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Essays on capitalizing Research and Development expenditures Huang, Ning 2009

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ESSAYS ON CAPITALIZING RESEARCH AND DEVELOPMENT EXPENDITURES  by  Ning Huang   B.A., Beijing Normal University, 1992 M.A., University of British Columbia, 2002   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  The Faculty of Graduate Studies  (Economics)   THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  JULY 2009  © Ning Huang, 2009  ii ABSTRACT  The next international version of the System of National Accounts (SNA) will recommend that Research and Development (R&D) expenditures be treated as capital formation instead of immediate expenses as in the present SNA 1993. This recommendation brings in challenges to many aspects of measuring R&D capital stocks. This dissertation focuses on measurement issues associated with capitalizing R&D expenditures.  In the first paper, we develop a simple model on production technology that allows for monopolistic competition. It enables estimation of annual depreciation rates for R&D capital. We treat R&D capital as a technology shifter instead of an explicit input factor. Both the R&D stock and the time trend are used to capture technological progress. The R&D depreciation rates and markup factors are estimated for the U.S. manufacturing sector and four U.S. knowledge intensive industries.  The second paper looks at the net benefits of a R&D project in the context of a very simple intertemporal general equilibrium model and suggests that R&D expenditures be amortized using the matching principle that has been developed in the accounting literature. This approach matches fixed costs of a project to a stream of future benefits. Of particular interest is the evaluation of the net benefits of a publicly funded project where the results are made freely available to the public.  In the third paper, we propose a new method of treating R&D expenditure in the growth accounting framework and investigate how to construct the associated measures for the stock of the R&D capital. We distinguish general knowledge from the more specific technologies applied in goods and services production, and treat them differently in the growth accounting framework. In addition, we derive a general formula to measure the newly created knowledge increments that depend not only on R&D expenditures but also on the existing general knowledge stock and knowledge productivity. We illustrate our new methodology for the treatment of R&D using U.S. manufacturing data for the last half century.  iii TABLE OF CONTENTS   ABSTRACT........................................................................................................................ ii TABLE OF CONTENTS................................................................................................... iii LIST OF TABLES .............................................................................................................. v LIST OF FIGURES ........................................................................................................... vi ACKNOWLEDGEMENTS.............................................................................................. vii DEDICATION ................................................................................................................. viii STATEMENT OF CO-AUTHORSHIP ............................................................................. ix 1 Introduction.................................................................................................................. 1 2 Estimation of R&D Depreciation Rates: A Suggested Methodology and a Preliminary Application.......................................................................................................................... 4 2.1 Introduction.................................................................................................... 4 2.2 Construction of the R&D Stock ..................................................................... 5 2.2.1 R&D Capital and Ordinary Physical Capital ................................... 6 2.2.2 Constructing the Stock of R&D Capital .......................................... 7 2.2.3 Reasons for R&D Capital Depreciation......................................... 10 2.3 The Estimation Framework...........................................................................11 2.3.1 The Basic Framework .....................................................................11 2.3.2 The Choice of the Functional Form for the Production Function.. 15 2.3.3 The Problem of Trending Elasticities ............................................ 17 2.3.4 Problems due to Non-Smooth Technical Progress......................... 19 2.4 Empirical Estimation and Results................................................................ 21 2.4.1 Estimation Methodology................................................................ 21 2.4.2 Data Construction .......................................................................... 23 2.4.3 Estimation Results ......................................................................... 25 2.5 Conclusion ................................................................................................... 30 3 Estimation of the Net Benefits of a R&D Project...................................................... 32 3.1 Introduction.................................................................................................. 32 3.2 Disembodied Technical Change and Growth Accounting ........................... 34 3.3 The New Technologies Production Function............................................... 37 3.4 R&D and Process Innovation: The Case of a Government Funded Project 39 3.5 R&D and Process Innovation: The Case of a Privately Funded Project...... 46 3.6 R&D and Product Innovation ...................................................................... 56 3.7 Towards a More Realistic Model of R&D Investment ................................ 57 3.8 Conclusion ................................................................................................... 70 4 Measuring the Stock of R&D Capital in a Growth Accounting Framework ............. 72 4.1 Introduction.................................................................................................. 72 4.2 The Treatment of R&D Capital in the Growth Accounting Framework ..... 75  iv 4.3 Conceptual Issues......................................................................................... 78 4.3.1 R&D Knowledge Investment and the Stock of R&D Knowledge Capital ........................................................................................................ 78 4.3.2 R&D Capital versus Knowledge Capital ....................................... 80 4.3.3 Measures of the R&D Capital Stock.............................................. 80 4.4 Measuring Stock of R&D Capital without Knowledge Diffusion............... 81 4.4.1 Measurement of the Knowledge Product....................................... 82 4.4.2 The Construction of the Two R&D Capital Stocks........................ 89 4.4.2.1 The Determination of the Relevancy Indexes......................... 91 4.4.2.2 The Determination of the Effective Efficiency Index............. 92 4.5 Empirical Results ......................................................................................... 95 4.5.1 Measuring the R&D Stock............................................................. 95 4.5.2 Productivity Analysis ................................................................... 105 4.6 Conclusion ..................................................................................................110 BIBLIOGRAPHY............................................................................................................112 APPENDICES ................................................................................................................ 120 Appendix A: The Industry’s Profit Maximization Problem.................................... 120 Appendix B: Break Points ...................................................................................... 123  v LIST OF TABLES  Table 2.1 Depreciation Rates Maximizing the Log-likelihood Function ............... 27 Table 2.2 Depreciation Rates and Markup Factors................................................. 28 Table 4.1 Regressions without R&D Capital as an Explanatory Variable .......... 106 Table 4.2 Estimates from Regressions Using the Traditional Approach ............. 106 Table 4.3 Estimates from Regressions Using the Proposed Approach................. 107 Table 4.4 Average Productivity Growth Rates Using the Conventional Approach and the Proposed Approach ................................................................................. 109 Table A.1 Break Points for Model III and Model IV ………………………………..…..123  vi LIST OF FIGURES  Figure 2.1 R&D Investments (1953-2000)............................................................. 9 Figure 2.2 R&D Stocks (1953-2000)..................................................................... 30 Figure 4.1 Utilization Path.................................................................................... 93 Figure 4.2 The Knowledge Product and the Two Types of Knowledge Stock . 99 Figure 4.3 Technology Knowledge Capital Stock and Conventional Knowledge Stocks ............................................................................................................... 99 Figure 4.4 Technological Knowledge Stocks with Different Rates of Knowledge Productivity ........................................................................................................... 101 Figure 4.5 Technological Knowledge Stocks with Different Service Lives..... 102 Figure 4.6 Technological Knowledge Stocks with Different Initial Stocks..... 102 Figure 4.7 Technological Knowledge Stocks with Different Initial Utilization Levels ............................................................................................................. 103 Figure 4.8 General Knowledge Stocks with Different Knowledge Productivities  104 Figure 4.9 General Knowledge Stocks with Different Service Lives .............. 104 Figure 4.10 Productivity Growth Trends Estimated by the Conventional Approach and the Proposed Approach (δ=0.15) ................................................ 108 Figure 4.11 Productivity Growth Trends Estimated by the Conventional Approach and the Proposed Approach (δ=0.20) ................................................ 108 Figure 4.12 Productivity Growth Trends Estimated by the Conventional Approach and the Proposed Approach (δ=0.25) ................................................ 109   vii ACKNOWLEDGEMENTS  I offer my enduring and extreme gratitude to my supervisor, Prof. W. Erwin Diewert for inspiring me to continue my work in this dissertation, broadening my vision of economics and providing guidance and advice for both my research and career.  I am also greatly grateful to other members of my supervisory committee, Profs. Shinichi Sakata and Hugh Neary for research direction and discussion.  I would like to thank the faculty, staff members and my colleagues at UBC for their valuable suggestions and support.  I owe special thanks to Prof. Alice Nakamura for her helpful comments and editorial advice.  I also thank the seminar participants at the 2006 and 2007 meetings of the Canadian Economics Association for their helpful comments.   viii DEDICATION  To my parents, my husband, and my son.  ix STATEMENT OF CO-AUTHORSHIP  The second and third chapters in this thesis are the joint research with Prof. W. Erwin Diewert.  For the second chapter, the author did virtually all the work, including developing the model, collecting data set and conducting empirical regressions.  For the third chapter, the author cooperated closely with Prof. Diewert on identifying the research question, developing the model and deriving formulas.     1 1 Introduction  In a recent paper presenting an innovative new indicator of technical change, Michelle Alexopoulos and Jon Cohen (2008) wrote that: “Although technical change is central in much of modern economics, our ability to identify empirically the factors that shape its pace, nature, and impact are constrained by data limitations.” The relevance of this statement is underlined by both the current imperative for national economies to achieve better productivity growth and new developments on the international scene in accounting for R&D in the System of National Accounts (SNA).  Although research and development (R&D) expenditures account for only a small portion of GDP, their importance in creating new technologies and promoting productivity growth is widely recognized. This necessitates the re-examination of measurement issues related to these expenditures that have broad implications for economists conducting analyses of the contribution of knowledge capital to endogenous growth and to the nation’s wealth, as well as for policy makers who may want to subsidize R&D investments in order to maximize economic growth.  A widely used definition of R&D is given in the 1993 Frascati Manual (OECD, 1994).1 In this document, R&D is defined as “creative work undertaken on a systematic basis in order to increase the stock of knowledge, including knowledge of man, culture and society, and the use of this stock of knowledge to devise new applications”. Three activities are covered: basic research, applied research, and experimental development. The key criteria that distinguish R&D from the other related activities are “the presence in R&D of an appreciable element of novelty and the resolution of scientific and/or technological uncertainty.”  The SNA 1993, which is still in effect as the international standard, does not treat R&D expenditures as investment. Thus, in the National Income and Product Accounts (NIPA’s) of United States, R&D expenditures are treated as an intermediate input for business and current consumption for the non-profit institutions and government. However, the SNA  1 This version of the Frascati Manual formed the methodological basis for the influential 2001 OECD STI Scoreboard. The version of the Frascati Manual currently in use is 2002 (OECD 2002). See http://library.certh.gr/libfiles/MOBILITY-PORTAL/MON-251-OECD-SCOREBOARD-2001-A92.pdf.  2 2008 that will soon supplant the SNA 1993 does recommend the capitalization of R&D expenditures. Successful R&D investments add to the stock of knowledge, and this stock in turn provides a flow of services over time, rather than in only one period. The generally accepted accounting principle regards current expenses as expenditures that generate revenues for the current period, whereas regards capital expenditures as expenditures that yield benefits for multiple periods. Since the benefits of R&D expenditures usually show up as increased profits (or as lower costs and prices) in future periods, R&D resembles investment more closely than intermediate input or current consumption. Thus, the current treatment is not appropriate and will generally understate current period income and overstate income in future periods.  The recommendation on capitalizing R&D expenditures will have big implications for the SNA. For example, R&D expenditures need to be taken out of primary and intermediate input expenses and capitalized; appropriate deflators for the R&D expenditure categories need to be found so that measures of real R&D effort can be constructed and new surveys will have to be designed to determine the average lengths of service life for R&D projects. Furthermore, the national accounts will no longer be able to proceed on the basis of the competitive model: monopoly profits have to be taken into account.  In order to incorporate the capitalization of R&D expenditures recommendation into the national accounts, many statistical agencies have already taken some actions. Both the Bureau of Economic Analysis of US and Statistics Canada have used their existing satellite accounts in order to illustrate the impact of treating R&D as investment on real GDP growth. Research on the output of R&D, on rates of return to R&D capital, on depreciation rates for R&D capital stocks and on R&D capital stock and productivity growth have been conducted in the US, Canada, Australia, UK and some other countries.  This thesis aims at investigating measurements issues associated with capitalizing R&D expenditures. We look at how to obtain scientific estimates for R&D capital depreciation rates, how to measure the welfare effects of a R&D project, especially of a publicly funded R&D project and how to incorporate the R&D capital stocks into a growth accounting framework.  A limitation of our analysis is that we will not discuss uncertainty about future revenues  3 generated by R&D projects. We also will not discuss spillovers generated by R&D projects.   4 2 Estimation of R&D Depreciation Rates: A Suggested Methodology and a Preliminary Application  2.1 Introduction  The 2008 version of SNA recommends the capitalization of R&D expenditures. The implementation of this idea requires the determination of depreciation rates for R&D capital. Due to the measurement difficulties involved, Nadiri and Prucha (1996) observe that, “researchers doing applied work typically assume an arbitrary depreciation rate of 10 to 15 percent to construct the stock of R&D capital using the perpetual inventory method.” The depreciation rates used by the Bureau of Economic Analysis (BEA) of US in the 2007 R&D Satellite Account were determined by literature survey and rates were industry specific: 18% was used for transportation equipment, 16.5% was used for computers and electronics, 11% was used for chemicals, and 15% was used for all the other industries. Statistics Canada assumed certain rates of depreciation based on the survey of expected service life of R&D. However, it would be preferable to have a scientific method for determining this depreciation rate. Statistics agencies are also interested in exploring possible methods of estimating R&D depreciation rates.  There have been some attempts to empirically measure R&D depreciation rates. Pakes and Shankerman (1981) estimated the rate of depreciation in the private value of patents using European data on patent renewal fees and rates of renewal; Nadiri and Prucha (1996) measured the depreciation rate of R&D stock for the U.S. manufacturing sector using a factor requirements function and restricted cost function, and treating the R&D capital as a “normal” reproducible capital input, as is also the case for Bernstein and Mamuneas (2005) who estimated R&D depreciation rates in an intertemporal cost minimization framework. However, the approaches adopted in previous studies generally do not allow for non-R&D effects that improve the technology, such as knowledge diffusion through education and learning by doing. Thus all of the technological  5 improvement is inappropriately attributed to R&D investments.2 Also, the estimation is typically conducted in a framework of competitive pricing behaviour. However, private R&D investments are often undertaken with the explicit goal of achieving short-run monopolistic advantages over competitors. Thus the assumption of competitive behaviour on output markets is unsuitable. Another weakness with most studies is that R&D investments are treated similarly to investments in ordinary physical capital, but R&D investments are quite different in their effects, as is explained subsequently.  In this paper, we propose to treat the stock of R&D capital not as an explicit input factor; instead we define the stock of R&D capital to be the technology index that locates the economy’s production frontier. An increase in the stock of R&D shifts the production frontier outwards. In our model, the R&D capital depreciation rate is estimated within a monopolistic competition framework using gross R&D investment data. We estimate R&D depreciation rates for the U.S. total manufacturing sector and four U.S. knowledge intensive industries: chemical products (SIC 28), non-electrical machinery (SIC 35), electrical products (SIC 36) and transportation equipment (SIC 37).  The rest of the paper is organized as follows. Section 2.2 explains the construction of the R&D stock and the reasons for R&D depreciation. Section 2.3 develops our basic model for estimating the rate of depreciation for R&D capital. Section 2.4 presents the estimation methodology and results. Section 2.5 concludes. Appendix A shows the derivation of the estimating equations. Different sets of break points used in the linear spline model and quadratic spline model are given in Appendix B.  2.2 Construction of the R&D Stock  According to the 1993 Frascati Manual definition, the objective of conducting R&D is to increase the stock of knowledge. Thus we define R&D capital as the knowledge asset created by R&D investment. Hence the stock of R&D capital can be regarded as a proxy for society’s technological level. R&D investments act as a mechanism for shifting outward society’s production possibility frontier. This treatment is the major distinguishing feature of our approach. The studies of others treat R&D capital as an explicit input in a manner that is similar to the treatment of ordinary physical capital.  2 The work of Bernstein and Mamuneas (2005) is not subject to this criticism.  6  2.2.1 R&D Capital and Ordinary Physical Capital  Physical capital like machinery, equipment and buildings wears out through use and the efficiency tends to decline over time. Successful R&D ventures create new knowledge for the firms conducting the R&D. This new knowledge can be either a new cost-saving process, or a new technology for an innovative or quality-improved product. R&D capital does not wear out because of its utilization and its absolute efficiency level does not change with the passage of time. However, R&D capital is subject to a type of “depreciation” because processes or products can become obsolete over time.  Although physical capital and R&D capital are both called “capital assets,” there are some fundamental differences. First, physical capital, such as machines and equipment, can be reproduced over multiple periods. With physical capital, reproducibility makes it possible for us to observe at the same point in time rental prices of different vintages, and also used asset prices of different vintages, of a capital asset, and this in turn allows us to estimate depreciation rates. In contrast, for R&D capital, once, say, a new blueprint has been produced, it can be made available to many economic units without further productive activity. Typically, we cannot collect price information for different vintages of R&D capital at the same point in time3. Secondly, physical capital and R&D capital support production in different ways. As Pitzer (2004) pointed out, R&D capital acts as if it were producing “recipes” while physical capital is regarded as one of the “productive” inputs4 that are “consumed” during a production process. Thirdly, the reasons for depreciation are different for these two types of capital assets. We return to this issue subsequently. Because of these differences, we believe that the treatment of R&D assets should necessarily be different from the treatment of reproducible capital assets.  3 However, we can sometime observe the market price for the rights to some new technology. If the same technology is again sold in a future period, then we could collect price information for different periods and infer a depreciation rate for the R&D investment. 4 Pitzer (2004) defines “productive input” as “Fundamental to production is the notion that inputs are proportional in some sense to outputs. Outputs are created by combining a particular collection of inputs in a particular manner. … If more outputs are desired, then more inputs are necessary, and usually, more of all inputs. It may not be necessary to double the inputs to double the outputs, but more inputs are necessary to produce more outputs.”  7  2.2.2 Constructing the Stock of R&D Capital  Lacking a good measure of R&D output, we use input information in the form of gross (real) R&D expenditures as a proxy measure. Because all the technologies, new or old, are created by R&D investments, the R&D stock can be written as a function of a series of past R&D investments. Thus, the R&D stock at the beginning of period t is defined as follows:  (2.1) ...5544332211 +++++= −−−−− ttttttttttt IIIIIR θθθθθ , where tnθ  is the period t efficiency index representing how much the R&D investment that was made n periods before period t contributes to the technology or knowledge stock in period t. The R&D lags and weights are all incorporated in these efficiency indexes. ntI −  is the R&D investment made in year t-n. We expect that the further back the R&D investment is made, the less it contributes to the prevailing knowledge. Thus the following relationships should hold among all the efficiency indexes5:  (2.2) ...54321 ttttt θθθθθ ≥≥≥≥  We add the superscript t because these efficiency indexes may vary with time. The speed of the technology upgrading is one factor that helps to determine the size of the efficiency indexes. If the new technology is created quickly, the past investment may become irrelevant at a fast pace with the newly updated-knowledge. For example, if there were more new knowledge created in year t compared to year t-1, the following inequalities  5 Another way to justify inequality (2.2) is follows. Suppose that the industry or firm aggregate output is not homogeneous but consists of a mix of new and old products. Over time, industry output shifts away from the older more obsolete products and towards the newer improved products. Hence the use of the old technology that produces the older obsolete products diminishes over time and the initial good effects of investments in technological improvements that were made many periods ago gradually wither away. Thus we have a justification for why the initial good efficiency effects of R&D investments diminish or depreciate over time. Note that all the output discussed in this paper is the aggregate output.  8 might hold:  (2.3) ... ;        ;           ;   and    144 1 33 1 22 1 11 −−−− <<<> tttttttt θθθθθθθθ These inequalities show us that the previous investments become less important at a faster speed as the technology improvement speeds up. The series of R&D stocks can be written as:  (2.4) 0 2 21 2 35 2 34 2 23 2 12 0 1 11 1 24 1 33 1 22 1 11 011332211 ... ... ... RIIIIR RIIIIR RIIIIR t t t tt t t t t t t t t t tt t t t t t t t t t tt t t t t t t − − − −− − − − − − − − − − −− − − − − − − −−−− +++++= +++++= +++++= θθθθθ θθθθθ θθθθθ  0R  is the initial knowledge stock. The efficiency index varies with R&D investment and technology updating frequency. To simplify our analysis, we assume that the efficiency indexes decline at a constant geometric rate; that is, we assume that the following relationships hold among the efficiency indexes: (2.5) 10        and             )1( 1 ≤≤−= − δ  nn δθ . Based on the above simplifications, the stock of R&D capital can be constructed as follows: (2.6) 11 )1( −− −+= ttt RIR δ , where δ  can be regarded as the R&D depreciation rate, which is assumed to be constant over time. From (2.6), we see that the stock of R&D capital in period t is constructed from the previous R&D investment ( 1tI − ) and the depreciated R&D stock of period t-1. This is a widely used method to construct capital stocks. However, it can be seen that we need a restrictive assumption about the efficiency index to end up with this simple equation. According to this equation, R&D capital accumulation depends on two opposite forces: the addition of the new knowledge stock, which is created by the current period R&D investments,6 and the depreciation of the old knowledge stock.  We use the stock of R&D capital as a technology index, indicating the position of the production frontier. If the newly created knowledge stock is at least as large as the  6 Equation (2.6) implies that one unit of R&D investment can create one unit knowledge. This can be regarded as the simplest functional form for a knowledge production function.  9 depreciation of the old knowledge stock, we will not have a backward shifting of the production frontier.  It is sometimes argued that the R&D depreciation rate is zero, since old knowledge is preserved. However, new innovations tend to make the innovations from pervious periods obsolete. Also, consumer preferences can shift over time, causing a drop in the demand for “old” products, and leading to obsolescence of past R&D investments. Hence the older R&D investments do suffer depreciation. Indeed, it is possible that the depreciation of old technologies outweighs the incremental effects of the new technologies and hence results in a net decrease of R&D capital stock.  The following figure shows the R&D investments in the US aggregate manufacturing sector and four knowledge intensive U.S. industries over the period 1953-2000:  Figure 2.1 R&D Investments (1953-2000) R&D Investments 0 2 4 6 8 10 12 14 16 18 195 3 195 6 195 9 196 2 196 5 196 8 197 1 197 4 197 7 198 0 198 3 198 6 198 9 199 2 199 5 199 8 SIC 28 SIC 35 SIC 36 SIC 37 Manuf   R&D investments in the Chemical Products (SIC 28) and the Non-electrical Machinery (SIC 35) categories have increased smoothly over the past 40 years, while R&D investments in Transportation Equipment (SIC 37) have fluctuated more. These different  10 trends in the R&D investments imply different frequencies for creating new technology and result in different depreciation rates for the four industries.  In the rest of this section, we discuss the reasons for R&D capital depreciation in more detail.  2.2.3 Reasons for R&D Capital Depreciation Two forces including obsolescence and deterioration can lead to the depreciation of a tangible asset. Generally speaking, we can use information on the price of a used asset to estimate its depreciation (provided that it is not a uniquely constructed tangible asset). If we treat R&D expenditures as investment, there exists a similar problem -- the decline in the utilization and efficiency of the knowledge capital. Although there is no apparent wear or tear of the intangible asset, we cannot assume that the knowledge capital has an infinite service life. In other words, for an intangible asset, it is not the physical deterioration of an asset but the obsolescence of a technology that results in depreciation.  Both price and quantity changes are responsible for changes in the value of R&D. R&D capital depreciation, under our investigation, means the real quantity change due to the change in efficiency and utilization of the knowledge asset. The relative efficiency of the knowledge capital will decline over time due to the following reasons:  • The obsolescence of the technology. When newer technologies are created by R&D investment, the old technology may be partly or entirely replaced by the newly created technologies and consequently, the relative efficiency and the utilization of the old knowledge would decline. • The changes in consumers’ preferences. Consumer’s tastes may change for different reasons, such as the implementation of new health restrictions, the emergence of new products, and changes in the consumer’s capabilities. Changing tastes may shift away the demand for some products that heavily rely on older technologies, and cause related market shrinkage. Responding to the shifting demand, firms would reduce the utilization level of the older technology that produces the outmoded products.  However, because there are very few observed market prices for old technologies, we cannot observe the depreciation on R&D capital as we can for a tangible asset that trades  11 on second hand markets. In the following section, we will set up a framework for estimating the R&D capital depreciation rate. Because of the specific feature of R&D capital depreciation, the estimated depreciation rates can be also interpreted as the “obsolescence rates” of the R&D capital. One more thing we want to point out here is that in the accounting literature both depreciation and amortization are methods that are used to allocate the cost of a specific type of asset to the asset's life. Usually, depreciation refers to distributing a tangible’s cost over that asset’s service life. Amortization, on the other hand, refers to the cost allocation of an intangible asset. In this paper, we use depreciation for both tangible asset and intangible asset.7  2.3 The Estimation Framework  We use a basic production function methodology as our starting point to estimate the depreciation rate of R&D capital. Each industry is treated as facing a monopolistic competition environment. The R&D stock is treated as a technology index for the position of the production frontier. We use an extension of a model due to Diewert and Lawrence (2005).  2.3.1 The Basic Framework  We allow for the possibility of increasing returns to scale in the industry and hence the assumption of competitive profit maximizing behaviour is not suitable for the modelling of the industry’s behaviour. In stead, we treat the industry as engaging in monopolistic profit maximization.  We assume that each industry has an aggregate production function f, with the form of ),,( tRxfy ttt = , so that f is a function that depends on the usual input vector x, the R&D stock R, and the time variable t that represents non R&D sources of technical change for  7 In Canada, depreciation and amortization can be used interchangeably to refer both tangible and intangible asset.  12 the production function.8 Thus both R and t shift the production function over time. Defining the production function in this way, we can avoid the overestimation of the effects of R&D capital on technological improvement, compared to production functions that have only an R&D variable R as a shift variable.  The aggregate demand function for the output of an industry (or sector such as manufacturing) in year t is represented by an inverse demand function: ),( tyPp tt = . Under this situation, each industry solves the following monopolistic profit maximization problem at each period by choosing inputs and the next period’s technological level: (2.8) t 1 , , x , 0 , 1 1   { ( , ) } subject to:   ( , , )   and     (1 ) ,         t 0,1,2.... t t t t t t t R t R t R t t t t t R t t P y t y w x P I y f x R t R I R Max β δ + ∞ = − − − ⋅ − = = + − = ∑  where tβ  is the period t discount factor, tw  is an input price vector, tRI ,  is R&D investment in period t, and tRP ,  is the corresponding price index. In this model, we assume that each industry maximizes the discounted future monopolistic profits with full information about future prices. Ignoring the uncertainty of future prices is not realistic, but it dramatically simplifies the problem.  The first order condition for the above maximization problem is: (2.9) 0,1,...T,         t,),,(]/),([),,( ==∇∂∂+∇ tttxttttxt wtRxfyytyPtRxfp (2.10) tRttR t t t tt PPR fp R f y Py ,1, 1 1 1 11 )1( )()()( βδβ = ⎭⎬ ⎫ ⎩⎨ ⎧ −+∂ ⋅∂+∂ ⋅∂ ∂ ⋅∂ + + + + ++ , where tp  is the output price and ),,( tRxf ttx∇  is the gradient vector of the production function with respect to the inputs x. Factoring tp  and ),,( tRxf ttx∇  out on the left hand side of (2.9), we obtain the following simplification of equation (2.9): (2.11) tttx tt tt t wtRxfyp ytyPp =∇×∂∂+× ),,() / /),(1( .  8 These non R&D sources of technical progress could include learning by doing effects, freely available research, information on new technologies made available at trade fairs and so on.  13 Applying similar algebraic rearrangements to equation (2.10), we have: (2.12) 1,, 11 ,11 11 11 1 )1( )1,( / /)( 1 + ++ ++ ++ ++ + −−=∂ +∂×⎥⎦ ⎤⎢⎣ ⎡ ∂⋅∂+× tRtr t t t tt tt tt t PPR tRxf yp yP p δβ β , where ttt r+=+ 1)/( 1ββ , and where tr  is the nominal interest rate prevailing at time t. The term yp yP / /)( ∂⋅∂  is the inverse of the price elasticity of demand, and reflects how output (demand) changes with respect to the price change. If we use ε to denote the price elasticity, we can define the inverse as the period t nonnegative markup, denoted by tm , that is: (2.13) 01 / /),( ≥−=∂∂−≡ εtt t t yp ytyPm and the markup factor tM  can be defined as follows: (2.14) tt t tt yp ytyP mM / /),( 1111 ∂∂+=+=−= ε If we assume that the markup factor is constant over time, then we can rewrite (2.11) and (2.12) as: (2.15) ]/),,([, ntttnt xtRxfMpw ∂∂××= ,   n=1,2,…,N, and (2.16) ]/)1,,([)1()1( 11111,, +++++ ∂+∂××=−−+ tttttRtRt RtRxfMpPPr δ , where n denotes the n-th factor in the input vector x. Details on derivation of these equations are given in appendix A.  The left hand side of equation (2.16) is the user cost of one unit of R&D investment purchased in period t9. Equations (2.15) and (2.16) form our system of estimating equations. Including equation (2.16) as an extra estimating equation is helpful for  9 It may be a be surprising initially that the net effect of purchasing an R&D investment in period t can be expressed in such a simple manner as is given in equation (2.16).  However, under our perfect foresight assumptions, the producer needs to purchase units of R&D in period t in order to adjust the stock of R&D to precisely the “right” level in period t+1; the R&D stock for period t+2 can be adjusted to the “right” level by purchasing additional units of R&D in period t+1 and so on.  14 distinguishing R and t. However we may also introduce some estimation problems by using anticipated variables in this equation, where the anticipations are formed in period t when we purchase tRI ,  at the price tRP , . To simplify our analysis, we use the actual data at period t+1 to approximate the predicted variables.  The left hand side user cost in (2.16) depends on the depreciation rate δ. In order to compare log likelihoods across alternative depreciation models, we need the left hand side variable to be constant across models. Thus we rewrite equation (2.16) as:  (2.16a) ]/)1,,([)1()1( 11111,, +++++ ∂+∂××+−=+ tttttRtRt RtRxfMpPPr δ .  Equation (2.16a) is still not ideal for econometric estimation because it involves a lagged dependent variable. However, if we divide both sides of equation (2.16a) by tRP , , this leads to the following estimating equation: (2.16b) ]/)1,,([)/()/)(1(1 111,1,1, +++++ ∂+∂××+−=+ ttttRttRtRt RtRxfMPpPPr δ From equations (2.15), we see that the role assumed for R&D capital in the production function determines our econometric estimation system. If we treat R&D capital as an ordinary input, we obtain a four inputs, one output aggregate production function model for each industry. However if we believe that R&D capital acts as a technology shifter, there will be only three input variables in (2.15), and R&D capital is treated similarly as the time variable in that the R&D stock shifts the technology like other sources of productivity improvement (proxied by the time variable).10  We treat R&D capital as a technology index that indicates the position of production frontier. Hence, in our model, we have labour, intermediate, and non-R&D capital service inputs. In order to identify some parameters in the model, we add the production function to the estimating system. Thus, our final estimating system includes the following five equations:   10 However, the R&D variable is different from the time variable because we regard the time effect as being entirely exogenous in our model whereas the R&D stock is endogenously determined by producers.  15 (2.17) 1, 1, ),,( t tt t t x tRxfM p w ∂ ∂= (2.18) 2, 2, ),,( t tt t t x tRxfM p w ∂ ∂= (2.19) 3, 3, ),,( t tt t t x tRxfM p w ∂ ∂= (2.20) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ +∂××⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛−=+ + ++++ 1 11 , 1 , 1, )1,,()1(1 t tt tR t tR tR t R tRxfM P p P P r δ (2.21) ),,( tRxfy tt= .  2.3.2 The Choice of the Functional Form for the Production Function  To specify estimating equations, we need to choose a functional form for the production function. As a starting point, we use the following variant of a normalized quadratic functional form: (2.22) )/()2/1( ),,( 33221121 332211332211332211 xxxSxxRete RxhRxhRxhtxgtxgtxgxcxcxcbtRxf T t tttt φφφ ++−++ +++++++++≡  where 1x  is the labour input, 2x  is the intermediate input, 3x  is the non-R&D capital input, and tR  is the stock of R&D capital. In addition, ]s[S ij=  is a 3 by 3 symmetric positive semi-definite substitution matrix of unknown parameters and the iφ is predetermined positive parameter. In our empirical work, we calculate the sample mean of the ix , called as * ix , and then set iφ  equal to )/( *3*2*1* xxxxi ++ .  This production function is a flexible functional form. The quadratic term in the function adds enough parameters to enable the functional form to approximate the second derivative of an arbitrary twice continuously differentiable functional form. The purpose  16 of dividing the quadratic term SxxT  by a normalizing factor, xTφ , is to make the above production function linearly homogeneous. With 1e  and 2e  equal to zero, the unknown parameter b determines the degree of returns to scale: if b = 0, we have constant returns to scale in production; if b is less than 0, then there are increasing returns to scale; and if b is greater than 0, there are decreasing returns to scale. The two parameters 1e and 2e  are technical progress parameters.  In order to identify all of the parameters and to reduce multi-collinearity, it is necessary to impose some linear restrictions on the matrix S. Our linear restrictions are as follows11:  (2.23) ∑ = =3 1 0j njs ,      n = 1, 2, 3.  The normalized quadratic production function defined by (2.22) and (2.23), with the parameters b, 1e , and 2e  equal to 0, is flexible in the class of constant return to scale production functions. The additional parameter b allows us to test the degree of returns to scale locally. The main advantage of choosing this flexible functional form is that the flexibility properties would not be destroyed by imposing curvature conditions; see Diewert and Wales (1987). In our example, imposing positive semi-definiteness conditions on the symmetric matrix S means that we can write S in term of the following matrix product: (2.24) TUUS = , where ][ ijuU ≡  is a 3 by 3 lower triangular matrix and UT is the transpose of U. The linear restrictions (2.23) on S can be imposed on U too; that is, we have:  11  Consider the functional form xSxxxCxg TTT φ/)( +≡  where S = ST. Set ])[2/1( TT ccAS φφ ++=  where A = AT. Then )/(/ xcxAxxxSxx TTTTT += φφ  and hence all of the parameters in g(x) would not be identified.  In order to prevent this exact multicollinearity problem, we impose the restrictions (2.23). Diewert and Wales (1987) show that these restrictions do not destroy the flexibility of the functional form.  17 (2.25) .0  and  ,0  ,0 333222312111 ==+=++ uuuuuu  We can find that there are only three independent parameters in the U matrix: 21u , 31u and 32u . The main diagonal parameters iiu  can be represented in terms of the off diagonal parameters iju .  Partially differentiating the production function defined by equation (2.22) with respect to inputs, 1x , 2x , and 3x , and to the next period’s R&D stock, 1tR + , and substituting the resulting derivatives into the estimating equations (2.17) to (2.20), we can rewrite the estimating equations as: (2.26) ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ×+−++= ∑ = 2 3 1 , , )(2 1/ t T t T t n t T j jtnj nnntnt x Sxx x xs rhtgcMpw φφφ  for n = 1, 2, 3, and (2.27) { } tR tR tR t t P P exhxhxhM P pr , 1, 2332211 , 1 )1(1 ++ −++++=+ δ where ∑ =≡ 3 1 ,j jtjtT xx φφ .  2.3.3 The Problem of Trending Elasticities  Diewert and Lawrence (2002) point out that, for the normalized quadratic functional form, the estimated elasticities often have strong trends when there are strong trends in the price and quantity data. They also suggest one way to solve this problem. We adopt their technique here.  In the initial functional form, the substitution matrix S is constant over time. To handle the trending elasticity problem, we let the production function be flexible at two sample points, which means the matrix S is allowed to change over time. We use the following weighted average substitution matrix:  18 (2.28) BTtATtS ×+×−= )/())/(1( ,  t = 0, 1, 2,…,T.  T is the total periods covered by the estimation data sample. We have data from 1953-2000, so T = 47. Using this weighted average substitution matrix, the technological progress captured by the time variable not only affects the constant terms in the estimating system, but also the substitution possibilities.  As in the basic case, we can impose the curvature conditions by setting A and B equal to the product of TUU  and TVV , where U and V are lower triangular matrices; that is, we set: (2.29) TT VVBUUA ==       and     .  Similarly, we can impose the following normalizations on these two matrices U and V: (2.30) 3333 01  and  01 == TT VU , where 31  and 30  are 3-dimensional vectors of 1’s and 0’s, respectively. With these constraints, we only add three additional independent parameters to the initial model; these new parameters are 323121 v and ,v,v . Making these changes, our production function can be written as follows: (2.31) )./(])/())/(1[()2/1( ),,( 21 t T t TTT t ttt T t T t T tt xxVVTtUUTtx ReteRxhtxgxcbtRxf φ×+×−− +++++≡ Here we write the production function in matrix form in order to simplify our notation. In equation (2.31), ),,( 321 cccc T = , ),,( 321 gggg T =  and ),,( 321 hhhhT = . In the estimation, if the trending elasticity problem exists, then we will see that the value of the likelihood function increases significantly as we add the parameters in the V matrix to those in the U matrix. We do see this for our results.    19 2.3.4 Problems due to Non-Smooth Technical Progress  Another problem related to the initial production function model is that it does not allow for non-smooth change in technical progress. We now add more features to the model in order to capture changes in the direction of technical progress over time.  Probably technological progress does not proceed smoothly. Thus, we add linear splines or quadratic splines in the time variable to allow the different change patterns of technological progress at different periods. The modified production function can be written as follows: (2.32) ),/(])47/())47/(1[()2/1( )()(),,( 21, 3 1 t T t TTT t ttt T tjj jt T tt xxVVtUUtx ReteRxhxtgxcbtRxf φ×+×−− +++++≡ ∑ = where )(1 te  and )(tg j  are linear spline functions of time t. The number of splines depends on the break points chosen by investigating the plots of preliminary estimations. A break point is a positive integer less than the maximum number of the time variable; so here, it is less than 47. We will illustrate how to define )(1 te  if we choose three break points, 470 321 <<<< ttt : (2.33) 47,...2,1tfor t       )()()( ,...2,1tfor t                              )()( ,...2,1tfor t                                                     )( 0,1,2,...tfor t                                                                           )( 3331423131212111 3222131212111 211112111 1111 ++=−+−+−+= ++=−+−+= ++=−+= =≡ tttettettete ttttettete ttttete tete   In equation (2.33), the je1  is the unknown parameter to be estimated. From the above example, we know that with n break points, there are n+1 parameters to be estimated. Similarly, we can generate linear splines for the function )(tg j . The subscript j means we allow different splines for different inputs j.  Adding linear spline induces, perhaps artificially, kinks in the direction of technical  20 change. For smooth change, the linear splines can be replaced by quadratic splines,12 in which case )(1 te  and )(tg j  are quadratic spline functions. With three break points, the quadratic spline functions can be defined as: (2.34) 2 315314213112113 2 23142131121123 2 12131121112 2 112111 3 2 315312311 2 214213112112 2 12131121112 2 112111 32 2 214212211 2 113112111 2 112111 21 2 113111111 1 2 121111 )()2/1())(( )()2/1())(( )()2/1())(()2/1( 47  t          )()2/1()/)()(()()( )()2/1())(( )()2/1())(()2/1(   t          )()2/1()/)()(()()( )()2/1())(()2/1(   t          )()2/1()/)()(()()( t    t                                        )2/1()()( tteteteteett tteteteett tteteetttete tttettetttete tteteteett tteteetttete ttttettetttete tteteetttete ttttettetttete tetetete ccd bbc aab a −++++−+ −+++−+ −++−++= ≤<−+∂∂−+== −+++−+ −++−++= ≤<−+∂∂−+== −++−++= ≤<−+∂∂−+== ≤+==   As in the linear spline case, the je1  is the unknown parameter to be estimated. If we choose n break points, then there are n+2 additional parameters that need to be estimated for each equation. Adding splines increases the flexibility of the functional form but at the cost of estimating more technical change parameters. In addition, choosing different break points will result in different estimates.  The following features distinguish our model from the previous literature:  • Instead of treating R&D capital as one explicit factor input like ordinary physical capital, we treat R&D capital as a technological index indicating the position of the production frontier. R&D capital, which is the knowledge asset created by the R&D investment, is not “consumed” like the physical capital in the production; it just indicates the technology level. Holding all the usual flow inputs constant, an increase of the stock of R&D capital would shift the production frontier outwards. Thus the R&D stock variable is treated in a manner similar to the time variable.  12 On setting up quadratic splines in a normalized quadratic model, see Diewert and Wales (1992). Normalized quadratic cost functions are modeled there whereas we are modeling normalized quadratic production functions.  21 • Both the time variable and R&D capital stock variable are present in the production function model. The R&D stock frequently grows in a roughly linear fashion, so the inclusion of the variables R and t in the regression equations can lead to a multi-collinearity problem. However, if the time variable, t, is dropped from the model, then the R variable becomes the only technical change shift variable, and frequently, the resulting rate of return to R&D investments is unrealistically large. Therefore, including both the time variable and the R&D variable in the model, we will generally not attribute all of the technological progress to R&D investments, and avoid overstatement of the effects of R&D investments on both technological progress and on productivity growth to some extent. • The model allows for the possibility of monopolistically competitive behaviour that is consistent with increasing return to scale.13 Thus our model estimates the markup factor and the degree of returns to scale along with the R&D depreciation rate. Because of these different features, different results are to be expected.  2.4 Empirical Estimation and Results  Here we describe our data and empirical results. We estimate R&D depreciation rates for the period of 1953-2000 for U.S. manufacturing and four U.S. technology intensive industries: chemical and allied products (SIC 28), non-electrical machinery (SIC 35), electrical products (SIC 36) and transportation equipment (SIC 37). In 1998, the R&D expenditures of these four industries accounted for 54.35% of the R&D expenditures of all industries and 76.37% of the manufacturing R&D expenditures.  2.4.1 Estimation Methodology The basic estimation system with curvature conditions (2.24) and linear restrictions (2.25) is given by equations (2.22), (2.26) and (2.27). To specify these estimating equations, we must define the normalized quantities and the differences for the normalized quantities. The nth normalized quantity, nq , is:  13 If the estimated markup factor M turns out to equal 1, then we have competitive behavior.  22 (2.35) x xq T n n φ=     n = 1, 2, 3 The differences between the normalized quantities can be defined in the following way: (2.36) 1221 qqq −= , 1331 qqq −= , and 2332 qqq −= .  Using the above definitions and substituting restrictions (2.24) and (2.25) into equation (2.26), we can express the first order necessary conditions for profit maximization with respect to the choice of the different inputs in the following way: (2.37) ⎭⎬ ⎫ ⎩⎨ ⎧ +++ +++++×= ])()[(5.0 ))(( / 2 3232 2 313121211 313121213121111 1, quququ ququuuRhtgc Mpw tt φ , (2.38) ⎪⎭ ⎪⎬⎫⎪⎩ ⎪⎨⎧ +++ ++−++×= ])()[(5.0 )()( / 2 3232 2 313121212 32 2 323131212121222 2, quququ quququuRhtgc Mpw tt φ , and (2.39) . ])()[(5.0 )()( / 2 3232 2 313121213 32 2 323131212131333 3, ⎪⎭ ⎪⎬⎫⎪⎩ ⎪⎨⎧ +++ −+−++×= quququ quququuRhtgc Mpw tt φ The production function is: (2.40) ].)())[((5.0 23232 2 3131212133221121 332211332211332211 quququxxxRete RxhRxhRxhtxgtxgtxgxcxcxcby ++++−++ +++++++++= φφφ The above four equations plus (2.27) form our basic estimating system with 16 parameters to be estimated.  Due to the non-linearity of the equations, we use non-linear maximum likelihood estimation option in the SHAZAM. To find the estimates for the R&D depreciation rates, we construct a grid of depreciation rates: δ = 0, δ = 0.01, δ = 0.02, δ = 0.03, …and δ = 1. Based on these depreciation rates, we build the initial stock of R&D capital using the following formula: (2.41) )/(0,0 RRIR γδ += ,     δ = 0, 0.01, 0.02, 0.03, 0.04,…1 where 0,RI  denotes the constant R&D investment at the first period, and Rγ  denotes the geometric growth rate of R&D investment over the sample period and can be calculated as: (2.42) )47/1(0,47, )/( RRR II=γ .  23  For the remaining periods, the R&D stock is calculated using equation (2.6) in the second section.  All together, we have 101 sets of R&D stocks, corresponding to the 101 possible choices for an R&D depreciation rate. Using these alternative R&D stock series, we can estimate the five equations. For each depreciation rate, we obtain the value of the log-likelihood function. Comparing these values of the log-likelihood function, we can locate the depreciation rate corresponding to the maximum value of the likelihood function. According to our estimating procedure, we believe that the depreciation rate that maximizes the value of log-likelihood is the best estimator for the R&D depreciation rate.  2.4.2 Data Construction  To conduct the estimation, we need quantity series and price series for industrial input and output, and price and quantity series for R&D investments. Industrial input and output data other than R&D related data are obtained from the Multifactor Productivity data sets provided by Bureau of Labour Statistics (BLS). R&D related data are derived from the website of National Science Foundation (NSF).  From the BLS, we obtained value series in current dollar and price index series in 1996 constant dollar14. For each industry, we have data on sectoral output, labour input (L), capital service input (K), energy input (E), non-energy materials (M) input, and purchased business services (S) input. The BLS uses the Törnqvist index number formula to construct the aggregate data. Labour is measured as the hours worked by all persons engaged in a sector. Capital input is defined as the flow of services from physical assets, which include equipment, structures, inventories and land. Service flows are assumed to be proportional to stocks. The description of the measures and the methodology for constructing all of these data sets are given in Chapter 10 and 11 of “BLS Hand Book of Methods”, and Gullickson and Harper (1987). Our measure of intermediate input is a Törnqvist aggregate of energy, material and purchased services. With value series and price series, we can construct the implicit quantity series. Industrial input and output data  14 We thank Mike Harper for providing us with manufacturing data not available on the BLS website.  24 sets are relatively well constructed over the R&D data sets, but we face a double counting problem when we try to explicitly model the role of R&D capital because R&D expenditures have already been included in the initial (BLS) input data.  The industrial R&D expenditure data are obtained from the website of NSF. For the years 1953-1998, the data are taken from the Industrial R&D Information System (IRIS) that uses the Standard Industrial Classification (SIC) to classify the industry. For other years, the data are obtained from “Research and Development in Industry”, for the year 2000 and 2001. These two publications use the North American Industrial Classification System (NAICS). We use the 1998 data as the bridge to link the series based on the different industrial classification and reclassify the data to conform to the SIC. Using SyearRI ,  to denote the data based on SIC and NyearRI ,  to denote the data based on NAICS, the constructed data we need for estimation can be obtained by using the following sort of formula: (2.43) 1998, 1998, 1999,1999, NR SR NRSR I I II ×= .  Equation (2.43) is for 1999. Similar adjustments can be made for 2000 and 2001.  The data on the cost components of R&D expenditures come from various issues of Research and Development in Industry. According to the NSF’s classification, the type of R&D expenditure includes: wage and salaries, materials, R&D depreciation, and other costs. There are two important problems associated with this data set: one is that the NSF does not use the above classification consistently over years; another problem is that data are not available for quite a few years. To deal with these problems, we must make assumptions and create approximate data. For example, for the years 1953 to 1961, we do not have data related to the type of cost. Thus, we assume that the cost structure for these years was the same as that in 1962. Similarly, for the period 1977 to 1997, we have data every two years. To obtain data for the missing years, we use moving average methods to determine the missing data. Finally, we group the R&D expenditures into three categories: Wage and Salaries (labour), Materials (intermediates) and Capital expenditure. Unfortunately, the cost category, overhead or other costs, accounts for a big portion of the expenditure. We allocate these expenditures to wage and salaries, materials and capital expenditure according to the BLS industry cost shares. From the above description, it  25 should be clear that assumptions made to fill in gaps in the R&D data can have a direct effect on the quality of the resulting data sets and on the results of analyses based on these data sets.  After constructing R&D cost component information, we can make adjustments to the initial BLS input data sets. R&D labour cost, wage and salaries, is subtracted from the total labour cost; the material component of R&D expenditure is subtracted from the total intermediate input cost; and the capital expenditure part of R&D is subtracted from the total capital service cost. Dividing these adjusted value series by an implicit price index, yields quantity series for labour, intermediate and capital service input. The quantity series of R&D investment is constructed by using the Törnqvist index formula. Price indexes for the three components of R&D expenditure are assumed to be same as industrial input price indexes.  Finally, we need nominal interest rates for the year 1953 to 2000 to construct the discount factors. Nominal interest rates are obtained from the on-line data of Federal Reserve System. We choose the long-term nominal interest rate: market yield on U.S. Treasury securities at 10-year constant maturities, quoted on an investment basis, to construct the series of discount factors.  2.4.3 Estimation Results  Maximum likelihood estimation, which is the non-linear option on SHAZAM, is sensitive to the choice of starting point, which also affects the number of iterations. We start from a simple regression with M, b, e1 and e2 in equation (2.27) and equations (2.36) to (2.40) set equal to zero initially. The estimated parameters from this regression are used as the starting point for the next regression, which adds additional parameters. As we proceed, we also check for “big jumps” in the log likelihood of the model (if the jumps are small, then the inclusion of the extra parameters is not warranted but in general, we obtain significant increases in the log likelihood as we add the extra parameters). We estimate the following four models: Model I: This is the basic model defined by equation (2.27) and equations (2.36) to (2.40). Without splines and without a weighted substitution matrix, this model may not properly reflect real world complexity. We expected relatively low values  26 of the log-likelihood for this class of models. Model II: This model adds a weighted substitution matrix to equations (2.36) to (2.40). It can deal with the possible trending elasticity problem. If our model does have this trending elasticity problem, we would expect to see a big increase in the value of the log-likelihood function. Model III: This model adds linear splines in the time variable based on the last model. Non-smooth change in technical progress can be captured by adding these splines. Model IV: This model adds quadratic splines for the time variable to Model II (rather than linear splines as in Model III).  The following table lists the estimates of the depreciation rate, the value of maximum log-likelihood, and the value of markup factor for the above 4 models for U.S. manufacturing and the four selected knowledge intensive industries.15  15 The break points for the spline models are reported in Appendix B.  27  Table 2.1 Depreciation Rates Maximizing the Log-likelihood Function   SIC 28 SIC 35 SIC 36 SIC 37 Manufactur ing Dep Rate 0 0.09 0 0.15 0.5 Markup Factor 0.92508 0. 92223 0.96331 0.97694 0.95618 Model 1 Log-likelihood 266.115 208.271 113.905 306.464 373.561 Dep Rate 0 0.06 0.18 0.12 0.49 Markup Factor 0.95645 0.90465 0.92672 0.98417 0.98372 Model 2 Log-likelihood 280.6107 221.700 148.229 308.473 394.441 Dep Rate 0.01 0.03** 0.07 0.06 0.26 Markup Factor 0.97094 0.84863 0.96381 0.90395 0.98041 Model 3(a) Log-likelihood 400.934 343.057 280.239 416.204 530.168 Dep Rate 0.02 0 0.12 0.27** 0.08 Markup Factor 0.985 0.9439 0.9087 0.96251 0.97252 Model 3(b) Log-likelihood 395.727 400.331 273.373 407.415 532.827 Dep Rate 0.01** 0 0.09 0.22 0.29** Markup Factor 0.96872 0.93698 0.86455 1.0087 0.94723 Model 3(c) Log-likelihood 395.907 395.578 236.828 413.723 549.806 Dep Rate 0 0 0.14** 0.34 0.32 Markup Factor 1.0124 1.1241 0.91897 0.88006 0.93338 Model 4(a) Log-likelihood 378.795 386.034 334.547 406.451 541.723 Dep Rate 0 0 0.1 0.3 0.38 Markup Factor 1. 0179 1.0955 0.93431 0.88437 0.93532 Model 4(b) Log-likelihood 380.397 395.822 285.808 381.842 539.3563 Dep Rate 0 0 0.05 0.32 0.28 Markup Factor 1.006 1.0898 0.94743 0.88176 0.94824 Model 4(c) Log-likelihood 379.044 397.954 271.822 388.501 544.687  From the above table, we can see that the value of the maximum log-likelihood generally improves considerably from Model I to Model II. This suggests we have trending elasticity problems. With linear splines or quadratic splines added to the model, the value  28 of maximum log-likelihood function increases dramatically. In comparison with the results obtained from the linear spline model (Model III) and the quadratic spline model (Model IV), the value of the log-likelihood function increases in Model IV for Electric Products, and decreases for Chemical Products and Transportation Equipment. For Non-electric Machinery and the manufacturing sector, the changes of the log-likelihood value from Model III to Model IV are mixed. We also try different break points for Model III and Model IV16.  Unfortunately, it turns out that the estimates are sensitive to the choice of the break points. Consequently, in order to choose an appropriate value for the depreciation rate of R&D capital, we have to use our subjective judgement. This is one drawback of our modeling strategy. In our example, we choose the depreciation rate for R&D based on values of both the markup factor and the log-likelihood function. According to the definition of markup factor given in the second section, a reasonable value should be less than 1. Our final choices of the depreciation rates and markup factors are given by table 2.2.  Table 2.2 Depreciation Rates and Markup Factors  SIC 28 SIC 35 SIC 36 SIC 37 Manufacturing Depreciation Rate 0. 01 (0.2866)1 0.03 (1.588) 0. 14* (3.0986) 0.27*** (15.3064) 0. 29*** (10.2534) Markup Factor 0. 96872 (0.0261)2 0. 84863 (0.0494) 0. 91897 (0.0554) 0. 96251 (0.0214) 0. 94723 (0.0191)  Note: 1: The values in the brackets are log-likelihood statistics, which is defined as: [ ])lg()lg(2 102 LL −×−=χ .   *: Statistically significant with 10% confidence level;   ***: Statistically significant with at least 1% confidence level.   2: The values in the brackets are the standard errors.  From the estimated markup factors, we can derive the markup for the total manufacturing sector and the four industries. The markup is 5.3 percent for the total manufacturing sector, 3.1 percent for chemical products, 15.1 percent for non-electrical machinery, 8.1 percent for electrical products, and 3.7 percent for transportation equipment. Our  16 Different break points corresponding to the estimations listed in Table 1 are given in the appendix B.  29 estimated markups are within the empirically plausible range. For example, Laitner and Stolyarov’s (2004) estimated markup is 9%-11%. The upper bound of the markup suggested by them is 20%. Jones and Williams (2000) suggest that empirical markups range from 5%—40%.  As shown in table 2.2, the estimated depreciation rates for SIC 28 and SIC 35 are not significantly different from zero. We may interpret this as the verification of some economists’ belief that the knowledge asset should not depreciate over time. Another possible explanation for the small depreciation rates is that our model does not fit the data well in these industries. In order to avoid exaggerating R&D’s contribution to the productivity growth, we have added a time trend to the model. Thus due to multicollinearity problems, we need some fluctuations in R&D investments in order to accurately estimate the R&D depreciation rate. If the R&D stock and the time trend are strongly proportional, or R&D investments increase smoothly overtime, as in the case of SIC 28 and SIC 35, our model cannot clearly distinguish between the time trend and growth in the R&D stock. This multicollinearity problem may be leading to an understatement of the benefits of R&D, and also to estimates of R&D depreciation rates that are too low. Our other estimates fall in the range of R&D depreciation rates that are found in the literature. Depreciation rates for SIC 37 and Manufacturing are very close to each other, which may due to the similar pattern in the changes of R&D investments in these two sectors.  Using our estimates of the depreciation rate for R&D capital stocks, we can construct the series of R&D capital stocks for manufacturing and the four knowledge intensive industries. Figure 2 shows how R&D stocks change over the period of 1953-2000.  30 Figure 2.2 R&D Stocks (1953-2000) R&D Stocks (1953-2000) 0 10 20 30 40 50 60 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 SIC 28 SIC 35 SIC 36 SIC 37 Manufacturing   The geometrical growth rate of R&D stock for manufacturing is 3.4%, and the geometric growth rates of the R&D stocks for the four knowledge intensive industries, namely SIC 28, SIC 35, SIC 36 and SIC 37, are 4.5%, 5.2%, 3.9% and 3.1%, respectively.  2.5 Conclusion  In this paper, we have developed a simple model based on a production function to estimate the depreciation rates of R&D capital for the U.S. total manufacturing sector and the four knowledge intensive industries, including chemical products (SIC 28), non-electric machinery (SIC 35), electric products (SIC 36) and transportation equipment (SIC 37). We treat R&D capital as a technology shifter instead of as an ordinary input in the model. Using both the R&D stock variable and time variable as technology shifters can avoid the overestimation of R&D capital’s contribution to productivity growth. Along with the estimation of the depreciation rate, we have estimated the markup factor for the U.S. manufacturing and the four selected industries. The estimated R&D depreciation rate is 29 percent for U.S. total manufacturing sector, 1 percent for chemical products, 3 percent for non-electrical machinery, 14 percent for electrical products and 27 percent for transportation equipment. The corresponding markup is 5.3 percent for the total manufacturing sector, 3.1 percent for chemical products, 15.1 percent for non-electrical machinery, 8.1 percent for electrical products, and 3.7 percent for transportation  31 equipment. Based on the estimated depreciation rate, the geometrical growth rate of the R&D stock is 3.4 percent for the manufacturing sector, 4.5 percent for chemical products, 5.2 percent for non-electrical machinery, 3.9 percent for electrical products, and 3.1 percent for transportation equipment.  The results reported here are preliminary. We have not incorporated some important features associated with R&D investment, such as the uncertainty of R&D investment and the externality of the created knowledge. Also, we have imposed some restrictive assumptions to simplify the problem, such as constant depreciation rates over years, constant markup factor, and full information about the future price. In addition, the robustness of the results should be checked against alternative functional forms for the production function and against alternative ways of constructing the stock of R&D capital.  The estimated R&D depreciation rates from this study are the average depreciation rates at industry level and based on information for the time period going from 1953 to 2001. As a longer time series become available, we might expect to have different numerical results. Although, these estimates might not be directly applied by statistical agencies, the methods used in this study will be a good reference for statistical agencies and interested economists to do further research.  32 3 Estimation of the Net Benefits of a R&D Project  3.1 Introduction  The present chapter looks at some of the accounting problems associated with capitalizing R&D expenditures. A highly simplified general equilibrium approach is taken as opposed to the usual partial equilibrium treatments of this topic. One reason for taking a general equilibrium approach is that we can deal with the case of a publicly funded R&D project where the results of the project are made freely available to the public.  The basic accounting problem associated with investments in R&D projects is that the expenditures required to develop a new technology are made now but the benefits of the new technology occur in subsequent periods. But what are the benefits of a new technology? From the viewpoint of a firm developing a new technology, the benefits would appear to be the discounted stream of (monopoly) profits that the new technology is expected to generate.17 The matching principle, a cornerstone of the accrual basis accounting, requires that sunk cost expenditures made in a prior period be amortized in subsequent periods to match the revenues generated by the investment in subsequent periods. If the sunk cost expenditures are simply charged to the period when the investment was undertaken, then profits for that period would be unduly depressed and in subsequent periods, profits would be unduly large. Thus investors, looking at annual accounting statements, would not be given an accurate picture of the ongoing profitability of the firm if all of the sunk cost expenditures were immediately expensed. Therefore, it seems appropriate to amortize the present period expenditures on R&D and spread these costs out to future periods so that costs can be better matched to benefits period by period. This is the point of view taken by Diewert (2005b), (2005c) and many others. Although based on the General Accepted Accounting Principle, private firms still treat R&D expenditures as current expense, e.g. as “cost of sale” in firms’ Cash-flow Statement, a new version of the System of National Accounts recommend the capitalization of R&D expenditures.  17 This is the point of view taken by Pitzer (2004), Diewert (2005b), (2005c) and Copeland, Medeiros and Robbins (2007).  33  Note that the amortization of R&D expenditures over future periods is conceptually different from wear and tear depreciation of a reproducible asset: the R&D expenditures are a sunk cost whereas wear and tear depreciation is a result of use of the reproducible asset during each period that the asset can deliver services. Thus, wear and tear depreciation within a period is a definite phenomenon that depends on use of the asset and can in principle be measured, whereas R&D amortization is largely arbitrary and depends on whatever accounting principle seems “reasonable” under the circumstances that will match costs to benefits in each period.  There are some problems with the above view on how to amortize R&D expenditures:  • The above approach to measuring the benefits of an R&D project sets the net benefits of the project equal to discounted monopoly profits that can be attributed to the project less the current period cumulated R&D expenditures. This view of the net benefits of the project is firm oriented and may not capture the social benefits of the project. • The above approach totally fails if the R&D project is a government financed project where the new technology that results from the project is made freely available to the public.  Thus there is a need to develop a welfare oriented perspective to evaluate the net benefits of an R&D project and this is what is done in the present paper. However, it must be noted that our model is rather crude and only represents a start on modeling the benefits of an R&D project.  In section 3.2, we lay out a simple intertemporal model and consider how the traditional growth accounting approach to R&D works in this highly simplified framework. In section 3.3, we briefly explain the knowledge production function that creates new technologies. Sections 3.4 and 3.5 assume that a new process innovation has been created and we look at the problems associated with measuring the welfare gains associated with the innovation. Section 3.4 looks at a publicly funded R&D project while section 3.5 looks at a privately funded R&D project and this is where the interesting accounting problems emerge. Section 3.6 extends our analysis to a product innovation (rather than a process innovation). Section 3.7 notes how our initial general equilibrium model can be generalized to account for obsolescence and section 3.8 concludes.  34 3.2 Disembodied Technical Change and Growth Accounting  In order to highlight the differences between the traditional Solow (1957), Jorgenson and Griliches (1967) (1972) growth accounting methodology and the R&D accounting methodology that will be developed in this paper, it is useful to review the traditional methodology in the case where the economy is producing only one output and using only one input.18  Thus let 0>ty  and 0>tx  denote the output produced and input used in period t for t = 0,1,... and let 0>tp  and 0>tw  denote the corresponding output and input prices. We assume that production is subject to constant returns to scale and there is competitive pricing in each period so that the value of outputs equals the value of inputs; i.e., we have:  (3.1) tttt xwyp =                                  t = 0,1,2, ... .  Assume that the output and input data for periods 0 and 1 can be observed. Then the period 0 and 1 productivity levels, 0a and 1a , can be defined as the output input ratio in each period; i.e., we have19  (3.2) 111000 /    ;/ xyaxya ≡≡ The Total Factor Productivity Growth of the economy going from period 0 to 1, 1,0τ , is  18 Index number complications are not present in this model. This one output and one input methodology is developed in Diewert (1992a), Balk (2003) and Diewert and Nakamura (2003). For extensions to many outputs and inputs, see Jorgenson and Griliches (1967) (1972), Caves, Christensen and Diewert (1982), Diewert and Morrison (1986), Kohli (1990) and Balk (1998). 19 Thus period 0 production function is xay 0=  and the period 1 production function is xay 1= where y is the output that can be produced by the amount of input x.  35 defined as (one plus) the rate of growth of productivity levels; i.e., we have:  (3.3) [ ] [ ] [ ] [ ]01010101011,0 /////// ppwwxxyyaa ==≡τ  where the last two equalities in (3.3) follow using (3.1) and (3.2). If 1a  is greater than 0a  (the usual case), then 1,0τ  is greater than one and we say that there has been a (total factor) productivity improvement going from period 0 to 1.  The above algebra captures the main aspects of TFP measurement and growth accounting. Note that the productivity improvement is not “earned”; it just happens! Thus we say that there is disembodied technical progress if 1,0τ  is greater than one.  In the following sections of this paper, we are going to relax the assumption of disembodied technical change and we will assume that productivity improvements require some effort in order to create new technologies. However, once we recognize that technological improvements generally require an investment of effort in a prior period before the benefits can be realized in subsequent periods, then a simple two period framework is no longer adequate in order to develop the welfare effects of an effort driven innovation. In order to prepare for this many period modeling effort, it will be useful to conclude this section with a multiple period disembodied technical change model.  Thus let [ ],...,, 210 yyyy ≡  be the sequence of expected outputs of the economy for periods 0, 1, 2, ... . We assume that all output is consumed in each period (so there are no durable outputs for simplicity). We need a Social Welfare Function (SWF) or intertemporal utility function, )(yW , that will enable us to evaluate the relative worth of different sequences of consumption. We choose the following very simple additively separable SWF to make these welfare evaluations:  (3.4) ...)1()1()( 22110 +++++≡ −− yryryyW   36 where 0>r  can be interpreted as a reference real interest rate.20  Let 0≥tx  be the amount of primary input that is expected to be available for use by the production sector of the economy in period t for t = 0, 1, 2,... . If the economy has only the period 0 technology available for all future periods, then ty  equal to txa0  is the output that can be expected to be produced in period t and the economy’s expected welfare using the period 0 technology will be 0W  defined as follows:  (3.5) 1000 33022010000 ...)1/()1/()1/( Xaxa rxarxarxaxaW += +++++++≡  where the future period discounted input aggregate 1X  is defined as follows:  (3.6) ...)1/()1/()1/( 332211 ++++++= rxrxrxX  Now suppose that starting in period 1, the economy has a new constant returns to scale technology that is defined by the production function xay 1=  where the new output input coefficient 1a  is strictly greater than 0a . Then if we use this new technology in period 1 and subsequent periods, the new level of expected social welfare that can be attained using the new technology is W1 defined as follows:  (3.7) (3.6) definition using )1/()1/()1/( 1100 33122111001 Xaxa rxarxarxaxaW += ++++++≡   Using the above definitions, we can calculate a benefit measure B which reflects the  20 Alternatively, )1/(1 r+  can be interpreted as a social discount factor. We assume that the tx  are bounded from above as are the one period technical coefficients at that define the period t production functions and so when we evaluate )( yW  using feasible input vectors and feasible technologies, )(yW is finite using the assumption that 0>r .  37 expected increase in discounted real consumption due to the disembodied productivity improvement going from the period 0 technology (represented by 0a ) to the period 1 technology (represented by 1a  which is assumed to be greater than 0a ):  (3.8) [ ][ ] 101 10001100 01 (3.7) and (3.5) using Xaa XaxaXaxa WWB −= +−+= −=   Recall that 1,0τ  equal to 01 / aa  is a traditional ratio type measure of a productivity improvement whereas 01 aa − is a difference type measure of a productivity improvement.21 We shall find that when we study the welfare effects of an R&D project, the difference approach is much more convenient.  The above disembodied model of technical progress assumed that the new technology dropped from heaven without requiring any sacrifices on the part of households and firms to create the new technology. In the following section, we will relax this assumption.  3.3 The New Technologies Production Function  We now recognize that in many cases, the creation of a new process technology cannot be done without some effort. Thus from the perspective of creating a new technology in period 0, we could think of a period 0 process innovation production function 0f  such that  (3.9) )(01 zfa =  21 The difference approach to productivity measurement is pursued by Balk (2007) and Diewert and Mizobuchi (2007) while the difference approach to the measurement of welfare change is pursued by Chambers (2001), Balk, Färe and Grosskopf (2004), Diewert (2005), Diewert and Fox (2005) and Fox (2006).  38 where 0≥z  is the expenditure in period 0 that is required to produce a new technology characterized by the output input coefficient 1a . We assume that 00 )0( af =  so if we expend no effort, we end up getting the same old period 0 technology, 0a , that we already have access to. We also assume that 0f  is nondecreasing (so that increased effort inputs cannot create worse technologies than we already have in period 0) and concave (so that there are constant or diminishing returns to scale in the creation of new technologies).  From the viewpoint of a central planner in period 0 who needs to decide how much of society’s period 0 input 0x  should be allocated to the creation of new technologies, the following social welfare maximization problem seems to be relevant:  (3.10) ( ) ( ){ }01000 0:max xzXzfzxaz ≤≤+− .  Thus increased investments in R&D (i.e., bigger levels of z) lead to lower period 0 consumption, )( 00 zxa − , but the efficiency of the economy in subsequent periods is increased: ( )zfa 01 =  is generally bigger than 0a  so that discounted future consumption, ( ) 10 Xzf , is generally bigger than 10 Xa , which is discounted future consumption using the old technology. Thus the diminished period 0 consumption is offset by increased consumption in subsequent periods and the 0z  which solves (3.10) balances these two effects. If 0z  is close to 0, there is no problem with this setup but if 0z  is equal to 0x  or is close to 0x  (so that it is enormously productive to invest in the creation of new technologies), then the maximization problem (3.10) is not reasonable from a practical point of view since the solution leads to starvation of the populace in period 0! Thus in this case, the upper bound to z in (3.10) will have to be reduced.  In the sections which follow, we will not assume that investment in new technologies is necessarily “optimal” in the sense that 0z  solves (3.10). We will simply assume that the government or private sector investors allocate the amount 0z  of input in period 0 to  39 create a new technology with output input coefficient ( ) 0001 azfa >≡  where 000 xz << .  It would be of some interest to investigate the “exogenous” determinants of the period 0 technology creation function ( )zf 0 ; i.e., this function will in general depend on the level of education in the economy under consideration, on the stocks of domestic “knowledge” that exist in period 0, on the access to the stocks of foreign “knowledge” and many other factors. However, this is not the focus of the present paper which has much narrower accounting goals in mind.22  In the following section, we will look at the effects of a government R&D project and in section 3.5, we will turn our attention to private sector R&D projects.  3.4 R&D and Process Innovation: The Case of a Government Funded Project  It turns out that the analysis of a government funded R&D project cannot be analyzed as a single case: we need to consider various alternative ways of financing the government R&D project. We will consider four different cases in this section. Privately funded projects will be considered in the following section.  Case 1: The Centrally Planned Economy  We first consider the case of a centrally planned economy since this is the easiest (and least relevant) case to consider. We suppose that the central planner allocates 0z  units of input to the creation of new technologies in period 0 where 000 xz << . This investment creates a new production function with output input coefficient 01 aa > . The resulting  22 There is a huge literature on endogenous growth models that looks at these questions; see Aghion and Howitt (1998) and Aghion and Durlauf (2005) for a sample of this literature.  40 value of expected discounted consumption is *1W  defined as follows:  (3.11) [ ][ ] 1 11000 33122111000*1 W (3.6) definition using ...)1/()1/()1/( < +−= +++++++−≡ Xazxa rxarxarxazxaW   where 1W  was the value of discounted consumption defined by (3.7) which assumed that the creation of the new technology was costless. Thus the present model is more realistic in assuming that technology creation is not generally costless.  Of course, we can now define a new net benefit measure B* associated with the costs and benefits of the creation of the new technology:  (3.12) [ ] [ ][ ][ ] B Xaa zaXaa XaxaXazxa WWB = −< −−= −−+−= −≡ 101 00101 100011000 0*1* (3.11) and (3.5) using   where the costless measure of benefit B was defined by (3.8). Thus not surprisingly, our new measure of the net benefits of a government investment in the creation of a new technology, B*, is less than our section 4.2 estimates of a disembodied (or costless) creation of a new technology.23   23 If our present (costly) model of technical progress is correct, then assuming incorrectly that the disembodied technical change model presented in section 4.2 is correct will not surprisingly generate errors. Thus using the framework in section 4.2 would lead to an estimated period 0 output input coefficient equal to ( ) 000000000*0 //)( aaxzaxzxaa <−=−≡ . Thus the disembodied model would incorrectly assume that the productivity growth rate going from period 0 to 1 was *01 / aa  which is greater than the actual rate 01 / aa .  41 Even if 1a  is greater than 0a , it may be the case that B* is negative if the fixed costs of creating the new technology, 00 za , are sufficiently large.24 In what follows, we will neglect the case of such an impoverishing investment and we will assume that 1a  is greater than 0a  and  (3.13) [ ] 0001010*1* >−−=−≡ zaXaaWWB  so that the R&D project generates an increase in discounted real consumption.25  We now turn our attention to the case of a market economy where the government has to raise the revenue required to fund the R&D project by taxing households.  Case 2: An Income Tax for Period 0  We now have to specify what is happening to prices in the economy in each period. We will assume that there is a central bank in the background that acts to stabilize the producer price of output in each period so that tp  is expected to equal 1 in each period. In period 0, primary inputs tx  are paid the value of what they produce (before any income taxes) so that 0w  is equal to 0a  in period 0 and tw  is equal to 1a  for periods t ≥ 1. As usual, 0y  equals 00 xa  in period 0 and ty  equals txa1  in subsequent periods. It is convenient to list these assumptions since we will draw on them in the cases to be developed later:  (3.14) Period 0: 000000   ; ;1 xayawp === ; Period t: tttt xayawp 11  ; ;1 ===  for t ≥ 1.  24 See Romer (1994) for a good discussion of this point in a slightly different context. 25 We will assume that (3.13) holds also for the case of a private R&D project.  42  The primary input prices tw  defined above are producer prices and are before any income taxes. We now assume that the government imposes an income tax in period 0 in order to finance the R&D investment. The size of the income tax, 0t , that is required to balance the government’s budget in period 0 is:  (3.15) 000 / xzt ≡ .  Thus primary input suppliers in period 0 face the after tax wage rate of )1( 00 tw −  so that their total period 0 after tax income is:  (3.16) ( )[ ] ( ) 000000000000000 /1)1()1( cpzxaxzxatxatxw =−=−=−=−  where ( )0000 zxac −≡  is period 0 consumption (recall that 10 =p ). The value of expected discounted consumption is still *1W  defined by (3.11) and the net benefit associated with the R&D project is still B* defined by (3.12). Thus there is no problem in attaining the planned economy level of welfare using an income tax in period 0 to finance the project in a decentralized market economy.  Case 3: A Consumption Tax for Period 0  Again, we assume that expected producer output and input prices are defined by (3.14). However, in this case, instead of assuming that the government imposes an income tax in period 0 in order to finance the R&D investment, we now assume that it imposes a consumption tax. The size of the consumption tax, *0t , that is required to balance the government’s budget in period 0 is defined by the following equation:  (3.17) ( )[ ] )1/(1/11 0100*0 txzt −=−≡+ −   43 where 0t  was the income tax defined by (3.15). In this case, households in period 0 have primary factor income equal to 00 xw  to spend on consumption which is priced at ( )*00 1 tp + . Thus the period 0 household expenditure equals income equation is:  (3.18) ( ) 000*00 1 xwctp =+  or ( ) [ ] 00000*01 xwzxat =−+  or  [ ] 000000 )1( xwtzxa −=−                    using (3.17)  But the last equation in (3.18) is equivalent to (3.16). Again, the value of expected discounted consumption is *1W  defined by (3.11) and the net benefit associated with the R&D project is still *B  defined by (3.12). Thus there is no problem in attaining the planned economy level of welfare using a consumption tax in period 0 to finance the project in a decentralized market economy.26  Case 4: Financing the Project by Foreign Borrowing  In the previous 3 cases, households who are alive in period 0 bear all of the burden of financing the government R&D project whereas subsequent generations get all of the benefits of the project without suffering any of the costs. Since we did not allow for durables in our model (or any other form of saving) and since we assumed that primary inputs were supplied inelastically in each period, we were unable to avoid this asymmetric bearing of the burden problem. We now relax our previous assumptions and assume that the government can borrow from abroad if it wishes to do so (at the same real interest rate r that appeared in our intertemporal utility function ( )yW  defined by (3.4) above). Thus we now assume that the government faces the following intertemporal balance of payments constraint:  26 Our model is too simple to record any difference in social welfare due to the imposition of a consumption tax versus an income tax. Of course, in more complex real life economies, there would be differences between the two methods for financing the government R&D project.  44  (3.19) ( ) ( ) 0...11 22110 =+++++ −− rbrbb  where tb  is the period t export (if tb  is positive) or import (if tb  is negative) of output for the economy in period t. Thus suppose the government imports the output commodity in period 0 so that 0b  is negative (and we can regard 0b−  as a capital import) and then repays the loan in the following period. Since capital made available in period 0 is more valuable in period 0 than in period 1, if the government repays the loan in period 1, then the government must export 1b  equal to ( )( )01 br −+  in order to repay the loan.  We will assume that the government borrows an amount in period 0 that is just sufficient to maintain the consumption level that would have resulted if the government R&D project were not implemented. Thus 0b  is defined as follows:  (3.20) 000 zab −= .  When we substitute (3.20) into the intertemporal balance of payments constraint (3.19), we could pick any pattern of future period tb  which satisfy the constraint; i.e., the repayment plan has many degrees of freedom. But fairness and ability to pay considerations might suggest that the burdens imposed by the loan repayment be proportional to the future period consumption levels. Consumption in period t before the imposition of any taxes will be equal to production ty  which is equal to txa1  for t ≥ 1. Thus we let period t exports tb  be proportional to period t production; i.e., we want to choose a fraction f such that27  27 We will obtain the same ct solution as is given by (3.25) if we make period t loan repayments proportional to the incremental consumption that is made possible by the innovation; i.e., we obtain the same pattern of consumption and social welfare if we set [ ] tt xaafb 01 −=  for 1≥t .  45  (3.21) ttt xfafyb 1== ;  t ≥ 1.  Now substitute (3.20) and (3.21) into (3.19) and we obtain the following equation:  (3.22) ( ) (3.6) definition using 1/ 11 1 100 Xfa rxfaza t tt = += ∑∞=  Solving (3.22) for f gives us:  (3.23) 1100 / Xazaf = ; (3.24) [ ]100 / Xxzab tt =                 for t ≥ 1 ; (3.25) [ ][ ][ ]  / (3.24) using              / 1for t 10011 1001 XxzaXa Xxzaxa byc t tt ttt −= −= ≥−=   Now define social welfare under this method of financing the project in the usual way as discounted consumption:  (3.26) ( ) ( ) [ ][ ]( ) [ ][ ] (3.11) using (3.6) using (3.25) using                1/ ...11 *1 11000 001100 1 1 001100 22110**1 W Xazxa zaXaxa rXxzaXaxa rcrccW tt t = +−= −+= +−+= +++++≡ −∞ = −− ∑   Thus under this borrowing from abroad financing scheme for the government funded R&D project, we attain the same level of net benefit, B* defined by (3.13), as was obtained in cases 1-3. However, the present scheme will be intertemporally much more equitable than the previous methods of financing the project, assuming that the inequality  46 (3.13) is satisfied.28  In order to implement the above equilibrium in a decentralized fashion, the government need only impose an income tax at the rate f  defined by (3.23) for periods t ≥ 1. Alternatively, the government could implement the above equilibrium by imposing a consumption tax for periods 1≥t  at the rate t* defined as follows:  (3.27) ( ) 11 1* −−≡ −ft  where f is defined by (3.23).  We now turn our attention to privately funded projects.  3.5 R&D and Process Innovation: The Case of a Privately Funded Project  Case 5: A Domestically Funded Project with No Outsourcing of Production  In this case, we assume that domestic investors forego consumption in period 0 in order to finance the R&D project and in return, these investors will get a stream of monopoly profits which they can spend on consumption in subsequent periods. The characteristics of the R&D project are the same as was the case in the previous section: the project uses up 0z  units of primary input in period 0 and it produces a new technology which is characterized by the output input coefficient, 1a , where 1a  is strictly larger than the pre-project technical coefficient, 0a . We will assume that this project is successful so that the inequality (3.13) is satisfied. In period 0, we assume that prices and quantities satisfy (3.14). In subsequent periods, we assume that the new technology immediately displaces the old technology and the monopolist produces output using the new  28  In order to ensure that future consumption with the loan repayment is greater than preproject consumption, we require that 1001 /1/ Xzaa +>  and this inequality is equivalent to (3.13).  47 technology. We assume that outputs and inputs, ty  and tx , and the corresponding producer prices, tp  and tw , satisfy assumptions (3.14). Producer prices are equal to final demand prices in period 0 but in subsequent periods, final demand output prices differ from the producer output prices described in (3.14) because the owners of the new technology can charge a monopolistic markup, m say, on the sales of the output.29 Thus for periods 1≥t , the owners charge households the final demand price, ( ) ( )mmpt +=+ 11 , instead of the producer price, 1=tp .30 We now have to determine the size of the markup.  After the introduction of the new technology, real wages will increase from 0w  equal to 0a  to tw  equal to 1a  for 1≥t . In order to produce one unit of output in period 1≥t using the old technology, an amount of input equal to 0/1 a  would have to be used since the equation:  (3.28) *01 xa=  has the solution x* equals 0/1 a . Thus potential competitors (using the old technology) to the monopolist would sell units of output at the period t price *tp  defined as follows in order to cover costs:  (3.29) ( ) ( ) ( ) 1         t                    11//1 0101** ≥+=+==== mpmaaaaxwp ttt  Thus the owners of the new technology can charge (in the limit) a monopolistic markup m defined as  29 Thus we are assuming that the new technology is proprietary and is not made available to other producers. 30 Recall that we are assuming that the central bank stabilizes the producer price of output in each period.  48  (3.30) ( ) 1/ 01 −= aam  Thus if the owners of the project fully exploit their monopoly position, households will have to pay the final demand prices *tp  defined by (3.29) for t ≥ 1.  With the above preliminaries out of the way, we can now look at the streams of factor incomes and expenditures at final demand prices and see if income equals expenditure in each period. The period by period sequence of consumption quantities, tc , is the following sequence:  (3.31) ( ) 1        t,  ; 10000 ≥=−= tt xaczxac .  The corresponding sequence of consumption expenditures at final demand prices is:  (3.32) ( ) ( ) ( ) 1         t,111   ; 1000 ≥+=+=+= tttt xamcmcmpccp .  Turning our attention to the quantities and values of inputs, the sequence of input quantities is the usual tx  for 0≥t . The corresponding primary input income sequence is given by:  (3.33) 1        t,  ; 10000 ≥== ttt xaxwxaxw .  However, primary input income does not exhaust income because for periods later than 0, there will be some monopoly profits that will be distributed back to the investors in the R&D project. This stream of monopoly profits is given by:  (3.34) 1       t,  ;0 1 ≥== tttt xmamccmp .  Comparing the stream of household expenditures given by (3.32) with the sum of the two  49 income streams defined by (3.33) and (3.34), it can be seen that total income from all sources will be equal to household consumption expenditures at final demand prices for all periods except for period 0. Thus for the periods beyond period 0, this monopoly model is perfectly consistent. However, in period 0, it can be seen that the value of consumption is 00cp  equal to ( )000 zxa − , which is less than the corresponding factor income, 00 xw  equal to 00 xa . The problem is that we have not accounted for the period 0 investment in developing the new technology. This investment is equal to 0z  units of input. We could revalue this input measure of investment into units of output that are foregone and thus we define the period 0 investment 0I  as follows:  (3.35) 000 zaI ≡ .  It can be seen that if we add period 0 investment 0I  to period 0 consumption 0c , then the period 0 value of outputs will equal the value of period 0 primary inputs.  Now we are ready to define social welfare under this private sector financing of the R&D project:  (3.36) ( ) ( ) ( ) ( ) ( ) (3.11) using                                                              W (3.6) using                                      a (3.31) using                1/ ...11 *1 11000 1 1000 22110***1 = +−= ++−= +++++≡ −∞ = −− ∑ Xazx rxazxa rcrccW t t t .  Thus somewhat surprisingly, the level of social welfare that is attained by this private sector model of R&D investment, ***1W , is exactly equal to the level of social welfare *1W  that was attained in the previous section with the various government financing schemes. Thus for this private sector financing model, we attain the same level of net benefit, B* defined by (3.13), as was obtained in Cases 1-4. However, at this point, we again must point to the inadequacies of our social welfare function, which does not distinguish between households who invest in the project and those who do not invest.  50 A more adequate social welfare function would take into account the increased income inequality between households that results from the monopoly profit stream and the fact that some households invested in the R&D project and some did not.  Our measurement problems are not quite over yet. We need to discuss the accounting problems that are associated with this privately funded R&D project. For accounting purposes, it is useful to break up the activities of the R&D project into two divisions:  • A production division which oversees the production of the output using the new technology and • An R&D management division which finances the expenditures for the project and collects the monopoly revenues from consumers.  The accounting for the production division is straightforward and need not be analyzed in detail here. The accounting problems associated with management division are much more interesting and will be discussed in more detail. The sequence of costs (in period 0) and revenues (in subsequent periods) associated with the R&D management division is the following one:  (3.37) 1    t,   ; 10000 ≥=−=− ttt xmamcpzazw .  The problem with the above sequence of net revenues earned by the management division is that all of the costs occur in the first period and all of the benefits occur in subsequent periods. Thus if we look at the net income earned by the division in any given period, it will not be “representative” for the project as a whole: the period 0 costs are not matched up with the period t revenues. Thus in order to construct more representative estimates of income for the management division, accountants have invented the matching principle for allocating costs that occur in period 0 but whose benefits occur in later periods.31 The basic idea can be explained as follows. Let { td : t = 0,1,....} be a sequence of cost allocations (imputed income or cost) that has the following property:  (3.38) ( ) ( ) ( ) 0...111 3322110 =+++++++ −−− rdrdrdd .  31 For references to the accounting literature on the matching principle, see Diewert (2005c).  51  If we add the period t imputation td  to the period t net revenues of the division for each t, and then take the present value of the above imputed net revenues to the actual net revenues of the firm, then the present value of the total revenue stream (including actual and imputed net revenues) will of course be equal to the present value of the actual net revenue stream. Thus we will attempt to choose a sequence of imputations td  that will result in more representative period by period net revenues; i.e., the resulting stream of period by period net revenues will better match the fixed costs of period 0 to the monopoly revenues that occur in subsequent periods.  We now proceed to an explicit application of the above “matching” model. In period 0, we create an imputed investment output 0d  defined to be equal to the period 0 cost of the R&D project:  (3.39) 00000 zazwd == .  We need to choose a sequence of cost allocations td  for t ≥ 1 which will satisfy equation (3.38) when we substitute (3.39) into (3.38). We will choose to make the period t cost allocation, td , proportional to the period t monopoly revenue, tmc . Letting f  be the factor of proportionality so that td  equals tfmc  for t ≥ 1, we want to solve the following equation for f:  (3.40) ( ) ( ) ( ) ( ) ( ) ( ){ } (3.31) and (3.6) using ...111 ...111 11 332211 33221100 Xfma rcrcrcfm rfmcrfmcrfmcza = ++++++= ++++++=− −−− −−−   Thus f is equal to:  (3.41) 1100 / Xmazaf −≡  52  and the sequence of period t cost allocations td  is given by  (3.42) [ ][ ][ ]100 11100 / (3.41) and (3.31) using                / 1   t Xxza xmaXmaza fmcd t t tt −= −= ≥≡  .  Thus period t net income tn  for the management division is equal to actual net income plus imputed net income in period t, td , which is actually an imputed cost for period 1≥t ; i.e., we have the following sequence of net incomes for the management division of the monopolist:  (3.43) 00 =n ; (3.44) [ ] ( )[ ] ( )[ ] ( ){ } 0 (3.30) using           /1/ /ma (3.42) and (3.31) using                           / 1,2,...      t 100101 1001 1001 > −−= −= −= =+≡ t t tt ttt xXzaaaa xXza Xxzaxma dmcn   where the above inequality follows from ( )[ ] ( )1001 /1/ Xzaa >− , which is equivalent to (3.13), and 01 aa > .  The point of all of these imputations is that the cost of the R&D research made in period 0 is now spread out over future periods, reducing the gross monopoly profits in period t of mct by the amount td−  equal to ( )100 / Xxza t  and the period 0 original negative income for the R&D management division of 00 za−  is increased to the zero level; i.e., the original fixed costs of the R&D project are intertemporally reallocated to subsequent periods in a way which matches costs to revenues in a “reasonable” manner. It is evident that td  could be interpreted as a period t depreciation allowance but it is more properly  53 interpreted as an amortization amount; it is simply an imputation that somewhat arbitrarily allocates the period 0 fixed cost to future periods. Note that the absolute value of td  is equal to the product of 00 za  (the fixed cost that is to be amortized over future periods) times tx divided by the future input aggregate 1X . Note also that  (3.45) ( )( ) 11/ 1 1 =+∑∞=t tt rXx .  Equations (3.42) and (3.45) mean that the sum of the amortization amounts, ( )∑∞= −1t td , will exceed the original period 0 R&D costs, 00 za , due to the discounting by the interest rate.32  It can be seen that accounting for R&D leads to an accounting framework which is, unfortunately, much more complex than the usual Solow-Jorgenson-Griliches growth accounting framework!  Case 6: A Domestically Funded Project with No Outsourcing of Production and with Foreign Financing of the Project  Domestic investors could decide to fund the R&D project by borrowing from abroad. Thus in this case, we will assume that domestic investors finance the period 0 costs of the R&D project by borrowing an amount of the consumption good that is equal to the period 0 consumption that is foregone by investing in the R&D project; this amount is 00 za , which we set equal to 0b− ; i.e., we define 0b  by (3.20) as in Case 4 where we considered the case of a government R&D project which was financed by foreign borrowing. As in our analysis of Case 4, we assume that the private investors in the R&D project face the intertemporal balance of payments constraint (3.19), where tb  for t ≥ 1 is the amount of the consumption good which must be exported in period t in order to  32 See section 11 in Diewert (2005b) and Diewert (2005c) for examples of how the matching methodology works.  54 repay the loan. As in Case 4, the R&D investors could pick any pattern of future period tb  which satisfy the constraint (3.19); i.e., the repayment plan has many degrees of freedom. But following the logic of the matching approach to the problem of amortizing the loan, it seems appropriate to make future period loan repayments proportional to the expected net benefit that the R&D project generates in period t, which is the amount of monopoly profits equal to txma1  for 1≥t . Thus tb  is defined as:  (3.46) ( )[ ] [ ]( ) 1  t                    ;/1/ 0101101 ≥−=−≡ ttt xaaaafxaaafb  where f is a parameter to be determined. Substituting (3.46) into the intertemporal balance of payments equation (3.19) with 0b  defined by (3.20) leads to the following equation:  (3.47) ( ) [ ]( ) ( ) [ ]( ) (3.6) definition using                                            / (3.46) using                        1// 1/ 10101 1 0101 1 00 Xaaaaf rxaaaaf rbza t tt t tt −= +−= += ∑ ∑ ∞ = ∞ =   Solving (3.47) for f gives us:  (3.48) ( ){ }0 0 1 0 1 0 1/ /f a z a a a a X⎡ ⎤= −⎣ ⎦ .  Substituting (3.48) into (3.46) gives us:33   33 We get the same answer for tb  if we note that the net social benefit of the project in period t is the extra production that the project makes possible which is [ ] txaa 01 −  and so we could set tb  to be proportional to this net social benefit so set [ ] tt xaafb 01 −=  and then solve equation (3.47) for f, etc. However, there is no reason for a monopolist to make cost allocations or loan repayments based on a consideration of social benefits.  55 (3.49) 1for t                / 100 ≥≡ Xxzab tt .  Equations (3.25) and (3.26) can be used in order to define period t consumption for the home economy, ttt byc −=  for t ≥ 1, and then (3.26) can be used to define the economy’s social welfare under this method for financing the project. Not surprisingly, we obtain the same level of social welfare that we have obtained in all of our previous cases.  We now calculate the period t monopoly revenue, txma1 , less the period t loan repayment, tb :  (3.50) ( )[ ] (3.44) using                                                       n (3.30) definition using         /1/ 1for t (3.49) using                         / t 100101 0011 = −−= ≥−=− Xxzaxaaa Xxzaxmabxma tt ttttt   where tn  was previously defined to be equal to the imputed net income of the management division of the monopolist. Thus in the present case where the R&D project is financed by a loan from abroad, our old imputed net income in period t is equal to actual monopoly revenue less actual loan repayments in period t. Thus the net income that accrues to the management division is no longer an imputed accounting income; in the present case, it is an actual income. This equality helps to justify the case for the imputed income concept that we developed in the previous case. However, note that for both the present case and the previous case, there are ambiguities in our estimates of period by period income; i.e., in Case 5, we had to decide how the fixed cost should be matched to future revenues and in Case 6, we had to decide exactly how the loan from abroad should be repaid.  Cases 7 and 8: A Domestically Funded Project with Licensing of Production (with domestic funding of the project or foreign borrowing to fund the project)  The setup here is exactly the same as in the previous two cases, except that now we assume that the developers of the new process license the technology to other producers.  56 If we assume that the license fees are proportional to the quantity of output produced by the new technology, then no new algebra needs to be developed: simply reinterpret the monopoly markup m in the previous case as the per unit output royalty that must be paid by independent producers for the right to use the new technology. Thus the welfare effects for these cases are exactly the same as in Cases 5 and 6. However, note that the accounting is slightly different in these licensing cases: we require a slightly augmented set of production accounts so that the flows generated by these royalty payments can be accommodated.  Case 9: A Foreign Funded Project  In this case, foreign investors fund the period 0 expenditures on primary inputs, 00 zw , in return for the stream of monopoly profits or licensing fees generated by the R&D project. In this case, the welfare effects of the R&D project are (finally!) different: the foreign investors will get all of the benefits of the R&D project, leaving domestic residents no better off than they would be if the project never took place.  This completes our discussion of a publicly or privately funded R&D project that develops a new process. In the following section, we turn our attention to R&D projects that develop a new product as opposed to a new process.   3.6 R&D and Product Innovation  Fortunately, it is not necessary to develop any new algebra for the case of a product innovation. We can adapt the analysis in the previous two sections very easily under two alternative sets of assumptions concerning the new product:  • The new product is simply a new mixture of characteristics that purchasers value and a hedonic regression methodology can be used to quality adjust a unit of the new product into an equivalent number of units of the existing product that is displaced. Once this has been done, then we look at the input requirements for producing one constant quality unit of the new and old products and this will give us the output input coefficients, 0a  and 1a , for producing units of the old and  57 new products. The rest of the analysis proceeds as in sections 3.4 and 3.5. • The new product has one or more really new characteristics so hedonic regression techniques may be problematic in this case, we may be forced to rely on the methodology developed by Hicks (1940) to value the contribution of new goods, except that we make the further restriction that the preferences of households over combinations of new and old products are linear. Thus we assume that one unit of primary input produces an amount of the “old” commodity which consumers value at 0v  and one unit of primary input produces an amount of the “new” commodity which consumers value at 01 vv > . Now let 0a  equal 0v  and 1a equal 1v  and we can use the analysis developed in the previous sections.  It should be noted that our analysis is not entirely satisfactory; i.e., in general, we have not modeled substitution effects in an adequate manner. On the production side of our model, we have essentially assumed Leontief no substitution type technologies in each period and on the consumer side of our model, we have assumed linear intertemporal preferences, which implies perfect substitutability between consumption at different points in time. Obviously, it would be desirable to relax these rather restrictive assumptions. However, even with our simplifying assumptions, accounting for R&D investments is rather complex.  3.7 Towards a More Realistic Model of R&D Investment  Another problem with our modeling of an R&D investment is that we have aggregated outputs into a single commodity and the R&D investment leads to a new technology which entirely displaces the old technology. This is obviously unrealistic. Thus in this section, we will develop a more realistic model of an R&D investment. In this new model, we will still have only one aggregate input but now we distinguish two outputs:  • Output 1 is general consumption and the technology that produces this output is unaffected by the R&D investment and • Output 2 is a specific commodity where the R&D investment generates a new technology that can produce this specific commodity more efficiently. This second commodity will typically make up only a small fraction of the entire  58 economy.  Since we now have two outputs, our old single output social welfare function, )(yW defined by (3.4), has to be modified. Our new welfare function will essentially be a discounted sum of period t (cardinal) utility levels, ),( 1 t t tt yyU , t = 0,1,... , where ty1  is the period t household consumption of the general commodity and ty2  is the period t consumption of the specific commodity whose production is affected by the R&D investment in period 0. We will assume that the period utility functions tU  are of the no substitution Leontief type; i.e., we define tU  as follows:  (3.51) { }tsyt yyyyU β/,min),( 21'21 ≡  where tβ  is a period t nonnegative34 taste parameter; i.e., the bigger tβ  is, the more important will be the specific commodity to the consumer in period t.35 Generally, we expect the sequence of tβ  parameters to decline over time as household tastes change and demand shifts away from the specific product until finally, for some period T > 1, the tβ  are all equal to 0 for t > T and for these distant future periods, the specific commodity is no longer demanded by any household.  Turning now to the production side of the model, we assume that each commodity is produced by a single common input.  Let 01 ≥tx  and 02 ≥tx  denote the amounts of  34 If 0=tβ , then we define 121 ),( yyyU t ≡ . Thus if tβ  equals zero, then the specific commodity is no longer desired by the consumer in period t. 35  The expenditure function ),,( 21 tttt ppuE  that is dual to tU  is { } [ ] ttttttttsytttt uppuyyUypypppuE 21212211'21 ),(:min),,( β+=≥+≡  for t = 0,1,2, ... .  59 input used to produce the period t outputs 01 ≥ty  and 02 ≥ty  respectively. If there is no R&D investment, then we assume that period t outputs and inputs are related by the following constant returns to scale production functions:  (3.52) 0     t;   ; 2 0 211 ≥== tttt xayxy where 00 >a  is the output input coefficient for sector 2. Note that we have chosen units so that output equals input for the production of the general consumption commodity; i.e., in sector 1, output in period t, ty1 , is equal to the amount of input used during period t, tx1 .  The total amount of input available to the economy in period t is 0>tx  as in the previous sections. Thus we have the following constraint in each period t on the allocation of input between the two sectors:  (3.53) 0      t          ;21 ≥=+ ttt xxx .  We now work out the (anticipated) competitive allocation of resources between each of the two sectors for each period under the assumption that there is no R&D project. Using definition (3.51) of the period t utility function tU , it can be seen that if tβ  is positive, then we have the following relationships between the period t level of utility, tu , and the period t consumption levels of the two commodities, ty1  and ty2 :  (3.54) 0  ,0                  ;/21 >≥== ttttt tyyu ββ .  The above equations imply that the following equations hold:  (3.55) 0     t          ;12 ≥= ttt yy β  60 and it can be verified that equations (3.55) hold even if tβ  equals 0. Substituting (3.52) and (3.55) into (3.53) leads to the following anticipated allocation of inputs for period t:  (3.56) ( )[ ] 0t                              ;1 1101 ≥+= −− ttt xax β (3.57) ( ) ( )[ ] 0        t          ;1 110102 ≥+= −−− tttt xaax ββ .  Substituting (3.56) and (3.57) into (3.52) gives us the anticipated production of each commodity in period t:  (3.58) ( )[ ] 0    t          ;1 11011 ≥=+== −− ttttt uxaxy β ; (3.59) ( )[ ] 0    t          ;1 110202 ≥+== −− ttttt xaxay ββ .  We turn our attention to the prices that are anticipated to prevail in each period t. We now assume that the central bank stabilizes the price of input tw  in each period and so we set tw  equal to unity for each t. Given assumptions (3.52) on the technology of the sectors, the price of output 1 in period t, tp1 , will be equal to the period t price of input so we will have tp1  equal to one in each period as well. For sector 2, we must have the period t value of outputs, tt yp 22 , equal to the corresponding value of input, ttt xxw 22 = , and this equation along with the second equation in (3.52) will imply that tp2  will equal the reciprocal of 0a . Putting this altogether, anticipated period t prices in the economy will be equal to the following specific values:  (3.60) 0   ;/1  ;1  ;1 021 ≥=== tappw ttt .  We could use the above information in order to calculate the discounted stream of period t utilities and this would lead to a measure of (anticipated) social welfare for the economy;  61 i.e., we could use the same social welfare function as was defined by (3.4) above except that period t utility tu  would replace our old period t consumption ty . However, we will find it more convenient to measure period t utility by period t household expenditure at constant prices; i.e., we will use money metric utility as our measure of period t utility.36 Let ),,( 21 ppuE t  be the expenditure function that is dual to the utility function ),( 21 yyU t  that is defined by (3.51). Then period t money metric utility te  is defined as follows:  (3.61) ( ) (3.53) using (3.59) and (3.58) using (3.60) using quantities household t period  theare y and y here         w 0,1,2,...for t             ),,( 21 2 10 1 t 2 t 12211 21 t tt tt tttt ttttt x xx yay ypyp ppuEe = += += += =≡ −  Thus period t money metric utility (or expenditure) te  is equal to period t aggregate input tx , which is reasonable, given our assumptions of no technical progress in the economy and our pricing conventions, (3.60).  Define the (anticipated) money metric level of social welfare, 0W , as discounted period by period money metric utility te :  (3.62) ( ) ( ) (3.61) using                      1 1 0 0 0 tt t tt t xr erW −∞ = −∞ = ∑ ∑ += +≡ .  Thus for our base case where there is no R&D investment in developing a new  36 The term money metric utility scaling is due to Samuelson (1974) but the idea of using an expenditure function with prices fixed to cardinalize utility can be traced back to Hicks (1942) and Allen (1949).  62 technology for sector 2, social welfare is equal to the discounted sum of period by period aggregate input availability for the economy.  We now consider the case of a government R&D project that develops a new technology for sector 2 in period 0 and is made freely available to the economy in subsequent periods. We suppose that the period 0 government R&D expenditures on primary input are equal to the quantity of input 0z , which is assumed to be less than the available total primary input for period 0, 0x . This R&D investment produces a new technology for sector 2 which has output input coefficient 1a , where as usual, we assume that 1a  is greater than 0a :  (3.63) 01 aa > .  We again assume (somewhat unrealistically) that the new technology immediately displaces the old technology used in sector 2 for all periods t greater than 0. The algebra associated with equations (3.52)-(3.59) can now be repeated for all periods t ≥ 1, except that 1a  replaces 0a . Letting *tix  and *t iy  denote the quantity of input used by sector i and output produced by sector i in period t for i = 1,2, it can be seen that we obtain the following equations:  (3.64) 1     t          ;y  ; *2 1*t 2 * 1 * 1 ≥== ttt xaxy ; (3.65) 1         t          ;*2 * 1 ≥=+ ttt xxx ; (3.66) 1         t          ;*1 * 2 ≥= ttt yy β ; (3.67) ( )[ ] 1     t                    ;1 111*1 ≥+= −− ttt xax β ; (3.68) ( ) ( )[ ] 1 t                    ;1 11111*2 ≥+= −−− tttt xaax ββ ; (3.69) ( )[ ] 1      t          ;1 *111*1*1 ≥=+== −− ttttt uxaxy β ; (3.70) ( )[ ] 1       t          ;1 111*21*2 ≥+== −− ttttt xaxay ββ .  63  In a similar manner, we obtain the following counterparts to equations (3.60); i.e., anticipated period t prices in the economy after the R&D investment will be equal to the following specific values:  (3.71) 1      t                    ;/1  ;1   ;1 1*2 * 1 * ≥=== appw ttt .  Using assumption (3.63) and our assumption that the period t taste parameter tβ  is nonnegative, we can establish the following inequalities:37  (3.72) ( )[ ] ( )[ ] 1       t          ;11 110111 ≥+≥+ −−−− tt aa ββ .  Using (3.58), (3.59), (3.69), (3.70) and (3.72), it can be shown that the following relationships hold between the period t anticipated outputs with no R&D investment, tiy , and the period t anticipated outputs with the period 0 R&D investment, *tiy , for i = 1,2:  (3.73) 1   t1,2;i                 ;* ≥=≥ titi yy .  It can also be shown that the two inequalities in (3.73) will hold strictly for any period t where the taste parameter tβ  is positive, so that the specific consumption commodity is demanded in that period. The inequalities (3.73) make good intuitive sense, since the increased productivity in the production of the specific commodity affected by the process innovation allows more of both commodities to be produced for any period where the second commodity is actually demanded by households.  We still need to calculate the period 0 allocation of resources between the two sectors after the R&D effort 0z  is subtracted off from total available primary input 0x  in period 0. If the required revenue to finance the government R&D project in period 0 is  37 If βt > 0, then inequality t in (3.72) holds strictly.  64 financed by an income tax in period 0, then the size of the required income tax 0t  is still equal to 00 / xz , the same tax rate that was defined by (3.15) above. The period 0 producer price for input will be 1*0 =w  and the corresponding consumer price will be )1( 0*0 tw − . The producer and consumer prices for the two outputs in period 0 will be defined as follows:  (3.74) 0*02 *0 1 /1    ;1 app ≡≡ .  The amounts of the two consumption goods that will be produced in the R&D equilibrium in period 0 turn out to be the following amounts:  (3.75) ( )[ ] [ ]001010*01 1 zxay −+= −− β ; (3.76) ( )[ ] [ ]0010100*02 1 zxay −+= −− ββ .  We will find it convenient to define the monopolistic markup m  that the new technology would allow if the technology were closely held by private investors (even though in the present case, the government makes the new technology freely available):  (3.77) [ ] 01/ 01 >−≡ aam  where the inequality follows from assumption (3.63).  We now calculate the value of consumption in each period for the new equilibrium, *te , but at the prices that prevailed in the equilibrium with no R&D investment. For period 0, this money metric utility level is given by the following expression:      65 (3.78) ( ) ( )[ ] [ ] ( ) ( )[ ] [ ] 00 00 1 0 10 0 1000 1 0 10 *0 2 10*0 1 *0 2 0 2 *0 1 0 1 0 2 0 1 *0*0 (3.76) and (3.75) using      11 (2.74) using ),,( zx zxaazxa yay ypyp ppuEe t −= −++−+= += += ≡ −−−−− − βββ .  Thus the value of consumption in period 0 (valued at the period 0 consumer prices corresponding to the no R&D equilibrium prices 0ip  which turn out to equal the period 0 consumer prices of the R&D equilibrium *0ip ), *0e , is equal to the value of primary input that is allocated to the production of consumer goods and services, [ ] [ ]0000*0 zxzxw −=− , where the equality follows since *0w  equals unity.  In a similar fashion, we calculate the period t money metric utility level *te , which is equal to the value of consumption in period t (valued at the period t consumer prices corresponding to the no R&D equilibrium consumer prices tip ):  (3.79) ( ) ( ) [ ] (3.65) using (3.77) using                                     1 (3.70) and (3.69) using (3.60) using 1        t                                        ),,( * 2 * 2 * 1 * 2 110* 1 * 2 10* 1 * 22 * 11 21 ** tt tt tt tt tttt ttttt mxx xmx xaax yay ypyp ppuEe += ++= += += += ≥≡ − −   Thus the value of consumption in period t (valued at the period t consumer prices corresponding to the no R&D equilibrium prices tip ), *te , is equal to the aggregate value of primary input that is available in period t, tx , plus the imputed monopoly markup over the old technology that the new technology generates, m, times the amount of primary input that is allocated to the new technology in period t, *2 tx . This is a rather  66 nice result.  We can use the above information in order to calculate an intertemporal money metric estimate of social welfare 1W  that the investment in the R&D project makes possible; i.e., define 1W  as the following discounted stream of period t money metric utility levels *te  defined above by (3.78) and (3.79):  (3.80) ( ) ( ) [ ] (3.79) and (3.78) using               1 1 * 21 00 0 *1 ttt t t tt mxxrzx erW +++−= +≡ −∞ = ∞ = − ∑ ∑   Finally, the intertemporal money metric estimates of social welfare defined by (3.62) and (3.80) can be differenced in order to give us an estimate of the expected net benefits B of the government R&D project:38  (3.81) ( ) [ ] ( ) (3.80) and (3.62) using    11 1 0 * 21 00 01 t t ttt t xrmxxrzx WWB −∞ = −∞ = ∑∑ +−+++−= −=  (3.82)  ( ) *210 1 ttt xrmz −∞=∑ ++−= (3.83) ( ) (3.71) and (3.70) using            1 *2*210 tttt yprmz −∞=∑ ++−= .  Expression (3.82) says that the net benefits of the R&D project are equal to the imputed monopolistic markup m times the discounted value of the input that is used by the new process, ( )∑∞= −+1 *21t tt xr ,39 less the amount of input that is allocated to the development of the new process in period 0, 0z . Expression (3.83) provides an interpretation of the net benefits of the project in terms of outputs rather than inputs and is a counterpart to our  38 It can be seen that the net benefit measure B is an intertemporal Hicksian equivalent variation; see Hicks (1945-46) and Diewert (1992b) for a discussion of the Hicksian variation measures. 39 This summation of terms will be a finite one if 0=tβ  for all t ≥ T.  67 earlier measure of net benefits *B  defined by (3.12). Expression (3.83) says that the net benefits of the R&D project are equal to the imputed monopolistic markup m times the discounted value of the anticipated output that will be produced by the new process, ( )∑∞= −+1 *2*21t ttt ypr , less the amount of input that is allocated to the development of the new process in period 0, 0z , which in turn is equal to the value of the output of commodity 1 that is foregone due to the R&D investment. The measure of the net benefits of the project defined by (3.83) is very close to the partial equilibrium measure of the net benefits of a privately funded R&D project that was suggested by Diewert (2005b) and (2005c).40  Obviously, we could go through various cases where the government funds the R&D project in various ways or we could look at the various cases where the private sector funds the R&D investment in our present more realistic model. However, the results of analyzing all of these cases will be similar to the cases that we analyzed in sections 3.4 and 3.5 above for our initial highly simplified model.  There are considerable measurement problems associated with implementing the R&D measurement model defined by the net benefit measure (3.82). It will generally be possible to obtain estimates of R&D effort in the current period, 0z , and it will be generally possible to make somewhat realistic estimates of the appropriate real interest rate r or the discount factor ( )r+1/1  but it will be very difficult to form estimates of the monopolistic markup factor m  defined by (3.77)41 or to form estimates of the amounts of input that the new technology will use in future periods.   40 The main difference between our present measure defined by (3.83) and Diewert’s suggested measure is that Diewert allowed for an erosion of the monopoly markups over time. However, it should be noted that Diewert’s partial equilibrium measure of the benefits of a privately funded project will usually understate the social benefits; see the discussion at the end of this section. 41 More realistically, we need to form estimates of the future expected sequence of monopolistic markup factors; i.e., m  is unlikely to be constant over the life of the project technology. For a publicly funded R&D project where the technology is made freely available, it will be particularly difficult to form estimates of the monopolistic markup m  or equivalently, of the output input coefficient 0a  that corresponds to the displaced old technology.  68 At this point, it is necessary to discuss some of the limitations of our present model. We have allowed for obsolescence of the technology due to changing tastes. However, there are other sources of obsolescence. There are a number of factors that determine the rate of obsolescence for a newly developed technology:  • Competitors develop new processes or products that erode the comparative advantage of the original new process or product. • Tastes change, leading to changes in demand for the products that the new technology produces over time. • Economic growth and nonunitary income elasticities change the demand for products over time; e.g., horses give way to bicycles that give way to motor bikes which in turn give way to automobiles which in turn, may give way to public transit!  Of the three factors listed above, we have modeled only the second factor; i.e., we have introduced exogenous changes in tastes that cause households to shift their demands away from the products that the new technology produces over time. The third factor could be accommodated but as soon as nonhomothetic period preferences are introduced, it becomes necessary to distinguish rich and poor household groups and our model would become very complex indeed.42  In order to model the first factor listed above in the context of a privately developed innovation, we could imagine that in period t, competitors have developed a competing technology that is characterized by the output input coefficient *ta  which satisfies the following inequalities:  (3.84) ;T1,2,...,     t          ; *1*0 =<≤ aaa t (3.85) *1* T   t                    ; >≥ aat .  Thus for periods between 1 and T*, the new technology that corresponds to the output  42 If each household has identical homothetic period preferences, our assumption of a single representative consumer is justified.  69 input coefficient 1a  will enjoy a monopolistic advantage with the imputed markup in period t, tm , defined as follows:  (3.86) [ ] **1 T1,2,...,      t                    01/ =>−≡ tt aam .  Thus instead of assuming a constant markup m which lasts forever, we could assume that there will be a sequence of expected monopoly markups tm  which will eventually become 0 after enough time has passed. However, note that the social benefit of the process innovation is still given by (3.82) or (3.83); i.e., in this case, the private benefit derived by the monopolist will be less than the social benefit. The difference between the social benefit of the innovation and the monopolist’s private benefit will show up as disembodied TPF growth!  In addition to the above difficulties with the model presented in this section, we note some of the other important limitations of our analysis:  • Our model can give only a first order approximation to the effects of an innovation; i.e., substitution effects, both on the consumer and producer sides of our model are absent.43 • Our social welfare function is very simple and highly aggregated. In particular, we did not distinguish how various segments of society would be affected by the innovation. This limitation is particularly important when studying the effects of a privately funded R&D project where the distribution of the gains generated by the project need not be spread evenly over all households in the economy. • Our model assumed that the new technology could be developed in a single period. In fact, many innovations (such as the development of a new drug) require many  43 Consumer substitution effects are eliminated because of our assumption of Leontief within period preferences and the assumption of linear intertemporal preferences between periods. Producer substitution effects are eliminated due to the high degree of aggregation in our producer production functions.  70 years to develop.44 • Our model assumed that the new technology immediately displaced the incumbent technology when in fact, the old technology will not be displaced immediately; i.e., the new technology will only gradually diffuse into the economy. • Our model assumed stable prices over time; i.e., we did not model how general inflation would affect accounting for the project. • Due to the high degree of aggregation in our model, many index number problems were suppressed; e.g., how exactly should the deflator for R&D effort be constructed and how exactly should the monopolistic markup factor be defined when there are many inputs in the economy so that the new technology cannot be summarized by a single output input coefficient?  3.8 Conclusion  Many details remain to be worked out when we account for R&D investments. We believe that the models presented in this paper may be particularly helpful in evaluating the benefits of publicly funded R&D projects, since this topic has proved to be very resistant to analysis. In order to make better decision on R&D investments, analysts in both government and private firms might apply the models developed in this study to estimate the net benefits of their R&D projects when the required data become available.  Some of the more important implications that emerge from our analysis are the following ones:  • The benefits of both privately and publicly funded R&D projects can only be evaluated in the context of an intertemporal general equilibrium model. • In the case of a publicly funded project where the results of the project are made freely available, there are a large number of alternative ways that the government can fund the project, giving rise to a large number of alternative accounting treatments of the R&D investment. In our highly simplified models, these  44 This limitation of our model is not too serious: we need only to cumulate the R&D expenditures over the development periods until the new technology is being used in production. Note that each period’s expenditures need to be carried forward at the rate given by one plus the interest rate.  71 alternative methods for funding the project did not affect social welfare but in more realistic disaggregated models, the alternative methods of funding will affect social welfare. • In the case of a privately funded R&D project, the discounted stream of monopoly profits generated as a result of the project is a lower bound to the social benefits of the project. In general, the social benefits will be considerably larger than the private benefits. • We suggested that the costs of an R&D project should be amortized over time according to the matching principle but this matching will inevitably be somewhat arbitrary, both for private and publicly funded projects. • A “correct” accounting treatment of an R&D project will lead to significant changes to the traditional Solow-Jorgenson-Griliches growth accounting paradigm. The production accounts in the current System of National Accounts will also require significant modifications. In particular, monopoly profits will have to be accommodated in a revised system of production accounts: an R&D “asset” is not at all like a depreciable capital stock asset.  As we indicated at the end of the previous section, there are some significant shortcomings in our modeling of the effects of an R&D project. However, we have provided a start on the development of a reasonable methodology for the treatment of R&D investments.  72 4 Measuring the Stock of R&D Capital in a Growth Accounting Framework  4.1 Introduction  The New Economy is characterized as a knowledge-based economy. Technical innovation, which is mainly attributed to firms’ R&D investments, is regarded as the major source of productivity and long-run economic growth. Economists have long been interested in identifying the quantitative relationship between R&D investments and productivity growth. In order to better reflect the role that R&D plays in productivity growth, the Canberra Group on Capital Measurement has recommended the capitalization of R&D expenditures for the next international version of the System of National Accounts (SNA). Treating R&D as capital formation has direct impacts on productivity analysis. In this paper, we are particularly interested in two issues associated with productivity analysis: namely how to treat R&D expenditures in the growth accounting framework and how to construct an appropriate measure for the R&D capital stock used in this framework and in productivity analysis.  The capitalization of R&D expenditures provides some new challenges for the standard growth accounting methodology45. The “standard” methodology treats R&D capital in a similar manner as physical capital. That is, R&D capital is treated as an additional input factor in the aggregate production function. The measure of the R&D capital stock is an input based measure and is constructed by using the Perpetual Inventory Method (PIM) based on a plausible assumption about the depreciation rate of R&D investments. However, Pitzer (2004) and Diewert (2005d) suggested that the treatment of R&D assets is not quite so straightforward as the “standard” methodology implies. They asserted that these R&D assets do not behave in the same manner as ordinary reproducible capital inputs. Investments in ordinary capital inputs, such as machine or structures, will increase the stocks of capital inputs used in production, which in turn will normally lead to a positive increment in production. On the contrary, an increase in R&D investments will generally not lead to an increase in current period output; the increased R&D investments in the current period will generally increase the stock of general knowledge and will  45 Refer to Diewert (2005b) (2005d) for a more detailed discussion.  73 create new technologies which will either enable producers to produce outputs at lower cost (a process innovation) or enable the production of new products (a product innovation). That is, R&D investments can be regarded as a knowledge production process, which creates new knowledge that can be employed in both goods and services production and in a future round of knowledge production. R&D capital is the output of R&D investments. It is not like a “normal” reproducible capital asset that depreciates with use. However, its value declines with the creation of even newer knowledge, the diffusion of knowledge and changes in consumer tastes.46 To sum up, we agree with Diewert (2005d), who pointed out that “there is a great deal of theoretical work that remains to be done in adapting the standard growth accounting methodology to deal with the complexities that are inherent in the treatment of R&D investments”.  In order to proceed with the capitalization of R&D in the national accounts, we need to construct an appropriate measure of the R&D capital stock. Constructing the right measure of the R&D stock is a critical step in implementing the idea of capitalizing the R&D expenditures. In addition, as a good proxy of the knowledge capital stock, the stock of R&D capital plays an important role in conducting economic research and analysis on many interesting issues, such as the sources of economic growth, the contribution of knowledge capital to productivity growth and the rate of return to knowledge capital. The credibility of the results derived from these various research efforts relies heavily on the proper measure of the R&D capital stock. If each economist constructs his “own” R&D stock for his research, it is obvious that the results based on these “self-made” R&D stocks will not be suitable for conducting comparative studies. Therefore, constructing a widely recognized and appropriate measure for the R&D capital stock is crucial for conducting reliable economic research and deriving meaningful policy implications. Given the importance of the R&D stock, it is worthwhile for us to investigate how to construct a proper measure for it.  Conventionally, the R&D capital stock is constructed by accumulating R&D expenditures based on the perpetual inventory method (PIM) and an arbitrarily chosen depreciation rate in the range from 10% to 15%. The R&D stock at time t can be calculated by adding the current period R&D investment to the depreciated R&D capital stock of the previous period. This PIM-based measure of the R&D capital stock is based on some restrictive assumptions, such as the constant productivity of R&D activity, a constant structure of  46 See first two chapters for more discussion on these points.  74 R&D by type of activity and a constant lag structure within types of activity47. These assumptions are problematic. For example, there is plenty of evidence indicating that the productivity of R&D activity does not remain constant over time. Because of these problematic assumptions, a new way to construct the measure of R&D capital stock should be developed in order to capitalize R&D expenditures.  In the literature, there are some papers discussing the construction of the “knowledge capital stock” based on R&D expenditures. Griliches (1979) and Jones and Williams (1998) used a “knowledge production function” to link current technological knowledge and all the current and past R&D expenditures. Alston, Craig and Pardey (1998) and Esposti and Pierani (2003) developed a more general “knowledge production function” model. The former group built the knowledge capital stock in terms of the sum of the depreciated old knowledge stock and the newly created knowledge. The latter just wrote the knowledge capital stock in terms of all the previous R&D investments. In short, all of these measures depend on R&D inputs. But there is uncertainty about the accuracy with which R&D expenditure represents delivery of R&D outputs. We need to be aware of the limitation of this input-based measure of R&D capital stock when we try to design a new approach to the construction of R&D capital stocks.48  The main objective of this paper is to propose a new treatment of R&D investments in the growth accounting framework and new measures of the stock of the R&D capital. The major issues we will deal with are the construction of the appropriate measures of both the general stock of knowledge and the more specific stock of technological knowledge.49 The key feature in section 4.2 is the introduction of two aggregate production functions. The first aggregate production function is the traditional macro production function that produces market goods and services as a function of aggregate labour input, aggregate (traditional) capital services input and it is also conditional on the stock of technological knowledge prevailing in the economy at the beginning of the period. The second production function describes how new technological knowledge is created using inputs of labour and capital and this second production function is conditional on the stock of  47 For more detail discussion, refer to Shanks and Zheng (2006). 48 However, in the end, our measures of knowledge stocks will end up being input based as well. 49 The general stock of knowledge includes mathematics, the laws of physics and all other forms of knowledge. The more specific stock of technological knowledge is a listing of all the methods for producing marketed goods and services. The latter stock can be thought of as a stock of blueprints or recipes.  75 general knowledge prevailing in the economy at the beginning of the period. In subsequent sections, we will show how this incremental knowledge feeds into the construction of both the general knowledge stock and the technological knowledge stock using a variant of the perpetual inventory method for constructing capital stocks.  The remaining sections of this chapter are structured as follows. Section 4.3 discusses some conceptual issues related to R&D capital and the corresponding R&D capital stocks. Section 4.4 develops a specific model to construct a measure of the general knowledge stock and the technology knowledge stock under the assumption of no knowledge diffusion. Section 4.5 attempts to implement our model by making plausible assumptions about some of the underlying parameters. The differences of our treatment in productivity analysis from the conventional treatment are presented in this section along with some sensitivity analyses on our basic assumptions. Section 4.6 provides a brief conclusion.  4.2 The Treatment of R&D Capital in the Growth Accounting Framework  The main objective of R&D ventures is to create new knowledge, either for future general knowledge creation or for the creation of new recipes or technologies that can produce market goods and services. In another words, the output of an R&D project can be additions to the general stock of knowledge as well as new technologies that can be immediately applied by market oriented producers. In general, we can assume that at least some production units will engage in these two types of knowledge creation. The knowledge production function gives the increment of new knowledge created as a function of the existing knowledge stock and current period inputs of intermediate inputs, labour and traditional capital. On the other hand, the ordinary product production function transforms intermediate inputs, labour and traditional capital into market goods and services. Over time, the stock of technological knowledge acts as a shifter in this traditional macro production function. Therefore, we can see that there are two distinct knowledge stocks (general knowledge and technological knowledge) and they play different roles in the two different production activities.  It is not only R&D activities that create new knowledge; other formal and informal  76 activities also create new knowledge of both types. However, in this chapter, we will concentrate on the role of R&D capital in creating new knowledge.50 It is difficult to decompose a given R&D project into contributions to both the general and the technology knowledge stocks. Later, we will make some rough guesses about this decomposition but for now, we assume that the investment in R&D in period t, tRI , , and previous periods feeds into the construction of both the end of period t general knowledge stock, KtR , and the technological capital stock, TtR , using a variant of the perpetual inventory method for constructing capital stocks. We assume that the economy (or a sector of the economy) has two types of aggregate output and two aggregate production functions. The two outputs are the creation of new general knowledge in period t, tI , and market goods and services produced by the sector during period t, tY .  Let tI  denote the new knowledge product created during period t and let G denote the general knowledge production function. Then we can write the knowledge asset created by R&D investments during period t in the following form:  (4.1) ),,( 1, tRIGI K ttRt −= where tRI ,  is the R&D investment made in period t (this is an input measure), K tR 1−  is the effective general knowledge stock available for the sector at the beginning of period t and t is a time variable that shifts the technology for knowledge creation in an exogenous manner.  We are also interested in the production of ordinary goods and services by the sector under consideration. Let tY  denote the aggregate output of market goods and services by  50 The effects of the other forms of knowledge creation will not be explicitly modeled but their effects will be captured by allowing the two production functions to depend on time t as well as the relevant knowledge stocks. It is empirically important to not attribute all shifts in the macro production function to R&D investments since this will lead to an overstatement of the rates of return earned on R&D investments; see Diewert (2005d) on this point.  77 the sector in period t and let F denote the corresponding production function. Then we have:  (4.2)  ),,,( 1 T tttt RtLCFY −=  where tC  and tL  are the inputs of traditional capital services and labour services into the sector that are used to produce market goods and services.51 The time variable t and the stock of technological knowledge TtR 1−  are variables that shift the production function as time moves on. Note that there are two separate knowledge stocks, KtR 1−  and TtR 1− , in the production functions which appear in (4.1) and (4.2).  Totally differentiating equation (4.2) with respect to time t and dividing both sides by tY , we have the following equation after some rearrangement:  (4.3) Y tF R R L L C C Y tF R R Y R R F L L Y L L F C C Y C C F Y Y T t T t RLC T t T t T t T t ∂∂+++= ∂∂+∂ ∂+∂ ∂+∂ ∂= − − − −− − / / 1 1 1 11 1 &&& &&&& ηηη  where dots indicate time derivatives and Xη  denotes the output elasticity with respect to component X. Equation (4.3) decomposes aggregate market output growth into different factors that explain this growth. Based on this equation, we can also conduct productivity analysis. The residual that results from regressing the output growth rate on the growth rates for capital input, labour input and the technology R&D stock is generally used as the measure of (unexplained) productivity growth in the growth accounting framework. We can also define Total Factor Productivity and Total factor Productivity Growth of the economy going from period t-1 to t as follows:    51 Intermediate inputs can be included with the labour inputs. Note that inputs that are used by the sector to produce knowledge do not appear in this list of inputs; rather, they appear in (4.1) as components of IR,t.  78 (4.4) tt XYTFP /=          and (4.5) 1 1 / / − −= tt tt TFP XX YYγ where Yt denotes the aggregate sectoral output excluding the knowledge products and Xt denotes the aggregate input used by the sector (excluding inputs used to produce knowledge products). How we treat the R&D capital stock in the production function; i.e., either as an input factor or as a technological shifter, would have some impact on the estimation of productivity. We will investigate this impact later in this paper.  The main difficulty in applying this growth accounting framework is the construction of proper measures of the two types of knowledge stocks: the R&D capital stocks for both general knowledge creation and for goods and services production, denoted by KtR  and T tR , respectively. In the following two sections of this paper, we will focus on how to construct the appropriate measures of these two types of R&D capital stocks.  4.3 Conceptual Issues  Before we go further into the basic methodology for constructing the appropriate measures of the R&D capital stocks, we need to clarify some conceptual issues in advance.  4.3.1 R&D Knowledge Investment and the Stock of R&D Knowledge Capital  R&D knowledge investment (capital) It (an output) is defined as the knowledge asset created by the R&D investments IR,t (an input measure) in each period. It is the incremental addition to the general knowledge stock created by R&D investments of resources. The stock of R&D capital is the accumulation of past R&D knowledge investments (output measures). It forms a good proxy for the general knowledge stock.   79 A major question is what should be included in the general stock of knowledge. Should we include all the knowledge created in history? This seems to be too broad a definition to be workable at this stage. Thus, we will adopt a narrower definition, based on the accepted definition of R&D capital. Note that R&D capital is defined as a type of asset. To be qualified as an asset, R&D capital should be able to generate economic benefits for its owner. The main benefit of an R&D investment is its ability to create a new technology. The new technology may be used to produce a new product or a quality-improved product that can be either a final good or an intermediate good; it can also be a cost-saving new process. With a successful R&D venture, the investor can benefit from newly created knowledge through the following channels:  (a) Commercializing the new technology through patenting. Thus, the innovator can benefit from selling or leasing the rights to use the new knowledge. This benefit is due to the exclusive right of the owner to the new knowledge asset and the high efficiency associated with the new technology. Because of the difficulty of maintaining the exclusive right and the obsolescence of knowledge, the incremental income accrued from the new knowledge will decrease as time goes by. (b) Employing new knowledge in the production process and earning monopolistic profits from the newly emerging market stimulated by the new knowledge, or enjoying a higher profit margin because of the improvement of efficiency. Monopolistic profits may shrink over time as the diffusion of knowledge proceeds. However, as long as the firm still uses the knowledge, it can still benefit from the utilization of its created knowledge.  Therefore, we propose that the stock of R&D capital should include all knowledge assets that are created by R&D investments and are still utilized by an economic entity. This means that the R&D capital stock should include patents and created new technologies that are kept confidential. The useful life of R&D capital depends on its efficiency and utilization level.     80 4.3.2 R&D Capital versus Knowledge Capital  As we have pointed out before, the stock of R&D capital can be regarded as a proxy for the stock of knowledge capital. However, R&D capital is not exactly equal to knowledge capital. Although R&D investment can create the lion’s share of new knowledge, other sources, such as “learning by doing” or “education” process can generate new knowledge too. Therefore, the stock of R&D capital can only be a good approximation to the knowledge stock; i.e., the scope of the knowledge stock is wider than that of the R&D stock. When we try to draw some implications about knowledge capital from the analysis of R&D capital, we should be aware of this limitation. In this paper, we only consider knowledge capital created by R&D investments. Without a particular specification, knowledge capital actually means R&D capital in our paper.  4.3.3 Measures of the R&D Capital Stock  In this paper, the knowledge stock of R&D capital for the sector under consideration, KR , is the accumulation of general knowledge created by past R&D investments. However, there is a narrower R&D stock concept, which is the stock of blueprints or recipes, used in general production for the sector under consideration that is also created by R&D projects. This is the R&D technology knowledge stock, TR . For the sector under consideration, it can be regarded as a technological index that indicates the efficiency level of technology for the sector. This knowledge stock is also an accumulation of past R&D investments. For both of these knowledge stocks, there are severe measurement issues in measuring the outputs: how should we aggregate different forms of general knowledge and how can we form aggregates of recipes in a meaningful way? Thus forming independent estimates of the values or prices of these two knowledge stocks will not be attempted here. Instead, we will follow national accounting conventions when there are difficulties in measuring outputs and measure outputs by using input estimates.  R&D capital belongs to the family of intangible assets. The unobservability of this type of asset causes many difficulties in its measurement. It would be preferable to directly measure these outputs. However there are difficulties in directly measuring the created  81 knowledge. The primary difficulty is that we lack direct measures of knowledge capital created by R&D investment. Patent data can provide a measure for some R&D outputs, but they have significant limitations. Shanks and Zheng (2006) pointed out that patent protection tended to be used more for product innovation than for process innovation and many new forms of knowledge that created new processes simply could not be patented. In addition some R&D performing industries avoid the use of patents in order to better preserve secrecy about proprietary technologies. A second possible method that could be used to value the outputs of R&D ventures is to value a project by the discounted expected stream of profits that the new technology will generate. However, these future expected yields accrued from a newly created knowledge are generally not observable due to the lack of future markets for these knowledge assets. Also, profitability does not fully capture the benefits of projects that are made freely available to users. Moreover, in practical terms, we generally do not know how fast obsolescence will set in. Because of the limitations on the information about project outputs and the special features of knowledge assets, we are forced to use the information about R&D expenditures to construct input-based measures of R&D capital stocks.  With the above clarifications in mind, we will move on to discuss how to construct appropriate measures for the two types of R&D capital stocks.  4.4 Measuring Stock of R&D Capital without Knowledge Diffusion  In this section, we discuss how to construct proper measures for the R&D capital stocks under the assumption that there is no knowledge diffusion. This means that each firm or industry has to rely on its own R&D activity to improve its technological efficiency and apply its own R&D capital in the production procedure. We still assume that each firm or industry conducts two different activities, namely knowledge production and product production. In order to construct appropriate measures of R&D capital stocks, we start with measuring the knowledge product or R&D investment asset, which is the direct output of knowledge production and the basic building block for constructing the stocks of R&D capital.   82 4.4.1 Measurement of the Knowledge Product  R&D output, tI , or additions to the R&D knowledge stock, is the output of knowledge production, which is a result of R&D investments, tRI , . Unlike ordinary physical capital, which can be employed without further productive activity (other than installation), R&D capital is created by R&D investments through a “production procedure”. The underlying technology of creating new knowledge can be represented by a knowledge production function, which is a function of general input factors, such as labour input, capital input and intermediate input. In addition, knowledge production also relies on the stock of ideas, the existing knowledge stock KtR 1− , which determines “technological opportunities”. The existing stock of knowledge capital plays a vital role in determining the productivity of new knowledge production. Letting tI  denote the new knowledge product created at the end of time t, we can write the knowledge asset created by R&D investments during period t in the following form:  (4.6) ),,( 1, tRIGI K ttRt −= where G(.) is the knowledge production function, tRI ,  is the R&D input investment made in period t, KtR 1− is the effective general knowledge stock available for the sector at the beginning of period t and t is a time variable that shifts the technology for knowledge creation in an exogenous manner. R&D input investments can be decomposed further into specific factors, such as labour inputs, capital inputs and material inputs. We use the input vector xt to represent these input factors of knowledge production function. Replacing IR,t by xt, we rewrite the knowledge production function as follows: (4.7) ),,( 1 tRxGI K ttt −= Based on our understanding of the knowledge creation process, we impose the following general assumptions on the knowledge production function:  • The knowledge production function is increasing in the inputs. For example, generally speaking, the cost of skilled labour accounts for a large portion of R&D  83 expenditures. Both the quality and quantity of skilled personnel are important elements determining the productivity of knowledge creation. The higher quality of the professional personnel (represented by a higher labour input), the higher is the possibility for the firm to create new knowledge. Similarly, a scientific project conducted by personnel that are more professional would be more likely to be successful. Therefore, we can say that increases in the components of the input vector xt would increase the likelihood of creating new knowledge.  • The knowledge production function is assumed to be increasing in the existing knowledge stock. The existing knowledge is another important factor determining the productivity of the knowledge production. However, the role it plays in the knowledge creation is subtle. Because the current researchers “stand on the shoulders” of the previous researchers, usually we would expect to see that the bigger is the R&D stock that is in position, the higher is the probability of creating new knowledge. However, as far as levels are concerned, it is likely that the chances of finding a new technology are small, even if the existing knowledge stock is large. To sum up, we assume that knowledge creation is increasing in the existing knowledge stock but the rate of increase is probably small.  • Knowledge production is concave in its elements. This implies constant or diminishing return to scale in the creation of new knowledge.  What is the functional form for the knowledge production function? In the literature, the Cobb-Douglas knowledge function is a popular choice. In this paper, we will choose a flexible functional form.  In the following part, we attempt to derive an explicit measure of the newly created knowledge product at each period using index number techniques. Every R&D activity performer is assumed to minimize the cost of creating new knowledge with a given output level. Then the total cost of knowledge production can be written as follows:  (4.8) }),,(:{min),,,( 11 t K ttttx K ttt ItRxGxwtRwIC =⋅≡ −− where wt is a positive vector of input prices.  We will consider two special cases of the general production function G which appears in  84 (4.7) and (4.8). Case 1 is the case where the new knowledge creation does not depend on the existing stock of general knowledge, KtR 1− .52 The resulting production function can be rewritten as follows:  (4.9) ),( txGI tt = where the period t input vector is xt. If we use a translog functional form as our knowledge production function, then the log of It can be written as follows:  (4.10) 2 1 ,0 1 1 ,, 1 ,0 lnlnln)2/1(ln ),(lnln txttxxx txGI N n ntn N i N j jtitij N n ntn tt γββααα +++++= = ∑∑∑∑ == ==   where ),( txGI tt =  is the new knowledge created during period t, and ( )Ntttt xxxx ,2,1, ,...,=  is a vector of inputs used by the firm during period t. With the following restrictions on the parameters,  (4.11) ∑ ∑∑ = == = ==== N n n N i ijjiij N n n 1 11 0 and  N1,2,...,jfor   ,0  ,  ,1 β αααα  G(.) defined by equation (4.10) is linearly homogeneous in x, and the resulting translog knowledge production function can provide a second-order approximation to an arbitrary twice continuously differentiable function of (x, t) that is linearly homogeneous in x.  Since equation (4.10) is a quadratic form in the components of log x and t, we can apply Diewert’s (1976; 118) Quadratic Identity and get the following equation:   52 In traditional approaches to modeling the effects of R&D, the R&D capital stock is treated just as if it were another “regular” input. For one of the best applications of this method, see Bernstein and Mamuneas (2005). However, this traditional approach assumes that knowledge is just another input whereas we believe that it shifts the production function rather than acts as a normal input.  85 (4.12) )( ),(ln),(ln)2/1( )ln(ln)],(lnˆ),(lnˆ)[2/1( ),(ln),(ln 01 0011 01000111 0011 tt t txG t txG xxtxGxtxGx txGtxG xx −⋅⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+∂ ∂+ −⋅∇+∇= −  where ≡1x̂  the vector 1x  diagonalized into a matrix, and ≡0x̂  the vector 0x diagonalized into a matrix. If we assume that knowledge developer faces the input price vector 0w >>0N, 1w >>0N during periods t0  , t1 and the producer competitively minimizes costs, then the observable quantity vectors in the two periods, 0x  and 1x , are the solutions to the corresponding cost minimization problems, and we can derive the following identities:  (4.13) ( )00000 /),(ln xwwtxGx ⋅=∇ ; ( )11111 /),(ln xwwtxGx ⋅=∇ Substituting (4.13) into (4.12) yields the following equation: (4.14) )(),(ln),(ln)2/1( )/ln(][lnln 01 0011 1 ,0,1,0,101 tt t txG t txG xxssII N n nnnntt −⋅⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+∂ ∂+ +=− ∑ = where ),( rrt txGK r =  and ( )rrnrnrnr xwxws ⋅= /,,,  for r = 0,1 and n = 1, 2,…,N. Rearranging and exponentiating equation (4.14), we can obtain the following relationship:  (4.15) ⎭⎬ ⎫ ⎩⎨ ⎧ −∂ ∂+∂ ∂= )](),(ln),(ln)[2/1(exp),,,( 0100111010 0 1 tt t txG t txGxxwwQ I I T t t where ),,,( 1010 xxwwQT  is the Törnqvist quantity index in R&D inputs. It is defined as follows:   (4.16) ⎭⎬ ⎫ ⎩⎨ ⎧ +∑= = )/ln()( 2 1exp),,,( ,0,1,0,1 1 1010 nnnn N n T xxssxxwwQ .  86  The exponential part of the right-hand side of equation (4.15) represents a theoretical expression for the cumulative effects of disembodied technical progress on knowledge production. Using equation (4.15), we can construct an input based measure of the incremental knowledge capital created during the period which is based on both R&D inputs and increases in the disembodied knowledge productivity. The problems associated with applying this equation are as follows: (a) we have ignored the effects of the existing knowledge stock in the derivation of equation (4.15), and (b) it is difficult to measure the disembodied technical progress term.  Case 2 of the general knowledge creation model defined by (4.7) will now be explained. Instead of treating the existing knowledge stock KtR 1−  as just another input factor or omitting it altogether, we now assume that KtR 1−  enters into the knowledge production function in the following separable way:  (4.17) ( ) )(),(,, 11 KttKttt RhtxgtRxGI −− == where ),( txg t  is the knowledge production function in terms of variable inputs, while h( KtR 1− ) is a function of the existing general knowledge stock, K tR 1− , which reflects the impact of the existing knowledge stock on new knowledge creation. Again we choose a translog functional form for ),( txg t . Note that (4.17) implies the following relation: (4.18) ( )Kttt RhItxg 1/),( −= . Thus if the producer solves the period t cost minimization problem (4.8), he or she will also solve the following period t cost minimization problem which conditions on tI  and K tR 1− : (4.19) ( ){ }1: ( , ) / Kx t t t t tMin w x g x t I h R −⋅ = . Again applying Diewert’s Quadratic Identity, we can derive the following relationship:   87 (4.20) )(),(ln),(ln)2/1(),,,(ln ),(ln),(ln 01 0011 1010 0011 tt t txg t txgxxwwQ txgtxg T −⋅⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+∂ ∂+= −  Substitute equation (4.18) into equation (4.20) and exponentiate both sides of equation (4.20) and we obtain the following relation:  (4.21) ⎭⎬ ⎫ ⎩⎨ ⎧ −∂ ∂+∂ ∂= − − )](),(ln),(ln)[2/1(exp),,,( )(/ )(/ 01 0011 1010 1 1 00 11 tt t txg t txgxxwwQ RhI RhI TK tt K tt Rearranging (4.21), we can derive the following input based measure of the newly created R&D knowledge asset:  (4.22) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⎭⎬ ⎫ ⎩⎨ ⎧ −∂ ∂+∂ ∂= − − )( )( )](),(ln),(ln)[2/1(exp),,,( 1 1 01 0011 1010 0 1 0 1 K t K t T t t Rh Rh tt t txg t txgxxwwQ I I From equation (4.22), we can see that the ratio of new knowledge assets created during two different periods is determined by the product of three parts: (a) a Törnqvist quantity index in “regular” inputs, which can be calculated using observable price and quantity data obtained from R&D expenditure information pertaining to the two periods under consideration; (b) an (unobserved) measure of disembodied technical progress on the creation of knowledge and (c) a measure of the rate of growth in the stock of general knowledge between the two periods, which measures the impact of the existing general knowledge stock on the knowledge creation and is typically difficult to measure.  In the existing literature, only the first term is taken into consideration. Usually, the effects of disembodied technical progress and of the stock of existing ideas are ignored or assumed to be constant. This treatment can simplify the problem, but may cause measurement errors. Even though the general measure of R&D capital defined by equation (4.22) reflects the reality of knowledge creation more closely, its implementation will be difficult due to data limitations. Difficulties associated with applying equation (4.22) include: (a) estimating the effects of disembodied technical progress, (b) constructing a realistic estimate of the stock of knowledge for the two periods, and (c) determining the functional form of the h function. We cannot obtain econometric estimates for these effects by estimating the knowledge production function due to the un-observability of the created new knowledge, tI  during each period. In  88 what follows, we will make some additional assumptions that will allow us to deal with these measurement difficulties.  If the time interval between t1 and t0 is short, we may not expect to see significant changes in the technology for producing new knowledge and thus we may assume that: (4.23) 0),(ln and ,0),(ln 0011 ≈∂ ∂≈∂ ∂ t txg t txg (4.24) 1 )( )( 1 1 0 1 ≈ − − K t K t Rh Rh  Thus, for two close periods, a useful approximation to (4.22) is:53 (4.25) ),,,( 1010 0 1 xxwwQ I I T t t ≈ This implies that if we use chained quantity indexes, our measure of R&D incremental knowledge growth may have a relatively small measurement error. That is we can construct a measure of newly created additions to the stock of R&D knowledge capital at each period using the following equations: (4.26) ... ),,,(),,,(),,,(),,,( ),,,(),,,(),,,( ),,,( 0101021213232232323 010102121121212 010101 00 VxxwwQxxwwQxxwwQIxxwwQI VxxwwQxxwwQIxxwwQI IxxwwQI VI TTTT TTT T == == = =  where V0 is the value of R&D expenditures in period 0.  In section 4.5 below, we will implement the more general model of knowledge creation that was defined by (4.22) above but we will make somewhat arbitrary assumptions about the magnitude of disembodied technical progress and about the functional form for the h function. The h function will be assumed to have the following functional form: (4.27) α/111 )()( 11 K t K t RRh −− = where α should be greater than 1. In fact, we will pick α to equal 2.   53 However, note that this approximation is essentially equivalent to a “traditional” approach, which we have just criticized.  89 The production function model for the creation of incremental knowledge tI  in period t defined by (4.7) above is the first step in constructing the measures of the two types of knowledge stocks (general knowledge stock and technological knowledge stock) which were discussed in section 4.2. In the following section, we will use these knowledge stock additions tI  to create the two knowledge stocks, K tR  and T tR , as (different) weighted sums of past period increments to knowledge.   4.4.2 The Construction of the Two R&D Capital Stocks  The addition to knowledge created by a sector’s R&D investments, tI , is the increment to the sector’s R&D general knowledge capital stock, KtR . This knowledge stock will be set equal to a weighted sum of past period increments to the knowledge stock. Below, we will discuss how these weights are determined. Some of the knowledge increments can be used as recipes or blueprints for producing goods and services. Therefore, the past period incremental knowledge outputs, 1−tI , 2−tI , ... , also determine the technological knowledge capital stock, TtR , which is the stock of recipes in use by the sector. Thus, this technology knowledge stock will also be set equal to a weighted sum of past period increments to the knowledge stock (but the weights may be different from the weights used to construct KtR ).  Again, we assume there are no knowledge spillover effects in the aggregation. Let tI denote the knowledge product created during period t and we assume that this incremental knowledge can be used by each firm in the sector at the end of time t (or at the beginning of time t+1). The set of past period increments to the knowledge stock can be written as { tI , 1−tI , 2−tI , 1I  and 0R }, where 0R  denotes the initial R&D stock available to the sector. Let KtR  and T tR denote the R&D stocks for general knowledge  90 and for technological use (i.e., for ordinary product production) at the end of time t, respectively. We assume that the general knowledge stock, KtR , can be constructed in the following way: (4.28) 0113322110 ... RIIIIIR t t t tt t t t t t t tK t μμμμμμ ++++++= −−−− where tiμ  is the relevancy index of i year old knowledge at period t. The relevancy index reflects to what extent the knowledge increments created in previous periods are related to the creation of new knowledge. Usually, we assume that the more recent the increment is, the more relevant it will be to new knowledge creation. Thus, we assume that the following relationship among the relevancy indexes holds:  (4.29) tt t t tttt μμμμμμ ≥≥≥≥≥ −13210 ...  The effective knowledge stock for the production of goods and services in the sector, the technology knowledge stock, TtR , can be constructed in the following way:  (4.30) Ttt t tt t t t t tT t RuIuIuIuIuR 01122110 ... +++++= −−− where tiu  denotes the time t effective efficiency index of the i-year-old knowledge asset and TR0  is the initial stock. A firm’s technological R&D stock is designed to represent the average efficiency level of its technology. The tiu s transform all the knowledge increments created at different time into effective technological knowledge with same efficiency level at time t. As a weighted sum of all the past knowledge investments, weighting by these effective efficiency indexes, TtR  can act as a technological shifter in the production of goods and services.  Theoretically, the stocks of general knowledge R&D capital and production recipes R&D capital can be constructed using equations (4.28) and (4.30). However, the problem is that the weights for the tI  in (4.28) and (4.30) (the relevancy and effective efficiency indexes) are not observable. In the following part of this section, we discuss how to  91 determine the relevancy indexes and effective efficiency indexes.  4.4.2.1 The Determination of the Relevancy Indexes  According to equation (4.28), the sequence of the general knowledge stocks can be written as follows: (4.31) ... ... ... ... 0 2 21 2 34 2 23 2 12 2 02 0 1 11 1 23 1 22 1 11 1 01 0113322110 RIIIIR RIIIIR RIIIIIR t t t tt t t t t tK t t t t tt t t t t tK t t t t tt t t t t t t tK t − − − −− − − − − − − − − − −− − − − − − − −−−− ++++= +++++= ++++++= μμμμμ μμμμμ μμμμμμ  The superscript of the relevancy index, t-i, i = 0, 1,2,…, indicates that the corresponding index is valid at period t-i. The subscript of the index indicates the age of the knowledge increment at period t-i. Note that the relevancy index is designed to reflect the relevancy of old knowledge to newly created knowledge. To simplify our problem, we assume that the relevancy index is independent of what would be created in the future; that is, the relevancy index is assumed to depend only on the time lag between the new knowledge and the existing old knowledge, or the age of the knowledge at a particular time. Based on this assumption, the following condition holds: (4.32) l t l t l ji μμμ ==  for ji tt and∀  where ti and tj denote two different points in time. Applying equation (4.32) to equation (4.31) and dropping the superscripts of the relevancy indexes, the sequence of the aggregate R&D capital stocks can be written as follow:  (4.33) ... ... ... ... 02134231202 01123221101 0113322110 RIIIIR RIIIIR RIIIIIR ttttt K t ttttt K t tttttt K t −−−−−− −−−−−− −−−− ++++= +++++= ++++++= μμμμμ μμμμμ μμμμμμ  How can we obtain estimates for these relevancy indexes? One possible way would be to use citation information. A well-established citation network might provide us relatively useful information for tracing the relationship between newly created knowledge and the  92 old knowledge. If we use tN  to denote the total number of citations made at time t, and itN ,  to denote the number of citations made at time t but referring to knowledge assets that are i years old, then the relevancy index, tiμ , could be estimated by the following formula: (4.34) t itt i N N ,=μ The above index is a time-specific relevancy index of i-year-old knowledge asset. The general time independent relevancy index, which is only related to the age of the knowledge, can be obtained by taking a weighted average over these time specific indexes; that is: (4.35) ∑∑= t it it t t ii N N , ,μμ We note that at present, it is difficult to get accurate citation information and so it will be difficult to implement the above model. However, over time, citation information may be improved and the above method could then be implemented. 4.4.2.2  The Determination of the Effective Efficiency Index  As defined in equation (4.30), the measure of the technology knowledge capital stock depends on the effective efficiency indexes. Generally speaking, we cannot observe these indexes of a specific knowledge directly. Then, how should we construct them? In the following part of this section, we attempt to explore possible ways to tackle this problem.  The estimates of the utilization level of a knowledge asset can be used as its effective efficiency index. In order to determine the utilization level of past R&D investments for the construction of the technology knowledge capital stock, we need to make some assumptions about the utilization path or life cycle of a particular knowledge investment. Before we specify these assumptions, we list some observable facts about the application of a new knowledge:  • Firms have a tendency of employing technologies that are more advanced as they are developed. This implies that the utilization level of an old technology would  93 be reduced by a certain portion due to the emergence of a new technology. • Usually, at the introductory stage of a new knowledge, the features of the new knowledge are not fully mastered by the workers using the technology. Thus, the utilization level of a new knowledge investment starts at an initial low level and gradually increases over time up to a certain point. • The utilization level of a knowledge investment will typically remain at a high level for some time until another new knowledge investment creates a better technology. • Following the emergence of a new technology, the utilization level of the old knowledge will start to decline for the rest of its service life until it is totally replaced by the new technology.  Based on the above “facts”, we may draw the following diagram to illustrate the utilization path of a specific technology:  Figure 4.1 Utilization Path            Although frequently, there might be a flat part on the utilization path, to simplify our analysis, we ignore this flat part. Thus, we assume that the utilization level of a newly created knowledge will increase at the early stage and it would start to decrease after reaching its peak point. The utilization of one technology would probably reach its maximum point when another new technology is developed, which will cause the existing technology to become obsolescent. Our assumptions imply that the utilization path exhibits an un-balanced bell shape. In summary, we need the following information to determine the whole path of the utilization:  94 (a) The initial utilization level of the R&D capital created at time t, tu0 . (b) The increasing rate of the utilization level at the introductory stage of the knowledge. (c) The position of the peak point. In order to locate the peak point, we need to know the frequency of creating new knowledge. (d) The decreasing rate of the utilization level after it reaches the peak point; (e) The length of the useful service life of the R&D capital, L, and the utilization level just before the old technology is fully displaced.  Due to data limitations, we do not have all the required information to determine the utilization path. In the remainder of this section, we make some simple assumptions to show how the utilization level can be determined if we have the required information. Our assumptions include: (a) The initial utilization index is 0u ; the maximum utilization level is 1; and the ending utilization level, which is the last positive utilization level just before an old knowledge fully displaced by a new knowledge, is Lu ; (b) The average time lag between the old technology and the new technology is γ ; (c) The useful life of the R&D capital is L; (d) The utilization level increases or decreases at a constant geometric rate, denoted by α and β, respectively;  With the above information, the geometric rate of increase can be derived by the following formula: (4.36) 1]/1[ )/1(0 −= γα u The geometric decrease rate can be calculated as follows: (4.37) )]/(1[][1 γβ −−= LLu  Using the above formulas, we can determine the utilization levels for the knowledge investments of different vintages. Let iu  denote the utilization level of the knowledge asset that is i year old, and then we can derive the utilization level by using the following formulas:  95 (4.38) γα /000 )/1()1( iii uuuu =+=       for i ≤ γ       )/()()()1( γγγβ −−− =−= LiLii uu              for γ < i ≤ L     0=iu                            For i > L  With all the required information in hand, theoretically we could construct the separate measures of the R&D capital stocks for general knowledge and for technological knowledge according to (4.28) and (4.30), respectively. In the following section, we will make some assumptions about the relevant parameters to show how our framework works.  4.5 Empirical Results  In this section, we will construct measures of the knowledge increments and the knowledge stocks applying both the conventional approach and our new approach. Due to data limitations, we will construct these measures by assuming different key parameters and check how sensitive of our measures to these different assumptions. We will then conduct productivity analysis using growth accounting framework based on the constructed measures.  4.5.1 Measuring the R&D Stock  Conventionally, the R&D stock is constructed based on the perpetual inventory method (PIM) using R&D input data. The essence of the PIM is to form an annual estimate of the knowledge stock by adding new R&D expenditures of the year to the existing stock and subtracting ‘depreciation’ or obsolescence of the existing stock. In other words, using the conventional approach, the series of R&D expenditures can be formed into measures of the R&D stock according to the following formula: (4.39) 1, )1( −−+= ttRt RIR δ where tR  is the stock of R&D capital at the end of period t, tRI ,  is the real R&D investment during period t, and δ is the geometric depreciation rate for the R&D stock.  96 This input-based measure of R&D stock is relatively easy to construct. However, because some key assumptions underlying the PIM, such as the fixed productivity of R&D activity and the constant rate of knowledge decay, do not hold in the real world, concerns about the measurement of R&D stock have been raised in both empirical and theoretical works54.  According to our proposed approach, we need to construct separate measures of general knowledge capital stock and technological knowledge capital stock. The measures of these two types of knowledge stocks rely on the output of knowledge production— the knowledge product, which in turn is affected by the existing knowledge stock employed in knowledge production. The knowledge product can be measured by applying equation (4.22). Because we do not have all the necessary information to derive estimates for the R&D productivity growth rate and for the impact of existing knowledge, we will try different values of the key parameters to test how R&D productivity and existing knowledge affect knowledge production. The following assumptions are our first set of assumptions used in order to construct measures of the knowledge product:  (4.40) ⎭⎬ ⎫ ⎩⎨ ⎧ −∂ ∂+∂ ∂ )](),(ln),(ln)[2/1(exp 010011 ttt txg t txg =1.005 (4.41) 2/111 )()( K t K t RRh −− = Equation (4.40) implicitly assumes that R&D productivity increases at a constant rate. Although this is not necessarily true, we can still examine how variations in R&D productivity affect the creation of knowledge, and the size of R&D stocks. As we have discussed in the previous section, the impacts of the existing R&D capital stock on the knowledge creation is subtle, and our choice of the h function is just our arbitrary assumption. However, our assumptions allow us to examine how the existing knowledge stock affects the production of new knowledge.  The stock of R&D capital for general knowledge production can be constructed based on equation (4.28). An important element of determining this measure is the “relevancy index”. In order to simplify the problem, we assume that the relevancy index starts from 1 and declines at a constant geometric rate. In addition, we assume that the average service  54 Refer to Shanks and Zheng (2006) and Parham (2007) for more detailed discussions.  97 life of general knowledge is 40 years, with the ending relevancy index at 0.0001. Using these assumptions, we can derive the relevancy index as follows: (4.42) 794.0)0001.0( 40/1 ==μ Based on the above assumptions, we can recursively build the measures of knowledge product and R&D stock for knowledge production by using the following equations:  (4.43) ... 794.0 005.1),,,( 794.0 005.1),,,( 794.0 005.1),,,( 233 2 2/1 1 2 223333 122 1 2/1 0 1 112222 011 2/1 1 0 0011111 KK K K KK K K K RIR I R RxwxwQI RIR I R RxwxwQI RIR R RxwxwQVI ×+= ×⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛××= ×+= ×⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛××= ×+= ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛×××= −   where V1 is the R&D expenditure at the first period. R0 is the initial knowledge stock. We construct our estimate of R0 using a well-established method that works as follows:  (4.44) )/(0,0 RRIR γδ += where 0,RI  denotes the R&D investment at the first period; δ denotes the depreciation rate of R&D capital; Rγ  denotes the geometric growth rate of R&D investment over the sample period. This growth rate can be calculated using the following equation: (4.45) )/1(0,, )/( T RTRR II=γ  where TRI ,  is the R&D investment in the last period of the sample period.  The measures of the technological knowledge stock can be constructed by using equation (4.30) based on the constructed measure of knowledge increments. Recall that the  98 effective efficiency index is the key factor for constructing the measures of this type of R&D stock. As we have discussed in the previous section, we need information about the frequency of knowledge creation and the path of knowledge utilization to estimate this index. Because of data limitations, we will make the following assumptions to derive the efficiency factor. First, we assume that there is a new knowledge asset created at each period. Second, the initial utilization index is assumed to be 0.8, and then it increases to 1 at the second period and starts to decline at a constant geometric rate thereafter55. Finally, the technological knowledge can still be used for 20 years after the maximum utilization level has been reached and the minimum utilization level is set to 0.0001. Therefore, we can calculate the constant geometric decline rate, β , defined by equation (4.37), using the following formula:  (4.46) 369.0)0001.0(1 20/1 =−=β  Then the technological knowledge stock can be constructed according to the following equations: (4.47) 011 0122 0 2 1233 4 2 3211 3 2 21 18.0 631.018.0 )631.0(631.018.0 ... ...)631.0(631.0118.0 ...)631.0(631.0118.0 RIR RIIR RIIIR IIIIR IIIIR T T T tttt T t tttt T t ×+×= ×+×+×= ×+×+×+×= +×+×+×+××= +×+×+×+××= −−−−− −−−   We use US manufacturing data from 1953 to 2001 to construct R&D stocks and to conduct a productivity analysis. In order to conduct the empirical work outlined above, we need quantity series and price series of input and output for both general production and R&D investments in US manufacturing. The input and output data other than R&D related data are obtained from the Multifactor Productivity data sets provided by the Bureau of Labour Statistics (BLS). R&D related data are derived from the website of  55  Generally speaking, general knowledge stock should depreciate more slowly than technological knowledge R&D stock.  99 National Science Foundation (NSF).56 The following figures show the main results of our constructed data.  Figure 4.2 The Knowledge Product and the Two Types of Knowledge Stock Knowledge Product and Two Types of Knowledge Stock 0 20 40 60 80 100 120 140 19 53 19 57 19 61 19 65 19 69 19 73 19 77 19 81 19 85 19 89 19 93 19 97 20 01 KnowledgeProduct General Knowledge Stock Technological Knowledge Stock  Figure 4.3 Technology Knowledge Capital Stock and Conventional Knowledge Stocks Technological Knowledge Stock and Conventional Knowledge Stocks 0 20 40 60 80 100 120 19 53 19 57 19 61 19 65 19 69 19 73 19 77 19 81 19 85 19 89 19 93 19 97 20 01 Technological Knowledge Stock Conventional R&D Stock(1) Conventional R&D Stock (2)    56 See Chapter 1 above for a detailed description of the data used here.  100 Figure 4.2 shows the trends of knowledge product output and the R&D stocks for both general knowledge and technological knowledge. It is obvious that the amount of general knowledge stock is greater than that of the technology knowledge stock. This is in line with our belief that general knowledge depreciates more slowly than specific technological knowledge.  Figure 4.3 shows the R&D capital stock calculated using the traditional approach and using our proposed approach. The traditional approach constructs the measure of R&D stock using the input-based methodology, i.e., R&D stock is constructed directly from the R&D expenditures using equation (4.39), with an assumed depreciation rate. We propose to incorporate the knowledge production process into the construction of R&D capital stock. Therefore, our measure of R&D capital stock is built indirectly from R&D expenditures. In addition, we take into account the impact of both the existing knowledge stock and knowledge productivity on our measures of R&D stock. In other words, we attempt to build a link between R&D inputs and R&D outputs via the knowledge production process. The three curves in the figure have the same initial starting stock. The conventional R&D stock57 curve in pink uses a smaller depreciation rate (δ=0.15) than our proposed approach and thus it generates the highest curve in the figure. The blue curve corresponds to a conventional R&D stock curve and it employs a depreciation rate that is equivalent to the utilization index used to construct the technological R&D stock. In other words, the assumed knowledge obsolescence rate is same for the two curves. Because the technological R&D stock is based on knowledge products other than simply the direct R&D expenditures, comparing equations (4.22) and (4.39), we can easily see why the blue conventional R&D stock curve is lower than the technological R&D stock curve.  Recall that the measures of the general knowledge stock and the technology R&D stock are constructed based on our assumptions about some key parameters. From now on, we will check the sensitivity of these two measures to the different assumptions. Figures 4.4-4.7 show how measures of the technological knowledge stock vary with our assumptions. In general, we can see that the R&D stock of technological knowledge is relatively sensitive to R&D productivity and the service life of technological knowledge, while it is not very sensitive to the initial knowledge stock and the starting utilization  57 The knowledge concept used in conventional R&D stock is wider than that used in the technological R&D stock.  101 level. Figures 4.8-4.9 show how the general knowledge stock varies with assumptions about knowledge productivity and the service life of the general knowledge.  Figure 4.4 Technological Knowledge Stocks with Different Rates of Knowledge Productivity Technological Knowledge Stock with Different Rates of Knowledge Productivity 0 50 100 150 200 250 19 53 19 56 19 59 19 62 19 65 19 68 19 71 19 74 19 77 19 80 19 83 19 86 19 89 19 92 19 95 19 98 20 01 0.0025 0.005 0.01 0.015 0.02 0.025 0.03   From Figure 4.4, we can see that the higher the knowledge productivity, the larger is the technological knowledge stock. When the exogenous growth rate of knowledge productivity equals to 3%, we have the highest technological knowledge stock in the figure. As the knowledge productivity gets bigger, the technological knowledge stock is getting more sensitive to the changes in the R&D productivity. Figure 4.5 shows how the technological knowledge stock varies with the service life of the technologies. If the average service life of knowledge assets is long, not surprisingly, more knowledge assets can be accumulated over time. Thus, the 30-year curve is highest one and the 5-year curve is the lowest one in the figure.  102  Figure 4.5 Technological Knowledge Stocks with Different Service Lives  Technological Knowledge Stocks with Different Service Lives 0 10 20 30 40 50 60 70 80 90 100 19 53 19 57 19 61 19 65 19 69 19 73 19 77 19 81 19 85 19 89 19 93 19 97 20 01 5 years 10 years 15 years 20 years 25 years 30 years   Figure 4.6 Technological Knowledge Stocks with Different Initial Stocks  Technological Knowledge Stock with Different Initial Stocks 0 10 20 30 40 50 60 70 80 90 100 19 53 19 56 19 59 19 62 19 65 19 68 19 71 19 74 19 77 19 80 19 83 19 86 19 89 19 92 19 95 19 98 20 01 0.05 0.1 0.15 0.2 0.25 0.3 0.35   103  Figure 4.7 Technological Knowledge Stocks with Different Initial Utilization Levels  Technological Knowledge Stocks with Different Initial Utilization Levels 0 10 20 30 40 50 60 70 80 19 53 19 56 19 59 19 62 19 65 19 68 19 71 19 74 19 77 19 80 19 83 19 86 19 89 19 92 19 95 19 98 20 01 0.6 0.65 0.7 0.75 0.8 0.85 0.9   Figures 4.6 and 4.7 represent how the initial conditions affect the accumulation of the technological knowledge capital. These two figures indicate that the technological knowledge stock is not very sensitive to changes in the both initial knowledge stock and the initial utilization level.  In Figure 4.6, we test how the technological knowledge stock responses to the arbitrary assumptions about the traditional R&D depreciation rate, allowing it only to affect the initial R&D stock through equation (4.44) in our approach. We can only observe the divergences among the technological knowledge stocks with different depreciation rates for the early period. A depreciation rate of 5% gives us the highest curve in the figure.  In general, a higher initial utilization level will lead to a higher technological knowledge stock. Examining Figure 4.7, we can see that the different initial utilization levels within the range from 0.6 to 0.9 do not result in substantial differences in the technological knowledge stocks. However, we can still see that the differences among the technological stocks increase slowly as time goes on. The bottom curve represents the technological knowledge stock curve with 0.6 as the initial utilization level.  104  Figure 4.8 General Knowledge Stocks with Different Knowledge Productivities General Knowledge Stock with Different Rates of Knowledge Productivity 0 50 100 150 200 250 300 350 19 53 19 56 19 59 19 62 19 65 19 68 19 71 19 74 19 77 19 80 19 83 19 86 19 89 19 92 19 95 19 98 20 01 0.0025 0.005 0.01 0.015 0.02 0.025 0.3  Figure 4.9 General Knowledge Stocks with Different Service Lives  General Knowledge Stocks with Different Service Lives 0 20 40 60 80 100 120 19 53 19 56 19 59 19 62 19 65 19 68 19 71 19 74 19 77 19 80 19 83 19 86 19 89 19 92 19 95 19 98 20 01 20 years 25 years 30 years 35 years 40 years 45 years 50 years    Figures 4.8 and 4.9 reflect how the general knowledge stock varies with assumptions  105 about two parameters: exogenous knowledge productivity and length of the service life of the general knowledge. As far as the sensitivity to these two factors is concerned, we find that the general knowledge stock behaves similarly to the technological knowledge stock (Figures 4.4 and 3.5). That is, the general knowledge stock increases with knowledge productivity and the length of the service life of the general knowledge capital. 4.5.2 Productivity Analysis  Based on the constructed measures of the R&D capital stock, we also conducted a simple productivity analysis using a growth accounting framework. We run a simple OLS regression on the following transformed form of equation (4.3): (4.48) 1 1 1 1 1 2 1 1 1 1 2 T T t t t t t t t t t t C L M R T t t t t t Y Y C C L L M M R R tfp Y C L M R η η η η− − − − − − − − − − − − − − − −= + + + +  where tM  is intermediate input. We can use the residuals from the above regression as a measure of the productivity growth. Although there are some econometric problems associated with this estimation, such as the multi-collinearity problem, we still can investigate the impacts of different treatments of R&D capital on productivity analysis. We do not focus on the statistical significance of the estimates; instead, we are particularly interested in the changes in the estimates caused by changes in our assumptions about the role of R&D capital stock.  In order to see how the treatment of R&D capital affects productivity analysis, we run preliminary regressions without explicit R&D capital in the equation. In the following table, Column (1), R&D expenditures are not separated from the inputs factors (R&D expenditures are implicitly included in the regression); Column (2) R&D expenditures are separated from the inputs factors but not treated as an explanatory variable in the regression (R&D expenditures are entirely excluded from the regression). From Table 4.1, we can see that ordinary capital explains more and intermediate inputs explain less output growth when R&D capital is not included in the regression.   106 Table 4.1 Regressions without R&D Capital as an Explanatory Variable   (1) (2) ηL 0.78865 (0.09866) 0.77959 (0.09928) ηM 0.30966 (0.1147) 0.25514 (0.1183) ηC 0.42129 (0.1132) 0.47960 (0.1186) ηL+ηM+ηC 1.5196 1.51433  Notes: Values in the brackets are standard errors.  We then run same regressions by including R&D capital as an explicit explanatory variable using both traditional approach and our proposed approach. The main difference between these two approaches lies in the method of constructing R&D stocks. The conventional R&D capital stock can be constructed based on equation (4.39). Our R&D stocks are constructed based on same amount of initial stock. The estimates obtained from the traditional approach are presented in the following table:  Table 4.2 Estimates from Regressions Using the Traditional Approach   δ=0.05 δ=0.1 δ=0.15 δ=0.2 δ=0.25 δ=0.30 δ=0.35 ηL 0.76818 (0.09643) 0.76397 (0.09563) 0.75795 (0.09423) 0.75158 (0.09321) 0.74893 (0.09325) 0.75143 (0.09486) 0.75638 (0.09628) ηM 0.27498 (0.1158) 0.28412 (0.1151) 0.29920 (0.1138) 0.31664 (0.1133) 0.32625 (0.1143) 0.32362 (0.1163) 0.31457 (0.1181) ηC 0.32673 (0.1391) 0.32168 (0.1379) 0.30480 (0.1320) 0.29767 (0.1290) 0.31617 (0.1262) 0.35173 (0.1241) 0.38590 (0.1226) ηR 0.10672 (0.05465) 0.13798 (0.06369) 0.18178 (0.07264) 0.21507 (0.07837) 0.21180 (0.7823) 0.17712 (0.07276) 0.13639 (0.06497)  Notes: Values in the brackets are standard errors.    107 The estimates obtained from our proposed approach are shown in Table 4.3:  Table 4.3 Estimates from Regressions Using the Proposed Approach   u=0.95 u =0.9 u =0.85 u =0.8 u =0.75 u =0.7 u =0.65 ηL 0.77040 (0.09528) 0.76646 (0.09403) 0.76053 (0.09204) 0.75309 (0.09028) 0.74764 (0.8990) 0.74701 (0.09102) 0.75028 (0.09275) ηM 0.25479 (0.1141 0.25940 (0.1125) 0.26913 (0.1101) 0.28519 (0.1082) 0.30197 (0.1080) 0.31195 (0.1099) 0.31193 (0.1124) ηC 0.31845 (0.1351) 0.30997 (0.1312) 0.288875 (0.1321) 0.27259 (0.1247) 0.27791 (0.1225) 0.30450 (0.1213) 0.33793 (0.1209) ηR 0.12058 (0.05464) 0.15647 (0.0628) 0.20483 (0.07043) 0.24567 (0.07530) 0.25458 (0.07582) 0.22889 (0.07229) 0.18842 (0.06641)  Notes: Values in the brackets are standard errors.  Comparing the above two tables we find that, in general, we obtain fairly similar estimated coefficients. The size of ηR is slightly smaller in Table 4.2 as compared to Table 4.3, while the size of ηC is smaller in Table 4.3. We also find that ηL is relatively stable in both tables as compared to the changes in ηC and ηM after we include R&D capital in the regression. In both tables, we do not observe any consistent relationship between the estimates and the decrease of the R&D stock.  We show the rates of productivity growth in the following set of figures. The lines with different colours in the figures indicate the different movements in productivity growth using two approaches. The general movements in the productivity growth rates estimated using both approaches exhibited similar patterns. This indicates that the traditional approach may be a satisfactory approximation to a theoretically superior model. Another possible explanation is that the assumptions used to derive our estimates, lead to results that are very similar to the outcomes generated by the conventional approach. For example, the factors that are most uncertain, including R&D productivity and the existing knowledge stock, have very small impacts on knowledge production.   108 Figure 4.10 Productivity Growth Trends Estimated by the Conventional Approach and the Proposed Approach (δ=0.15) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 19 55 19 59 19 63 19 67 19 71 19 75 19 79 19 83 19 87 19 91 19 95 19 99 CGTFP-0.15 PGTFP-0.15    Figure 4.11 Productivity Growth Trends Estimated by the Conventional Approach and the Proposed Approach (δ=0.20) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 19 55 19 59 19 63 19 67 19 71 19 75 19 79 19 83 19 87 19 91 19 95 19 99 CGTFP-0.2 PGTFP-0.2   109  Figure 4.12 Productivity Growth Trends Estimated by the Conventional Approach and the Proposed Approach (δ=0.25) -0.06 -0.04 -0.02 0 0.02 0.04 19 55 19 59 19 63 19 67 19 71 19 75 19 79 19 83 19 87 19 91 19 95 19 99 CGTFP-0.25 PGTFP-0.25   Table 4.4 Average Productivity Growth Rates Using the Conventional Approach and the Proposed Approach  Average Productivity Growth Rate Conventional Approach Proposed Approach δ=0.15 0.342% 0.317% δ=0.20 0.326% 0.288% δ=0.25 0.34% 0.287% δ=0.30 0.372% 0.311% δ=0.35 0.404% 0.344%  Table 4.4 shows the average productivity growth rates with different obsolescence rates for the sample period using both the conventional approach and the proposed approach. Our approach yields lower growth rates than the conventional approach. However, note that we only show that the proposed approach and the conventional approach generate different point estimates for productivity growth rates; i.e., we do not test whether the differences between these estimates are statistically significant.   110  4.6 Conclusion  Implementing capitalization of R&D expenditures in the national accounts provides some new challenges to the measurement of R&D expenditures and productivity analysis. In this paper, we investigate how to incorporate R&D capital into the growth accounting framework and how to construct an appropriate measures for the stock of R&D capital.  Traditionally, R&D capital is treated as an additional input factor in the macro production function and the stock of R&D capital is formulated using R&D expenditure data based on the PIM. There is uncertainty about the accuracy with which R&D expenditures represent the creation of R&D outputs. In our paper, we attempt to incorporate the knowledge creation procedure into the construction of measures of the knowledge stock. R&D outputs are linked with R&D inputs via a knowledge production process. We assume that each firm conducts two different types of activities: goods or services production and knowledge production. Because knowledge capital plays different roles in these two different activities, we propose to distinguish general knowledge from the more specific knowledge about how to manufacture or provide goods and services. Our measure of the knowledge stock is constructed as a weighted sum of knowledge increments, and these knowledge increments are created by R&D investments using a knowledge production technology. The proposed measures of knowledge increments and knowledge stocks rely heavily on the quality of the R&D related data. We expect that statistical agencies will provide more accurate R&D measures with improvements in data collecting techniques.  The empirical results show that our proposed approach generates different results from the traditional approach in many aspects. In terms of the size of the R&D stocks, using same knowledge obsolescence rates, the estimates of R&D stock obtained from the proposed methodology is bigger than those obtained from the conventional approach. We also find that the output elasticity with respect to R&D capital stock is, on average, bigger using our approach. The average growth rate of total factor productivity is smaller in the proposed approach than in the traditional approach. However, we do not find large differences in the patterns of productivity growth between the two approaches. All our comparisons are based on same initial R&D stock and similar knowledge obsolescence rates.  111  Our new approach of incorporating R&D capital in the growth accounting framework, distinguishing the general knowledge R&D stock and the technological knowledge R&D stock, seems to be a new idea which more accurately reflects how R&D investments contribute to productivity growth, and is a fresh idea in the literature. The associated measures of the R&D capital stocks are established indirectly based on the input data, using knowledge production as a link and taking the impact of knowledge productivity and existing knowledge stock into account. This method also gives us a new way to deal with the measurement of R&D capital. Providing some new thoughts on measuring R&D capital is the main contribution of this chapter. Although the data for fully implementing all these new ideas are not ready, the new ideas presented in this paper would add to the general knowledge of measuring R&D capital, and hopefully would have some impacts on data collection in the future.  Our current work only deals with the R&D measurement problems in an economy without knowledge diffusion. In the future, we will attempt to extend this work to an economy with knowledge spillovers, and deal with the measurement issues associated freely available knowledge. In addition, more work is needed to better understand the relationships between knowledge accumulation, R&D investments, knowledge capital and human capital.   112 BIBLIOGRAPHY   [1] Aghion, P. and Durlauf S.N. (eds), Hand book of Economic Growth, Volume 1A, Amsterdam North-Holland, 2005.  [2] Aghion, P. and Howitt, P.W. (1998), Endogenous Growth Theory, Cambridge MA: MIT Press, 1998.  [3] Alexopoulos, M. and Cohen, J., Volumes of Evidence: Examining Technical Change Last Century Through a New Lens. Conference paper presented at the 2008 World Congress on National Accounts and Economic Performance Measures for Nations, 2008. http://www.indexmeasures.com/dc2008/papers/Volumes%20of%20Evidence-confere nceversion.pdf  [4] Allen, R.G.D., The Economic Theory of Index Numbers, Economica, 16:197-203, 1949.  [5] Alston, J.M., Craig, B.J. and Pardey, P.G., Dynamics in the Creation and Depreciation of Knowledge and Returns to Research. Environment and Production Technology Division, Disccusin paper, No.35, 1998.  [6] Australian Bureau of Statistics (ABS), Capitalizing Research and Development in the National Accounts’, Paper prepared by the National Accounts Research Section for the Canberra II Group meeting, Washington, 17-19 March, 2004.  [7] Baldwin, J., Beckstead, D. and Gellatly, G., Canada’s Expenditures on Knowledge Capital, Economic Analysis Research Paper Series, Statistics Canada, Ottawa, 2005.  [8] Balk, B.M., Industrial Price, Quantity and Productivity Indices. Boston: Kluwer Academic Publishers, 1998.  [9] Balk, B.M., The Residual: on Monitoring and Benchmarking Firms, Industries and  113 Economies with Respect to Productivity. Journal of Productivity Analysis, 20:5-47, 2003.  [10] Balk, B.M., Measuring Productivity Change without Neoclassical Assumptions: A Conceptual Analysis. Conference paper presented at the Sixth Annual Ottawa Productivity Workshop, Bank of Canada, May 14-15, 2007.  [11] Balk, B.M., Färe, R. and Grosskopf, S., The Theory of Economic Price and Quantity Indicators. Economic Theory, 23:149-164, 2004.  [12] Bernstein, J.I. and Mamuneas, T. R&D Depreciation, Stocks, User Costs and Productivity Growth for US R&D Intensive Industries. Structural Change and Economic Dynamics ,17(1):70-98, January, 2005.  [13] Cameron, G., On the Measurement of real R&D—Divisia Price Indices for UK Business Enterprise R&D. Research Evaluation, 6:215-219, December, 1996.  [14] Carson, C.S., Grimm, B.T. and Moylan, C.E., A Satellite Account for Research and Development. Survey of Current Business, Bureau of Economic Analysis, Washington, November, 1994.  [15] Caves, D.W., Christensen, L.R. and Diewert, W.E., The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity. Econometrica, 50:1392-1414, 1982.  [16] Chambers, R.G., Consumers’ Surplus as an Exact and Superlative Cardinal Welfare Indicator. International Economic Review, 41:105-119, 2001.  [17] Copeland, A.M., Medeiros, G.W. and Robbins, C.A. Estimating Prices for R&D Investment in the 2007 R&D Satellite Account. Bureau of Economic Analysis/National Science Foundation 2007 R&D Satellite Account Background Paper, Bureau of Economic Analysis, Washington D.C., 2007. http://www.bea.gov/papers/pdf/Estimation_of_prices_final.pdf  [18] Diewert, W.E., Exact and Superlative Index Numbers. Journal of Econometrics 4:115–146, 1976.  114  [19] Diewert, W.E (1992a), The Measurement of Productivity, Bulletin of Economic Research, 44 (3):163-198, 1992.  [20] Diewert, W.E. (1992b), Exact and Superlative Welfare Change Indicators. Economic Inquiry, 30:565-582, 1992.  [21] Diewert, W. Erwin, Harmonized Indexes of Consumer Prices: Their Conceptual Foundations. Swiss Journal of Economics and Statistics, 547-637, 2002.  [22] Diewert, W.E. (2005a), Index Number Theory Using Differences Instead of Ratios. The American Journal of Economics and Sociology, 64 (1):311-360, 2005.  [23] Diewert, W.E. (2005b), “Issues in the Measurement of Capital Services, Depreciation, Asset Price Changes and Interest Rates.” In: Measuring Capital in the New Economy, C. Corrado, J. Haltiwanger and D. Sichel (eds.), Studies in Income and Wealth Volume 65, NBER, Chicago: University of Chicago Press. 479-542, 2005. http://www.econ.ubc.ca/discpapers/dp0411.pdf  [24] Diewert, W.E. (2005c), “Constructing a Capital Stock for R&D Investments”, Chapter 4 in The Measurement of Business Capital, Income and Performance, Tutorial presented at the University Autonoma of Barcelona, Spain, September 21-22, 2005; revised December.  http://www.econ.ubc.ca/diewert/barc4.pdf  [25] Diewert, W.E. (2005d), Productivity Perspectives in Australia: Conclusions and Future Directions. Discussion Paper, No.05-05, Department of Economics, The University of British Columbia, Vancouver, Canada, V6T 1Z1, 2005. http://www.econ.ubc.ca/discpapers/dp0505.pdf  [26] Diewert, W.E. and Fox, K.J., On Measuring the Contribution of Entering and Exiting Firms to Aggregate Productivity Growth. Discussion Paper 05-02, Department of Economics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z1, 2005.  [27] Diewert, W. E. and Lawrence, D., “The Deadweight Costs of Capital Taxation in Australia” In Efficiency in the Public Sector, Fox, K.J. (ed.), Boston: Kluwer  115 Academic Publishers, 103-167, 2002.  [28] Diewert, W.E. and Lawrence, D., Estimating Aggregate Productivity Growth for Australia: The Role of Information and Communications Technology. Occasional Economic Paper, Department of Communications, Information Technology and the Arts, Canberra: Government of Australia, 2005.  [29] Diewert, W.E. and Mizobuchi, H. (2007), “An Economic Approach to the Measurement of Productivity Growth Using Differences Instead of Ratios”, forthcoming.  [30] Diewert, W.E. and Morrison, C.J., Adjusting Output and Productivity Indexes for Changes in the Terms of Trade. The Economic Journal, 96:659-679, 1986.  [31] Diewert, W.E. and Nakamura, A.O., Index Number Concepts, Measures and Decompositions of Productivity Growth. Journal of Productivity Analysis 19:127-159, 2003.  [32] Diewert, W. Erwin and Wales, T.J., Flexible Functional Forms and Global Curvature Conditions. Econometrica, 55 (1):43-68, 1987.  [33] Diewert, W.E. and Wales, T.J (1988a), Normalized Quadratic Systems of Consumer Demand Functions. Journal of Business and Economic Satistics, 6:303-312, 1988.  [34] Diewert, W.E. and Wales, T.J (1988b), A Normalized Quadratic Semiflexible Functional Form. Journal of Econometrics, 37:327-42, 1988.  [35] Diewert, W.E. and Wales, T.J., Quadratic Spline Model for Producer’s Supply and Demand Functions. International Economic Review, 33:705-722, 1992.  [36] Epstein, L.G. and Denny, M., Endogenous Capital Utilization in a Short Run Production Model: Theory and Empirical Application. Journal of Econometrics 12 (2):189-207, 1980.  [37] Esposti, R. and Pierani, P. Building the Knowledge Stock: Lags, Depreciation and Uncertainty in R&D Investment and Link with Productivity Growth, Journal of  116 Productivity Analysis, 19, 33-58, 2003.  [38] Eurostat, World Bank, International Monetary Fund, Organization for Economic Cooperation and Development, and United Nations. System of National Accounts 1993, United Nations, New York, 1993.  [39] Fixler, Dennis, Accounting for R&D in the National Accounts. Bureau of Economic Analysis working paper, presented at the ASSA meetings in San Francisco, January, 2009.  [40] Fraumeni, B.M. and Okubo, S., R&D in the National Income and Product Accounts: a First Look at its Effect on GDP. Conference paper presented at the CRIW “Conference on Measuring Capital in the New Economy”, 2003.  [41] Fraumeni, B.M. and Okubo, S., R&D in U.S. National Accounts. Conference Paper prepared for the 28th General Conference of the International Association for Research in Income and Wealth, Cork, Ireland, August 22-28, 2004.  [42] Fox, K.J., A Method for Transitive and Additive Multilateral Comparisons: A Transitive Bennet Indicator.  Journal of Economics, 87:73-87, 2006.  [43] Griliches, Z, Issues in Assessing the Contribution of Research and Development to Productivity Growth. The Bell Journal of Economics, 10 (1):92-116, 1979.  [44] Griliches, Z, R&D, Education, and Productivity: A Retrospective. Harvard University Press, 2000.  [45] Gullickson, W. and Harper, M.J., Multifactor Productivity in U.S. Manufacturing, 1949-83. Monthly Labour Review, 110 (1):18-28, 1987.  [46] Hall, B.H., Discussion on ‘R&D in the National Income and Product Accounts: A First Look at its Effect on GDP’ by Barbara Fraumeni and Sumiye Okubo, 2003. http://www.nber.org/books/CRIW02/hall-bronwyn8-23-04comment.pdf  [47] Harhoff, D., Scherer, F.M. and Vopel, K., Citations, Family Size, Opposition and the Value of Patent Rights. Research Policy, 32:1343-1363, 2003.  117  [48] Hicks, J.R., The Valuation of the Social Income. Economica, 7:105-140, 1940  [49] Hicks, J.R., Consumers’ Surplus and Index Numbers. The Review of Economic Studies,  9:126-137, 1942.  [50] Hicks, J.R., The Generalized Theory of Consumers’ Surplus. The Review of Economic Studies, 13:68-74, 1945-1946.  [51] Jaffe, S., A Price Index for Deflation of Academic R&D Expenditure. National Science Foundation, NSF 72-310, 1973.  [52] Jankowski, J.E. Jr., Do we need a price index for industrial R&D? Research Policy, 22 :195-205, 1993.  [53] Jones, C.I. and Williams, J.C., Measuring The Social Return to R&D. The Quarterly Journal of Economics, 13 (4):1119-1135, 1998.  [54] Jorgenson, D.W. and Griliches, Z., The Explanation of Productivity Change. The Review of Economic Studies, 34:249–283, 1967.  [55] Jorgenson, D.W., and Griliches, Z.,Issues of Growth Accounting: A Reply to Edward F. Denison. Survey of Current Business, 55(5) part II:65–94, 1972.  [56] Kendrick, J.W., The Formation and Stocks of Total Capital, New York: Columbia University Press, for the National Bureau of Economic Research, 1976.  [57] Kohli, U., Growth Accounting in the Open Economy: Parametric and Nonparametric Estimates. Journal of Economic and Social Measurement, 16:125-136, 1990.  [58] Nadiri, M.I. and Prucha, I.R., Estimation of the depreciation rate of physical and R&D capital in the U.S. Total Manufacturing Sector. Economic Inquiry, 34 (1):43-56, 1996.  [59] Organization for Economic Co-operation and Development (OECD), Main Definitions and Conventions for the Measurement of Research and Experimental  118 Development (R&D)—A Summary of the Frascati Manual 1993. Paris, OECD, 1994.  [60] Organization for Economic Co-operation and Development (OECD), Frascati Manual 1993: The Measurement of Scientific and Technological Activities -- Proposed Standard Practice for Surveys on Research and Experimental Development, Paris, OECD, 1994.  [61] Organization for Economic Co-operation and Development (OECD), Frascati Manual 2002: The Measurement of Scientific and Technological Activities -- Proposed Standard Practice for Surveys on Research and Experimental Development, 2002. http://www.oecdbookshop.org/oecd/display.asp?CID=&LANG=EN&SF1=DI&ST1= 5LMQCR2K61JJ  [62] Pakes, A. and Schankerman. M., “The Rate Obsolescence of Patents, Research Gestation Lags, and the Private Rate of Return to Research Resources.” In R&D, Patents, and Productivity, edited by Griliches, Z. Chicago, IL: University of Chicago Press, for the National Bureau of Economic Research, 73–88,1984.  [63] Parham, D., Empirical Analysis of the Effects of R&D on Productivity: Implications for Productivity Measurement? Conference paper presented at OECD Workshop on Productivity Measurement and Analysis, Bern Switzerland, 2007.  [64] Pitzer, J.S., Intangible Produced Assets. Conference paper presented at the London Meeting of the Canberra II Group: On the Measurement of Non-Financial Assets, September 1-3, 2004.  [65] Rapoport, J. “The Anatomy of the Product-Innovation Process: Cost and Time” In Research and Innovation in the Modern Corporation, edited by Mansfield, E., New York: W.W. Norton, 110-135, 1971.  [66] Romer, P., New Goods, Old Theory and the Welfare Costs of Trade Restrictions. Journal of Development Economics, 43:5-38, 1994.  [67] Samuelson, P.A., Complementarity—An Essay on the 40th Anniversary of the  119 Hicks-Allen Revolution in Demand Theory. Journal of Economic Literature, 12:1255-1289, 1974.  [68] Sanders, B., Rossman, J., and Harris, L., The Economic Impact of Patents. The Patent and Trademark Copyright Journal, 2 (2):340-362, 1958.  [69] Shanks, S. and Zheng, S., Econometric Modeling of R&D and Australia’s Productivity. Staff Working Paper, Productivity Commission, Canberra, April, 2006. http://www.pc.gov.au/research/swp/economicmodelling/index.html  [70] Solow, R.M., Technical Change and the Aggregate Production Function. Review of Economics and Statistics, 39:312-320, 1957.   120 APPENDICES  APPENDICES FOR CHAPTER 1  Appendix A: The Industry’s Profit Maximization Problem  Each industry’s profit maximization problem can be written as follows: (A.1) { }∑ = −−∞+ 0 ),( ,,'1,' t IPxwytyPRMax tRtRttttttststx β subject to   ),,( tRxfy ttt =  and 11, )1( −− −+= ttRt RIR δ . Substituting the constraints into the objective function, we have:  (A.2)  ... )})1(()2,,()2),2,,(({ )})1(()1,,()1),1,,(({ )})1((),,()),,,(({', 232,22223222 121,11111111 1,1' + −−−−++++ −−−−++++ −−−− +++++++++++ +++++++++++ ++ tttRtttttttt tttRtttttttt tttRtttttttttstx RRPxwtRxfttRxfP RRPxwtRxfttRxfP RRPxwtRxfttRxfPsRMax δβ δβ δβ  Therefore the first order necessary conditions with respect to vector tx  can be written as: (A.3) 0,1,...T,         t,),(]/),([),,( , ==∇∂∂+∇ tttxttttxt wtRxfyytyPtRxfp where tp  is the output price and ),,( tRxf ttx∇  is the vector of first order partial derivatives of the period t production function with respect to the components of the input vector x. Factoring out the output price tp  and ),,( tRxf ttx∇  on the left hand side of equation (A.3), we have the following simplified form for equation (A.3):  121  (A.4) tttx tt tt t wtRxfyp ytyP p =∇×⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂∂+× ),,( / /),( 1 The first order necessary conditions with respect to R&D stock variable 1+tR  are as follows: (A.5) ( ) ( ) ( ) tRttR t t t tt PPR fp R f y Py ,1, 1 1 1 11 )1( βδβ =⎭⎬ ⎫ ⎩⎨ ⎧ −+∂ ⋅∂+∂ ⋅∂ ∂ ⋅∂ + + + + ++  After some rearrangement, we can rewrite the above equation as follows: (A.6) [ ] 1,,11 1111 111 )1()1,,(/ /)(1 +++ ++++ +++ −−=∂ +∂×∂⋅∂+× tRtrtt tttt ttt PPtR tRxfyP yPp δββ Assuming that ( ) ttt r+=+ 1/ 1ββ  where tr  is the nominal interest rate, the above equation can be written as:  (A.7) [ ] 1,,1 1111 111 )1()1()1,,(/ /)(1 ++++++ +++ −−+=∂ +∂×∂⋅∂+× tRtRtt tttt ttt PPrR tRxfyP yPp δ Define period t non-negative markup as follows:  (A.8) 0 1 / /),( ≥−=∂∂−≡ εtt t t yp ytyP m The corresponding markup factor tM  can be defined as: (A.9) tt t tt yp ytyP mM / /),( 1 1 11 ∂∂+=+=−= ε If we assume that markup factors are constant over time, we can rewrite our system of first order conditions as: (A.10) ),,( tRxfMpw ttxtt ∇= , and (A.11) 1 11 11,, )1,,( )1()1( + ++ ++ ∂ +∂=−−+ t tt ttrtRt R tRxf MpPPr δ Moving the 2nd term to the right hand side of equation (A.11) and dividing through by  122 tRP , , we obtain: (A.12) tR tR t tt tR t t P P R tRxf MP p r , 1, 1 11 , 1 )1( )1,,( 1 + + +++ −+∂ +∂=+ δ  Equations (A.10) and (A.12) form our final system of estimating equations.   123  Appendix B: Break Points  Table A.1 Break Points for Model III and Model IV   Eq. SIC 28 SIC 35 SIC 36 SIC 37 MANF 1 9, 21, 29, 34, 44 4, 31, 39 20, 38, 46 13, 27, 32, 42 14, 21, 26, 29, 34, 38 2 15, 21, 32, 40 29, 40 9, 19, 27, 39 12, 27, 30, 38 9, 21, 23, 36, 40 3 13, 28, 35, 41 8, 13, 22, 32, 42 12, 22, 38 12, 28, 32, 38 13, 28, 36, 41, 47 Model 3(a) and Model 4(a) 4 21, 28, 35, 41 30, 39 26, 40 12, 28, 35, 40 11, 21, 29, 36, 40 1 15, 21, 29, 34, 44 31, 39 20, 32, 38 9, 21, 30, 43 14, 21, 29, 34, 38 2 15, 21, 43 10, 19, 30, 38 9, 22, 36, 41 6, 22, 30,39 9, 21, 23, 36, 40 3 13, 28, 35, 41 8, 13, 22, 38, 42 12, 22, 30, 40, 45 13, 27, 32, 38 13, 22, 36, 41, 47 Model 3(b) and Model 4(b) 4 21, 28, 35, 41 19, 30, 39 24, 37 6, 12, 24, 28, 33, 40 11, 21, 29, 36, 40 1 9, 21, 29, 34, 44 31, 39 20, 38 9, 22, 30, 43 15, 21, 29, 34, 39, 44 2 15, 21, 30, 39 10, 19, 30, 38 8, 22, 41 7, 22, 30,37 9, 21, 23, 28, 39, 46 3 13, 28, 35, 45 8, 14, 22, 37, 42 12, 22, 30, 45 12, 28, 32, 38 13, 28, 36, 41, 45 Model 3(c) and Model 4(c) 4 21, 28, 35, 41 19, 30, 39 24, 37 13, 28, 35, 39 9, 21, 29, 36, 40  Note:  Equations (1)-(3) are estimating equations for the three inputs; equation (4) is the production function.       MANF: denotes “Manufacturing”.

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