The Nature and Uses of the

Forest Sci., Vol. 30, No. 3, 1984, pp. 761-773 Copyright 1984, by the Society of American Foresters The Nature and Uses of the Timber Production Function: Eucalyptus grandis in Brazil J. C. NAUTIYAL LAERCIO COUTO ABSTRACT. The nature of a timber production function is discussed and one such function is derived for Eucalyptus grandis W. Hill ex Maiden grown in Brazil. It estimates growth rather than yield, and age appears as a surrogate factor of production in addition to water, nutrients, solar radiation, and growing stock. A polynomial was judged the best of several alternative growth functions for the data used. Potential uses of a production function in intensive forest management are discussed. FOREST SCI. 30:761-773. A PRODUCTION FUNCTION is a description of the production process and has been considered (Duerr 1960, 1979) as the first essential step in production planning. Economists have paid much attention to the specification and estimation of production functions in manufacturing industries (Walters 1963), while biological production processe such as timber growing have received little or no attention. Agricultural economists have used econometric techniques to estimate crop responses to irrigation and fertilization (Hexem and Heady 1978), and to predict livestock production for feeding treatments (Epplin and others 1980). In forestry the closest one comes to the concept of a production function is in growth and yield studies (for example, see MacKinney and others 1937, Spurr 1952, Buckman 1962, Clutter 1963, Moser 1967, and Pienaar and Turnbull 1973). The yield function can be taken as the production function with only one variable input (time), and the growth function can be taken as the marginal product of time. The other inputs in these studies are usually lumped together in the parameter "site." While growth and yield studies are useful, their capability to help the forester in intensive forest management is limited because they do not explicitly take into account the basic factors responsible for biological growth, namely, water, nutrients, and solar radiation, some of which can be manipulated by the manager. This paper reports an attempt to derive a production function for timber based on independent variables which are under the control of foresters. Data for Eu- 'calyptus grandis W. Hill ex Maiden from Brazil are used in an example to demonstrate its uses in intensive forest management. METHODOLOGY Differences Between Timber and Mechanistic Production.--Walters (1963) distinguished two categories of production functions: (i) those derived from engineering The authors are with the Faculty of Forestry, University of Toronto, Toronto, Ontario M5S IA1, Canada, and the Universidade Federal de Vicosa, Brazil. They thank Professor D. K. Foot of the Department of Economics, University of Toronto, for help in the development of the model presented, and Mr. J. K. Rawat, graduate student in Forestry, for assistance in preparation of the manuscript. Financial support received from the Natural Science and Engineering Research Council of Canada through Grant A6205 is also gratefully acknowledged. Manuscript received 12 January 1983. VOLUME 30, NUMBER 3, 1984 / 761

and biological data which do not explicitly consider the entrepreneurial ability of the producer and do not include the nontechnical process of selling the product, and (ii) those which cover all the activities of the firm. According to Walters, those in the first category should be called process functions to distinguish them from the more comprehensive production functions of the firms. This paper deals with the first category of production functions, the biological timber production process of a hectare of forest land. As mentioned by Gregory (1972) and Teeguarden (1979), there are a number of distinctive peculiarities of primary timber production. Perhaps the most important difference between a timber and a mechanistic production process is the role played by the variable time. In a defined mechanistic process, there will be no nonrandom difference in output level from one production period to another if the same levels of inputs are maintained. In timber, or any biological production process, this is not true. Hence, time should be treated as a surrogate factor of production. This will explain the variations which occur in the output level from one production period to another, while the input levels are maintained constant. The second significant difference is the dual nature of the product. It is impossible to remove the new growth of a stand without liquidating a part of the factory or growing stock represented by the trees harvested. Therefore, in the short run it is possible for the manager to obtain output greater than current growth simply by cutting more deeply into the growing stock or accumulated inventory. In terms of the timber production function the implication of this peculiarity is that the "output" in the function should be the growth in any duration of the production process rather than the yield. It also follows that some measure of the existing growing 'stock will have to be included as an independent variable for explaining growth as a dependent variable. Thirdly, the degrees of control which can be exerted on mechanistic and biological processes are different. In an industrial process it is generally possible to keep the levels of output and inputs within specific ranges to operate within the region corresponding to the rational stage of production. In a biological process, such as timber growing, it is difficult to control the environment (and thus, the levels of some inputs). Negative marginal returns are therefore not uncommon in timber production. Still another difference is that in the timber production process it is often not possible to change the levels of the inputs by incurring relatively small marginal costs (e.g., in water and solar radiation). Sometimes adjustment is possible only in one direction (e.g., in basal area and nutrients). A Timber Production Model.--On the basis of the theories of investment, and capital accumulation, the production process of timber can be expressed in the following manner: E -- E- + XXE (1) where Vt is the capital stock at the end of the time period (t); Vt_: represents the capital stock at the end of the time period (t-1); and AVt is the net growth--a flow variable over the period (t). This flow variable is the difference between the gross growth Gt and the mortality Dt and hence gt gt-i ' - Gt -Dt. (2) Mortality, defined as the initial volume of trees dying between two consecutive measurements (Beers 1962), may be considered as a constant proportion of the capital stock at the beginning of each production period (see for example Jorgenson and Stephenson 1967). Thus, 762 / FOREST SCIENCE

Dt--- 3 t_ (3) where 3 is the mortality parameter of the equation. However, in the context of mechanistic processes, this approach was challenged by Feldstein and Foot (1971) who provided evidence that there could be substantial variation from year to year in the ratio of replacement investment to the capital stock. Thus, replacement investment could vary around some average nonzero level according to short-run economic forces. If it is assumed that replacement investment is equal to depreciation then their approach would imply the following modification of (3) D e = 'YtVt_l. In timber production, neither of the above two approaches would be satisfactory for trees below the lower boundary of the zone of imminent competition mortality as defined by Drew and Fiewelling (1979). Here, mortality is independent of stand density, erratic, and very difficult to predict. Previous research (Schonau 1970) and available biological data for Eucalyptus grandis indicate that mortality is not proportional to the timber stock in young plantations (up to 6.5 years) at spacings from 3 to 12 square meters per tree. In Brazil, mortality in eucalypt stands ranges from 5 to 20 percent (Ferreira and others 1977) and is likely to be similar to that observed by Schonau (1970). Consequently, it would be best included in our model of juvenile plantations by using average values computed from remeasured sample plots for each locality. Thus, Dt = fit (4) where/ t is determined outside the system, as an extraneous estimate (see Theil 1978). The general growth function, involving all production periods of a rotation, can now be postulated as the age-site-density type such as presented by Curtis (1967): G -- d. G(Age, Site, Density) (5) where d is the duration of growth which can, for convenience, be made equal to one year. Such models are well known in forest mensuration. Age and duration of growth (d) compose the analytical boundary separating each production period from the entire process (Georgescu-Rogen 1971). Site depends on water, nutrients, and solar radiation available to the trees on location and density can be represented by basal area per hectare. The growth in one year on each hectare of forest can, then, be written as G = G(B, M, N, R, A), (6) where B is the basal area at the beginning of the production period; M, N, and R are water, nutrient, and solar radiation available to the crop over the growth period; and A is the mean age of the crop during the period. THE DATA The data used in this study are from a research program involving 36 species of eucalypts established in different regions of Brazil (Golfari 1976) by the "Instituto Brasileiro de Desenvolvimento Florestal" (IBDF). Details of plot establishment, data collection and compilation can be found in Golfaft (1976), and in Moura and others (1980). Site preparation consisted of bulldozing, ploughing, and harrowing the soil. Each plot was planted with 25 seedlings (4-5 months old and 25 VOLUME 30, NUMBER 3, 1984 / 763

cm high) at a 3 x 2 m initial spacing. Fertilization at that time consisted of 70 g per planting hole of a mixture of phosphates described in Golfaft (1976). Aldrin was also applied to protect the roots of the seedlings against damage by termites. Measurements in each plot were taken annually from age 1.5 to 6.5 years on a per tree basis, and consisted of diameter at breast height (outside bark), total height, survival, and a series of other physiological variables. Three hundred of these plots, 0.15 ha each, spread over Bioclimatic Regions 3, 6, and 7 of Minas Gerais (see Golfaft 1975), which cover major portions of the state, provided the required data from regular remeasurements. Two hundred and twenty-five plots were randomly selected to estimate the model parameters and 75 were retained for evaluating its accuracy. Definition and Measurement of Output and Inputs.--Output and inputs in the timber production function could be defined in many ways. However, because of the limitations of the data, and based on the relevant literature, the following were chosen. Details can be seen in Couto (1982). (i) The timber output (G) The total volume of tree stems was chosen as the most meaningful representation of output in a production period. A volume table was prepared from data in the region of study, the best representation being V - 0.000l 18 aø-197964d 2.526110, (7) where v is the total outside bark volume of a tree in cubic meters, a is the age in years and d is the diameter at breast height in centimeters. The per hectare volumes were computed from measurements of diameter and count of number of trees of known age in each plot. (ii) The growing stock input (B) Basal area in square meters per hectare at the beginning of the production period was taken to represent the growing stock input in each production period as it is the most commonly used representation for stand density (Curtis 1967). TABLE 1. Univariate statistics for the data set used for parameter estimation. Vari- Standard Extremes able Mean deviation CV Lowest Highest Range G 31.9 10.8 34.0 6.6 49.8 43.2 B 9.8 6.2 63.4 0.3 23.4 23.1 ETA 908.0 60.0 6.6 847.0 1,031.0 184.0 RET 0.95 0.04 4.0 0.83 0.99 0.16 P 6.8 3.3 48.5 3.1 12.3 9.2 K 60.4 33.4 55.3 23.3 126.8 103.5 h 2,455.0 262.0 10.6 2,060.0 2,927.0 867.0 Ro 15,486,353.0 924,748.0 6.0 14,032,711.0 17,015,441.0 2,982,730.0 Rs 2,937,792.0 12,951.0 0.4 2,920,705.0 2,954,935.0 34,230.0 G = gross growth in m s per hectare per year. B = basal area in m 2 per hectare. ETA actual evapotranspiration in mm of water per year. RET relative evapotranspiration. P phosphorus in kg per hectare. K potassium in kg per hectare. h hours of sunshine per year. Ro extraterrestrial solar radiation in Kcal m -2 per year. Rs terrestrial solar radiation in Kcal m -2 per year. 764 / FOREST SCIENCE

TABLE 2. the models. Univariate statistics for the data set used for testing the accuracy of Vail- Standard Extremes able Mean deviation CV Lowest Highest Range G 30.3 9.6 31.5 7.6 49.8 42.2 B 10.0 6.3 63.3 0.3 22.7 22.4 ETA 914.0 61.0 6.7 847.0 1,031.0 184.0 RET 0.97 0.04 3.9 0.83 0.99 0.16 P 6.4 3.2 50.1 3.1 12.3 9.2 K 56.8 32.3 56.9 23.3 126.8 103.5 h 2,470.0 264.0 10.7 2,060.0 2,927.0 867.0 Ro 15,528,463.0 929,324.0 6.0 14,032,711.0 17,015,441.0 2,982,730.0 Rs 2,938,885.0 12,906.0 0.4 2,920,705.0 2,954,935.0 34,230.0 G = gross growth in m 3 per hectare per year. B = basal area in m e per hectare. ETA = actual evapotranspiration in mm of water per year. RET = relative evapotranspiration. P = phosphorus in kg per hectare. K = potassium in kg per hectare. h = hours of sunshine per year. Ro = extraterrestrial solar radiation in Kcal m-: per year. Rs = terrestrial solar radiation in Kcal m -2 per year. (iii) The water input (3//) The literature on this subject suggests that the following two alternative representations could be used: (a) actual evapotranspiration ETA (Thornthwaite and Mather 1957), (b) relative evapotranspiration RET (Minhas and others 1974). Both alternatives were tried. (iv) The nutrient input (N) Available research (Van Goor and Nascimento 1970, Mello and others 1970) indicates that the most important nutrient for eucalypts in Brazil is phosphorus (P) followed by potassium (K). Therefore, the levels of P and K in the soil determined by using a double extractant (H2SO 4 0.025 N and HC1 0.05 N as reported in Anonymous 1978) were used as alternative representations of the nutrient input. (v) The solar radiation input (R) Three alternative representations were tried: (a) hours of sunshine (h) during the production period, (b) extraterrestrial solar radiation (Ro) in Kcal m -2 during the production period (Villa Nova and Salati 1977), and (c) amount of solar radiation (Rs) in Kcal m -2 during the production period (Angstrom 1924). Summaries of the univariate statistics of the data sets used for estimation of parameters and for testing the accuracy of the models are given in Tables 1 and 2. ANALYSIS AND RESULTS Exploratory Data Analysis.- The main objectives of this initial exploratory analysis were: (i) to select the best variables to represen the inputs in the regression analysis; (ii) to get some information about multicollinearity between the explanatory inputs; and (iii) to verify the effect of possible transient factors (due to location of plots in three different regions) on the growth of the stands. VOLUME 30, NUMBER 3, 1984 / 765

Where an input could be represented by more than one variable, as in the case of M, N, and R, the correlation coefficients with the output were examined. The following values were obtained ETA* 0.59 RET 0.05 P* 0.64 K 0.40 h* 0.62 Ro 0.58 Rs 0.40. Therefore, the variable exhibiting the highest correlation (marked with an asterisk) was chosen for preliminary regression runs. That these variables were the appropriate ones was confirmed again after model selection. Multicollinearitf between inputs was checked by examination of the coefficients of simple correlation between pairs of inputs. Foot and North (1977) mentioned that if the absolute value of the coefficient is greater than 0.90 then harmful multicollinearity may be expected. For our data the highest coefficient was 0.79 (between P and h) and so it was assumed that perhaps no major problems exist in estimation of parameters due to multicollinearity. The effect of transient factors was tested by using dummy variables. A joint F-statistic test was used and it was concluded that there was no evidence of significant effect of transient factors on growth. The dummy variables were eliminated at this stage and the F-test carried out again after selection of the best model to confirm that transient factors were not significant. Selection of the Model.-- Numerous algebraic forms have been used to represent Multicollinearity is not easy to detect and standard error of the estimated parameter can also be used as a guide in this matter. Large standard error and low t-statistic may signal existence of multicollinearity though their existence alone is not enough to conclude its presence. An alternative and better method for detecting multicollinearity is by using the Farrar-Glauber equations (Judge and others 1980). They consider the more general relationship between each predictor and all the remaining independent variables. However, in this study we did not consider it necessary to use this approach. Still another approach that could have been used is principal component approach which uses eigen values. As the component variables would be very hard to interpret we decided that it would not be worthwhile to pursue this approach either. TABLE 3. Statistics for the polynomial model Variables Parameter Estimate Standard error t-statistic Intercept /50-9.216005 3.122038-2.95 A 2 /5 1.369210 0.205835 6.65 B2 /52-0.218207 0.023956-9.11 3/2 /53-0.370767 0.054807-6.76 M.N /sn 0.009760 0.001819 5.36 N'R /55 0.001208 0.000467 2.58 B'M'R /56 0.000002 0.0000001 15.31 A'B'M'R /57-0.0000002 0.00000004-5.63 A'B'N'R /58 0.000058 0.000006 9.54 A'M'N'R /59-0.000001 0.0000002-9.73 R 2 = 0.79 SEE = 5.022125 I = 5.022125 766 / FOREST SCIENCE

biological production functions. 2 A review of the history of production functions in agricultural science and analysis of the important algebraic forms are given in Heady and Dillon (1961), Heady (1965), and Dillon (1977). A study of these functions and those used in growth and yield models in forestry suggested that seven alternative specifications were suitable for representing the annual growth of a forest crop. Actual equations estimated by regression analysis are given below: (i) (ii) (iii) (iv) (v) (vi) the polynomial model: G = -9.2 + 1.4A 2-0.2B 2-0.4/V 2 q- 0.01M. N + 0.001N.R + O.000002B.M.R - O.0000002A.B.M.R + O.00006A.B.N.R - O.000001A.M.N.R; (8) the reciprocal model: G = 45.5 + 39.7/A + 4.7/B 2 + 51,288.8/(A.R) - 46,932.4/(M.N) + 396,367.4/(A. B. R) - 377,369,008.3/(B. M. R); (9) the square-root model: G = 101.3-99.3X/' - 22.1A + 22.0X/ - 3.8B - 8.5%/ + O.08V'(B.N.R); (10) the semilogarithmic model: In G = 3.9-0.0lB 2-0.05A.B - 0.14A.N - 0.0000006M. R + 0.00005N. R + O.004A.B.N- O.00002A.B.R + O.00003A.N.R + O.00000009B.M.R; (11) the logarithmic-reciprocal model: In G = 3.6 + 0.2/B 2 + 1,658.2/(A.R) + 53.6/(A.B.N) - 15,359.1/(B.M.N) - 84,037,484.4/(A.B.M.N.R); (12) the transcendental model: In G = 99.1 + 0.2A - 1.1 In A - 0.02B (vii) the double-logarithmic model: + 0.6 In B + 0.01M - 10.3 In M- 0.1N - 0.1N + 0.6 In N + 0.002R - 5.5 In R; (13) lng=-0.2-0.71na + 0.51rib + 0.3 In M + 0.04 In N + 0.2 In R. (14) To compare the models and select the most suitable, the following criteria were used: (i) consistency of the model with a priori biological information; (ii) statistical significance of the individual parameters; (iii) goodness of fit; and (iv) accuracy and consistency of the models for practical applications. The statistical significance of the coefficients associated with each independent variable was determined by means of a two-tail t-test at 95 percent level of significance. On the basis of the first criterion for selection, the square-root model (10) was rejected at once. It did not show water as an important input either independently 2 The algebraic forms of the comprehensive production functions of firms and industries are not being considered here. For them, the reader may see Cobb and Douglas (1928), Arrow and others (1961), Uzawa (1962), McFadden (1963), Mukerji (1963), $ato (1967) and Revankar (1971) which give the most important algebraic forms used in production studies for manufacturing industries. For a review of the algebraic forms of the same category of production studies in the agriculture industry Heady and Dillon (1961) may be referred to. VOLUME 30, NUMBER 3, 1984 / 767

or interacting with any other input(s). In the transcendental model (13) some of the parameters were not significant and it could not include all the inputs in its final form on the basis of the second and third criteria. It was therefore rejected. The double-logarithmic model (14) was also rejected on the same grounds. Of the remaining models the polynomial was the best fitting as Furnival's (1961) index for it was the lowest (I -- 5.022125). The semilogarithmic model was the poorest (I -- 5.74282). The R 2 was highest for the semilogarithmic and the logarithmic-reciprocal models (0.80), followed closely by the polynomial (0.79), and was lowest for the reciprocal model (0.73). Therefore, the polynomial, for which statistics are shown in Table 3, was chosen as the best model for expressing growth as a function of the inputs considered here. The accuracies of models (8), (9), (11), and (12) were tested using a Chi-square, as proposed by Freese (1960) and modified by Rennie and Wiant (1978), which was applied to the 75 observations set aside for the purpose. The results confirmed that the polynomial (8) was better than any other specification considered. To confirm the initial findings that ETA, P, and h were the best representations of the water, nutrient, and solar radiation inputs, the selected polynomial was tested with the alternative variables. It was found that the initially selected representations were better than others. Similarly, the joint F-statistic test was again carried out by using dummy variables and it was confirmed that transient factors were not significant. By using the selected polynomial growth function (8), the corresponding timber yield function can be written as where v,-- Vo + oj- z>j o5) j=l Gj-- -9.2 + 1.40' + 1) 2-0.2Bfi - 0.4Nil + 0.01M.N + 0.001,.Rj + 0.000002Bj. Mr& - 0.00000020 + + 0.00006(j + 1).B7.N.R - O.000001(j + 1).M/N.R (16) is the growth function and the suffix j refers to the value of the variable in the period j, i.e., in the year that elapsed between age (j - 1/2) and (j + 1/2) years. In the use of this yield function, however, one must recognize that basal area obtained from the growth of the first period, if used for estimating the yield at the end of second and subsequent years, has a potential for recursire bias. Therefore, the function is most useful for predicting one year at a time. The Input-Output and Input-Input Relationships.--A number of input-output and input-input relationships (Griliches and Ringstad 1971) can now be determined for the various growth periods. They are defined as shown below: (i) (ii) (iii) (iv) average products of the inputs apb = G/B, apm = G/M,... marginal products of the inputs mpb = OG/OB, mpm-- OG/OM,... elasticities of production eb = mpb/apb, em-- mpm/apm,... elasticity of scale s=eb+ em+en+er, 768 / FOREST SCIENCE

TABLE 4. Input-output and input-input relationships for all production periods computed with average levels of inputs. Rela- Production period (average age) tionship 1 (2) 2 (3) 3 (4) 4 (5) 5 (6) apb 6.266 5.989 5.998 6.293 6.873 apm 0.068 0.065 0.065 0.068 0.074 apn 9.030 8.631 8.644 9.069 9.905 apr 0.025 0.024 0.024 0.025 0.027 rapb 5.170 5.726 6.282 6.837 7.393 rapm 0.073 0.052 0.030 0.009-0.013 rapn 4.524 3.738 2.952 2.167 1.381 rapr 0.017 0.013 0.009 0.005 0.001 eb 0.825 0.956 1.047 1.086 1.076 em 1.081 0.798 0.465 0.126-0.174 en 0.501 0.433 0.341 0.238 0.139 er 0.667 0.532 0.366 0.192 0.031 s 3.074 2.720 2.220 1.644 1.072 rtsbm - 0.014-0.009-0.005-0.001 0.002 rtsbn -0.875-0.653-0.470-0.317-0.187 rtsbr - 0.003-0.002-0.001-0.001-0.0001 rtsmn -61.880-72.445-98.110-252.302 106.945 rtsmr -0.228-0.247-0.292-0.561 0.067 rtsnr -0.004-0.003-0.003-0.002-0.0006 esbm - 1.310-0.835-0.444-0.116 0.162 esbn -0.607-0.453-0.326-0.220-0.130 esbr -0.809-0.557-0.350-0.176-0.029 esmn -0.463-0.542-0.735-1.889 0.801 esmr -0.617-0.667-0.788-1.517 0.180 esnr - 1.332-1.229-1.073-0.803-0.225 (v) (vi) the rates of technical substitution between pairs of inputs rtsbm = - (mpm/mpb), rtsbn = - (mpn/mpb)... elasticities of substitution between pairs of inputs esbm = rtsbm/(b/m), esbn = rtsbn/(b/n)... It will be noted that elasticity of substitution is defined here not according to Allen (1938) but according to Dillon (1977). Table 4 shows these quantities computed for each of the production periods. They provide an idea of the sensitivity of the growth function over time for Eucalyptus grandis stands. These have been computed from equation (23) by using the average levels of inputs. It is seen that, for the input and output levels considered, there are increasing returns to scale (s > 1) for all production periods, the scale parameter s being highest in the first production period and decreasing as the stand becomes older. Perhaps in juvenile stands the value of s can be expected to remain above unity. Except for growing stock and water the other inputs show diminishing returns in all five production periods. Growing stock has increasing returns in the third, fourth, and fifth periods (eb > 1) and water has increasing returns only in the first production period (em > 1). The elasticity of production with respect to a particular input over time indicates the trend of the relative importance of such input in the timber production and VOLUME 30, NUMBER 3, 1984 / 769

process. For example, in the first production period, the most important input is water, followed by growing stock, solar radiation, and nutrient. In the subsequent production periods, the growing stock is the most important, followed in the second and third periods by water, solar radiation, and nutrient; and finally in the fourth and fifth periods by nutrient, solar radiation, and water. It should be noted that in the fifth production period, the elasticity of production of water is negative. This is linked to the fact that the water input is represented in the model by actual evapotranspiration and therefore as the stand becomes older more energy is channelled into maintaining respiration instead of the growth processes (Reed 1980). The trend presented by eb increasing from the first to the fourth production period, and then decreasing in the fifth (and perhaps subsequent) production period(s), suggests a possible full occupancy of site by the eucalypt trees at an age around 5.5 years. Regarding the water input, it should be noted that em decreases from the first to the last production period, which may be an indication that as the stand becomes older the roots of the trees explore deeper layers of the soil, and are less dependent on current year precipitation. However, the most plausible explanation is that the increase in respiration is adversely affecting the growth of the stand. The results regarding nutrients show that this input is important in the early stages of growth and its importance decreases as the stand becomes older. This is consistent with the a priori information about nutrient requirements of Eucalyptus grandis and the recycling of phosphorus in the ecosystem as the eucalypt stand becomes older (see Poggiani 1976). The decreasing trend of the output elasticity of the solar radiation input from the first to the fifth production period suggests that its relative importance decreases with age because, as the stand becomes older, growth will be concentrated more on the dominant and codominant trees of the stand. These changes in the relative importance of the inputs from one production period to another differentiate the biological process of timber production from a mechanistic production process. In the latter, if the same levels of inputs were considered, there would be no change in their relative importance from one period to another. USES The production function presented here can be used in intensive forest management for prediction, control, and economic decisionmaking. Prediction.--Knowing the basal area, actual evapotranspiration, amount of phosphorus, and hours of sunshine available to a crop in each growth period it is possible to estimate the growth in various years. The estimates can be revised and finely tuned if the inputs applied change from year to year due to changes in climatic factors or silvicultural operations. Control and Stand Manipulations.--The forest manager may find that due to poor rainfall the growth in one year has fallen short of the expected and so 3 or 4 years hence the yield may not meet expectations. The production model presented can be used to determine ways of increasing the growth, e.g., the application of fertilizers to make good the loss in the year of poor rainfall. In the same way losses due to fires, insects, and diseases can be made up. The reductions in growing stock for meeting unexpected temporary demands can also be compensated for by use of the model. Economic Decisionmaking.--Perhaps the most useful application of the produc- 770 / FOREST SCIENCE

tion function is in the determination of the optimal amounts of inputs to be applied in each production period and the optimal age to which the crop should be grown. An example showing the determination of optimal stocking and rotation can be seen in Nautiyal and Couto (1984) and another dealing with nutrients is in Couto and Nautiyal (1983). For planning purposes the water and solar radiation inputs in each period can be assumed to be at average levels. The combinations ofnand A that yield the maximum soil expectation value indicate the management regime best suited to a plantation. For two plantations in Bom Despacho, Minas Gerais, results indicated that using shorter rotations than presently practiced and more fertilizer application would increase the soil expectation values greatly (Couto 1982). CONCLUSIONS On the basis of the results obtained in this study, the following conclusions can be drawn: 1. If adequate data are available the timber production process can be estimated by using traditional methods of production theory in economics, while explicitly taking environmental inputs such as water, nutrient, and solar radiation into consideration in conjunction with stand density. 2. Age must enter as a surrogate factor of production in the timber production function and for estimation purposes the output must be growth. 3. Even if the predictive ability of the timber production models is, in some cases, lower than that of the traditional growth and yield models, the former are more useful for practicing intensive forest management on a rational basis, because the latter cannot provide the necessary input-output and input-input relationships. 4. For Eucalyptus grandis in Minas Gerais, the best functional form to represent annual timber growth was a polynomial including second order terms for growing stock and nutrient inputs, and also terms of interaction amongst the input variables. While the functional form and parameters of the estimated production function are expected to vary with age, location, and species, the theoretical approach outlined in this study can be applied to any species and situation. SUGGESTIONS FOR FURTHER RESEARCH Further research on this subject should attempt to use data spread over a longer period of crop ages. The longer time period will allow examination of both the juvenile and adult growth processes. In the current study juvenile growth alone was studied. If laying new sample plots is contemplated, wider range of nutrient applications and combinations of inputs over their entire ranges should be considered. In the model itself attempts to take into account lagged effects of the basic variables of the timber production function should be made. Representation of growth as relative rate of growth rather than in absolute terms may also yield better results and should therefore be tried. Also density related mortality functions should be developed since they will improve the predictive properties of the model. Another related area in need of research is the effect that silvicultural practices such as thinning, disking, irrigation, drainage, weeding, pruning, and defoliation (due to application of herbicides) have on growth through their effect on the levels of evapotranspiration. Treating pests and weeds as negative factors of production is perhaps a fruitful avenue of research as control of these factors becomes increasingly important in intensive forestry and determining the economic threshold of pest or weed damage is essential. Finally, alternative representations for the basic inputs and for the output, not considered in this study, could also be examined. VOLUME 30, NUMBER 3, 1984 / 771

LITERATURE CITED ALLEN, R. G.D. 1938. Mathematical analysis for economists. MacMillan and Co., London. 548 p. ANGSTROM, A. 1924. Solar and terrestrial radiation. Q J R Met Soc 50:121-126. ANomc ous. 1978. Recomenda oes para o uso de corretivos e fertilizantes em Minas Gerais. Comissao de Fertilidade do Solo do Eshado de Minas Gerais. Belo Horizonte. 80 p. (In Portuguese.) ARROW, K. J., H. g. CHENERY, g. S. MINHAS, and R. M. SOLOW. 1961. Capihal-labor substitution and economic efficiency. Rev Econ Shat 43(3):225-250. BEERS, T.W. 1962. Components of forest growth. J For 4:245-248. BUCKMAN, R. E. 1962. Growth and yield of red pine in Minnesota. U S Dep Agric Tech Bull 1272, 50 p. CLUTTER, J.L. 1963. Compatible growth and yield models for loblolly pine. Forest Sci 9:354-371. COBB, C. W., and P. H. DOUGLAS. 1928. A theory of production. Am Econ Rev 18:(suppl):139-165. COUTO, L. 1982. The nature of the timber production function: Eucalyptus grandis W. Hill ex Maiden in Brazil. Unpubl Ph D thesis, Dep Forestry. Univ Toronto. 146 p. COUTO, L., and J. C. NAUTIYAL. 1983. A nutrient-related time-dependentimber production function for Eucalyptus grandis Hill ex Maiden in southeastern Brazil. Unpubl Pap, Faculty of Forestry, Univ Toronto. 19 p. CURTIS, R.O. 1967. A method of estimation of gross yield of Douglas-fir. Forest Sci Monogr 13. 24 p. DILLON, J.L. 1977. The analysis of response in crop and livestock production. 2nd ed. Pergamon Press, Elmsford, New York. 213 p. DREw, T. J., and J. W. FLEWELLING. 1979. Stand density management: an alternative approach and its application to Douglas-fir planhations. Forest Sci 25(3):518-532. DUEP., W.A. 1960. Fundamentals of forestry economics. McGraw-Hill, New York. 579 p. DUERR, W.A. 1979. Choices and their prediction. In Forest resource management: decision-making principles and cases (W. A. Duerr, D. E. Teeguarden, N. B. Christiansen, and S. Gutenberg, eds), p 35-45. W. B. Saunders Co., Toronto. EPPLIN, F., S. BHIDE, and E. O. HEADY. 1980. Empirical investigation of beef grain roughageconcentrate substitution. Am J Agric Econ 62(3):468-477. FELDSTEIN, M. S., and D. K. FOOT. 1971. The other half of gross investment: replacement and modernization expenditures. Rev Econ Star 53:49-58. FERREIRA, F. A., M. S. REIS, A. C. ALFENAS, and A. R. DA SILVA. 1977. Niveis de incidencia natural de cancro em Eucalyptus spp. no nordeste do estado do Espirito Santo. Amore 1(2):89-106. (In Portuguese.) FREESE, F. 1960. Testing accuracy. Forest Sci 6:139-145. FOOT, D. K., and A. NORTH. 1977. The use and misuse of econometrics. Inst for Policy Analysis, Univ Toronto, Toronto. 256 p. FURNIVAL, G.M. 1961. An index for comparing equations used in constructing volume tables. Forest Sci 7(4):337-341. GEORGE$cu-RoGEN, N. 1971. The entrophy law and the economic process. Harvard Univ Press. 457 p. GOLFARI, L. 1975. Zoneamento ecologico do estado de Minas Gerais para reflorestamento. PNUD/ FAO/IBDF-BRA/71/545. Belo Horizonte. Seria Tecnica 3.65 p. (In Portugese.) GOLFARI, L. 1976. As introdu9oes de Especies/Procedencias Eucalyptus Realizadas pelo C.P.F.R.C.--Resulhados iniciais. Brasilia. PNUD/FAO/IBDF/BRA-45. Serie Divulga9ao 11.75 p. (In Portuguese.) GOLFARI, L. 1978. Zoning for reforestation in Brazil and trials with tropical eucalypts and pines in central region. Brasilia. PNUD/FAO/IBDF/BRA-45. Tech Rep 12. 25 p. GREGORY, G.R. 1972. Forest resource economics. The Ronald Press, New York. 548 p. GRILICHES, Z., and V. RINGSTAD. 1971. Economies of scale and the form of the production function: an econometric study of the Norwegian manufacturing establishment data. In Contributions to economic analysis. (J. Johnston, D. W. Jorgenson, J. Waelbroeck, and J. Tinbergen, eds). North- Holland Publ Co., Amsterdam. 204 p. HEADY, E. O. 1965. Economics of agricultural production and resource use. 6th ed. Prentice-Hall, Englewood Cliffs. 850 p. HEADY, E. O., and J. L. DILLON. 1961. Agricultural production functions. Iowa State Univ Press, Ames. 667 p. 772 / FOREST SCIENCE

HEXEM, R. W., AND E. O. HEADY. 1978. Water production functions for irrigated agriculture. Iowa State Univ Press, Ames. 215 p. JORGENSON, D. W., and J. A. STEPHENSON. 1967. Investment behaviour in US manufacturing, 1947-1960. Econometrica 35:169-220. JUDGE, G. G., W. E. GRIFFITH, R. C. HILL, and T-C. LEE. 1980. The theory and practice of econometrics. John Wiley and Sons, New York. 793 p. MACKINNEY, A. L., F. X. SCHUMACHER, and L. E. CHAIKEN. 1937. Construction of yield tables for nonnormal 1oblolly pine stands. J Agric Res 54:531-545. McFADDEn, D. 1963. Constant elasticity of substitution production functions. Rev Econ Stud 30: 73-83. MELLO, H. A., J. MASCARENHASOBRINHO, J. W. SIMOES, and H. T. DO COUTO. 1970. Resultados da Aplicacao de Fertilizantes Minerais na Producao de Madeira de Eucalyptus saligna Sm. em Solos de Cerrado do Estados de S. Paulo. Piracicaba. IPEF 1:7-26. (In Portuguese.) M NHAS, B. S., K. S. PAR KH, and T. N. SRIN vasan. 1974. Toward the structure of a production function for wheat yields with dated inputs of irrigation water. Water Resour Res 10(3):383-392. MosER JR, J.W. 1967. Growth and yield models for uneven-aged forest stands. Ph D thesis, Purdue Univ, West Lafayette, Ind. 166 p. MouPo, V. P. G., R. L. CASER, F. C. ALBINO, D. P. GUIMARAES, J. T. DE MELO, and S. A. COMASTRI. 1980. Avaliacao de especies e procedencias de Eucalyptus em Minas Gerais e Espirito Santo: Resultados parciais. Brasilia. EMBRAPA-CPAC. Boletim de Pesquisa N 1. 104 p. (In Portuguese.) MUKERJI, V. 1963. A generalized SMAC function with constant ratios of elasticity of substitution. Rev Econ Stud 30:233-236. NAUTIYAL, J. C., and L. CouTo. 1982. The use ofproduction-functionanalysis in forest management: eucatypts in Brazil, a case study. Can J Forest Res 12(2):452-458. PIENAAR, L. V., and K. J. TURNBULL. 1973. The Chapman-Richard genaralization ofvon Bertalanffy's growth model for basal area growth and yield in even-aged stands. Forest Sci 19(1):2-22. POGGIANI, F. 1976. Ciclo de nutrientes e produtividade de florestas implantadas. Silvicultura 3:45-49. (In Portuguese.) REED, H. L. 1980. An ecological approach to modelling growth of forest trees. Forest Sci 26(1): 33-50. RENNIE, J. C., and H. V. WIANT, JR. 1978. Modification of Freese's Chi-square test of accuracy. USDI, Bur Land Manage, Denver, Colo, Resour Inventory Notes. 3 p. REvANKAR, N.S. 1971. A class of variable elasticity of substitution production functions. Econometfica 39(1):61-71. SATO, K. 1967. A two level constant elasticity of substitution production function. Rev Econ Stud 34:201-218. SCHONAU, A. P.G. 1970. Planting, spacement and pruning of Eucalyptus grandis on a low quality site. South African For J 73:11-15. SPURR, S.H. 1952. Forest inventory. The Ronald Press, New York. 476 p. TEEGUARDEN, D. E. 1979. Managing timber resources. In Forest resource management: decisionmaking principles and cases (W. A. Duerr, D. E. Teeguarden, N. B. Christiansen, and S. Guttenberg, eds), p 173-214. W. B. Saunders Co., Toronto. THEIL, H. 1978. Introduction to econometrics. Prentice-Hall Inc, New Jersey. 447 p. THORNTHWAITE, C. W., and R. J. MATHER. 1957. Instructions and tables for computing potential evapotranspiration and the water balance. Publ Climatol 10(3):311 p. UZAWA, H. 1962. Production functions with constant elasticity of substitution. Rev Econ Stud 20: 291-299. VAN OOOR, C. P., and R. NASCIMENTO. 1970. Aduba ao em planta oes florestais. Brasil Florestal 1:35-42. (In Portuguese.) VILLA NOVA, N. A., and E. SALATI. 1977. Radia ao solar no Brasil. ESALQ-CNEN-USP. 33 p. (In Portuguese.) WALTERS, A.A. 1963. Production and cost functions: an econometric survey. Econometrica 31: 1-65. VOLUME 30, NUMBER 3, 1984 / 773