ESSAYS ON FORESTRY PRODUCTS INDUSTRY: SAWMILL PRODUCTIVITY AND INDUSTRIAL TIMBERLAND OWNERSHIP Except where the reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classified information. __________________________ Yanshu Li Certificate of Approval: __________________________________ _________________________________ David Laband Daowei Zhang, Chair Professor Professor Forestry and Wildlife Sciences Forestry and Wildlife Sciences ____________________________________ _________________________________ Gregory Traxler T. Randolph Beard Professor Professor Agricultural Economics and Rural Sociology Economics ____________________________________ Stephen L. McFarland Dean Graduate School ESSAYS ON FORESTRY PRODUCTS INDUSTRY: SAWMILL PRODUCTIVITY AND INDUSTRIAL TIMBERLAND OWNERSHIP Yanshu Li A Dissertation Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Auburn, Alabama May 11, 2006 iii ESSAYS ON FORESTRY PRODUCTS INDUSTRY: SAWMILL PRODUCTIVITY AND INDUSTRIAL TIMBERLAND OWNERSHIP Yanshu Li Permission is granted to Auburn University to make copies of this dissertation at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. ____________________________________ Signature of Author _______________________________ Date of Graduation iv VITA Yanshu Li, daughter of Zhigong Li and Biandeng Wang, was born on November 23, 1975, in Xinzhou, Shanxi Province, People?s Republic of China. She entered Department of Mathematics at Shanxi University in Taiyuan, Shanxi, in September 1993, and graduated with a Bachelor of Science degree in Accounting in July 1997. She entered Graduate school, China Agricultural University, Beijing, in September 1997 and graduated with a Master of Science degree in Agricultural Economics and Management in July 2000. In August, 2000, she entered Graduate School, Purdue University, Indiana, and graduated with a Master of Science degree in Agricultural Economics in May 2003. She married Zhenchuan Fan, son of Mingde Fan and Xiue Guo, on July 30, 2002. v DISSERTATION ABSTRACT ESSAYS ON FORESTRY PRODUCTS INDUSTRY: PRODUCTIVITY AND INDUSTRIAL TIMBERLAND OWNERSHIP Yanshu Li Doctor of Philosophy, May 11, 2006 (M.S., Purdue University, 2003) (M.S., China Agricultural University, 2000) (B.S., Shanxi University, 1997) 115 typed pages Directed by Daowei Zhang In this dissertation, two topics of forest products industry were investigated: inter- regional productivity comparison of sawmilling industries in the North America, and the relationship between industrial timberland ownership and corporate financial performance in the U.S. The first study used nonparametric programming approach to estimate technical efficiency and total factor productivity (TFP) growth of sawmill industries in the U.S. and Canada between 1963 and 2001 The results showed that the U.S. sawmill industry was more likely to be on the industry frontier than Canada during 1990-2001 although the Canadian sawmill industry was shown more efficient compared to the U.S. counterpart during 1963-1989. The weighted annual productivity growth of sawmill industry was 2.5% for the U.S. and 1.3% for Canada. Regional differences in technical efficiency and vi TFP growth existed. All regions were shown to have a trend of moving towards the industry frontier. Assumption of Hicks neutrality in production was rejected for both countries. Bootstrap results suggested that both countries experienced statistically significant productivity growth and the U.S. had a higher rate of growth during the whole study period although the estimates may be sensitive to outliers. The second study presented an empirical analysis of the relationship between industrial timberland ownership and financial performance of forestry products companies in the U.S. A three stage least square (3SLS) model system was used for estimation. The results showed that generally timberland holding may improve a forest products company?s profitability in terms of return on asset (ROA) and return on equity (ROE) as well as its ability of response of rate of returns to uncertainty. However, higher capital expense and debt/asset ratio were shown associated with timberland holding. Forest product companies may divest some of their timberland to ease the financial burden. vii ACKNOWLEDGEMENTS The author wishes to express her sincere gratitude to her advisor, Dr. Daowei Zhang, for his constant guidance, support and encouragement during her entire Ph.D study. The author would like to express her deep appreciation to her committee members: Dr. David Laband, Dr. T. Randolph Beard, and Dr. Gregory Traxler for their many help and advice. In addition, the author appreciates the great help from Dr. Beverly Marshall in data collection. The author would like to thank her parents who have always loved her and encouraged her in her study. Special thanks also go to her parents-in-law for their love and support. Finally, the author would like to thank her husband. This dissertation would not be possible without his endless love. viii Style manual or journal used: Forest Science Computer software used: Word and Excel for Windows XP Version XP, GAMS 2.50, SAS 8.2, R 2.2.0, FEAR 0.912, Limdep 8.0. ix TABLE OF CONTENTS LIST OF TABLES???????????????????????????..ix LIST OF FIGURES??????????????????????????xii I. INTRODUCTION?????????????????????????.1 II. PRODUCTIVITY IN THE SAWMILLING INDUSTRIES OF THE UNITED STATES AND CANADA: A NONPARAMETRIC ANALYSIS...............................2 INTRODUCTION????????????????????????...2 METHODOLOGY: DISTANCE FUNCTION AND THE MALMQUIST PRODUCTIVITY INDICES????????????????????...5 DATA????????????????????????????..10 LABOR INPUTS ?????????????????.???????11 CAPITAL INPUT ????????????????.???????...12 ENERGY INPUT????????????????.????????.13 WOOD INPUT ????????????????????????...14 SOFTWOOD AND HARDWOOD LUMBER OUTPUTS??????.?????15 WOODCHIPS ?????????????????????????.15 RESULTS AND DISCUSSIONS??????????????????..16 TECHNICAL EFFICIENCY ????????????????????..16 x F?RE MALMQUIST PRODUCTIVITY INDEX AND COMPONENTS?????..?21 BIAS COMPONENTS OF TECHNICAL CHANGE ??????...??????26 x COMPARISON WITH THE T?RNQVIST-THEIL INDEX APPROACH AND OTHER STUDIES??????????????????????..????..29 SENSITIVITY ANALYSIS ????????????.????????...31 CONCLUSION?????????????????????????..41 III. AN EMPIRICAL ANALYSIS OF TIMBERLAND OWNERSHIP AND CORPORATE FINANCIAL PERFORMANCE FOR FORESTRY INDUSTRIES IN THE U.S????????????????????????????...44 INTRODUCTION????????????????????????.44 CURRENT SITUATION OF INDUSTRIAL TIMBERLAND HOLDINGS?..????.47 LITERATURE REVIEW ?????????????????????.56 INDUSTRIAL TIMBERLAND HOLDINGS: BENEFITS AND COSTS?????..?.56 STUDIES ON DETERMINANTS OF COMPANIES? FINANCIAL PERFORMANCE??61 DATA AND METHODOLOGY??????????????????...64 DEPENDENT VARIABLES??????????????????...64 INDEPENDENT VARIABLES?????????????????..66 EMPIRICAL MODEL?????????????????????.70 DATA ANALYSIS AND RESULTS?????????????????71 DESCRIPTIVE STATISTICS??????????????????71 ECONOMETRIC MODEL??????????????????75 DISCUSSIONS AND CONCLUSIONS???????????????...80 IV. SUMMARY????..???????????????????????.82 V. REFERENCES??????????????????????????84 VI. APPENDIX A?..??..???????????????????????91 xi VII. APPENDIX B???????????????????????????95 VIII.APPENDIX C?..???.?.?????????????????????99 IX. APPENDIX D??????????????????????????103 xii LIST OF TABLES Table 2.1. Concordance between SIC242 and NAICS for the U.S. ????????.11 Table 2.2. Percentage of time on the industry frontier over different periods???.?..20 Table 2.3.F?re productivity index, efficiency change, and technical change, 1964- 2001)???????...???????????????????..22 Table 2.4. F?re productivity index in different periods ????????????24 Table 2.5. Annual biased technical change, 1964-2001???????????..27 Table 2.6. F?re productivity index, efficiency change, and technical change for 1964- 2001 (without wood chips)??.????????????????..33 Table 2.7. Bootstrap results of F?re productivity index by year ??...???????37 Table 3.1. Industrial timberland area by region and year ??????...????..?48 Table 3.2. Trend of selected forest product company timberland ownership by region .49 Table 3.3. Industrial timberland holdings by corporate and year .?. ???????..50 Table 3.4. The U.S. industrial timberland concentration ratio (1981, 1994, 2003)??54 Table 3.5. Selected wood self-sufficiency rate????.???????????55 xiii Table 3.6. Descriptive statistics of the sample: Dependent variables?...?????71 Table 3.7. Descriptive statistics of the sample: Independent variables???...???72 Table 3.8. Comparison of financial performance among firms owning timberland and owning no timberland ??????????.????. ?????75 Table 3.9. Cross model correlation??????????.????. ??????75 Table 3.10. Estimated results of the system of equations???????????77 xiv LIST OF FIGURES Figure 2.1. Output distance functions in two periods (two outputs)???????.?6 Figure 2.2. Percentage of time on industry frontier for selected states/provinces in the U.S. and Canada over 1963-2001????????????????18 Figure 2.3. Annual F?re productivity index for the U.S. and Canada, 1964-2001...??.25 Figure 2.4. Cumulated F?re productivity index for the U.S.&Canada, 1964-2001??..26 Figure 2.5. Annual biased technical change effects, the U.S. ??????.???..28 Figure 2.6. Annual biased technical change effects, Canada ?????????..29 1 I. INTRODUCTION This dissertation addresses two issues related to forest products industries in the US. Productivity comparisons in the North American sawmilling industries have been of concern for decades as they play an important role in regional resource allocation and relative competitiveness among regional counterparts. Although costs of inputs affect relative competitiveness in the short run, competitiveness in the long run will be determined by technical efficiency and productivity growth. Chapter II presents a study of productivity analysis of sawmill industries in the U.S. and Canada by using non- parametric programming method or Data Envelope Analysis approach. The second issue concerns industrial timberland ownership of forest products companies in the US. forest products company restructuring involves decisions about industrial timberland holdings. However, the patterns of timberland holdings are far from uniform. There have been quite a few theories explaining timberland holding behavior of forest products companies have been proposed, favorable return and financial success among them. However, there has been no empirical analysis of this hypothesis. To fill this gap, an econometric analysis of timberland ownership and corporate financial performance is performed using cross-sectional data for 36 publicly-traded U.S. forest products companies from 1988 to 2003. Results are reported in Chapter III. 2 II. PRODUCTIVITY IN THE SAWMILLING INDUSTRIES OF THE UNITED STATES AND CANADA: A NONPARAMETRIC ANALYSIS INTRODUCTION Productivity measures the efficiency with which inputs are transformed into outputs. Higher productivity occurs when larger quantities of outputs are produced with given inputs. Among various techniques to estimate the performance of industries, total factor productivity (TFP) provides a simple yet comprehensive measurement. TFP, the ratio of an index of aggregate output to an index of aggregate input, is a measure taking into account the contribution of all inputs. Productivity comparisons in the North American sawmilling industries have been of concern for decades as they play an important role in regional resource allocation and relative competitiveness among regional counterparts. Although costs of inputs affect relative competitiveness in the short run, competitiveness in the long run will be determined by technical efficiency and productivity growth. In the ongoing U.S.-Canada softwood lumber dispute, the Canadian industry uses relatively higher productivity as an argument to explain their increasing share of the U.S. lumber market. However, this argument is refuted by the U.S. lumber industry. While many studies have been devoted to the productivity growth of the sawmill industry in the U.S and Canada, the results are 3 mixed. Some studies suggest that there has been little or no technical progress in Canada, and productivity growth in the Canadian sawmill industry is lower than the U.S. counterpart (Constantino and Haley 1989, Ghebremichael et al. 1990, Abt et al. 1994, Nagubadi and Zhang 2004). At one extreme, Meil and Nautiyal (1988) reported negative TFP growth for all four Canadian regions over 1950-1983. On the other hand, Gu and Ho (2000) estimated that TFP growth of lumber & wood products industry increased by 0.62% per year in Canada while decreasing by 0.21% annually in the U.S. between 1961 and 1995. Different approaches adopted by these studies may contribute to the differences in the results. Often, either an index approach or an econometric model is used to estimate productivity growth and technical change. Both approaches assume that all firms in the industries operate efficiently, which may not be the case in the reality, and some specific forms of cost or profit functions have to be assumed for econometric analysis. As a more flexible approach, a nonparametric programming approach (or called data envelopment analysis) has been used recently in the area of agricultural and industrial productivity analysis (e.g., F?re et al. 1994, Granderson and Linvill 1997, Preckel et al. 1997, Arnade 1998, Yin 1998, 1999, 2000, Hailu and Veeman 2001, Nin, Arndt, and Preckel 2003, Nin, Arndt, Hertel, and Preckel 2003, Umetsu et al. 2003). This method, proposed by F?re et al. (1994) involves estimating an input or output based Malmquist index (Caves et al. 1982). Compared to other methods, the nonparametric programming approach has the advantage of imposing no a priori restrictions on the functional form of the underlying technology and allowing for inefficiency in production (Varian 1984, F?re et al. 1994). This approach is also capable of decomposing productivity growth into 4 changes in technical efficiency over time and shifts in technology over time. Requiring only quantity data, the nonparametric programming approach may avoid distortions due to estimated errors in price data and fluctuations in exchange rate. Until recently, however, the nonparametric programming approach has rarely been used in sawmill productivity analysis. Nyrud and Baardsen?s (2003) analysis of Norwegian sawmill productivity is one of the few exceptions. This study attempts to expand the analytic scope of the technical efficiency and productivity trends of sawmill industries in the North America by using the nonparametric programming approach. In doing so, it answers the following questions: Which state/province, region or country is on average the most efficient in sawmill production in the North America? What is the pattern of TFP growth for each state/province, region or country? Decomposition of productivity growth can also shed light on the sources of the growth as a shift in the production frontier or movement towards or away from the production frontier, and bias in technical change: input or output oriented, which assists policy makers and managers make decisions. Are estimates from the nonparametric estimation different from those obtained by using other estimation methods? In this chapter, distance functions and the nonparametric Malmquist index will be reviewed. Then, the data will be described followed by the results. Comparisons between the results from this study and other previous relevant studies as well as the results using other approach (T?rnqvist-Theil index approach) are made. Finally, conclusions and suggestions for future research will be presented. 5 METHODOLOGY: DISTANCE FUNCTION AND THE MALMQUIST PRODUCTIVITY INDICES As in Caves et al. (1982), the productivity change of the sawmilling industry over time is estimated as the geometric mean of two output-based Malmquist productivity indices, developed based on distance functions. Suppose that for each time period Tt ,...,1= the feasible production set of the industry is: t S ={(x t ,y t ): x t can produce y t } [2.1] Where, x t ? N + and y t ? M + are input and output quantity vectors from N and M dimensional real number spaces; and N and M are the total number of inputs and outputs. t S is assumed to be closed, bounded, convex and to satisfy strong disposability 1 of outputs and inputs. Following Shepherd (1970), the output-based distance function at t is defined as the reciprocal of the maximum proportional expansion of output vector y t given input x t : ),( 0 ttt D yx = ? ? ? ? ? ? ? t t t S),(:inf ? ? y x = { } 1 )),(:(sup ? ? ttt Syx ?? . [2.2] The distance function measures how far the production function of interest is from the frontier of the whole industry in period t. Figure 2.1 shows the case of two outputs (y 1 and y 2 ). The frontier at t is developed by production unit B, C, and D. For production unit A, the distance function at t can be 1 Which means if ttt S?),( yx ,then ttt S?) ~ , ~ ( yx for all ) ~ , ~ ( tt yx such that tt xx ? ~ and tt yy ? ~ . 6 expressed t t ttt OP OA D =),( 0 yx . And its distance function at t+1 is 1 1 + + t t OP OA . ),( 0 ttt D yx equals 1 when production unit is on the frontier, or technically efficient. On the other hand, ),( 0 ttt D yx is less than 1 when production is technically inefficient. The greater its value is, the closer is the production unit to the efficient production frontier. The distance function provides a complete characterization of the production technology. Figure 2.1. Output distance functions in two periods ),( 0 ttt D yx can be obtained by solving the following linear programming model: * , k k Maximize ?? (),( 0 ttt D yx ) -1 = t k * ? Subject to: t k t mk t km K k k yy ** 1 ?? ? ? = m=1,?,M A t O P t B t C t D t B t+1 C t+1 D t+1 A t+1 y 2 y 1 P t+1 e f 7 t nk t kn K k k xx * 1 ? ? = ? n=1,?,N [2.3] 0? k ? k=1,?,K where m indexes outputs; n indexes inputs; k indexes production regions ( * k is a particular region of interest); k ? is the weight on the kth region data; t k * ? is the efficiency index, or the reciprocal of the distance function for region * k . The inequalities for inputs and outputs make free disposability possible. Non-negativity of k ? allows the model to exhibit constant returns to scale. In the same way, the distance from the production point in t relative to the frontier in t+1 can be defined as ),( 1 0 ttt D yx + ( Oe OA t in Figure 2.1). Two simple Malmquist indices can be defined depending on the technology reference of time periods by using distance functions. Using the technology at t as the reference, the period t-based Malmquist index is defined as: ),( ),( 0 11 0 0 ttt ttt t D D M yx yx ++ = [2.4] Using the technology at t+1 as the reference, the period t+1-based Malmquist index is: ),( ),( 1 0 111 01 0 ttt ttt t D D M yx yx + +++ + = [2.5] In Figure 2.1, for production unit A, t M 0 is t tt OP OA Of OA 1+ , and 1 0 +t M is Oe OA OP OA t t t 1 1 + + . A Malmquist index of greater than 1 implies positive productivity growth, 8 or technical progress. As F?re et al. (1997) noted, however, these two measures may not provide consistent results in some cases. The estimate of productivity growth may vary depending on the choice of Malmquist indices. Based on Caves et al. (1982), F?re et al. (1994) suggested the use of a geometric mean of t M 0 and 1 0 +t M as the output-based Malmquist index ( 0 M ). That is: 2 1 2 1 ),( ),( ),( ),( ][ 1 0 111 0 0 11 01 000 ? ? ? ? ? ? ?=?= + +++++ + ttt ttt ttt ttt tt D D D D MMM yx yx yx yx [2.6] Improvement in productivity yields a F?re Malmquist index value greater than 1 while deterioration in performance over time is associated with an index value less than 1. Furthermore, F?re et al. (1994) show that 0 M can be decomposed into an efficiency change component and a technical change component. Thus, Equation [2.6] is equivalent to: 2 1 ),( ),( ),( ),( ),( ),( 1 0 0 111 0 11 0 0 111 0 0 ? ? ? ? ? ? ??= ++++ +++++ ttt ttt ttt ttt ttt ttt D D D D D D M yx yx yx yx yx yx [2.7] where, the first part on the right hand side is defined as efficiency change (EFFCH) or ?catch up?, which measures the change in how far the observed production unit is from the potential production frontier between period t and period t+1. The second part is defined as technical change (TECH) or ?innovation?, which captures the shift in technology between two periods. In Figure 2.1, EFFCH is t t t t OP OA OP OA 1 1 + + , and TECH is t t OP Oe Of OP 1+ for A. 9 Nin, Arndt, and Preckel (2003) show that these three Malmquist indices ( t M 0 , 1 0 +t M and 0 M ) have the same efficiency change. The potential differences stem from the estimate of technical change. When technical change is biased (either input or output biased) the estimates of technical change from the three indices will be different. F?re et al. (1997) decompose the technical change component of 0 M into three parts: output- biased technical change (OBTECH), input-biased technical change (IBTECH), and the magnitude of technical change under input and output neutrality (MATECH). 43421 4444434444421444444344444421 MATECH ttt ttt IBTECH ttt ttt ttt ttt OBTECH ttt ttt ttt ttt ttt ttt ttt ttt D D D D D D D D D D D D D D TECH ),( ),( ),( ),( ),( ),( ),( ),( ),( ),( ),( ),( ),( ),( 1 0 0 1 0 0 11 0 1 0 11 0 1 0 111 0 11 0 1 0 0 111 0 11 0 2 1 2 1 2 1 yx yx yx yx yx yx yx yx yx yx yx yx yx yx ++++ + ++ + +++ ++ ++++ ++ ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ?= [2.8] It should be noted that this decomposition is valid only under constant returns to scale (CRS) (F?re et al. 1997). OBTECH measures the output bias of technical change by the ratio of the magnitude of technical change along a ray through 1+t y to the magnitude of technical change along a ray through t y holding input vector fixed at 1+t x . IBTECH captures the input bias of technical change by providing the ratio of the magnitude of technical change along a ray through 1+t x to the magnitude of technical change along a ray through t x holding output vector fixed at t y . MATECH measures the magnitude of technical change along a ray through period t. OBTECH=1 implies neutral output technical change, and IBTECH=1 is associated with neutral input technical change. 10 DATA A time-series dataset of sawmills and planing mills 2 covering 1963-2001 for 26 states in the U.S. 3 and 8 provinces 4 in Canada is used. The selection of state/province is mainly based on data availability and each state?s share in national lumber production. In 2001, selected states accounted for 96.8% of softwood lumber production and 93.2% of hardwood lumber production in the U.S. And selected Canadian provinces accounted for about 99% of both national softwood and hardwood lumber production. Since state-level lumber production data prior to 1963 are not available, the study period was selected from 1963-2001. For purpose of regional comparison, selected states of the U.S. were classified into three regions (West, South, North). Canadian provinces were classified into British Columbia, Ontario, Quebec and Others mainly based on their shares of lumber production. Main data sources for the U.S are the Annual Survey of Manufactures (ASM) and the Census of Manufacturing (CM). Data for Canada are from the Annual Census of Manufactures (ACM), principal statistics from the Canadian Forest Service, and the CANSIM II database. In 1997, the new industry classification system, North American Classification System (NAICS), was introduced and replaced the Standard Industrial Classification (SIC) system. For this study, we used the industry definition based on the 2 1987 Standard Industry Classification (SIC) System 242 for U.S. and 251 for Canada, concordances between SIC and NAICS are made to assemble the data after 1996. 3 Selected U.S. western states: California (CA), Idaho (ID), Montana (MT), Oregon (OR), Washington (WA). Selected U.S. northern states: Indiana (IN), Maine (ME), Michigan (MI), Missouri (MO), New York (NY), Ohio (OH), Pennsylvania (PA), Wisconsin (WI), West Virginia (WV). Selected U.S. southern states: Alabama (AL), Arkansas (AR), Florida (FL), Georgia (GA), Kentucky (KY), Louisiana (LA), Mississippi (MS), North Carolina (NC), South Carolina (SC), Tennessee (TN), Texas (TX), Virginia (VA). 4 Alberta (AB), British Columbia (BC), Manitoba (MB), New Brunswick (NB), Nova Scotia (NS), Ontario (ON), Quebec (QC) and Saskatchewan (SK). 11 1987 SIC system. A bridge between SIC and NAICS was constructed based on value of shipments, number of employees, and annual payrolls in 1997. All principal production data 5 in NAICS were converted based on Table 2.1. For example, the sum of 85% of value of shipments under 3211 and 19% under 3219 are estimated as the value of shipment for 242. Canadian series were merged using average proportions developed from data reported for the same years 1990-1997 under NAICS and SIC classifications. Table 2.1. Concordance between SIC242 and NAICS for the U.S. used in this study NAICS Value of Shipment (%) # of Employee (%) Annual Payroll (%) 3211 85 91 91 3219 19 16 15 Five inputs and three outputs were used to estimate the Malmquist index. The construction of each variable is described as follows. LABOR INPUTS This study used two types of labor input: production labor and non-production labor. Manufacturing-related labor is measured in terms of hours worked for the American states and in terms of hours-paid for the Canadian provinces, which includes paid vacation. Abt et al. (1994) suggested that this may lead to a slight downward bias of the productivity estimate for the Canadian industry. Labor not related to manufacturing is measured in terms of the number of employees who are not production workers. 5 Employee number, production hours and production worker number are converted based on the concordance of # of employee. Employee wages and production worker wages are converted based on the concordance of annual payroll. All others are converted based on the concordance of value of shipment. 12 CAPITAL INPUT Capital stock in 1997 constant U.S. dollars is estimated using the perpetual inventory method (PIM). As in Ahn and Abt (2003) investment on plants and structures was depreciated over 28 years, and machinery and equipment was depreciated over 16 years. Annual capital stock estimates for different asset types were aggregated as a total capital stock for each state/province. Estimate of capital stock for any given state/province s at the end of year t is calculated as: ? ? ? ? ? ? = ? = tsts IK , 0 , [2.9] where, ? is age of asset; ? ? is the relative efficiency function at age? ; and I is investment. The hyperbolic efficiency function is: ? ? ? <