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Essays in economic measurement with application to Japan Mizobuchi, Hideyuki 2009

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ESSAYS IN ECONOMIC MEASUREMENT WITH APPLICATION TO JAPAN by Hideyuki Mizobuchi B.A., Tohoku University, 2000 M.A., Tohoku University, 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)  September, 2009 © Hideyuki Mizobuchi, 2009  Abstract The objective of this dissertation is to improve our understanding of economic measurement, in particular, index number theory and its application to the Japanese economy. Two different approaches exist in index number theory: the first decomposes a value ratio of costs or profits into the product of a price index times a quantity index. The second approach decomposes a value difference in costs or profits into the sum of a price indicator plus a quantity indicator. This thesis will examine both approaches. Following the ratio approach, our first essay will investigate the origins of the growth of the Japanese standard of living. This growth is attributed to technical progress, changes in output prices and input quantities. In many ways, improvements in the terms of trade are synonymous with technical progress, because they make it possible to obtain more for less. We compare two distinct impacts of changes in the terms of trade and technical progress on the Japanese standard of living, and will show that while technical progress made the most contribution, the impact of the changes in the terms of trade were negligible. The second and third essays follow the difference approach in index number theory. The second essay deals with the producer model and proposes a productivity analysis based on the difference approach in index number theory. We show that following the difference approach, change in real income per unit primary input can be additively decomposed into explanatory factors such as technical progress, changes in relative output prices and deflated input quantities. The third essay deals with the consumer model, introducing the concept of the exact and superlative indicator into the difference approach; we will show that the Bennet indicator is an exact and superlative indicator. The second essay applies the new decomposition result to the Japanese market sector for the years 1955–2006. The third essay uses Japanese aggregate consumption data and compares different real expenditures based on various distinct indexes and indicators.  ii  Table of Contents Abstract................................................................................................................................. ii Table of Contents ................................................................................................................ iii List of Tables......................................................................................................................... v List of Figures..................................................................................................................... vii Acknowledgement ............................................................................................................. viii Co-Authorship Statement................................................................................................... ix Chapter 1. Objective and Overview.............................................................................. 1 1.1 Objective .......................................................................................................................... 1 1.2 Overview....................................................................................................................... 3 Chapter 2. On Measuring the Productivity and the Standard of Living in Japan, 1955-2006 ...................................................................................................................... 4 2.1 Introduction...................................................................................................................... 4 2.2 The Theoretical Framework............................................................................................. 8 2.3 Data Construction .......................................................................................................... 20 2.4 Results............................................................................................................................ 27 2.5 Conclusion ..................................................................................................................... 43 Chapter 3. An Economic Approach to the Measurement of Productivity Growth Using Differences Instead of Ratios............................................................................ 45 3.1 Introduction.................................................................................................................... 45 3.2 The Production Theory Framework ............................................................................... 47 3.3 The Theoretical Explanation of Per Unit Primary Input Real Income Growth using Differences .................................................................................................................. 52 3.4 The Normalized Quadratic Income Function and Bennet Indicators of Price, Quantity and Productivity Change............................................................................................. 56 3.5 An Application to the Japanese Economy for 19552006............................................. 66 3.6 Conclusion ..................................................................................................................... 79 Chapter 4. Exact and Superlative Price and Quantity Indicators ........................... 81 4.1 Introduction.................................................................................................................... 81 4.2 Exact and Superlative Price and Quantity Indexes ........................................................ 84 4.3 Value Differences, Variations and Indicators of Price and Quantity Change ................ 92 iii  4.4 Superlative Price and Quantity Indicators in the Homothetic Preferences Case ........... 99 4.5 Strongly Superlative Price and Quantity Indicators..................................................... 103 4.6 The Decomposition Properties of the Bennet Indicators ..............................................111 4.7 The Bennet Indicators using Japanese Data..................................................................117 4.8 Conclusion ................................................................................................................... 120 Chapter 5. Summary and Conclusion....................................................................... 122 5.1 Summary of Contributions........................................................................................ 122 5.2 Directions for Further Research................................................................................ 123 Bibliography ..................................................................................................................... 126 Appendix........................................................................................................................... 134 Appendix A. Data Construction......................................................................................... 134 Appendix B. Tables of Prices, Quantities, and Value for Inputs and Outputs ................... 166 Appendix C. On the Flexibility of the Normalized Quadratic Income Function............... 168 Appendix D. Measures for the Effects of Individual Price and Quantity Changes ........... 172 Appendix E. Japanese Price and Quantity Data................................................................. 179 Appendix F. On the Flexibility of the Translation Homothetic Normalized Quadratic Cost Function. ...................................................................................................................... 191 Appendix G. The Japanese Consumption Data.................................................................. 199  iv  List of Tables Table 2-1: List of Capital Services, Inventory Services, and Land Services ............... 24 Table 2-2: Chained Fisher Indexes of Output, Input and Productivity Growth and the Balancing Real Rate of Return in the Japanese Economy, 1956-2006................ 29 Table 2-3: The Decomposition of Market Sector Real Income Growth into Translog Productivity, Real Output Price Change and Input Quantity Contribution Factors.................................................................................................................. 31 Table 2-4: The Decomposition of Market Sector Real Income Level into Cumulative Productivity, Real Output Price Change and Input Quantity Contribution Factors.................................................................................................................. 33 Table 2-5: The Quantity and Price of Gross Investment, Depreciation, Net Investment, Capital Services, Waiting Capital Services (Billion Yen) ................ 38 Table 2-6: The Decomposition of Market Sector Net Real Income Growth into Translog Productivity, Real Output Price Change, and Input Quantity Contribution Factors ............................................................................................ 40 Table 2-7: The Decomposition of Market Sector Net Real Income Cumulative Growth into Productivity, Real Output Price Change, and Input Quantity Contribution Factors using the Translog Net Product Approach ......................... 42 Table 3-1: Decomposition of Growth Rate of Real Income Per Unit Labour (%) ...... 70 Table 3-2: Decomposition of the Cumulative Change in Real Income Per Unit Labour (in 1955 yen) ........................................................................................... 72 Table 3-3: Decomposition of Growth Rate of Net Real Income Per Unit Labour (%) 76 Table 3-4: Decomposition of the Cumulative Change in Net Real Income Per Unit Labour (in 1955 yen) ........................................................................................... 78 Table 4-1: Real Expenditure Differences and Bennet Indicators, 1981-2006 ............118 Table 4-2: Annual Averages of Real Expenditure Differences and Bennet Indicators118 Table 4-3: Comparison of Japanese Real Consumption, 1980-2006 ......................... 120 Table B-1: Market Sector Output and Input Prices for Japan 1955-2006.................. 166 Table B-2: Market Sector Output and Input Quantities for Japan 1955-2006 (Billion Yen) .................................................................................................................... 167 Table E-1: Prices of Main Aggregates ....................................................................... 179 v  Table E-2: Quantities of Main Aggregates................................................................. 180 Table E-3: Prices of Investment Goods...................................................................... 181 Table E-4: Prices of Capital Services......................................................................... 182 Table E-5: Prices of Inventory Services, Land Services and Labour Inputs.............. 183 Table E-6: Prices of Depreciations............................................................................. 184 Table E-7: Prices of Waiting Capital Services ........................................................... 185 Table E-8: Quantities of Investment Goods............................................................... 186 Table E-9: Quantities of Capital Services .................................................................. 187 Table E-10: Quantities of Inventory Services, Land Services and Labour Inputs..... 188 Table E-11: Quantities of Depreciations .................................................................... 189 Table E-12: Quantities of Waiting Capital Services .................................................. 190 Table G-1: Prices of Consumption Goods ................................................................. 200 Table G-2: Quantities of Consumption Goods........................................................... 200 Table G-3: Real Prices of Consumption Goods ......................................................... 201  vi  List of Figures Figure 2-1: The Real Rate of Return for the Years 1955-2006 .................................... 27 Figure 2-2: Comparison of Real Rate of Return for the Years 1955-2006 .................. 29 Figure 2-3: Comparison of Total Factor Productivity for the Years 1955-2006 .......... 30 Figure 2-4: Decomposition of Market Sector Real Income Level into Factors........... 33 Figure 2-5: Price of Gross Investment, Depreciation, Net Investment, Capital Services and Waiting Capital Services................................................................. 38 Figure 2-6: Quantity of Gross Investment, Depreciation, Net Investment, Capital Services and Waiting Capital Services (Billion Yen)........................................... 39 Figure 2-7: Decomposition of Market Sector Net Real Income Level into Factors .... 42 Figure 3-1: Real Income Change Per Unit Labour (in 1955 yen)................................ 72 Figure 3-2: Net Real Income Change Per Unit Labour (in 1955 yen)......................... 78 Figure F-1: Leontief Translation Homothetic Preferences with No Inferior Goods.. 194 Figure F-2: Translation Homothetic Preferences with No Inferior Goods ................ 195 Figure F-3: Leontief Translation Homothetic Preferences with an Inferior Good .... 197  vii  Acknowledgement I would like to express my gratitude to my supervisory committee, Erwin Diewert, Kevin Milligan and Brian Copeland for their support at each stage of this thesis. I am particularly grateful to Erwin Diewert for his suggestions and encouragements, which led me to think more deeply about the subject and helped me to complete various projects. I especially appreciate his insightful lectures, which introduced me to the field. I would also like to thank my external examiner Melvin Fuss for carefully reading this dissertation and making useful comments. The support of follow students and friend over the years is deeply acknowledged with appreciation. Finally, I also would like to thank my family: my wife, Yukie and my parents, Mitsusada and Ikuko for their support over years.  viii  Co-Authorship Statement Chapter 2 includes joint work with Erwin Diewert and Koji Nomura. I started working on this chapter in order to improve Diewert, Mizobuchi and Nomura (2005). Erwin Diewert and I discussed the direction the chapter should be improved. I collected all the data which is necessary to construct input and output data for the Japanese market sector. Exceptions are the investment data and initial capital stocks which Koji Nomura provided. The novelty of our database is to extract the residential structures and land used by the market sector from the economy-wide residential structure and land. Since it is very difficult to divide residential structures and land between the market sector and the non-market sector, even the study on the market sector often include the entire part of residential structure and land in the list of the inputs of the market sector or exclude it from the list of the input of the market sector. I collected annual data of newly constructed owner-occupied houses and land formation in order to estimates of the residential structure and land used by the market sector. This chapter was initially drafted by me and revised by Erwin Diewert. Chapter 3 includes joint work with Erwin Diewert. I proposed the idea of productivity analysis based on the difference approach in index number theory and wrote a paper with proofs of justifying the productivity analysis based on Bennet indicators. However, there is a problem that a part of my analysis is based on the income function which is not linearly homogeneous with respect to prices. Introducing a new functional form and looking at real prices, Erwin Diewert showed that even if we assume the usual income function, which is linearly homogeneous with respect to prices, we can still justify the productivity analysis based on Bennet indicator. I collected the input and output data for Japanese market sector and calculated the origins of the Japanese labour productivity growth for the period 1955-2006. My initial manuscript has been revised by Erwin Diewert. Chapter 4 includes joint work with Erwin Diewert. I proposed the idea of introducing the concept of exact and superlative indicators into the difference approach in index number theory and wrote a paper with proofs of Bennet indicators as superlative indicators. However, there is a problem that a part of my analysis is based on the cost function which is not linearly homogeneous with respect to prices. Introducing a new functional form and looking at real prices, Erwin Diewert showed that even if we assume ix  the usual cost function, which is linearly homogeneous with respect to prices, we can still show that Bennet indicators are exact and superlative indicators. I collected the Japanese aggregate consumption data and calculated real expenditure by applying several indexes and indicators to collected data. My initial manuscript has been revised by Erwin Diewert.  x  Chapter 1.  Objective and Overview  1.1 Objective Economic analysis is concerned with modelling the supply and demand for individual goods and services (commodities) by individual economic agents. However, given the enormous numbers of commodities in real life economies, we must use aggregated data. How can this aggregation best be accomplished? Index number theory studies this type of problem and asks how individual price data should be summarised, or aggregated, in a single aggregate price level (price index or price indicator), and how individual data on quantities should be summarised in a single aggregate quantity level (quantity index or quantity indicator). Of the two approaches in index number theory, the ratio approach decomposes the value ratio of expenditures (in the case of consumer), or costs and profits (in the case of producer), between the current and the reference periods into the product of a price index and a quantity index. On the other hand, the difference approach decomposes the value difference in expenditures, costs and profits between the current and the reference periods into the sum of a price indicator and a quantity indicator. In this thesis, we examine the theoretical and empirical problems of these two approaches. In practice, the Laspeyres and Paasche indexes, which are based on the ratio approach, are widely used by statistical agencies all over the world. However, index number theory recommends superlative indexes, such as the Fisher and Törnqvist indexes, because they are free of the substitution bias from which the Laspeyres and Paasche indexes suffer. In the difference approach, the Bennet indicator plays a central role. We recommend its use in the context of consumer as well as producer. Chapter 2 looks at the contribution of the market sector to changes in Japanese living standards between 1955 and 2006. Initially, we take a conventional Total Factor Productivity (TFP) growth approach, in which TFP growth is measured as a year-to-year Fisher quantity index of gross outputs, divided by a Fisher quantity index of primary inputs. We document the slowdown in Japanese TFP performance in the post-bubble period. This chapter also examines what happens when inventories and land are omitted from the list of primary inputs. The remainder of this chapter looks at the market sector’s contribution to the growth in Japanese living standards, and decomposes this growth into three contribution factors: factors for technical progress, changes in real output prices (including changes in the terms of trade) and growth in primary input 1  quantities. We adapt the decomposition results developed by Diewert and Morrison (1987), as well as Kohli (1990) (2004), to this real income context, and observe that technical progress and the growth in capital services were the main contributors to real income growth; changes in the terms of trade, on average, had a minimal effect on real income. Finally, the chapter changes gears from a gross output approach to a theoretically preferable net output approach. In the net output context, we discover that the role of capital deepening as a contributor to higher living standards diminished, and the role of technical progress and labour growth became more important. The index number approach makes a weaker assumption on production technology and preferences, but its data requirements are more stringent. We then construct prices and quantities for inputs and outputs of the Japanese market sector for the period between 1955 and 2006. This database will be used in Chapter 3. While Chapter 2 follows the traditional ratio approach in index number theory, wherein TFP is defined as the ratio of the output and input quantity indexes, Chapter 3 takes an alternative difference approach in index number theory: the approach decomposes a value difference into the sum of a price difference plus a quantity difference. Applying this difference approach, we decompose the growth of a new measure of labour productivity into additive contribution factors: factors for technical progress, changes in real output prices (including changes in the terms of trade) and growth in relative input quantities. This new measure of labour productivity takes into account changes in the terms of trade. We then use this methodology to investigate the growth in real income per unit labour in the Japanese economy from 1955 to 2006. Chapter 3 also introduces a new flexible functional form for an income function based on the normalised quadratic functional form pioneered by Diewert and Wales (1987). Chapter 4 discusses the ratio approach in index number theory in the context of the consumer problem. As previously explained, the traditional economic approach to index number theory is based on a ratio concept. The Konüs true cost of living index is a ratio of cost functions evaluated at the same utility level, but with the prices of the current period in the cost function appearing in the numerator, and the prices of the base period in the cost function in the denominator. The Allen quantity index is also a ratio of cost functions in which the utility levels vary, but the price vector is held constant in the numerator and denominator. A corresponding theory initiated by Hicks (1946) exists for differences in cost functions; this chapter develops this approach. Diewert 2  (1976) has defined superlative price and quantity indexes as observable indexes that were exact for a ratio of unit cost functions, or for a ratio of linearly homogeneous utility functions. Chapter 4 looks for counterparts of his results in the difference context, for both flexible homothetic and flexible nonhomothetic preferences. The Bennet indicators of price and quantity change turn out to be superlative for the nonhomothetic case; the underlying preferences are of the translation homothetic form, as discussed by Balk, Färe and Grosskopf (2004), Chambers (2001) and Dickenson (1980). Using Japanese aggregate consumption data from national accounts, we calculate real expenditures based on different types of indexes and indicators. We show that real expenditures based on superlative indexes and superlative indicators are very close in magnitude. 1.2 Overview This thesis is organized as follows. Chapter 2 conducts the decomposition of the change in Japanese standard of living into several components for the years between 1955 and 2006. Chapter 3 proposes the new measure for labour productivity and applies it to the Japanese economy for the years between 1955 and 2006. Chapter 4 examines the ratio approach to index number theory in the consumer context; here we apply different indexes and indicators to Japanese aggregate consumption data. Chapter 5 concludes this thesis. A detailed description of Chapter 2’s data construction for the Japanese market sector between 1955 and 2006 is given in Appendix A. The price series and the quantity series for 13 aggregated inputs and outputs are listed in Appendix B. New functional forms are introduced in Chapters 3 and 4. Appendixes C and F include the proofs that they are also flexible functional form.  3  Chapter 2.  On Measuring the Productivity and the Standard of Living in Japan, 1955-20061  2.1 Introduction The Japanese economy has experienced tremendous growth in the past half century, but there has been a pronounced slowdown for more than 10 years since the bubble economy collapsed. 2 This unique experience of the post-war Japanese economy has been attracting the interest of numerous researchers. However, most of these studies have been concerned with the growth of production and the productivity of the Japanese economy. These studies quantify the various sources of Japanese economic growth and measure the productive efficiency of the Japanese economy.3 In this chapter, we are primarily concerned with increases in the standard of living in the post-war Japanese economy. In particular, we focus on the contribution of Japan’s market sector to improvements in Japanese living standards over the period 1955–2006. We have excluded the general government sector from our analysis because this sector may provide its outputs without charging for them, or it may sell them for prices that are not economically significant. Our analysis will rely on the assumptions of revenue-maximising and cost-minimising behaviour, assumptions that are generally not applicable for the government sector. There are other measurement difficulties associated with the government sector. Because it is difficult to measure multiple outputs in this sector, the System of National Accounts 19934 (see the UN (1993)) recommends measuring the value of these difficult-to-measure outputs of the general government as the cost of producing the outputs, and recommends setting the price of these outputs as equal to an input cost index. This is not a proper way of measuring  1  It includes joint work with W. Erwin Diewert and Koji Nomura.  2  Although the Japanese economy has been in a period of economic recovery since 2002, its growth is still  modest. 3  See Hayashi and Nomura (2005), Hayashi and Prescott (2002), Jorgenson and Nishimizu (1978),  Jorgenson, Kuroda and Nishimizu (1987), Jorgenson and Nomura (2005)(2007), Nishimizu and Hulten (1978) and Nomura (2004). 4  Hereafter called 1993 SNA.  4  output from a theoretical point of view.5 The household sector is also a producer in the 1993 SNA. In particular, this sector produces imputed rent from owner-occupied houses. However, this is a special type of production where productivity improvements cannot take place, and so we have decided to exclude this sector from our concept of the market production sector. Therefore, we completely net out the general government and the household sector from our framework in order to avoid the above-mentioned difficulties. Real Gross Domestic Product (GDP) is often used as a proxy for a country’s living standard. However, real GDP is a measure of the level of production, and it is well known that real GDP can be a very misleading indicator of a country’s welfare in the face of changing terms of trade.6 Real (Gross) Income, which deflates nominal GDP by the price of domestic consumption goods, is more appropriate as a measure of economic welfare. Economic welfare comes from consumption. Real income captures how much consumption people can purchase for their income. For example, if the nominal output is constant over two consecutive periods, an improvement in the terms of trade should increase economic welfare by enhancing the purchasing power of domestic households. Real Income always increases in this situation. However, the conventional Laspeyres type real GDP may decrease in this case. We shed new light on the measurement of the standard of living in Japan for the years 1955–2006 by focusing on the real income generated by the market sector. We attempt to measure the determinants of real income growth in Japan by adapting the analytic framework for productivity measurement, developed by Diewert and Morrison (1986) and Kohli (1990) (2004), to the real income growth context. We thus find that the main determinants of growth in real income generated by the market sector of the economy are:   5  Technical progress or improvements in Total Factor Productivity (TFP);  There is an additional difficulty with the SNA treatment of input cost in the government sector: the user  cost of capital in the government sector is just depreciation, with no imputation for the financial cost of capital. 6  Thus, if a government building is sold to the private sector, GDP will increase!  Hamada and Iwata (1984) show that real GNP (equivalent to real GDP in this chapter) may overestimate  national welfare when the terms of trade deteriorate for a country. Kohli (2004) demonstrated that an improvement in the terms of trade can actually lead to a fall in real GDP.  5    Growth in non-consumption real domestic output prices, growth in the real prices of exports and falls in the real prices of imports; and    Growth in primary inputs.  Using our adapted decomposition formula, all three of these determinants can be calculated using only the observable price and quantity data over the years 1955–2006. A difficulty in applying this decomposition formula is uncertainty as to what is the “correct” user cost of capital. In particular, should an exogenous rate of return be used in the formula, or should an endogenous balancing rate of return that makes the value of inputs equal to the value of outputs be used? Schreyer (2007) surveys several different procedures for estimating rates of return. We use ex post balancing (real) rates of return. These balancing real rates of return seem to capture business conditions in the post-war Japanese economy. GDP in a closed economy with no government is simply consumption plus gross investment plus changes in inventory. Economists have argued for a long time that Net Domestic Product (NDP), which equals consumption and net investment, is a more appropriate welfare measure than GDP.7 The problem with the gross concept is that it gives us a measure of output that is not sustainable. In order to move to a more theoretically appropriate NDP concept using our framework for the determinants of real income growth, it is necessary to treat depreciation as an offset to gross investment. Thus, we take the depreciation term in the user cost formula out of costs and treat it as a negative output that will act as an offset to gross investment. Our theoretical results, which explain the growth of gross real income generated by the market sector, can still be applied to the net real income concept. Our dataset of the market sector has been constructed from different data sources. For net output data, we heavily depend on data in Japanese national accounts.8 For capital stocks and investment, we also made extensive use of the investment and asset data in the Keio Economic Observatory (KEO) database. For labour input, we constructed total 7  See Marshall (1890), Pigou (1924), Samuelson (1961) and Weitzman (1976).  More recent papers that  argue for the net product framework are Diewert and Fox (2005), Oulton (2004), and Weitzman (2003). 8  In 1978, the Japanese system of national accounts was revised to comply with the guidelines proposed by  the United Nations System of National Accounts (1968 SNA).  In 2000, it was revised to comply with the  guidelines newly proposed by the United Nations System of National Accounts (1993 SNA).  6  hours of work for three different types of workers: employees, the self-employed, and family workers from the Japanese Labour Force Survey and the Japanese national accounts. One limitation of our study which must be noted is that we have no industry detail in our database, and thus we cannot locate the contributions of TFP growth in individual industries to the aggregate market sector TFP growth.9 The remainder of this chapter is organized as follows. In section 2.2, we outline a framework for measuring productivity and decomposing real income growth into several explanatory factors. This framework is an adaptation of a methodology introduced by Diewert and Morrison (1987). In section 2.3, we explain our data construction. The rate of return is endogenously determined from observed data. The procedure for determining the rate of return is explained there. In section 2.4 of the chapter, we present a conventional TFP growth accounting for the market sector of the Japanese economy for the years 1955–2006. There have been several recent studies that do more or less the same thing, so one might question the value of yet another study of Japanese TFP growth.10 However, all of these alternative studies cover a much shorter period (with the exception of Nomura (2004)), and there are other significant differences. In the present study, we consider only the market sector of the Japanese economy, where productivity improvements are possible under current national income accounting conventions. Moreover, our focus is not on TFP growth rates per se but rather the contribution of TFP growth to real income growth, i.e. the existing studies do not follow our treatment of changes in the terms of trade. In section 2.4.2, the gross income methodology developed in section 2.2 is implemented using our Japanese market sector database for the years 1955–2006, and the net income methodology is implemented in section 2.4.3. In section 2.5, we conclude the chapter.  9  Moreover, we cannot measure the contributions to aggregate TFP growth of shifts in labour resources  from less productive to more productive sectors. labour data.  This difficulty comes from the aggregate nature of our  Since we consider three types of labour (employees, the self-employed, family workers), we  can only capture the contribution of shifts in labour resources within these three types.  On the other hand,  industry price and quantity data are notoriously unreliable because of the lack of detailed surveys on gross outputs and intermediate inputs, particularly for service industries.  A final reason for our use of aggregate  market sector data is that reliable breakdowns of exports and imports by industry are not available. 10  Some of the recent studies are Hayashi and Nomura (2005), Hayashi and Prescott (2002), Jorgenson and  Nomura (2005) (2007) and Miyagawa, Ito and Harada (2004).  7  2.2 The Theoretical Framework In this section, we present the production theory framework which will be used in the remainder of the chapter. The main references are Diewert and Morrison (1986) and Kohli (1990).11 Initially, we assume that the market sector of the economy produces quantities of M (net) 12 outputs, y  [y1,...,yM], which are sold at the positive producer prices P  [P1,...,PM]. We further assume that the market sector of the economy uses positive quantities of N (primary)13 inputs, x  [x1,...,xN] which are purchased at the positive primary input prices W  [W1,...,WN]. In period t, we assume that there is a feasible set of output vectors y that can be produced by the market sector if the vector of primary inputs x is utilized by the market sector of the economy; denote this period t production possibilities set by St. We assume that St is a closed convex cone that exhibits a free disposal property.14 11  The theory also draws on Samuelson (1953), Diewert (1974; 133-141) (1980) (1983; 1077-1100), Fox  and Kohli (1998), Kohli (1978) (1991) (2003) (2004a) (2004b), Morrison and Diewert (1990), Samuelson (1953) and Sato (1976).  This chapter is essentially an extended version of Diewert, Mizobuchi and  Nomura (2005) using newly available data.  The theoretical framework explained in this section was  recently used by Diewert and Lawrence (2006). 12  If the mth commodity is an import (or other produced input) into the market sector of the economy, then  the corresponding quantity ym is indexed with a negative sign.  We will follow Kohli (1978) (1991) and  Woodland (1982) in assuming that imports flow through the domestic production sector and are “transformed” (perhaps only by adding transportation, wholesaling and retailing margins) by the domestic production sector. 13  The recent textbook by Feenstra (2004; 76) also uses this approach.  Primary inputs only include labour and capital services from reproducible assets, inventories and land.  Intermediate inputs fall into the category of (net) outputs. 14  For a more explanation for the meaning of these properties, see Diewert (1973) (1974; 134), Woodland  (1982) or Kohli (1978) (1991). constant returns to scale.  The assumption that St is a cone means that the technology is subject to  This is an important assumption since it implies that the value of outputs should  equal the value of inputs in equilibrium.  In our empirical work, we use an ex post rate of return in our  user costs of capital, which forces the value of inputs to equal the value of outputs for each period. The function gt is known as the GDP function or the national product function in the international trade literature (see Kohli (1978) (1991), Woodland (1982) and Feenstra (2004; 76). economics literature by Samuelson (1953).  It was introduced into the  Alternative terms for this function include: (i) the gross profit  8  Given a vector of output prices P and a vector of available inputs x, we define the period t market sector GDP function, gt(P,x), as follows:15 (2.1) gt(P,x)  max y {Py : (y,x) belongs to St} ;  t = 0,1,2, ... .  Thus, market sector GDP depends on t (which represents the period t technology set St), on the vector of output prices P that the market sector faces and on x, the vector of inputs that is available to the market sector. If Pt is the period t output price vector and xt is the vector of inputs used by the market sector during period t and if the GDP function is differentiable with respect to the components of P at the point Pt,xt, then the period t vector of market sector outputs yt will be equal to the vector of first order partial derivatives of gt(Pt,xt) with respect to the components of P; i.e., we will have the following equations for each period t:16 (2.2) yt = P gt(Pt,xt) ;  t = 0,1,2, ... .  If the GDP function is differentiable with respect to the components of x at the point Pt,xt, then the period t vector of input prices Wt will be equal to the vector of first order partial derivatives of gt(Pt,xt) with respect to the components of x; i.e., we will have the following equations for each period t:17 (2.3) Wt = x gt(Pt,xt) ;  t = 0,1,2, ... .  The constant return to scale assumption on the technology sets St implies that the value of function; see Gorman (1968); (ii) the restricted profit function; see Lau (1976) and McFadden (1978); and (iii) the variable profit function; see Diewert (1973) (1974) (1993). 15  The function gt(P,x) will be linearly homogeneous and convex in the components of P and linearly  homogeneous and concave in the components of x; see Diewert (1973) (1974; 136). Notation: Py  m=1M Pmym. 16  These t  t  t  relationships t  t  are  due  to  Hotelling  (1932;  594).  Note  that  P gt(Pt,xt)    t  [g (P ,x )/P1, ...,g (P ,x )/PM]. 17  These relationships are due to Samuelson (1953) and Diewert (1974; 140).  [gt(Pt,xt)/x1, ...,gt(Pt,xt)/xN].  9  Note that xgt(Pt,xt)   outputs will equal the value of inputs in period t; i.e., we have the following relationships: (2.4) gt(Pt,xt) = Ptyt = Wtxt ;  t = 0,1,2, ... .  The above material will be useful in what follows. However, our focus is not on GDP; instead, our focus is on the income generated by the market sector or more precisely, on the real income generated by the market sector. However, since market sector GDP (the value of market sector production) is distributed to the factors of production used by the market sector, nominal market sector GDP will be equal to nominal income of the market sector. As an approximate welfare measure that can be associated with market sector,18 we will choose to measure the real income generated by the market sector in period t in terms of the number of consumption bundles that the nominal income could purchase in period t.  Therefore, we define t as follows:  (2.5) t  Wtxt/PCt ; = wtxt = ptyt = gt(pt,xt)  t = 0,1,2, ...  where PCt > 0 is the period t consumption expenditures deflator and the market sector period t real output price pt and real input price wt vectors are defined as the corresponding nominal price vectors deflated by the consumption expenditures price index; i.e., we have the following definitions:19  18  We focus on the income produced by the market sector.  national incomes from other sources.  Thus, we do not take into account domestic or  For example, we dismiss the net income received from abroad (e.g.  the balance of primary incomes and current transfer in relation to the rest of the world) and social benefits from the government.  Moreover, our suggested welfare measure has a problem as welfare measure such  as it is not sensitive to the distribution of the income that is generated by the market sector. of domestic welfare generated by the market sector is only an approximate one.  Our measure  Thus, our analysis in this  chapter is a first step toward the more comprehensive analysis on Japanese living standard. 19  Our approach is similar to the approach advocated by Kohli (2004b; 92), except he essentially deflates  nominal GDP by the domestic expenditures deflator rather than just the domestic (household) expenditures deflator; i.e., he deflates by the deflator for C+G+I, whereas we suggest deflating by the deflator for C. Another difference in his approach compared to the present approach is that we restrict our analysis to the  10  (2.6) pt  Pt/PCt ; wt  Wt/PCt ;  t = 0,1,2, ... .  The first and last equalities in (2.5) imply that period t real income, t, is equal to the period t market sector GDP function, evaluated at the period t real output price vector pt and the period t input vector xt, gt(pt,xt). Thus the growth in real income over time can be explained by three main factor; t (Technical progress or Total Factor Productivity growth), the changes in real output prices and the growth of primary input quantities. We will shortly give formal definitions for these three growth factors. Using the linear homogeneity properties of the GDP functions gt(P,x) in P and x separately, we can show that the following counterparts to the relations (2.2) and (2.3) hold using the deflated prices p and w:20 (2.7) yt = p gt(pt,xt) ; (2.8) wt = x gt(pt,xt) ;  t = 0,1,2, ... t = 0,1,2, ... .  Now we are ready to define a family of period t productivity growth factors (p,x,t): (2.9) (p,x,t)  gt(p,x)/gt1(p,x) ;  t = 1,2, ... .  Thus, (p,x,t) measures the proportional change in the real income produced by the market sector that is induced by the technical change going from period t – 1 to t, facing the reference real output prices p and using reference input quantities x. We can choose the reference vectors for the measure of technical progress defined by (2.9) from the current year or the previous year: a Laspeyres type measure Lt that chooses the period t1 reference vectors pt1 and xt1 and a Paasche type measure Pt that chooses the period t reference vectors pt and xt:  market sector GDP, whereas Kohli deflates all of GDP (probably due to data limitations).  Our treatment  of the balance of trade surplus or deficit is also different. 20  If producers in the market sector of the economy are solving the profit maximization problem that is  associated with gt(P,x), which uses the original output prices P, then they equivalently solve the profit maximization problem that is associated with gt(p,x), which uses the normalized output prices p P/PC; i.e., Therefore, their behaviour can be described by using either gt(P,x) or gt(p,x).  11  (2.10) Lt  (pt1,xt1,t) = gt(pt1,xt1)/gt1(pt1,xt1) ; (2.11) Pt  (pt,xt,t) = gt(pt,xt)/gt1(pt,xt) ;  t = 1,2, ... ; t = 1,2, ... .  Since both measures of technical progress are equally valid, it is natural to average them to obtain an overall measure of productivity growth. If we want to treat the two measures in a symmetric manner and we want the measure to satisfy the time reversal property from index number theory21 (so that the estimate going backwards is equal to the reciprocal of the estimate going forwards), then the geometric mean will be the best simple average to take.22 Thus we define the geometric mean of (2.10) and (2.11) as follows:23 (2.12) t  [Lt Pt]1/2 ;  t = 1,2, ... .  At this point, it is not clear how we will obtain empirical estimates for the theoretical productivity growth indexes defined by (2.10)–(2.12). One obvious way would be to assume a functional form for the GDP function gt(p,x), collect data on output and input prices and quantities for the market sector for a number of years (and for the consumption expenditures deflator), add error terms to equations (2.7) and (2.8) and use econometric techniques to estimate the unknown parameters in the assumed functional form. However, econometric techniques are generally not completely straightforward: different econometricians will make different stochastic specifications and will choose different functional forms.24 Moreover, as the number of outputs and inputs grows, it will be impossible to estimate a flexible functional form. Thus we will suggest methods for implementing measures like (2.12) in this chapter that are based on exact index number techniques. We turn now to the problem of defining theoretical indexes for the effects on real income 21  See Fisher (1922; 64).  22  See the discussion in Diewert (1997) on choosing the “best” symmetric average of Laspeyres and  Paasche indexes that will lead to the satisfaction of the time reversal test by the resulting average index. 23  The theoretical productivity change indexes defined by (2.10)–(2.12) were first defined by Diewert and  Morrison (1986; 662-663). See Diewert (1993) for properties of symmetric means. 24  “The estimation of GDP functions such as (2.19) can be controversial, however, since it raises issues  such as estimation technique and stochastic specification. ... We therefore prefer to opt for a more straightforward index number approach.” Ulrich Kohli (2004a; 344).  12  due to changes in real output prices. t1  t  change factors (p ,p ,x,s):  Define a family of period t real output price  25  (2.13) (pt1,pt,x,s)  gs(pt,x)/gs(pt1,x) ;  s = 1,2, ... .  Thus (pt1,pt,x,s) measures the proportional change in the real income produced by the market sector that is induced by the change in real output prices going from period t1 to t, using the reference technology that is available during period s and using the reference input quantities x. Thus, each choice of the reference period for technology s and the reference input vector x will generate a possibly different measure of the effect on real income of a change in real output prices going from period t1 to period t. Again, we can choose the reference vectors for the measure of output price change defined by (2.13) from the current year or the previous year: a Laspeyres type measure Lt that chooses the period t1 reference technology and reference input vector xt1 and a Paasche type measure Pt that chooses the period t reference technology and reference input vector xt: (2.14) Lt  (pt1,pt,xt1,t1) = gt1(pt,xt1)/gt1(pt1,xt1) ; (2.15) Pt  (pt1,pt,xt,t) = gt(pt,xt)/gt(pt1,xt) ;  t = 1,2, ... ; t = 1,2, ... .  Since both measures of real output price change are equally valid, it is natural to average them to obtain an overall measure of the effects on real income of the change in real output prices:26 The simple geometric mean is justified from the same reason as that for the technical progress shift function. (2.16) t  [Lt Pt]1/2 ; 25  t = 1,2, ... .  This measure of real output price change was essentially defined by Fisher and Shell (1972; 56–58),  Samuelson and Swamy (1974; 588–592), Archibald (1977; 60–61), Diewert (1980; 460–461) (1983; 1055) and Balk (1998; 83-89).  Readers who are familiar with the theory of the true cost of living index will note  that the real output price index defined by (2.13) is analogous to the Konüs (1924) true cost of living index which is a ratio of cost functions, say C(u,pt)/C(u,pt1) where u is a reference utility level: gs replaces C and the reference utility level u is replaced by the vector of reference variables x. 26  The indexes defined by (2.13)–(2.16) were defined by Diewert and Morrison (1986; 664) in the nominal  GDP function context.  13  Finally, we look at the problem of defining theoretical indexes for the effects on real income due to changes in real output prices. Define a family of period t input quantity growth factors (xt1,xt,p,s):27 (2.17) (xt1,xt,p,s)  gs(p,xt)/gs(p,xt1) ;  s = 1,2, ... .  Thus (xt1,xt,p,s) measures the proportional change in the real income produced by the market sector that is induced by the change in input quantities going from period t1 to t, using the technology that is available during period s and facing the reference real output prices p. Thus, each choice of the reference period for technology s and the reference real output price vector p will generate a possibly different measure of the effect on real income of a change in input quantities going from period t1 to period t. Again, We can choose the reference vectors for the measure of input quantity change defined by (2.13) from the current year or the previous year: a Laspeyres type measure Lt that chooses the period t1 reference technology and reference real output price vector pt1 and a Paasche type measure Pt that chooses the period t reference technology and reference real output price vector pt: (2.18) Lt  (xt1,xt,pt1,t1) = gt1(pt1,xt)/gt1(pt1,xt1) ; (2.19) Pt  (xt1,xt,pt,t) = gt(pt,xt)/gt(pt,xt1) ;  t = 1,2, ... ; t = 1,2, ... .  Since both measures of real input growth are equally valid, it is natural to average them to obtain an overall measure of the effects of input growth on real income:28 The simple geometric mean is justified from the same reason as that for the technical progress shift function. (2.20) t  [Lt Pt]1/2 ;  t = 1,2, ... .  We provide all the theoretically motivated measures such as Lt, Lt, Lt, Pt, Pt, and Pt. 27  This type of index was defined as a true index of value added by Sato (1976; 438) and as a real input  index by Diewert (1980; 456). 28  The theoretical indexes defined by (2.17)–(2.20) were defined in Diewert and Morrison (1986; 665) in  the nominal GDP context.  14  Now, we consider how to calculate these measures from the observed data. On the first sight, the effort is hopeless. Under the constant returns to scale assumption on the technology set St, we have (2.21) pt yt = gt(pt,xt) It means that we observe values for the denominator of Laspeyres type measure such as Lt, Lt, Lt and for the numerator of Paasche type measure such as Pt, Pt, Pt. However, since we only know the denominator or the numerator of indicators, we cannot calculate even any indicators of Lt, Lt, Lt, Pt, Pt, and Pt. For example, the numerator of Laspeyres type measure technical progress gt(pt1,xt1) is the hypothetical revenue that would result from using the period t technology with the period t−1 input quantities and real output prices. Fortunately, these hypothetical revenues can be inferred from observed data if the period t revenue function follows the Translog functional form. The following Translog functional form is known as the flexible functional form which approximates the arbitrary revenue function by the second order:29 (2.22) lngt(p,x)  a0t + m=1M amt lnpmt + (1/2) m=1Mk=1M amk lnpmt lnpkt + n=1N bnt lnxnt + (1/2)n=1Nj=1N bnj lnxnt lnxjt + m=1Mn=1M cmn lnpmt lnxnt ; t = 1,2, ... . Note that the coefficients for the quadratic terms are assumed to be constant over time. The coefficients must satisfy the following restrictions in order for gt to satisfy the linear homogeneity properties that we have assumed in section 2.2 above:30 (2.23) m=1M amt = 1 for t = 0,1,2, ...; (2.24) n=1N bnt = 1 for t = 0,1,2, ...; (2.25) amk = akm for all k,m ; 29  This functional form was first suggested by Diewert (1974; 139) as a generalization of the translog  functional form introduced by Christensen, Jorgenson and Lau (1971).  Diewert (1974; 139) indicated that  this functional form was flexible. 30  There are additional restrictions on the parameters which are necessary to ensure that gt(p,x) is convex in  p and concave in x.  15  (2.26) bnj = bjn for all n,j ; (2.27) k=1M amk = 0 for m = 1,...,M ; (2.28) j=1N bnj = 0 for n = 1,...,N ; (2.29) n=1N cmn = 0 for m = 1,...,M ; (2.30) m=1M cmn = 0 for n = 1,...,N . Theorem 1: Adaptation of Diewert and Morrison (1986)31: If gt1 and gt are defined by (2.22)–(2.30) above and there is competitive profit maximizing behaviour on the part of all market sector producers for all period t, we have (2.31) t = ptyt/(t t pt-1yt-1) ;  t = 1,2, ...  where (2.32) lnt = Σm=1(1/2)[(pmt1ymt1/pt1yt1) + (pmtymt/ptyt)] ln(pmt/pmt1) ; (2.33) lnt = Σn=1(1/2)[(wnt1xnt1/wt1xt1) + (wnt xnt/wtxt)] ln(xnt/xnt1); The above theorem shows how we can calculate the theoretically motivated measures such as t, t, and t from observed data. Another contribution of the theorem is to show that the real income growth can be exactly decomposed into the three explanatory factors: productivity growth, real output price change, and input quantity change. For some purposes, it is convenient to decompose the aggregate period t contribution to real income growth due to changes in all real output prices t into separate effects for the change in each real output price. Similarly, it can sometimes be useful to decompose the aggregate period t contribution to real income growth due to changes in all input quantities t into separate effects for the change in each input quantity. We indicate how this can be done, making the same assumptions on the technology that we have made so far. We first model the effect of the change in the real output price of output m, say pm, going from period t1 to t or the period t real price change factor for output m. 31  Counterparts  Diewert and Morrison (1986) established their proof using the nominal GDP function gt(P,x). However,  it is easy to rework their proof using the deflated GDP function gt(p,x) using the fact that gt(p,x) = gt(P/PC,x) = gt(P,x)/PC using the linear homogeneity property of gt(P,x) in P. for theorem 2 and 3.  16  This argument is also true  to the theoretical Laspeyres and Paasche type price indexes defined by (2.14) and (2.15) above for all the outputs are the following Laspeyres type measure Lmt that chooses the period t1 reference technology, holds other real output prices (other than output m) constant at their period t1 levels, and holds input quantities constant at their period t1 levels and a Paasche type measure Pmt that chooses the period t reference technology, holds other real output prices constant at their period t levels, and holds input quantities constant at their period t levels; (2.34) Lmt  gt1(p1t1,...,pm1t1,pmt,pm+1t1,..., pMt1,xt1)/gt1(pt1,xt1) ;  m = 1,...,M; t = 1,2, ... ;  (2.35) Pmt  gt(pt,xt)/gt(p1t ,...,pm1t,pmt1,pm+1t,..., pMt,xt) ;  m = 1,...,M; t = 1,2, ... .  Since both measures of real output price changes are equally valid, it is natural to average them to obtain an overall measure of the effects on real income of the change in the real price of output m:32 (2.36) mt  [Lmt Pmt]1/2 ;  m = 1,...,M ; t = 1,2, ... .  Under the assumption that the deflated GDP functions gt(p,x) have the Translog functional forms as defined by (2.22)–(2.30), the arguments of Diewert and Morrison (1986; 666) provide the following exact decomposition of the period t aggregate real output price contribution factor t into a product of separate price contribution factors mt: Theorem 2: Adaptation of Diewert and Morrison (1986): If gt1 and gt are defined by (2.22)-(2.33) above and there is competitive profit maximizing behaviour on the part of all market sector producers for all periods t, then we have (2.37) t = 1t2t... Mt ;  t = 1,2, ... .  where  32  The indexes defined by (2.34)–(2.36) were defined by Diewert and Morrison (1986; 666) in the nominal  GDP function context.  17  (2.38) lnmt = (1/2)[(pmt1ymt1/pt1yt1) + (pmtymt/ptyt)] ln(pmt/pmt1) ;  m = 1,...,M.  For example, we consider the case where there are four net outputs:      1: Domestic sales; 2: Investment; 3: Exports and; 4: Imports.  Since commodities 1, 2, and 3 are outputs, y1, y2, and y3 will be positive but since commodity 4 is an input into the market sector, y4 will be negative. Hence an increase in the real price of exports will increase real income but an increase in the real price of imports will decrease the real income generated by the market sector, as is evident by looking at the contribution terms defined by (2.38) for m = 3 (where ymt > 0) and for m = 4 (where ymt < 0). The above decomposition (2.38) is useful for analyzing the impacts that the changes in the real price of exports (i.e., a change in the price of exports relative to the price of domestic consumption) and in the real price of imports make on the real income generated by the market sector. We now model the effects of the change in the quantity of input n, say xn, going from period t1 to t or the period t quantity growth factor for input n. Individual counterparts to the overall theoretical Laspeyres and Paasche type quantity indexes defined by (2.18) and (2.19) above for all the inputs are the following Laspeyres type measures for input n Lnt that chooses the period t1 reference technology and holds constant real output prices at their period t1 levels and holds other input quantities (other than input n) constant at their period t1 levels and the Paasche type measures for input n Pnt that chooses the period t reference technology and hold constant real output prices at their period t levels and hold other input quantities constant at their period t levels; (2.39) Lnt  gt1(pt1,x1t1,...,xn1t1,xnt,xn+1t1,..., xNt1)/gt1(pt1,xt1) ;  n = 1,...,N; t = 1,2, ... ;  (2.40) Pnt  gt(pt,xt)/gt(pt,x1t ,...,xn1t,xnt1,xn+1t,..., xNt) ;  n = 1,...,N; t = 1,2, ... .  Since both measures of input change are equally valid, as usual, we average them to obtain an overall measure of the effects on real income of the change in the quantity of 18  input n:33 (2.41) nt  [Lnt Pnt]1/2 ;  n = 1,...,N ; t = 1,2, ... .  Under the assumption that the deflated GDP functions gt(p,x) have the Translog functional forms as defined by (2.22)–(2.30), the arguments of Diewert and Morrison (1986; 666) can be adapted to provide the following exact decomposition of the period t aggregate input growth contribution factor t into a product of separate input contribution factors nt: Theorem 3: Adaptation of Diewert and Morrison (1986): If gt1 and gt are defined by (2.22)–(2.30) above and there is competitive profit maximizing behaviour on the part of all market sector producers for all periods t, then we have (2.42) t = 1t2t... Nt ;  t = 1,2, ... .  where (2.43) lnnt = (1/2)[(wnt1xnt1/wt1xt1) + (wnt xnt/wtxt)] ln(xnt/xnt1) ;  n = 1,...,N.  Substituting the results in Theorems 2 and 3 into the decomposition in Theorem 1 and rearranging the terms, we obtain the following formula; (2.44) ρt/ρt-1 = τt∏m=1Mαmt∏n=1Nβnt;  t = 1,2, ... .  Thus the growth in real income over time can be explained by three main factors: productivity growth, the growth of real output prices, and the growth of input quantities. Rather than look at explanatory factors for the growth in real income, it is sometimes convenient to express the level of real income in period t in terms of cumulated growth factors. Thus (2.45) below defines a period t index of the technology level or of Total Factor Productivity in period t relative to period 0, Tt; (2.46) defines a period t cumulative index of the real price change factor for output m since period 0, Amt; and (2.47) defines a period t cumulated index of the quantity growth factor for input n since 33  The indexes defined by (2.33)–(2.41) were defined by Diewert and Morrison (1986; 667) in the nominal  GDP function context.  19  period 0, Bnt: (2.45) T0 ≡ 1; Tt ≡ Tt-1τt; (2.46) Am0 ≡ 1; Amt ≡ Amt1 αmt; (2.47) Bn0 ≡ 1; Bnt1 ≡ Bn0βnt;  t = 1,2,…,; m = 1,2,…,M and t = 1,2,…,; n = 1,2,…,N and t = 1,2,…,;  Using the appropriate equalities (2.44) for the chain links that appear in (2.45)–(2.47), we can establish the following exact relationship for the level of real income in period t, ρt, relative to its counterpart in period 0, 0, and the cumulated growth factors defined by (2.45)–(2.47): (2.48) ρt/ρ0 = Tt∏m=1MAmt∏n=1NBnt;  t = 1,2, ... .  2.3 Data Construction 2.3.1 The Definition of the Market Sector In this chapter, we focus on the production of the market sector within the entire economy. This is the sector that plays a critical role for economic growth. Following the 1993 SNA, we classify the producers for the entire economy into the following five mutually exclusive institutional sectors:       IU1; Non-financial corporations; IU2; Financial corporations; IU3; General government; IU4; Households34; IU5; Non-profit institutions serving households.  In our definition, the market sector consists of IU1, IU2, and IU5. Thus, we subtract the government’s and the households’ outputs and inputs from aggregate output, capital services and labour input in order to perform the growth accounting exercise explained in 34  According to the UN (1993), the unincorporated enterprises are parts of the household sectors.  However, we added the unincorporated enterprises to IU1 or IU2 based on the characteristics of the enterprise.  Therefore, in our classification, the household sector only includes household own-account  producers.  20  the previous section. There are two types of transactions between the market sector and the general government sector: (1) the sales of goods and services of the general government sector to the market sector and (2) the purchases of intermediate inputs by the general government sector from the market sector. Note that the sales of goods and services by the general government sector are considered as inputs for the market sector, while the purchases of intermediate inputs by the general government sector are regarded as outputs by the market sector. Thus, the difference between (1) and (2) is the net output by the market sector, which has been purchased by the general government sector. In the Japanese national accounts, the imputed rent of the owner-occupied dwellings is the major output by the household sector.35 The households sector produces imputed rent by using capital services from the stock of owner-occupied houses and land services from the residential land under the owned houses. Therefore, we need to subtract the imputed rent from the consumption output, the stock of owner-occupied houses from the total stock of residential buildings, and the residential land under the owned houses from the total residential land.36 2.3.2 The Database for Inputs and Outputs Our final dataset consists of price and quantity series for the variables listed below. They are the inputs and outputs for the market sector. We also followed the conventions on the treatment of indirect taxes that Jorgenson and Griliches introduced, i.e. we adjusted prices for tax wedges whenever possible, so that the adjusted prices reflect the prices that producers face.37 We also included the services of inventories and land (utilized by the market sector) as primary inputs. The net outputs in our most disaggregated database are as follows: 35  We ignore other own-account production by households.  36  The household sector purchases goods and services for the maintenance of owner-occupied houses.  The expenses on these goods and services are produced by the market sector and thus are part of our market sector output. 37  Thus our suggested treatment of indirect commodity taxes in an accounting framework that is suitable  for productivity analysis follows the example set by Jorgenson and Griliches, who advocated the following treatment of indirect taxes: “In our original estimates, we used gross product at market prices; we now employ gross product from the producers’ point of view, which includes indirect taxes levied on factor outlay, but excludes indirect taxes levied on output.” Jorgenson and Griliches (1972; 85).  21    C; Domestic final consumption expenditure of households (excluding the imputed rent of owner-occupied dwellings);    N; Final consumption expenditure of private non-profit institutions serving households (NPISHs);    G; Net purchases of goods and services by the general government from the market sector;    X; Exports of goods and services (including direct purchases in the domestic market by non-resident households since this is a source of revenue to the market sector);    M; Imports of goods and services (excluding direct purchases abroad by resident households, since this is not a source of revenue to the Japanese market sector);     I1 to I95; Gross investments for 95 asset categories; IV1 to IV4; Change in inventories for 4 types of inventory asset.  The primary inputs in our most disaggregated database are as follows:      K1 to K95; Capital services from 95 classes of fixed assets; KIV1 to KIV4; Inventory services for 4 classes of inventory assets; LD1 to LD4; Market sector land services and LB1 to LB3; Labour input for 3 types of labour.  Dividing the prices of inputs and outputs by the price of domestic final consumption expenditure of households PC, we constructed the corresponding real input and output prices. The net outputs C, N, G, X, and M have been constructed as aggregates of the net output components listed above, and are essentially based on the data of the Japanese national accounts. Since the Japanese national accounts experienced several revisions, there are different data series even for the same variable. We chose the reference data series from the most recent publication and extended these series by using the growth rates of other data series from earlier publications.38 A more detailed explanation of our data construction methods is provided in Appendix A. Quantities of capital services, inventory services, and land services are proportional to 38  The data from the Japanese national accounts are taken from publications such as the Annual Report on  National Income Statistics 1975, Report on National Accounts from 1955 to 1998, Annual Report on National Accounts of 2000 and Annual Report on National Accounts of 2004-2008. explanation.  22  See for further  capital stocks of fixed assets, inventories, and land at the beginning of the year. Capital stocks have been constructed by applying the perpetual inventory method, using the gross investment data (I1 to I95) to the initial stocks of 1955, based on asset-specific depreciation rates suggested by Nomura (2004). Changes in Inventories IV1–IV4 are calculated by using the difference between the inventory stock in the current period and the inventory stock in the previous period. The initial stocks for fixed assets, inventory stocks and land, and the price and quantity series for investments have been taken from the KEO database. This is a comprehensive productivity database for the Japanese economy constructed by KEO at Keio University.39 The detailed procedures used to construct prices and quantities for capital stocks and services are explained in Nomura (2004). For residential structures and residential land, we use other data sources in order to net out the stocks that the household sector holds, which correspond to the stocks of residential land and structures used by owner-occupied households. Our procedures will be explained in the following subsection. Prices of capital services for the 95 types of reproducible capital services K1–K95, for the 4 types of inventory services KIV1–KIV4 and the 4 types of land services LD1–LD4 are calculated by applying a user cost formula that we will explain in the next section. Estimates of the quantities of labour services LB1–LB3 are based on hours of work. There are three different types of workers: employees, the self-employed, and family workers. After we calculate the average labour income and the hours of work for each type of worker, we aggregate these three types of labour into a labour aggregate using a superlative index number formula. Investments I1–I95 correspond to the capital services K1–K95. Changes in inventories IV1–IV4 correspond to the inventory services KIV1–KIV4. Table 2-1, which follows, lists the names of all capital services, inventory services, and land services.  39  This database also includes labour inputs.  Kuroda, Shimpo, Nomura, and Kobayashi (1997) explain  the data construction of labour inputs in detail.  In this chapter, we use only their relative wages for  employees, the self-employed and family workers from the KEO database.  23  Table 2-1: List of Capital Services, Inventory Services, and Land Services Name K1  Trees (growth)  K53  Other ships  K2  Livestock (growth)  K54  Railway vehicles  K3  Textile products  K55  Aircraft  K4  Wooded products  K56  Bicycles  K5  Wooden furniture and fixtures  K57  Transport equipment for industrial use  K6  Metallic furniture and fixtures  K58  Other transport equipment  K7  Nuclear fuel rods  K59  Camera  K8  Metallic products  K60  Other photographic and optical instruments  K9  Boilers and turbines  K61  Watches and clocks  K0  Engines  K62  Physics and chemistry instruments  K11  Conveyors  K63  Analytical and measuring instruments and testing machines  K12  Refrigerators and air conditioning apparatus  K64  Medical instruments  K13  Pumps and compressors  K65  Miscellaneous manufacturing products  K14  Sewing machines  K66  Residential construction (wooden)  K15  Other general industrial machinery and equipment  K67  Residential construction (non-wooden)  K16  Mining, civil engineering and construction machinery  K68  Non-residential construction (wooden)  K17  Chemical machinery  K69  Non-residential construction (non-wooden)  K18  Industrial robots  K70  Road construction  K19  Metal machine tools  K71  Street construction  K20  Metal processing machinery  K72  Bridge construction  K21  Agricultural machinery  K73  Toll road construction  K22  Textile machinery  K74  River improvement  K23  Food processing machinery  K75  Erosion control  K24  Sawmill, wood working, veneer and plywood machinery  K76  Seashore improvement  K25  Pulp equipment and paper machinery  K77  Park construction  K26  Printing, bookbinding and paper processing machinery  K78  Sewer construction  K27  Casting equipment  K79  Sewage disposal facilities  K28  Plastic processing machinery  K80  Waste disposal facilities  K29  Other special industrial machinery, nec  K81  Harbor construction  K30  Other general machines and parts  K82  Fishing port construction  K31  Office machines  K83  Airport construction  K32  Vending, amusement and other service machinery  K84  Agricultural construction  K33  Electric audio equipment  K85  Forest road construction  K34  Radio and television sets  K86  Forestry protection  K35  Video recording and playback equipment  K87  Railway construction  K36  Household electric appliance  K88  Electric power facilities  K37  Electronic computer and peripheral equipment  K89  Telecommunication facilities  K38  Wired communication equipment  K90  Other civil engineering and construction  K39  Radio communication equipment  K91  Plant engineering  K40  Other communication equipment  K92  Mineral exploration  K41  Applied electronic equipment  K93  Custom software  K42  Electric measuring instruments  K94  Pre-packaged software  K43  Generators  K95  Own-account software  K44  Electric motors  KIV1  Finished-goods inventory  K45  Relay switches and switchboards  KIV2  Work-in-process inventory  K46  Other industrial heavy electrical equipment  KIV3  Work-in-process inventory for cultivated assets  K47  Electric lighting fixtures and apparatus  KIV4  Material inventory  K48  Passenger motor vehicles  LD1  Land for agricultural use  K49  Trucks, buses and other vehicles  LD2  Land for industrial use  K50  Two-wheel motor vehicles  LD3  Land for commercial use  K51  Motor vehicle parts  LD4  Land for residential use  K52  Steel ships  The following subsection explains how we constructed the price, quantity and value 24  series for the capital inputs. found in Appendix A.  The details on how our labour inputs were constructed are  2.3.3 User Costs and Real Rates of Return Prices of capital services, inventory services and land services are estimated as user costs of capital. User costs are constructed for each of our capital stock components. The general formula for a user cost for a capital stock component in year t, ut, is as follows: (2.49) ut = PKt − (1 − )PKt(1 + it)/(1 + rt), where PKt is the beginning of year t asset price for the capital stock component,  is the geometric depreciation rate that applies to the asset,40 it is the amount of asset-specific price change that is expected to occur over the course of year t and rt is the nominal rate of return (or opportunity cost of capital) that producers face at the beginning of year t. Thus, the user cost of a durable input is equal to its purchase cost or opportunity cost at the beginning of the year, PKt, less the discounted expected value of the depreciated asset at the end of the year.41 The nominal rate of return rt can be decomposed into a real rate of return component rt* and an expected general inflation component, it*, by using the following formula: (2.50) 1 + rt = (1 + rt*)(1 + it*). The practical problem for the economic statistician with a user cost formula of the type defined by (2.49) and (2.50) is that there is uncertainty about how exactly to estimate the depreciation rate , the relevant real rate of return rt* and the two anticipated inflation rates, it* (general inflation rate) and it (asset-specific inflation rate). If we estimate it* and it as actual ex post inflation rates, we will almost certainly generate user costs ut that are negative for some years, which is not sensible in our context since we want our user costs to closely approximate market rental rates for the assets and since these rates would not be negative. Even if we estimate it* and it by smoothing the ex post values for these variables or by using a forecasting model, with Japanese data, we will inevitably generate some negative user costs for land components because of the very rapid land price 40  We assume that the depreciation rates for the inventory stocks and land components are 0.  41  We have temporarily neglected tax factors for the sake of simplicity.  25  inflation that occurred in Japan during the 1980s. We decided to avoid these negative user cost problems by assuming that producers expect the asset-specific inflation rate it to equal the general inflation rate it*, i.e. we make the following assumption: (2.51) it = it*. If we substitute (2.50) and (2.51) into (2.49), we find that our user cost formula simplifies as follows:42 (2.52) ut = PKt − (1 − )PKt/(1 + rt*) = [rt* + ]PKt /(1 + rt*). We approximate the beginning of the year asset price PKt by the corresponding year t investment price PIt. Thus, our final user cost formula has the following generic form, where t is the business tax rate:43 (2.53) ut  [rt* + t + ]PIt/(1 + rt*). There remains the problem of finding a suitable real interest rate series, rt*, which will be discussed below. Once rt* has been determined, the user cost defined by (2.53) can be calculated for each of our 95 types of capital services, 4 types of inventory services, and 4 types of land services. The corresponding beginning-of-year capital stocks have already been described and tabled. We normalize the resulting user costs, so that they are 1 in 1955, and make offsetting normalizations are made to our capital quantity series. The resulting input prices are WK1t–WK95t, WKIV1t–WKIV4t and WLD1t–WLD4t, and the resulting quantities are xK1t–xK95t, xKIV1t–xKIV4t and xLD1t–xLD4t. The price series and the quantity series for 13 aggregated inputs and outputs are listed in Appendix B.44 Finally, we discuss how we obtained a suitable real interest rate series rt*. The rate we chose is an economy-wide ex post real rate of return. If we use the user cost formula (2.53) to form prices for capital services, we can rearrange the value of outputs equals the 42  For discussions about alternative assumptions for user cost formulae, see Schreyer (2001) (2007) (2008)  and Diewert (1980) (2004) (2005b). 43  For business structures and land, we need to add the appropriate specific property tax rates to the general  tax rate t. 44  More detailed data will be provided to the interested reader.  26  value of primary inputs equation into a single (linear) equation involving an unknown real rate of return. The solutions to these equations (one for each year) are the series of ex post real interest rates, rt*, listed below in Figure 2-1. 8% 7% 6% 5% 4% 3% 2% 1% 0% 1955  1960  1965  1970  1975  1980  1985  1990  1995  2000  2005  Figure 2-1: The Real Rate of Return for the Years 1955-2006 The average rate for the years 1955–2006 is 3.25%. This rate is consistent with “the long run observed economy-wide real rates of return for most OECD countries which fall into the 3 to 5 percent range” (Diewert (2004)). The performance of the Japanese economy over the years 1955–2006 can be divided into 5 periods:       Period 1: 1955–1973: Rapid economic growth; Period 2: 1974–1979: Stagnation between oil shocks; Period 3: 1980–1990: Stable economic growth; Period 4: 1991–2001: Long recession and stagnation; Period 5: 2002–2006: Modest economic recovery.  The average real rates of return for the above periods are 5.379% (Period 1), 2.86% (Period 2), 2.214% (Period 3), 1.483% (Period 4) and 1.885% (Period 5). Even though the Japanese economy is currently in a period of economic recovery, its average real rate of return is still significantly smaller than the average real rate of return of the period of rapid economic growth (Period 1). 2.4 Results 2.4.1 Japanese Productivity Growth: A Conventional Approach 27  In this section, we measure the productivity growth of the market sector of the Japanese economy using a conventional chained Fisher index number approach. Depending on whether inventory services or land services are included in the list of primary inputs, we can consider three different cases:     Case 1: inventory services and land services are in the list of primary inputs; Case 2: land services are not in the list of primary inputs; and Case 3: inventory services and land services are both excluded from the list of primary inputs.  Since we endogenously determine the real rate of return from the zero-profit condition, the composition of primary inputs affects the real rate of return. Basically, the conventional measure of productivity growth is set equal to a chained Fisher output quantity index divided by a chained Fisher input quantity index. Thus, the composition of primary inputs also affects the productivity growth. Thus in Table 2, under the above three assumptions, we list three sets of the Fisher year to year output and input growth factors, yt/yt−1, xt/xt−1, respectively, along with their ratios, τt  [yt/yt−1]/[xt/xt−1] and the balancing year t real interest rates rt*. Figure 2 plots three sets of the real rate of return, and Figure 3 plots three sets of Total Factor Productivity.45  45  Total Factor Productivity Tt is calculated from the ratios of output growth factors and input growth  factors τt using equation (2.45).  28  Table 2-2: Chained Fisher Indexes of Output, Input and Productivity Growth and the  Balancing Real Rate of Return in the Japanese Economy, 1956-2006. Case 1 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Average 1956-2006 1956-1973 1974-1979 1980-1990 1991-2001 2002-2006  Case 2  Case 3  yt/yt-1 1.11124 1.09559 1.05114 1.13732 1.17782 1.15667 1.07165 1.10261 1.13972 1.05337 1.13090 1.13551 1.13640 1.14133 1.10251 1.03639 1.08462 1.10733 0.98945 1.00794 1.06272 1.03519 1.05398 1.08746 1.02811 1.02026 1.02889 1.01990 1.04942 1.05245 1.02548 1.04535 1.06680 1.06722 1.05502 1.03178 1.01006 0.99101 1.01093 1.01713 1.02595 1.01392 0.97028 0.99587 1.02453 0.99262 1.00694 1.00865 1.03540 1.03493 1.03959  xt/xt-1 1.04614 1.04684 1.03596 1.03339 1.04742 1.04835 1.06155 1.04217 1.04800 1.05219 1.06055 1.05547 1.04436 1.05589 1.05400 1.04883 1.03835 1.05480 1.02220 1.00881 1.04473 1.03237 1.02280 1.02771 1.03101 1.02825 1.02404 1.03434 1.01686 1.03022 1.02563 1.02963 1.03324 1.03107 1.02909 1.02859 1.01805 1.01111 1.00959 1.01180 1.01054 1.00811 1.00632 1.00156 1.01694 0.99708 0.99907 1.00271 1.00793 1.00521 1.01539  tt 1.06223 1.04657 1.01466 1.10057 1.12450 1.10332 1.00951 1.05800 1.08753 1.00113 1.06634 1.07583 1.08813 1.08092 1.04602 0.98814 1.04457 1.04980 0.96796 0.99914 1.01722 1.00273 1.03048 1.05814 0.99719 0.99224 1.00474 0.98604 1.03202 1.02158 0.99985 1.01528 1.03249 1.03507 1.02520 1.00310 0.99215 0.98012 1.00133 1.00526 1.01525 1.00576 0.96418 0.99431 1.00746 0.99553 1.00788 1.00593 1.02726 1.02957 1.02383  rt* 0.04108 0.04185 0.03895 0.04630 0.05528 0.06227 0.05742 0.06053 0.06418 0.05963 0.06342 0.06543 0.06830 0.06596 0.06122 0.04787 0.04237 0.03816 0.03022 0.02485 0.02923 0.02665 0.02824 0.03241 0.02798 0.02463 0.02265 0.01999 0.02257 0.02432 0.02258 0.01992 0.01981 0.02050 0.01859 0.01636 0.01782 0.01506 0.01479 0.01474 0.01637 0.01668 0.01289 0.01343 0.01351 0.01150 0.01548 0.01721 0.02056 0.01977 0.02121  yt /yt-1 1.11130 1.09565 1.05135 1.13733 1.17783 1.15657 1.07167 1.10258 1.13972 1.05338 1.13091 1.13552 1.13642 1.14127 1.10246 1.03592 1.08456 1.10783 0.98892 1.00775 1.06273 1.03508 1.05394 1.08740 1.02829 1.02022 1.02889 1.01990 1.04940 1.05234 1.02588 1.04535 1.06678 1.06723 1.05505 1.03175 1.01007 0.99101 1.01093 1.01716 1.02596 1.01389 0.97030 0.99592 1.02452 0.99263 1.00693 1.00868 1.03538 1.03491 1.03949  xt /xt-1 1.05025 1.05395 1.04498 1.03941 1.06034 1.06481 1.08370 1.05847 1.06508 1.06821 1.06332 1.06088 1.06490 1.06961 1.07714 1.07085 1.05161 1.05966 1.03300 1.01496 1.04880 1.03481 1.02587 1.03195 1.03747 1.03468 1.02819 1.03807 1.02096 1.03605 1.03199 1.03581 1.03803 1.03660 1.03562 1.03598 1.02277 1.01390 1.01095 1.01322 1.01163 1.00926 1.00708 1.00232 1.01671 0.99741 0.99939 1.00235 1.00847 1.00565 1.01556  tt 1.05812 1.03957 1.00610 1.09421 1.11080 1.08617 0.98890 1.04168 1.07008 0.98611 1.06356 1.07035 1.06716 1.06699 1.02351 0.96738 1.03133 1.04546 0.95733 0.99290 1.01328 1.00026 1.02736 1.05373 0.99115 0.98602 1.00068 0.98250 1.02786 1.01572 0.99408 1.00921 1.02770 1.02956 1.01876 0.99592 0.98758 0.97742 0.99999 1.00388 1.01416 1.00458 0.96348 0.99362 1.00768 0.99520 1.00754 1.00631 1.02669 1.02909 1.02356  rr* 0.10607 0.11285 0.11178 0.14250 0.18621 0.20866 0.19320 0.20565 0.21995 0.20023 0.21862 0.23938 0.25819 0.26210 0.24603 0.19496 0.18738 0.17564 0.12274 0.09538 0.10476 0.09700 0.10472 0.11403 0.09910 0.09013 0.08504 0.07831 0.08678 0.09176 0.08965 0.08714 0.09075 0.09508 0.09000 0.07772 0.07115 0.05873 0.05543 0.05368 0.05584 0.05302 0.04171 0.04096 0.04145 0.03631 0.04101 0.04266 0.04775 0.04688 0.04833  yt/yt-1 1.11130 1.09565 1.05135 1.13733 1.17783 1.15657 1.07167 1.10258 1.13972 1.05338 1.13091 1.13552 1.13642 1.14127 1.10246 1.03592 1.08456 1.10783 0.98892 1.00775 1.06273 1.03508 1.05394 1.08740 1.02829 1.02022 1.02889 1.01990 1.04940 1.05234 1.02588 1.04535 1.06678 1.06723 1.05505 1.03175 1.01007 0.99101 1.01093 1.01716 1.02596 1.01389 0.97030 0.99592 1.02452 0.99263 1.00693 1.00868 1.03538 1.03491 1.03949  xt/xt-1 1.04580 1.04972 1.04278 1.04126 1.05894 1.06578 1.08216 1.06334 1.06744 1.07048 1.06617 1.06338 1.06546 1.07158 1.07974 1.07397 1.05762 1.06523 1.03444 1.01541 1.05231 1.03634 1.02822 1.03460 1.03642 1.03458 1.02921 1.03906 1.02248 1.03720 1.03286 1.03763 1.03888 1.03738 1.03581 1.03654 1.02342 1.01450 1.01161 1.01374 1.01160 1.00913 1.00699 1.00248 1.01731 0.99748 0.99981 1.00259 1.00881 1.00601 1.01503  tt 1.06263 1.04375 1.00823 1.09227 1.11228 1.08518 0.99031 1.03690 1.06772 0.98403 1.06072 1.06784 1.06660 1.06503 1.02104 0.96457 1.02548 1.03999 0.95600 0.99246 1.00990 0.99878 1.02501 1.05104 0.99215 0.98612 0.99969 0.98156 1.02633 1.01459 0.99324 1.00744 1.02686 1.02878 1.01857 0.99539 0.98696 0.97685 0.99933 1.00337 1.01419 1.00471 0.96356 0.99345 1.00709 0.99514 1.00712 1.00607 1.02634 1.02872 1.02410  rr* 0.14230 0.15657 0.15762 0.20277 0.27052 0.30588 0.28022 0.29516 0.31324 0.27948 0.30559 0.33723 0.36665 0.36999 0.33981 0.26161 0.24757 0.22985 0.15718 0.12091 0.13064 0.11946 0.12730 0.13687 0.11903 0.10787 0.10114 0.09257 0.10153 0.10619 0.10290 0.09902 0.10267 0.10683 0.10086 0.08660 0.07857 0.06449 0.06052 0.05836 0.06072 0.05759 0.04533 0.04446 0.04484 0.03918 0.04391 0.04559 0.05092 0.04998 0.05163  yt/yt-1 1.05642 1.10956 1.03946 1.04172 1.00764 1.02510  xt/xt-1 1.02934 1.04857 1.02644 1.02849 1.01088 1.00606  tt 1.02596 1.05821 1.01261 1.01288 0.99677 1.01889  rt* 0.03241 0.05446 0.02860 0.02214 0.01483 0.01885  yt /yt-1 1.05641 1.10957 1.03930 1.04176 1.00765 1.02507  xt /xt-1 1.03613 1.06151 1.03156 1.03395 1.01284 1.00628  tt 1.01926 1.04542 1.00748 1.00757 0.99486 1.01864  rt* 0.11577 0.19274 0.10644 0.08943 0.05327 0.04533  yt/yt-1 1.05641 1.10957 1.03930 1.04176 1.00765 1.02507  xt/xt-1 1.03707 1.06282 1.03355 1.03468 1.01316 1.00645  tt 1.01834 1.04414 1.00553 1.00685 0.99455 1.01847  rt* 0.15054 0.27011 0.13206 0.10369 0.05824 0.04841  40% 35%  Case1  30%  Case2  25%  Case3  20% 15% 10% 5% 0% 1955  1960  1965  1970  1975  1980  1985  1990  1995  2000  2005  Figure 2-2: Comparison of Real Rate of Return for the Years 1955-2006  29  4.0 3.5 3.0 2.5 2.0  Case1 Case2  1.5  Case3 1.0 1955  1960  1965  1970  1975  1980  1985  1990  1995  2000  2005  Figure 2-3: Comparison of Total Factor Productivity for the Years 1955-2006 The above table and figures show that it is extremely important to include land and inventory services in the list of primary inputs. If they are omitted from a productivity analysis, the results will be distorted. The growth rates for primary inputs will be excessively high, and thus productivity growth will be underestimated. The exclusion of land services from the primary inputs significantly changes our estimate. This exclusion bias becomes more serious during periods when the price of land is very high. Thus, productivity growth when land is included (Case 1) is almost 50% greater than when it is excluded during the years 1980–1990 (Cases 2 and 3). This period includes the bubble era at the end of the 1980s. Now, we turn to the theoretical framework presented in section 2.2 and determine the factors that explain real income growth in the Japanese economy using a traditional value added framework as opposed to the net framework that will be discussed later in section 2.4.3. 2.4.2 The Decomposition of Deflated GDP Growth We implement the Diewert and Morrison type approach using our Japanese database. The chain link information on period by period changes in real income that corresponds to (2.44) is listed in Table 2-3.  30  Table 2-3: The Decomposition of Market Sector Real Income Growth into Translog Productivity, Real Output Price Change and Input Quantity Contribution Factors 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  t  t  rt  tt  αN  αG  1.13349 1.09829 1.05396 1.12932 1.17609 1.15450 1.04941 1.07386 1.13395 1.03110 1.12652 1.13975 1.12644 1.14055 1.09177 1.02189 1.08106 1.11654 0.95963 0.96594 1.03629 1.02215 1.05758 1.07995 0.99400 1.01436 1.02273 1.01495 1.04747 1.04710 1.03404 1.04380 1.06827 1.06921 1.05126 1.02965 1.00706 0.98677 1.00652 1.01522 1.01852 1.00570 0.97011 0.98762 1.01821 0.98803 1.00473 1.00378 1.03283 1.02784 1.04007  1.06212 1.04655 1.01464 1.10058 1.12450 1.10336 1.00952 1.05803 1.08754 1.00114 1.06633 1.07582 1.08816 1.08095 1.04605 0.98815 1.04460 1.04975 0.96786 0.99916 1.01722 1.00273 1.03049 1.05812 0.99719 0.99224 1.00473 0.98603 1.03202 1.02152 0.99987 1.01526 1.03249 1.03507 1.02520 1.00311 0.99215 0.98012 1.00133 1.00526 1.01525 1.00576 0.96418 0.99431 1.00746 0.99554 1.00788 1.00593 1.02726 1.02958 1.02384  1.00065 1.00033 1.00004 1.00006 1.00024 1.00011 1.00000 0.99995 1.00052 0.99985 1.00024 1.00031 1.00009 1.00031 1.00058 1.00042 1.00052 1.00039 1.00021 0.99999 1.00012 1.00006 0.99996 1.00022 0.99997 0.99983 0.99999 0.99995 1.00000 1.00003 1.00004 1.00004 1.00005 1.00005 1.00024 1.00009 0.99997 0.99990 1.00004 1.00006 1.00004 1.00005 1.00008 0.99993 1.00000 1.00009 0.99983 0.99993 1.00007 1.00029 1.00040  1.00110 1.00032 0.99874 0.99919 0.99918 0.99889 0.99811 0.99764 0.99889 0.99816 0.99941 0.99990 0.99905 0.99920 0.99883 0.99810 0.99821 0.99937 1.00525 0.99241 0.99908 0.99884 0.99813 1.00047 1.00187 1.00031 0.99900 0.99958 0.99933 1.00130 0.99747 0.99866 0.99896 1.00011 1.00026 0.99935 0.99945 0.99951 0.99920 0.99957 0.99968 1.00019 0.99991 0.99966 1.00004 1.00009 1.00009 0.99977 1.00019 1.00053 1.00084  t  t  t  t  αX  αM  αI  αIV  1.00404 0.99667 0.99183 0.99872 0.99979 0.99297 0.99005 0.99478 0.99828 0.99082 0.99500 0.99643 0.99443 0.99730 0.99558 0.99451 0.99212 0.99809 1.00805 0.99079 0.98907 0.98364 0.98503 1.00610 1.00201 0.99792 1.00205 0.99164 0.99876 0.99178 0.97995 0.99459 0.99760 1.00178 0.99912 0.99462 0.99537 0.99131 0.99630 0.99810 1.00399 1.00090 1.00235 0.98883 0.99535 1.00440 1.00063 0.99627 1.00032 1.00462 1.01021  0.99244 0.99924 1.02114 1.00485 1.00329 1.00584 1.00977 1.00610 1.00149 1.00920 1.00369 1.00537 1.00567 1.00066 1.00529 1.01105 1.01061 0.99228 0.96430 1.00263 1.00535 1.01502 1.02459 0.97331 0.96453 1.00624 0.99970 1.01139 1.00703 1.01037 1.03974 1.00739 1.00282 0.99552 0.99558 1.00677 1.00639 1.00801 1.00393 1.00152 0.99364 0.99481 1.00247 1.00867 0.99765 0.99634 0.99950 1.00023 0.99439 0.98584 0.98415  1.01873 1.00742 0.99164 0.99045 0.99757 1.00429 0.98409 0.97719 0.99622 0.98204 0.99855 1.00280 0.99305 1.00166 0.99146 0.98340 0.99523 1.01663 0.99443 0.97304 0.98143 0.99028 0.99612 1.01299 0.99932 0.99041 0.99334 0.99268 0.99313 0.99195 0.99217 0.99790 1.00197 1.00447 1.00130 0.99725 0.99587 0.99698 0.99610 0.99887 0.99539 0.99600 0.99502 0.99475 1.00087 0.99437 0.99780 0.99903 1.00264 1.00186 1.00489  1.00309 0.99853 0.99958 0.99973 0.99847 0.99608 0.99725 0.99821 0.99954 0.99881 0.99925 0.99893 0.99897 1.00019 0.99856 0.99910 1.00020 1.00128 0.99792 0.99944 1.00000 0.99992 1.00012 1.00035 0.99909 0.99965 0.99998 1.00005 0.99997 0.99970 1.00001 0.99999 1.00001 0.99996 0.99990 0.99992 1.00001 1.00009 1.00009 0.99999 0.99999 0.99991 0.99999 0.99996 0.99995 1.00014 0.99997 0.99994 0.99995 1.00006 1.00027  t  t  βK  βKIV  1.00105 1.00899 1.01509 1.01405 1.01589 1.02537 1.03126 1.02886 1.02677 1.02821 1.02267 1.02417 1.02842 1.03111 1.03248 1.03310 1.02578 1.02340 1.02382 1.01696 1.01332 1.01237 1.01230 1.01476 1.01497 1.01418 1.01316 1.01180 1.01107 1.01902 1.01695 1.01835 1.01614 1.01766 1.01948 1.01904 1.01729 1.01235 1.00732 1.00543 1.00697 1.00884 1.00896 1.00571 1.00482 1.00656 1.00595 1.00334 1.00428 1.00473 1.00691  1.00232 1.00279 1.00245 1.00074 1.00248 1.00288 1.00431 1.00170 1.00238 1.00242 1.00161 1.00188 1.00300 1.00278 1.00256 1.00200 1.00033 1.00021 1.00127 1.00108 0.99973 1.00030 0.99988 0.99979 1.00116 1.00072 1.00020 1.00012 0.99984 1.00022 1.00015 0.99986 1.00009 1.00010 1.00033 1.00019 1.00009 1.00001 0.99989 0.99991 1.00012 1.00017 1.00015 1.00000 0.99978 1.00002 0.99984 0.99991 0.99986 0.99985 1.00035  t  t  βLD  βLB  1.00037 1.00086 1.00105 1.00179 1.00130 1.00514 1.00515 1.00495 1.00521 1.00700 1.01541 1.01505 1.00627 1.01529 1.00760 1.00624 1.00664 1.01344 1.00826 1.00648 1.00374 1.00573 1.00386 1.00324 1.00347 1.00178 1.00243 1.00202 1.00049 1.00296 1.00052 1.00182 1.00224 1.00183 1.00226 1.00106 1.00223 1.00194 1.00166 1.00098 1.00139 1.00130 1.00145 1.00049 1.00091 1.00066 1.00035 1.00080 0.99989 1.00012 1.00098  1.04235 1.03376 1.01703 1.01651 1.02714 1.01424 1.01969 1.00621 1.01297 1.01375 1.01967 1.01338 1.00614 1.00580 1.01054 1.00692 1.00522 1.01685 0.98908 0.98451 1.02743 1.01363 1.00660 1.00972 1.01112 1.01134 1.00810 1.02010 1.00539 1.00785 1.00786 1.00939 1.01446 1.01122 1.00681 1.00811 0.99843 0.99683 1.00070 1.00544 1.00202 0.99781 0.99580 0.99539 1.01136 0.98990 0.99296 0.99866 1.00389 1.00051 1.00709  t  αT  0.99644 0.99591 1.01280 1.00356 1.00308 0.99876 0.99972 1.00085 0.99977 0.99994 0.99867 1.00179 1.00007 0.99796 1.00085 1.00550 1.00265 0.99039 0.97206 0.99340 0.99437 0.99841 1.00925 0.97925 0.96646 1.00414 1.00175 1.00293 1.00578 1.00207 1.01890 1.00193 1.00041 0.99729 0.99470 1.00136 1.00173 0.99925 1.00022 0.99961 0.99761 0.99571 1.00482 0.99739 0.99302 1.00072 1.00013 0.99651 0.99471 0.99040 0.99420  Average  rt  tt  αNt  αGt  αXt  αMt  αIt  αIVt  βKt  βKIVt  βLDt  βLBt  αTt  1956-2006 1956-1973 1974-1979 1980-1990 1991-2001 2002-2006  1.05000 1.10436 1.02026 1.03702 1.00304 1.02185  1.02596 1.05821 1.01260 1.01287 0.99677 1.01890  1.00012 1.00026 1.00009 1.00002 1.00002 1.00010  0.99944 0.99902 0.99903 0.99971 0.99970 1.00028  0.99657 0.99563 0.99378 0.99611 0.99741 1.00241  1.00192 1.00489 0.99753 1.00366 1.00184 0.99282  0.99622 0.99625 0.99138 0.99624 0.99650 1.00124  0.99965 0.99921 0.99962 0.99984 1.00000 1.00004  1.01591 1.02315 1.01559 1.01571 1.00939 1.00504  1.00086 1.00216 1.00034 1.00025 1.00003 0.99996  1.00369 1.00660 1.00522 1.00198 1.00128 1.00043  1.00858 1.01601 1.00516 1.01033 1.00016 1.00062  0.99842 1.00048 0.99112 0.99967 0.99922 0.99519  The growth rate of real income ρt is decomposed into the product of a productivity growth factor τt, several factors for changes in real output prices αCt (domestic final consumption), αNt (non-profit institution final consumption), αGt (net government purchases from the market sector), αXt (exports), αMt (imports), αIt (investments in reproducible capital) and αIVt (inventory changes); and several factors for growth in input quantities βKt (capital services), βKIVt (inventory services), βLDt (land services) and βLBt (labour input). The last column in Table 3, αTt, is the effect of changes in the terms of trade, and is simply the product of the real price change factors of exports and imports αXt and αMt. Based on the average contribution factors, real income ρt grew at a very high average annual rate of 5%. On average, productivity growth τt accounted for 2.596% of 31  the real income growth, labour input growth βLBt for 0.858% and capital services growth βKt for 1.591%, while the contribution factors for changes in real output prices were 0.378% due to investment prices falling faster than consumption prices αIt; 0.343% per year due to export prices falling faster than consumption prices αXt; and 0.192% per year due to import prices falling more rapidly than consumption prices αMt. Thus, the effect of changes in the terms of trade on living standards αTt was very small for Japan on average over the entire period 1955–2006: an overall negative contribution of 0.151% per year. However, during shorter periods of time, it had a greater impact. It accounted for 0.888% of the average real income growth rate of 2.026% during the stagnation period between the oil shocks (1973–1979), and it also accounted for 0.481% of the average real income growth rate of 2.185% during the period of economic recovery (2002–2006). Productivity growth contributed the most to the overall growth in real income, and the growth in capital services made the second largest contribution on average over the years 1955–2006. However, the average contribution after 1973 of the capital services growth factor (1.196% per year) was larger than that of the productivity growth factor (0.837% per year). We can see that the importance of the contribution of capital services growth relative to the contribution of productivity growth increases over time. However, this is not the end of the story. During the period of economic recovery 2002–2006, the average contribution factor for productivity growth of 1.889% became greater than the average contribution factor of capital services growth, which was 0.504%. Thus, we can observe that the recent economic recovery was mostly boosted by productivity growth rather than by the deepening of capital. The annual change information in the previous table can be converted into cumulative changes using equations (2.48). Table 2-4 and Figure 2-4, which follow, give this cumulative growth information.  32  Table 2-4: The Decomposition of Market Sector Real Income Level into Cumulative Productivity, Real Output Price Change and Input Quantity Contribution Factors 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  ρt/ρ1955 1.00000 1.13349 1.24491 1.31208 1.48175 1.74267 2.01192 2.11133 2.26726 2.57097 2.65093 2.98631 3.40364 3.83398 4.37284 4.77412 4.87863 5.27410 5.88872 5.65097 5.45848 5.65659 5.78190 6.11481 6.60372 6.56409 6.65838 6.80973 6.91155 7.23964 7.58065 7.83873 8.18206 8.74062 9.34557 9.82460 10.11592 10.18736 10.05262 10.11815 10.27214 10.46240 10.52201 10.20747 10.08113 10.26470 10.14186 10.18978 10.22826 10.56405 10.85811 11.29320  Τt 1.00000 1.06212 1.11155 1.12782 1.24126 1.39579 1.54005 1.55472 1.64493 1.78893 1.79097 1.90976 2.05456 2.23569 2.41666 2.52795 2.49798 2.60939 2.73920 2.65117 2.64895 2.69456 2.70193 2.78430 2.94613 2.93783 2.91503 2.92883 2.88792 2.98041 3.04453 3.04412 3.09058 3.19099 3.30290 3.38613 3.39665 3.36999 3.30300 3.30738 3.32479 3.37550 3.39494 3.27334 3.25472 3.27901 3.26437 3.29009 3.30958 3.39979 3.50035 3.58378  ANt  AGt  AXt  AMt  AIt  AIVt  BKt  1.00000 1.00065 1.00098 1.00101 1.00107 1.00131 1.00142 1.00142 1.00137 1.00188 1.00173 1.00198 1.00229 1.00239 1.00269 1.00328 1.00370 1.00422 1.00461 1.00483 1.00481 1.00494 1.00500 1.00496 1.00518 1.00515 1.00498 1.00497 1.00492 1.00492 1.00495 1.00498 1.00503 1.00508 1.00513 1.00537 1.00546 1.00543 1.00533 1.00537 1.00543 1.00547 1.00552 1.00559 1.00552 1.00552 1.00561 1.00543 1.00536 1.00543 1.00573 1.00613  1.00000 1.00110 1.00142 1.00015 0.99934 0.99852 0.99742 0.99553 0.99319 0.99209 0.99026 0.98967 0.98958 0.98864 0.98784 0.98669 0.98481 0.98305 0.98243 0.98759 0.98009 0.97919 0.97806 0.97623 0.97669 0.97852 0.97883 0.97785 0.97743 0.97678 0.97805 0.97557 0.97427 0.97326 0.97336 0.97361 0.97298 0.97245 0.97197 0.97119 0.97078 0.97047 0.97065 0.97056 0.97024 0.97028 0.97037 0.97046 0.97023 0.97042 0.97093 0.97174  1.00000 1.00404 1.00069 0.99252 0.99124 0.99103 0.98406 0.97427 0.96918 0.96751 0.95863 0.95383 0.95043 0.94514 0.94259 0.93842 0.93327 0.92592 0.92415 0.93159 0.92301 0.91293 0.89799 0.88455 0.88994 0.89172 0.88987 0.89169 0.88424 0.88314 0.87588 0.85832 0.85368 0.85162 0.85314 0.85239 0.84780 0.84388 0.83654 0.83345 0.83187 0.83519 0.83594 0.83791 0.82854 0.82469 0.82832 0.82884 0.82575 0.82601 0.82983 0.83831  1.00000 0.99244 0.99168 1.01265 1.01756 1.02091 1.02687 1.03690 1.04323 1.04479 1.05440 1.05829 1.06397 1.07000 1.07071 1.07638 1.08827 1.09982 1.09133 1.05236 1.05513 1.06078 1.07671 1.10318 1.07374 1.03565 1.04212 1.04181 1.05367 1.06108 1.07209 1.11469 1.12293 1.12609 1.12104 1.11609 1.12364 1.13082 1.13988 1.14436 1.14609 1.13880 1.13289 1.13569 1.14553 1.14284 1.13866 1.13809 1.13836 1.13197 1.11595 1.09826  1.00000 1.01873 1.02629 1.01772 1.00799 1.00555 1.00987 0.99379 0.97113 0.96746 0.95009 0.94871 0.95137 0.94475 0.94632 0.93824 0.92267 0.91827 0.93354 0.92834 0.90331 0.88654 0.87792 0.87451 0.88588 0.88527 0.87678 0.87094 0.86457 0.85863 0.85172 0.84506 0.84328 0.84494 0.84872 0.84982 0.84749 0.84398 0.84144 0.83816 0.83721 0.83335 0.83002 0.82589 0.82155 0.82227 0.81764 0.81583 0.81504 0.81719 0.81871 0.82271  1.00000 1.00309 1.00161 1.00119 1.00091 0.99938 0.99547 0.99273 0.99095 0.99049 0.98931 0.98857 0.98751 0.98649 0.98668 0.98525 0.98437 0.98456 0.98583 0.98377 0.98322 0.98322 0.98314 0.98326 0.98361 0.98271 0.98236 0.98234 0.98239 0.98235 0.98206 0.98207 0.98206 0.98207 0.98203 0.98192 0.98184 0.98185 0.98194 0.98202 0.98201 0.98200 0.98191 0.98190 0.98187 0.98182 0.98195 0.98192 0.98186 0.98181 0.98187 0.98213  1.00000 1.00105 1.01005 1.02528 1.03968 1.05620 1.08300 1.11686 1.14909 1.17985 1.21313 1.24063 1.27062 1.30672 1.34737 1.39113 1.43717 1.47422 1.50872 1.54466 1.57086 1.59177 1.61147 1.63129 1.65537 1.68014 1.70397 1.72640 1.74676 1.76610 1.79968 1.83018 1.86376 1.89384 1.92728 1.96483 2.00223 2.03686 2.06202 2.07711 2.08840 2.10296 2.12154 2.14055 2.15278 2.16316 2.17735 2.19032 2.19764 2.20705 2.21749 2.23280  BKIVt 1.00000 1.00232 1.00512 1.00758 1.00832 1.01083 1.01373 1.01810 1.01983 1.02225 1.02472 1.02637 1.02831 1.03139 1.03425 1.03689 1.03896 1.03930 1.03952 1.04084 1.04196 1.04168 1.04199 1.04187 1.04165 1.04286 1.04361 1.04382 1.04394 1.04377 1.04400 1.04415 1.04401 1.04410 1.04421 1.04456 1.04476 1.04486 1.04486 1.04475 1.04466 1.04479 1.04497 1.04512 1.04512 1.04489 1.04491 1.04475 1.04465 1.04450 1.04434 1.04470  BLDt 1.00000 1.00037 1.00123 1.00227 1.00406 1.00537 1.01053 1.01574 1.02076 1.02608 1.03326 1.04918 1.06497 1.07165 1.08803 1.09630 1.10313 1.11045 1.12538 1.13467 1.14203 1.14631 1.15288 1.15733 1.16107 1.16510 1.16717 1.17001 1.17237 1.17295 1.17642 1.17702 1.17916 1.18180 1.18396 1.18663 1.18789 1.19053 1.19284 1.19482 1.19599 1.19766 1.19921 1.20094 1.20152 1.20262 1.20341 1.20384 1.20480 1.20466 1.20480 1.20598  BLBt 1.00000 1.04235 1.07754 1.09589 1.11398 1.14422 1.16051 1.18336 1.19071 1.20615 1.22273 1.24678 1.26347 1.27122 1.27859 1.29207 1.30101 1.30780 1.32983 1.31531 1.29494 1.33047 1.34860 1.35751 1.37070 1.38595 1.40167 1.41302 1.44142 1.44920 1.46057 1.47205 1.48587 1.50736 1.52427 1.53466 1.54710 1.54466 1.53977 1.54085 1.54923 1.55236 1.54896 1.54245 1.53535 1.55279 1.53711 1.52629 1.52425 1.53017 1.53095 1.54180  AT t 1.00000 0.99644 0.99236 1.00507 1.00865 1.01175 1.01050 1.01022 1.01108 1.01084 1.01078 1.00943 1.01123 1.01130 1.00924 1.01009 1.01565 1.01834 1.00855 0.98037 0.97390 0.96841 0.96687 0.97582 0.95557 0.92352 0.92735 0.92897 0.93169 0.93708 0.93902 0.95677 0.95862 0.95901 0.95641 0.95134 0.95263 0.95427 0.95356 0.95377 0.95340 0.95112 0.94703 0.95160 0.94912 0.94249 0.94317 0.94329 0.94000 0.93503 0.92605 0.92068  12 Real income Productivity growth  10  Investments Capital services Land services  8  Labour services T erms of Trade  6  4  2  0 1955  1960  1965  1970  1975  1980  1985  1990  1995  2000  2005  Figure 2-4: Decomposition of Market Sector Real Income Level into Factors 33  The level of real income ρt/ρ1955 is decomposed into the product of the level of productivity Tt; the levels of several real output prices ACt, ANt, AGt, AXt, AMt, AIt and AIVt; and the levels of several input quantities BKt, BKIVt, BLDt and BLBt. The last column in Table 2-4, ATt, is the cumulative effect of changes in the terms of trade, and is simply the product of the levels of real export and import prices AXt and AMt. Over the 52 year period, real income (from the gross domestic product point of view) grew over eleven times (ρ2006/ρ1955 = 11.2932), which was spectacular.46 From Table 2-4 above, it can be seen that productivity growth contributed the most to the overall growth in real income (T2006 = 3.58378), and that the growth in capital services made the next largest contribution (BK2006 = 2.23281), followed by the growth in labour input (BLB2006 = 1.5418). The change in real investment prices made a negative contribution (AI2006 = 0.82271), as did the change in real export price (AX2006 = 0.83831), and the change in real import prices made a modest positive contribution (AM2006 = 1.09826). Thus, the 2006 change in the terms of trade made a modest negative contribution (AT = 0.92068). Figure 2-4 plots real income and the main factors contributing to its growth. 2.4.3 The Decomposition of Deflated NDP Growth There is a severe flaw with all of the analysis presented in the previous sections. The problem is that depreciation payments are part of the user cost of capital for each asset, but depreciation does not provide households with any sustainable purchasing power. Hence our real income measure defined by (2.5) above is overstated. To see why Gross Domestic Product overstates income, consider the model of production that is described by the following researchers: “We must look at the production process during a period of time, with a beginning and an end. It starts, at the commencement of the Period, with an Initial Capital Stock; to this there is applied a Flow Input of labour, and from it there emerges a Flow Output called Consumption; then there is a Closing Stock of Capital left over at the end. If Inputs are the things that are put in, the Outputs are the things that are got out, and the production of the Period is considered in isolation, then the Initial Capital Stock is an Input. A Stock Input to the Flow Input of labour; and further (what is less well recognized in the tradition, but is equally clear when we are strict with translation), the Closing Capital Stock is an Output, a Stock Output to 46  However, note that real income grew just over ten times by 1991, and in the 15 years since then, it has  grown by only 11.6%.  34  match the Flow Output of Consumption Goods. Both input and output have stock and flow components; capital appears both as input and as output” John R. Hicks (1961; 23).  “The business firm can be viewed as a receptacle into which factors of production, or inputs, flow and out of which outputs flow...The total of the inputs with which the firm can work within the time period specified includes those inherited from the previous period and those acquired during the current period. The total of the outputs of the business firm in the same period includes the amounts of outputs currently sold and the amounts of inputs which are bequeathed to the firm in its succeeding period of activity.” Edgar O. Edwards and Philip W. Bell (1961; 71–72).  Hicks and Edwards and Bell obviously had the same model of production in mind: in each accounting period, the business unit combines the capital stocks and goods in process that it has inherited from the previous period with “flow” inputs purchased in the current period (such as labour, materials, services and additional durable inputs) to produce current period “flow” outputs as well as end of the period depreciated capital stock components, which are regarded as outputs from the perspective of the current period (but will be regarded as inputs from the perspective of the next period).47 All of the “flow” inputs that are purchased during the period and all of the “flow” outputs that are sold during the period are the inputs and outputs that appear in the usual definition of cash flow. These are the flow inputs and outputs that are very familiar to national income accountants. But this is not the end of the story: the firm inherits an endowment of assets at the beginning of the production period and at the end of the period, the firm will have the net profit or loss that has occurred because of its sales of outputs and its purchases of inputs during the period. In addition, it will have a stock of assets that it can use when it starts production in the following period. Just focusing on the flow transactions that occur within the production period will not give a complete picture of the firm’s productive activities. Hence, to get a complete picture of the firm’s production activities over the course of a period, it is necessary to add the value of the closing stock of assets less the beginning of the period stock of assets to the cash flow that accrued to the firm from its sales and purchases of market goods and services during the accounting period. 47  For more on this model of production and additional references to the literature, see the Appendices in  Diewert (1977) (1980).  The usual user cost of capital can be derived from this framework if depreciation  is independent of use.  35  We illustrate the theory described above by considering a very simple two output, two input model of the market sector. One of the outputs is output in year t, Yt, and the other output is an investment good, It. One of the inputs is the flow of non-capital primary input Xt and the other input is Kt, capital services. Suppose that the average prices during period t of a unit of Yt, Xt and It are PYt, PXt and PIt, respectively. Suppose further that the interest rate prevailing at the beginning of period t is rt. The value of the beginning of period t capital stock is assumed to be PIt, the investment price for period t. The user cost of capital is calculated as ut = (rt + t + t)PIt/(1 + rt). As usual, it represents the price of capital services input. Thus, the period t profit of the market sector is expressed as follows: (2.54) t = PYt Yt + PIt It  PXt Xt  [(rt* + )PIt/(1 + rt*)]Kt. Under the assumption of constant returns to scale, a zero profit condition should be satisfied such as t = 0. Using this condition, we obtain the following value of output equals value of input equation: (2.55) PYt Yt + PIt It = PXt Xt + [(rt* + )PIt/(1 + rt*)]Kt. Equation (2.55) is essentially the closed economy counterpart to the (gross) value of outputs equals (gross) value of primary inputs equation (2.4), Ptyt = Wtxt, that we have been using thus far in this study. We now come to the point of this rather long digression: the (gross) payments to primary inputs that is defined by the right hand side of (2.55) is not income, in the sense of Hicks.48 The owner of a unit of capital cannot spend the entire period t gross rental income (rt* + )PIt/(1+rt*) on consumption during period t because the depreciation portion of the rental, PIt/(1+rt*), is required in order to keep his or her capital intact. Thus the owner of a new unit of capital at the beginning of period t loans the unit to the market sector and gets the gross return (rt* + )PIt at the end of the period, plus the depreciated unit of the initial capital stock, which is worth only (1  )PIt. 48  Thus, PIt of this gross return must be set aside in order to restore the lender  We will use Hicks’ third concept of income here: “Income No. 3 must be defined as the maximum  amount of money which the individual can spend this week, and still be able to expect to spend this week, and still be able to expect to spend the same amount in real terms in each ensuing week.” (1946; 174).  36  J.R. Hicks  of the capital services to his or her original wealth position at the beginning of period t. This means that period t Hicksian market sector income is not the value of payments to primary inputs, PXt Xt + [(rt* + )PIt/(1 + rt*)]Kt; instead it is the value of payments to labour PXt Xt, plus the reward for waiting, [rt* PIt/(1 + rt*)]Kt. Using this definition of market sector (net) Hicksian income, we can rearrange equation (2.55) as follows: (2.56) Hicksian market sector income  PXt Xt + [rt*PIt/(1 + rt*)]Kt = PYt Yt + PIt It − [PIt/(1 + rt*)]Kt = Value of consumption + value of gross investment − value of depreciation. Thus, in this Hicksian net income framework, our new output concept is equal to our old output concept less the value of depreciation. We take the price of depreciation to be the corresponding investment price PIt/(1 + rt*), and the quantity of depreciation is taken to be the depreciation rate times the beginning of the period stock, Kt. Hence the overstatement of income problem that is implicit in the approaches used in the previous sections can readily be remedied: all we need to do is to take the user cost formula for an asset and decompose it into two parts:   One part that represents depreciation and foreseen obsolescence, [PIt/(1 + rt*)] and    The remaining part that is the reward for postponing consumption, [rt*PIt/(1 + rt*)].  Thus, in this section, we split up each user cost times the beginning of the period stock Kt into the depreciation component [tPIt/(1 + rt*)]Kt and the remaining term [rt*PIt /(1 + rt*)] Kt. We regard the second term as a genuine income component, but the first term is treated as an intermediate input cost for the market sector and is an offset to gross investment made by the market sector during the period under consideration. Thus, in this section, we use a net product approach instead of a gross product approach. Using the chained Törnqvist indexes, we construct prices and quantities of net investment PNI and yNI and “reward for waiting” capital service WKW and xKW. Waiting capital services is compared with the gross user cost concept that was used in the previous section, and net investment is also compared with the gross investment that was used in the previous section in Table 2-5 and Figures 2-5 and 2-6.  37  Table 2-5: The Quantity and Price of Gross Investment, Depreciation, Net Investment, Capital Services, Waiting Capital Services (Billion Yen) Price 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  Quantity  PIt  PDEPt  PNIt  W Kt  W KWt  1.00000 1.07799 1.14107 1.10683 1.10406 1.12828 1.21002 1.23850 1.24410 1.26346 1.28731 1.33289 1.38861 1.43391 1.49307 1.56173 1.59771 1.66686 1.93381 2.35584 2.45112 2.55621 2.66758 2.75138 2.94163 3.17559 3.23328 3.24736 3.23912 3.24600 3.22555 3.16648 3.14775 3.17448 3.26989 3.36079 3.40881 3.41944 3.41916 3.39038 3.36617 3.32070 3.31842 3.24115 3.16839 3.15573 3.06595 2.99292 2.95534 2.94031 2.92203 2.92197  1.00000 1.08491 1.14181 1.11037 1.09317 1.09662 1.14280 1.16964 1.16251 1.17169 1.19228 1.21074 1.24330 1.27105 1.32124 1.37852 1.41885 1.45957 1.64631 2.00876 2.09258 2.14654 2.23783 2.27503 2.37953 2.53657 2.58179 2.60505 2.60486 2.59707 2.58060 2.53286 2.51094 2.52204 2.57824 2.63630 2.67085 2.66346 2.66361 2.63181 2.60287 2.55734 2.55981 2.53578 2.47610 2.45212 2.38618 2.32119 2.27919 2.26154 2.24905 2.25761  1.00000 1.06922 1.13803 1.10165 1.10949 1.14554 1.24282 1.27208 1.28334 1.30732 1.33273 1.39186 1.45860 1.51184 1.57474 1.64856 1.68260 1.76613 2.07349 2.52401 2.62432 2.76375 2.88593 3.00240 3.25045 3.53519 3.60019 3.60435 3.58846 3.61110 3.58866 3.52345 3.51065 3.55162 3.67878 3.79504 3.85260 3.88374 3.88287 3.86525 3.85509 3.82112 3.81126 3.65180 3.57614 3.59639 3.48448 3.42590 3.42691 3.43047 3.40409 3.37541  1.00000 1.08191 1.15677 1.08155 1.15227 1.27205 1.43460 1.41665 1.44748 1.50767 1.48527 1.56819 1.64780 1.72926 1.77701 1.80976 1.69396 1.68648 1.88426 2.16568 2.12267 2.28986 2.36134 2.50163 2.72584 2.82205 2.79355 2.76155 2.70685 2.80907 2.85633 2.79678 2.75369 2.79600 2.87215 2.87945 2.83715 2.80031 2.70041 2.66357 2.65553 2.67333 2.65758 2.50101 2.45188 2.47580 2.37012 2.36613 2.36658 2.46403 2.48954 2.51931  1.00000 1.07906 1.17091 1.05376 1.21048 1.44628 1.72758 1.66188 1.73453 1.85201 1.77918 1.93720 2.07154 2.21700 2.25834 2.25524 1.93093 1.85512 2.05008 2.18505 1.95938 2.27639 2.31200 2.58908 2.96687 2.96381 2.82885 2.71767 2.58656 2.84527 2.98583 2.91406 2.84163 2.93024 3.03092 2.95053 2.77940 2.69507 2.42947 2.38827 2.41817 2.54540 2.49934 2.12572 2.10171 2.20944 2.04429 2.15363 2.23309 2.53391 2.62787 2.69424  yIt 1618.6 1988.9 2412.8 2535.9 3008.9 4052.4 5030.0 5640.8 6310.8 7337.8 7589.2 8685.0 10288.5 12292.3 14428.7 16754.7 17458.3 19078.7 21349.8 20170.2 19936.6 20780.6 21330.6 23135.9 24388.7 24266.1 24825.2 24953.0 24683.6 25793.8 28001.8 29879.8 32170.8 36216.4 39761.6 43024.0 44379.5 43490.6 42076.7 41384.3 41473.8 43423.7 43709.9 40835.5 40556.0 40715.9 40621.5 38912.1 38698.7 39369.2 40884.9 41477.0  yDEPt 1014.0 1010.7 1048.4 1122.9 1201.9 1298.5 1475.0 1725.0 1985.3 2255.2 2574.3 2855.8 3200.8 3669.4 4263.0 4976.3 5821.7 6557.8 7317.4 8134.3 8722.9 9209.9 9690.8 10218.0 10897.7 11596.0 12297.3 13006.2 13686.6 14373.0 15621.3 16865.7 18221.6 19513.5 21067.9 22885.1 24716.1 26456.4 27694.6 28380.9 28871.3 29593.2 30619.2 31679.7 32380.2 32980.4 33842.7 34627.7 35050.2 35619.2 36236.7 37180.6  yNIt 604.6 979.7 1367.4 1416.0 1810.0 2748.4 3540.9 3905.8 4319.5 5070.4 5027.5 5832.9 7066.5 8573.8 10103.6 11711.1 11668.4 12586.9 14101.7 12352.5 11665.3 12067.0 12202.2 13459.0 14093.8 13477.4 13476.4 13081.3 12345.5 12849.0 13935.3 14728.6 15812.4 18513.9 20576.9 22203.4 22132.6 20147.5 18053.4 16975.6 16720.6 17931.2 17492.4 14245.4 13511.9 13240.1 12566.8 10532.6 10062.1 10262.0 11153.9 11037.2  xKt 2070.8 2082.1 2153.2 2283.2 2417.9 2584.6 2873.6 3267.5 3679.8 4109.1 4615.6 5071.3 5612.7 6327.7 7218.9 8287.8 9543.7 10678.6 11840.4 13105.6 14071.4 14889.3 15693.4 16534.2 17572.5 18658.7 19745.2 20819.3 21837.9 22836.7 24610.0 26285.3 28206.7 29998.1 32068.4 34502.6 37050.8 39485.2 41286.1 42385.3 43217.6 44299.0 45691.6 47134.2 48066.2 48854.5 49938.8 50928.0 51479.1 52176.3 52938.7 54059.6  xKWt 1056.8 1071.4 1104.9 1160.3 1216.2 1288.7 1410.5 1571.3 1740.2 1918.4 2128.0 2320.5 2543.6 2831.9 3186.3 3609.0 4094.7 4548.3 5006.6 5511.4 5928.3 6292.8 6648.4 6997.2 7404.6 7841.9 8275.5 8688.3 9070.1 9427.0 10041.4 10568.1 11232.7 11828.6 12467.2 13223.6 14069.8 14880.8 15526.7 15996.1 16383.1 16793.5 17224.4 17664.8 17926.4 18141.2 18395.7 18631.3 18782.7 18947.0 19138.9 19394.4  4.5 Investments  4.0  Depreciations Net investments  3.5  Capital services  3.0 Waiting capital services  2.5  2.0  1.5  1.0 1955  1960  1965  1970  1975  1980  1985  1990  1995  2000  2005  Figure 2-5: Price of Gross Investment, Depreciation, Net Investment, Capital Services and Waiting Capital Services  38  60000  (B illion Y en) Investments  50000  Depreciations Net investments  40000  Capital services Waiting capital services  30000  20000  10000  0 1955  1960  1965  1970  1975  1980  1985  1990  1995  2000  2005  Figure 2-6: Quantity of Gross Investment, Depreciation, Net Investment, Capital Services and Waiting Capital Services (Billion Yen) Note that the price of net investment increases much more rapidly than that of the gross investment. The explanation for this fact is as follows. Machinery and equipment investment has increased much more than structures investment in Japan, similar to the situation in most countries. But due to the computer chip revolution, the price of machinery and equipment investment declined much more rapidly than the price of structures (which tended to increase). But structures have a low depreciation rate, and machinery and equipment items have high depreciation rates. Therefore, when we subtract depreciation from gross investment, the weight of machinery and equipment is reduced in the net investment aggregate relative to the weight of structures, thus leading to a much higher rate of overall price increase in the net investment aggregate. Note that the price of waiting capital services increases much more rapidly than the other investment prices. This is mainly due to the fact that land services are included in the capital services, but that there is no investment in land. Hence, the situation is explained by the fact that land prices in Japan have been increasing much more rapidly than the prices of investment goods over most of the sample period. Note that gross investment grew 25.625 times over the entire period 1955–2006, whereas net investment grew only 17.667 times. Note that (traditional) capital services input grew 26.15 times over the entire period 1955–2006, whereas waiting capital services grew only 18.343 times. All of the analysis presented in Section 2.4.2 above applies to the new situation with 39  obvious modifications. The counterpart to Table 2-3 in the previous section, which uses the new framework, is Table 2-6 below. Table 2-6: The Decomposition of Market Sector Net Real Income Growth into Translog Productivity, Real Output Price Change, and Input Quantity Contribution Factors 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Average 1956-2006 1956-1973 1974-1979 1980-1990 1991-2001 2002-2006  rt 1.14124 1.10415 1.05559 1.14475 1.19189 1.15857 1.04195 1.07459 1.13610 1.02527 1.13169 1.14275 1.12786 1.13784 1.08637 1.00920 1.07891 1.11472 0.94066 0.96030 1.04438 1.02200 1.06224 1.07981 0.98564 1.01146 1.01944 1.01201 1.05171 1.04489 1.03065 1.03911 1.06749 1.06643 1.04517 1.02175 0.99798 0.97592 1.00578 1.01641 1.02106 1.00156 0.95572 0.98354 1.01889 0.98271 1.00280 1.00394 1.03544 1.02869 1.03834  tt 1.07188 1.05360 1.01679 1.11487 1.14038 1.11541 1.01063 1.06498 1.09806 1.00128 1.07439 1.08478 1.09843 1.09042 1.05145 0.98620 1.05062 1.05726 0.96333 0.99854 1.02005 1.00301 1.03517 1.06743 0.99692 0.99092 1.00553 0.98365 1.03755 1.02509 0.99980 1.01800 1.03844 1.04162 1.03003 1.00368 0.99056 0.97585 1.00163 1.00647 1.01867 1.00702 0.95599 0.99298 1.00931 0.99441 1.00991 1.00747 1.03433 1.03721 1.02988  α Nt  αGt  αXt  αMt  αIt  αDEPt  αIVt  βKW t  βKIVt  βLDt  βLBt  α Tt  1.00075 1.00038 1.00004 1.00007 1.00027 1.00012 1.00000 0.99994 1.00057 0.99984 1.00027 1.00035 1.00010 1.00034 1.00065 1.00047 1.00059 1.00044 1.00024 0.99999 1.00014 1.00007 0.99995 1.00026 0.99997 0.99980 0.99999 0.99994 1.00000 1.00003 1.00005 1.00005 1.00006 1.00006 1.00028 1.00010 0.99996 0.99988 1.00005 1.00007 1.00005 1.00006 1.00009 0.99991 1.00000 1.00011 0.99978 0.99991 1.00009 1.00037 1.00050  1.00127 1.00037 0.99856 0.99908 0.99908 0.99877 0.99790 0.99737 0.99877 0.99794 0.99934 0.99989 0.99894 0.99911 0.99870 0.99786 0.99798 0.99929 1.00603 0.99117 0.99894 0.99866 0.99784 1.00054 1.00217 1.00036 0.99883 0.99951 0.99921 1.00151 0.99704 0.99843 0.99878 1.00013 1.00030 0.99923 0.99934 0.99940 0.99901 0.99948 0.99961 1.00023 0.99989 0.99958 1.00006 1.00011 1.00012 0.99971 1.00024 1.00066 1.00103  1.00466 0.99618 0.99066 0.99854 0.99976 0.99219 0.98893 0.99418 0.99808 0.98975 0.99441 0.99603 0.99381 0.99700 0.99508 0.99382 0.99107 0.99784 1.00926 0.98930 0.98731 0.98108 0.98273 1.00703 1.00233 0.99758 1.00239 0.99021 0.99855 0.99045 0.97655 0.99363 0.99717 1.00209 0.99896 0.99361 0.99443 0.98941 0.99546 0.99768 1.00486 1.00110 1.00289 0.98602 0.99417 1.00552 1.00079 0.99529 1.00040 1.00578 1.01266  0.99128 0.99912 1.02422 1.00551 1.00369 1.00648 1.01088 1.00681 1.00167 1.01028 1.00412 1.00598 1.00630 1.00074 1.00589 1.01245 1.01204 0.99124 0.95904 1.00306 1.00622 1.01740 1.02846 0.96931 0.95905 1.00727 0.99965 1.01336 1.00823 1.01207 1.04671 1.00871 1.00332 0.99474 0.99479 1.00806 1.00770 1.00978 1.00484 1.00186 0.99228 0.99369 1.00304 1.01087 0.99706 0.99540 0.99937 1.00030 0.99294 0.98235 0.98041  1.02165 1.00852 0.99045 0.98917 0.99728 1.00477 0.98231 0.97458 0.99579 0.97997 0.99838 1.00312 0.99228 1.00185 0.99050 0.98134 0.99459 1.01892 0.99362 0.96871 0.97845 0.98875 0.99550 1.01498 0.99918 0.98880 0.99219 0.99143 0.99197 0.99064 0.99083 0.99753 1.00232 1.00525 1.00153 0.99673 0.99503 0.99632 0.99521 0.99862 0.99441 0.99514 0.99388 0.99342 1.00109 0.99293 0.99722 0.99877 1.00332 1.00233 1.00605  0.98704 0.99689 1.00386 1.00637 1.00319 1.00173 1.00507 1.00877 1.00209 1.00623 1.00279 1.00073 1.00322 0.99967 1.00279 1.00451 1.00352 0.99828 1.00179 1.01126 1.01086 1.00443 1.00393 0.99795 1.00229 1.00444 1.00241 1.00330 1.00448 1.00421 1.00417 1.00161 0.99971 0.99917 1.00033 1.00163 1.00348 1.00179 1.00331 1.00162 1.00390 1.00223 1.00020 1.00435 1.00075 1.00414 1.00276 1.00232 0.99837 0.99823 0.99489  1.00357 0.99831 0.99951 0.99969 0.99829 0.99565 0.99695 0.99800 0.99948 0.99867 0.99916 0.99881 0.99885 1.00021 0.99839 0.99899 1.00023 1.00146 0.99761 0.99935 1.00000 0.99991 1.00014 1.00041 0.99895 0.99959 0.99997 1.00006 0.99996 0.99965 1.00002 0.99998 1.00001 0.99995 0.99988 0.99990 1.00001 1.00011 1.00010 0.99999 0.99998 0.99989 0.99999 0.99996 0.99994 1.00017 0.99996 0.99992 0.99993 1.00008 1.00033  1.00187 1.00489 1.00726 1.00669 1.00844 1.01385 1.01674 1.01555 1.01494 1.01572 1.01295 1.01375 1.01607 1.01744 1.01792 1.01704 1.01301 1.01138 1.01133 1.00819 1.00660 1.00623 1.00592 1.00706 1.00743 1.00677 1.00587 1.00501 1.00454 1.00791 1.00649 1.00771 1.00656 1.00674 1.00743 1.00753 1.00671 1.00500 1.00343 1.00279 1.00301 1.00316 1.00301 1.00170 1.00144 1.00169 1.00158 1.00107 1.00125 1.00155 1.00210  1.00268 1.00321 1.00280 1.00084 1.00278 1.00320 1.00479 1.00189 1.00265 1.00270 1.00180 1.00210 1.00333 1.00309 1.00286 1.00225 1.00037 1.00024 1.00145 1.00125 0.99969 1.00035 0.99986 0.99976 1.00135 1.00083 1.00023 1.00014 0.99981 1.00026 1.00017 0.99984 1.00011 1.00012 1.00039 1.00023 1.00011 1.00001 0.99987 0.99990 1.00015 1.00021 1.00018 1.00000 0.99973 1.00003 0.99980 0.99988 0.99982 0.99981 1.00043  1.00043 1.00098 1.00120 1.00203 1.00145 1.00570 1.00573 1.00553 1.00580 1.00782 1.01723 1.01677 1.00697 1.01701 1.00848 1.00702 1.00753 1.01529 1.00948 1.00754 1.00435 1.00664 1.00446 1.00373 1.00402 1.00207 1.00284 1.00237 1.00058 1.00346 1.00061 1.00214 1.00265 1.00216 1.00268 1.00127 1.00268 1.00236 1.00203 1.00120 1.00170 1.00159 1.00178 1.00060 1.00114 1.00082 1.00044 1.00101 0.99986 1.00014 1.00123  1.04905 1.03884 1.01950 1.01876 1.03045 1.01582 1.02194 1.00694 1.01446 1.01537 1.02200 1.01491 1.00682 1.00645 1.01177 1.00780 1.00592 1.01916 0.98747 0.98201 1.03195 1.01580 1.00763 1.01122 1.01289 1.01322 1.00946 1.02359 1.00632 1.00918 1.00922 1.01107 1.01709 1.01328 1.00809 1.00970 0.99810 0.99615 1.00086 1.00666 1.00247 0.99732 0.99481 0.99426 1.01421 0.98737 0.99117 0.99832 1.00489 1.00064 1.00890  0.99590 0.99530 1.01466 1.00404 1.00345 0.99863 0.99969 1.00095 0.99974 0.99993 0.99851 1.00199 1.00007 0.99773 1.00094 1.00619 1.00300 0.98909 0.96792 0.99233 0.99346 0.99816 1.01070 0.97612 0.96128 1.00482 1.00204 1.00344 1.00677 1.00240 1.02216 1.00228 1.00048 0.99682 0.99376 1.00161 1.00208 0.99908 1.00027 0.99953 0.99709 0.99478 1.00594 0.99674 0.99125 1.00090 1.00017 0.99559 0.99334 0.98803 0.99282  rt 1.04857 1.10575 1.01823 1.03400 0.99830 1.02184  tt 1.02964 1.06563 1.01459 1.01523 0.99605 1.02376  α Nt  αGt  αXt  αMt  αIt  αDEPt  αIVt  βKW t  βKIVt  βLDt  βLBt  α Tt  1.00014 1.00029 1.00011 1.00002 1.00003 1.00013  0.99936 0.99890 0.99886 0.99966 0.99963 1.00035  0.99605 0.99511 0.99279 0.99544 0.99683 1.00298  1.00216 1.00548 0.99725 1.00435 1.00223 0.99108  0.99564 0.99586 0.99000 0.99561 0.99571 1.00154  1.00229 1.00204 1.00504 1.00238 1.00249 0.99931  0.99961 0.99912 0.99957 0.99982 1.00000 1.00005  1.00765 1.01253 1.00755 1.00659 1.00359 1.00151  1.00097 1.00242 1.00039 1.00030 1.00004 0.99995  1.00421 1.00739 1.00603 1.00232 1.00156 1.00054  1.00983 1.01811 1.00601 1.01213 1.00017 1.00078  0.99812 1.00054 0.98978 0.99966 0.99902 0.99399  The growth rate of net real income ρt is decomposed into the product of a productivity growth factor τt; several factors for changes in real output prices αCt, αNt, αGt, αXt, αMt, αIt, αDEPt and αIVt; and several factors for growth in input quantities βKWt, βKIVt, βLDt and βLBt. The new results are quite interesting. While the average growth rate of real income ρt was 5% per year in the gross product approach, the average growth rate of net real income ρt now decreased by 0.143 percentage points per year to 4.857%. More importantly, there were some big shifts in the explanatory factors. Productivity growth 40  τt now accounts for 2.964% of the average net real income growth, compared to 2.596% of the average real income growth, an increase of 0.368 percentage points per year. Capital services growth βKt accounted for 1.591% of the average real income growth, while waiting capital services growth βKWt accounted for 0.765% of the average net real income, a decrease of 0.826 percentage points per year. The average contribution of labour input growth βLBt has marginally increased from 0.858% per year to 0.983% per year. The average contributions of changes in real export prices αXt and real import prices αMt remain quite similar estimates in the previous product approach. Thus, as we stated in the previous analysis, the effect of changes in the terms of trade on living standards αTt was negligible for Japan on average over the entire period 1955–2006: an overall negative contribution of −0.188%. However, during shorter periods of time, it had greater impacts. It accounted for −1.022% of the average growth rate of net real income of 1.823% during the stagnation period between oil shocks (1973–1979), and it also accounted for −0.601% of the average growth rate of net real income 2.184% during the period of economic recovery (2002–2006). The negative contribution of the change in real investment prices αIt equal to −0.436% was offset by the positive contribution of the change in real depreciation prices αDEPt equal to 0.231%. Finally, we note that the productivity recovery in the period 2002–2006 is quite striking. Using the previous gross product approach, the average contribution of productivity growth during this period was 1.89% per year, and using the current net output model, its average contribution increases to a very respectable 2.376% per year. The annual change information in the previous table can be converted into cumulative changes using equations (2.48). Table 2-7 and Figure 2-7 below give this cumulative growth information.  41  Table 2-7: The Decomposition of Market Sector Net Real Income Cumulative Growth into Productivity, Real Output Price Change, and Input Quantity Contribution Factors using the Translog Net Product Approach 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  ρt/ρ1955 1.00000 1.14124 1.26010 1.33015 1.52268 1.81487 2.10265 2.19086 2.35427 2.67468 2.74226 3.10339 3.54640 3.99986 4.55120 4.94427 4.98977 5.38352 6.00110 5.64498 5.42086 5.66141 5.78595 6.14605 6.63654 6.54124 6.61617 6.74479 6.82582 7.17875 7.50102 7.73095 8.03333 8.57546 9.14513 9.55820 9.76607 9.74634 9.51160 9.56654 9.72349 9.92824 9.94376 9.50347 9.34707 9.52364 9.35896 9.38517 9.42213 9.75600 10.03593 10.42070  Τt 1.00000 1.07188 1.12932 1.14828 1.28019 1.45990 1.62838 1.64569 1.75262 1.92449 1.92695 2.07031 2.24582 2.46688 2.68994 2.82834 2.78932 2.93053 3.09832 2.98470 2.98035 3.04010 3.04926 3.15650 3.36935 3.35898 3.32848 3.34690 3.29217 3.41580 3.50150 3.50081 3.56383 3.70083 3.85485 3.97062 3.98523 3.94760 3.85227 3.85854 3.88351 3.95602 3.98379 3.80847 3.78172 3.81693 3.79559 3.83320 3.86183 3.99442 4.14307 4.26684  ANt  A Gt  AXt  AMt  AIt  ADEPt  AIVt  BKWt  BKIVt  1.00000 1.00075 1.00112 1.00117 1.00123 1.00150 1.00163 1.00162 1.00156 1.00214 1.00197 1.00224 1.00259 1.00270 1.00304 1.00369 1.00416 1.00476 1.00520 1.00545 1.00543 1.00558 1.00565 1.00560 1.00586 1.00583 1.00563 1.00562 1.00555 1.00555 1.00558 1.00563 1.00568 1.00574 1.00580 1.00609 1.00619 1.00616 1.00603 1.00609 1.00616 1.00620 1.00626 1.00636 1.00626 1.00626 1.00638 1.00616 1.00607 1.00616 1.00653 1.00702  1.00000 1.00127 1.00164 1.00019 0.99927 0.99835 0.99713 0.99503 0.99241 0.99119 0.98915 0.98850 0.98839 0.98734 0.98646 0.98518 0.98307 0.98108 0.98038 0.98629 0.97759 0.97655 0.97524 0.97313 0.97366 0.97577 0.97613 0.97498 0.97450 0.97374 0.97521 0.97232 0.97079 0.96960 0.96972 0.97002 0.96927 0.96863 0.96805 0.96709 0.96659 0.96621 0.96644 0.96633 0.96592 0.96597 0.96608 0.96620 0.96591 0.96615 0.96678 0.96778  1.00000 1.00466 1.00082 0.99147 0.99003 0.98979 0.98207 0.97119 0.96554 0.96369 0.95381 0.94848 0.94471 0.93887 0.93605 0.93145 0.92569 0.91743 0.91544 0.92392 0.91403 0.90244 0.88536 0.87007 0.87619 0.87823 0.87610 0.87819 0.86959 0.86833 0.86004 0.83987 0.83451 0.83215 0.83389 0.83302 0.82770 0.82309 0.81437 0.81067 0.80878 0.81271 0.81360 0.81596 0.80455 0.79986 0.80428 0.80492 0.80113 0.80145 0.80608 0.81628  1.00000 0.99128 0.99041 1.01440 1.01998 1.02374 1.03038 1.04159 1.04868 1.05043 1.06123 1.06560 1.07197 1.07873 1.07952 1.08588 1.09940 1.11263 1.10288 1.05771 1.06095 1.06755 1.08613 1.11704 1.08275 1.03841 1.04595 1.04559 1.05956 1.06828 1.08118 1.13168 1.14153 1.14532 1.13929 1.13336 1.14249 1.15128 1.16253 1.16816 1.17032 1.16128 1.15396 1.15747 1.17005 1.16661 1.16124 1.16051 1.16086 1.15266 1.13232 1.11015  1.00000 1.02165 1.03035 1.02051 1.00945 1.00671 1.01151 0.99362 0.96836 0.96428 0.94496 0.94343 0.94637 0.93906 0.94080 0.93186 0.91447 0.90952 0.92673 0.92081 0.89200 0.87277 0.86296 0.85907 0.87194 0.87123 0.86147 0.85475 0.84742 0.84062 0.83275 0.82511 0.82308 0.82498 0.82931 0.83058 0.82787 0.82376 0.82073 0.81679 0.81567 0.81110 0.80716 0.80222 0.79694 0.79781 0.79217 0.78997 0.78900 0.79162 0.79346 0.79826  1.00000 0.98704 0.98397 0.98777 0.99406 0.99723 0.99895 1.00401 1.01281 1.01493 1.02125 1.02410 1.02485 1.02815 1.02781 1.03068 1.03532 1.03896 1.03717 1.03902 1.05072 1.06213 1.06683 1.07103 1.06883 1.07128 1.07604 1.07863 1.08219 1.08705 1.09163 1.09618 1.09794 1.09763 1.09671 1.09707 1.09886 1.10268 1.10465 1.10831 1.11011 1.11444 1.11692 1.11714 1.12200 1.12284 1.12749 1.13060 1.13322 1.13137 1.12937 1.12360  1.00000 1.00357 1.00187 1.00138 1.00108 0.99936 0.99502 0.99198 0.99000 0.98949 0.98817 0.98734 0.98617 0.98503 0.98524 0.98366 0.98266 0.98289 0.98432 0.98197 0.98133 0.98133 0.98124 0.98137 0.98177 0.98074 0.98034 0.98031 0.98037 0.98033 0.97999 0.98001 0.97999 0.98000 0.97995 0.97983 0.97973 0.97974 0.97985 0.97995 0.97994 0.97993 0.97982 0.97981 0.97976 0.97970 0.97987 0.97983 0.97976 0.97969 0.97977 0.98009  1.00000 1.00187 1.00677 1.01408 1.02086 1.02948 1.04374 1.06121 1.07771 1.09381 1.11100 1.12539 1.14087 1.15920 1.17941 1.20055 1.22101 1.23689 1.25096 1.26513 1.27550 1.28391 1.29191 1.29956 1.30873 1.31845 1.32738 1.33518 1.34186 1.34796 1.35861 1.36743 1.37797 1.38701 1.39635 1.40672 1.41732 1.42682 1.43396 1.43888 1.44290 1.44724 1.45181 1.45618 1.45866 1.46075 1.46322 1.46553 1.46710 1.46893 1.47121 1.47429  1.00000 1.00268 1.00589 1.00871 1.00955 1.01236 1.01560 1.02046 1.02239 1.02511 1.02788 1.02972 1.03188 1.03532 1.03851 1.04148 1.04382 1.04420 1.04445 1.04597 1.04728 1.04695 1.04732 1.04717 1.04692 1.04833 1.04921 1.04945 1.04959 1.04939 1.04967 1.04985 1.04968 1.04979 1.04992 1.05033 1.05057 1.05069 1.05070 1.05056 1.05045 1.05061 1.05083 1.05102 1.05102 1.05073 1.05075 1.05055 1.05042 1.05023 1.05003 1.05049  1.00000 1.00043 1.00141 1.00261 1.00464 1.00610 1.01184 1.01764 1.02326 1.02920 1.03725 1.05512 1.07282 1.08030 1.09867 1.10799 1.11577 1.12417 1.14135 1.15217 1.16086 1.16591 1.17365 1.17889 1.18328 1.18803 1.19049 1.19387 1.19670 1.19739 1.20153 1.20226 1.20483 1.20802 1.21063 1.21387 1.21541 1.21867 1.22155 1.22403 1.22549 1.22758 1.22953 1.23172 1.23247 1.23387 1.23489 1.23543 1.23668 1.23650 1.23668 1.23820  BLDt  1985  1990  1995  2000  2005  BLBt 1.00000 1.04905 1.08979 1.11104 1.13188 1.16635 1.18481 1.21080 1.21920 1.23683 1.25584 1.28347 1.30261 1.31149 1.31995 1.33549 1.34590 1.35387 1.37980 1.36251 1.33800 1.38075 1.40256 1.41326 1.42912 1.44754 1.46667 1.48054 1.51547 1.52504 1.53904 1.55324 1.57043 1.59728 1.61849 1.63159 1.64741 1.64429 1.63795 1.63936 1.65028 1.65435 1.64992 1.64137 1.63194 1.65513 1.63423 1.61979 1.61707 1.62497 1.62600 1.64047  A Tt 1.00000 0.99590 0.99122 1.00575 1.00981 1.01329 1.01190 1.01159 1.01254 1.01228 1.01221 1.01070 1.01271 1.01278 1.01049 1.01144 1.01770 1.02076 1.00962 0.97724 0.96974 0.96340 0.96162 0.97190 0.94870 0.91196 0.91636 0.91823 0.92138 0.92762 0.92985 0.95046 0.95263 0.95308 0.95005 0.94411 0.94564 0.94760 0.94673 0.94699 0.94654 0.94379 0.93887 0.94445 0.94136 0.93313 0.93396 0.93412 0.93000 0.92380 0.91274 0.90619  12 Net real income Productivity growth  10  Investments Depreciations  8  Waiting capital services Labour services Terms of Trade  6  4  2  0 1955  1960  1965  1970  1975  1980  Figure 2-7: Decomposition of Market Sector Net Real Income Level into Factors 42  The current level of net real income compared to its level in 1955, ρt/ρ1955, is decomposed into the product of the level of productivity factor Tt; the levels of several real output price factors ACt, ANt, AGt, AXt, AMt, AIt, ADEPt and AIVt; and the levels of several input quantity factors BKWt, BKIVt, BLDt and BLBt. Over the 52 year period, net real income grew about 10.4 times (ρ2006/ρ1955 = 10.42067). From Table 2.7 above, it can be seen that productivity growth contributed the most to the overall growth in real income (T2006 = 4.26684), and that the growth in labour input made the next largest contribution (BLB2006 = 1.64047), followed by the growth in waiting capital services (BKW2006 = 1.05049) and the growth in land input (BLD2006 = 1.2382). There were smaller effects due to the changes in real output prices, such as the contributions of the changes in depreciation prices (ADEP2006 = 1.12359), real export prices (AX2006 = 0.81628) and real import prices (AM2006 = 1.11014). The combined effects of cumulative changes in the output prices relative to the price of household consumption were negligible (0.94832) over the entire sample period. Figure 2-7 plots net real income and the main factors contributing to its growth. 2.5 Conclusion On a theoretical level, the results of Diewert and Morrison (1986) were modified to give an exact decomposition of the growth in real incomes generated by the market sector. Empirically, we analyzed the contribution to Japanese living standards according to the income generated by the market sector. First, we constructed a database of inputs and outputs of the market sector. Second, we took a measured conventional TFP growth approach by applying Fisher quantity indexes. We calculated traditional TFP in the case in which land and inventories were included as primary inputs, and two other cases in which at least one of these two inputs was excluded. Our results showed the importance of the inclusion of land and inventories when computing TFP growth rates.49 Third, we applied the exact decomposition results to decompose the growth of Japanese real income into the contributions of changes in real output prices and changes in primary input quantities. We observed that productivity growth and the growth of capital services are the main contributors to real income growth when a traditional user cost approach to the pricing of primary inputs is used. We also observed that changes in the terms of trade had very small effects on real income on average. Fourth, we moved to our theoretically 49  Nomura (2004) also made this observation.  We noted also that balancing real rates of return were  greatly exaggerated when inventory and land inputs were omitted.  43  preferred measure of real net income. We applied our theoretical results to decompose the growth of Japanese real net income into the contributions of changes in real output prices and changes in input quantities. We observed that in this net approach, productivity growth was still the largest contributor (and was an even more important factor than before). However, the contribution of capital services was greatly reduced in this net product approach, becoming smaller than the contribution of labour input. A few problems with our approach should be mentioned:   Our labour aggregate has not been sufficiently disaggregated to capture changes in the average quality of labour input over time. Characteristics such as education, sex, age and experience should be taken into account when constructing a measure of aggregate labour input.    We included non-profit institutes serving households (NPISHs) as part of our market sector. However, goods and services produced by NPISHs are traded free or at prices that are not economically significant. Since our theoretical approach relies on competitive profit maximizing behaviour, NPISHs should be netted out of the market sector.    We have not dealt with intangible assets and the problems associated with accounting for R&D investments.    We were not able to provide sectoral contributions analysis. Since a primary focus of our chapter was to look at the effects of changes in the prices of exports and imports on living standards, we could not extend our analysis to industrial sectors because reliable data on exports by industry and imports used by industry are not available.  However, it would be straightforward to extend our analysis to datasets in which a more detailed breakdown of exports and imports by commodity classification is available. This would enable researchers to give more precise estimates of the effects on the income produced by the market sector of an oil shock or any other unusual movement in the prices of internationally traded goods.  44  Chapter 3.  An Economic Approach to the Measurement of Productivity Growth Using Differences Instead of Ratios50  3.1 Introduction The recent boom in the prices of natural resources and the low prices of manufactured goods produced by some developing countries has stimulated interest in the effects of changes in the prices of exports and imports on living standards for a country. An improvement in a country’s terms of trade has much the same effect as an improvement in a country’s productivity growth. Diewert (1983), Diewert and Morrison (1986), Morrison and Diewert (1990), Kohli (1990) (1991) (2003) (2004a) (2004b) (2006) (2007), Diewert, Mizobuchi and Nomura (2005) and Diewert and Lawrence (2006) have all developed production theory methodologies which enable one to obtain exact index number estimates of the contributions of productivity growth and changes in a country’s terms of trade. The present chapter is yet another contribution to this exact index number literature. Many observers use labour productivity (real output divided by labour input) as an approximate welfare measure. However, improvements in a country’s terms of trade (an increase in the price of exports relative to the price of imports) do not show up in labour productivity measures, because the effects of changes in output and intermediate input prices are removed from the measure of output growth. The primary contribution of the present chapter is a proposed improved measure of labour productivity that will allow us to assess the relative contributions to welfare of an improvement in Total Factor Productivity and of changes in real international prices. Our proposed measure is equal to the nominal income generated by the market sector of the economy divided by the product of the price of consumption times the quantity of labour input.51 This proposed measure can be modeled using production theory and exact index number techniques. This approach will be implemented for the Japanese business sector in section 3.5 of this chapter. The main determinants of growth for this measure are: 50  It includes joint work with W. Erwin Diewert.  51  The analysis presented in sections 3.2–3.4 below is somewhat more general.  Instead of deflating  nominal income by a single price, we deflate by a fixed weight price index of outputs and instead of deflating nominal income by labour input, we deflate by a fixed weight quantity index of inputs. in our empirical work in section 3.5, we will specialize these indexes as indicated.  45  However,    Technical progress or improvements in the Total Factor Productivity of the market sector of the economy;    Changes in domestic output prices or the prices of internationally traded goods and services relative to the price of consumption; and    The effects of capital deepening; i.e., growth in market sector capital input relative to the growth of market sector labour input.  Section 3.2 introduces the market sector nominal output function. With a constant returns to scale technology, the value of output is distributed to the primary inputs that produced the market sector outputs. Thus the nominal output function can also be interpreted as a nominal income function. In section 3.3, this income function is used to provide theoretical definitions of the effects of real output price and relative input quantity changes on deflated market sector real income. A formal definition of productivity change is also provided. Section 3.4 introduces the normalized quadratic income function. Using this functional form to represent the technology of the market sector in each period enables us to obtain empirically observable exact measures of the effects of real output price and relative input quantity changes on deflated market sector real income. Appendix C shows that this functional form is a flexible functional form. It turns out that our empirically observable measures of price, quantity and productivity change are equal to measures of price and quantity change that were originally suggested by Bennet (1920). Thus our chapter is also a contribution to the recent literature on the Bennet indicators of price and quantity change.52 In section 3.4, our decomposition of real income per unit labour input change into explanatory factors largely parallels the corresponding decomposition of nominal income change obtained by Diewert and Morrison (1986) and Kohli (1990), who derived exact results using the Translog functional form. An advantage of the present approach over the Translog approach is that our present approach is valid even if some individual prices or quantities are zero whereas the Translog approach fails if an exogenous price or quantity approaches zero. Since zero prices and quantities do occur empirically, applied 52  Our results in section 3.4 are similar in part to the results obtained by Balk, Färe and Grosskopf (2004).  See Diewert (1992) (2005c), Chambers (2001) (2002) and Balk (2003) (2007) for additional material on the Bennet indicators of price and quantity change.  46  welfare economists may find our present approach useful in these situations. Another advantage of our suggested difference approach is that it is valid even if value subaggregates (such as net exports or inventory change) change sign over the two periods being compared where as the traditional ratio approach to index number theory breaks down under these conditions. Section 3.5 applies our methodology based on the difference approach to the analysis of the market sector in Japanese economy. We decompose changes in the (gross) real income per unit labour and the net real income per unit labour input for the years 1955–2006. We found that productivity growth and the growth in capital services were the two main contributors to the growth in real income per unit labour and net real income per unit labour. It is also shown that changes in terms of trade had smaller effects on them on average. 3.2 The Production Theory Framework In this section, we outline the economic approach to production theory which will be used in the remainder of the chapter.53 The main reference is Diewert and Morrison (1986).54 The economic approach to production theory relies on the assumption of (competitive) optimizing behaviour on the part of producers. In our empirical work, we will apply the economic approach to the market sector of the Japanese economy. Thus we will consider only that part of the Japanese economy that is motivated by profit maximizing behaviour.55  53  This material is drawn from Diewert, Mizobuchi and Nomura (2005) and Diewert and Lawrence (2006).  54  The theory also draws on Samuelson (1953), Fisher and Shell (1972), Diewert (1974; 133-141) (1980)  (1983; 1077-1100), Archibald (1977), Fox and Kohli (1998), Kohli (1978) (1990) (1991) (2003) (2004a) (2004b) (2006) (2007) and Morrison and Diewert (1990). 55  The Japanese market sector excludes all of the general government sectors such as schools, hospitals,  universities, defence and public administration where no independent measures of output can be obtained. For owner occupied housing, output is equal to input and hence no productivity improvements can be generated by this sector according to SNA conventions. residential housing services in our model.  47  However, we do include the consumption of  We assume that the market sector of the economy produces quantities of M (net)56 outputs, Y  [Y1,...,YM], which are sold at the positive producer prices, P  [P1,...,PM]. We further assume that the market sector uses positive quantities of N primary inputs, X  [X1,...,XN], which are purchased at the positive primary input prices W  [W1,...,WN]. In period t, we assume that there is a feasible set of output vectors Y that can be produced by the market sector if the vector of primary inputs X is utilized by the market sector; denote this period t production possibilities set by St. We assume that St is a closed convex cone that exhibits a free disposal property.57 Given a vector of output prices P and a vector of available primary inputs X, we define the period t market sector income function, gt(P,X), as follows:58 (3.1) gt(P,X)  max Y {PY : (Y,X) belongs to St} ;  t = 0,1,2, ... .  Thus market sector nominal income depends on t (which represents the period t 56  If the mth commodity is an import (or other produced input) into the market sector of the economy, then  the corresponding quantity ym is indexed with a negative sign.  We will follow Kohli (1978) (1991) and  Woodland (1982) in assuming that imports flow through the domestic production sector and are “transformed” (perhaps only by adding transportation, wholesaling and retailing margins) by the domestic production sector. 57  The recent textbook by Feenstra (2004; 76) also uses this approach.  For a more explanation for the meaning of these properties, see Diewert (1973) (1974; 134) or  Woodland (1982) or Kohli (1978) (1991). The assumption that St is a cone means that the technology is subject to constant returns to scale.  This is an important assumption since it implies that the value of  outputs should equal the value of inputs in equilibrium.  In our empirical work, we use an ex post rate of  return in our user costs of capital, which forces the value of inputs to equal the value of outputs for each period. 58  The function gt is known as the GDP function or the national product function in the international trade  literature (see Kohli (1978) (1991), Woodland (1982) and Feenstra (2004; 76)). economics literature by Samuelson (1953).  It was introduced into the  Alternative terms for this function include: (i) the gross profit  function; see Gorman (1968); (ii) the restricted profit function; see Lau (1976) and McFadden (1978); and (iii) the variable profit function; see Diewert (1973) (1974).  However, we will call it the (nominal)  income function, since it also defines the amount of income that is distributed to the vector of primary inputs that is used by the market sector.  The function gt(P,X) will be linearly homogeneous and convex in  the components of P and linearly homogeneous and concave in the components of X; see Diewert (1973) (1974; 136). Notation: PY  m=1M PmYm.  48  technology set St), on the vector of output prices P that the market sector faces and on X, the vector of primary inputs that is available to the market sector. If Pt is the period t output price vector and Xt is the vector of inputs used by the market sector during period t and if the income function is differentiable with respect to the components of P at the point Pt,Xt, then the period t vector of market sector outputs Yt will be equal to the vector of first order partial derivatives of gt(Pt,Xt) with respect to the components of P; i.e., we will have the following equations for each period t:59 (3.2) Yt = P gt(Pt,Xt) ;  t = 0,1,2, ... .  Thus the period t market sector supply vector Yt can be obtained by differentiating the period t market sector income function with respect to the components of the period t output price vector Pt. If the income function is differentiable with respect to the components of X at the point Pt,Xt, then the period t vector of input prices Wt will be equal to the vector of first order partial derivatives of gt(Pt,Xt) with respect to the components of X; i.e., we will have the following equations for each period t:60 (3.3) Wt = X gt(Pt,Xt) ;  t = 0,1,2, ... .  Thus the period t market sector input prices Wt paid to primary inputs can be obtained by differentiating the period t market sector income function with respect to the components of the period t input quantity vector Xt. The constant returns to scale assumption on the technology sets St implies that the value of outputs will equal the value of inputs in period t; i.e., we have the following relationships: (3.4) gt(Pt,Xt) = PtYt = WtXt ; 59  These t  t  t  relationships t  t  are  due  t = 0,1,2, ... . to  Hotelling  (1932;  594).  Note  that  Pgt(Pt,Xt)    t  [g (P ,X )/P1, ...,g (P ,x )/PM]. 60  These relationships are due to Samuelson (1953) and Diewert (1974; 140).  [gt(Pt,Xt)/X1, ...,gt(Pt,Xt)/XN].  49  Note that Xgt(Pt,Xt)   The above material will be useful in what follows. Note that our focus is not on the value of outputs generated by the market sector; instead our focus is on the amount of nominal income generated by the market sector. Since the value of market sector production is distributed to the factors of production used by the market sector, nominal market sector output will be equal to nominal market sector income; i.e., from (3.4), we have g(Pt,Xt) = PtYt = WtXt. We will choose to measure the real income generated by the market sector in period t, rt, in terms of the number of fixed basket consumption bundles (with weights represented by the nonnegative, nonzero vector  > 0M) that the nominal income generated by the market sector could purchase in period t61; i.e., we define rt as follows: (3.5) rt  WtXt/Pt = (1/Pt)gt(Pt,Xt) = gt(Pt/Pt,Xt) = gt(pt,Xt)  t = 0,1,2, ... using (3.4) using the linear homogeneity of gt(P,X) in P using definition (3.6) below  = ptYt  using (3.4) and (3.6)  where Pt > 0 is a period t consumption expenditures deflator62 and the market sector period t real output price pt and real input price wt vectors are defined as the corresponding nominal price vectors deflated by the consumption expenditures deflator; i.e., we have the following definitions:63  61  This measure can be interepreted as an approximate welfare measure associated with market sector  production. Since some of the primary inputs used by the market sector can be owned by foreigners, our measure of domestic welfare generated by the market production sector is only an approximate one. Moreover, our suggested approximate welfare measure is not sensitive to the distribution of the income that is generated by the market sector. 62  In our empirical work, we will form 7 subaggregates of Japanese net outputs where the first  subaggregate is consumption.  We will choose our 7 dimensional  vector to be the first unit vector.  Thus we simply deflate the period t Pt and Wt price vectors by P1t, the price of consumption in period t. 63  Our approach to measuring real income is similar to the approach advocated by Kohli (2004b; 92),  except he essentially deflates nominal GDP by the domestic expenditures deflator rather than just the domestic (household) expenditures deflator; i.e., he deflates by the deflator for C+G+I, whereas we suggest deflating by the deflator for C.  Another difference in his approach compared to the present approach is  50  (3.6) pt  Pt/Pt ; wt  Wt/Pt ;  t = 0,1,2, ... .  The first and last equality in (3.5) imply that period t real income, rt, is equal to the period t income function, evaluated at the period t real output price vector pt and the period t input vector Xt, Gt(pt,Xt). Thus the growth in real income over time can be explained by three main factors: t or technical progress or Total Factor Productivity growth (shifts in gt), growth in real output prices (changes in pt) and growth in primary input quantities (changes in Xt). However, rather than find an exact decomposition for the change in real income over time into explanatory factors, the methodology to be developed in the following section only allows us to find an exact decomposition for the change in real income divided by an index of primary inputs. Thus define the real income generated by the market sector in period t per unit primary input, t, as our previous real income measure rt divided by a period t index of primary inputs used, Xt (with weights represented by the nonnegative, nonzero vector  > 0N); i.e., define t as follows: (3.7) t  WtXt/[(Pt)( Xt)] = (Wt/Pt)(Xt/ Xt) = wtxt = ptyt = gt(pt,xt)  t = 0,1,2, ... rearranging terms using definitions (3.6) and (3.8) using (3.5) and (3.8)  where the last equality follows using (3.4) and the linearly homogeneity of gt(P,X) in both P and X and where the market sector period t relative output quantity and input quantity vectors yt and xt are defined as the corresponding quantity vectors Yt and Xt deflated by the primary input index deflator Xt  64  ; i.e.,  we have the following definitions:  that we restrict our analysis to the market sector GDP, whereas Kohli deflates all of GDP (probably due to data limitations). 64  Our treatment of the balance of trade surplus or deficit is also different.  In our empirical work, we will form the following aggregates of Japanese primary inputs: K (capital  services), KIV (inventory services), LD (land services) and LB (labour input).  We will choose our 4  dimensional  vector to be the first unit vector.  Thus we simply deflate the period t quantity vectors Yt  and Xt by XLt, the quantity of labour in period t.  We regard the resulting measure t defined by (3.7) as an  improved measure of labour productivity since it takes into account changes in the prices of investment,  51  (3.8) yt  Yt/Xt ; xt  Xt/Xt ;  t = 0,1,2, ... .  Using the linear homogeneity properties of the income function gt(P,X) in P and X separately, we can show that the following counterparts to the relations (3.2) and (3.3) hold using the real prices pt and wt defined by (6) and the deflated quantities yt and xt defined by (3.8):65 (3.9) yt = p gt(pt,xt) ; (3.10) wt = x gt(pt,xt) ;  t = 0,1,2, ... t = 0,1,2, ... .  In the following section, we will define various explanatory factors that will be used to explain the change in the real income generated by the market sector in period t per unit primary input over the previous period, t  t1, into explanatory factors which are also differences. There will be three sets of explanatory factors that are associated with:     Changes in real output prices pt; Changes in relative primary input quantities xt and Changes in technology; i.e., shifts in the income functions gt.  3.3 The Theoretical Explanation of Per Unit Primary Input Real Income Growth using Differences Now we are ready to define a family of period t productivity growth factors or technical progress shift factors (p,x,t) using the difference approach as opposed to the usual ratio  exports and imports relative to the price of consumption.  Traditional measures of labour productivity  cannot take into account such price changes; in particular, they cannot take into account changes in the country’s terms of trade. 65  If producers in the market sector of the economy are solving the profit maximization problem that is  associated with gt(P,X), which uses the original output prices P and the original primary input vector X, then they will also solve the profit maximization problem that uses the deflated output prices p P/P and the deflated primary input vector x  X/X; i.e., they will also solve the revenue maximization problem defined by gt(p,x).  52  approach:66 (3.11) (p,x,t)  gt(p,x)  gt1(p,x) ;  t = 1,2, ... .  Thus (p,x,t) measures the change in the real income per unit primary input produced by the market sector at the reference real output prices p and reference relative input quantities used by the market sector x where the first term in the right hand side of (3.11) uses the period t technology represented by gt and the second term in (3.11) uses the period t1 technology gt1. Thus each choice of reference vectors p and x will generate a possibly different measure of the shift in technology going from period t1 to period t. Note that we are using the chain system to measure the shift in technology. It is natural to choose special reference vectors for the measure of technical progress defined by (3.11): a Laspeyres type measure Lt that chooses the period t1 reference vectors pt1 and xt1 and a Paasche type measure Pt that chooses the period t reference vectors pt and xt:67 (3.12) Lt  (pt1,xt1,t) = gt(pt1,xt1)  gt1(pt1,xt1) ; (3.13) Pt  (pt,xt,t) = gt(pt,xt)  gt1(pt,xt) ;  t = 1,2, ... ; t = 1,2, ... .  Since both measures of technical progress are equally valid, it is natural to average them to obtain an overall measure of technical change. If we want to treat the two measures in a symmetric manner and we want the measure to satisfy the time reversal property from the difference approach to index number theory68 (so that the estimate going backwards is equal to the negative of the estimate going forwards), then the arithmetic mean will be the best simple average to take in this context. Thus we define the arithmetic mean of (3.12) and (3.13) as follows: 66  The corresponding ratio type measure, (p,x,t)  gt(p,x)/gt1(p,x) is due to Diewert and Morrison (1986;  662). 67  A special case of it was defined earlier by Diewert (1983; 1063).  Diewert and Morrison (1986; 662-663) introduced the ratio counterparts to (3.12) and (3.13) in the  nominal GDP context 68  Diewert (2005c; 366) developed the axiomatic approach to index number theory using differences and  introduced this time reversal test, which is the counterpart to the usual time reversal test that can be found in Fisher (1922; 64).  Balk (2003; 29) also emphasized the importance of a symmetric treatment of time.  Balk (2007) further developed the axiomatic approach using differences.  53  (3.14) t  (1/2)[Lt + Pt] ;  t = 1,2, ... .  At this point, it is not clear how we will obtain empirical estimates for the theoretical productivity growth indexes defined by (3.12)(3.14). One obvious way would be to assume a functional form for the nominal income function gt(P,X), collect data on output and input prices and quantities for the market sector for a number of years, add error terms to equations (3.2) and (3.3) and use econometric techniques to estimate the unknown parameters in the assumed functional form. However, econometric techniques are generally not completely straightforward: different econometricians will make different stochastic specifications and will choose different functional forms. 69 Moreover, as the number of outputs and inputs grows, it will be impossible to estimate a flexible functional form. Thus in the following section, we will suggest methods for estimating productivity change measures like (3.14) that are based on exact index number techniques. We turn now to the problem of defining theoretical indexes for the effects on real income per unit primary input due to changes in real output prices. Define a family of period t real output price change factors (pt1,pt,x,s):70 (3.15) (pt1,pt,x,s)  gs(pt,x)  gs(pt1,x) ;  s = 1,2, ... .  Thus (pt1,pt,x,s) measures the difference in the real income per unit primary input produced by the market sector that is induced by the change in real output prices going from period t1 to t, using the technology that is available during period s and using the reference input quantities x. Thus each choice of the reference technology s and the reference input vector x will generate a possibly different measure of the effect on real income per unit primary input of a change in real output prices going from period t1 to period t. 69  “The estimation of GDP functions such as (3.19) can be controversial, however, since it raises issues  such as estimation technique and stochastic specification. ... We therefore prefer to opt for a more straightforward index number approach.” Ulrich Kohli (2004a; 344). 70  This measure of real output price change is the difference version of the usual ratio concept due to Fisher  and Shell (1972; 56-58), Samuelson and Swamy (1974; 588-592), Archibald (1977; 60-61), Diewert (1980; 460-461) (1983; 1055) and Balk (1998; 83-89).  54  Again, it is natural to choose special reference vectors for the measures defined by (3.15): a Laspeyres type measure Lt that chooses the period t1 reference technology and reference input vector xt1 and a Paasche type measure Pt that chooses the period t reference technology and reference input vector xt: (3.16) Lt  (pt1,pt,xt1,t1) = gt1(pt,xt1)  gt1(pt1,xt1) ; (3.17) Pt  (pt1,pt,xt,t) = gt(pt,xt)  gt(pt1,xt) ;  t = 1,2, ... ; t = 1,2, ... .  Since both measures of real output price change are equally valid, it is natural to average them to obtain an overall measure of the effects on real income per unit primary input of the change in real output prices: (3.18) t  (1/2)[Lt + Pt] ;  t = 1,2, ... .  Finally, we look at the problem of defining theoretical indexes for the effects on real income per unit primary input due to growth in relative input quantities going from period t1 to t. (xt1,xt,p,s):  Define a family of period t relative input quantity growth factors  (3.19) (xt1,xt,p,s)  gs(p,xt)  gs(p,xt1) ;  s = 1,2, ... .  Thus (xt1,xt,p,s) measures the difference in the real income per unit primary input produced by the market sector that is induced by the change in input quantities relative to the index of primary inputs used by the market sector going from period t1 to t, using the technology that is available during period s and using the reference real output prices p. Thus each choice of the reference technology s and the reference real output price vector p will generate a possibly different measure of the effect on real income of a change in relative input quantities going from period t1 to period t. Again, it is natural to choose special reference vectors for the measures defined by (3.19): a Laspeyres type measure Lt that chooses the period t1 reference technology and reference real output price vector pt1 and a Paasche type measure Pt that chooses the period t reference technology and reference real output price vector pt: (3.20) Lt  (xt1,xt,pt1,t1) = gt1(pt1,xt)  gt1(pt1,xt1) ; 55  t = 1,2, ... ;  (3.21) Pt  (xt1,xt,pt,t)  = gt(pt,xt)  gt(pt,xt1) ;  t = 1,2, ... .  Since both measures of (relative) input quantity change are equally valid, it is natural to average them to obtain an overall measure of the effects of relative input change on real income: (3.22) t  (1/2)[Lt + Pt] ;  t = 1,2, ... .  Recall that market sector real income for period t was defined by (3.5) as rt which is equal to nominal period t factor payments WtXt deflated by an index of fixed weight consumption prices, Pt. Recall also that rt was further deflated by the fixed weight index of primary inputs, Xt, in order to obtain the period t real income per unit primary input measure t defined by (3.7). Recall also that using the linear homogeneity properties of the income function gt(P,X) in P and X, we showed that t is equal to gt(pt,xt). It is convenient to define t as the absolute amount of growth in real income per unit primary input going from period t1 to t: (3.23) t  t  t1 ;  t = 1,2, ... .  In the following section, we will show that under certain functional form assumptions on the income functions gt, t is exactly equal to the sum of (3.14), (3.18) and (3.22) respectively.  t, t and t defined above by  3.4 The Normalized Quadratic Income Function and Bennet Indicators of Price, Quantity and Productivity Change Suppose that the period t nominal net revenue or income function gt has the following normalized quadratic functional form:71 (3.24) gt(P,X)  atPX + ctXP + (1/2) PAP [X/P] 71  This functional form is a generalization to many primary inputs of the normalized quadratic unit profit  function introduced by Diewert and Wales (1992; 707) which in turn is an adaptation of the normalized quadratic functional form used by Diewert and Wales (1987) (1988a) (1988b) in a variety of contexts.  It  is also a generalization of the normalized quadratic profit function introduced by Diewert and Ostensoe (1988; 44).  56  + PBX + (1/2) XCX [P/X] ;  t = 0,1,2,...  where at and ct are M and N dimensional vectors of unknown parameters which can be different for each time period t, A = [amk] is an M by M positive semidefinite symmetric matrix of unknown parameters, B = [bmn] is an M by N matrix of unknown parameters, C = [cni] is an N by N negative semidefinite symmetric matrix of unknown parameters and  and  are the same known vectors of parameters that appeared in definitions (3.6) and (3.8) above. Note that with our curvature restrictions on the matrices A and C, gt(P,X) will be convex (and linearly homogeneous) in the components of P and concave (and linearly homogeneous) in the components of X. Note that by allowing the parameter vectors at and ct to change arbitrarily with time, we are allowing for very general forms of technical progress. However, the theory to be developed below does require that the parameter matrices A, B and C to be fixed over time. In Appendix C below, we show that the gt defined by (3.24) is a flexible functional form. Now evaluate (3.24) at the data for period t, Pt, Xt.  Dividing both sides of the resulting  equation by Xt times Pt gives us the following equation for period t real income per unit primary input, t: (3.25) t = gt(pt,xt) = atpt + ctxt + (1/2) ptApt + ptBxt + (1/2) xtCxt ;  t = 0,1,2,...  where pt  Pt/Pt and xt  Xt/Pt. Differentiating (3.24) with respect to the components of P and evaluating the resulting derivatives at the data pertaining to period t leads to the following equations using (3.2): (3.26) Yt = Pgt(Pt,Xt) = atXt + ctXt + APt[Xt/Pt]  (1/2)PtAPtXt[Pt]2 + BXt + (1/2)[Xt]1XtCXt  ; t = 0,1,2,... . Now divide both sides of (3.26) by Xt , define pt  Pt/Pt, xt  Xt/Xt and yt  Yt/Xt and equations (3.26) become the following equations: (3.27) yt = Pgt(pt,xt) = at + ctxt + Apt  (1/2)ptApt + Bxt + (1/2)xtCxt ; t = 0,1,2,... . Now premultiply both sides of equation t in (3.27) by the transpose of pt. Pt/Pt = 1, the resulting equations become: 57  Using pt =  (3.28) ptyt = atpt + ctxt + (1/2) ptApt + ptBxt + (1/2) xtCxt = t = gt(pt,xt) ;t = 0,1,2,... . Recall definition (3.16) for the Laspeyres period t real output price change factor, Lt. Using the gt functions defined by (3.25), we have: (3.29) Lt  gt1(pt,xt1)  gt1(pt1,xt1) t = 1,2, ... t1 t t1 t1 t t t t1 t1 t1 t1 t1 = [a p + c x + (1/2) p Ap + p Bx + (1/2) x Cx ]  p y where we have used definition (3.24) in order to evaluate gt1(pt,xt1) and we have also used equation t1 in (3.28) to obtain gt1(pt1,xt1) equal to pt1yt1. Similarly, using definition (3.17) for the Paasche period t real output price change factor, Pt, we have: (3.30) Pt  gt(pt,xt)  gt(pt1,xt) = ptyt  [atpt1 + ctxt + (1/2) pt1Apt1 + pt1Bxt + (1/2) xtCxt].  t = 1,2, ...  Using (3.29) and (3.30), it can be seen that the sum of the above two real output price change factors is equal to the following expression: (3.31) 2t = Lt + Pt t = 1,2, ... = ptyt  pt1yt1 + [at1pt + ct1xt1 + (1/2) ptApt + ptBxt1 + (1/2) xt1Cxt1]  [atpt1 + ctxt + (1/2) pt1Apt1 + pt1Bxt + (1/2) xtCxt]. Unfortunately, the expressions on the right hand sides of (3.29)(3.31) are not observable without a knowledge of the unknown parameters in the gt functions defined by (3.24). However, the Bennet (1920)72 indicator of real output price change, PB(pt1,pt,yt1,yt),  72  Bennet noticed that the value aggregate difference ptyt  pt1yt1 is exactly equal to the sum of the price  change term, PB(pt1,pt,yt1,yt) defined by (3.32) and the corresponding quantity change term, QB(pt1,pt,yt1,yt) defined as (1/2)[pt1 + pt][yt  yt1].  Diewert (1992) termed PB and QB the Bennet  indicators of price and quantity change for the value aggregate; i.e., he introduced the term indicator as the difference counterpart to the price and quantity index concepts in traditional ratio type index number theory. Diewert (2005c) developed the axiomatic or test approach to price and quantity indicators and showed that  58  defined by (3.32) is empirically observable: (3.32) PB(pt1,pt,yt1,yt)  (1/2)[yt1 + yt][pt  pt1] ;  t = 1,2, ...  where the observable output quantity vectors (deflated by the index of primary inputs) yt  Yt/Xt were defined earlier by (3.8) and the deflated output price vectors pt  Pt/Pt were defined earlier by (3.6). We now show that if there is competitive net revenue maximizing behaviour in the market sector in each period t and the market sector income functions gt are defined by (3.24) above, then the Bennet indicator of real price change, PB defined by (3.32), is exactly equal to t defined by (3.18), the arithmetic average of the Laspeyres and Paasche real output price change factors. Using definition (3.32), we have: (3.33) 2PB(pt1,pt,yt1,yt) = [yt1 + yt][pt  pt1] ; t = 1,2, ... t t t1 t1 t t1 t1 t = p y  p y + p y  p y = ptyt  pt1yt1 + pt[at1 + ct1xt1 + Apt1  (1/2)pt1Apt1 + Bxt1 + (1/2)xt1Cxt1]  pt1[at + ctxt + Apt  (1/2)ptApt + Bxt + (1/2)xtCxt] using (3.27) t t t1 t1 = p y  p y + [ptat1 + ct1xt1 + ptApt1  (1/2)pt1Apt1 + ptBxt1 + (1/2)xt1Cxt1]  [pt1at + ctxt + pt1Apt  (1/2)ptApt + pt1Bxt + (1/2)xtCxt] using pt = 1 = ptyt  pt1yt1 + [at1pt + ct1xt1 + (1/2) ptApt + ptBxt1 + (1/2) xt1Cxt1]  [atpt1 + ctxt + (1/2) pt1Apt1 + pt1Bxt + (1/2) xtCxt] simplifying = Lt + Pt using (3.31) t = 2 using definition (3.18). Thus under our assumptions on technology, the theoretical measure of period t per unit primary input market sector real income change due to change in real output prices, t defined by (3.18), is exactly equal to the observable Bennet indicator of real price change, PB defined by (3.32).73 the Bennet indicators were the difference counterparts to the Fisher price and quantity indexes in terms of their axiomatic properties. 73  Balk (2007) also looked at the axiomatic properties of the Bennet indicators.  Equations (3.33) show that the theoretical measure of change in gt due to changes in real output prices pt,  t defined by (3.18), is equal to m=1M (1/2)[ymt1 + ymt][pmt  pmt1].  59  In Appendix D below, we show that  The above analysis can be modified to give us an observable estimator for the theoretical measure of the effects of relative input quantity change on market sector real income per unit primary input, t defined by (3.22) above. However, it is first necessary to differentiate the normalized quadratic functions gt(P,X) defined above by (3.24) with respect to the components of X and then use Samuelson’s Lemma (3.3) in order to obtain expressions for the period t input price vectors Wt. Differentiating (3.24) with respect to the components of X and evaluating the resulting derivatives at the data pertaining to period t leads to the following equations using (3.3): (3.34) Wt = Xgt(Pt,Xt) = atPt + Ptct + (1/2)[Pt]1PtAPt + BTPt + CXt [Pt/Xt]  (1/2)[Xt]2XtCXtPt.  t = 0,1,2,...  Now divide both sides of (3.34) by Pt, define pt  Pt/Pt, xt  Xt/Xt and wt  Wt/Pt and equations (3.34) become the following equations: (3.35) wt = atpt + ct + (1/2)ptApt + BTpt + Cxt  (1/2)xtCxt ;  t = 0,1,2,.. .  Now premultiply both sides of equation t in (3.35) by the transpose of xt. Using xt = Xt/Xt = 1, the resulting equations become: (3.36) wtxt = atpt + ctxt + (1/2) ptApt + ptBxt + (1/2) xtCxt ; = gt(pt,xt)  t = 0,1,2,... using (3.28).  Now recall definition (3.20) for the Laspeyres period t relative input quantity growth factor, Lt.  Using the gt functions defined by (3.24), we have:  (3.37) Lt  gt1(pt1,xt)  gt1(pt1,xt1) t = 1,2, ... = [at1pt1 + ct1xt + (1/2) pt1Apt1 + pt1Bxt + (1/2) xtCxt]  pt1yt1. Similarly, using definition (3.21) for the Paasche period t relative input quantity growth each term in this summation can be interpreted as an approximate theoretical measure of the change in gt due to the change in a single real price pmt.  60  factor, Pt, we have, using (3.25) and (3.28): (3.38) Pt  gt(pt,xt)  gt(pt,xt1) = ptyt  [atpt + ctxt1 + (1/2) ptApt + ptBxt1 + (1/2) xt1Cxt1].  t = 1,2, ...  Thus using (3.37) and (3.38), it can be seen that the sum of the above two relative input quantity growth factors is equal to the following expression: (3.39) 2t = Lt + Pt t = 1,2, ... t t t1 t1 t1 t1 t1 t t1 t1 t1 t = p y  p y + [a p + c x + (1/2) p Ap + p Bx + (1/2) xtCxt]  [atpt + ctxt1 + (1/2) ptApt + ptBxt1 + (1/2) xt1Cxt1]. Unfortunately, the expressions on the right hand sides of (3.37)(3.39) are not observable without a knowledge of the unknown parameters in the gt functions defined by (3.24). However, the Bennet (1920) indicator of relative input quantity change, QB(wt1,wt,xt1,xt), defined by (3.40) is empirically observable: (3.40) QB(wt1,wt,xt1,xt)  (1/2)[wt1 + wt][xt  xt1] ;  t = 1,2, ...  where the observable real input price vectors wt  Wt/ Pt were defined earlier by (3.6) and the input quantity vectors (deflated by the index of primary inputs) xt  Xt/Xt were defined earlier by (3.8). We now show that if there is competitive profit maximizing behaviour in the market sector in each period t and the market sector income functions gt are defined by (3.24) above, then the Bennet indicator of relative input quantity change, QB defined by (3.40), is exactly equal to t defined by (3.22), the arithmetic average of the Laspeyres and Paasche relative input quantity growth factors. Using definition (3.40), we have: (3.41) 2QB(wt1,wt,xt1,xt) = [wt1 + wt][xt  xt1] ; t = 1,2, ... = wtxt  wt1xt1 + wt1xt  wtxt1 = ptyt  pt1yt1 + wt1xt  wtxt1 using (3.38) and (3.36) = ptyt  pt1yt1 + xt[at1pt1 + ct1 + (1/2)pt1Apt1 + BTpt1 + Cxt1  (1/2)xt1Cxt1]  xt1[ atpt + ct + (1/2)ptApt + BTpt + Cxt  (1/2)xtCxt] using (3.35) = ptyt  pt1yt1 61  + [at1pt1 + ct1xt + (1/2)pt1Apt1 + pt1Bxt + xtCxt1  (1/2)xt1Cxt1]  [atpt + ct xt1 + (1/2)ptApt + ptBxt1+ xt1Cxt  (1/2)xtCxt] using xt1 = 1 = ptyt  pt1yt1 + [at1pt1 + ct1xt + (1/2)pt1Apt1 + pt1Bxt  (1/2)xt1Cxt1]  [atpt + ct xt1 + (1/2)ptApt + ptBxt1  (1/2)xtCxt] using xtCxt1 = xt1Cxt = (1/2)[ Lt + Pt] using (3.39) = t using definition (3.22). Thus under our assumptions on technology, the theoretical measure of period t per unit primary input market sector real income change due to change in relative input quantities, t defined by (3.22), is exactly equal to the observable Bennet indicator of relative input quantity change, QB defined by (3.40).74 We now turn our attention to developing an observable measure of technical progress. Recall the (unobservable) theoretical measures of technical progress Lt, Pt and t defined by (3.12)(3.14) respectively. If the income functions gt are defined by (3.24), then by substituting these definitions for the gt into definitions (3.12)(3.14), we obtain the following (unobservable) expressions for these technical progress measures: (3.42) Lt  gt(pt1,xt1)  gt1(pt1,xt1) ; = [at  at1]pt1 + [ct  ct1]xt1 ; (3.43) Pt  gt(pt,xt)  gt1(pt,xt) ; = [at  at1]pt + [ct  ct1]xt ; (3.44) t  (1/2)[Lt + Pt] ; = (1/2)[pt1 + pt][at  at1] + (1/2)[xt1 + xt][ct  ct1] .  t = 1,2, ... t = 1,2, ... t = 1,2, ...  Now look at the period t change in real income per unit primary input over the previous period, t  t1 equal to ptyt  pt1yt1. Subtract the Bennet indicator of real price change PB(pt1,pt,yt1,yt) defined by (3.32) and subtract the Bennet indicator of relative input quantity change QB(wt1,wt,xt1,xt) defined by (3.40) from this income difference and evaluate the resulting expression using the gt defined by (3.24). We obtain the 74  Equations (3.41) show that the theoretical measure of change in gt due to changes in relative input  quantities xt, t defined by (3.22), is equal to n=1N (1/2)[wnt1 + wnt][xnt  xnt1].  In Appendix D below,  we show that each term in this summation can be interpreted as an approximation to a theoretical measure of the change in gt due to the change in a single deflated input quantity xnt.  62  following identity: (3.45) ptyt  pt1yt1  (1/2)[yt1 + yt][pt  pt1]  (1/2)[wt1 + wt][xt  xt1] t = 1,2, ... t1 t t1 t1 t t t t1 t1 t1 =  (1/2)[a p + c x + (1/2) p Ap + p Bx + (1/2) x Cx ] + (1/2)[atpt1 + ctxt + (1/2) pt1Apt1 + pt1Bxt + (1/2) xtCxt]  (1/2)[at1pt1 + ct1xt + (1/2)pt1Apt1 + pt1Bxt  (1/2)xt1Cxt1] + (1/2)[atpt + ct xt1 + (1/2)ptApt + ptBxt1  (1/2)xtCxt] using (3.33) and (3.41) = (1/2)[pt1 + pt][at  at1] + (1/2)[xt1 + xt][ct  ct1] = t  cancelling terms using (3.44).  Thus the first line in (3.45) gives us an observable exact estimator for the theoretical technical progress measure t. Using (3.32), (3.40) and (3.45), it can be seen that under the assumption that the income functions gt are defined by (3.24), we have the following exact decomposition for the change in real income per unit primary input, t  t1: (3.46) t  ptyt  pt1yt1 = t  t1 = t + t + t ;  t = 1,2, ... .  The above equation says the change in real income per unit primary input is equal to the sum of a change in real output prices factor t plus a change in relative primary input quantities factor t plus a change in technical efficiency term t where all three explanatory factors can be estimated using observable price and quantity data pertaining to periods t and t1.75 Rather than look at explanatory factors for the difference in real income per unit primary input between the adjacent periods, it is sometimes convenient to express the difference in real income between the current period t and the reference year 0 in terms of the difference in the indicator of the technology level Tt, of the change in the level of real output prices in period t, At, and of the difference in the level of primary input quantities in period t, Bt. Thus, we use the growth factors τt, αt, and βt as follows to define the (3.47) T0 0; Tt = Tt-1 + τt; (3.48) A0 0; At = At-1 + α t; 75  t = 1,2, ... t = 1,2, ...  The decomposition (3.46) is a difference counterpart to the ratio decomposition of nominal income  growth obtained by Diewert and Morrison (1986; 663665) and Kohli (1990).  63  (3.49) B0 0; Bt = Bt-1 + βt;  t = 1,2, ... .  Using the chain links that appear in (3.47)(3.49), we can establish the following exact relationship for the cumulative change in real income per unit primary input going from period 0 to period t: (3.50) t  0 = At + Bt + Tt;  t = 1,2, ... .  Instead of using the first line in (3.45) to define the technical progress term that turns out to be equal to t, we can obtain some alternative expressions for technical progress (or the change in Total Factor Productivity) using Bennet (1920) identity for the decomposition of the change of a value aggregate into price and quantity components; i.e., Bennet showed that the following exact identity holds: (3.51) ptyt  pt1yt1 = (1/2)[yt1 + yt][pt  pt1] + (1/2)[pt1 + pt][yt  yt1] . Substituting (3.51) into (3.45) leads to the following alternative exact expression for the technical progress term t: (3.52) t =  (1/2)[pt1 + pt][yt  yt1]  (1/2)[wt1 + wt][xt  xt1] ;  t = 1,2, ... .  Thus the absolute changes in the (deflated) output quantities, yt  yt1, are weighted by the average of the real output prices for periods t1 and t, (1/2)[pt1 + pt], and then we subtract the absolute changes in the (deflated) primary input quantities, xt  xt1, weighted by the average of the real input prices for periods t1 and t, (1/2)[wt1 + wt]. We call the right hand side of (52) the primal Bennet measure of technical progress.76 It is a difference counterpart to the following primal Fisher index of productivity growth or index of technical progress:77 (3.53) t  [ptyt pt1yt/ptyt1 pt1yt1]1/2/[wtxt wt1xt/wtxt1 wt1xt1]1/2 .  76  Balk (2003; 29) (2007), Diewert (2005c; 353) and Diewert and Fox (2005; 8) all suggested variants of  this Bennet indicator of real profit change as a measure of efficiency improvement. 77  See Diewert and Nakamura (2003) for material on the Fisher (1922) index and its use in the traditional  ratio approach to productivity measurement.  64  However, we can obtain a third expression for t, which is also instructive. Under our assumptions (3.4), it can be seen that ptyt is equal to wtxt for each t. Thus we have: (3.54) ptyt  pt1yt1 = wtxt  wt1xt1 = (1/2)[xt1 + xt][wt  wt1] + (1/2)[wt1 + wt][xt  xt1] where we have applied the Bennet value difference decomposition to obtain the second equality in (3.54). Substituting (3.54) into the first line of (3.45) leads to the following exact expression for t: (3.55) t = (1/2)[xt1 + xt][wt  wt1]  (1/2)[yt1 + yt][pt  pt1] . Thus if real input prices increase faster than real output prices, there will be positive technical progress. We call the right hand side of (3.55) the dual Bennet measure of technical progress. It is a difference counterpart to the following dual Fisher index of productivity growth or index of technical progress:78 (3.56) τFt  [wtxt wtxt1/wt1xt wt1xt1]1/2/[ptyt ptyt1/pt1yt pt1yt1]1/2 . The above difference approach can be converted into a more traditional growth rate approach. The period t rate of growth of real income per unit primary input is equal to the period t change in real income generated by the market sector, t  t1, divided by last period’s real income, t1. Using (3.40), we have: (3.57) [t  t1]/t1 = [t + t + t]/t1;  t = 1,2, ... .  Thus the rate of growth of market sector real income per unit primary input is explained by a sum of three additional explanatory factors, t/t1 (the contribution of real output price change), plus t/t1 (the contribution of input quantity growth relative to the  78  The fact that primal indexes of Total Factor Productivity Growth (an index of output quantity growth  divided by an index of input quantity growth) could be also written in dual form (an index of input prices divided by an index of output prices) dates back to the pioneering contributions of Jorgenson and Griliches (1967).  65  average growth of primary inputs), plus t/t1 (the contribution of technical change).79 Using the results derived in Appendix D, the overall period t real price change term t is equal to the sum of M individual real price change terms, m=1M mt, and the overall relative input quantity change term t is equal to the sum of N individual relative input quantity change terms, n=1N nt. Moreover, these individual price and quantity terms can be calculated empirically without econometric estimation. Using these results from Appendix D, (3.57) can be rewritten as follows: (3.58) [t  t1]/t1 = [m=1M mt + n=1N nt + t]/t1 ;  t = 1,2, ... .  The decomposition of market sector real income growth (per unit primary input) given by (3.58) is comparable to the decomposition of real income growth that was derived by Diewert, Mizobuchi and Nomura (2005) and Diewert and Lawrence (2006) using the Translog methodology that was originally developed by Diewert and Morrison (1986) and Kohli (1990). However, the present normalized quadratic methodological approach has an advantage over the earlier Translog approach in that the present approach allows individual prices and quantities to be zero, whereas the Translog approach fails if any (exogenous) price or quantity becomes zero. Since a great deal of R&D effort is devoted to the development of new goods and services, it is useful to have a methodology that is able to deal with the creation of new products. A second advantage of the present approach is that when we specialize the fixed weight input index to be labour input, a “better” decomposition of labour productivity into explanatory factors is obtained; i.e., in our methodological approach, real output is replaced by real income and hence the effects on real income per unit labour input of changes in the terms of trade can be modeled using our present approach. 3.5 An Application to the Japanese Economy for 19552006 3.5.1 The Japanese Data We apply our methodology to a modified version of the Japanese productivity database 79  If there are only two primary inputs, labour and capital, and the primary input weighting vector  is the  unit vector (1,0) so that the input aggregate collapses down to labour input, then the term t/t1 is the contribution of capital deepening; i.e., of the growth of capital input relative to the growth of labour input.  66  developed by Chapter 2. This database consists of the quantity, price, and value series for eleven main classes of outputs and inputs for the Japanese market sector, with some further detailed breakdowns for the investment outputs and capital service inputs; see Appendix E for a detailed listing of the data. We briefly mention the data construction procedure of Chapter 2. We followed the conventions introduced by Jorgenson and Griliches on the treatment of taxes; i.e., we adjusted prices for tax wedges whenever possible so that the adjusted prices reflect the prices that producers face.80 We also included the services of inventories and land as additional capital inputs. A listing of the outputs and inputs follows. The 7 main classes of net outputs are:  80    C; Domestic final consumption expenditure of households (excluding imputed rent for owner-occupied houses);    N; Final consumption expenditure of private non-profit institutions serving households (NPISHs);    G; Net sales of goods and services by the market sector to the general government sector; i.e., the value aggregate is equal to minus government sales of goods and services to the market production sector plus purchases of intermediate inputs from the market sector;    X; Exports of goods and services (excluding direct purchases in the domestic market by non-resident households) and    M; Imports of goods and services (excluding direct purchases abroad by resident households);    I; Investment which consists of the following 12 subaggregates: I1: Animals and plants; I2: Construction; I3: Textile products; I4: Wood products; I5: Furniture and fixtures; I6: Metallic products; I7: General machinery; I8: Electric machinery; I9:  Thus our suggested treatment of indirect commodity taxes in an accounting framework that is suitable  for productivity analysis follows the example set by Jorgenson and Griliches who advocated the following treatment of indirect taxes: “In our original estimates, we used gross product at market prices; we now employ gross product from the producers’ point of view, which includes indirect taxes levied on factor outlay, but excludes indirect taxes levied on output.” Dale W. Jorgenson and Zvi Griliches (1972; 85).  All  other taxes such as taxes on financial assets and poll taxes which do not affect the producers’ behaviour are ignored in this study.  67  Automobiles; I10: Other transportation; I11: Precision machinery; I12: Other investment products;   IV; Change in Inventories which consists of the following 4 subaggregates: IV1: Finished goods inventory change; IV2: Work in progress inventory change; IV3: Work in progress inventory change for cultivated assets; IV4: Change in materials inventory.  The 4 main classes of primary inputs are:   K; Capital service which consists of the following 12 subaggregates: K1: Animals and plants; K2: Construction; K3: Textile products; K4: Wood products; K5: Furniture and fixtures; K6: Metallic products; K7: General machinery; K8: Electric machinery; K9: Automobiles; K10: Other transportation; K11: Precision machinery; K12: Other investment products;    KIV; Inventory service which consists of the following 4 components: KIV1: Finished good inventory services; KIV2: Work in progress inventory services; KIV3: Work in progress inventory services for cultivated assets; KIV4: Materials inventory services;    LD; Land service which consists of the following 4 subaggregates: LD1: Agricultural land services; LD2: Industrial land services; LD3: Commercial land services; LD4: Residential land services for renters;    LB; Labour input which consists of the following 3 subaggregates: LB1: Labour input of the self-employed; LB2: Labour input of family workers; LB3: Labour input of employees in the market sector;  Prices and quantities for the net output aggregates YC,YN,YG,YX,YM have been constructed using data in the national accounts based on 1968 SNA and 1993 SNA81 and National Income Statistics which was the national accounting system prior to the introduction of 1968 SNA. We took the numbers constructed on the basis of 1993 JSNA as our standard. We extended the data series backwards by using data from 1968 JSNA and the earlier national income statistics. The capital stocks are stocks at the beginning of the year.  Estimates for the reproducible  capital stocks during 19552006 have been constructed by applying the perpetual 81  We call Japanese national accounts based on 1968 (1993) SNA simply 1968 (1993) JSNA, hereafter.  68  inventory method to the initial stocks in 1955 and using the investment data and asset specific depreciation rates. The initial capital stocks, the investment data and the change in inventory data have been taken from capital and investment data in the KEO database. This is a comprehensive productivity database for the Japanese economy constructed at Keio University. The detailed procedures used to construct these capital data are explained in Nomura (2004). It should be noted that the KEO data base of price and quantity data for investments and the corresponding capital stocks for 95 classes of asset. This chapter has aggregated these 95 asset classes into 12 classes for reproducible capital. Estimates of the quantities of labour services XLB are based on hours of work. There are three different types of workers; the self employed, family workers and employees. Hours of works for each type of worker are aggregated into the quantity of aggregate labour input by applying a Fisher (1922) price index. Price and quantity data for market sector net outputs and primary inputs are listed in Tables E1 (prices) and E2 (quantities) below. The detailed data on investments, changes in inventory, capital stocks, inventory stocks and land are listed in Tables E3E7 (prices) and E8E12 (quantities). 3.5.2 The Decomposition of Real Income Growth into Explanatory Factors Substituting the estimates defined in (3.32), (3.41) and (3.45) into equation (3.52), we can decompose the growth rate of real income per unit labour (ρt – ρt-1)/ρt-1 = γt/ρt-1 into the contribution of technical progress τt/ρt-1, changes in real output prices αCt/ρt-1 (domestic final consumption), 82 αNt/ρt-1 (non-profit institution final consumption), αGt/ρt-1 (net government purchases from the market sector), αXt/ρt-1 (exports), αMt/ρt-1 (imports), αIt/ρt-1 (investments in reproducible capital) and αIVt/ρt-1 (inventory changes) and growth in relative input quantities βKt/ρt-1 (capital services), βKIVt/ρt-1 (inventory services), βLDt/ρt-1 (land services) and βLBt/ρt-1 (labour input).83 The chain link information on period by period changes in real income per unit labour input that corresponds to (3.52) is given in Table 3-1. The effect of changes in the terms of trade is αXMt/ρt-1 and is simply the sum of the contributions of real price changes in exports and imports αXt/ρt-1 and αMt/ρt-1. 82  Since we divided market sector nominal income by the price of consumption, Ct/t1 will be identically  equal to zero and hence it is not listed. 83  Since we divided market sector nominal income by the quantity of labour input, LBt /t1 will be  identically equal to zero and hence it is not listed.  69  Table 3-1: Decomposition of Growth Rate of Real Income Per Unit Labour (%) γt/rt-1  tt/rt-1  aNt/rt-1  aGt/rt-1  aXt/rt-1  aMt/rt-1  aXMt/rt-1  aIt/rt-1  aIVt/rt-1  b Kt/rt-1  b KIVt/rt-1  b LDt/rt-1  1956  4.44867  6.16182  0.06632  0.11226  0.41197  -0.77573  -0.36376  1.89131  0.31518  -2.17691  -0.04565  -1.51192  1957  2.87446  4.57677  0.03315  0.03240  -0.33890  -0.07741  -0.41631  0.74566  -0.14922  -0.81207  0.04929  -1.18521  1958  2.00107  1.40635  0.00374  -0.12786  -0.82820  2.09237  1.26417  -0.84716  -0.04260  0.74947  0.13084  -0.53588  1959  9.40710  10.00776  0.00620  -0.08480  -0.13447  0.50801  0.37354  -1.00889  -0.02934  0.67686  -0.03759  -0.49664  1960  11.39564  12.25692  0.02537  -0.08627  -0.02266  0.34772  0.32506  -0.26007  -0.16189  0.45406  0.05441  -1.21196  1961  12.02089  9.87093  0.01186  -0.11711  -0.74467  0.61915  -0.12553  0.45644  -0.42145  2.43849  0.18021  -0.27296  1962  0.65867  1.02717  -0.00037  -0.19034  -1.00636  0.97419  -0.03217  -1.61067  -0.27202  2.01110  0.27188  -0.54592  1963  6.00309  5.81813  -0.00565  -0.24375  -0.53866  0.62781  0.08915  -2.37507  -0.18648  2.60020  0.12646  0.18011  1964  10.38152  8.83887  0.05422  -0.11601  -0.18209  0.15700  -0.02509  -0.39846  -0.04880  2.08023  0.15132  -0.15475  1965  0.26884  0.14921  -0.01467  -0.18490  -0.92682  0.91727  -0.00955  -1.81352  -0.11846  2.08152  0.15052  0.02870  1966  8.30842  6.67815  0.02531  -0.06157  -0.52277  0.38307  -0.13970  -0.15269  -0.07812  1.36975  0.03927  0.62803  1967  10.90512  7.79238  0.03298  -0.01017  -0.37574  0.56626  0.19052  0.29416  -0.11398  1.74651  0.10816  0.86455  1968  11.20952  8.93386  0.00975  -0.10101  -0.59204  0.59657  0.00453  -0.73902  -0.10851  2.62481  0.27313  0.31197  1969  12.66910  8.31488  0.03252  -0.08557  -0.28794  0.07037  -0.21756  0.18416  0.01995  2.89693  0.25461  1.26919  1970  6.81164  4.69959  0.06019  -0.12126  -0.45890  0.54747  0.08857  -0.88636  -0.14864  2.72774  0.20001  0.19178  1971  0.80235  -1.13843  0.04221  -0.19201  -0.55358  1.10380  0.55021  -1.68052  -0.08855  2.84638  0.16764  0.29540  1972  7.04619  4.57045  0.05415  -0.18580  -0.81746  1.09228  0.27482  -0.49672  0.02079  2.34018  0.01274  0.45557  1973  8.21062  5.16147  0.04033  -0.06564  -0.19834  -0.80638  -1.00472  1.71536  0.13194  1.61309  -0.03921  0.65799  1974  -2.10599  -3.30126  0.02096  0.52592  0.78894  -3.55585  -2.76691  -0.55100  -0.20618  2.79465  0.16055  1.21729  1975  -0.80101  -0.05185  -0.00123  -0.78528  -0.92131  0.26167  -0.65964  -2.72388  -0.05333  2.24790  0.14923  1.07707  1976  -0.91568  1.70858  0.01237  -0.09118  -1.09543  0.53152  -0.56391  -1.86509  -0.00081  0.26413  -0.09106  -0.28872  1977  -0.04042  0.28968  0.00604  -0.11593  -1.65183  1.48929  -0.16254  -0.97772  -0.00704  0.68083  -0.00065  0.24692  1978  4.61772  3.07762  -0.00407  -0.19174  -1.53785  2.47558  0.93773  -0.40146  0.01203  0.98133  -0.02657  0.23283  1979  6.24388  5.79355  0.02299  0.04859  0.62635  -2.77360  -2.14725  1.32970  0.03592  1.13219  -0.04378  0.07197  1980  -2.46584  -0.23679  -0.00283  0.18431  0.19778  -3.56576  -3.36798  -0.06985  -0.08939  0.97508  0.09021  0.05138  1981  -0.47804  -0.75036  -0.01721  0.03119  -0.20801  0.62034  0.41233  -0.96505  -0.03501  0.90058  0.04771  -0.10223  1982  0.91102  0.50508  -0.00083  -0.10075  0.20521  -0.02997  0.17524  -0.67400  -0.00256  0.94956  0.00435  0.05493  1983  -1.76589  -1.32994  -0.00522  -0.04176  -0.83239  1.12161  0.28922  -0.72845  0.00494  0.30348  -0.02194  -0.23621  1984  3.82543  3.21652  -0.00034  -0.06847  -0.12652  0.71447  0.58795  -0.70254  -0.00412  0.89396  -0.02571  -0.07181  1985  3.34933  2.14525  0.00298  0.13122  -0.83450  1.04168  0.20719  -0.81838  -0.02988  1.58949  0.00965  0.11181  1986  2.04506  0.05804  0.00398  -0.25710  -2.03370  3.88014  1.84644  -0.79428  0.00312  1.32266  0.00234  -0.14014  1987  2.73782  1.58341  0.00412  -0.13556  -0.54988  0.19763  -0.21337  1988  4.22853  3.25248  0.00553  -0.10609  -0.24517  0.28680  0.04163  0.20014  0.00079  0.99066  -0.01045  -0.14616  1989  4.86549  3.54667  0.00518  0.01110  0.18092  -0.45688  -0.27596  0.45343  -0.00443  1.25601  -0.00436  -0.12214  1990  3.88612  2.57180  0.02425  0.02607  -0.08930  -0.44836  -0.53766  0.13151  -0.01065  1.61561  0.02519  0.04000  1991  1.54626  0.35054  0.00875  -0.06525  -0.54072  0.67575  0.13503  -0.27589  -0.00826  1.48972  0.01028  -0.09865  0.74751  -0.00167  1.37929  -0.02728  -0.04875  1992  0.97480  -0.83497  -0.00305  -0.05479  -0.46585  0.63862  0.17277  -0.41528  0.00096  1.83890  0.01108  0.25918  1993  -0.80061  -1.92132  -0.00987  -0.04899  -0.87092  0.79541  -0.07552  -0.30133  0.00918  1.28832  0.00345  0.25547  1994  0.53597  0.11707  0.00443  -0.08087  -0.37287  0.39526  0.02239  -0.39280  0.00852  0.71471  -0.01116  0.15367  1995  0.62974  0.52496  0.00554  -0.04283  -0.19063  0.15194  -0.03869  -0.11381  -0.00090  0.30076  -0.01266  0.00738  1996  1.51755  1.53595  0.00386  -0.03194  0.39919  -0.63814  -0.23895  -0.46359  -0.00133  0.59592  0.01089  0.10674  1997  0.93056  0.54779  0.00482  0.01910  0.09015  -0.51950  -0.42934  -0.40024  -0.00896  1.01264  0.01890  0.16585  1998  -2.32846  -3.52030  0.00747  -0.00915  0.23108  0.24313  0.47422  -0.49102  -0.00100  0.99101  0.01754  0.20277  1999  -0.50530  -0.55889  -0.00740  -0.03375  -1.12645  0.86508  -0.26138  -0.52810  -0.00355  0.77766  0.00263  0.10747  2000  -0.01529  0.69245  -0.00019  0.00444  -0.46803  -0.23583  -0.70386  0.08725  -0.00511  -0.00730  -0.02959  -0.05339  2001  0.42896  -0.40427  0.00900  0.00880  0.44108  -0.36885  0.07223  -0.56832  0.01433  1.09798  0.00850  0.19071  2002  1.62364  0.78536  -0.01750  0.00953  0.06361  -0.05021  0.01340  -0.22275  -0.00316  0.95067  -0.01162  0.11970  2003  0.59777  0.53901  -0.00713  -0.02316  -0.37656  0.02356  -0.35300  -0.09819  -0.00613  0.45800  -0.00841  0.09678  2004  2.61774  2.66161  0.00727  0.01929  0.03238  -0.57112  -0.53874  0.26765  -0.00546  0.28555  -0.01758  -0.06185  2005  2.69552  2.91441  0.02956  0.05311  0.46561  -1.43705  -0.97143  0.18762  0.00639  0.48674  -0.01571  0.00481  2006  2.76460  2.28553  0.03999  0.08351  1.01583  -1.59688  -0.58104  0.48778  0.02672  0.38950  0.02927  0.00336  Average  γt/rt-1  tt/rt-1  aNt/rt-1  aGt/rt-1  aXt/rt-1  aMt/rt-1  aXMt/rt-1  aIt/rt-1  aIVt/rt-1  b Kt/rt-1  b KIVt/rt-1  b LDt/rt-1  1956-2006  3.27008  2.60627  0.01235  -0.05800  -0.35132  0.18932  -0.16200  -0.38418  -0.03620  1.20032  0.04493  0.04659  1956-1973  6.96794  5.84035  0.02653  -0.10163  -0.45098  0.49688  0.04590  -0.38789  -0.08223  1.57046  0.11378  -0.05733  1974-1979  1.16642  1.25272  0.00951  -0.10160  -0.63185  -0.26190  -0.89375  -0.86491  -0.03657  1.35017  0.02462  0.42623  1980-1990  1.92173  1.32383  0.00178  -0.02962  -0.39414  0.35560  -0.03854  -0.38008  -0.01535  1.10694  0.00816  -0.05539  1991-2001  0.26493  -0.31555  0.00212  -0.03047  -0.26127  0.18208  -0.07919  -0.35119  0.00035  0.91821  0.00271  0.11793  2002-2006  2.05985  1.83719  0.01044  0.02846  0.24018  -0.72634  -0.48616  0.12442  0.00367  0.51409  -0.00481  0.03256  Looking at Table 3-1, it can be seen that there are five different periods: the rapid economic growth for 19551973, the slowdown of economic growth between two oil shocks for 19741979, the revival of steady economic growth for 19801990, the long recession for 19912001 and the modest economic recovery for 20022006. Over the 52 period, real income per unit labour ρt grew on average by 3.27008% annually. Productivity growth τt contributed the most to the overall annual growth in real income per unit labour (2.60627%), the growth in the capital services βKt contributed the second largest amount to the overall annual growth in real income per unit labour (1.20032%), 70  declining real import prices αMt contributed 0.18932% per year and the changes in land services βLDt contributed 0.04659% per year. The largest negative contributing factor to the growth of real income per unit labour over the sample period was the fall in real investment prices αIt (0.38418% per year) while falls in real export prices αXt contributed 0.35132% per year. The remaining contributions were very small. Thus, the effect of changes in the terms of trade on real income per unit labour αXMt was very small for Japan on average over the entire period 19552006: an overall negative contribution of 0.162% per year. We can see that the importance of the contribution of capital services growth relative to the contribution of productivity growth increases over time. The average contribution after 1973 of capital services growth βKt (1.1964% per year) was larger than that of productivity growth τt (0.8368% per year). However, this is not the end of story. During the period of economic recovery 2002-2006, the average contribution of productivity growth τt of 1.83719% became bigger than the average contribution of capital services growth βKt, which was 0.51409%. Thus we can observe that the recent increase of real income per unit labour was mostly boosted by productivity growth τt rather than by capital deepening βKt. The annual change information in the previous table can be converted into cumulative changes using equations (3.47)(3.49). The difference between the current level of net real income per unit labour and its level in 1955, ρt – ρ1955 is decomposed into the sum of the level of productivity factor Tt, the levels of several real output price factors ACt, ANt, AGt, AXt, AMt, AIt and AIVt, and the levels of several input quantity factors BKt, BKIVt, BLDt and BLBt. The cumulative effect of changes in the terms of trade is ATt and is simply the product of the levels of real export and import prices AXt and AMt. Following Table 3-2 and Figure 3-1 give this cumulative growth information.  71  Table 3-2: Decomposition of the Cumulative Change in Real Income Per Unit Labour (in 1955 yen) rt - r1955 0.08816 0.14765 0.19026 0.39458 0.66536 0.98356 1.00309 1.18226 1.51073 1.52012 1.81106 2.22466 2.69616 3.28880 3.64780 3.69297 4.09282 4.59157 4.45314 4.40159 4.34314 4.34059 4.63253 5.04551 4.87223 4.83947 4.90161 4.78006 5.03872 5.27386 5.42224 5.62495 5.94659 6.33234 6.65543 6.78898 6.87448 6.80358 6.85066 6.90628 7.04116 7.12513 6.91308 6.86813 6.86678 6.90474 7.04902 7.10300 7.34081 7.59210 7.85678  Tt 0.12211 0.21684 0.24678 0.46414 0.75539 1.01668 1.04714 1.22079 1.50045 1.50566 1.73951 2.03505 2.41084 2.79979 3.04748 2.98340 3.24275 3.55628 3.33928 3.33595 3.44501 3.46333 3.65791 4.04110 4.02446 3.97303 4.00749 3.91594 4.13344 4.28404 4.28825 4.40548 4.65289 4.93408 5.14790 5.17817 5.10494 4.93478 4.94507 4.99144 5.12795 5.17738 4.85679 4.80708 4.86836 4.83259 4.90238 4.95105 5.19285 5.46455 5.68336  ANt  AGt  AXt  AMt  AXMt  AIt  AIVt  BKt  BKIVt  BLDt  0.00131 0.00200 0.00208 0.00221 0.00282 0.00313 0.00312 0.00295 0.00467 0.00415 0.00504 0.00629 0.00670 0.00822 0.01140 0.01377 0.01685 0.01929 0.02067 0.02059 0.02138 0.02176 0.02151 0.02303 0.02283 0.02165 0.02159 0.02123 0.02121 0.02142 0.02171 0.02201 0.02243 0.02284 0.02486 0.02562 0.02535 0.02448 0.02486 0.02535 0.02570 0.02613 0.02681 0.02615 0.02614 0.02693 0.02538 0.02474 0.02540 0.02815 0.03198  0.00222 0.00290 0.00017 -0.00167 -0.00372 -0.00682 -0.01246 -0.01974 -0.02341 -0.02987 -0.03202 -0.03241 -0.03666 -0.04066 -0.04705 -0.05786 -0.06840 -0.07239 -0.03782 -0.08835 -0.09417 -0.10150 -0.11362 -0.11041 -0.09746 -0.09532 -0.10219 -0.10507 -0.10970 -0.10049 -0.11914 -0.12918 -0.13725 -0.13637 -0.13420 -0.13984 -0.14464 -0.14898 -0.15608 -0.15987 -0.16271 -0.16098 -0.16182 -0.16482 -0.16442 -0.16364 -0.16280 -0.16489 -0.16314 -0.15819 -0.15019  0.00816 0.00115 -0.01649 -0.01941 -0.01995 -0.03966 -0.06950 -0.08557 -0.09134 -0.12370 -0.14201 -0.15626 -0.18116 -0.19463 -0.21882 -0.24998 -0.29637 -0.30842 -0.25656 -0.31584 -0.38577 -0.49024 -0.58747 -0.54604 -0.53214 -0.54640 -0.53240 -0.58970 -0.59825 -0.65684 -0.80439 -0.84511 -0.86375 -0.84941 -0.85684 -0.90354 -0.94440 -1.02153 -1.05428 -1.07112 -1.03564 -1.02751 -1.00646 -1.10666 -1.14808 -1.10905 -1.10340 -1.13740 -1.13446 -1.09105 -0.99380  -0.01537 -0.01697 0.02758 0.03861 0.04687 0.06326 0.09215 0.11089 0.11586 0.14789 0.16130 0.18278 0.20788 0.21117 0.24002 0.30216 0.36414 0.31516 0.08142 0.09826 0.13219 0.22639 0.38290 0.19945 -0.05113 -0.00861 -0.01065 0.06655 0.11486 0.18799 0.46952 0.52486 0.54668 0.51045 0.47318 0.53154 0.58755 0.65800 0.69272 0.70614 0.64942 0.60255 0.62469 0.70164 0.68077 0.64813 0.64367 0.64579 0.59391 0.45994 0.30706  -0.00721 -0.01583 0.01109 0.01921 0.02693 0.02361 0.02265 0.02531 0.02452 0.02419 0.01929 0.02652 0.02671 0.01653 0.02120 0.05218 0.06777 0.00674 -0.17514 -0.21758 -0.25358 -0.26386 -0.20457 -0.34659 -0.58327 -0.55501 -0.54305 -0.52315 -0.48339 -0.46885 -0.33488 -0.32024 -0.31708 -0.33896 -0.38366 -0.37200 -0.35684 -0.36353 -0.36156 -0.36498 -0.38622 -0.42496 -0.38177 -0.40502 -0.46731 -0.46092 -0.45973 -0.49161 -0.54055 -0.63111 -0.68674  0.03748 0.05291 0.03487 0.01296 0.00678 0.01886 -0.02890 -0.09978 -0.11239 -0.17573 -0.18107 -0.16992 -0.20100 -0.19239 -0.23910 -0.33371 -0.36189 -0.25769 -0.29391 -0.46919 -0.58824 -0.65008 -0.67546 -0.58751 -0.59242 -0.65857 -0.70454 -0.75468 -0.80219 -0.85964 -0.91727 -0.93307 -0.91784 -0.88189 -0.87096 -0.89479 -0.93121 -0.95790 -0.99240 -1.00246 -1.04366 -1.07977 -1.12449 -1.17146 -1.16374 -1.21403 -1.23382 -1.24269 -1.21837 -1.20088 -1.15419  0.00625 0.00316 0.00225 0.00161 -0.00223 -0.01339 -0.02146 -0.02702 -0.02857 -0.03270 -0.03544 -0.03976 -0.04433 -0.04339 -0.05123 -0.05621 -0.05503 -0.04702 -0.06057 -0.06400 -0.06405 -0.06450 -0.06374 -0.06136 -0.06764 -0.07004 -0.07022 -0.06988 -0.07015 -0.07225 -0.07203 -0.07215 -0.07209 -0.07244 -0.07333 -0.07404 -0.07396 -0.07314 -0.07239 -0.07247 -0.07259 -0.07340 -0.07349 -0.07381 -0.07426 -0.07299 -0.07327 -0.07383 -0.07432 -0.07373 -0.07117  -0.04314 -0.05995 -0.04399 -0.02929 -0.01850 0.04605 0.10568 0.18329 0.24911 0.32180 0.36977 0.43601 0.54642 0.68193 0.82569 0.98593 1.11872 1.21671 1.40041 1.54506 1.56192 1.60498 1.66702 1.74191 1.81043 1.87215 1.93692 1.95781 2.01826 2.12985 2.22581 2.32793 2.40329 2.50287 2.63719 2.76586 2.92714 3.04124 3.10403 3.13059 3.18355 3.27492 3.36517 3.43434 3.43370 3.53085 3.61533 3.65669 3.68263 3.72801 3.76530  -0.00090 0.00012 0.00290 0.00209 0.00338 0.00815 0.01621 0.01998 0.02477 0.03003 0.03140 0.03551 0.04700 0.05891 0.06945 0.07888 0.07961 0.07723 0.08778 0.09738 0.09157 0.09153 0.08985 0.08695 0.09329 0.09656 0.09686 0.09535 0.09361 0.09429 0.09446 0.09244 0.09164 0.09130 0.09339 0.09428 0.09525 0.09556 0.09458 0.09346 0.09443 0.09613 0.09773 0.09796 0.09535 0.09610 0.09506 0.09431 0.09271 0.09124 0.09405  -0.02996 -0.05449 -0.06590 -0.07669 -0.10549 -0.11271 -0.12890 -0.12353 -0.12842 -0.12742 -0.10543 -0.07264 -0.05952 -0.00014 0.00996 0.02659 0.05244 0.09241 0.17243 0.24174 0.22331 0.23892 0.25364 0.25840 0.26201 0.25501 0.25875 0.24250 0.23764 0.24549 0.23532 0.23171 0.22059 0.21091 0.21424 0.20572 0.22845 0.25107 0.26457 0.26522 0.27471 0.28968 0.30814 0.31770 0.31298 0.32985 0.34049 0.34923 0.34361 0.34406 0.34438  8 7 6 5 4 3 2 1 0 -1  Real income per unit of labour Terms of trade Capital services  2006  2004  2002  2000  1998  1996  1994  1992  1990  1988  1986  1984  1982  1980  1978  1976  1974  1972  1970  1968  1966  1964  1962  1960  1958  -2 1956  1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  Productivity growth Investments Land services  Figure 3-1: Real Income Change Per Unit Labour (in 1955 yen) 72  Over the 52 year period, real income per unit labour grew by 7.85678 yen at the prices of 1955. From Table 3-2, it can be seen that productivity growth contributed the most to the overall growth in real income per unit labour (T2006 = 5.68336 yen, 72.34 % of the overall change in real income per unit labour), the growth in capital services made the second largest contribution (BK2006 = 3.7653 yen, 47.92 % of the overall change in real income per unit labour), the change in real investment prices made the third largest contribution in magnitude, (AI2006 = 1.15419 yen, 14.69 % of the overall change in real income per unit labour), the change in terms of trade made the forth largest contribution (AXM2006 = 0.68674 yen, 8.74 % of the overall change in real income per unit labour) followed closely by the growth in land services (BLD2006 = 0.34438 yen, 4.38 % of the overall change in real income per unit labour). The change in the real price of the consumption of NPISHs and the growth in inventory services have very small impact on the growth in the real income per unit labour (less than one percent of the overall change in real income per unit labour). Figure 3-1 plots real income per unit labour and main factors contributing to its growth. 3.5.3 The Decomposition of Net Real Income Growth into Explanatory Factors In the previous subsection 3.5.2, we focus on the real income per unit labour. Deflating the income of the market sector by the price of household consumption, we obtain the (gross) real income. Real income captures how much consumption people can purchase for their income. Since economic welfare comes from consumption, while real GDP is the measure of output, real income is the measure of welfare. However, it is well known that net real income is the better measure of welfare, because it captures the sustainable level of welfare. Net income is the gross income net of the value of depreciated assets in the production period. By deducting depreciation from the income, we come closer to a measure of income that could be consumed in the present period without impairing production possibilities in future period. Deflating the net income of the market sector by the price of household consumption, we obtain the net real income. This subsection analyses the net real income per unit labour. By applying our methodology to the Japanese economy, we can decompose the change in net real income per unit labour into explanatory factors. In this section, we consider the production model based on net output concept. We illustrate the theory by considering a very simple two output, two input model of the market sector. One of the outputs is output in year t, Yt and the other output is an 73  investment good, It. One of the inputs is the flow of noncapital primary input Xt and the other input is Kt, capital services. Suppose that the average prices during period t of a unit of Yt, Xt and It are PYt, PXt and PIt respectively. Suppose further that the real interest rate prevailing at the beginning of period t is rt*. The value of the beginning of period t capital stock is assumed to be PIt, the investment price for period t. The user cost of capital is calculated such as ut = (rt* + t + t)PIt/(1 + rt*). As usual, it represents price of capital services input. Thus, the period t profit of the market sector is expressed as follows:84 (3.59) t = PYt Yt + PIt It  PXt Xt  [(rt* + )PIt/(1 + rt*)]Kt Under the assumption of constant returns to scale, a zero profit condition should be satisfied such as t = 0. Using this condition, we obtain the following value of output equals value of input equation: (3.60) PYt Yt + PIt It = PXt Xt + [(rt* + )PIt/(1 + rt*)]Kt. Equation (3.60) is essentially the closed economy counterpart to the (gross) value of outputs equals (gross) value of primary inputs equation (3.4), PtYt = WtXt. The (gross) payment to primary inputs that is defined by the right hand side of (3.60) is not income, in the sense of Hicks. Net income in this chapter is the same concept as Hicks’ third concept of income: “Income No. 3 must be defined as the maximum amount of money which the individual can spend this week, and still be able to expect to spend this week, and still be able to expect to spend the same amount in real terms in each ensuing week.” (Hicks, 1946). The owner of a unit of capital cannot spend the entire period t gross rental income (rt* + )PIt/(1 + rt*) on consumption during period t because the depreciation portion of the rental, PIt/(1 + rt*), is required in order to keep his or her capital intact. Thus the owner of a new unit of capital at the beginning of period t loans the unit to the market sector and gets the gross return (rt* + )PIt at the end of the period plus the depreciated unit of the initial capital, which is worth only (1  )PIt. Thus PIt of this gross return must be set aside in order to restore the lender of the capital services to his or her original wealth position at the beginning of period t. This means that period t Hicksian market sector 84  We have temporarily neglected tax factors for the sake of simplicity.  74  income is not the value of payments to primary inputs, PXt Xt + [(rt* + )PIt/(1 + rt*)]Kt; instead it is the value of payments to labour PXt Xt plus the reward for waiting, [rt*PIt/(1 + rt*)]Kt. Using this definition of market sector net income, we can rearrange equation (3.60) as follows: (3.61) Hicksian market sector income  PXt Xt + [rt*PIt/(1 + rt*)]Kt = PYt Yt + PIt It – [PIt/(1 + rt*)]Kt = Value of consumption + value of gross investment  value of depreciation. Thus in this net income framework, our new output concept is equal to our old output concept less the value of depreciation. Hence the overstatement of income problem that is implicit in the approaches used in previous subsections can readily be remedied: all we need to do is to take the user cost formula for an asset and decompose it into two parts:    One part that represents depreciation and foreseen obsolescence, PIt/(1 + rt*) and The remaining part that is the reward for postponing consumption, rt*PIt/(1 + rt*).  Thus, in this subsection, we split up each user cost times the beginning of the period stock Kt into the depreciation component [tPIt /(1 + rt*)]Kt and the remaining term [rt*PIt /(1 + rt*)]Kt. The first term is considered as an intermediate input cost for the market sector and is an offset to gross investment made by the market sector during the period under consideration. We regard the second term as a genuine income component and call it waiting capital services. We take the price of depreciation PDEPt to be the corresponding investment price and the quantity of depreciation YDEPt is taken to be the depreciation rate times the beginning of the period stock such as PDEPt=PIt/(1 + rt*) and YDEPt=ntKt. We take the price of waiting capital services WKWt to be the corresponding investment price times the real rate of return and the quantity of waiting capital services XKWt is taken to be the beginning of the period stock such as WKWt=rt*PIt/(1 + rt*), and XKWt=Kt.85  85  These data have already been constructed in Chapter 2.  75  Table 3-3: Decomposition of Growth Rate of Net Real Income Per Unit Labour (%) t  t-1  γ /r  t  t-1  t /r  t  t-1  aN /r  1956  5.16234  7.09905  0.07684  1957  3.42361  5.31072  1958  2.15837  1.71104  1959  10.90201  1960 1961 1962  t  t-1  aG /r  t  t-1  aX /r  t  t-1  t  aM /r  t-1  t  t-1  t-1  t  t-1  t  t-1  t  t-1  t  t-1  aIV /r  b KW /r  b KIV /r  -1.34163  0.36516  -1.13236  -0.05289  -1.75163  -0.47904  0.85802  -0.31773  -0.17171  -0.54500  0.05672  -1.36380  1.44694  -0.96964  0.38901  -0.04875  0.23544  0.14976  -0.61336  0.42688  -1.15297  0.66740  -0.03353  0.25145  -0.04296  -0.56757  0.39202  0.36647  -0.29321  0.33740  -0.18251  0.08207  0.06134  -1.36637  0.68878  -0.13964  0.50777  0.18408  -0.46884  0.98403  0.20048  -0.30365  1.07994  -0.03566  -1.78551  0.50639  -0.30155  1.01286  0.30139  0.47728  -0.89871  -0.42143  0.03815  0.03728  -0.38997  -0.08907  0.00428  -0.14635  -0.94794  2.39487  11.44312  0.00709  -0.09691  -0.15368  0.58056  12.89222  13.95568  0.02860  -0.09726  -0.02555  12.41574  11.56860  0.01319  -0.13028  -0.82842  -0.05697  1.06168  -0.00041  -0.21100  -1.11560  aI /r  t  aDEP /r  2.19117  aXM /r  0.13006  b LD /r  -0.60518  1963  6.07559  6.47743  -0.00631  -0.27214  -0.60141  0.70094  0.09953  -2.65175  0.90162  -0.20820  1.39313  0.14119  0.20109  1964  10.59010  9.83337  0.06050  -0.12943  -0.20317  0.17517  -0.02799  -0.44457  0.22028  -0.05445  1.13624  0.16883  -0.17266  1965  -0.29847  0.13606  -0.01634  -0.20590  -1.03213  1.02149  -0.01064  -2.01958  0.62043  -0.13192  1.12983  0.16762  0.03196  1966  8.80586  7.47759  0.02834  -0.06895  -0.58548  0.42902  -0.15646  -0.17101  0.29033  -0.08750  0.74617  0.04399  0.70337  1967  11.19742  8.56801  0.03677  -0.01134  -0.41889  0.63129  0.21240  0.32795  0.07765  -0.12706  1.02863  0.12058  0.96383  1968  11.35070  9.90224  0.01085  -0.11232  -0.65829  0.66333  0.00504  -0.82172  0.34066  -0.12065  1.49602  0.30370  0.34688  1969  12.40148  9.17208  0.03611  -0.09503  -0.31975  0.07815  -0.24160  0.20451  -0.03339  0.02215  1.64448  0.28275  1.40943  1970  6.28300  5.16961  0.06701  -0.13498  -0.51082  0.60941  0.09860  -0.98664  0.28873  -0.16545  1.51001  0.22264  1971  -0.44911  -1.37424  0.04722  -0.21479  -0.61928  1.23479  0.61551  -1.87996  0.44914  -0.09906  1.48908  0.18754  0.33046  1972  6.83319  5.11441  0.06134  -0.21046  -0.92597  1.23728  0.31130  -0.56265  0.36379  0.02355  1.20144  0.01443  0.51604  0.21348  1973  8.03421  5.79734  0.04577  -0.07450  -0.22512  -0.91524  -1.14036  1.94693  -0.17785  0.14975  0.78480  -0.04450  0.74682  1974  -4.04094  -3.71652  0.02382  0.59790  0.89691  -4.04248  -3.14557  -0.62641  0.17587  -0.23439  1.31797  0.18253  1.38388  1975  -1.38038  -0.10221  -0.00142  -0.91075  -1.06851  0.30348  -0.76504  -3.15909  1.11363  -0.06186  1.08412  0.17308  1.24916  1976  -0.14292  1.96380  0.01443  -0.10636  -1.27792  0.62007  -0.65785  -2.17580  1.07874  -0.00094  0.18410  -0.10623  -0.33681  1977  -0.05550  0.30838  0.00699  -0.13420  -1.91210  1.72395  -0.18815  -1.13178  0.44191  -0.00814  0.36441  -0.00075  0.28582  1978  5.07852  3.52999  -0.00471  -0.22198  -1.78042  2.86607  1.08565  -0.46479  0.40251  0.01393  0.49912  -0.03076  0.26956  1.53269  -0.21115  0.04141  0.53332  -0.05046  1979  6.22923  6.69302  0.02650  0.05600  0.72197  -3.19702  -2.47505  1980  -3.28620  -0.29272  -0.00327  0.21248  0.22800  -4.11068  -3.88267  -0.08052  0.22556  -0.10305  0.47476  0.10400  0.05923  1981  -0.76339  -0.89638  -0.02001  0.03627  -0.24183  0.72121  0.47938  -1.12197  0.44163  -0.04070  0.42176  0.05547  -0.11885  1982  0.58638  0.56313  -0.00097  -0.11747  0.23926  -0.03494  0.20432  -0.78584  0.24192  -0.00298  0.41516  0.00508  0.06404  1983  -2.05034  -1.61699  -0.00610  -0.04885  -0.97365  1.31195  0.33830  -0.85207  0.32618  0.00578  0.10538  -0.02567  -0.27630  1984  4.24524  3.74457  -0.00040  -0.08033  -0.14842  0.83815  0.68972  -0.82416  0.45662  -0.00484  0.37844  -0.03016  -0.08424  1985  3.13115  2.52050  0.00348  0.15332  -0.97501  1.21708  0.24207  -0.95617  0.42739  -0.03491  0.63356  0.01127  0.13064  1986  1.71035  0.03128  0.00466  -0.30103  -2.38115  4.54306  2.16191  -0.92998  0.42064  0.00365  0.48056  0.00274  -0.16408  1987  2.27648  1.81145  0.00484  -0.15924  -0.64595  0.23215  -0.25064  0.16284  -0.00196  0.56634  -0.03204  -0.05727  1988  4.15239  3.84301  0.00653  -0.12519  -0.28930  0.33842  0.04912  0.23617  -0.02917  0.00093  0.35578  -0.01234  -0.17247  1989  4.59268  4.16851  0.00611  0.01310  0.21364  -0.53952  -0.32587  0.53544  -0.08498  -0.00523  0.43499  -0.00515  -0.14423  1990  3.28436  3.02015  0.02871  0.03086  -0.10573  -0.53083  -0.63656  0.15570  0.03319  -0.01261  0.58774  0.02982  0.04735  1991  0.76672  0.38625  0.01042  -0.07770  -0.64391  0.80471  0.16080  -0.32854  0.16323  -0.00984  0.56734  0.01224  -0.11747  0.87810  0.08295  1992  0.06414  -0.94522  -0.00366  -0.06576  -0.55905  0.76638  0.20733  -0.49836  0.34787  0.00115  0.69645  0.01330  0.31103  1993  -1.89217  -2.39426  -0.01195  -0.05932  -1.05467  0.96322  -0.09145  -0.36490  0.17705  0.01111  0.52801  0.00418  0.30937  1994  0.46194  0.16668  0.00542  -0.09902  -0.45656  0.48398  0.02742  -0.48096  0.33154  0.01044  0.32593  -0.01367  0.18816  1995  0.74731  0.64555  0.00679  -0.05249  -0.23359  0.18617  -0.04741  -0.13945  0.16285  -0.00110  0.17906  -0.01552  0.00904  1996  1.77005  1.85391  0.00473  -0.03909  0.48858  -0.78103  -0.29245  -0.56740  0.39290  -0.00163  0.27511  0.01333  0.13065  0.11007  -0.63425  1997  0.51581  0.70358  -0.52418  -0.48864  0.22306  -0.01094  0.35816  0.02308  0.20248  1998  -3.77675  -4.38280  0.00916  -0.01121  0.28329  0.29806  0.58135  -0.60195  0.01909  -0.00122  0.34076  0.02150  0.24858  1999  -0.91622  -0.70282  -0.00920  -0.04199  -1.40173  1.07648  -0.32525  -0.65716  0.43183  -0.00441  0.25578  0.00327  0.13373  2000  0.05146  0.91206  -0.00024  0.00555  -0.58482  -0.29467  -0.87950  0.10902  0.07476  -0.00639  -0.06015  -0.03697  -0.06671  2001  -0.11205  -0.54598  0.01124  0.01099  0.55078  -0.46058  0.09020  -0.70966  0.41341  0.01789  0.35110  0.01061  0.23814  2002  1.42882  0.97646  -0.02197  0.01196  0.07986  -0.06304  0.01682  -0.27966  0.27758  -0.00396  0.31590  -0.01459  0.15028  2003  0.61393  0.73102  -0.00897  -0.02914  -0.47366  0.02963  -0.44403  -0.12351  0.23225  -0.00772  0.15285  -0.01058  0.12174  2004  2.87675  3.39291  0.00914  0.02427  0.04073  -0.71828  -0.67756  0.33661  -0.16542  -0.00686  0.06357  -0.02211  -0.07779  2005  2.78109  3.69571  0.03708  0.06663  0.58412  -1.80279  -1.21867  0.23537  -0.18021  0.00802  0.15082  -0.01970  0.00603  2006  2.59345  2.98263  0.05012  0.10467  1.27331  -2.00163  -0.72832  0.61141  -0.51884  0.03349  0.01738  0.03669  0.00421  Average  t  t-1  γ /r  t  t-1  t /r  0.00588  t  t-1  aN /r  0.02332  t  t-1  aG /r  t  t-1  aX /r  t  t-1  t  t-1  t  t-1  t-1  t-1  t-1  t  t-1  t  t-1  b KIV /r  b LD /r  0.23084  -0.04036  0.54517  0.05016  0.05351  0.55633  0.05166  -0.42794  0.20924  -0.09114  0.80269  0.12681  -0.07116  -0.28766  -1.02434  -1.00420  0.50025  -0.04167  0.66384  0.02790  0.48909  -0.04074  -0.44310  0.23835  -0.01781  0.44132  0.00937  -0.06511  0.21895  -0.09938  -0.42982  0.24887  0.00046  0.34705  0.00321  0.14427  -0.91122  -0.61035  0.15604  -0.07093  0.00459  0.14010  -0.00606  0.04090  0.01416  -0.06624  -0.40356  1956-1973  7.09563  6.57910  0.02994  -0.11357  -0.50468  1974-1979  0.94800  1.44608  0.01094  -0.11990  -0.73668  1980-1990  1.62537  1.53605  0.00214  -0.03510  -0.46183  0.42109  1991-2001  -0.21089  -0.39119  0.00260  -0.03697  -0.31833  2002-2006  2.05881  2.35575  0.01308  0.03568  0.30087  aIV /r  t  b KW /r  -0.44215  2.97005  aDEP /r  t  aXM /r  -0.19234  3.12280  aI /r  t  aM /r  0.21122  1956-2006  Thus in the net production model of this subsection, we add depreciations to the original list of net outputs and use waiting capital services instead of capital services among the original list of primary inputs. Substituting the estimates defined in (3.32), (3.41) and (3.45) into equation (3.52), we can decompose the growth rate of real income per unit labour (ρt – ρt-1)/ρt-1 = γt/ρt-1 into the contribution of technical progress τt/ρt-1 changes in real output prices αCt/ρt-1, Nt/t1, Gt/t1, Xt/t1, Mt/t1, It/t1, DEPt/t1 and  76  IVt/t1,86 and growth in relative input quantity changes KWt /t1, KIVt/t1, LDt/t1 and LBt/t1.87 The chain link information on period by period changes in net real income per unit labour input that corresponds to (3.52) is given in Table 3-3. The new results are quite interesting. While the average growth rate of real income per unit labour ρt was 3.27008% per year in the gross output model, the average growth rate of net real income per unit labour ρt has now decreased by 0.14728 percent points per year to 3.1228%. More importantly, there are some big shifts in the explanatory factors. Productivity growth τt now accounts for 2.97005% of the overall annual growth in net real income per unit labour compared to 2.60627% of the overall annual growth in real income per unit labour, an increase of 0.36378 percentage points per year. The growth in capital services βKt accounted for 1.20032% of the overall annual growth in real income per unit labour while the growth in waiting capital services βKWt accounts for 0.54517% of the overall annual growth in net real income per unit labour, a decrease of 0.65515 percentage points per year. The average contributions of changes in real export prices αXt and real import prices αMt remain quite similar estimates in the previous gross output model. Thus, as we stated in the previous analysis, the effect of changes in the terms of trade on living standards αTt was negligible for Japan on average over the entire periods 19552006: an overall negative contribution of -0.19234%. The negative contribution of the change in real investment prices αIt equal to 0.44215% was offset by the positive contribution of the change in real depreciation prices αDEPt equal to 0.23084%. Finally, we note that the productivity recovery in the period 2002-2006 is quite striking. Using the previous gross output model, the average contribution of productivity growth during this period was 1.83719% per year and using the current net output model, its average contribution increases to a very respectable 2.35575% per year.  86  Since we divided market sector nominal income by the price of consumption, Ct/t1 will be identically  equal to zero and hence it is not listed. 87  Since we divided market sector nominal income by the quantity of labour services, LBt /t1 will be  identically equal to zero and hence it is not listed.  77  Table 3-4: Decomposition of the Cumulative Change in Net Real Income Per Unit Labour (in 1955 yen) 1955  r -r 0.08830 0.14988 0.19004 0.39723 0.66896 0.96438 0.96286 1.12528 1.42559 1.41622 1.69156 2.07250 2.50189 3.02428 3.32177 3.29917 3.64148 4.07147 3.83783 3.76124 3.75342 3.75039 4.02772 4.38516 4.18485 4.13984 4.17415 4.05349 4.29819 4.48633 4.59231 4.73580 5.00347 5.31182 5.54246 5.59807 5.60275 5.46438 5.49752 5.55138 5.67992 5.71804 5.43749 5.37200 5.37564 5.36770 5.46883 5.51291 5.72071 5.92738 6.12546  t  AN  T 0.12143 0.21695 0.24879 0.46626 0.76041 1.03567 1.06407 1.23723 1.51608 1.52035 1.75415 2.04564 2.42023 2.80659 3.05136 2.98221 3.23842 3.54869 3.33380 3.32813 3.43559 3.45244 3.64520 4.02926 4.01142 3.95857 3.99152 3.89637 4.11220 4.26365 4.26559 4.37976 4.62749 4.90736 5.11944 5.14746 5.07838 4.90328 4.91524 4.96177 5.09640 5.14840 4.82282 4.77258 4.83718 4.79849 4.86760 4.92009 5.16517 5.43980 5.66761  t  AG  0.00131 0.00200 0.00208 0.00221 0.00282 0.00313 0.00312 0.00295 0.00467 0.00415 0.00504 0.00629 0.00670 0.00822 0.01140 0.01377 0.01685 0.01929 0.02067 0.02059 0.02138 0.02176 0.02151 0.02303 0.02283 0.02165 0.02159 0.02123 0.02121 0.02142 0.02171 0.02201 0.02243 0.02284 0.02486 0.02562 0.02535 0.02448 0.02486 0.02535 0.02570 0.02613 0.02681 0.02615 0.02614 0.02693 0.02538 0.02474 0.02540 0.02815 0.03198  t  t  0.00222 0.00290 0.00017 -0.00167 -0.00372 -0.00682 -0.01246 -0.01974 -0.02341 -0.02987 -0.03202 -0.03241 -0.03666 -0.04066 -0.04705 -0.05786 -0.06840 -0.07239 -0.03782 -0.08835 -0.09417 -0.10150 -0.11362 -0.11041 -0.09746 -0.09532 -0.10219 -0.10507 -0.10970 -0.10049 -0.11914 -0.12918 -0.13725 -0.13637 -0.13420 -0.13984 -0.14464 -0.14898 -0.15608 -0.15987 -0.16271 -0.16098 -0.16182 -0.16482 -0.16442 -0.16364 -0.16280 -0.16489 -0.16314 -0.15819 -0.15019  t  t  AX  AM  AXM  0.00816 0.00115 -0.01649 -0.01941 -0.01995 -0.03966 -0.06950 -0.08557 -0.09134 -0.12370 -0.14201 -0.15626 -0.18116 -0.19463 -0.21882 -0.24998 -0.29637 -0.30842 -0.25656 -0.31584 -0.38577 -0.49024 -0.58747 -0.54604 -0.53214 -0.54640 -0.53240 -0.58970 -0.59825 -0.65684 -0.80439 -0.84511 -0.86375 -0.84941 -0.85684 -0.90354 -0.94440 -1.02153 -1.05428 -1.07112 -1.03564 -1.02751 -1.00646 -1.10666 -1.14808 -1.10905 -1.10340 -1.13740 -1.13446 -1.09105 -0.99380  -0.01537 -0.01697 0.02758 0.03861 0.04687 0.06326 0.09215 0.11089 0.11586 0.14789 0.16130 0.18278 0.20788 0.21117 0.24002 0.30216 0.36414 0.31516 0.08142 0.09826 0.13219 0.22639 0.38290 0.19945 -0.05113 -0.00861 -0.01065 0.06655 0.11486 0.18799 0.46952 0.52486 0.54668 0.51045 0.47318 0.53154 0.58755 0.65800 0.69272 0.70614 0.64942 0.60255 0.62469 0.70164 0.68077 0.64813 0.64367 0.64579 0.59391 0.45994 0.30706  -0.00721 -0.01583 0.01109 0.01921 0.02693 0.02361 0.02265 0.02531 0.02452 0.02419 0.01929 0.02652 0.02671 0.01653 0.02120 0.05218 0.06777 0.00674 -0.17514 -0.21758 -0.25358 -0.26386 -0.20457 -0.34659 -0.58327 -0.55501 -0.54305 -0.52315 -0.48339 -0.46885 -0.33488 -0.32024 -0.31708 -0.33896 -0.38366 -0.37200 -0.35684 -0.36353 -0.36156 -0.36498 -0.38622 -0.42496 -0.38177 -0.40502 -0.46731 -0.46092 -0.45973 -0.49161 -0.54055 -0.63111 -0.68674  t  t  AI  ADEP  0.03748 0.05291 0.03487 0.01296 0.00678 0.01886 -0.02890 -0.09978 -0.11239 -0.17573 -0.18107 -0.16992 -0.20100 -0.19239 -0.23910 -0.33371 -0.36189 -0.25769 -0.29391 -0.46919 -0.58824 -0.65008 -0.67546 -0.58751 -0.59242 -0.65857 -0.70454 -0.75468 -0.80219 -0.85964 -0.91727 -0.93307 -0.91784 -0.88189 -0.87096 -0.89479 -0.93121 -0.95790 -0.99240 -1.00246 -1.04366 -1.07977 -1.12449 -1.17146 -1.16374 -1.21403 -1.23382 -1.24269 -1.21837 -1.20088 -1.15419  -0.02295 -0.02866 -0.02143 -0.00874 -0.00163 0.00275 0.01629 0.04040 0.04664 0.06610 0.07518 0.07782 0.09071 0.08930 0.10297 0.12557 0.14380 0.13428 0.14445 0.20623 0.26526 0.28941 0.31139 0.29927 0.31302 0.33906 0.35321 0.37240 0.39872 0.42440 0.45047 0.46073 0.45885 0.45315 0.45548 0.46732 0.49274 0.50569 0.52948 0.54121 0.56975 0.58623 0.58765 0.61852 0.62381 0.65311 0.67275 0.68943 0.67748 0.66409 0.62446  AIV  t  0.00625 0.00316 0.00225 0.00161 -0.00223 -0.01339 -0.02146 -0.02702 -0.02857 -0.03270 -0.03544 -0.03976 -0.04433 -0.04339 -0.05123 -0.05621 -0.05503 -0.04702 -0.06057 -0.06400 -0.06405 -0.06450 -0.06374 -0.06136 -0.06764 -0.07004 -0.07022 -0.06988 -0.07015 -0.07225 -0.07203 -0.07215 -0.07209 -0.07244 -0.07333 -0.07404 -0.07396 -0.07314 -0.07239 -0.07247 -0.07259 -0.07340 -0.07349 -0.07381 -0.07426 -0.07299 -0.07327 -0.07383 -0.07432 -0.07373 -0.07117  t  t  BKW  BKIV  -0.01937 -0.02917 -0.02479 -0.02001 -0.01828 0.00513 0.03222 0.06947 0.10169 0.13712 0.16045 0.19544 0.25204 0.32131 0.39280 0.46774 0.52793 0.56993 0.64613 0.70628 0.71636 0.73627 0.76352 0.79413 0.82307 0.84793 0.87222 0.87842 0.90023 0.93830 0.96808 1.00378 1.02671 1.05592 1.09719 1.13834 1.18924 1.22785 1.25124 1.26414 1.28412 1.31059 1.33590 1.35419 1.34993 1.37481 1.39717 1.40814 1.41273 1.42394 1.42527  -0.00090 0.00012 0.00290 0.00209 0.00338 0.00815 0.01621 0.01998 0.02477 0.03003 0.03140 0.03551 0.04700 0.05891 0.06945 0.07888 0.07961 0.07723 0.08778 0.09738 0.09157 0.09153 0.08985 0.08695 0.09329 0.09656 0.09686 0.09535 0.09361 0.09429 0.09446 0.09244 0.09164 0.09130 0.09339 0.09428 0.09525 0.09556 0.09458 0.09346 0.09443 0.09613 0.09773 0.09796 0.09535 0.09610 0.09506 0.09431 0.09271 0.09124 0.09405  7 6 5 4 3 2 1 0 -1  Net real income per unit of labour Terms of trade Depreciations  2006  2004  2002  2000  1998  1996  1994  1992  1990  1988  1986  1984  1982  1980  1978  1976  1974  1972  1970  1968  1966  1964  1962  1960  1956  -2 1958  t  1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  Productivity growth Investments Waiting capital services  Figure 3-2: Net Real Income Change Per Unit Labour (in 1955 yen) 78  t  BLD  -0.02996 -0.05449 -0.06590 -0.07669 -0.10549 -0.11271 -0.12890 -0.12353 -0.12842 -0.12742 -0.10543 -0.07264 -0.05952 -0.00014 0.00996 0.02659 0.05244 0.09241 0.17243 0.24174 0.22331 0.23892 0.25364 0.25840 0.26201 0.25501 0.25875 0.24250 0.23764 0.24549 0.23532 0.23171 0.22059 0.21091 0.21424 0.20572 0.22845 0.25107 0.26457 0.26522 0.27471 0.28968 0.30814 0.31770 0.31298 0.32985 0.34049 0.34923 0.34361 0.34406 0.34438  The annual change information in the previous table can be converted into cumulative changes using equations (3.47)(3.49). The difference between the current level of net real income per unit labour and its level in 1955, ρt – ρ1955 is decomposed into the sum of the level of productivity factor Tt, the levels of several real output price factors ACt, ANt, AGt, AXt, AMt, AIt, ADEPt, and AIVt, and the levels of several input quantity factors BKWt, BKIVt, BLDt, and BLBt. Table 3-4 and Figure 3-2 give this cumulative growth information. Over the 52 year period, net real income per unit labour grew by about 6.12546 yen at the price of 1955. From the above Table 3-7, it can be seen that productivity growth contributed the most to the overall growth in net real income per unit labour (T2006 = 5.66761, 92.53% of the overall growth in net real income per unit labour) and the growth in waiting capital services made the next largest contribution (BKW2006 = 1.42527, 23.27% of the overall growth in net real income per unit labour) followed by the change in real investment price in magnitude (AI2006 = –1.15419, –18.84% of the overall growth in net real income per unit labour). There were smaller effects due to the changes in real output prices such as the contributions of the changes in real depreciation prices (ADEP2006 = 0.62446, 10.19% of the overall growth in net real income per unit labour), real export prices (AX2006 = –0.9938, –16.22% of the overall growth in net real income per unit labour), and real import prices (AM2006 = 0.30706, 5.01% of the overall growth in net real income per unit labour). The change in the real price of the consumption of NPISHs and the growth in inventory services had very small impact on the growth in net real income per unit labour (less than one percent of the overall change in real income per unit labour). Figure 3-2 plots net real income per unit labour and main factors contributing to its growth. 3.6 Conclusion In this chapter, we derived a decomposition for changes in real income per unit primary input into explanatory factors that is exact for a flexible functional form; i.e., we showed that the change in real income generated by the market sector per unit primary input is equal to the sum of a productivity growth term, plus terms due to changes in real output prices, plus terms due to changes in relative primary input quantities where all three sets of explanatory factors can be calculated using observable price and quantity data. The above difference approach can be converted into a growth rate approach. However, the present approach has an advantage over earlier exact Translog approaches in that the present approach allows individual prices and quantities to be zero. Our present 79  approach also allows value subaggregates (such as inventory change or net exports) to change sign from period to period. If prices or quantities are zero or a value aggregate changes sign, the Translog approach fails, so our present approach offers some clear advantages in these situations. We applied our methodology to analyze changes in the amount of real income per unit labour generated by the market sector of the Japanese economy for the years 1955-2006. The main findings emerging from this application is that, taken over the entire time period of 52 years, productivity growth and the growth of reproducible capital stocks and their resulting services are the two main contributors to the growth of real income per unit labour. We also observed that changes in the terms of trade had very small effects on real income per unit labour on average. We moved to our theoretically preferred measure of net real income per unit labour. We applied our methodology to analyze changes in the amount of net real income per unit labour generated by the market sector of the Japanese economy. We observed that in the net approach, productivity growth was still the largest contributor (and was an even more important factor than before). However, the contribution of capital services was greatly reduced.  80  Chapter 4.  Exact and Superlative Price and Quantity Indicators88  4.1 Introduction Traditional index number theory adopts a theoretical framework based on a ratio concept. In this approach, the ratio of the value aggregate between two periods is decomposed into the product of a price index and a quantity index. The price index, a function of the price and quantity data pertaining to the two periods under consideration, is interpreted as the ratio of the current price of the aggregate to the aggregate price in the base period. The quantity index, another function of the price and quantity data pertaining to the two periods, is interpreted as the ratio of the current period quantity aggregate to the base period quantity aggregate. In the economic approach to index number theory, it is assumed that the consumer has preferences over the individual quantities in the aggregate that can be represented by a utility function which has a dual cost function. This cost function is used to define consumer’s family of Konüs (1939) price indexes or true cost of living indexes and the consumer’s family of Allen (1949) quantity indexes. If the consumer’s preferences are homothetic (so that they can be represented by a linearly homogeneous utility function), then the family of Konüs price indexes collapses to a ratio of unit cost functions and the family of Allen quantity indexes collapses to a ratio of utility functions, where these functions are evaluated at the data of say period 1 in the numerator and the data of period 0 in the denominator. If preferences are homothetic, then Konüs and Byushgens (1926), Afriat (1972) and Pollak (1983) showed that certain numerical index number formula were exactly equal to the underlying theoretical economic indexes, provided that the consumer’s utility function or dual unit cost function had certain functional forms. Diewert (1976) took this theory of exact indexes one step further and looked for indexes that were exact for flexible functional forms, for either the linearly homogeneous utility function or for the dual unit cost function and he called such indexes that were exact for flexible functional forms superlative. However, empirically, it has been shown that consumer preferences are generally not homothetic and hence the relevance of Diewert’s concept of a superlative index is somewhat doubtful, at least in the consumer context. But Diewert (1976; 122) 88  It includes joint work with W. Erwin Diewert. A version of this chapter has been accepted for  publication. Diewert W.E. and Mizobuchi H. (2009) Exact and Superlative Price and Quantity Indicators. Macroeconomic Dynamics, Volume 13.  81  did implicitly develop a stronger concept for a superlative index in the context of general nonhomothetic preferences and we will formalize his idea in the present chapter in section 4.2 below where we will define strongly superlative indexes. Section 4.2 will also review the standard definitions for exact and superlative indexes in the case of homothetic preferences. In section 4.3, we switch from the traditional economic approach to index number theory, which is based on ratios, to an economic approach pioneered by Hicks (1942) (1943) (1945–46) which is based on differences. In the traditional approach to index number theory, a value ratio is decomposed into the product of a price index times a quantity index whereas in the difference approach, a value difference is decomposed into the sum of a price indicator (which is a measure of aggregate price change) plus a quantity indicator (which is a measure of aggregate quantity change). The difference analogue to a theoretical Konüs price index is a Hicksian price variation and the difference analogue to an Allen quantity index is a Hicksian quantity variation such as the equivalent or compensating variation. For normal index number theory, the theoretical Konüs and Allen indexes are defined using ratios of cost functions but in the difference approach to index number theory, the theoretical price and quantity variation functions are defined in terms of differences of cost functions. In the difference approach, the counterparts to price and quantity index number formulae are price and quantity indicator functions.89 Both index number formulae and indicator functions are known functions of the price and quantity data pertaining to the two periods under consideration. In section 4.3, we provide a definition for an exact price or quantity indicator function. In sections 4.4 and 4.5, we develop further the difference approach to index number theory. In section 4.4, we will define a given price or quantity indicator function to be superlative if it is exactly equal to a corresponding theoretical price or quantity variation under the assumption that the consumer has homothetic preferences that are represented by a flexible linearly homogeneous utility function or which are dual to a flexible unit cost function. We draw on the theory of superlative price and quantity indexes to exhibit many superlative indicator functions. The theory that we develop in section 4.4 for the case of homothetic preferences turns out to be a variant of the theory of superlative indicators developed earlier by Diewert (2005c).  89  This indicator terminology was introduced by Diewert (1992a) (2005).  82  In section 4.5, we will define a given price or quantity indicator function to be strongly superlative if it is exactly equal to a corresponding theoretical price or quantity variation, under the assumption that the consumer has (general) preferences which are dual to a flexible cost function that is subject to money metric utility scaling. The term money metric utility scaling is due to Samuelson (1974) and it is simply a convenient way of cardinalizing a utility function. It proves to be much more difficult to find strongly superlative price or quantity indicator functions but in section 4.5, we show that the Bennet (1920) indicator functions are strongly superlative. Our results require that the consumer’s preferences be represented by a certain translation homothetic cost function that is a variant of the normalized quadratic cost function introduced by Diewert and Wales (1987) (1988a) (1988b). The flexibility of this functional form is shown in Appendix A. Our work draws on the earlier work on translation homothetic preferences (or linear parallel preferences) by Blackorby, Boyce and Russell (1978), Dickinson (1980), Chambers and Färe (1998), Chambers (2001; 111) and Balk, Färe and Grosskopf (2004). The practical usefulness of the difference approach to the measurement of price and quantity change is illustrated at the end of section 4.5 where we show that under certain conditions including the assumption that each household faces the same prices in each period, it is possible to exactly measure the arithmetic average of the economy’s sum of the individual household equivalent and compensating variations using only aggregate data since this aggregate measure of welfare change is exactly equal to the Bennet quantity indicator using aggregate quantity data. In other words, the difference approach to the measurement of aggregate price and quantity change has better aggregation properties than the traditional ratio approach. In section 4.6, we provide economic interpretations for each term in the sum of terms that make up the Bennet price and quantity indicators. The decomposition results developed here are analogues to similar results obtained by Diewert and Morrison (1986) and Kohli (1990) in the traditional approach to index number theory. In section 4.7, we illustrate the use of the difference approach to measure aggregate Japanese consumption and we contrast the traditional ratio approach to the measurement of real consumption to our difference approach. Section 4.8 concludes. 83  4.2 Exact and Superlative Price and Quantity Indexes In preparation for the difference approach to aggregate price and quantity measurement, in this section, we review the standard ratio approach to the measurement of price and quantity change. Thus we will define exact price and quantity indexes and present two definitions for a superlative price index. In the following sections, we will attempt to adapt these standard index number theory concepts to the difference context. The starting point for the economic approach to index number theory is the consumer’s cost or expenditure function C. Thus suppose that the consumer has preferences that are defined by the utility function f(q) over all nonnegative N dimensional quantity vectors q  [q1,...,qN]  0N. 90 In addition, suppose that f is a nonnegative, increasing, 91 continuous and quasiconcave function over the nonnegative orthant   {q : q  0N}. Now suppose that the consumer faces the positive vector of commodity prices p >> 0N and suppose that the consumer wishes to attain the utility level u belonging to the range of f as cheaply as possible. Then the consumer will solve the following cost minimization problem and the consumer’s cost function, C(u,p), will be the minimum cost of achieving the target utility level u: (4.1) C(u,p)  min q {pq : f(q)  u ; q  0N}. It can be shown92 that C(u,p) will have the following properties: (i) C(u,p) is jointly continuous in u,p for p >> 0N and uU where U is the range of f and is a nonnegative function over this domain of definition set; (ii) C(u,p) is increasing in u for each fixed p and (iii) C(u,p) is nondecreasing, linearly homogeneous and concave function of p for each uU.93 Conversely, if a cost function is given and satisfies the above properties, then the utility function f that is dual to C can be recovered as follows.94 For uU and q >> 0N, define the function F(u,q) as follows: 90  Notation: q  0N means each component of q is nonnegative; q >> 0N means each component of q is  positive and q > 0N means q  0N but q  0N where 0N denotes an N dimensional vector of zeros. T  Also pq  N  denotes the inner product of the vectors p and q; i.e., pq = p q  n=1 pnqn. 91  Thus if q2 >> q1  0N, then f(q2) > f(q1).  92  See Diewert (1993; 124).  93  Call these conditions on the cost function Conditions I.  94  See Diewert (1974; 119) (1993; 129) for the details and for references to various duality theorems.  84  (4.2) F(u,q)  max p {C(u,p) : pq  1 ; p  0N}. Now solve the equation: (4.3) F(u,q) =1 for u* and this solution u* will equal f(q). The utility function f(q) and the dual cost function C(u,p) are used in order to define the consumer’s family of Konüs (1939) true cost of living indexes, PK(p0.p1,f(q)), where p0 and p1 are the vectors of positive commodity prices that the consumer faces in periods 0 and 1 respectively and u = f(q) is a positive reference level of utility: (4.4) PK(p0.p1,f(q))  C(f(q),p1)/C(f(q),p0). Thus for each reference quantity vector q that gives rise to a positive utility level, u = f(q) > 0, the consumer’s aggregate price index for that reference level of utility is the ratio of C(u,p1) to C(u,p0). The consumer’s utility and cost functions can be used in order to define the consumer’s family of Allen (1949) quantity indexes, QA(q0.q1,p), where q0 and q1 are the observed consumption vectors that the consumer chose in periods 0 and 1 respectively and p >> 0N is a strictly positive vector of reference prices: (4.5) QA(q0,q1,p)  C(f(q1),p)/C(f(q0),p). The meaning of (4.5) is that if the consumer faces the reference price vector p, then his or her period t utility, f(qt), is set equal to the minimum cost of achieving this utility level using the reference prices p, C(f(qt),p), for t = 0,1 and the consumer’s quantity index is set equal to the ratio C(f(q1),p)/C(f(q0),p). Samuelson (1974) called this type of cardinalization of utility, money metric utility.95 However, note that different choices of p will generate different cardinalizations of utility and different Allen quantity indexes.  95  The basic idea can be traced back to Hicks (1942).  85  It is useful to specialize the above definitions for price and quantity indexes for the case where the consumer’s preferences are homothetic96 or neoclassical. We say that a utility function is neoclassical if it satisfies the following properties over the positive orthant: (i) f is a positive function; i.e., f(q) > 0 if q >> 0N; (ii) f is positively linearly homogeneous; i.e., f(q) = f(q) for all  > 0 and q >> 0N and (iii) f is concave; i.e., for 0 <  < 1, q0 >> 0N and q1 >> 0N, we have f(q0 + (1)q1)  f(q0) + (1)f(q1). It turns out that a concave function defined over the positive orthant is also continuous over this domain of definition. Furthermore, f defined over the positive orthant has a continuous extension to the nonnegative orthant97 and this extended f will also satisfy properties (ii) and (iii) above. The extended f(q) will also be nondecreasing in its variables q over the nonnegative orthant.98 If the consumer’s preferences are neoclassical, then it turns out that the corresponding cost function defined by (4.1) above has the following representation: (4.6) C(u,p) = c(p)u where c(p)  C(1,p) is the consumer’s unit cost function. It also turns out that the unit cost function, c(p), is also a neoclassical function, i.e., it is a positive, nondecreasing, continuous, concave and linearly homogeneous function of p over the positive orthant. Finally, the consumer’s utility function f can be recovered from a knowledge of the unit cost function as follows:99 for q >> 0N, (4.7) f(q) = 1/max p {c(p) : pq = 1; p  0N}. The assumption that the consumer has neoclassical (or homothetic) preferences greatly 96  Preferences are homothetic if the consumer’s utility function can be written as G[f(q)] where f is  neoclassical and G is a continuous increasing function of one variable.  Note that the homothetic  preferences G[f(q)] can be represented by the neoclassical utility function f.  Thus, at times in what  follows, we will sometimes refer to neoclassical preferences as homothetic preferences. The concept of homotheticity is due to Shephard (1953). 97  See Fenchel (1953; 78) or Rockafellar (1970; 85).  98  See Diewert (1974; 111).  99  This is a version of the Samuelson (1953) Shephard (1953) duality theorem; see also Diewert (1974;  110-112) and Samuelson and Swamy (1974).  86  simplifies index number theory. Under the assumption of neoclassical preferences, for each reference q such that f(q) is positive, we have100 (4.8) PK(p0,p1,f(q))  C(f(q),p1)/C(f(q),p0) = c(p1)f(q)/c(p0)f(q) = c(p1)/c(p0).  using definition (4.4) using (4.6)  Thus under the assumption of neoclassical preferences, the Konüs price index is equal to the unit cost ratio, c(p1)/c(p0), and is independent of the reference utility level. Similarly, under the assumption of neoclassical preferences, for each positive reference price vector p, we have (4.9) QA(q0,q1,p)  C(f(q1),p)/C(f(q0),p). = c(p)f(q1)/c(p)f(q0) = f(q1)/f(q0).  using definition (4.5) using (4.6)  Thus under the assumption of neoclassical preferences, the Allen quantity index is equal to the utility ratio, f(q1)/f(q0), and is independent of the reference price vector p. Now suppose that the consumer has homothetic preferences (which we represent by a neoclassical utility function f(q) or the dual unit cost function c(p)) and he or she faces prices pt >> 0N in period t and minimizes the cost of achieving the utility level ut in period t for t = 0,1. Let qt be the consumer’s observed quantity vector for period t so that ut = f(qt) for t = 0,1. written as follows:  Then the consumer’s observed period t cost, ptqt can be  (4.10) ptqt = C(f(qt),pt) = c(pt)f(qt) ;  t = 0,1.  Under these assumptions, the consumer’s ratio of period 1 expenditures to period 0 expenditures satisfies the following equations:  100  See Shephard (1953) (1970), Pollak (1983) and Samuelson and Swamy (1974).  Shephard in particular  realized the importance of the homotheticity assumption in conjunction with separability assumptions in justifying the existence of subindexes of the overall cost of living index.  87  (4.11) p1q1/p0q0 = [c(p1)f(q1)]/[c(p0)f(q0)] using (4.10) 1 0 1 0 = [c(p )/c(p )][f(q )/f(q )] = PK(p0,p1,f(q))QA(q0,q1,p) for arbitrary reference q and p using (4.8) and (4.9). Thus under the assumption of homothetic preferences and cost minimizing behavior on the part of the consumer for the two periods under consideration, the consumer’s observed expenditure ratio is equal to the product of the Konüs price index for arbitrary reference vector q and the Allen quantity index for arbitrary reference vector p. Note that in general, without a knowledge of the consumer’s preferences, the Konüs price index and the Allen quantity index are not directly observable; i.e., they are theoretical indexes as opposed to the “practical” bilateral price and quantity formulae, say P(p0,p1,q0,q1) and Q(p0,p1,q0,q1), that are known functions of the observed consumer data pertaining to the two periods being compared. We assume that the bilateral index number formulae P and Q satisfy the following product test for all strictly positive price and quantity vectors:101 (4.12) p1q1/p0q0 = P(p0,p1,q0,q1)Q(p0,p1,q0,q1). Diewert (1976; 117) defined a quantity index Q(p0,p1,q0,q1) to be exact for a neoclassical utility function f if under the assumption that the consumer minimizes the cost of achieving the utility level ut = f(qt) in period t for t = 0,1, we have (4.13) Q(p0,p1,q0,q1) = f(q1)/f(q0) ; i.e., the quantity index Q(p0,p1,q0,q1) is exactly equal to the utility ratio which in turn is equal to the theoretical Allen quantity index under the assumption of neoclassical preferences.102 Under the same assumptions of cost minimizing behavior and assuming that the preferences of the consumer can be represented by the dual unit cost function c(p), then Diewert (1976; 134) defined a price index P(p0,p1,q0,q1) to be exact for c(p) if we have 101  This is Fisher’s (1922) weak factor reversal test.  102  Diewert (1976) gave many examples of exact index number formulae drawing on the earlier work of  Konüs and Byushgens (1926), Pollak (1983) (originally written in 1971) and Afriat (1972).  88  (4.14) P(p0,p1,q0,q1) = c(p1)/c(p0) ; i.e., the price index P(p0,p1,q0,q1) is exactly equal to the ratio of unit costs which in turn is equal to the theoretical Konüs price index under the assumption of neoclassical preferences. Suppose the index number pair P(p0,p1,q0,q1) and Q(p0,p1,q0,q1) satisfy the product test (4.12) and either P is exact for c(p) or Q is exact for f(q).103 Then Diewert (1976) defined P and Q to be superlative indexes if either c or f could provide a second order approximation to an arbitrary twice continuously differentiable neoclassical unit cost function c*(p) or to an arbitrary twice continuously differentiable neoclassical utility function f*(q).104 Thus the advantage of superlative price and quantity indexes is that they can generate reasonably accurate price and quantity aggregates without having to undertake any econometric estimation of preferences, which becomes difficult or impossible as the number of commodities in the aggregate increases. Examples of superlative price index formulae105 are the Fisher (1922) ideal price index PF and the Törnqvist (1936) (1937) Theil (1967) index PT defined as follows: (4.15) PF(p0,p1,q0,q1)  [p1q0/p0q0]1/2 [p1q1/p0q1]1/2 ; (4.16) ln PT(p0,p1,q0,q1)  n=1N (1/2)[sn0 + sn1] ln [pn1/pn0] where the period t expenditure share on commodity n is defined as snt  pntqnt/ptqt for n = 1,...,N and t = 0,1. This completes our summary of the existing theory for superlative indexes in the case of homothetic preferences. Unfortunately, if the consumer’s preferences are homothetic, then all income elasticities of demand are equal to unity but Engel’s Law and other econometric evidence strongly suggests that income elasticities are not equal to one and 103  Of course, if P is exact for c, P and Q satisfy (4.12) and f is dual to c, then Q is exact for f (and vice  versa). 104  Blackorby and Diewert (1979) showed that if c is a differentiable flexible functional form and has a  differentiable dual f(q), then f is also flexible in the class of neoclassical utility functions and vice versa. 105  See Diewert (1976) for the details.  89  hence consumer preferences are not homothetic. Thus while the theory of exact and superlative indexes may be very useful when we wish to construct subaggregate prices and quantities, it seems that superlative indexes may not be appropriate when constructing overall aggregate consumer price and quantity indexes. Thus we need to determine whether we can find indexes which are exact for more general nonhomothetic preferences. Fortunately, this can be done. Suppose the consumer has general preferences defined by the utility function f(q) and the general cost function C(u,p) is dual to f. As usual, let pt and qt be the observed price and quantity data pertaining to period t and define the period t level of utility ut  f(qt) for t = 0,1. We assume that the consumer is minimizing the cost of achieving the utility level ut in period t so we have: (4.17) ptqt = C(f(qt),pt) ;  t = 0,1.  Under the above assumptions, we say that the bilateral price index number formula, P(p0,p1,q0,q1), is exact for the cost function C if there exists a u* such that u* is between u0 and u1 so that (4.18) either u0  u*  u1 or u1  u*  u0 and (4.19) P(p0,p1,q0,q1) = C(u*,p1)/C(u*,p0)  PK(p0,p1,u*). Thus P is an exact index number formula if under the assumption of cost minimizing behavior, P(p0,p1,q0,q1) is exactly equal to the Konüs theoretical price index PK(p0,p1,u*) where u* is an intermediate reference level of utility. The requirement that the reference level of utility be between the period 0 and 1 utility levels (or possibly equal to one of these levels) is a natural one: we do not want the reference utility level to be too far from the two levels actually experienced by the consumer during the two periods under consideration. Initially, we define P to be a strongly superlative index number formula if it is exact according to the definition immediately above and in addition, the cost function C(u,p) that P is exact for can approximate an arbitrary cost function to the second order. Diewert (1976; 122) showed that the Törnqvist Theil index PT defined by (4.16) is exact for a general Translog cost function where the reference level of utility u* is equal to [u0 90  u1]1/2, the square root of the product of the period 0 and 1 utility levels. Since the general translog cost function is a fully flexible functional form, this shows that PT is a strongly superlative price index. Since the scaling of utility is arbitrary up to an increasing transformation of an initial representation of the utility function, we will find it convenient to impose money metric utility scaling on the underlying utility function f and its dual cost function C. Thus let p* >> 0N be an arbitrary positive price vector. We will assume that the consumer’s utility is scaled so that the dual cost function C satisfies the following equation:106 (4.20) C(u,p*) = u for all uU. Thus our final definition for a strongly superlative index number formula P is that it is exact according to the above definition (4.18) and (4.19) and in addition, the cost function C(u,p) that P is exact for can approximate an arbitrary cost function (that satisfies the money metric utility scaling property (4.20)) to the second order. An analogous definition of exactness can be made for a quantity index. Thus we say that the bilateral quantity index number formula, Q(p0,p1,q0,q1), is exact for the cost function C if there exists a reference price vector p*  [p1*,...,pN*] such that p* is between p0 and p1 so that (4.21) either pn0  pn*  pn1 or pn1  pn*  pn0 for n = 1,...,N and (4.22) Q(p0,p1,q0,q1) = C(f(q1),p*)/C(f(q0),p*)  QA(q0,q1,p*). Thus Q is an exact index number formula if under the assumption of cost minimizing behavior, Q(p0,p1,q0,q1) is exactly equal to the Allen theoretical quantity index QA(q0,q1,p*) where p* is a vector of intermediate reference prices. The requirement that the reference price vector p* be between the period 0 and 1 price vectors that the consumer faced (or possibly equal to one of these two vectors) is again a natural one: we do not want the money metric cardinalizing vector of reference prices to be too far from the two price vectors actually faced by the consumer during the two periods under consideration. 106  If the cost function C(u,p) satisfies Conditions I and in addition, satisfies the money metric utility  scaling conditions (4.20), then we will say that C satisfies Conditions II.  91  Finally, we define Q to be a strongly superlative index number formula if it is exact according to the definition immediately above and in addition, the cost function C(u,p) that is dual to the utility function f can approximate an arbitrary cost function (that has the money metric utility scaling property (4.20)) to the second order. The above material summarizes the theory of exact and superlative indexes which is based on decompositions of the value ratio into price and quantity components that multiply together. In the following section, we will review and extend the companion theory that is based on decompositions of the value difference into a sum of a price change component and a quantity change component. 4.3 Value Differences, Variations and Indicators of Price and Quantity Change Assume that the consumer’s cost function, C(u,p), satisfies Conditions I and the dual utility function is f(q) as usual. Throughout this section, we will assume that pt and qt are the observed price and quantity data pertaining to period t and we define the consumer’s period t observed level of utility ut  f(qt) for t = 0,1. We assume that the consumer is minimizing the cost of achieving the utility level ut in period t so that conditions (4.17) hold; i.e., we have ptqt = C(f(qt),pt) for t = 0,1. Our task in the present section is to decompose the consumer’s observed value change over the two periods under consideration, p1q1  p0q0, into the sum of two terms, one of which is the part of the value change that is due to price change and the other part due to quantity change. This is the difference approach to explaining a change in a value aggregate as opposed to the usual ratio approach used in index number theory.107 The difference counterpart to the Allen (1949) quantity index explained in the previous section is the following Hicks Samuelson quantity variation QS: for each strictly positive reference price vector p >> 0N, define QS(q0,q1,p) as follows:108 (4.23) QS(q0,q1,p)  C(f(q1),p)  C(f(q0),p). 107  Hicks (1942) seems to have been the first to explore the similarities between the two approaches.  108  Samuelson (1974) recognized that C(f(q),p) was a valid cardinalization of utility for any reference price  vector p and thus (4.23) is a valid cardinal measure of the utility difference between periods 0 and 1. Hicks on the other hand only considered the special cases (4.24) and (4.25) defined below.  92  Just as the Allen quantity index QA(q0,q1,p) defined by (4.5) was an entire family of indexes (one for each reference price vector p), so too is the family of quantity variations, QS. Two special cases of (4.23) are of particular importance, the equivalent and compensating variations, QE and QC, defined as follows:109 (4.24) QE(q0,q1,p0)  QS(q0,q1,p0) = C(f(q1),p0)  C(f(q0),p0) ; (4.25) QC(q0,q1,p1)  QS(q0,q1,p1) = C(f(q1),p1)  C(f(q0),p1) . Thus the equivalent variation uses the period 0 price vector p0 as the reference price vector while the compensating variation uses the period 1 price vector p1 as the reference price vector. Generalizing Hicks (1939; 40–41) (1946; 331–332), we will define a family of Hicksian price variation functions PH(p0,p1,f(q)) as follows: for each nonnegative reference quantity vector q, define PH(p0,p1,f(q)) as follows: (4.26) PH(p0,p1,f(q))  C(f(q),p1)  C(f(q),p0). Just as the Konüs price index, PK(p0.p1,f(q)), defined by (4.4) was an entire family of indexes (one for each reference quantity vector or reference utility level u  f(q)), so too is the family of Hicksian price variations. Two special cases of (4.26) are of particular importance, the Laspeyres and Paasche price variation functions, PHL and PHP,  109  Henderson (1941; 120) introduced these variations in the N = 2 case and Hicks (1942) introduced them  in the general case, although his exposition is difficult to follow. to Henderson (1941; 118) and  The term compensating variation is due  the term equivalent variation is due to Hicks (1942; 128).  Hicks (1939;  40–41) initially defined the compensating variation as a measure of price change: “As we have seen, the best way of looking at consumer’s surplus is to regard it as a means of expressing, in terms of money income, the gain which accrues to the consumer as a result of a fall in price.  Or better, it is the  compensating variation in income, whose loss would just offset the fall in price and leave the consumer no better off than before.”  However, later, Hicks (1942; 127-128), following Henderson (1941; 120) defined  (geometrically) the compensating variation as C(u1,p1)  C(u0,p1) and the equivalent variation as C(u1,p0)  C(u0,p0), which are measures of welfare (or quantity) change.  93  defined as follows:110 (4.27) PHL(p0,p1,f(q0))  PH(p0,p1,f(q0)) = C(f(q0),p1)  C(f(q0),p0) ;111 (4.28) PHP(p0,p1,f(q1))  PH(p0,p1,f(q1)) = C(f(q1),p1)  C(f(q1),p0) .112 Thus the Laspeyres price variation uses the period 0 quantity vector q0 as the reference quantity vector while the Paasche price variation uses the period 1 quantity vector q1 as the reference quantity vector. Let M0  p0q0 be the consumer’s nominal “income” or expenditure on the N commodities in period 0. Then PHL(p0,p1,f(q0)) is the amount of nominal income that must be added to the period 0 income M0 in order to allow the consumer, facing period 1 prices p1, to achieve the same utility level as was achieved in period 0, which is u0  f(q0). Similarly, let M1  p1q1 be the consumer’s nominal “income” in period 1. Then PHP(p0,p1,f(q1)) is the amount of nominal income that must be subtracted from the period 1 income M1 in order to allow the consumer, facing period 0 prices p0, to achieve the same utility level as was achieved in period 1, which is u1  f(q1). Note that the equivalent quantity variation defined by (4.24) matches up with the Paasche price variation defined by (4.28) in order to provide an exact decomposition of the value change going from period 0 to 1; i.e., using these definitions and assumptions (4.17), it can be seen that: (4.29) p1q1  p0q0 = C(f(q1),p1)  C(f(q0),p0) = QE(q0,q1,p0) + PHP(p0,p1,f(q1)). Similarly, the compensating quantity variation defined by (4.25) matches up with the Laspeyres price variation defined by (4.27) in order to provide another exact 110  In the index number literature, C(u0,p1)/C(u0,p0) is known as the Laspeyres Konüs (1939; 17) true cost  of living index or price index and C(u1,p1)/C(u1,p0) is known as the Paasche Konüs theoretical price index; see Pollak (1983).  It can be seen that (4.27) and (4.28) are the difference counterparts to these ratio type  indexes. 111  Hicks (1945–46; 68) called this measure the ‘price compensating variation’ and distinguished this  measure from a quantity compensating variation, which he did not define in a very clear manner. also considered price and quantity variations in Hicks (1943). 112  Hicks (1945–46; 69) called this measure the “price equivalent variation”.  94  Hicks  decomposition of the value change going from period 0 to 1: (4.30) p1q1  p0q0 = C(f(q1),p1)  C(f(q0),p0) = QC(q0,q1,p1) + PHL(p0,p1,f(q0)). A problem with the quantity variations defined by (4.24) and (4.25) and the price variations defined by (4.27) and (4.28) is that they asymmetrically single out a reference price or quantity vector that pertains to a single period. Since both measures are equally valid and if a single measure of price or quantity change is required, then for some purposes, it may be useful to take an arithmetic average of the equivalent and compensating variations defined by (4.24) and (4.25) (denote the resulting average quantity variation as QA(q0,q1,p0,p1)) and to take an arithmetic average of the price variations defined by (4.27) and (4.28) (denote the resulting average price variation as PHA(p0,p1,q0,q1)). It can be seen that these average price and quantity variations will also provide an additive decomposition of the value change; i.e., we have: (4.31) p1q1  p0q0 = C(f(q1),p1)  C(f(q0),p0) = QA(q0,q1,p0,p1) + PHA(p0,p1,f(q0),f(q1)). All of the price and quantity variations defined above cannot be evaluated in general using observed price and quantity data pertaining to the two periods under consideration. Thus we now turn our attention to the problem of finding observable approximations to the above theoretical variation functions. Looking at definition (4.24) for the equivalent variation, it can be seen that the term C(f(q0),p0) is equal to period 0 expenditure on the N commodities, p0q0, and hence this term is observable. The remaining term, C(f(q1),p0), is not observable but we can use Shephard’s (1953; 11) Lemma in order to obtain the following first order approximation to this term: (4.32) C(f(q1),p0)  C(f(q1),p1) + pC(f(q1),p1)[p0  p1] = C(f(q1),p1) + q1[p0  p1] = p1q1 + p0q1  p1q1 = p0q1.  using Shephard’s Lemma using (4.17) for t = 1  Using (4.17) for t = 0, (4.32) and definition (4.24), we obtain the following first order approximation to the equivalent variation: 95  (4.33) QE(q0,q1,p0)  p0q1  p0q0 = p0[q1  q0]  VL(p0,p1,q0,q1) where the observable Laspeyres indicator of quantity change, VL(p0,p1,q0,q1), is defined as p0[q1  q0], the inner product of the base period prices p0 with the quantity change vector, q1  q0. In a similar fashion, it can be shown that a first order approximation to the term C(f(q0),p1) is p1q0 and so a first order approximation to the compensating variation QC(q0,q1,p1) defined by (4.25) is:113 (4.34) QC(q0,q1,p1)  p1q1  p1q0 = p1[q1  q0]  VP(p0,p1,q0,q1) where the observable Paasche indicator of quantity change, VP(p0,p1,q0,q1), is defined as p1[q1  q0], the inner product of the current period prices p1 with the quantity change vector, q1  q0. Note that VL and VP are the difference counterparts to the ordinary Laspeyres and Paasche quantity indexes, QL, and QP, defined as follows: (4.35) QL(p0,p1,q0,q1)  p0q1/p0q0 ;  QP(p0,p1,q0,q1)  p1q1/p1q0.  We now turn our attention to the problem of finding observable approximations for the Laspeyres and Paasche price variation functions defined by (4.27) and (4.28) above. An observable first order approximation to the term C(f(q0),p1) in (4.27) is (4.36) C(f(q0),p1)  C(f(q0),p0) + pC(f(q0),p0)[p1  p0] = C(f(q0),p0) + q0[p1  p0] = p0q0 + p1q0  p0q0 = p1q0.  using Shephard’s Lemma using (4.17) for t = 0  Using (4.17) for t = 0, (4.36) and definition (4.27), we obtain the following first order 113  The first order approximations (4.33) and (4.34) were obtained by Hicks (1942; 127–134); see also  Diewert (1992a; 568).  96  approximation to the Laspeyres price variation: (4.37) PHL(p0,p1,f(q0))  p1q0  p0q0 = q0[p1  p0]  IL(p0,p1,q0,q1) where the observable Laspeyres indicator of price change, IL(p0,p1,q0,q1), is defined as q0[p1  p0], the inner product of the base period quantity vector q0 with the price change vector, p1  p0. In a similar fashion, it can be shown that a first order approximation to the term C(f(q1),p0) is p0q1 and so a first order approximation to the Paasche price variation PHP(p0,p1,f(q1)) defined by (4.28) is: (4.38) PHP(p0,p1,f(q1))  p1q1  p0q1 = q1[p1  p0]  IP(p0,p1,q0,q1) where the observable Paasche indicator of price change, IP(p0,p1,q0,q1), is defined as q1[p1  p0], the inner product of the current period quantity vector q1 with the price change vector, p1  p0.114 Note that IL and IP115 are the difference counterparts to the ordinary Laspeyres and Paasche price indexes, PL, and PP, defined as follows: (4.39) PL(p0,p1,q0,q1)  p1q0/p0q0 ;  PP(p0,p1,q0,q1)  p1q1/p0q1.  114  The first order approximations (4.37) and (4.38) were obtained by Hicks (1945–46; 72–73) (1946; 331).  115  Hicks (1942; 128) (1945–46; 71) called IL and IP the Laspeyres and Paasche variations but we will  reserve the term “variation” for the (unobservable) theoretical measures of price and quantity change defined by (4.23) for changes in quantities and by (4.26) for changes in prices.  We will follow Diewert  (1992a; 556) (2005; 313) and use the term “indicator” to denote a given function of the price and quantity data pertaining to the two periods under consideration so that the term indicator becomes the difference theory counterpart to an index number formula in the ratio approach to the measurement of price and quantity change.  Since P and Q are usually used to denote price and quantity indexes, a different notation  is required to denote price and quantity indicators.  Using I to denote a price indicator and V to denote a  quantity (or volume) indicator follows the conventions used by Diewert (2005c). income accountants use the term “volume index” to denote a quantity index.  97  Note that national  In the usual approach to index number theory, it proves to be useful to take the geometric average of the Laspeyres and Paasche price indexes, leading to the Fisher price index PF defined by (4.15), since the Fisher index has very good properties from the viewpoint of the test or axiomatic approach to index number theory; see Diewert (1992b) and Balk (1995). However, in the axiomatic approach116 to price and quantity measurement in the difference context, it proves to be better to take the arithmetic average of the Paasche and Laspeyres indicators. This leads to the Bennet (1920) indicators of price and quantity change defined as follows: (4.40) IB(p0,p1,q0,q1)  (1/2)IL(p0,p1,q0,q1) + (1/2)IP(p0,p1,q0,q1) = (1/2)[q0+q1][p1p0] ; (4.41) VB(p0,p1,q0,q1)  (1/2)VL(p0,p1,q0,q1) + (1/2)VP(p0,p1,q0,q1) = (1/2)[p0+p1][q1q0]. Note that Hicks (1942; 134) (1945-46; 73) obtained the Bennet quantity indicator VB as an approximation to the arithmetic average of the equivalent and compensating variations and he also identified VB as a generalization to many markets of Marshall’s consumer surplus concept. It can be verified that the Laspeyres, Paasche and Bennet price and quantity indicators can be used in order to obtain the following exact decompositions of the value change in the aggregate over the two periods under consideration: (4.42) p1q1  p0q0 = IL(p0,p1,q0,q1) + VP(p0,p1,q0,q1) ; (4.43) p1q1  p0q0 = IP(p0,p1,q0,q1) + VL(p0,p1,q0,q1) ; (4.44) p1q1  p0q0 = IB(p0,p1,q0,q1) + VB(p0,p1,q0,q1) . We conclude this section by defining indicator counterparts to our index number definitions of exactness in the case of nonhomothetic preferences. As usual, we assume that the consumer minimizes cost in periods 0 and 1 so that the consumer has the utility function f(q) that satisfies the usual regularity Conditions I and has the dual cost function C(u,p) so that equations (4.17) are satisfied. Recall that the price index number formula P(p0,p1,q0,q1) was defined to be exact for the cost function C if conditions (4.18) and (4.19) were satisfied. The price indicator counterpart to this definition is as follows: 116  See Diewert (2005c) and Balk (2007) on the axiomatic approach to measures of price and quantity  change using differences.  98  I(p0,p1,q0,q1) is exact for the cost function C if there exists a u* such that u* is between u0  f(q0) and u1  f(q1) so that (4.45) either u0  u*  u1 or u1  u*  u0 and (4.46) I(p0,p1,q0,q1) = C(u*,p1)  C(u*,p0) = PH(p0,p1,u*). Thus I(p0,p1,q0,q1) is exact for the preferences that are dual to C(u,p) if under the assumption of cost minimizing behavior on the part of the consumer, I(p0,p1,q0,q1) is exactly equal to the theoretical Hicksian price variation function PH(p0,p1,u*) defined by (4.26) for a reference utility level u* that is between the period 0 and 1 utility levels attained by the consumer. Recall that the quantity index number formula Q(p0,p1,q0,q1) was defined to be exact for the cost function C if conditions (4.21) and (4.22) were satisfied. The quantity indicator counterpart to this definition is as follows: V(p0,p1,q0,q1) is exact for the cost function C if there exists a reference price vector p*  [p1*,...,pN*] such that p* is between p0 and p1 so that (4.47) either pn0  pn*  pn1 or pn1  pn*  pn0 for n = 1,...,N and (4.48) V(p0,p1,q0,q1) = C(f(q1),p*)  C(f(q0),p*) = QS(q0,q1,p*). Thus V(p0,p1,q0,q1) is exact for the preferences that are dual to C(u,p) if under the assumption of cost minimizing behavior on the part of the consumer, V(p0,p1,q0,q1) is exactly equal to the theoretical Hicks Samuelson quantity variation function QS(q0,q1,p*) defined by (4.23) for a reference price vector p* that is between the period 0 and 1 price vectors faced by the consumer.. In the following section, we will assume that the consumer has homothetic preferences and we will attempt to find price and quantity indicators that are exact and superlative in this case. In section 4.5, we will drop the assumption of homothetic preferences and we will attempt to find superlative indicators in this more general context. 4.4 Superlative Price and Quantity Indicators in the Homothetic Preferences Case We now suppose that the consumer’s utility function f(q) is neoclassical and the dual unit cost function is c(p). Under these conditions, using (4.6), we have 99  (4.49) C(f(q),p) = c(p)f(q). Thus the family of Hicks Samuelson quantity variations QS defined by (4.23) and the family of Hicksian price variations PH defined by (4.26) have the following structures under the assumption of neoclassical preferences: (4.50) QS(q0,q1,p)  C(f(q1),p)  C(f(q0),p) = [f(q1)  f(q0)]c(p) ; (4.51) PH(p0,p1,f(q))  C(f(q),p1)  C(f(q),p0) = [c(p1)  c(p0)]f(q) . It turns out that if we choose the vector of reference prices p in (4.50) to be equal to p0 or p1, then we can find exact quantity indicator functions V(p0,p1,q0,q1) and if we choose the reference quantity vector q in (4.51) to be equal to q0 or q1, then we can find exact price indicator functions I(p0,p1,q0,q1), by drawing on exact index number theory in the case of homothetic preferences. Thus let P(p0,p1,q0,q1) and Q(p0,p1,q0,q1) be an exact pair of price and quantity indexes; i.e., they satisfy (4.12), (4.13) and (4.14) in section 4.2. Now let the reference price vector p in (4.50) above equal the period 0 price vector, p0. Then QS(q0,q1,p0) becomes the equivalent variation QE(q0,q1,p0) and thus (4.50) becomes the following equation: (4.52) QE(q0,q1,p0) = [f(q1)  f(q0)]c(p0) = [{f(q1)/f(q0)}  1]c(p0)f(q0) = [Q(p0,p1,q0,q1)  1]p0q0  VE(p0,p1,q0,q1).  using (4.13) and (4.10) for t = 0  Thus the observable function of the data, VE(p0,p1,q0,q1), defined to be equal to [Q(p0,p1,q0,q1)  1]p0q0, is exactly equal to the equivalent variation, QE(q0,q1,p0), and hence is an exact quantity indicator function. If in addition, Q is exact for a flexible neoclassical utility function f, then we say that the corresponding VE(p0,p1,q0,q1) is a superlative quantity indicator. Now let the reference price vector p in (4.50) above equal the period 1 price vector, p1. Then QS(q0,q1,p1) becomes the compensating variation QC(q0,q1,p1) and thus (4.50) becomes the following equation: (4.53) QC(q0,q1,p1) = [f(q1)  f(q0)]c(p1) 100  = [1  {f(q0)/f(q1)}]c(p1)f(q1) = [1  Q(p0,p1,q0,q1)1]p1q1  VC(p0,p1,q0,q1).  using (4.13) and (4.10) for t = 1  Thus the observable function of the data, VC(p0,p1,q0,q1), is exactly equal to the compensation variation, QC(q0,q1,p0), and hence is an exact quantity indicator function. If in addition, Q is exact for a flexible neoclassical utility function f, then we say that the corresponding VC(p0,p1,q0,q1) is a superlative quantity indicator. Thus each superlative quantity index function, Q(p0,p1,q0,q1), generates two superlative quantity indicator functions, VE(p0,p1,q0,q1) defined in (4.52) which is exact for the theoretical equivalent variation, and VC(p0,p1,q0,q1) defined in (4.53) which is exact for the theoretical compensating variation. Since there are an infinite number of superlative quantity indexes117, there are an infinite number of superlative quantity indicators. The above analysis can be repeated with some modifications in order to find superlative price indicator functions. Thus again let P(p0,p1,q0,q1) and Q(p0,p1,q0,q1) be an exact pair of price and quantity indexes. Now let the reference quantity vector q in (4.51) above equal the period 0 quantity vector, q0. Then PH(p0,p1,f(q0)) becomes the Laspeyres price variation PHL(p0,p1,f(q0)) defined by (4.27) and thus (4.51) becomes the following equation: (4.54) PHL(p0,p1,f(q0))  = [c(p1)  c(p0)]f(q0) = [{c(p1)/c(p0)}  1]c(p0)f(q0) = [P(p0,p1,q0,q1)  1]p0q0  IHL(p0,p1,q0,q1).  using (4.14) and (4.10) for t = 0  Thus the observable function of the data, IHL(p0,p1,q0,q1), defined to be equal to [P(p0,p1,q0,q1)  1]p0q0, is exactly equal to the Laspeyres price variation, PHL(p0,p1,f(q0)), and hence is an exact price indicator function. If in addition, P is exact for a flexible unit cost function c, then we say that IHL(p0,p1,q0,q1) is a superlative price indicator. Now let the reference quantity vector q in (4.51) above equal the period 1 quantity vector, q1. Then PH(p0,p1,f(q1)) becomes the Paasche price variation PHP(p0,p1,f(q1) and thus 117  See Diewert (1976).  101  (4.51) becomes the following equation: (4.55) PHP(p0,p1,f(q1)) = [c(p1)  c(p0)]f(q1) = [1  {c(p0)/c(p1)}]c(p1)f(q1) = [1  P(p0,p1,q0,q1)1]p1q1  IHP(p0,p1,q0,q1).  using (4.14) and (4.10) for t = 1  Thus the observable function of the data, IHP(p0,p1,q0,q1), defined to be equal to [1  P(p0,p1,q0,q1)1]p1q1, is exactly equal to the Paasche price variation, PHP(p0,p1,f(q1)), and hence is an exact price indicator function. If in addition, P is exact for a flexible unit cost function c, then we say that IHP(p0,p1,q0,q1) is a superlative price indicator. Again, since there are many superlative price index functions P(p0,p1,q0,q1), there will be many superlative price indicator functions.118 There is one more detail to be settled in this analysis of superlative price and quantity indicator functions that are generated by traditional index number formulae: we want the sum of the price indicator and quantity indicator to be exactly equal to the value difference. Thus suppose that we are given bilateral index number formulae P and Q that satisfy the product test (4.12) and we use these indexes to define the quantity indicators VE(p0,p1,q0,q1) by (4.52) and VC(p0,p1,q0,q1) by (4.53) and the price indicators IHL(p0,p1,q0,q1) by (54) and IHP(p0,p1,q0,q1) by (4.55). Then using (4.12), it can be shown that numerically, the following equations will hold: (4.56) p1q1  p0q0 = IHP(p0,p1,q0,q1) + VE(p0,p1,q0,q1) ; (4.57) p1q1  p0q0 = IHL(p0,p1,q0,q1) + VC(p0,p1,q0,q1) . Thus the equivalent variation indicator VE(p0,p1,q0,q1) generated by Q needs to be matched up with the Paasche price variation indicator IHP(p0,p1,q0,q1) generated by P and 118  Diewert (2005c; 333–337) also defined superlative price and quantity indicators in the case where  consumer preferences were homothetic. 0  1  0  Diewert’s (2005; 336) superlative economic indicator of price  1  change was defined as IE (p ,p ,q ,q ) = (1/2)IHL(p0,p1,q0,q1) + (1/2)IHP(p0,p1,q0,q1) where IHL and IHP are defined by the third equation in (4.54) and (4.55) respectively where the index number formula P(p0,p1,q0,q1) is superlative.  Thus our present definition of a superlative price or quantity indicator is a  variation of Diewert’s earlier definition.  It should be noted that Fox (2006) generalized Diewert’s (2005)  bilateral approach to multilateral comparisons.  102  the compensating variation indicator VC(p0,p1,q0,q1) generated by Q needs to be matched up with the Laspeyres price variation indicator IHL(p0,p1,q0,q1) generated by P in order for the value difference to equal the sum of a price and quantity indicator. This completes our discussion of superlative indicators when the consumer’s preferences are homothetic. In the following section, we address the much more difficult task of finding superlative indicators in the nonhomothetic case. 4.5 Strongly Superlative Price and Quantity Indicators The holy grail of applied welfare economics is to obtain a quantity variation indicator that is exact for fully flexible preferences. To our knowledge, no one yet has succeeded in this quest.119 In this section, we will show that the Bennet quantity indicator is exact for fully flexible preferences, subject to the money metric cardinalization of utility defined by (20), except that normalized prices that are adjusted for general inflation between the two periods must be used in place of the original prices facing the consumer. Since our focus is on quantity variations, this scaling of prices does not seem to be too serious a drawback to our suggested indicator of quantity change. We now distinguish the original (unscaled) price vector Pt  [P1t,...,PNt] >> 0N that the consumer faces in period t for t = 0,1 from the scaled or normalized price vector pt which is proportional to Pt and will be defined shortly. As in previous sections, the consumer’s observed quantity vector in period t is qt for t = 0,1. Let the consumer’s utility function f(q) satisfy Conditions II (which are the usual nonhomothetic assumptions plus the assumption of money metric utility scaling (4.20) for some strictly positive reference prince vector P* >> 0N) and let the corresponding dual cost function be C(u,P). We assume that the consumer’s cost function has the following translation homothetic 119  Recent attempts by Weitzman (1988) and Diewert (1992a) ended up making homotheticity assumptions  or in the case of Diewert’s (1992a) Theorems 2 and 4, unrealistic assumptions relating the parameters of preferences to utility levels were made.  Chambers and Färe (1998) and Chambers (2001) also came close  but their preference classes fell short of being fully flexible; Chambers (2001; 111) explained the problem with his class of preferences.  Diewert (1976; 123-124) had a fully flexible result but his result was exact  for a Malmquist (1953) quantity index which is not an exact result for a quantity variation and moreover, the Malmquist index does not have the convenient aggregation properties that a Hicks-Samuelson quantity variation possesses.  103  normalized quadratic functional form, 120 which is a special case of translation homothetic preferences:121 (4.58) C(u,P)  bP + (1/2) (P)1PBP + cPu where  > 0N, b and c are N dimensional parameter vectors and B is parameter matrix. These parameter vectors and matrix satisfy the following restrictions, where P* >> 0N is the reference vector which appears in (4.20), the definition for C to satisfy money metric utility scaling at the reference prices P*: (4.59) B = BT so that B is symmetric and B is negative semidefinite; (4.60) BP* = 0N ; (4.61) bP* = 0 and (4.62) cP* = 1. Using the techniques in Diewert and Wales (1987), it can be shown that (4.59) implies that the C defined by (4.58) is globally concave. In the Appendix F, we show that this functional form is flexible in the class of preferences satisfying the money metric utility scaling restrictions in (4.20) for any predetermined parameter vector  > 0N; i.e., given any  > 0N, we can find vectors b and c and a matrix of parameters B such that the restrictions (4.59)–(4.62) are satisfied and the resulting C defined by (4.58) is flexible at the arbitrary point (u*,P*). However, in general, this flexible functional form may not satisfy Conditions II for all u > 0 and all P >> 0N. In the Appendix F, we will define the 120  Diewert and Wales (1987) (1988a) (1988b) introduced the normalized quadratic cost function which  can be defined as C(u,P)  bP + [(1/2) (P)1PBP + cP]u  where b, c and B satisfy (59)-(62) and they  showed that this functional form was flexible in the class of cost functions that satisfy the money metric utility scaling restrictions (20) for any predetermined parameter vector  > 0N.  The advantage of this  functional form is that it contains a flexible unit cost function as a special case (just set b = 0N).  However,  since preferences are generally nonhomothetic, this advantage is not necessarily a huge one. 121  Chambers and Färe (1998; 640) and Chambers (2001; 111) introduced the term “translation homothetic  preferences” and studied these preferences in some detail and noted their importance for the measurement of welfare change; see also Balk, Färe and Grosskopf (2004).  Blackorby, Boyce and Russell (1978; 348)  introduced this class of preferences and Dickinson (1980; 1713) referred to this class of preferences as linear parallel preferences.  Dickinson (1980; 1715–1717) exhibited several examples of this class of  preferences that were flexible.  104  region of prices and utility levels where the functional form satisfies the required regularity conditions for a cost function. Assuming that the consumer’s preferences can be represented by the cost function defined by (4.58)–(4.62) for the two periods under consideration, then assuming cost minimizing behavior on the part of the consumer, the following equations will hold: (4.63) Ptqt = C(f(qt),Pt) = bPt + (1/2) (Pt)1PtBPt + cPt f(qt) ;  t = 0,1.  Using Shephard’s Lemma, the consumer’s observed period t demand vector qt is equal to the following expression: (4.64) qt = PC(f(qt),Pt) = b + (Pt)1 BPt  (1/2)(Pt)2 PtBPt  + c f(qt) ;  t = 0,1.  If there is a great deal of general inflation between periods 0 and 1, then the compensating variation will be much larger than the equivalent variation simply due to this general inflation and taking an average of these two variations will be difficult to interpret due to the change in the scale of prices. In order to eliminate the effects of general inflation between the two periods being compared, it will be useful to scale the prices in each period by a fixed basket price index of the form P where   [1,...,N] > 0N is a nonnegative, nonzero vector of price weights.122 Thus, having chosen the price 122  A reasonable “standard” choice for the weighting vector  is   q0/P0q0. t  t  For this choice of , the  t  vector of period t normalized prices, p  P /P , can be interpreted as a period t vector of “real” prices using a fixed base Laspeyres price index to do the deflation of nominal prices.  Diewert (2005c; 340-341)  commented on the general inflation problem as follows: “The above quotation alerts us to a potential problem with our treatment of value changes; namely, if there is a great change in the general purchasing power of money between the two periods being compared, then our indicators of volume change may be “excessively” heavily weighted by the prices of the period that has the highest general price level.  Put  another way, the units that quantities are measured in do not require any comparisons with other quantities but the dollar price of a quantity is the valuation of a unit of a commodity relative to a numeraire commodity, money.  Thus the indicators of price change that we have discussed in this chapter encompass  both general changes in the purchasing power of money as well as changes in inflation adjusted prices. Thus if there is high inflation between periods 0 and 1 and quantities have increased, then the use of symmetric in prices and quantities indicators (like the Bennet and Montgomery indicators) will shift some of the inflationary increase in values over to the indicator of volume change.”  105  Diewert (2005c; 341)  weighting vector , the period t real prices that the consumer faces pt are defined as follows: (4.65) p0  P0/P0 ;  p1  P1/P1 .  Note that these real price vectors will satisfy the following restrictions: (4.66) pt = Pt/Pt = 1 ;  t = 0,1.  Divide both sides of equation (4.63) by Pt and using definitions (4.65), the resulting equations become: (4.67) ptqt = C(f(qt),pt) = bpt + (1/2)ptBpt + cptf(qt) ;  t = 0,1.  Similarly, substituting equations (4.65) and (4.66) into equations (4.64) leads to the following equations relating the consumer’s period t quantity vectors qt to the real price vectors pt: (4.68) qt = PC(f(qt),pt) = b + Bpt  (1/2)ptBpt + cf(qt) ;  t = 0,1.  With the above preliminaries out of the way, we are ready to state our first Proposition which relates the Bennet quantity indicator defined earlier by (4.41), VB(p0,p1,q0,q1)  (1/2)[p0+p1][q1q0], to the theoretical equivalent and compensating variations defined by (4.24) and (4.25), QE(q0,q1,p0)  C(f(q1),p0)  C(f(q0),p0) and QC(q0,q1,p1)  C(f(q1),p1)  C(f(q0),p1), where we are using the scaled real price vectors pt defined by (4.65) as reference price vectors in place of the original nominal price vectors Pt. Proposition 1: Let the consumer’s observed period t data be (Pt,qt) and suppose that the consumer minimizes the cost of achieving the period t utility level for each period t = 0,1. Let  > 0N be a given vector of price weights that are used in order to construct the period suggested deflating the prices of the second period by a general index of inflation going from period 0 to 1 whereas our solution is more specific in that we choose a Laspeyres type index to do the deflation. Diewert (1992a; 566) discussed other normalizations that have been used historically by various authors in order to construct suitable real prices for use in the measurement of welfare change by volume or quantity indicators.  106  t real price vectors, pt  Pt/Pt for t = 0,1. Suppose a consumer has preferences f(q) which are dual to the translation homothetic normalized quadratic cost function C(u,P) defined by (4.58)–(4.62) and define ut  f(qt) for t = 0,1. Then the Bennet quantity indicator defined by (4.41) using the real prices defined by (4.65) is exactly equal to the arithmetic average of the equivalent and compensating variations defined by (4.24) and (4.25) using the real price vectors as reference prices rather than the original nominal price vectors; i.e., we have (4.69) VB(p0,p1,q0,q1) = (1/2)QE(q0,q1,p0) + (1/2)QC(q0,q1,p1). Proof: (4.70) 2VB(p0,p1,q0,q1) = [p0 + p1][q1  q0] using definition (4.41) 0 1 0 0 1 1 1 0 = p q  C(f(q ),p ) + C(f(q ),p )  p q using (4.67) = p0[b + Bp1  (1/2)p1Bp1 + cf(q1)]  C(f(q0),p0) + C(f(q1),p1)  p1[b + Bp0  (1/2)p0Bp0 + cf(q0)] using (4.68) = p0b +p0Bp1  (1/2)p1Bp1 + p0cf(q1)  C(f(q0),p0) + C(f(q1),p1)  [p1b + p1Bp0  (1/2)p0Bp0 + p1cf(q0)]using (4.66) = [p0b + (1/2)p0Bp0 + p0cf(q1)]  C(f(q0),p0) + C(f(q1),p1)  [p1b + (1/2)p1Bp1 + p1cf(q0)] using (4.59) = C(f(q1),p0)  C(f(q0),p0) + C(f(q1),p1)  C(f(q0),p1) using (4.67) = QE(q0,q1,p0) + QC(q0,q1,p1) using definitions (4.24) and (4.25) which is equivalent to (4.69).  Q.E.D.  Corollary 1: Under the conditions of the above Proposition, the following equality holds: (4.71) VB(p0,p1,q0,q1) = C(f(q1),(1/2)[p0+p1])  C(f(q0),(1/2)[p0+p1])  QS(q0,q1,(1/2)[p0+p1]). Proof: From (4.70), we have the following equality: (4.72) 2VB(p0,p1,q0,q1) = C(f(q1),p0)  C(f(q0),p0) + C(f(q1),p1)  C(f(q0),p1) = p0c[f(q1)  f(q0)] + p1c[f(q1)  f(q0)] using definition (4.67) = [p0 + p1]c[f(q1)  f(q0)] = C(f(q1),p0+p1)  C(f(q0),p0+p1) 107  where the last equality follows adding and subtracting terms and using definition (4.67) for C. Using the linear homogeneity property of C(u,p) in p, it can be seen that (4.72) implies (4.71). Q.E.D. The equality (4.71) shows that the Bennet quantity indicator, VB(p0,p1,q0,q1), is a strongly superlative indicator, since it is exact for the theoretical quantity variation, QS(q0,q1,(1/2)p0 + (1/2)p1), using reference prices that are between p0 and p1, namely the arithmetic average reference prices (1/2)p0 + (1/2)p1. There is a counterpart to Proposition 1 for the Bennet price indicator.  Proposition 2  relates the Bennet price indicator defined earlier by (4.40), IB(p0,p1,q0,q1)  (1/2)[q0+q1][p1p0], to the theoretical Laspeyres and Paasche price variation functions defined by (4.27) and (4.28), PHL(p0,p1,f(q0))  C(f(q0),p1)  C(f(q0),p0) and PHP(p0,p1,f(q1))  C(f(q1),p1)  C(f(q1),p0), where again we use the scaled real price vectors pt defined by (4.65) as reference price vectors in place of the original nominal price vectors Pt. Proposition 2: Under the hypotheses listed in Proposition 1, the Bennet price indicator defined by (4.40) using the real prices defined by (4.65) is exactly equal to the arithmetic average of the Laspeyres and Paasche price variations defined by (4.27) and (4.28) using the real price vectors as reference prices rather than the original nominal price vectors; i.e., we have (4.73) IB(p0,p1,q0,q1) = (1/2) PHL(p0,p1,f(q0)) + (1/2)PHP(p0,p1,f(q1)). Proof:123 (4.74) 2IB(p0,p1,q0,q1) = [q0 + q1][p1  p0] using definition (4.40) 1 0 0 0 1 1 0 1 = p q  C(f(q ),p ) + C(f(q ),p )  p q using (4.67) = p1[b + Bp0  (1/2)p0Bp0 + cf(q0)]  C(f(q0),p0) + C(f(q1),p1)  p0[b + Bp1  (1/2)p1Bp1 + cf(q1)] using (4.68) 123  Our technique of proof is closely related to the techniques used by Balk, Färe and Grosskopf (2004;  160-161) but our functional form assumptions are different and they do not establish a flexibility result for the class of functional forms that they use in their proofs.  108  = p1b +p1Bp0  (1/2)p0Bp0 + p1cf(q0)  C(f(q0),p0) + C(f(q1),p1)  [p0b + p0Bp1  (1/2)p1Bp1 + p0cf(q1)]using (4.66) = [p1b + (1/2)p1Bp1 + p1cf(q0)]  C(f(q0),p0) + C(f(q1),p1)  [p0b + (1/2)p0Bp0 + p0cf(q1)] using (4.59) = C(f(q0),p1)  C(f(q0),p0) + C(f(q1),p1)  C(f(q1),p0) using (4.67) = PHL(p0,p1,f(q0)) + PHP(p0,p1,f(q1)) using definitions (4.27) and (4.28) which is equivalent to (4.73). Q.E.D. Corollary 2: Under the conditions of the above Proposition, the following equality holds: (4.75) IB(p0,p1,q0,q1) = C((1/2)f(q0)+(1/2)f(q1),p1)  C((1/2)f(q0)+(1/2)f(q1),p0)  PH(p0,p1,(1/2)f(q0)+(1/2)f(q1)) . Proof: From (4.74), we have the following equality: (4.76) 2IB(p0,p1,q0,q1) = C(f(q0),p1)  C(f(q0),p0) + C(f(q1),p1)  C(f(q1),p0) = [p1b + (1/2)p1Bp1 + p1cf(q0)]  [p0b + (1/2)p0Bp0 + p0cf(q0)] + [p1b + (1/2)p1Bp1 + p1cf(q1)]  [p0b + (1/2)p0Bp0 + p0cf(q1)] using (4.58) = 2[p1b + (1/2)p1Bp1 + p1c(1/2){f(q0)+f(q1)}]  2[p1b + (1/2)p1Bp1 + p1c(1/2){f(q0)+f(q1)}] rearranging terms 0 1 1 0 1 0 = 2C((1/2)f(q )+(1/2)f(q ),p )  2C((1/2)f(q )+(1/2)f(q ),p ) using definition (4.58) 0 1 0 1 = 2 PH(p ,p ,(1/2)f(q )+(1/2)f(q )) using definition (4.26) which is equivalent to (4.75).  Q.E.D.  The equality (4.75) shows that the Bennet price indicator, IB(p0,p1,q0,q1), is a strongly superlative indicator, since it is exact for the theoretical Hicksian price variation, PH(p0,p1,(1/2)u0+(1/2)u1), using the arithmetic average of the period 0 and 1 utility levels, u0 and u1, as the reference utility level. Bennet (1920) showed that the sum of the Bennet price and quantity indicators, IB(p0,p1,q0,q1) plus VB(p0,p1,q0,q1), is numerically equal to the value difference, p1q1  p0q0; recall (4.44) above. The above two Propositions show that the Bennet indicators have strong economic interpretations if we use real prices instead of nominal prices when 109  calculating these indicators; i.e., they are both strongly superlative indexes.124 Another advantage of the Bennet quantity indicator is that it has a nice aggregation over households property. Thus let  > 0N and suppose that there are H households in the economy and household h has normalized quadratic translation homothetic preferences fh(q) that are dual to the following cost function Ch for h = 1,...,H: (4.77) Ch(uh,P)  bhP + (1/2) (P)1PBhP + chPuh where bh, ch and Bh satisfy the restrictions (4.59)–(4.62) for h = 1,...,H. Let qht be household h’s observed consumption vector for period t and let household h face the price vector Pht in period t for h = 1,...,H and t = 0,1. Define the vector of real prices that household h faces in period t, pht, as follows: (4.78) pht  Pht/Pht ;  t = 0,1 ; h = 1,...,H.  Now make the hypotheses in Proposition 1 for each household and we find that the sum over households of the Bennet quantity indicators VB(ph0,ph1,qh0,qh1) for each household h is equal to the average of the sum of the household h equivalent and compensating variations, QEh(qh0,qh1,ph0) and QCh(qh0,qh1,ph1); i.e., using Proposition 1, we have: (4.79) h=1H VB(ph0,ph1,qh0,qh1)  h=1H (1/2)[ph0 + ph1][qh1  qh0] = (1/2)h=1H QEh(qh0,qh1,ph0) + (1/2)h=1H QCh (qh0,qh1,ph0) = h=1H QSh(qh0,qh1,(1/2)[ph0+ph1]) using Corollary 1 where for h = 1,...,H, QSh(qh0,qh1,(1/2)[ph0+ph1]) is the Hicks Samuelson theoretical quantity variation for household h using the vector of average real prices facing household h for the two periods under consideration, (1/2)ph0 + (1/2)ph1, as the reference price vector. Thus if individual household price and quantity data are available, the sum of these theoretical quantity variations can be calculated as the sum of the observable Bennet quantity indicators. 124  Diewert (2005c) and Balk (2007) indicated that the Bennet indicators had excellent axiomatic  properties as well.  Thus the Bennet indicators seem to be the difference counterparts to the Fisher indexes  in normal ratio index number theory, since the Fisher indexes also have strong economic and axiomatic properties.  110  If in addition, each household faces the same vector of prices p0 in period 0 and p1 in period 1, then (4.79) simplifies as follows: (4.80) VB(p0,p1,q0,q1)  (1/2)[p0 + p1][q1  q0] = (1/2)h=1H QEh(qh0,qh1,p0) + (1/2)h=1H QCh (qh0,qh1,p0) = h=1H QSh(qh0,qh1,(1/2)[p0+p1]) where the aggregate period t quantity vectors qt are defined as the sum of the individual household quantity vectors: (4.81) q0  h=1H qh0 ; q1  h=1H qh1 . Thus under the assumptions of Proposition 1 and the assumption that each household faces the same prices in each period, the aggregate Bennet indicator of quantity change, VB(p0,p1,q0,q1) defined by the first line in (4.80), is exactly equal to the arithmetic average of the sum of the individual household equivalent variations, h=1H QEh(qh0,qh1,p0), plus the sum of the individual compensating variations, h=1H QCh (qh0,qh1,p0). Under these hypotheses, the aggregate Bennet indicator of quantity change is also exactly equal to the sum over households of the Hicks Samuelson theoretical quantity variations using the vector of average real prices facing household h for the two periods under consideration, h=1H QSh(qh0,qh1,(1/2)[p0+p1]).125 4.6 The Decomposition Properties of the Bennet Indicators In the production context, Diewert and Morrison (1986) and Kohli (1990) (1991) developed a methodology that enables one to obtain exact decompositions of various Törnqvist indexes into explanatory factors for each price or quantity change using the assumption of a translog technology.126 It would be useful if we could provide a similar decomposition result for the Bennet indicators but we are not able to accomplish this task. However, Diewert and Morrison (1986; 674-676) developed an average of first order 125  This result is analogous to Chambers’ (2001; 114) exact result for an aggregate normalized Bennet  quantity indicator in the context of Chambers’ benefit function framework. 126  Diewert (2002) also developed some decomposition results for the Fisher indexes but these results lack  the simplicity of the Törnqvist decomposition results.  111  approximations methodology which gave very similar results to their translog methodology127 and so we will use this second approach below in order to provide economic interpretations for each separate term in the Bennet indicators. In this section, we will not make any specific parametric assumptions; we will assume only that the consumer’s cost function C(u,P) satisfies conditions I and in addition, C(u,P) and the dual f(q) are once differentiable in a neighbourhood around the observed period t real price and quantity vectors, pt  Pt/Pt and qt, and around the observed period t utility levels, ut  f(qt), for t = 0,1. Hence the following equations will be satisfied by the data under the assumption that the consumer minimizes costs in each period: (4.82) ptqt = C(f(qt),pt) ; (4.83) qt = P C(f(qt),Pt) = p C(f(qt),pt) ;  t = 0,1; t = 0,1.  The first set of equalities in (4.83) follows from Shephard’s Lemma and the second set follows from the proportionality of the real prices pt to the corresponding nominal prices Pt and the linear homogeneity of the cost function C(u,P) in the components of P so that the partial derivative functions C(u,P)/Pn are homogeneous of degree 0 in their price variables. Define the nth partial Bennet price and quantity indicators, In(pn0,pn1,qn0,qn1) and Vn(pn0,pn1,qn0,qn1), as follows: (4.84) IBn(pn0,pn1,qn0,qn1)  (1/2)[qn0 + qn1][pn1  pn0] ; (4.85) VBn(pn0,pn1,qn0,qn1)  (1/2)[pn0 + pn1][qn1  qn0] ;  n = 1,...,N; n = 1,...,N.  Note that the above partial indicators using real prices sum up to the overall Bennet indicators using real prices; i.e., we have: (4.86) IB(p0,p1,q0,q1) = n=1N IBn(pn0,pn1,qn0,qn1) ; (4.87) VB(p0,p1,q0,q1) = n=1N VBn(pn0,pn1,qn0,qn1). We will relate the above observable partial indicators to theoretical partial indicators: for 127  See Morrison and Diewert (1990) and Diewert and Lawrence (2006).  112  each n, define the Laspeyres and Paasche partial price variations, Ln and Pn, as follows:128 (4.88) Ln  C(f(q0),p10,...,pn10, pn1, pn+10,...,pN0)  C(f(q0),p0) ; (4.89) Pn  C(f(q1),p1)  C(f(q1),p11,...,pn11, pn0, pn+11,...,pN1) ;  n = 1,...,N ; n = 1,...,N .  Thus the nth partial price Laspeyres variation, Ln, is the difference in real expenditure that would result if the standard of living of the consumer were held constant at the period 0 utility level, u0  f(q0), and all real prices are also held constant at their period 0 levels except that we allow the nth real price to increase from the period 0 level, pn0, to the period 1 level, pn1. The nth partial price Paasche variation, Pn, has a similar interpretation except that the reference utility level is held constant at the period 1 level, u1  f(q1), and all real prices are held constant at their period 1 levels and as before, we allow the nth real price to increase from the period 0 level, pn0, to the period 1 level, pn1. It is possible to adapt the first order approximation methods used to derive the approximations (4.36) and (4.38) in the present context. Thus first order approximations to the unobservable terms in (4.88) and (4.89) can be obtained as follows: for n = 1,...,N, we have: (4.90) C(f(q0),p10,...,pn10, pn1, pn+10,...,pN0)  C(f(q0),p0) + [C(f(q0),p0)/pn][pn1  pn0] = C(f(q0),p0) + qn0[pn1  pn0] using Shephard’s Lemma (4.83) ; (4.91) C(f(q1),p11,...,pn11, pn0, pn+11,...,pN1)  C(f(q1),p1) + [C(f(q1),p1)/pn][pn0  pn1] = C(f(q1),p1) + qn1[pn0  pn1] using Shephard’s Lemma (4.83). Substituting (4.90) and (4.91) into (4.88) and (4.89) leads to the following observable first order approximations, aLn and aPn to the Laspeyres and Paasche partial price variations, Ln and Pn:129 128  These variations are difference counterparts to the partial indexes defined in Diewert and Morrison  (1986) and Kohli (1990). 129  Using simple feasibility arguments for the cost minimization problems defined by the left hand sides of  (4.90) and (4.91), it can be shown that aLn  Ln  and aPn  Pn so that the Laspeyres partial price indicators  aLn generally biased upwards for the true partial Laspeyres price indexes Ln and the Paasche partial price indicators aPn generally biased downwards for the true partial Paasche price indexes Pn; i.e., these partial  113  (4.92) Ln  qn0[pn1  pn0]  aLn ; (4.93) Pn  qn1[pn1  pn0]  aPn ;  n = 1,...,N ; n = 1,...,N .  Thus using definitions (4.84) for the Bennet partial price indicators, IBn(pn0,pn1,qn0,qn1), it can be seen that they are exactly equal to the arithmetic average of the Laspeyres and Paasche partial price indicators, (1/2)aLn + (1/2)aPn, which in turn approximate the average of theoretical Laspeyres and Paasche partial price variations, (1/2)Ln + (1/2)Pn, to the first order; i.e., we have: Proposition 3: (4.94) IBn(pn0,pn1,qn0,qn1) = (1/2)aLn + (1/2)aPn  (1/2)Ln + (1/2)Pn ;  n = 1,...,N.  The above results are nonparametric; i.e., the approximations given by (4.92)–(4.94) are first order Taylor series approximations that are valid no matter what (once differentiable) preferences the consumer holds. However, if we assume that the consumer has preferences that can be represented by the translation homothetic normalized quadratic cost function C(u,P) defined by (4.58)–(4.62), then we can obtain an exact expression for the gap between the Bennet partial indicator on the left hand side of (4.94) and the average of the theoretical partial variations on the right hand side of (4.94); i.e., we can obtain the following expression for the bias BBPn in the nth Bennet partial price indicator; i.e., we have: (4.95) IBn(pn0,pn1,qn0,qn1) = (1/2)Ln + (1/2)Pn + BBPn ; (4.96) BBPn  (1/2)n[p0Bp0 + p1Bp1][pn1  pn0]  n = 1,...,N;  where n is the nth component in the weighting vector  that is used to form real prices. Since n=1N npnt = 1 for t = 0,1, it can be seen that: (4.97) n=1N BBPn = 0 so that the sum of the bias terms in the Bennet partial indicators IBn(pn0,pn1,qn0,qn1) sums price indicators will generally have some substitution bias, which will tend to cancel out when we take their averages.  114  to zero.130  Let P* be the money metric utility scaling vector which appears in (60)-(62)  and define its real counterpart by p*  P*/P*. If p0 is proportional to p*, then p0Bp0 is equal to 0 and if p1 is proportional to p*, then p1Bp1 is equal to 0 and under these conditions, it can be seen that all of the bias terms BBPn will be equal to 0 as well. Hence if p0 and p1 are close to each other, then we can choose the reference price vector p* to be close to p0 and p1 and the bias terms will all be close to 0. Finding economic interpretations for the Bennet partial quantity indicators, VBn(pn0,pn1,qn0,qn1), is more difficult. For each n, we first define the theoretical Laspeyres and Paasche partial quantity variations, Ln and Pn, as follows: (4.98) Ln  C(f(q10,...,qn10, qn1, qn+10,...,qN0),p0)  C(f(q0),p0) ; (4.99) Pn  C(f(q1),p1)  C(f(q11,...,qn11,qn0,qn+11,...,qN1),p1) ;  n = 1,...,N; n = 1,...,N.  Thus the nth partial quantity Laspeyres variation, Ln, is the difference in real expenditure that would result if the real prices of the consumer are held constant at their period 0 levels p0 and all quantities are also held constant at their period 0 except that we allow the nth quantity to increase from the period 0 level, qn0, to the period 1 level, qn1. The nth partial quantity Paasche variation, Pn, has a similar interpretation except that the reference prices are held constant at their period 1 levels p1 and all quantities are also held constant at their period 1 levels except that we allow the nth quantity to increase from the period 0 level, qn0, to the period 1 level, qn1. In order to obtain observable first order approximations to the theoretical quantity variations defined by (4.98) and (4.98), it is first necessary to develop some preliminary material. Define the function ht(q) for q’s in a neighborhood of qt as follows: (4.100) ht(q)  C(f(q),pt) ;  t = 0,1.  Under our assumptions, ht(q) is once differentiable at qt and we can calculate the vector of first order partial derivatives as follows: (4.101) qht(qt) = [C(f(qt),pt)/u] qf(qt) ;  130  t = 0,1.  This must be the case in order for Proposition 2 to hold.  115  Under our assumptions, qt solves the cost minimization problem defined by C(f(qt),pt) for t = 0,1 and since f(q) is differentiable at qt, there exists a nonnegative Lagrange multiplier t such that the following first order necessary conditions for the period t cost minimization problem are satisfied:131 (4.102) pt = t qf(qt) ;  t = 0,1.  But Samuelson (1947) showed that the period t Lagrange multiplier t which appears in (4.102) is also equal to the period t marginal cost around the equilibrium point so that we have: (4.103) t = C(f(qt),pt)/u ;  t = 0,1.  Substituting (4.102) and (4.103) into (4.101) gives us the following simple expression for the derivatives of the function ht(q) defined by (4.100): (4.104) [C(f(qt),pt)/u]qf(qt)  qht(qt) = pt ;  t = 0,1.  Equations (4.104) seem to have been first derived by Balk (1989; 166) so we can call these relationships Balk’s Lemma. With the above preliminary material out of the way, we can now proceed to the task of finding first order approximations to the theoretical partial quantity variations Ln and Pn defined by (4.98) and (4.99). Thus a first order approximation to the unobservable term C(f(q10,...,qn10, qn1, qn+10,...,qN0),p0) in (4.98) is: (4.105) C(f(q10,...,qn10, qn1, qn+10,...,qN0),p0)  C(f(q0),p0) + [h0(q0)/qn][qn1  qn0] = C(f(q0),p0) + pn0[qn1  qn0]  n = 1,...,N using (4.100) for t = 0 using (4.104) for t = 0.  Similarly, a first order approximation to the unobservable term C(f(q11,...,qn11,qn0,qn+11,...,qN1),p1) in (4.99) is: (4.106) C(f(q11,...,qn11,qn0,qn+11,...,qN1),p1)  C(f(q1),p1) + [h1(q1)/qn][qn0  qn1] 131  n = 1,...,N using (4.100) for t = 1  Strictly speaking, we require qt >> 0N to ensure that conditions (102) are satisfied and later we will also  require that marginal cost be positive so that C(f(qt),pt)/u > 0 for t = 0,1.  116  = C(f(q1),p1) + pn1[qn0  qn1]  using (4.104) for t = 1.  Substituting (4.105) and (4.106) into (4.98) and (4.99) leads to the following observable first order approximations, bLn and bPn to the Laspeyres and Paasche partial quantity variations, Ln and Pn:132 (4.107) Ln  pn0[qn1  qn0]  bLn ; (4.108) Pn  pn1[qn1  qn0]  bPn ;  n = 1,...,N ; n = 1,...,N .  Thus using definitions (4.85) for the Bennet partial quantity indicators, VBn(pn0,pn1,qn0,qn1), it can be seen that they are exactly equal to the arithmetic average of the Laspeyres and Paasche partial quantity indicators, (1/2)bLn + (1/2)bPn, which in turn approximate the average of the theoretical Laspeyres and Paasche partial price variations, (1/2)Ln + (1/2) Pn, to the first order; i.e., we have: Proposition 4: (4.109) VBn(pn0,pn1,qn0,qn1) = (1/2)bLn + (1/2)bPn  (1/2) Ln + (1/2) Pn ;  n = 1,...,N.  This completes our theoretical discussion of the properties of the Bennet indicators. the following section, we illustrate the use of these indicators for a Japanese data set.  In  4.7 The Bennet Indicators using Japanese Data In this section, we apply our methodology to Japanese consumption data. These data were constructed from the Japanese national accounts for 12 classes of expenditure for the period 1980-2006. The prices for each commodity class were normalized to equal one in 1980; see Tables G-1 and G-2 in Appendix G for a listing of the data. We chose  132  Using simple feasibility arguments for the cost minimization problems defined by the left hand sides of  (4.102) and (4.103), it can be shown that bLn  Ln  and bPn  Pn so that the Laspeyres partial quantity  indicators bLn generally biased upwards for the true partial Laspeyres quantity indexes Ln and the Paasche partial quantity indicators bPn generally biased downwards for the true partial Paasche quantity indexes Pn; i.e., these partial quantity indicators will generally have some substitution bias, which will tend to cancel out when we take their averages.  117  food and non-alcoholic beverages to be our numeraire commodity 133 Aggregate expenditures evaluated in terms of real prices are 127753 billion yen in 1980 and 232679 in 2006. Therefore, household expenditures evaluated in real prices increased by 104927 billion yen over the last 27 years. We calculate Bennet indicators of quantity changes and real price changes to decompose the expenditure difference for every year. Table 4-1 lists value the real expenditure differences and the Bennet indicators for the period 1981–2006. Table 4-2 lists their annual averages. It tells us that the effects of real price changes are much smaller than the effects of quantity changes. However, the impact of real price changes has been significant for the last decade. Table 4-1: Real Expenditure Differences and Bennet Indicators, 1981-2006 Year  Difference  Bennet Quantity Indicator  Bennet Price Indicator  1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  2657.4 8980.9 3191.4 2352.7 6652.0 6622.4 9040.9 8122.2 8089.1 5992.3 1489.5 6674.9 2324.7 5214.8 6675.9 4329.8 1297.4 -4934.2 1424.7 4120.6 3656.7 1932.1 4.3 480.4 4579.9 3953.4  1966.8 6252.6 4054.7 3456.0 6002.3 4996.6 6538.3 7847.4 8398.1 8497.3 5275.9 4586.8 2438.6 5253.6 3224.2 5314.7 1683.8 -2125.0 1188.5 1851.2 3969.5 2188.8 1433.0 3708.4 3944.9 6167.5  690.7 2728.3 -863.3 -1103.2 649.7 1625.7 2502.6 274.8 -309.0 -2505.0 -3786.4 2088.1 -113.9 -38.8 3451.7 -984.9 -386.3 -2809.2 236.3 2269.3 -312.8 -256.7 -1428.7 -3228.0 635.1 -2214.1  Table 4-2: Annual Averages of Real Expenditure Differences and Bennet Indicators  133  Year  Difference  Bennet Quantity Indicator  Bennet Price Indicator  1980-2006 1981-1990 1991-2000 2001-2006  4035.6 6170.1 2861.8 2434.5  4158.3 5801.0 2869.2 3568.7  -122.6 369.1 -7.4 -1134.2  This choice of deflator means that we used the weighting vector α = (1,0,…0)T.  118  Our focus is on real consumption that measures the overall utility or volume of aggregate consumption. Real consumption can be computed throughout either the traditional ratio approach to quantity indexes or by the difference approach as outlined in this chapter. However, if we use the ratio approach, the choice of specific index number formula could matter for the value of real consumption. Therefore, we use the difference approach as well as alternative index number formulae in order to evaluate the performance of the difference approach relative to that of the ratio approach. Real consumption coincides with the corresponding nominal value at the reference year. Setting 1980 to be the reference year, we calculate different versions of real consumption for all years using the ratio approach and the difference approach. Fixed base and chained quantity indexes were computed using the Laspeyres, Paasche, Fisher and Törnqvist-Theil formulae. 134 The results are listed in Table 4-3 below. The last column of Table 4-3 lists the corresponding Bennet estimate of total consumption. The first entry in this column is simply the 1980 measure of Japanese total consumption expenditures divided by the price of food; i.e., the first entry in the second column of the Table. The next entry in the Bennet column just adds the Bennet measure of quantity change or volume change VB defined by (4.41) above where the real price vectors and quantity vectors pertaining to the years 1980 and 1981 are used in the formula. The 1982 entry in the Bennet column is just the 1981 entry plus the Bennet measure of quantity change going from 1981 to 1982 and so on. Looking at Table 4-3, it can be seen that all of the index number estimates of real Japanese consumption are very close to each other with the exception of the fixed base Laspeyres and Paasche estimates. This lack of correspondence is normal since these indexes are known to differ from their superlative counterparts when a fixed base is used. The superlative chained indexes are particularly close to each other. But how do these chained superlative indexes compare to the corresponding Bennet estimates of real consumption listed in the last column of Table 4-3? It can be seen that the Bennet measures are always equal to or greater than their chained superlative counterparts but 134  Two versions of the Törnqvist-Theil were computed for both fixed base and chained indexes: one that  constructed the price index first using the usual formula (and then the quantity index was defined by dividing real expenditures by this direct price index) and the other was constructed by directly comparing a share weighted average of log quantity changes and exponentiating.  These two quantity indexes are  superlative as is the Fisher index; see Diewert (1976) for the formula details.  119  the differences are not very large: on average, the Bennet estimate exceeds its chained Fisher counterpart by 0.74% per year, with a maximum deviation of 1.1%. Table 4-3: Comparison of Japanese Real Consumption, 1980-2006 Difference Approach  Ratio Approach Fixed Base Index Year  1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006  Total Expenditure  127752.7 130410.1 139391.1 142582.5 144935.2 151587.2 158209.6 167250.5 175372.7 183461.8 189454.1 190943.7 197618.6 199943.2 205158.0 211834.0 216163.8 217461.2 212527.0 213951.7 218072.3 221729.0 223661.1 223665.4 224145.8 228725.7 232679.2  Chained Index  Laspeyres Paasche Fisher Tornqvist Quantity Index Quantity Index Quantity Index Quantity Index 127752.7 129723.2 135942.2 139930.0 143397.3 149444.7 154316.2 160639.7 168160.1 176183.0 184481.0 189745.3 194323.3 196770.9 202159.1 205423.2 210890.5 212945.5 210785.1 212406.8 214655.4 219124.7 221564.5 223647.4 228274.4 233194.6 241317.8  127752.7 129705.2 135820.7 139723.9 143032.0 148913.2 153786.8 160067.6 167493.2 175390.8 183247.0 188051.9 192481.4 194760.7 199719.7 202857.5 207390.9 208639.4 206006.4 206864.1 208394.7 211575.8 213242.1 213901.5 216395.4 219024.9 223461.4  127752.7 129714.2 135881.4 139826.9 143214.5 149178.7 154051.3 160353.4 167826.3 175786.5 183863.0 188896.7 193400.1 195763.2 200935.7 204136.3 209133.4 210781.4 208382.1 209617.1 211501.9 215317.2 217363.5 218720.2 222255.5 225998.7 232218.0  127752.7 129714.5 135885.0 139829.8 143216.7 149179.0 154058.6 160365.5 167841.9 175810.7 183935.2 188939.3 193406.1 195728.0 200849.2 204059.7 209019.5 210627.0 208221.2 209381.3 211186.4 214843.1 216757.6 218000.7 221340.7 224784.3 230683.0  Implicit Implicit Laspeyres Paasche Fisher Tornqvist Tornqvist Tornqvist Quantity Index Quantity Index Quantity Index Quantity Index Quantity Index Quantity Index 127752.7 129714.1 135881.7 139825.9 143211.9 149173.6 154048.9 160354.6 167828.7 175789.6 183864.1 188886.1 193384.4 195727 200883 204090 209048.8 210645.4 208186.6 209366.7 211206.1 214905.3 216910.3 218185.9 221580.2 225100.2 230972  127752.7 129723.2 135909.6 139890.4 143312.0 149246.3 154151.3 160474.4 168006.2 176072.5 184306.9 189504.2 194039.0 196436.0 201593.5 204745.7 209921.7 211585.7 209497.1 210687.9 212530.9 216449.2 218589.8 220041.5 223775.6 227751.1 234019.0  127752.7 129705.2 135830.6 139778.2 143160.8 149063.8 153940.0 160251.2 167755.6 175781.1 183951.3 189110.7 193622.0 196010.0 201165.2 204290.4 209399.4 211004.4 208936.5 210084.2 211863.0 215674.6 217802.2 219154.9 222753.6 226625.5 232665.4  127752.7 129714.2 135870.1 139834.3 143236.4 149155.0 154045.6 160362.8 167880.8 175926.7 184129.0 189307.3 193830.4 196222.9 201379.3 204517.9 209660.4 211294.8 209216.6 210385.9 212196.7 216061.6 218195.6 219597.8 223264.0 227187.6 233341.2  127752.7 129714.5 135871.1 139835.3 143237.4 149156.0 154047.0 160364.3 167882.7 175928.8 184132.2 189310.6 193833.8 196226.7 201383.0 204521.8 209665.1 211298.6 209219.0 210386.6 212196.6 216061.0 218194.5 219596.3 223261.8 227184.5 233336.2  127752.7 129714.1 135870.4 139834.5 143236.6 149155.2 154045.8 160363.1 167881.1 175927.1 184129.5 189307.8 193830.9 196223.3 201379.6 204518.2 209660.4 211294.5 209216.1 210385.3 212195.9 216060.5 218194.6 219596.3 223261.8 227184.7 233336.5  Bennet Quantity Indicator 127752.7 129719.5 135972.1 140026.9 143482.8 149485.1 154481.7 161020.0 168867.4 177265.5 185762.8 191038.7 195625.6 198064.1 203317.7 206541.9 211856.6 213540.4 211415.4 212603.8 214455.0 218424.6 220613.4 222046.3 225754.8 229699.6 235867.2  What are we to conclude from the above results? For the Japanese data, it seems that a standard superlative index number approach to measuring aggregate real consumption will be fairly close to the results generated by the theoretically preferable Bennet approach, which has better aggregation over consumer properties and is consistent with nonhomothetic preferences. However, there seem to be small but significant differences between the Bennet estimates and those generated by chained superlative indexes. 4.8 Conclusion This chapter has established satisfactory difference theory counterparts to the standard results on exact and superlative indexes in the ratio approach to the aggregation over commodities problem. The counterpart to a superlative index number formula is a superlative indicator formula. We found that the Bennet indicators of price and quantity change were (strongly) superlative and thus we recommend their use in practical applications of cost benefit analysis when ex post variations must be calculated. In section 4.7 above, we found that, somewhat surprisingly, the results using the Bennet indicator of quantity change are rather close to the quantity aggregates generated by a 120  superlative quantity index. This is somewhat reassuring in that the ratio and difference approaches to economic aggregation seem to give more or less the same answer, at least for our Japanese data set. Finally, we mention one strong advantage of the difference approach over the ratio approach: the ratio approach fails if the quantity aggregate has a value equal to zero in the base period whereas the difference approach is unaffected by this complication. This observation is important if labour supply enters the consumer’s utility function (negatively rather than positively) since in this case, zero or negative value aggregates can readily occur. Although we did not formally model this situation, we are confident that our techniques can be generalized to cover this situation.  121  Chapter 5.  Summary and Conclusion  5.1 Summary of Contributions This thesis has addressed several important theoretical and empirical problems in economic measurement. Each of these problems was addressed with reference to both of the existing approaches in index number theory: the ratio approach and the difference approach. In our analysis of the Japanese living standard, we focused on real income and net real income in the Japanese market sector as appropriate measures. We adopted the decomposition formula developed by Diewert and Morrison (1986), as well as Kohli (1990) (2004), based on the ratio approach in index number theory. Applying their decomposition formula, we decomposed the growth in real income and net real income into contribution factors such as technical progress, changes in real output prices and growth in input quantities. The key roles of technical progress and capital services growth were observed in an analysis of real income as well as net real income, and the role of capital services significantly decreased in the net approach. Construction of a new data set of price and quantity of input and output for Japanese market sector is a notable feature of this chapter. We applied the difference approach to a producer model as well as a consumer model; in the producer model, we were concerned with a new approach to productivity measurement based on the difference approach. In this new approach, productivity growth is measured by the difference between output quantity differences and the input quantity differences instead of the ratio of output quantity ratio to input quantity ratio. First, we proposed theoretical measures of contribution factors for technical progress, changes in real output prices and growth in relative input quantities. Second, we showed that Bennet indicators coincided with these theoretical measures by introducing an income function based on the normalized quadratic functional form pioneered by Diewert and Wales (1987). Bennet indicators add up to the change in the real income per unit primary input. Thus, our theoretical result would enable us to decompose the growth in real income per unit primary input into additive contribution factors. We applied our result to investigate the growth in real income and net real income per unit labour in the Japanese economy from 1955 to 2006, and observed that productivity growth and capital services were the two main contributors to growth in real income and 122  net real income per unit labour. In the consumer model, we addressed the price and quantity aggregation problem with multiple consumption goods in the difference approach, and investigated the best method for calculating representative price and quantity difference. Diewert (1976) has defined superlative price and quantity indexes as observable indexes that are exact for a ratio of unit cost functions or for a ratio of linearly homogeneous utility functions. We looked for counterparts of his results in the difference context of index number theory for both flexible homothetic and nonhomothetic preferences. First, we proposed theoretical measures for the aggregate price and quantity indicator. Second, we showed that Bennet indicators coincided with these theoretical measures by introducing a cost function with the translation homothetic normalized quadratic functional form discussed by Balk, Färe and Grosskopf (2004), Chambers (2001) and Dickenson (1980). Using Japanese aggregate consumption data from the national accounts, we calculated real expenditures based on different types of indexes and indicator 5.2 Directions for Further Research We conclude this thesis by noting three unsolved problems remaining for future research: (1) an industrial analysis of the Japanese standard of living, (2) a productivity measurement based on alternative types of the rate of return and (3) an investigation of the theoretical foundation of the arithmetic mean of the sum of compensating variations and that of equivalent variations. In our analysis of real income and net real income, we provided an economy-wide analysis but did not provide an industry contributions analysis. The primary focus of our analysis was on the effects of changes in the terms of trade on living standards. We would like to extend our analysis by introducing two new components into our database. First, we will introduce the industry production accounts consisting of inputs and outputs. Second, we will introduce a more detailed breakdown of exports and imports by commodity classification. Changes in the terms of trade reflect price changes in international commodities, and industries respond to price changes in different ways. Thus, the extended database would enable us to decompose the impact of changes in the terms of trade on living standards into industries and commodities. We are especially interested in how changes in crude oil prices have affected the industries’ incomes. Increased oil price would more severely affect the incomes of the industries that use oil 123  heavily than those of other industries. Through such a study, we could investigate the channels through which changes in crude oil price affect a country’s standard of living. The measurement of prices and quantities of capital services is indispensable for growth accounting, which this thesis’ analysis adopts. The quantity of capital services is proportional to the productive capital stock, and their growth rates coincide with each other. Capital stock is estimated based on the perpetual inventory method. On the other hand, the price of capital services from a fixed asset, which is called a user cost or a rental price, consists of three main components: rate of return, depreciation rate and change in its asset’s price. Among them, the choice of the rate of return is an important element in the construction of the price of capital services. Two types of rate of return exist: the ex-post rate of return and the ex-ante rate of return. Under an ex-post approach, the rate of return is computed by imposing the condition that estimated capital services exactly correspond to gross operating surplus and the capital element of gross mixed income. Under an ex-ante approach, the rate of return is chosen as to best reflect economic agents’ expectations of the investment’s required return. A typical example is a government one-year bond rate. It is thus unlikely that there would be exact equality between the value of capital services and gross operating surplus, and the capital element of gross mixed income. The ex-post approach has stood the test of time and is still in use by many researchers; this thesis also used the ex-post rate of return. However, the ex-ante approach has several advantages, two of which are as follows: (1) the theoretical assumptions needed are less restrictive than for the ex-post approach and (2) the ex-ante approach provides a means of splitting mixed income between income and labour, and income and capital. By following the ex-ante approach, we avoid the problem of imputing a wage for the self-employed and family workers. We would like to conduct growth accounting by constructing the price of capital services based on an ex-ante rate of return. Our concerns are the differences in productivity growth and wages for the self-employed and family workers between estimates based on an ex-ante rate of return and an ex-post rate of return. We show that the Bennet quantity indicator coincides with the arithmetic mean of the sum of compensating variations and that of equivalent variations. Positive sums of compensating variations and equivalent variations must enable the gainers to compensate the losers and have something left over for themselves. Using the sum of compensating 124  variations and equivalent variations for a cost–benefit test is often justified by appealing to this line of reasoning. We would like to investigate the possibility of using the arithmetic mean of the sum of compensating variations, as well as that of equivalent variations, for a cost–benefit test. But there are practical and theoretical problems associated with the sums of compensating variations and equivalent variations. The practical problem is that we need to estimate consumers’ cost functions or indirect utility functions in order to calculate compensating and equivalent variations. If these functions vary across consumers, we need to estimate the distinct cost functions and distinct indirect utility functions for each consumer, and this is a practically impossible task. The theoretical problem is the so-called Boadway paradox (Blackorby and Donaldson (1990)). A move from one Walrasian equilibrium to another Walrasian equilibrium always yields a positive sum of compensating variations and a non-positive sum of equivalent variations. This leads to the violation of transitivity in social decision making. On the other hand, the arithmetic means of the sum of compensating variations and the sum of equivalent variations clearly has the advantage of easy implementation over the sums of compensating variations and the sums of equivalent variations. As this thesis shows, it can be calculated from observable prices and quantities by using a Bennet indicator formula. However, we have no clear answer for whether or not the arithmetic mean of the sum of compensating variations and the sum of equivalent variations violates transitivity. When we are able to show that it is free from the Boadway paradox, we will be able to clearly justify its use in a cost–benefit analysis.  125  Bibliography Afriat, S.N. (1972), “The Theory of International Comparisons of Real Income and Prices”, pp. 13-69 in International Comparisons of Prices and Outputs, D.J. Daly (ed.), Chicago: University of Chicago Press. Allen, R.G.D. (1949), “The Economic Theory of Index Numbers”, Economica 16, 197-203. Archibald, R.B. 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Data Construction In this appendix, we explain our methods for data construction in detail. The main data sources are the Japanese national accounts and the KEO database and these sources will often not be explicitly acknowledged in what follows. 1. Final Demand Components other than Gross Capital Formation There are five price and quantity series for final demand components other than investments and inventory changes in our database:   C; Domestic final consumption expenditure of households (excluding the imputed rent of owner-occupied dwellings);    N; Final consumption expenditure of private non-profit institutions serving households;     G; Net purchases of goods and services by the general government; X; Exports of goods and services (including direct purchases in the domestic market by non-resident households);    M; Imports of goods and services (excluding direct purchases abroad by resident households).  1.1 Data Sources Our output data series (other than for investments and inventory changes) are based on Japanese official national accounts (JSNA), which has been published by the Economic and Social Research Institute of the Cabinet of Office and the Economic Research Institute of the Economic Planning Agency of the Japanese Government. We call Japanese national accounts JSNA. In 1978, the Japanese system of national accounts was revised to comply with the guidelines proposed by the United Nations System of National Accounts (1968 SNA). In 2000, it was revised to comply with the guidelines newly proposed by the United Nations System of National Accounts (1993 SNA). The Economic and Social Research Institute and Economic Research Institute published separate historical data series based on 1968 SNA and then later based on the 1993 SNA (but not extending back to 1955). Our basic strategy is to start with the reference data 134  series in the JSNA based on the 1993 SNA135 and extend this series backwards by using the growth rates of other data series in the 1968 JSNA. Before the 1968 JSNA was introduced, the Japanese national accounts were called National Income Statistics. Data from the National Income Statistics are used only for constructing the imputed rent of owner-occupied houses. Publications we used are as follows; National Income Statistics   Annual Report on National Report on National Income Statistics 1975, Economic Planning Agency 1968 JSNA   Report on National Accounts from 1955 to 1998, Economic and Social Research Institute, Cabinet Office   Annual Report on National Accounts of 2000, Economic Planning Agency 1993 JSNA   Annual Report on National Accounts of 2004-2008, Economic and Social Research Institute, Cabinet Office  1.2 Laspeyres Quantity Indexes and Subindexes The Laspeyres quantity index (and therefore, a corresponding Paasche price index) is applied for constructing quantity indexes everywhere in the Japanese national accounts. Therefore, we can implicitly derive the price and quantity of a component from data series for the aggregate and all other components. Suppose that 0 is the base year and there are two products, B and C. Further suppose that the statistical agency forms a fixed base Laspeyres quantity index that aggregates these two products into the aggregate A say. The base period expenditure shares of products B and C are (A1) sB0  PB0QB0/[PB0QB0 + PC0QC0] = VB0/[VB0 + VC0] ; sC0  PC0QC0/[PB0QB0 + PC0QC0] = VC0/[VB0 + VC0]  = 1  sB0  where VB0 = PB0QB0 is the value of expenditures on product B in period 0 and PB0 and QB0 are the corresponding price, etc. The price of A in period 0 can be set equal to 1 and the quantity of A in period 0 can be set equal to the expenditure on B and C in period 0. 135  In the future, we will abbreviate “the JSNA based on the 1993 SNA” to “the 1993 JSNA” and “the  JSNA based on the 1968 SNA” to “the 1968 JSNA”.  135  The value of the aggregate A in period t is set equal to the sum of the expenditure values on B and C, VBt + VCt, so that putting this all together, we have: (A2) PA0 = 1;  VA0  VB0 + VC0  QA0 ; VAt  VBt + VCt.  Use the Laspeyres quantity index formula to determine the period t quantity aggregate QAt and the corresponding price PAt as follows: (A3) QAt/QA0 = sB0(QBt/QB0) + sC0(QCt/QC0) ; PAt = (VAt/VA0)/(QAt/QA0). Now suppose that we have value series for A, B and C for periods 0 and t (so that in particular we can calculate the shares defined by (A1) above), we know the statistical agency has used the equations in (A3) in order to calculate the price and quantity of A in period t, and we know the prices and quantities for product A and B for periods 0 and t. Our problem is to calculate prices and quantities for product C. We can set the price of product C in period 0 to 1 so that PC0 = 1 and set the quantity of product C in period 0 to the period 0 value, so that QC0 = VC0. Now use the first equation in (A3) to solve for QCt, which is a straightforward linear equation in one unknown. Once QCt has been determined, the corresponding price PCt can be set equal to (VCt/VC0)/(QCt/QC0). We found this data recovery technique to be extremely useful in practice. 1.3 Domestic Final Consumption Expenditure of Households (C) For the years 1955-1998, current and constant yen series for Domestic Final Consumption Expenditure of Households are found in the Report on National Accounts from 1955 to 1998 (1968 JSNA); see Part 1 Flows, 1 Figures of Calendar Year,  Time Series Tables, 1 Arranged for Main Figures; Calendar Year, in billions of yen. The constant yen series is at the market prices of 1990. We use these series to construct price and quantity series for Domestic Final Consumption Expenditure of Households. However, this aggregate consumption is the sum of market sector sales to households (our C) plus direct purchases abroad by resident households. Therefore, we need to use the technique described in section 1.2 above to remove the latter series. Our target consumption aggregate also excludes the imputed expenditures on owner-occupied housing and so we have to deduct these expenditures from the national accounts consumption aggregate as well. We follow two steps in order to accomplish 136  this task. First, we construct price and quantity series for total housing consumption (rental expenses plus imputed owner-occupied expenses) from the data of the 1968 JSNA series. Second, for the years prior to 1980, we divided the total housing expenses between the owner-occupied houses and the rental houses based on their relative floor space.136 We will explain how we estimated these relative floor spaces in the following section. For the years 1970-1998, we use the current and constant yen series for the sum of rental and owner occupied housing using unpublished data from the 1968 JSNA.137 For the years 1955-1973, current and constant yen series for total housing expenses can be found in the Annual Report on National Income Statistics 1975; see Part 1 Basic Accounts and Main Tables, 2 Figures of Calendar Year, Account 1 Gross National Product and Expenditure. From these two sources, we can construct price and quantity series for the years 1955-1998. We linked these two data series at the time when data series based on the 1968 JSNA started which was 1970. 138 Thus we extended backwards the series in the 1968 JSNA using the growth rates of housing expenditures tabled in the National Income Statistics for the years 1955-1970.139 As explained in section 2.3 below, we constructed estimates for relative floor spaces of owner-occupied houses and rental houses for the years 1955-2006. Thus, we could decompose our already constructed data series for the total quantity of housing services into quantity components for imputed rent and market rent by using the ratios of floor spaces. Thus we have obtained price and quantity series for the imputed rent of the owner-occupied houses. We then constructed price and quantity series for C for the years 1955-1998 by applying chained Fisher indexes to the price and quantity series for Domestic Final Consumption Expenditure of Households of the 1968 JSNA and our price and quantity series for the imputed expenditures on owner-occupied housing, with the quantity of owner occupied housing entering the index number formula with a negative sign. The resulting series will be linked to the more recent 1993 JSNA household consumption 136  This is the procedure which had been adopted by Japanese national accounts until 2006.  137  Shuji Hasegawa at the Japanese Cabinet Office has made available to us these data series.  138  Some variables in JSNA based on 1968 SNA are revised back to 1955.  However, total housing  expenses in the 1968 JSNA are available only after 1970. 139  There is another database consisting of gross outputs, exports, and imports for the years 1951-1968.  This database was constructed at the joint project between Japan Center of Economic Research and Keio Economic Observatory in the early 1970s, headed by Professor Iwao Ozaki, Keio University.  We used  the estimates of households’ housing expenditure from this database in order to extrapolate the JSNA data based on the 1968 SNA backwards for the years 1955-1968.  137  series to be described in the next paragraph. For the years 1980-2002, current and constant yen series for (1) Final Consumption Expenditures of Household (excluding imputed rents for the services of owner-occupied dwellings), (2) Direct Purchases Abroad by Resident Households, and (3) Direct Purchases in the Domestic Market by Non-Resident Households can be found in the Annual Report on National Accounts of 2004 (1993 JSNA), and these data series are further updated for the years 1980-2003 (constant yen series only for the years 1994-2003) using the Annual Report on National Accounts of 2005 (1993 JSNA) and for the years 1994-2006, using the information posted on the Cabinet Office website: Annual Report on National Accounts of 2008 (1993 JSNA); see Part 1 Flow Accounts; 1.Integrated Accounts; (1) Gross Domestic Product Account (Production and Expenditure Approach); Calendar Year, in billions of yen. The constant yen series in the Annual Report on National Accounts of 2004 is at the market prices of 1995. The constant yen series in the Annual Report on National Accounts of 2005 is at the market prices of 2000. The constant yen series in the Annual Report on National Accounts of 2008 is at the market prices of 2000 (chained).140 From these data series, we constructed price and quantity series for each of the three variables listed above for the years 1980-2006. One difference between the 1968 JSNA and the 1993 JSNA is in the treatment of social benefits in kind. 141 It was a part of Domestic Final Consumption Expenditure of Households instead of a part of Government Final Consumption Expenditure in the 1968 JSNA. However, it became a part of Government Final Consumption Expenditure instead of a part of Domestic Final Consumption Expenditure of Households in the 1993 JSNA. We consider social benefits in kind as a part of Domestic Final Consumption Expenditure of Households following the treatment of the 1968 JSNA.142 For the years 140  Constant yen series based on chained indexes have been introduced in the Annual Report on National  Accounts of 2006.  Since a chained index avoids the substitution bias that the usual fixed based index is  likely to show, we use constant yen series based on chained indexes as much as possible. 141  Social benefits in kind consist of transfers made by government units to households.  It consists of  subsidies for medical treatment and for text books which are supposed to be delivered to all the students without any cost. 142  However, these goods are also produced by the production sector.  Following the same treatment of social benefits in kind as in the 1968 JSNA, we can construct price and  quantity series for C for a longer period than we can construct following the same treatment in the 1993 SNA.  138  1990-2003, current and constant yen series for Social Benefits in Kind etc. are found in the Annual Report on National Accounts of 2005 (1993 JSNA), and these data series are further updated for the years 1996-2006 in the Annual Report on National Accounts of 2008 (1993 JSNA); see Part 1 Flow Accounts; 5. Supporting Tables, (8) Final Consumption Expenditure of General Government classified by Purpose; Fiscal Year, in billions of yen. The constant yen series in the Annual Report on National Accounts of 2005 are at the market prices of 1995. The constant yen series of the updated data series in the Annual Report on National Accounts of 2008 is at the market prices of 2000. Since these annual data series are based on a fiscal year, we transformed these data series of Social Benefits in Kind etc. based on fiscal years into those based on calendar years by linear interpolation.143 From these data series, we constructed price and quantity series for Social Benefits in Kind etc. based on calendar years for the years 1991-2006. For the years 1991-2006, we constructed price and quantity series for our consumption aggregate C by applying chained Fisher indexes to the above data series, which were essentially equal to an aggregate of: (1) Final Consumption Expenditures of Household (excluding imputed rent services of owner-occupied dwellings), (2) (less) Direct Purchases Abroad by Resident Households, plus (3) Direct Purchases in the Domestic Market by Non-Resident Households and plus (4) Social Benefits in Kind. We constructed two data series for C: one data series constructed from the data of the 1968 JSNA for the years 1955-1998 and the other data series for C constructed from the data of the 1993 JSNA for the years 1991-2006. We linked these two data series at 1991, the time when the data series for social benefits in kind became available in the 1993 JSNA. There is a final adjustment to our concept for the Domestic Final Consumption Expenditures of Households, C. Goods and services purchased for the maintenance of owner-occupied houses (in other words, maintenance expenses) are part of the imputed rent of owner-occupied houses. Since they are produced by the market sector, we must  143  The fiscal year in Japan is from April for the current year to May for the next year.  series of fiscal year t are vt.  Suppose that data  Then, data series of calendar year t, Vt is calculated as follows: Vt = (1/4)   vt-1 + (3/4)  vt.  139  add these maintenance expenses to our aggregate C. 144 We aggregated these two variables by applying chained Fisher indexes. The operating surplus from the owner-occupied houses is the imputed rent minus the maintenance expenses for owner-occupied houses. Thus, once we obtain the imputed rent and the operating surplus from the owner-occupied houses, we can implicitly construct maintenance expenses.145 For the years 1980-2003, current yen series for Operating surplus (imputed services of owner-occupied dwellings) can be found in the Annual Report on National Accounts of 2005 (1993 JSNA) and this data is updated for the years 1996-2006 using the Annual Report on National Accounts of 2008 (1993 JSNA); see Part 1 FLOWS, 2. Income and Outlay Accounts classified by Institutional Sectors, (5) Households (Including Private Unincorporated Enterprises). For the years 1955-1979, we assume the ratio of the imputed rent to maintenance expenses is constant and is the same as the average ratio over the five years 1980-1984. By utilizing this ratio and the imputed rent of owner-occupied houses, we can extrapolate the maintenance expenses backwards. 1.4 Consumption Expenditures of Non-Profit Institutes (N) For the years 1955-1998, current and constant yen series for Final Consumption Expenditure of Private Non-Profit Institutions Serving Households (N) is found in the Report on National Accounts from 1955 to 1998 (1968 JSNA); see Part 1 Flows, 1 Figures of Calendar Year,  Time Series Tables, 1 Arranged for Main Figures; Calendar Year, in billions of yen. The constant yen series is at the market prices of 1990. For the years 1980-2003, current and constant yen series for Final Consumption Expenditure of Private Non-Profit Institutions Serving Households N is found in the Annual Report on National Accounts of 2004 (1993 JSNA), and these data series are further updated for the years 1980-2003 (constant yen series only for the years 1994-2003) in the Annual Report on National Accounts of 2005 (1993 JSNA), the years 1994-2006 on the Cabinet Office website, Annual Report on National Accounts for 2008 (1993 JSNA); see Part 1 Flow Accounts; 1.Integrated Accounts; (1) Gross Domestic 144  It is possible to make a separate data series of the goods and services for the maintenance.  However,  since its value is relatively small (about 5 % of the imputed rent), we treat it as a part of household consumption. 145  We assume that the deflators for the maintenance expenses and the operating surplus of the  owner-occupied houses are the same as that of the imputed rent of the owner-occupied houses.  140  Product Account (Production and Expenditure Approach); Calendar Year, in billions of yen. Thus we have two data series for N. We link these two data series at 1980, the time when data series for the 1993 JSNA starts. 1.5 Net Purchases of Goods and Services by the Government (G) We define net purchases of goods and services by the general government (G) as the purchases of intermediate inputs by the general government (G1) minus the sales of goods and services of the general government to the market sector (G2):     G (= G1 – G2): Net purchases of goods and services by the general government; G1: Purchases of intermediate inputs by the general government; G2: Sales of goods and services of the general government.  By definition, sales of goods and services of the general government (G1) is equal to government final consumption expenditures (G3) minus the gross output of general government (G4).     G2: (= G3 − G4) Sales of goods and services by the general government; G3: Gross output of the general government; G4: Government final consumption expenditures.  For the years 1955-1998, current and constant yen series for (1) Purchases of intermediate inputs by the general government (G1), (2) Gross output of the general government (G3), and (3) Government final consumption expenditures (G4) are found in the Report on National Accounts from 1955 to 1998 (1968 JSNA); see Part 1 Flows, 2 Figures of Calendar Year, I Time Series Tables 1, Arranged for Main Figures. The constant yen series is at the market prices of 1990. From these data series, we constructed price and quantity series for G1, G3, and G4. Thus, we can implicitly construct price and quantity series for G2 by using the data series of G3, and G4. In the end, we constructed data series of G for the years 1955-1998 by applying chained Fisher indexes to the price and quantity series of G1 and G2.146 146  When we aggregated these two data series, we put a negative sign in front of the quantity series G2.  141  In the 1993 JSNA, current and constant yen series for G1 and G2 are available for the years 1990-2006. For the years 1990-2003, current and constant yen series for Commodity and Non-Commodity Sales of the General Government (G2) are found in the Annual Report on National Accounts of 2005 (1968 JSNA); see Part 1 Flows, 5. Supporting Tables, Table 8, Final Consumption Expenditure of General Government classified by Purpose, Fiscal Year, in billions of yen. These data series are further updated for the years 1996-2006 in the Annual Report of National Accounts of 2008 (1993 JSNA). The constant yen series in the Annual Report on National Accounts of 2005 is at the market prices of 1995. The constant yen series of the updated data series in the Annual Report on National Accounts of 2008 is at the market prices of 2000 (chained). We transform the data series for G2 based on fiscal years into those based on calendar years by linear interpolation.147 We construct price and quantity series for G1 from the data of the 1993 JSNA for the years 1990-2006. For the years 1990-2003, current and constant yen series for Purchases of intermediate inputs by the general government (G1) are found in the Annual Report on National Accounts of 2005 (1993 JSNA), and these data are updated for the years 1996-2006 in the Annual Report on National Accounts of 2008 (1993 JSNA); see Part 1 Flow, 5. Supporting Tables, (2) Gross Domestic Product and Factor Income classified by Economic Activities; Calendar year, in billions of yen. The constant yen series in Annual Report on National Accounts of 2005 is at the market prices of 1995. The constant yen series of the updated data series in Annual Report on National Accounts of 2008 is at the market prices of 2000 (chained). From these data series, we construct price and quantity series for G1 for the years 1990-2006. Thus we have two data series for G: one data series constructed using the data of the 1968 JSNA for the years 1955-1998 and the other data series for G constructed from the data of the 1993 JSNA for the years 1991-2006. We linked these two data series at 1991.  147  G2 based on fiscal years is available for the years 1990-2006.  However, since we transform data  based on fiscal yeas to data based on calendar years by using linear interpolation as we explained before, G2 based on calendar years is available only for the years 1991-2006.  142  1.6 Exports and Imports of Goods and Services (X and M) For the years 1955-1998, current and constant yen series on the 1968 JSNA basis are available for the following four variables; (1) Exports of goods and services (S1), (2) Imports of goods and services (S2), (3) Direct purchases abroad by resident households (S3), and (4) Direct purchases in the domestic market by non-resident households, (S4). These series are found in the Report on National Accounts from 1955 to 1998 (1968 JSNA); see Part 1 Flows, 1 Figures of Calendar Year,  Time Series Tables, 1 Arranged for Main Figures; Calendar Year, in billions of yen. The constant yen series are at the market prices of 1990. For the years 1980-2003, current and constant yen series for the above variables such as S1, S2, S5, and S6 can be found in the Annual Report on National Accounts of 2005 (1993 JSNA), and these data are further updated for the years 1994-2006 in the Annual Report on National Accounts of 2008; see Part 1 Flow Accounts; 1.Integrated Accounts, (1) Gross Domestic Product Account (Production and Expenditure Approach), Calendar Year, in billions of yen. The constant yen series in the Annual Report on National Accounts of 2005 is at the market prices of 1995. The constant yen series of the updated data series in the Annual Report on National Accounts of 2008 is at the market prices of 2000 (chained). We have two data series for S1, S2, S3, and S4 individually. As usual, we linked these two data series at 1980, the time when the data of the 1993 JSNA start. Finally, we constructed data series of X by applying chained Fisher indexes to the price and quantity series of S1 and S4 and constructed data series for M by applying chained Fisher indexes to the price and quantity series for S2 and S3.148 2. Capital Services and Investments There are price and quantity series for investments and capital stocks for 95 asset classes; 90 tangible assets and 5 intangible assets. These data are taken from the KEO data base.149  148  When we constructed these aggregates, we put negative sign in front of the quantity series for S3 and  S4. 149  Nomura (2004) gives the detailed explanation of the construction of capital data of KEO data base.  143     I1 to I95: Investments for asset classes; K1 to K95: Capital services for the asset classes.  We have already listed the 95 asset classes in Table 1. The numbering of investments corresponds to that of capital services. Quantities of capital services are proportional to capital stocks at the beginning of the year. Thus, we first describe how we constructed the capital stocks. 2.1 Capital Stocks For constructing capital services, we make use of the following data from the KEO database:   Gross fixed capital formation by asset class for the whole country for the years 1955-2006;    Gross fixed capital formation by asset class for the market sector for the years 1955-2006;150      Asset price indexes for the years 1955-2007; Capital stocks by asset class at the end of 1955; Depreciation rates.  Time series for Gross Fixed Capital Formation (GFCF) and for asset price indexes for each asset class are available in the latest version of the KEO database. By subtracting the investment in 1955 from the corresponding capital stock in 1955, we can construct capital stocks at the beginning of 1955. If the initial constructed capital stock of a good becomes negative, we set the resulting capital stock to be zero. These constructed capital stocks at the beginning of 1955 are our initial benchmark estimate of the capital stocks. Now, all the requirements to apply the Perpetual Inventory Method (PIM) are ready and we can apply the method to estimate the capital stocks. Since asset price indexes are available for the years 1955-2006, we obtain nominal and real capital stocks for 95 asset classes for the years 1955-2006. By applying the user cost formula (53) in the main text, we can construct prices and quantities for the capital services of these 95 asset classes.  150  Nominal and real GFCF for the market sector are available at KEO database.  144  2.2 Investments All investment goods are produced by the market sector. However, the bit of GFCF that is used by the government sector must be subtracted from the total economy estimates for GFCF by asset. Fortunately, the required information is available for constructing market sector investment by asset class and is available on the KEO data base. Thus we could obtain price and quantity data on investments for 95 asset classes for the market sector over the years 1955-2006 based on the KEO database.151 However, the stock of residential structures includes the stocks held by owner occupiers of houses. We need to extract the residential capital held by owner occupiers from the total stock of residential structures in order to obtain the market sector’s residential capital that is used to produce market rental housing. We now turn to the problem of estimating the owner’s portion of the residential housing capital stock. 2.3 The Decomposition of Residential Structures Capital into Two Components Overall residential structures can be classified into owner-occupied houses and rental houses. The owner-occupied houses are attributed to household sector. The household sector earns imputed rent by providing the services of its owned houses to itself. Therefore, the residential structures held by owner occupiers should be subtracted from the total stock of residential housing. In order to accomplish this task, we first calculate the total floor space of owner-occupied houses and rental houses. Second, we divide the total stock of residential structures into owner-occupied houses and rental houses according to their relative total floor spaces. 2.3.1 Data sources  151    Housing Survey of Japan Volume 1, Results for the Whole of Japan (1968) (1973) (1978) (1983) (1988) (1993), Statistics Bureau;    Revised Report on National Income Statistics (1951-1973), Economic Planning Agency;    Housing Survey of Japan, volume 1, results (1968)(1973)(1978)(1983)(1988), Statistics Bureau;  These data are used for constructing capital services for the market sector.  145  for  whole  Japan    Housing and Land Survey of Japan volume 1, results for whole Japan (1993)(1998)(2003), Statistics Bureau;    Monthly of Construction Statistics (1955-2006), Ministry of Land, Infrastructure and Transport  2.3.2 Total Floor Space Estimates for Owner-Occupied Houses and Rental Houses Point estimates for total floor space for owned and rented houses are available every five years from 1968 to 2003 in the Housing survey of Japan (1968) (1973) (1978) (1983) (1988) (1993) and the Housing and land survey of Japan (1998) (2003). We use these data as benchmarks. We extend these benchmarks by utilizing the information on the net investment of the housing stock every year. Annual data for residential investment and loss of residential buildings are available from Monthly of Construction Statistics (for the years 1960-2006). Annual investment (loss) is the floor space which has been newly added to (subtracted from) the existing stock of housing. The following two data series from Monthly of Construction Statistics are used:   New dwellings started: new construction starts of dwellings by owner occupant relation (floor area) for the years 1955-2006.    Loss of residential buildings: changes in total area and in dwelling units, by cause of loss for the years 1955-2006.  New dwellings started provide estimates for the area of new construction of the following four different kinds of houses:   (1) Owned houses; Dwellings which are owned by the households occupying them.    (2) Rented houses; Rented houses which are owned and administrated by the local government or public corporations.    (3) Issued houses; Dwellings which are owned or administered by private companies, public bodies, etc and rented to their employees or officials in order to meet the needs of their work or issued as a part of salaries and wages regardless of rent being paid.    (4) Houses built for sales; Dwellings constructed by the public or private sectors in order to be sold with the site under the house included in the sale. 146  We classify the types of overall houses as owned houses or rental houses. Since the data in the Monthly of Construction Statistics have a more detailed classification than we are using, we regroup the four different kinds of overall houses listed there into two categories; owned houses and rental houses. We regard the floor area of new construction of owned houses (1) and houses built for sales (4) as additions (new investment) to the floor stock of the owned houses. We regard the floor area of new construction of rental houses (2) and issued houses (3) as additions (new investment) to the floor stock of the rented houses.152 Statistics on the loss of residential buildings show the destroyed floor space of total residential building by aging, natural disaster, and fires. These statistics include the loss of the owner-occupied houses and the loss of rental houses. We distribute the area of the floor space losses between the owner-occupied houses and the rental houses in proportion to the areas of their stocks and we obtain net investment series (in terms of floor space) for the two types of house. We adjust these net investments every 5 years between benchmarks by a linear interpolation method so that the accumulated floor spaces coincide with the benchmark floor spaces. The earliest benchmark floor space is the one in 1968. For the years 1955-1967, we use the same adjustment coefficient as the one used between 1968 and 1973. We apply the Perpetual Inventory Method to estimate floor space of housing for the years 1955-1967. Now, we calculate floor spaces of overall residential structures, owner-occupied houses, and rental houses. The last step is to make this database of floor space more consistent with the 1993 JSNA data. Until 2003, housing expenses have been divided between owner-occupied houses and rental houses according to their floor space in JSNA. The 1993 JSNA based data for imputed rents for owner-occupied houses and for market rents gives us another estimate for the ratio of the floor space of owner-occupied houses and the floor space of rental houses for the years 1981-2006. These 1993 JSNA based estimates are quite similar to our estimates which have just been described. However, for the years 1981-2006, we use the 1993 JSNA based estimates as the primary estimate 152  This assumption will only be approximately correct since some of the houses built for sale can be sold  to home owners who lend their houses for tenants.  As will be seen shortly, our assumptions here do not  have to be precisely correct. We just need them to be approximately correct so that we can construct reasonable estimates of floor space by type of house between censuses.  147  and link our earlier series to the 1993 JSNA data. This enables us to provide a breakdown of the total stock of residential structures into owner-occupied stocks and market sector rental stocks. 3. Inventory Services and Changes in Inventories We regard the change in inventories as outputs and the stock of inventories as inputs to production of the market sector.153 There are price and quantity series for 4 types of changes in inventory and 4 types of inventory services in our database.      IV1: Changes in finished-goods inventories; IV2: Changes in work-in-process inventories; IV3: Changes in work-in-process inventories for cultivated assets; IV4: Changes in materials inventories.       KIV1: Finished-goods inventory services; KIV2: Work-in-process inventory services; KIV3: Work-in-process inventory services for cultivated assets; KIV4: Material inventory services.  The quantities of inventory services are proportional to inventory stocks held at the beginning of the year. Thus, first of all, it is necessary to construct prices and quantities for the inventory stocks. For constructing inventory stocks, we make use of the following data from the KEO database. The following nominal and real inventory stocks are available at the KEO database:     (1) Finished-goods inventories at the end of the years 1955-2004; (2) Work-in-process inventories at the end of the years 1955-2004; (3) Work-in-process inventories for cultivated assets at the end of the years 1955-2004;    (4) Material inventories at the end of the years 1955-2004.  Since they are stock at the end of the years 1955-2004, we use them as stocks at the beginning of the years 1956-2005. We extend these data series backward to 1955 and 153  This follows the treatment advocated by Diewert (2005a).  148  forward to 2007. For the years 1997-2007, current yen series for inventory stocks (1), (2), and (4) are found in the Annual Report on National Accounts of 2008 (1993 JSNA); Part 2 STOCKS, 5. Supporting Tables, (1) Closing Stocks of Assets/Liabilities for the Nation. For the years 2006-2007, we extrapolate the data series in the KEO data base by using the growth rate in the corresponding data series of the JSNA. Since there is no information on the inventory stocks (3), we assumed that the growth rate of the nominal inventory stocks (3) is the same as that of the nominal inventory stocks (4). For the same periods 1955 and 2005-2006, we extend prices of these four types of inventory stocks using the average growth rate over the recent 5 years. The change in inventories over a year for each of our four types of inventory is regarded as an (investment) output of the market sector and it is equal to the difference between the stock of the inventory class under consideration at the beginning of current year and the stock of the same inventory class at the beginning of the next year. Since we have already prepared estimates for our 4 types of inventory stock, it is straightforward estimates of inventory change for the years 1956-2006. For 1955, we extrapolate the change in inventory backward by using the average growth rate over the following 5 years. Adding change in inventory in 1955 to the inventory stock at the beginning of 1956, we obtain inventory stock at the beginning of 1955. In the end, we obtain estimates for the 4 types of changes in inventories IV1-IV4 for the years 1955-2006. We also obtain estimates for the four types of nominal and real inventory stocks (1)-(4). By applying the user cost formula (53), we can construct prices and quantities of four types of inventory services KIV1-KIV4 for the years 1955-2006. 4. Land Services There are price and quantity series for four types of land in our database:    154  LD1: Agricultural land services;154 LD2: Industrial land services; LD3: Commercial land services;  Our estimates for agricultural land only include the land used for agricultural production. Other types of  land such as waste and fields are excluded from land for agricultural use.  149    LD4: Residential land services.  Quantities of land services are proportional to land stocks at the beginning of the year. Thus, first of all, we have to construct land stocks. 4.1 Four Types of Land For measuring land stocks, we make use of the following data from the KEO database, where estimates for nominal and real land stocks are available for the following types of land:       (1): Land for agricultural use; (2): Land for industrial use; (3): Land for commercial use; (4): Land for residential use; (5): Land for general government.  KEO estimates are supplemented by data taken from the following websites.   Website of Land Use Survey, http://www.mlit.go.jp/hakusyo/tochi/tochi_.html, Ministry of Land, Infrastructure, Transport and Tourism;    Website of Prefectural Land Price Survey, http://tochi.mlit.go.jp/english/index.html, Ministry of Land, Infrastructure, Transport and Tourism  Since the KEO estimates are for stocks at the end of the years 1955-2004, we use them as estimates for the land stocks at the beginning of the years 1956-2005. We extend these data series backward to 1955 and forward to 2006. For the stocks of 1955, we assume that they are equal to the corresponding stocks of 1956. For stocks of 2006, we estimate the real land stocks (1)-(4) by using the growth rates for the area space information found in the website of the Land Use Survey and we also extrapolate nominal land stock (1)-(4) by using the growth rate of nominal land stocks calculated from the real land stocks and land prices that are found in the website of the Prefectural Land Price Survey. For information on land for the general government sector, current yen estimates for land for general government for the years 1996-2006 can be found in the Annual Report on National Accounts of 2008 (1993 JSNA); see Part 2 Stocks, 2 Accounts classified by 150  Institutional Sectors, (3) General Government; Calendar Year in billion yen. We obtain an estimate for the nominal land stock (5) in 2006 by using the growth rate of nominal land for general government using the JSNA just described. The price of land for residential use (4) in 2006 is obtained by using the rate of growth in the land price for general government (5) over 2005-2006. We attribute land for general government as land that is being used for commercial uses. It is not quite true but it is a reasonable approximation. Thus, by subtracting land for general government from land for commercial use, we obtain final estimates for commercial land. Since the stocks of land in the KEO database are stocks at the end of the period, we convert these stocks to beginning of period stocks. We assume that between 1955 and 1956, the area of each type of land is constant and the price of each land changes at the average growth rate over the years 1956-1960. There remains one problem that land for residential use also includes land being used for owner-occupied houses. Since the imputed rent of owner-occupied houses is an output by household sector, it is excluded from the output produced by the market sector. Therefore, the services of the land and residential building stocks for owner-occupied houses should also be excluded from the input of market sector. We will explain how to subtract land for owner-occupied houses from land for residential use in the following section. 4.2 Residential Land Residential land is the land that lies under residential structures. We divide the residential land into the owned residential land and the rental residential land in proportion to the land areas utilized by the two types of residential structures. We calculate the area of the site under owner occupied and rental houses.155 The nominal and real residential land is divided into owned residential land and rental residential land according to their areas of use. The owned residential land will be excluded from land for residential use of the market sector. 4.2.1 Data Sources   155  Housing Survey of Japan, Volume 1, Results for the Whole of Japan (1968) (1973) (1978) (1983) (1988) (1993), Statistics Bureau;  Unfortunately, the land under owner-occupied houses is not always owned land.  151    Housing and Land Survey of Japan, Volume 1, Results for the Whole of Japan (1998) (2003), Statistics Bureau;    Monthly of Construction Statistics (1955 to 2004), Ministry of Land, Infrastructure and Transport;  4.2.2 The Land under the Owned Houses and the Site under the Rental Houses For the years 1968-2006, we use benchmark land areas under owned houses and rental houses, which are available every five years; see Housing and Land Survey of Japan and Housing Survey of Japan. We interpolate these areas between benchmarks using annual “Areas of finished housing sites” reported annually by the Monthly Statistics of Construction. The area of a finished housing site is the area of the site which is currently transformed into residential land. There are four types of site:      (1) Housing sites for a public housing complex; (2) Housing sites for individual houses; (3) Housing sites for re-developm