achieve this objective, the study will . focus oh economic decisions, particularly the ones that are associated with the intensity of thinning and length of rotation cycle. Thinning the standing stock of timber can be beneficial for the following reasons: (I) to eliminate undesirable trees and thereby concentrate growth on the desirable trees which produce a high value of timber, (2) to prevent excessive competition on site resources among trees which would result in wasting energies needed for total growth, (3) to obtain revenues before final harvest. The fourth reason often advanced is to increase the net growth of . timber. However, evidence on this aspect of thinning is quite inconclusive, at least for most timber species. Timber production involves a large lapse of time.between planting and harvesting. Hence time is an important variable in management decisions of forestry. In a sense, decisions must be made in the forestry enterprise at each moment in time over the entire planning 25 horizon of the firm. A mathematical model adapted to this kind of problem is commonly called "control theory." 3.2. Formulation of General Control Problem The essence of an optimal control problem is to choose a feasible time path for the control (decision) variables which will maximize the objective function. The choice of time paths for the control variables implies a set of differential equations called equations of motion, and also determines time paths for certain additional variables called state variables which describe the state of the system. Employing this concept in the forest management problem, let us consider a forest production system which can be described by a set of state variables , x^, x̂ _. Hence the state of the system at any given time t can be represented by the state vector X(t) = [x^(t)^ (t), ...,‘x (t)]. Suppose the objective of forest management is to maximize total dis­ counted net benefits over a given planning horizon. In order to achieve this objective-, a sequence of decisions is required that will maximize the objective function. Let us designate the set of managerial decisions by decision variables u^, u^, ..., ug, so that at any time t, the related decision is represented by the vector of decision U(t) = [u^(t), u2(t), ..., ug(t)]. From the definition of state and decision variables, there should exist a relation such that the rate of change in a state variable at 26 any time t is a function of the present state of the system, the date, and the decision taken which is implied by the decision vector. This relation can be expressed in the form of a system of differential equa­ tions : * riyx(t) = ^ = w(x(t),u(t),t). (17) Expression (17) constitutes the equations of motion which can trace values of the state variables in a dynamic procress. Decisions influence the system in two different ways: (I) the rate at which net returns are earned at that moment in time, and (2) the rate at which the growing stock of timber is changing, or in more general terms, the future path of the state vector. Let us designate R(X(t),U(t),t) as the marginal, return with respect to time generated from a stand of timber at time t, where the state is described by state vector X(t), and U(t) implies the decision taken at that time. Further we define H(X(T),U(T),T) as net harvest value at the end of rotation cycle T resulting from the decision implied by U(T), and when the state of the system is described by X(T). The objective of timber management can be translated into the following problem: 00 . T Max J(X ,d) = Max £ e ' ^ f / R(X(t) ,U(t).,t)e“rtdt d T,U(t) i=0 0 + H(X(T),U(T),T)e~rT] 27 = Max I/ R(X(t) ,U(t) ,t)e^rtdt T,U(t) O + H(X(T),U(T),T)e ]/(l - e"rt). (18) Both T and the vector function of time-U(t) are variables in the max­ imization subject to the following constraints: X(t) = oi(X(t) ,U(t) ,t) h(t) E I-(XCt)). (19) If the initial: state of the control problem is included in the system and the state vector is confined to a given region, then X(O) = X — — o (ZO) X(t) E S ' where X is the initial state of the standing timber system and S is.-o the set of possible states of the system. , The problem of (18) subject to (19) and (20) is a model in optimal control theory which can be applied to timber management decisions in : the continuous time case 0 t <_<». Theoretically this problem can be solved by either the classical calculus of variations or the maximum principle of Pontryagin et_ al (see Intriligrator, . 1971). An application of the maximum principle to solve the optimal thinning and rotation prob­ lem in timber management has been presented by Naslund [1969], and an 28 excellent interpretation of the principle in an economic context, has been given by Dorfman [1969]. Mathematically, this principle is a generalization of the Lagrange multiplier method to solve optimization problems.over time. Derivation of the necessary conditions for an optimal solution according to this principle boils down to finding the Hamiltonian function, so to speak, which underlies the basic theory of . the principle. 6/ However, for the discrete time situation, any optimal control, prob­ lem can be formulated in a dynamic programming framework, because in that situation, both the maximum principle and dynamic programming give equivalent solutions. Hence for practical purposes in which the deci­ sion is taken periodically like the one in timber management decisions, dynamic programming often has some advantages, especially in getting numerical solutions. The following section will discuss a more concrete and detailed formulation of optimal control to approach the investment problem in timber management. The numerical procedure for the problem will be described in a dynamic programming framework. 6/ The name of the function is after the astronomer-physicist W. R. Hamilton who first used such a function to unify mechanics and optics. See D. J. Wilde and C. S. Brightler,'Foundations of Optimization (Englewood Cliffs, N. J.: Prentice-Hall, Inc.,. 1967) , p. 431. 3.3. Dynamic Programming Solution In order to formulate optimal control in a dynamic programming context, first we. have to specify a stage of the decision process which in timber management may be taken as a year or some longer interval of time. A stage of one year gives the most sensitive decision rule, but little precision is lost in timber management applications by using a . somewhat longer period. Next we have to determine the state variables which are capable of describing the future state of the system completely. Since the deci­ sion process of our problem is growing a stand of timber, age of the stand and a density index can be considered as state variables, because once we know the present age of the stand and the stand density, the future condition (state) of the stand can be predicted by. the present state of growing timber. However, any exogenous factors such as the disease of dwarf mistletoe which influences growth conditions of the stand will change the future state of the forest system also. Hence, the level of disease infestation will constitute another state variable. Then we have to determine the appropriate decision variables that are going to be used to control future states of the standing timber system. Since we know that there is a relationship between thinning action and the future level of mistletoe infestation, and of course the stand density (see Hawksworth and Hinds, 1964; Baranyay and Safranyik, 1970) , the level of thinning constitutes a decision or control variable. Another decision variable which will change the future state of the forest system is the harvest variable which is a dichotomous variable which specifies to cut or let the stand grow. In fact, both of these decision variables can be considered as one decision variable because the harvest variable is an extreme case of the thinning decision. One important aspect of the dynamic programming method.requires estimation of biological relationships which express each state variable in period t+1 as a function of state and control variables in the .. previous period t. These relationships constitute first-order differ­ ence equations in the state variables and are discrete counterparts of the differential equations in (17). In order to be more specific > . let us introduce the following nota­ tion: x^ = age of the stand; % 2 = stand density index; X^ = dwarf mistletoe rating; and u = thinning level, including harvest as an extreme. The set of dynamic relationships will be three in number: Xl,t+1 = f(ut’Xlt’X2t5X3t) = Xlt+1 X2, t+1 g^ V Xlt,X2t’X3t̂ X3,t+1 = h("t'=lt'X2t'X3t) V (21) The equation of x^ happens to be extremely simple, so its specific algebiaic form was. written out in the above set of equations. The relationships in the last two equations of (21) can be estimated by some kind of regression analysis, if adequate data could be found. These three equations in (21) also constitute a discrete version of the equa­ tions of motion which trace the dynamic process over time, starting from some initial state. Now we define a strategy or conditional decision rule (not neces­ sarily optimal) which states how to control, the standing stock system at any given stage for each set of values of the state variables. Mathe­ matically, this rule expresses each decision variable as a function of all state variables, at any given stage. . Following the above notation and applying it to our problem, this rule can be expressed as follows: 31 dt(=lt'*2t'*3t)- (22) If we could estimate the relationship of (21) and set up the rule of (22) , we would be able to trace the value of each state variable through time starting from an initial,state of the process Xq by iterating the equations in (21). Therefore, the strategy expressed in (22) can be evaluated at any stage.and state of the process. Complete enumeration of all possible decision rules given by (22) is riot feasible except in the simplest of problems; dynamic programming pro­ vides a powerful method to systematically determine the Optimal deci­ sion rule.. Since some of the state variables are subjected to random fluctuation, the decision rule of (22) which is a function of random 32 variables is also a random variable. The fact that the decision rule is random, makes it difficult to evaluate a strategy by the above iterative method. In cases where a decision process involves some random state variable, a Markov decision model solved by dynamic programming can overcome the difficulty. In addition, stochastic dynamic programming provides a systematic method to obtain the optimal policy, i.e., the one that maximizes the criterion function. Since the decision to be made involves some random variables, the appropriate model for optimal control in the discrete case.will, be a stochastic decision model solved by dynamic programming to obtain the optimal policy. The following discussion will first present the deter­ ministic version of dynamic programming and then modify it into a stochastic framework by incorporating a Markov chain into the model. A characteristic of the dynamic programming method is that decisions are made sequentially in a multi-stage process. One important aspect of this multi-stage decision process is the effect of a decision at a given stage upon.the state variables in the immediately following stage. A particular decision within the set of admissible alterna­ tives will change the state of the process in the passage from one stage to the next. If the optimal decision to be made at a particular stage of the process depends only on the state of the process at that 33 stage, then the decision process.satisfies the Markovian requirement. Jj. In a deterministic case, the Markovian requirement is equivalent to the difference equation of (21) being first order instead of second or higher order difference equations, and also assuming that all relevant state variables have been included in the set. The implied change of the state of the process from one stage, to the next resulting from a decision is referred to as state transition. Transition.from one state to another can be made deterministically (with certainty) or stochastically— according to a probability distri­ bution of the future state of the process. The latter is called a stochastic decision process. Now let us introduce dynamic programming with a deterministic model. In order to be able to formulate the dynamic decision model, the first order difference equation of (21) must be shifted into dynamic programming notation in terms of stages as follows: xI1I-I ' £(un’xln>x2n-x3n) ' xIn'1 xZ1D-I " s0V xIn-xZn-xSnV xS1H-I " h(un-xln-x2n-x3n) ( 21') Jj A formal definition of the Markovian requirement can be found in Richard E. Bellman, Adaptive Control Process (New Jersey: Princeton University Press, 1961), p. 54. 34 where n refers to the number of stages remaining in the planning horizon The relation between stages and chronological time can be illustrated by the following diagram: end of the planning _____________________________ horizon n n-1 0 Figure I. The Relation Between Stages and Chronological Time. Bellman's principle of optimality states "an optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision" (Bellman,1957). 8/ By application of this principle, the entire problem can be summarized by the following recurrence relation: Fn (Xl9X2JX3) = Max [Rn(U1XlJX2 JX3) + g F ^ ^ (x^l) ,g ( ),h( )}] (23) where: In (X1 sX 2 5 X3) is present value of total net returns from soil site in the next n stages under an'Optimal policy, when the initial state is that implied by the variables X1, X2, X3. time — 0 stage — N 8/ A direct mathematical derivation of the recurrence relation.as an alternative to the application of the principle of optimality can be found in R. E. Bellman and. S. E. Dreyfus, Applied Dynamic Programming (New Jersey: Princeton University Press, 1962), p. 15. 35 R^CujX^jX^jX^) is the immediate net return resulting from decision u in stage n. F { (x-1) ,'g( ),h( ) } is the future net returns from the site : n-l I in the next, h-1 stages to go. The functions g( ) and h( ) are evaluated with their arguments set equal to values of the variables in stage n, i.e., the arguments are as given in (2lf). 3 = I/(1+r) is a discount factor with interest rate r. The above procedure describes how to formulate the multi-stage decision model to control the production system in growing timber in a deterministic framework. If we want to incorporate stochastic elements into the model which would more closely represent real world phenomena, the dynamic equation (21') and the decision model (23) need slight modifications as follows: x2,„-l' s(V xln’X2„'x3n-en) .> x3,n-l * K(un’xln-x2=’x3n-.<) where e and e' are random variables.. discounted net returns function: (24) Also (23) becomes an expected Fn(XljX 2 lX3) = Max E[Rn(U1Xl)X3 lX3) + SF ^ 1C (X1-I) ,g( ) 1h( )}] u Max [Rn (U1 XllX 3 lX3) + 3EFn_1{(X1-I),g( ),h( )}] U (25) 36 where E is the mathematical expectation operator. The function R^(u,x^,X2 ,Xg) is now defined as an expected value measure of immediate net returns and F^(x^,X2 ,Xg) is an expected value measure of discounted net returns. Since the decision model (25) is a dynamic stochastic decision model, it can be approximated as a Markov decision model with a finite number of states. The stochastic difference equations in (24) deter­ mine the random transition from state to state in the Markov process. The Markov process can be expressed in matrix form, where the matrix is composed of transition probabilities of changing from any stage i to another state j when the stage of the process moves from the n-th stage to the (n-l)-th stage. The state of a forest system can be repre­ sented by some meaningful set of state variables, such as the ones that were defined in (21) for the problem. Let us define the probability of going from state i to state, j at the n-th stage under decision alternative k as p^.. Then let us con- sider a decision rule which specifies k for any given state i, and let that rule be denoted as k(i). This decision rule is a discrete variable approximation to the functional relationship (22). If there are M states possible, the M x M matrix with element is the transition1J matrix of a Markov chain. . . 37 Because a Markov chain decision process can be considered as a discrete approximation to the stochastic difference equations in (24), Howard [i960] was able to formulate the Markov decision model in a dynamic programming framework. This approximation to the decision model of (25) is expressed as follows: M (n) = Max [R^(n) + B £ P^jFj(n-1)] , n = I, 2, .... N (26) . 3=1 Fj(O) = 8j where: F\(n) is the maximum expected present value of total net returns from soil site when the process is initially in state i and n stages remain in the planning horizon. (n) is expected immediate net returns,given the i-th state, k-th decision alternative, and Chq^nHfhyStage of the process. The {0 } are terminal values of the decision process which can be set equal to zero if the planning horizon is sufficiently long. There are two solution methods for obtaining the optimal policy, i.e., the value iterative and policy iterative methods (Howard,1960). However, the value iterative method appears to be the most feasible and efficient for problems with a large number of states. The value iterative method solves the optimization problem by starting at the end of the planning horizon with only one stage remaining; then works back to the present by utilizing the principle of optimality at. each step. Applying this method, the general formula of (26) then becomes: F (I) = Max- [Rj(I) + 8 I P^ F (O) = Max [R^(I) + g I pJ.eJj. k . J=Iljj k 1 j=l 13 J Or if 6. =0, j = I, 2.....M, we haveJ kF .(I) = Max R.(I). k 1 By using the results of F^(I) we can calculate the following: k kF (2) = Max-[Rf(2) + g } pf.F (I)] . k 1 j=l ij J F (3) = Max [R,(3) + 8 I pf.F (2)] . (27) • k j=l 13 J . . . . . V MF (n+1) = Max [R?(n+1) + g I pf F ,(n) ]. k j=l J J The iterative procedure of (27) could be used as a computational algorithm to solve the optimal control problem of a production system in growing timber. A FORTRAN program and guide for its application using the above algorithm is now available and was written by Burt et al. (undated). k ■If R ( ) and consequently R^(n) are independent of the stage n as n"*00, then it can be shown that the recurrence relation in (26) con­ verges to a constant decision rule for large n. Therefore the solu­ tion of the problem for n large enough that convergence has taken place provides an optimal policy for all values of n. A criterion to. 38 39 verify convergence of the optimal policy- is given in Burt and Allison [1963]. Hence the computational burden and time for obtaining an optimal policy could possibly be reduced. By using the above algorithm at each stage of the decision process an optimal decision rule could be obtained with respect to the level of thinning control and final harvest of a timber stand, which underlies the basic principle of forest investment decisions, for every possible state of the timber production system. Derivation and application of an optimal control policy to a disease infected stand of timber in a particular forest region requires detailed specification of forest conditions in that region. If the forest region consists of several management blocks which in turn consist of several forest stands, then a different optimal decision rule must be derived for each of the various stand conditions which have.different characteristics. In order to be able to accommodate changes in parameter values and changes in price and cost structure for making economic projections, a sensi­ tivity analysis in dynamic programming can be applied without much difficulty. ‘ The general model of dynamic programming expressed in (26) con­ sists of two major ingredients for obtaining optimal policies. Specifically it requires information related to expected immediate returns {R^(n)} and the transition probabilities {p^ }. The estimate of expected immediate net returns can be obtained via estimation of 40 conditional expected yield of timber. This expected yield should adequately describe total cubic feet of yields by class, of timber so that the conversion can be made, to market value as a function of all related decision and state variables involved in the model. In economic jargon this relationship constitutes a multi-product joint production function where generalized input variables are represented by decision and state variables, arid output is comprised of the various classes of timber products which have market value. The second kind of information heeded for the model pertains to . stochastic behavior of the standing timber system, which can be repre­ sented by two interacting random variables, dwarf mistletoe rating and'., stand density index. A third interacting variable in the system is age of the stand,. but this variable is non-stochastic. Hence, estimated •transition probabilities should be based on the joint probability dis­ tribution of two state variables with age entering the joint density function as a parameter. Let us first discuss estimation of the conditional yield relation­ ship to estimate expected immediate returns. In the estimation we assume wood yield to be similar to that.in the past. Yield varies because of many factors; among them are disease, insects, and climatic factors which behave.stochastically. Another group.of factors which have more "systematic" influence are site characteristics and age of the stand at harvest. The relationship of yield and the above factors 41 is assumed to be independent of the stage of the process; hence this relationship will constitute a function of state and decision variables at any given stage if all factors enter the function as either state or decision variables. Let us simplify the discussion by assuming that a single variable (total marketable volume if you please) can be used to measure timber production, and thus we are dealing with only one equation which describes yield in terms of various factors mentioned above. Factors which remain unchanged over the growth cycle of timber, site charac­ teristics in particular, do not need to be entered in the timber yield equation because their values can be treated as merely parameters in the dynamic optimization problem; Certain stochastic variables which one might want to delete for the sake of simplicity cause no problem, if they are not stochastically dependent with any of the included factors of the yield equation, or at least the dependency should be quite weak. For example, a study such as this one which focuses on the influence of dwarf mistletoe upon timber growth would be simplified a great deal if we could delete climatic and insect variables from the timber yield equation, and essentially pool the random influences of these variables in the error term of the regression equation. The regression equation for this simplified view of the timber yield relationship would take the following form: J, qt. =. *(ut,Xlt,X2t’X3t̂ + et , t = 0, 1, 2, ..., T E(e. ) = 0 E(e )̂ = a2 , E(et,et+T) = 0 T ^ (28) where: is the quantity of merchantable volume of timber in cubic feet for period t. ■ . is thinning level, including harvest. x is age of the stand for period t. it: X^t is stand density index for period t. X^t is dwarf mistletoe rating for period, t. et is a random disturbance which represents climatic factors and other factors which individually are relatively minor. The above physical relationship would be used for estimating expected net returns at each period resulting from the decision implied by u .■ t Expected immediate net returns are defined as the difference between value of gross returns, and variable costs which can be . expressed as follows: ■ where: R(ut,X^fc,Xgfc,Xgt) is expected immediate net returns in period t. 43 G(ufc,x^^,X2 t, i s expected immediate gross returns in period t. ' C(u^,x^^,X2 ^,Xg^) is expected variable cost of thinning for period t. Expected timber yield can be obtained by estimating the conditional expectation of yield from equation (28) for any decision implied by u at. period t. This conditional expected yield can be expressed as., follows: t E(qt|ut'xl f x2 f x3t) " *("t'xlt'x2t'x3f> <30) where E is.the expectation operator. Assuming perfect competition in the output market, expected immediate net returns can be calculated as follows: R(uf xlt’x2 f x3t) ' P»E t, also reflects some level of disease, depending on y^, ^.. yj-r In a sense the objective function of the control process in (35) can be considered"as a generalization of the Faustmann criterion. Since the maximization of (35) has to be done with respect to two sets of arguments, T and {y.}, the maximization operation can be carried outJ in two phases. First, we maximize the criterion function (35) with, respect to T which gives the following: Max TT(T,yt, y ^ , ... , Y ^ X t) = • • •■» T Then, in the second phase we maximize with respect to y^, > • yT to obtain the final solution, ", Hence, in short we have: (36) f(Xfc) = Max Y(yt,yt+1, ..., yT ,Xt) yi i=t to T Max [Max ^(T.y , y , ..., y ,X )] Y1 T i=t to T Max ir(T,y , y - , ..., yT ,X ) T,y i=t to T (37) The above maximization.operation.is essentially what we use in the *computer program for this study. First we find K , then .'V'(yt >. » i 62 yT,X ) for a limited set of thinning policies, and finally we maximize tover thinning policies to get f(X ). Each of the last two maximiza­ tions must be made with initial disease condition, stand age, etc.. taken as fixed. CHAPTER V APPLICATION TO LODGEPOLE PINE 5.1. Data Description and Assumptions The present value functions in (34) and (35) for determining optimal thinning and rotation policies consist of two important components to be estimated; (I) net returns associated with timber management decisions at each period, and.(2) the discount rate associated with opportunity cost of capital invested in the project. In order to obtain numerical solutions to the investment problem, we have to■quantify'and evaluate these variables. To derive estimates of net returns, the physical yields, of wood, generated by the computer program in a yield table for each age period of the timber, are multiplied by an appropriate price of wood products which then gives the stream of gross returns. The.stream of gross returns adjusted by variable costs related to a particular timber manage­ ment decision then gives the stream of net returns.. In order to clarify the calculations involved, let us use the following notation: tR (y ,X ) = expected net returns associated with managing a timber t stand where the decision is. implied by yt and the state of the system at period t is described by Xt. t 'G (yt,X ) = expected gross returns associated with managing a timber t stand where the decision is implied by yt and the state of the system at period t is described by Xt. 64 £ct(yt>X ) = expected variable costs associated with managing a timber stand where the decision is implied by yt and the state of the system at period t is described by Xt- EC^tlyt’— = T is expected physical wood yield whose estimate can be generated by a computer program in a yield table associated with the management decision implied by y when the state of the sys­ tem at period t is described by Xt. Assuming a perfectly competitive wood product market, the expected net returns for any period t can be calculated by the following method: Rt(y^ x t) = PwECqt Iyt9Xtjt) - CtCyfcjXt) = P j Cyfc, A t ) - C tCytA ) . (38) Assume further that the price of wood products, Pw , is determined by interaction of supply and demand forces in the wood product market out­ side the timber production system. Then an estimate of wood product price can be based on time series of wood price data for related wood species which reflects the equilibrium prices that have been* prevailing in the market as the result of the interacting forces. The most closely related wood product price for timber production decisions is stumpage price. The stumpage price in a perfectly compe­ titive market at any specific location will equal the price of sawlogs at the mill gate minus the cost of availability (including harvesting and transfer costs). A general model of stumpage price formation has been described by Gregory [1972] and a stumpage price response to 65 changes in volume of timber sold has been studied by Hamilton [1970]. The former author explained the uniqueness of the s.tumpage price, forma­ tion where the location factor has an important role in determining the equilibrium price. The second author concluded that in the short-run, stumpage prices have fluctuated considerably. Moreover, an examination of the region's stumpage market indicates that shifts in demand for wood products, rather than quantity of stumpage offered for sale, are the primary determinants of this short-term price fluctuation. One source of wood price data used in this study originated from stumpage price data which was compiled by Holt [1973] at the Pacific Northwest forest research station. The published stumpage price data are categorized into several subgroups according to wood species and the regions where they are formed. One of the wood species, in the publica­ tion is pertinent to this study, namely lodgepole pine (Pihus concorte) and the regional coverage of the price report includes the Rocky Mountain region. From this price data we can determine the current stumpage price, the minimum price which has ever prevailed, and the highest price projected for the future. These different levels of estimated price are used in making sensitivity analyses in the simulation study. Another component of the stream of net returns in (38) is cost variables. The cost of timber production has time and spatial dimen­ sions, where the costs may vary from period, to period and from one stand to another. Hence, the forest manager should be aware of possible cost 66 variations associated with each timber stand during each period of stand management. Therefore, an attempt must be made to obtain representative costs which can accommodate these cost differences. Whenever possible, cost should be related to physical input, state of the standing timber system, and the decision taken at each timber management period. In relation to the cost structure, we know that production economics distinguishes two kinds of costs, i.e., fixed and variable costs, as components of total cost. However, the costs that directly relate to production decisions are variable costs; that is, the costs that vary with the state of the timber stand system, investment decision, and age period. A source of cost information that virtually meets the above requirement has been developed by Wikstrom and Alley [1967]. These authors derived predictive cost equations, originally, for the purpose of cost control in timber management decisions, by using multiple regression techniques. They constructed cost equations related to each timber management activity that incurred costs such as site preparation . costs, including slash piling, burning, and land scarification costs, planting costs— in case natural regeneration failed to,establish a new stand— and thinning costs. By applying proper adjustments to local conditions, labor wage and equipment costs, this study employes these cost equations in combination-with the computer simulation program which, at its final stage, is capable of generating a performance measure based on the economic criteria as discussed in the last chapter. 67 The most difficult parameter to be estimated in measuring present value of net returns in an investment model is the discount rate which reflects the opportunity cost of capital, either as viewed by private enterprise, or social opportunity cost as viewed by public decision making agencies. Although we are dealing with investment decisions related to timber land which belongs to the public domain, private investment criteria may help throw some light on public investment decisions related to forest land resources (see Lutzes, 1951; Masse, 1962; and Hirschleifer, 1970). In. discussing difficulties of assessing social opportunity cost of capital, Feldstein [1964] concluded that the only way to assess the social opportunity, cost is to discount at social time preference. But the search for a perfect formula to specify a social rate of time preference is futile. It is often the case that the rate is set solely as a matter of. social policy, taking into account certain ethical principles as well as economic opportunities. Hence, various lines of approach attempting to find a rational choice for the discount rate all appear to be highly subjective. Therefore for purposes of.the analysis of this study, the discount rate used in the investment decision.is specified at several values rather than a single value. Another advantage of selecting a set of different values of discount rates is that it enables us to evaluate their impact on the optimal solution of the investment problem. 6 8 5.2. Model Experimentation The purpose of model experimentation with a simulation program is to evaluate the impact of changes in parameters and initial states on the optimal decision rule for the problem. This analysis sometimes is called sensitivity or parameterization analysis. The computer simula­ tion model provides a basis for making such analysis and to evaluate those changes on the performance measure of the syste