Index Number Theory and Price Statistics: Sonderausgabe Heft 6/Bd. 230 (2010) Jahrbücher für Nationalökonomie und Statistik 9783110511123, 9783828205314

Citation preview

Jahrbücher für Nationalökonomie und Statistik Journal of Economics and Statistics

Begründet von

Bruno Hildebrand

Fortgeführt von

Johannes Conrad, Ludwig Elster Otto v. Zwiedineck-Südenhorst Gerhard Albrecht, Friedrich Lütge Erich Preiser, Knut Borchardt Alfred E. Ott und Adolf Wagner

Herausgegeben von

Peter Winker Wolfgang Franz, Werner Smolny Peter Stahlecker, Adolf Wagner Joachim Wagner, Dietmar Wellisch

Band 2 3 0

Lucius & Lucius Stuttgart 2010

© Lucius & Lucius Verlagsgesellschaft mbH • Stuttgart • 2010 Alle Rechte vorbehalten Satz: Mitterweger 8c Partner Kommunikationsgesellschaft mbH, Plankstadt Druck und Bindung: Neumann Druck, Heidelberg Printed in Germany

Index Number Theory and Price Statistics

Herausgegeben v o n Peter M . v o n der Lippe W. E r w i n D i e w e r t

M i t Beiträgen v o n Auer, Ludwig von, Trier, Germany Balk, Bert M., Rotterdam, NL deHaan, Jan, The Hague, NL Diewert, W. Erwin, Vancouver, Canada Greenlees, John, Washington, USA Linz, Stefan, Wiesbaden, Germany Lippe, Peter von der, Duisburg-Essen, Germany

Lucius &c Lucius • Stuttgart 2 0 1 0

Mehrhoff, Jens, Frankfurt a.M., Germany Nishimura, Kiyohiko G., Tokyo, Japan Shimizu, Chihiro, Chiba, Japan Watanabe, Tsumotu, Tokyo, Japan Williams, Elliot, Washington, USA

Anschriften der Herausgeber des Themenheftes Prof. Dr. Peter Michael von der Lippe Universität Duisburg-Essen Campus Duisburg Mercator School of Business Lotharstrasse 65, LB 146 4 7 0 5 7 Duisburg E-Mail: [email protected] W. Erwin Diewert Professor of Economics Department of Economics The University of British Columbia #997-1873 East Mall Vancouver, BC V 6 T 1Z1 Canada E-Mail: [email protected]

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar ISBN 9 7 8 - 3 - 8 2 8 2 - 0 5 3 1 - 4

© Lucius Sc Lucius Verlagsgesellschaft mbH Stuttgart 2 0 1 0 Gerokstraße 51, D - 7 9 1 8 4 Stuttgart Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlags unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen und Mikroverfilmungen sowie die Einspeicherung und Verarbeitung in elektronischen Systemen.

Satz: Mitterweger & Partner Kommunikationsgesellschaft mbH, Plankstadt Druck und Bindung: Neumann Druck, Heidelberg Printed in Germany

Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 2010) Bd. (Vol.) 230/6

Inhalt des 230. Bandes Abhandlungen/Original Papers Ahlfeldt, Gabriel M., Wolfgang Maennig, Impact of Non-smoking Ordinances on Hospitality Revenues: The Case of Germany Ahlfeldt, Gabriel M., Wolfgang Maennig, Hanno Scholz, Erwartete externe Effekte und Wahlverhalten: Das Beispiel der Münchner Allianz-Arena Expected External Effects and Voting: The Case of the Munich Allianz-Arena. Arntz, Melanie, Stephan L. Thomsen, Are Personal Budgets a Financially Sound Reform Option for the German Long-Term Care Insurance? Aschhoff, Birgit, Who Gets the Money? The Dynamics of R & D Project Subsidies in Germany Auer, Ludwig von, Drobisch's Legacy to Price Statistics Balk, Bert M., Lowe and Cobb-Douglas Consumer Price Indices and their Substitution Bias Baismeier, Benjamin, Achim Buchwald, Heiko Peters, Auswirkungen von Mehrfachmandaten deutscher Vorstands- und Aufsichtsratsvorsitzender auf den Unternehmenserfolg The Impact of Multiple Board Memberships of CEOs and Chairmen of Supervisory Boards on Corporate Performance in Germany Belke, Ansgar, Die Auswirkungen der Geldmenge und des Kreditvolumens auf die Immobilienpreise - Ein ARDL-Ansatz für Deutschland Money, Credit and House Prices - An ARDL-Approach for Germany Braakmann, Nils, Fields of Training, Plant Characteristics and the Gender Wage Gap in Entry Wages Among Skilled Workers - Evidence from German Administrative Data Burgstaller, Johann, Bank Lending and Monetary Policy Transmission in Austria de Haan, Jan, Hedonic Price Indexes: A Comparison of Imputation, Time Dummy and'Re-Pricing'Methods Diewert, Erwin, User Costs versus Waiting Services and Depreciation in a Model of Production Diewert, Erwin, Peter von der Lippe, Notes on Unit Value Index Bias Dreger, Christian, Reinhold Kosfeld, Do Regional Price Levels Converge? Eickmeier, Sandra, Analyse der Übertragung US-amerikanischer Schocks auf Deutschland auf Basis eines FAVAR A FAVAR-based Analysis of the Transmission of US Shocks to Germany Görlitz, Katja, The Development of Employers' Training Investments Over Time A Decomposition Analysis Using German Establishment Data Greenlees, John S., Elliot Williams, Reconsideration of Weighting and Updating Procedures in the US CPI Gudehus, Timm, Logik des Marktes. Marktordnung, Marktverhalten und Marktergebnisse Logic of Markets. Market Rules, Behaviour of Actors, and Market Outcome . Horbach, Jens, The Impact of Innovation Activities on Employment in the Environmental Sector - Empirical Results for Germany at the Firm Level Jahn, Elke J., Reassessing the Pay Gap for Temps in Germany Jochem, Axel, International Financial Competitiveness and Incentives to Foreign Direct Investment

506-521

2-26 378-402 522-546 673-689 726-740

547-570 138-162 27-41 163-185 772-791 759-771 690-708 274-286 571-600 186-207 741-758

601-629 403-419 208-233 42-58

IV • Inhalt des 230. Bandes

Kholodilin, Konstantin A., Jan-Oliver Menz, Boriss Siliverstovs, What Drives Housing Prices Down? Evidence from an International Panel Linz, Stefan, Regional Consumer Price Differences Within Germany Mehrhoff, Jens, Aggregate Indices and Their Corresponding Elementary Indices . Mohrenweiser, Jens, Uschi Backes-Gellrter, Die Wirkung des Betriebsverfassungsgesetzes am Beispiel der Freistellung von Betriebsräten - ein Beitrag zur Rechtstatsachenforschung The Effect of the Works Council Act on Paid Leave of Absence of Works Councillors Pierdzioch, Christian, Georg Stadtmann, Herdenverhalten von Wechselkursprognostikern? Herd Behavior of Exchange Rate Forecasters? Pfeifer, Christian, Work Effort During and After Employment Probation: Evidence from German Personnel Data Reitz, Stefan, Jan C. Rülke, Georg Stadtmann, Regressive Oil Price Expectations Toward More Fundamental Values of the Oil Price Shimizu, Chihiro, Kiyohiko G. Nishimura, Tsutomu Watanabe, Housing Prices in Tokyo: A Comparison of Hedonic and Repeat Sales Measures Stapleford, Thomas, The Cost of Living in America. A Political History of Economic Statistics, 1880-2000 Stops, Michael, Thomas Mazzoni, Matchingprozesse auf beruflichen Teilarbeitsmärkten Job Matching on Occupational Labour Markets Südekum, Jens, Human Capital Externalities and Growth of High- and Low-Skilled Jobs Tamm Marcus, Child Benefit Reform and Labor Market Participation Tenhofen, Jörn, Guntram B. Wolff, Kirsten H. Heppke-Falk, The Macroeconomic Effects of Exogenous Fiscal Policy Shocks in Germany: A Disaggregated SVAR Analysis Thiele, Silke, Erhöhung der Mehrwertsteuer für Lebensmittel: Budget- und Wohlfahrtseffekte für Konsumenten Increase of the Value Added Tax (VAT): Budgetand Welfare-Effects for Consumers Woessmann, Ludger, Institutional Determinants of School Efficiency and Equity: German States as a Microcosm for OECD Countries Wiibker, Ansgar, Dirk Sauerland, Achim Wübker, Beeinflussen bessere Qualitätsinformationen die Krankenhauswahl in Deutschland? Does Better Quality Information Affect Hospital Choice in Germany? Ziegler, Andreas, Z-Tests in Multinomial Probit Models under Simulated Maximum Likelihood Estimation: Some Small Sample Properties

59-76 814-831 709-725

420-435 436-453 77-91 454-466 792-813

287-312 92-114 313-327

328-355 115-130 234-270

467-490 630-652

Inhalt des 230. Bandes- V

Buchbesprechungen / Book Reviews Baker, Dean, Taking Economics Seriously Berman, Eli, Radical, Religious and Violent: The New Economics of Terrorism . Brenke, K., K. F. Zimmermann (Hrsg.), Die Wirtschaft in Ostdeutschland 20 Jahre nach dem Fall der Mauer - Rückblick, Bestandsaufnahme, Perspektiven Buccirossi, Paolo (ed.), Handbook of Antitrust Economics Gnos, Claude, Louis-Philippe Rochon (eds.), Monetary Policy and Financial Stability - A Post-Keynesian Agenda Grab, Johannes, Econometric Analysis in Poverty Research Klein, Lawrence R. (ed.), The Making of National Economic Forecasts Mandler, Martin, Geldpolitische Reaktionsfunktionen und makroökonomische Unsicherheit Manzur, Meher (ed.), Purchasing Power Parity Neck, Reinhard (Hrsg.), Die Österreichische Schule der Nationalökonomie . . . . Neumark, David, William L. Wascher, Minimum Wages Nowotny, E., P. Mooslechner, D. Ritzberger-Grünwald (eds.), The Integration of European Labour Markets Parisi, Francesco, Charles K. Rowley (eds.), The Origins of Law and Economics Poot, Jacques, Brigitte Waldorf, Leo van Wissen (eds.), Migration and Human Capital Schmidtchen, D., C. Koboldt, J. Helstroffer, B. Will, G. Haas, S. Witte, Transport, Welfare and Externalities: Replacing the Polluter Pays Principle with the Cheapest Cost Avoider Principle Schneider, Friedrich, Johannes Kepler (eds.), The Economics of the Hidden Economy Schulze, Günther G. (Hrsg.), Reformen für Deutschland. Die wichtigsten Handlungsfelder aus ökonomischer Sicht Stapleford, Thomas, The Cost of Living in America. A Political History of Economic Statistics, 1880-2000 Stiglitz, Joseph, Im freien Fall. Vom Versagen der Märkte zur Neuordnung der Weltwirtschaft Wagner, Adolf (Hrsg.), Empirische Wirtschaftsforschung heute Werding, Martin, Robert Jäckle, Christian Holzner, Marc Piopiunik, Ludger Wössmann, Humankapital in Deutschland: Wachstum, Struktur und Nutzung der Erwerbseinkommenskapazität von 1984 bis 2006 Wickstrom, Bengt-Ame (Hrsg.), Finanzpolitik und Schattenwirtschaft Wiegmann, Jochen Gert Arend, Produktivitätsentwicklung in Deutschland Winkler, Othmar W., Interpreting Economic and Social Data - A Foundation of Descriptive Statistics Wray, L. Randall, Mathew Forstater (eds.), Keynes and Macroeconomics after 70 Years: Critical Assessments of The General Theory

655 356 653 357 131 832 833 835 359 360 364 271 491 493 495 132 367 838 496 498 369 371 133 373 500

VI • Gutachter zum 230. Jahrgang (2010)

Die Gutachter zum 230. Jahrgang der Jahrbücher für Nationalökonomie und Statistik (01.01.2010 bis 31.12.2010) Im N a m e n der Herausgeber danke ich allen Wissenschaftlerinnen und Wissenschaftlern, die in diesem Zeitraum bereit waren, f ü r die Jahrbücher f ü r Nationalökonomie und Statistik Manuskripte zu begutachten. M i t ihrer Hilfe sind wir dem Z i e l , eine möglichst schnelle Entscheidung über die Publikation der Einreichungen herbeizuführen, ziemlich nahe gekommen. Die Autoren konnten die detaillierten Verbesserungsvorschläge aufnehmen, und davon hat die Qualität der Manuskripte stark profitiert. Peter Winker Amisano, Gianni, European Central Bank, Frankfurt a . M . Andreou, Panayiotis, D u r h a m Business School Arni, Patrick, University of Lausanne Arnold, Lutz, Universität Regensburg Balázs, Egert, O E C D , Paris Balk, Bert M . , Statistics Netherlands, The H a g u e Beckmann, Michael, Universität Basel Bender Stefan, LAB Nürnberg Beninger, Denis, Z E W , M a n n h e i m Bergemann, Annette, Universität M a n n h e i m Biewen, M a r t i n , Universität Tübingen Bode, Eckhardt, The Kiel Institute for the World Economy, Kiel Bohl, Martin T., Universität Münster B r a a k m a n n , Nils, Leuphana Universität Lüneburg Breitung, J ö r g , Universität Bonn Breuer, Claus Christian, Universität DuisburgEssen Brueggemann, R a l f , Universität Konstanz Buscher, Herbert, I W H Halle Büttner Thiess, Friedrich-Alexander-Universität Erlangen-Nürnberg Clots Figueras, Irma, Universidad Carlos III de M a d r i d C o r r a d o , Carol, A . , The Conference B o a r d , N e w York C r o u x , Christophe, K . U . Leuven, Belgium D ' A m b r o s i o , Conchita, Università Bocconi, Milano d ' E r a s m o , Pablo, University of M a r y l a n d de H a a n , J a n , Statistics Netherlands, The H a g u e Dées, Stephane, European Central Bank, Frankfurt a. M . Demary, M a r k u s , Institut der deutschen Wirtschaft, Köln D i e w e r t , W. Erwin, University of British Columbia, Vancouver, C a n a d a Dilger, A l e x a n d e r Westfälische WilhelmsUniversität Münster Eckwert, Bernhard, Universität Bielefeld Eichhorn, Wolfgang, Karlsruhe Fehr, H a n s , Universität Würzburg

Fendel, R a l f , W H U - Otto Beisheim School of Management Fischer, Christoph, Deutsche Bundesbank, Frankfurt a . M . Flaig, Gebhard, Ludwig-MaximiliansUniversität München Fürnkranz-Prskawetz, A l e x i a , Vienna University of Technology Gasche, M a r t i n , M E A , Universität M a n n h e i m Geis, W i d o , I f o Institut, München Genser, Bernd, Universität Konstanz Görlitz, K a t j a , R W I Essen Götz, Georg, Justus-Liebig-Universität Gießen G r a b k a , M a r k u s , D I W Berlin Gradin, Carlos M . , Universidade de Vigo, Spain Graf L a m b s d o r f f , J o h a n n , Universität Passau G r a m m i g , J o a c h i m , Universität Tübingen Gregg, Paul, University of Bristol Heilemann, Ullrich, Universität Leipzig Helmedag, Fritz, T U Chemnitz Henke, Klaus-Dirk, T U Berlin Herzog-Stein, Alexander; Hans-Böckler-Stiftung, Düsseldorf H o f f m a n n , Johannes, Deutsche Bundesbank, Frankfurt a . M . Holm-Hadulla, European Central B a n k , Frankfurt a . M . Holzner, Christian, ifo Institut f ü r Wirtschaftsforschung, München Huber; Peter, W i F O , Wien Hübler, O l a f , Leibniz Universität H a n n o v e r Hülsewig, Oliver, Hochschule München Janz, Norbert, F H Aachen Jirjahn, U w e , Universität Trier Kajuth, Florian, Deutsche Bundesbank, Frankfurt a . M . Kappler, M a r c u s , Z E W , Mannheim K i f m a n n , Mathias, Universität Augsburg Kluve, Jochen, R W I Essen K o c h , Susanne, I A B Nürnberg K o h n , Karsten, K f W Bankengruppe, Frankfurt a. M . Kolmar, M a r t i n , Universität St. Gallen Köthenbürger, M a r k o , University of Copenhagen

Gutachter zum 230. Jahrgang (2010) • VII

Kreyenfeld, Michaela, Universität Rostock Kuglet Peter, Universität Basel Liesenfeld, Roman, Christian-Albrechts-Universität zu Kiel Löschel, Andreas, ZEW Mannheim Maiterth, Ralf, Leibniz Universität Hannover Meckl, Jürgen, Justus-Liebig-Universität Gießen Mehrhoff, Jens, Deutsche Bundesbank, Frankfurt a. M. Menkhoff, Lukas, Leibniz Universität Hannover Merz, Joachim, Leuphana Universität Lüneburg Meyer, Mark, GWS, Osnabrück Mohrenweiser, Jens, ZEW Mannheim Mühlenweg, Andrea, Z E W Mannheim Nakamura, Alice, University of British Columbia, Vancouver, Canada Overesch, Michael, Universität Mannheim Paha, Johannes, Justus-Liebig-Universität Gießen Pakko Michael, UALR Little Rock, USA Pedersen, Michael, Banco Central de Chile, Santiago de Chile Peters, Heiko, Sachverständigenrat, Wiesbaden Pijoan-Mas, Josep, CEMFI, Madrid Pohlmeier, Winfried, Universität Konstanz Ragnitz, Joachim, ifo Institut, NL Dresden Rao, D. S. Prasada, University of Queensland, Brisbane Reimers, Hans-Eggert, Hochschule Wismar Reinhold, Steffen, Universität Mannheim Reiß, Winfried, Universität Paderborn Riebe, Martin, Statistics Sweden, Stockholm Rodriguez-Pose, Andres, London School of Economics Rotfuss, Waidemac, ZEW, Mannheim Rottmann, Horst, Hochschule Amberg-Weiden Schellhorn, Martin, Christian-AlbrechtsUniversität zu Kiel Schimmelpfennig, Jörg, Ruhr-Universität Bochum Schindler, Felix, ZEW Mannheim

Schreyer, Paul, OECD, Paris Schröder, Carsten, Christian-AlbrechtsUniversität zu Kiel Schwager, Robert, Universität Göttingen Silver Mick, International Monetary Fund, Washington Spiess, Martin, Universität Hamburg Staal, Klaas, Universität Bonn Stadler, Manfred, Universität Tübingen Stancanelli, Elena, Université Cergy-Pontoise, Ile de France Steger, Thomas, Universität Erfurt Steinet; Viktor, DIW Berlin Tamm, Marcus, RWI Essen Tillmann, Peter, Justus-Liebig-Universität Gießen Kaplan, Todd, R., University of Exeter Trenkler, Carsten, Universität Mannheim van der Grient, Heymerik, Statistics Netherlands, The Hague van der Ploeg, Rick, University of Oxford von der Lippe, Peter, Universität Duisburg-Essen Voronkova, Svitlana, ZEW Mannheim Weichenrieder, Alfons, Universität Frankfurt a.M. Westerhoff, Frank, Universität Bamberg Wigger, Berthold, Karlsruher Institut für Technologie, Karlsruhe Winter, Joachim, Ludwig-MaximiliansUniversität München Wolf, Elke, Hochschule München Wolswijk, Guido, EZB, Frankfurt a. M. Wolter; Stefan C., Universität Bern Wrede, Matthias, Universität Marburg Zanola, Roberto, Università degli Studi del Piemonte Orientale, Alessandria Zietz, Joachim, Middle Tennessee State University Zimmermann, Volkes KFW Bankengruppe, Frankfurt a . M . Zweimüller, Martina, Johannes Kepler University Linz Zwick, Thomas, Ludwig-MaximiliansUniversität München

Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 2010) Bd. (Vol.) 230/6

Inhalt / Contents Introduction to the Special Issue on Index Number Theory and Price Statistics

660-672

Abhandlungen/Original Papers Aueri Ludwig von, Drobisch's Legacy to Price Statistics Diewert, W.Erwitt, Peter von der Lippe, Notes on Unit Value Index Bias. . . Mehrhoff, Jens, Aggregate Indices and Their Corresponding Elementary Indices Balk, Bert M., Lowe and Cobb-Douglas Consumer Price Indices and their Substitution Bias Greenlees, John S., Elliot Williams, Reconsideration of Weighting and Updating Procedures in the US CPI Diewert, W. Erwin, User Costs versus Waiting Services and Depreciation in a Model of Production de Haan, Jan, Hedonic Price Indexes: A Comparison of Imputation, Time Dummy and 'Re-Pricing' Methods Shimizu, Chihiro, Kiyohiko G. Nishimura, Tsutomu Watanabe, Housing Prices in Tokyo: A Comparison of Hedonic and Repeat Sales Measures . . Linz, Stefan, Regional Consumer Price Differences Within Germany

673-689 690-708 709-725 726-740 741-758 759-771 772-791 792-813 814-831

Buchbesprechungen / Book Reviews Grab, Johannes, Econometric Analysis in Poverty Research Klein, Lawrence R. (ed.), The Making of National Economic Forecasts . . . Mandler, Martin, Geldpolitische Reaktionsfunktionen und makroökonomische Unsicherheit Stapleford, Thomas, The Cost of Living in America. A Political History of Economic Statistics, 1880-2000 Bandinhalt des 230. Jahrgangs der Zeitschrift für Nationalökonomie und Statistik Contents of Volume 230 of the Journal of Economics and Statistics

832 833 835 838

Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 2010) Bd. (Vol.) 230/6

Introduction to the Special Issue on Index Number Theory and Price Statistics By Peter von der Lippe, Duisburg-Essen, and W. Erwin Diewert, Vancouver* 1

Introduction

Index theory as well as price statistics has undergone a remarkable change in the last twenty years. Some of the influential publications were the System of National Accounts 1993,1 the famous Reports of the Boskin 2 and Schultze 3 Commissions, the ongoing creation of a number of Manuals and recommendations of best practice, created by international groups of experts and edited by international organisations such as the ILO, IMF, OECD, UNECE, Eurostat, or the World Bank. Thus we now have best practice Manuals for Productivity Measurement, Consumer Prices, Producer Prices, Export and Import Prices, the Measurement of Capital, the Measurement of Non-Market Production, 4 and additional Manuals or Handbooks for House Prices (or Residential Property Prices) and for International and Interregional Comparisons of Consumer Prices are in progress. Such initiatives have not only contributed enormously to a fundamental reorientation of index number theory but also to the creation of new indices in official price statistics in a process which is from the outset based on an intense international cooperation. A most prominent project of this kind was to establish the Harmonised Index of Consumer Prices (HICP) in the European Union and there are a number of other new indices already existing or gradually emerging (e. g. price indices for labour costs, certain services and house prices). The interesting point is that new results in index number theory were explicitly taken into account in these efforts to improve official statistics in the last twenty or so years. Moreover we now not only experience more co-operation between the two spheres of activity which both are the subject of this special issue, that is index number theory and price statistics, but we may also note that the scope of index theory is now much broader than before. In former days, the focus of index number theory was very much confined to the determination of the appropriate formula for comparing prices (or quantities) related to two periods in time. However, we now encounter mathematical and conceptual studies related to many other economic measurement problems such as quality adjustment, productivity measurement, asset pricing, problems of aggregation and deflation, interspatial comparisons and other problems inspired by the practice of price statistics. * The authors thank Bert Balk for helpful comments. The financial support of the S S H R C of C a n a d a is gratefully acknowledged. 1 See Eurostat (1993) 2 See Boskin et al. (1996). 3 See Schultze and Mackie (2002). 4 See Schreyer (2001), the I L O (2004), I M F (2004), I M F (2009) and Schreyer (2009, 2010).

Introduction to the Special Issue on Index Number Theory and Price Statistics • 6 6 1

It is appropriate to give an account of some of the impacts we owe to the above mentioned international activities because with this background material, it may be easier to understand why certain topics are addressed in this special issue. The System of National Accounts 1993 recommended a move to chain indices both in inflation measurement and the deflation of aggregates so that volumes could be derived from these deflated values. This turned out to be quite a fundamental change particularly when contrasted with the German price index tradition, which recommended fixed base indices. Furthermore, the Boskin Report strongly recommended that a consumer price index (CPI) should be patterned after the model of the (true) cost-of-living index (or better known as COLI) as it was developed in the economic approach to index number theory, as opposed to the fixed basket (and not chained) Laspeyres price index. The modified Laspeyres (1871) or Lowe (1823) index used to be the prevailing type of Consumer Price Index in large parts of Europe. For many years the COLI seemed to be not eligible for a target index in official statistics due to the problems involved in the notion of operationalizing the concept of utility in an index number context. This changed, however, when W. Erwin Diewert (1976, 1978) succeeded in showing that certain symmetrically weighted price index formulas (using the quantities in the base period as well as in the current reference period as weights in a symmetric fashion) like the indices of Fisher (1922), Törnqvist (1937) or Walsh (1901, 1921) are superlative indices in that they approximate the COLI under fairly general conditions (and also have favourable axiomatic properties). Two papers in this special issue of the Jahrbücher für Nationalökonomie und Statistik5 explicitly refer to the COLI concept of a CPI. In addition to vigorously advocating the COLI paradigm, the Boskin Report also emphasized some important practical aspects of index methodology. Two examples are the hedonic approach to quality adjustment and the choice of the "best" unweighted elementary index, which is an index for the lowest level of aggregation as compared to upper level aggregation where expenditure weights are available. Both hedonics and index compilation in two stages are comparatively new topics of index theory, and they are both addressed in contributions to this special issue. Elementary indices such as the formula of Dutot (1738), Carli (1764) or Jevons (1865) are unweighted indices while in the second stage of aggregation, when traditional formulas like Laspeyres (1871) and Paasche (1874) indices are calculated, quantities are introduced as weights for the prices. Another possibility, examined in the paper of Diewert and von der Lippe in this special issue, is to use unit values rather than the unweighted elementary price indices used in the first stage of a two stage index compilation. This special issue is another example of a productive and prolific international co-operation with authors coming from various countries and being influenced by different schools and traditions of thinking in economics and statistics. We hope that the benefits of a joint multinational effort will become visible in this issue and we also appreciate that this journal, with its long tradition in index number theory, was open to devoting an issue to such often neglected topics like index number theory and price statistics. In the following sections, we will provide brief introductions to the papers collected for this special issue of the Jahrbücher für Nationalökonomie und Statistik (the Journal of Economics and Statistics). 5

See the papers by Balk and by Greenlees and Williams in this issue.

662 • P. von der Lippe and W.E. Diewert

2

Drobisch, unit value price indexes and elementary indexes

This journal takes pride in the fact that many famous German index number theorists have published their seminal papers in this very journal. Not only Laspeyres and Paasche did so but also the less well known Moritz Wilhelm Drobisch. It is therefore no coincidence that this special issue of the journal starts with the paper of Ludwig von Auer, Drobisch's Legacy to Price Statistics, which is a sketch of life and work of a sadly neglected German scholar who among other things also contributed to index theory. According to von Auer, Drobisch (1871) proposed among other index formulas his unit value index (UVI), defined as the ratio of two all-items unit values, as his favourite price index. This index may be used as an index at lower levels of aggregation, where only closely related items are aggregated, but it is not a useful price index at higher levels of aggregation, where quantity units cannot be summed in a meaningful manner. However for Drobisch, his price index was intended to be an upper aggregation level index and he thought that his index stood on the same footing as the formula of Laspeyres (1871) or Paasche (1874). Unfortunately, the term unit value index is used for both situations; both when aggregating over similar items as well as when aggregating over very different products. This leads to some terminological problems, which had to be solved in the paper of Diewert and von der Lippe. These authors preferred to call the unit value index for the second, more problematic case of an all-item index, the Drobisch price index. In a rather formal manner, the indices (throughout this issue we use "indices" and "indexes" as synonyms) of Laspeyres and Paasche, PL and Pp respectively, could be regarded as special cases of Drobisch's index (they emerge if either the base period quantity vector qo or the current period quantity vector q, are used in both the numerator and denominator of the Drobisch formula). However, Drobisch rejected the formulas PL and Pp which later were to become famous. Moreover Drobisch did not effectively realise the axiomatic shortcomings of his formula. According to von Auer there must have been a vigorous academic dispute between Drobisch and Laspeyres and of course, Drobisch came out on the losing end of this dispute. Loosing also seems to have made Drobisch feel to have been robbed of the merits of being the inventor of both the Laspeyres and the Paasche formula. In our view (with which von Auer might perhaps disagree), Drobisch's complaint for lack of academic recognition as compared to Laspeyres, who acquired a lot of posthumous fame, is understandable though possibly not justified since he examined the formulas PL and Pp and he explicitly rejected them in favour of his quite objectionable and rightly much less acceptable formula. Such disputes may have been responsible for the fact that Drobisch's interest in index theory was not lasting, and that he never properly responded to the valid criticism of his formula and therefore failed to become as prominent as Laspeyres for example (who by the way also was unable to fully realise in his lifetime the relevance of the formula which later was to bear his name). It is just because Drobisch fell into oblivion that von Auer's interesting narrative of life and work of this almost unknown German mathematician and philosopher is a valuable contribution. An extensive analysis of the just defined Drobisch price index PD can be found in W. Erwin Diewert and Peter von der Lippe in their paper Notes on Unit Value Index Bias. The problem with the index PD is that it is affected by the changing structure

Introduction to the Special Issue on Index Number Theory and Price Statistics • 663

of the quantity weights in the numerator and denominator of the Drobisch index. The weights Sr in the period t unit value P( = p t -(q t /2 , n= iq n t ) = p'-S' therefore reflect a changing structural component in the numerator and denominator of the Drobisch price index, PD = PVP° = p'-S'/p^S 0 . Thus the Drobisch index can change even if prices do not change; i. e., we can have PD ^ 1 even if p° = p'. 6 Another unfavourable axiomatic property of the Drobisch index when aggregation takes place over very heterogeneous items is that it is not invariant to changes in the units of measurement. 7 However, the focus of this paper is not on the axiomatic properties of the Drobisch price index PD, but on determining the bias of PD when aggregating over "reasonably" homogeneous (but not completely homogeneous) products as compared to standard indexes, like PL, Pp and Pp It turns out that the bias in PD relative to these indexes can be defined in terms of covariances between prices in the base period 0 and various measures of the change in the (structure of) quantities between base and current periods. Another topic discussed in this paper is an index which is called hybrid Paasche index (PHP) (and its Laspeyres counterpart) by Diewert and von der Lippe and hitherto only rarely examined in detail in the literature on unit value indexes. 8 It is clear that the Drobisch index requires that quantities of items can meaningfully be added up to a total quantity (for which a common unit of measurement across all items is a necessary but not sufficient condition) and that this condition is not fulfilled for an aggregate as wide as an all items CPI for example. It is, however, reasonable to make use of ratios of unit values as building blocs on the lower aggregation level of an index compiled in two stages. The PHP index uses unit values in the first stage and the Paasche formula in the second stage of aggregation (correspondingly, the hybrid Laspeyres index PHL uses the Laspeyres formula in the second stage). Again a bias formula is derived, now for the bias of PHP (rather than PD) relative to Pp and again it turns out that the bias is determined by a covariance between base period prices and changes in the quantities. One of conditions for a vanishing covariance suggests the following rule for choosing how to construct the subaggregates: in order to minimize bias (relative to the Paasche price index), use unit value aggregation over products that sell for the same price in the base period. On the basis of the results concerning PD and PHP the paper also considered the question: does disaggregation of a unit value into more homogeneous subgroups reduce the unit value bias? The answer seems to be: probably yes. Jens Mehrhoff in his paper Aggregate Indices and Their Corresponding Elementary Indices also deals with the problem of price indices compiled in two stages. This paper explicitly takes into account, that in practice, price indices are in general compiled in a two stage procedure. The problem Mehrhoff addressed is to find for the three second stage or weighted indices of Laspeyres (PL), Paasche (Pp) and Fisher (PF) the "best" corresponding elementary indices (first stage indices). Here the first stage uses well known unweighted price indices; that is, the formulas of Carli and Jevons are considered (or more precisely: all indices that can be derived as special cases of the so called power means or generalised means or means of order r). 9 The practical implication of this might be that there can be situations which do not permit calculation of PL, P p and PF, perhaps 6 7 8 9

See the paper by Diewert and von der Lippe for definitions of the notation. See Balk (1995, 1998) and (2008) on the axiomatic approach to index number theory. Parniczky (1974) initiated this line of research and Balk (2008) significantly added to it. See Hardy et al. (1934) for the properties of means of order r.

664 • P. von der Lippe and W.E. Diewert

due to missing information about quantities, and it then may be reasonable to approximate the desired index by an unweighted price index which uses prices only. In which sense does a first stage unweighted (elementary) index correspond to a second stage weighted index in Mehrhoff's study? According to Mehrhoff, corresponding is defined as best numerically approximating the target index PL, P p or PF respectively. This type of methodology was also used by Shapiro and Wilcox (1997). In Mehrhoff's view, the distinctive feature of his approach to assess elementary indices is an attempt to give an (intuitive or economic) interpretation to the notion of correspondence. His interpretation rests on assuming that the joint probability distribution of prices and quantities in both periods, 0 and t is a bivariate log-normal distribution (LND). This possibly doubtful LND assumption provides a framework in which the relationship between elementary (unweighted) and second stage (weighted) indices can be studied because the LND assumption determines both the distribution of the price relatives (the raw material of elementary indices) and the distribution of the expenditure shares (as weights for the second stage). His assumption may also help to make the choice of the unweighted index on the first stage data-driven. Note that the power mean can take on any value between the lowest and the highest price relative depending on the power r (-00 < r < +oo) of the mean of order r. Thus the task of finding the elementary index (equal to a mean of order r) that corresponds to PL, P p or PF now boils down to determination of the correct power r of the generalised mean. Mehrhoff succeeded in showing that the expectations of PL and P p and the order r of the generalised mean can all be expressed in terms of LND-moments. Next Mehrhoff attempts to give an interpretation to the moments. For this purpose Mehrhoff introduced a partial adjustment model (PAM) which allows him to give an elasticity interpretation to the moments of the LND. It should be borne in mind, however, that his elasticity (/? in the paper) is not a price elasticity of demand or supply because there is no demand or supply function involved, but his parameter is associated with an equation describing an adjustment process. The economic meaning of the PAM model and the consistency of the assumptions underlying the LND and the PAM may be called in question. Nonetheless the ideas of Mehrhoff are innovative and worthy of notice and discussion. 3

Problems associated with the construction of a Consumer Price Index

In this section, we briefly describe the two papers in this special issue that deal with measurement problems associated with the construction of a Consumer Price Index (CPI). The first paper is by Bert Balk, who holds positions at Statistics Netherlands and the Rotterdam School of Management at Erasmus University. He is one of the most prolific writers on index number theory of all time. The title of his paper is Lowe and Cobb Douglas Consumer Price Indices and their Substitution Bias. Many statistical agencies use the Lowe (1823) index as the basis for their CPI. This index uses an implicit quantity vector pertaining to household expenditures in a base year b in order to price out this basket at the prices of the current month t and at the prices of a base month 0 and then the ratio of these expenditures is defined to be the monthly CPI for the country. Chapter 17 of the Consumer Price Index Manual10 written by Balk and 10

See the ILO (2004).

Introduction to the Special Issue on Index Number Theory and Price Statistics • 665

Diewert works out the likely bias of the Lowe index relative to a Koniis (1924) True Cost of Living Index. In the present paper, Balk undertakes a similar exercise where he attempts to determine the bias of a Cobb-Douglas price index 11 relative to a Cost of Living index. The Cobb-Douglas index is a weighted geometric mean of the monthly price relatives where the weights are the household expenditure shares in the base year b. This index is just as easy to construct as the Lowe index and hence it is a practical alternative that statistical agencies should consider using as their CPI concept. In the present paper, Balk reworks and improves the earlier analysis on the bias in the Lowe index and he develops some new formulae for the bias in the Cobb-Douglas index. Unfortunately, as Balk notes towards the end of his paper, it is difficult to draw a firm conclusion about the magnitude of the likely bias in the Cobb-Douglas or Geometric Young index relative to a Cost of Living index. The second paper in this section that deals with possible bias in the Lowe and Geometric Young indexes as approximations to a Cost of Living index is by two researchers in the U. S. Bureau of Labor Statistics (BLS), John Greenlees and Elliot Williams. The title of their paper is Reconsideration of Weighting and Updating Procedures in the US CPI. In 2002, the BLS made two important changes to their CPI, which was previously based on a Lowe index: (i) they increased the frequency of expenditure weight updates in their headline CPI, the CPI-U, from approximately 10 years to 2 years and (ii) they introduced a new supplemental CPI using the superlative formula Tornqvist formula, 12 the C-CPI-U. This new index was introduced to provide a closer approximation to a Cost of Living Index (COLI). The availability of eight calendar years of final C-CPI-U data along with expenditure data from four consecutive two-year CPI-U base periods gave Greenlees and Williams the opportunity to analyze both the effects of more frequent basket updating and the change to a superlative formula. Thus Greenlees and Williams compute several alternative indexes over the period 1 9 9 9 - 2 0 0 7 including several variants of the Lowe type CPI-U that vary the updating period as well as the superlative Tornqvist and Fisher indexes and they also compute the Geometric Young index which was studied in Balk's paper. The authors note that it is not possible to compute a superlative index in real time and hence the issue of whether more frequent updating of the annual basket will move the resulting index closer to a superlative index is important, since the CPI-U is widely used in government tax and indexed bond programs, largely because unlike the superlative C-CPI-U it is not subject to revision. Thus Greenlees and Williams note that although the BLS and many economists believe that the superlative C-CPI-U is a closer approximation to a COLI, improvements to the CPI-U are of the utmost practical importance. The first important result that Greenlees and Williams establish is that the Fisher and Tornqvist indexes are quite close together in every year but one, 2000, although the yearto-year changes in the Tornqvist are always slightly higher. The authors attribute the apparent divergence in the first year of their sample period to the greater sensitivity of the Fisher index to extreme data points, which is a very interesting observation. Greenlees and Williams also computed chained Laspeyres and Paasche indexes and they find that over their sample period, the chained Laspeyres index rises about 2.9 percentage points more than the chained Paasche. On a per year basis, they found that the 11 12

This index is also known as a geometric Young index. See Tornqvist (1936) and Tornqvist and Tornqvist (1936).

666 • P. von der Lippe and W.E. Diewert

chained Laspeyres index rises about 0.18 percentage point per year more rapidly, and the chained Paasche 0.18 percentage point more slowly, than the corresponding chained Fisher. In their Figure 2, Greenlees and Williams show that the effects of changing the annual basket more frequently does cause the corresponding Lowe type CPI-U to come closer to the chained Tornqvist index but that more frequent basket updates cannot fully close the gap. In their Figure 3, they compare a Lowe index with 5 year basket updating with the CPI-U (a Lowe index with 2 year basket updating) and with the C-CPI-U (a superlative index) and finally with a Geometric Lowe index. The results are perhaps as one might expect, the Lowe indexes lie well above the corresponding superlative index and the Geometric Young index lies somewhat below the superlative index. Thus for U. S. data over this period, it appears that the Geometric Young index is closer to a COLI than the corresponding Lowe indexes. This seems to be an important result that other statistical agencies might want to look at very closely.13 We conclude this section noting that not both editors agree on the significance of this paper. The paper supports the conclusion that as a rule, more frequent updating of weights seems to be a good thing in that the result of this more frequent updating causes the fixed base index to come closer to a superlative index. This result is not much to the liking of one of us (Peter von der Lippe) who is an ingrained opponent of chain indices and strict adherent to the "pure price comparison" school of thought, according to which a price index should preferably be reflective of price movements only. 14 Thus while Diewert sees a superlative index as a reasonable target index for a CPI, von der Lippe sees a fixed base basket type index as the more reasonable target index. While Diewert recognizes the attraction of making fixed base basket type comparisons (they are very natural and easy to explain), there are some practical considerations which lead him to generally prefer chained indexes (at least for annual data): • If statistical agencies took the von der Lippe point of view, then instead of providing a single time series of fixed base price and volume movements, in order to suit the needs of all users, logically, they should provide matrices of comparisons, where every year is compared with every other year (since different users might have different ideas of what the right base year is and how many years the fixed base should be held fixed). This could be done but many users would not be comfortable with this matrix presentation. • Bilateral comparisons between distant periods become more problematic for two reasons: (i) individual products cannot be matched to disappearing obsolete products and the introduction of new products and (ii) Paasche and Laspeyres spreads tend to become larger using direct comparisons as opposed to chained comparisons when comparing distant periods. 15 Thus using the chain principle (for annual data) will tend to diminish the importance of the choice of the index number formula. 13 14 15

A Geometric Young index can be constructed in real time just as easily as a Lowe index. See von der Lippe (2001, 2007) for a more complete exposition of his position on chaining. Paasche and Laspeyres spreads can be narrowed between any two points by picking a path of bilateral comparisons based on the similarity of relative prices between pairs of observations. For examples of this methodology, see Hill (2004, 2009) and see Diewert (2009) for materials on how to measure the similarity of relative prices.

Introduction to the Special Issue on Index Number Theory and Price Statistics • 667

However, Diewert does concede that the use of the chain principle does not necessarily work when dealing with subannual data where price and quantity changes can be very large.16 Thus this issue is not completely resolved. For a balanced viewpoint see Balk (2010b).

4

User costs, waiting services and the treatment of depreciation

W. Erwin Diewert in his paper, " User Costs versus Waiting Services and Depreciation in a Model of Production", addresses a problem that occurs when measuring the Multifactor or Total Factor Productivity growth of a production unit. Productivity is defined as the output of a production unit over a period of time divided by the input used and Productivity Growth is defined as the rate of growth of outputs divided by the rate of growth of inputs. This definition of Productivity Growth means that it could be measured by index number techniques; i. e., it could be set equal to an output quantity index divided by an input quantity index. However, such a gross output productivity index is not entirely suitable for making cross sectional comparisons for similar production units where the focus is on the amount of value added (gross outputs minus intermediate inputs) per unit of primary input (primarily labour and capital services). Thus Jorgenson and Griliches (1967) developed their Divisia index methodology to measure the value added productivity growth of an economy and they used the ratio of Tornqvist indexes of value added and primary inputs as approximations to their productivity concept. Diewert and Morrison (1986), using duality theory, showed how a slight variant of the Jorgenson and Griliches methodology could be justified using (flexible) translog functional forms and exact index numbers. The resulting methodology measures the growth of value added of the production unit per unit of primary input, where primary input is equal to labour and capital services. However, a component of capital services is depreciation or deterioration, which is not really a primary input. Primary input services should include the services of labour, land, waiting services and natural resource and other environmental services. Put otherwise, the user cost of capital includes a term reflecting the decline in capital services due to the aging of the asset; i. e., due to depreciation of the asset. A better measure of the effectiveness of a production unit in producing useful goods and services per unit of primary input would take depreciation out of the user cost of capital and treat it as an offset to gross investment. The resulting amended value added concept could be defined as a net value added concept as opposed to the usual gross value added concept. 17 If depreciation can be considered to be a decision variable by the production unit, then it turns out that the analysis of Diewert and Morrison (1986) can be adapted to this net value added framework and a rigorous productivity growth framework based on production theory can be readily be implemented using index number theory.18 This brings us to the contribution of the Diewert paper in this special issue: he argues that depreciation is indeed a decision variable and can be extracted from the usual user cost of capital and treated as a separate offset to gross investment.

16

17

18

See Ivancic et al. ( 2 0 1 0 ) and H a a n and van der Grient ( 2 0 1 0 ) for some methods that combine fixed base indexes with chain indexes that can deal with large fluctuations in period to period price and quantity data. Balk ( 2 0 1 0 a ) discusses these concepts in more detail; see also the earlier discussion by Schreyer ( 2 0 0 1 ) . If the production unit is the entire economy, net value added is equivalent to net domestic product and value added is equivalent to gross domestic product. See Diewert and Lawrence ( 2 0 0 6 ) .

668 • P. von der Lippe and W.E. Diewert

Diewert's paper develops an extension of a one period model of production involving beginning and end of the period capital stocks along with output and input flows. This model is due to the economist Hicks (1961) and the accountants Edwards and Bell (1961). Diewert's generalized Austrian model of production takes into account that end of the period capital stocks result from: (i) purchases of new investment goods; (ii) internal construction of firm capital stock components and (iii) holdings of (depreciated) capital goods that were held by the firm at the beginning of the period. These different methods of creating end of period holdings of capital stocks generally have different resource requirements and hence the one period production possibilities set is more complex than the usual one. This general model of production is used to justify the decomposition of the Jorgensonian (1963) user cost of capital into separate waiting services and depreciation components. 5

Hedonic regressions

In this section, we briefly describe the two papers in this special issue that deal with hedonic regressions. At the lowest level of aggregation for a CPI, prices are collected on a sample of products and the ratio of the price of an identical product over two time periods is a basic building block in the construction of the CPI. However, in today's dynamic economy, new products appear and obsolete products disappear. This product churning creates problems for the matched model methodology that is the basis for a Consumer Price Index. A technique for dealing with this problem was developed and popularized by Court (1939) and Griliches (1971). Basically, the price of a product that is subject to rapid technological progress (or a transformation of the price) is regressed on the important characteristics of the product. Then the price of a product that was available in a previous period but is no longer available in the current period can be predicted using this hedonic regression for the current period. Thus hedonic regression techniques can be viewed as a method for obtaining missing prices that can be inserted into a Consumer Price Index. These techniques are particularly useful in the housing context where it is difficult to price identical houses in the same condition in multiple periods. The first paper in this section is by the Dutch economist, Jan de Haan, who works at Statistics Netherlands. The title of his paper is Hedonic Price Indexes: A Comparison of Imputation, Time Dummy and 'Re-Pricing' Methods. The author is a master of hedonics: his paper explains in easy to understand language 19 the many variants for a hedonic regression and evaluates the strengths and weaknesses of these alternative methods. Price statisticians will find this paper to be very useful when they are faced with the problems posed by quality change. The second paper in this section involving hedonic regression methods is by the Japanese economists, Chihiro Shimizu, Kiyohiko G. Nishimura and Tsumotu Watanabe. The title of their paper is Housing Prices in Tokyo: A Comparison of Repeat-Sales and Hedonic Measures. The authors' data set contains more than 4 7 0 , 0 0 0 listings of house and condominium prices in Tokyo between 1986 and 2 0 0 8 . In addition to developing this unique and rich data set, there are at least three additional important contributions in the paper:

19

In particular, de H a a n explains the differences between the hedonic models recently discussed by Diewert et al. ( 2 0 0 9 ) in a manner that price statisticians will find easy to follow.

Introduction to the Special Issue on Index Number Theory and Price Statistics • 669

• They provided a generalization of the repeat sales model that made an adjustment for the depreciation of the structure between the two periods of the sale of a property; • They generalized the usual idea of a dummy variable adjacent period hedonic regression to a rolling window dummy variable hedonic regression model. • They find that there exists a substantial discrepancy in terms of turning points in their time series on house prices between their hedonic and repeat sales indexes. All of the above contributions are important ones; this paper is sure to be very influential in the housing literature.

6

Regional price indexes in Germany

The guest editors aimed at having a good mix of contributors to this special issue of the Jahrbücher für Nationalökonomie und Statistik, both academic index theorists as well as price statisticians in National Statistical Institutes (or experts in both fields). Thus we are glad to have in addition to representatives of Statistics Netherlands and the US Bureau of Labor Statistics (BLS) also Stefan Linz, a price statistician at the Federal Statistical Office of Germany. The title of his paper is Regional Consumer Price Differences Within Germany. Linz reports on the efforts to construct regional CPIs (or RPIs) for Germany in response to requests from various institutions in Germany. Demands for RPIs have come from universities, stimulated by the "new economic geography" and "spatial econometrics" as well as from the German Constitutional Court in a decision about regionally adjusted cost-of-living allowances. The task now for the German Statistical Office is to find ways to satisfy this demand for a family of high quality RPIs in a way which avoids if possible additional surveys, which would entail new response burdens and extra costs. Initially, it was thought that the RPI methodology could follow the national CPI methodology and regional price quotes could simply be drawn from the national sampling frame. However, Linz demonstrated that this easy solution is not advisable for two reasons: • The number of prices collected for the national CPI is adequate to give an accurate answer but the number of regional quotes is not generally adequate. • For a national index, it is acceptable to compare goods which are not strictly comparable across towns as long as they remain more or less the same over time. However, this is not acceptable if it is desired to construct regional price indexes that are comparable across regions. However, if we nonetheless make use of CPI data as much as possible for the compilation of an RPI, there are two methods to consider. The first consists in an "ex-post selection" of comparable (homogeneous) goods and the second is to run hedonic regressions (imputations) in order to quantify the "corrections" needed to provide truly comparable prices. In his paper Stefan Linz gave a report on an empirical study focusing on the first method and carried out by the German Federal Statistical Office for a small selection of goods and towns. We encounter here again the since long well known trade-off in price statistics between comparability (requiring a tight product specification) and representativity for the region in question (which allows for more latitude for price collectors). Regressions helped to establish the relative importance (for "comparability") of the product characteristics (as determinants of the price) and it turned out that even if a rather loose spe-

670 • P. von der Lippe and W.E. Diewert

cification limited to only few characteristics is applied, the ex-post selection would mean that we have to put up with an enormous reduction of the effective sample size. This is true even for such simple goods like noodles. To sum up, it can be seen that to satisfy demands for regional price data will be costly and a methodological challenge. 7

Conclusion

Index number theory and price statistics have many more interesting and controversial topics that could be addressed in this special measurement issue. Some examples of interesting topics are: the use of scanner data, the measurement of core inflation, the measurement of the price and quantity of nonmarket activities, measuring the subjective perception (on the part of private households) of inflation, new methods of deflation in the National Accounts or in making international comparisons, to name only a few of these difficult issues. However, we are confident that all of the papers in this special issue will be useful to other researchers in the area of economic measurement. References Balk, B. M. (1995), Axiomatic Price Index Theory: A Survey. International Statistical Review 63: 69-93. Balk, B. M. (1998), On the Use of Unit Value Indices as Consumer Price Subíndices. In: W. Lane (ed.), Proceedings of the Fourth Meeting of the International Working Group on Price Indices. Washington, DC: Bureau of Labour Statistics. Balk, B. M. (2008), Price and Quantity Index Numbers. New York: Cambridge University Press. Balk, B. M. (2010a), An Assumption Free Framework for Measuring Productivity Change. The Review of Income and Wealth 56, Special Issue 1: 224-255. Balk, B. M. (2010b), Direct and chained indices: a review of two paradigms. In: W. E. Diewert, B.M. Balk, D. Fixler, K.J. Fox, A.O. Nakamua (eds.), Price and Productivity Measurement: Volume 6 - Index Number Theory. Tratfort Press (www.vancouvervolumes.com and www.indexmeasures.com). Boskin, M. J. (Chair), E. R. Dullberger, R.J. Gordon, Z. Griliches, D. W. Jorgenson (1996), Final Report of the Commission to Study the Consumer Price Index. U. S. Senate, Committee on Finance, Washington DC: US Government Printing Office. Carli, G.-R. (1804), Del valore e della proporzione de' metalli monetati. Pp. 297-366 in: G. G. Destefanis, Scrittori classici italiani di economía política. Volume 13, Milano (originally published in 1764). Court, A. T. (1939), Hedonic Price Indexes with Automotive Examples. Pp. 98-117 in: The Dynamics of Automobile Demand. New York: General Motors Corporation, de Haan, J., H. van der Grient (2010), Eliminating Chain Drift in Price Indexes Based on Scanner Data. Journal of Econometrics, forthcoming. Diewert, W. E. (1976), Exact and Superlative Index Numbers. Journal of Econometrics 4: 114145. Diewert, W. E. (1978), Superlative Index Numbers and Consistency in Aggregation. Econometrica 46: 883-900. Diewert, W. E. (2009), Similarity Indexes and Criteria for Spatial Linking. Pp. 183-216 in: D. S. Prasada Rao (ed.), Purchasing Power Parities of Currencies: Recent Advances in Methods and Applications. Cheltenham UK: Edward Elgar. Diewert, W. E., S. Heravi, M. Silver (2009), Hedonic Imputation versus Time Dummy Hedonic Indexes. Pp. 161-196 in: W. E. Diewert, J. Greenlees, C. Hulten (eds.), Price Index Concepts and Measurement, NBER Studies in Income and Wealth. Vol. 70, Chicago: University of Chicago Press.

Introduction to the Special Issue on Index N u m b e r Theory and Price Statistics • 671

Diewert, W. E., D. Lawrence (2006), Measuring the Contributions of Productivity and Terms of Trade to Australia's Economic Welfare. Consultancy Report to the Productivity Commission, Australian Government, Canberra, March. Diewert, W. E., C.J. Morrison (1986), Adjusting Output and Productivity Indexes for Changes in the Terms of Trade. Economic Journal 96: 659-679. Drobisch, M. W. (1871), Uber die Berechnung der Veränderungen der Waarenpreise und des Geldwerths. Jahrbücher für Nationalökonomie und Statistik 16: 143-156. Dutot, Ch. (1738), Réflexions politiques sur les finances et le commerce. Volume 1, La Haye: Les frères Vaillant et N . Prévost. Edwards, E. O., P. W. Bell (1961), The Theory and Measurement of Business Income. Berkeley: University of California Press. Eurostat (1993), System of National Accounts 1993. Eurostat, IMF, OECD, UN and World Bank, Luxembourg, Washington, D. C., Paris, New York, and Washington, D. C. Fisher, I. (1922), The Making of Index Numbers. Boston: Houghton-Mifflin. Griliches, Z. (1971), Introduction: Hedonic Price Indexes Revisited. Pp. 3-15 in: Z. Griliches (ed.), Price Indexes and Quality Change. Cambridge MA: Harvard University Press. Hardy, G. H., J. E. Littlewood, G. Polya (1934), Inequalities. Cambridge: Cambridge University Press. Hicks, J. R. (1961), The Measurement of Capital in Relation to the Measurement of Other Economic Aggregates. Pp. 18-31 in: F. A. Lutz, D. C. Hague (eds.), The Theory of Capital. London: Macmillan. Hill, R.J. (2004), Constructing Price Indexes Across Space and Time: The Case of the European Union. American Economic Review 94: 1379-1410. Hill, R. J. (2009), Comparing Per Capita Income Levels Across Countries Using Spanning Trees: Robustness, Prior Restrictions, Hybrids and Hierarchies. Pp. 217-244 in: D. S. Prasada Rao (ed.), Purchasing Power Parities of Currencies: Recent Advances in Methods and Applications. Cheltenham UK: Edward Elgar. ILO, IMF, OECD, UNECE, Eurostat and World Bank (2004), Consumer Price Index Manual: Theory and Practice. Ed. by P. Hill, ILO: Geneva. IMF, ILO, OECD, Eurostat, UN and the World Bank (2004), Producer Price Index Manual: Theory and Practice. Ed. By Paul Armknecht, Washington: International Monetary Fund. IMF, ILO, OECD, UNECE and World Bank (2009), Export and Import Price Index Manual. Ed. by M. Silver, IMF: Washington, D. C. Ivancic, L., W. E. Diewert, K.J. Fox (2010), Scanner Data, Time Aggregation and the Construction of Price Indexes. Journal of Econometrics, forthcoming. Jevons, W. S., (1865), The Variation of Prices and the Value of the Currency since 1782. Journal of the Statistical Society of London 28: 294-320; reprinted in Investigations in Currency and Finance (1884), London: Macmillan and Co.: 119-150. Jorgenson, D.W. (1963), Capital Theory and Investment Behaviour. American Economic Review 53: 247-259. Jorgenson, D.W., Z. Griliches (1967), The Explanation of Productivity Change. The Review of Economic Studies 34: 249-283. Konüs, A. A. (1924), The Problem of the True Index of the Cost of Living. Translated in Econometrica 7: 10-29 (1939). Laspeyres, E. (1871), Die Berechnung einer mittleren Waarenpreissteigerung. Jahrbücher für Nationalökonomie und Statistik 16: 296-314. Lowe, J. (1823), The Present State of England in Regard to Agriculture, Trade and Finance. Second edition, London: Longman, Hurst, Rees, Orme and Brown. Paasche, H. (1874), Uber die Preisentwicklung der letzten Jahre nach den Hamburger Börsennotirungen. Jahrbücher für Nationalökonomie und Statistik 12: 168-178. Pârniczky, G. (1974), Some Problems of Price Measurement in External Trade Statistics. Acta Oeconomica 12: 229-240. Schreyer, P. (2001 ), OECD Productivity Manual: A Guide to the Measurement of Industry-Level and Aggregate Productivity Growth. Paris: OECD.

672 • P. von der Lippe and W.E. Diewert

Schreyer, P. (2009), Measuring Capital, Statistics Directorate, National Accounts. STD/ N A D ( 2 0 0 9 ) 1 , Paris: O E C D . Schreyer, P. (2010), Towards Measuring the Volume of O u t p u t of Education and Health Services: A H a n d b o o k . Statistics Directorate Working Paper N o . 31, STD/DOC(2010)2, Paris: O E C D , April 28. Schultze, C. L., C. Mackie (eds.) (2002), At W h a t Price? Conceptualizing and Measuring Costof-Living Indexes. Washington D. C.: National Academy Press. Shapiro, M . D . , D . W . Wilcox (1997), Alternative Strategies for Aggregating Prices in the CPI. Federal Reserve Bank of St. Louis Review, May/June: 113-125. Törnqvist, L. (1936), The Bank of Finland's Consumption Price Index. Bank of Finland Monthly Bulletin 10: 1-8. Törnqvist, L., E. Törnqvist (1937), Vilket är förhällandet mellan finska markens och svenska kronans köpkraft? Ekonomiska Samfundets Tidskrift 39: 1-39 reprinted as pp. 121-160 in: Collected Scientific Papers of Leo Törnqvist, Helsinki: The Research Institute of the Finnish Economy, 1981. von der Lippe, P.M. (2001), Chain Indices, A Study in Index Theory. Spectrum of Federal Statistics, Stuttgart: Metzler-Poeschel. von der Lippe, P.M. (2007), Index Theory and Price Statistics, Frankfurt a . M . : Peter Lang. Walsh, C. M . (1901), The Measurement of General Exchange Value. N e w York: Macmillan and Co. Walsh, C . M . (1921), The Problem of Estimation. London: P.S. King & Son.

Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 2010) Bd. (Vol.) 230/6

Abhandlungen / Original Papers Drobisch's Legacy to Price Statistics By Ludwig von Auer, Trier* JEL C43, E31, E52 Price, inflation, index theory, measurement.

Summary This paper attempts to establish a greater awareness among researchers for the noteworthy contributions to price index theory made by Moritz Wilhelm Drobisch (1802-1896), a German mathematician and philosopher at the Universität Leipzig. Few economists and statisticians are aware of the fact that neither Etienne Laspeyres nor Hermann Paasche originally devised the well-known price indices that presently carry their names. Moritz Wilhelm Drobisch was the first to publish them in 1871 in a treatise and, shortly thereafter, in an abridged version that appeared in this very journal. He rejected them, however, because in his view they were inappropriate measures of inflation. Instead, he devised the unit value index, which he regarded as superior to all other price index formulas. This paper contains a description of his pioneering scientific achievements together with a synopsis of his personal and professional life. Its purpose is to give credit where credit is due, but more importantly, it attempts to recognize these seminal contributions in light of the factors that have tarnished them in the recorded annals of price index history. It attempts to put them into their proper perspective.

1

Introduction

If asked to name the two price index formulas that are the most renowned, economists and statisticians alike would probably answer in unison, the Laspeyres and Paasche indices. The references most often quoted are their famous articles published in 1871 and 1874 in this very journal. Unfortunately, few researchers are aware, however, that neither of these two scholars, Etienne Laspeyres (1834-1913) nor H e r m a n n Paasche (18511925), devised these t w o index formulas. 1 Instead, they first appeared in a publication authored by Moritz Wilhelm Drobisch (1802-1896), a mathematician and philosopher at the Universität Leipzig. In addition, he also developed the unit value index. These three price indices are the key to the official inflation measurements made around the world today. Therefore, this distinguished scholar's contribution to price statistical research should be appropriately acclaimed and acknowledged.

* I was motivated to write this article by the retirement of a highly esteemed colleague, Peter von der Lippe at the Universität Duisburg-Essen. His enthusiasm for price statistics and their originators was infectious. This paper is testimony to that fact. I am indebted to Nick Barton and Claudia Haller for excellent research assistance. John Brennan's generous advice greatly improved the readability of this paper. Two anonymous referees and the editors of this journal provided several invaluable comments. 1 Notable exceptions are Balk (2008: 7 and 8, footnotes 11 and 13) and Diewert (2004: 265, footnotes 7 and 8).

674 • L. von Auer

M.W. Drobisch

E.L.E. Lasperyres

H. Paasche

Figure 1 Three German Price Statisticians Source: M.W. Drobisch: Reproduced from Wiemers (2003) courtesy of Gerald Wiemers. E.L.E. Laspeyres and H. Paasche: Wikipedia.

In 1871, the Königliche Sächsische Gesellschaft der Wissenschaften (Royal Scientific Society of Saxony) published a treatise written by its founder Drobisch (1871a). Later that year, Drobisch (1871b) wrote an abridged version ofthat treatise, which appeared in this journal. Today it is known that these two publications provide the scientific basis for modern applied price statistics. In the original paper, Drobisch (1871a: 37ff.) proposed three index formulas: the unit value index and a pair of index formulas that subsequently became known as the Laspeyres and Paasche indices. Among these three formulas, he clearly expressed a preference for the unit value index, which he thought was an optimal price index formula. At that time, many prominent economists and statisticians considered these articles. Not surprisingly, Etienne Laspeyres and Hermann Paasche were drawn to them as well. In his response to Drobisch's (1871a) publication, Laspeyres (1871: 306) suggested the index formula that was later to carry his name. This suggestion elicited an immediate and pointed response from Drobisch (1871c: 423): "However, also this formula is not new. As Laspeyres could have seen in my first paper, which he had available, I initially tried this approach myself to calculate price increases and I described the reasons that persuaded me not to continue with this approach." This crucial phrase verbalizes the undeniable facts concerning this unfortunate historical incident. In the end, the scientific recognition for the discovery of the two most widely known price indices today was erroneously bestowed. It brings to light, however, Moritz Wilhelm Drobisch's culpability in this scholarly dispute as well. He rejected these price indices as inappropriate for inflation measurements and instead formulated the unit value index, the index formula he considered to be optimal. The tragedy of this episode lies in his failure to recognize the inherent inconsistencies present in the unit value index due to the manner in which he formulated it. Everything considered, however, even though his foray into the field of price statistics was a brief one, the lasting contributions he made are indeed noteworthy.

Drobisch's Legacy to Price Statistics • 675

The purpose of this paper is to convey information and, additionally, to produce a greater awareness concerning this important scholar. It attempts to give credit where credit is due, but more importantly, it tries to put this unfortunate historical incident into its proper perspective. Consequently, the paper proceeds as follows. Section 2 contains a short description of the pioneering contributions to price index theory made by Moritz Wilhelm Drobisch. Economists and statisticians alike know little about his personal and professional life and, therefore, a brief summary is provided in Sections 3 through 5. Section 6 contains a eulogy and the concluding remarks are contained in Section 7. 2

Contribution

Wilhelm Georg Friedrich Roscher (1817-1894), ordentlicher Professor (Professor) at the Faculty of Philosophy of Universität Leipzig, was the person who aroused Drobisch's interest in price index theory. Roscher was the founder of an approach to academic economics that is known as the Ältere Historische Schule der Ökonomie (early Historical School of Political Economy). He was considering the problem: How should the mean price change of N different items between some base period, t = 0, and a comparison period, t = 1, be calculated, if it is assumed that the same N items are sold in the marketplace in both periods? He was undecided whether the price index formula proposed by Carli (1764),

or the one proposed by Jevons (1863), N

,hl

where p' is the price of item i (i = 1 , . . . , N) observed in period t (t = 0,1), would yield the most suitable result. 2 Therefore, he asked his senior colleague Drobisch (1871a: 4 4 , 1871c: 4 1 7 , footnote 3) for some guidance. Drobisch was lecturing in the areas of mathematics and philosophy at the time and his interest in the subject was stimulated by this request. He agreed to look into the issue. In his treatise, Drobisch (1871a: 32f.) recognized that various items have different purchase relevancies among consumers. Accordingly, a price index formula should give stronger weights to the price developments of the highly relevant items and lesser weights to the others. Therefore, a proper price index formula must take into account the quantities transacted in the base, qf, and comparison periods, qj. This led him to reject both the Carli and Jevons indices in search of a more appropriate solution to the problem. He asked himself: How should the relevance of an item be represented in a price index formula? In a preliminary step, Drobisch (1871a: 35) stated that all of the quantities should be measured in a common unit of weight (Zentner = 50 kilogramme) and, consequently, 2

For an exposition of the early history of price index research see Balk (2008, Chapter 1) and Diewert (1993).

676 • L. von Auer

the prices of the items involved would need to be adjusted accordingly. This could be accomplished by multiplying the original quantity data, q\, and dividing the original price data, p', by some transformation factor, z,. Drobisch did not explicitly use transformation factors but directly employed the converted quantities, q\ = q'zj, and prices, p\ = p\/z, (Note: p\q\ = p'tf). He approached the measurement problem from the standpoint of a simple scenario where the quantities transacted remained constant over time, qf = qj = q,. For such a scenario, Drobisch (1871a: 36) proposed the following index formula: E M i z

m

where ^ is used to denote E j l i . Unbeknownst to him, however, Lowe (1822: Appendix 94f.) had proposed this formula half a century earlier. Presently, it is known as the Lowe index, PLo. He then progressed to a scenario with variable quantities, qf ^ qj. For this scenario, Drobisch (1871a: 37f.) suggested two alternative formulas:

L

E MI

roxo '

and

Subsequently, these two formulas have become known as the Laspeyres, P i , and Paasche, Pp, indices. Drobisch stated that with unchanging quantities these two formulas are equal to the previous formula, P i 0 . Furthermore, Drobisch (1871a: 39) recognized that due to their inherent symmetry, both formulas are theoretically equivalent. He regarded this equivalence as a serious weakness. Drobisch (1871c: 4 2 5 ) indicated, however, that the arithmetic mean of these indices could be used as a measure of price change:

Even though he considered this formula as unsatisfactory, some price statisticians presently denote it as the Drobisch index, Pp. His preferred price index, however, was a somewhat different one. In the derivation of this index, Drobisch (1871a: 39) took as the point of departure the unit value formula:

EM E

3

for f = 0 , 1 .

(4)

In the previous year, Segnitz (1870: 184) had proposed the unit value formula (4) in an article published in this journal, a fact that Drobisch (1871a) failed to mention. Segnitz cautioned, however, that the use of this formula should be limited to homogenous items and the example he gave was different quantities of the cereal grain, rye. When heterogeneous items are considered the situation becomes much more complicated. These items

Drobisch's Legacy to Price Statistics • 677

are denominated in a variety of different units of measure and, as a result, the summation, q y i e l d s a meaningless number. Drobisch was aware of this problem and thought he had the appropriate solution in hand. He believed that by measuring the quantities in a common unit of weight, the Zentner, and adjusting the prices accordingly, he had solved the problem. Consequently, Drobisch (1871a: 39) defined the unit value index, Puv, as the ratio of two unit values:

Puv-n~v'Emm-

(5)

Drobisch (1871a: 39, 1871c: 422f.) pointed out that with constant quantities, q0 = q\ = qh this formula simplifies to his previous proposal (1) and also to his other proposals (2) and (3). Drobisch regarded his unit value index (5) as superior to all other price index formulas. 3 He recognized that combining heterogeneous items into an aggregate quantity presents a problem but he was confident that his weight-related conversion scheme had solved it. Unfortunately, he failed to recognize that this weight-related conversion method involves a summation, q', that could require combining some very different items, for example, a Zentner of butter and a Zentner of indigo dyestuff. Although these items have equal weight, they have extremely different monetary values. This invalidates the unit values calculated by formula (4) and, therefore, the unit value index (5) as well. Moreover, this weight-related conversion of prices and quantities is not even possible when intangibles such as services are involved. In their responses to Drobisch's (1871a) proposals, Laspeyres (1871: 307) and Paasche (1874: 172) both expressed their reservations regarding the unit value index (5). Laspeyres saw no problems involved with the proposal to transform all of the data into a common weight based unit of measure. He criticized Drobisch, however, on the grounds that the unit value index formula allowed the quantities to change over time. He suggested that the base period quantities, q°t, should prevail during the comparison period as well, which yields index formula (2). This index today bears his name. Furthermore, Laspeyres (1871: 308) pointed out that the unit value index (5) violates the Identity Axiom. This axiom postulates that with constant prices, a price index should equal one regardless of the changes in the quantities that might take place. Paasche (1874: 172), on the other hand, suggested taking the comparison period quantities, qj, as fixed. Consequently, he advocated formula (3). Today this index is known as the Paasche Index. Drobisch (1871a) first proposed it as well, although, he subsequently rejected it. Paasche (1874) did not mention Drobisch's contribution at all, while Laspeyres (1871: 305) did recognize that the written recommendations expounded by Drobisch (1871a: 30) would lead to the correct index formula. From today's perspective, the price statistical research of Moritz Wilhelm Drobisch seems to contain a surprising lack of consistency. The astute research qualities he demonstrated in the derivation and analysis of the Laspeyres and Paasche indices stand in stark 3

Lippe ( 2 0 0 7 : 18, footnote 4 3 ) points out that Drobisch's unit value index should not be confused with the unit value index used in some statistics of foreign trade.

678 • L. von Auer

contrast to his failure to recognize the obvious inconsistencies of his weight-related unit value index (5) when it is applied to heterogeneous items. Nevertheless, the legacy that he leaves behind in the field of official price measurement cannot be overlooked. As a rule, statistical agencies compute inflation estimates by comparing prices in a base period to those in a comparison period utilizing a two-stage process. The elementary level is concerned with the price changes of individual items. An item is a narrowly defined group of products that should be as homogeneous as possible. In the Consumer Price Index Manual (a joint publication of the ILO, IMF, OECD, UNECE, Eurostat, and The World Bank), Boldsen and Hill (2004: 164) recommended the use of Drobisch's unit value index formula (5) to compute these price changes. The upper level of price measurement is concerned with aggregating the unit values and quantities computed at the elementary level into an overall price change. This is most often accomplished utilizing a Laspeyres or Paasche type price index. Moritz Wilhelm Drobisch proposed both of these index formulas. Therefore, the complete process of official price measurement relies upon concepts that he devised. His contribution, however, goes even further. Drobisch (1871a) considered his unit value index (5) to be appropriate for the upper level of price aggregation as well. Some fourteen years later, Lehr (1885) studied his work and concluded that the unit value index formula (5) is a useful concept but the weight-related conversion relies on an incorrect basic premise. In order to illustrate Lehr's consideration, formula (5) can be expressed in an alternative form. Utilizing the transformation factors, z,, the original quantities, q\, and the original prices, p\, formula (5) can be expressed in the following manner:

Lehr (1885: 38f.) recognized that in order to make the summation, ^ q]z„ meaningful, the units in which the quantities, q\z¡, are measured had to be of comparable value and not simply of comparable weight. Therefore, he proposed the following scheme for the calculation of the transformation factors: Z

=

M

±

M

,

for

i =

1,2,...,N.

(7)

Hi + 1 i

Substituting this result into formula (6) yields the Lehr index. A number of years later, Davies (1924: 185) made a similar proposal. Instead of formula (7), he proposed the following formulation for the transformation factors: Zi

=

\fpfpj,

for i = 1,2, . . . , N .

Substituting this result into formula (6) yields the Davies index. Auer (2010) demonstrated with a systematic analytical elaboration that the proposal of Davies (1924: 185) is a member of a specific set of price indices. Auer named this set the generalized unit value indices and demonstrated that the Laspeyres and Paasche indices are also members of this particular family. Drobisch's (1871a,b,c) short excursion into price index theory resulted in a considerable boon for this area of research. He developed the unit value index (5), as well as the Laspeyres (2) and Paasche (3) indices. Who was this distinguished German scholar? Why is it

Drobisch's Legacy to Price Statistics • 6 7 9

that he is not better known among researchers within the field? Fortunately for us, eight years after his passing one of his grandsons, Walther Neubert-Drobisch (1902), wrote an illuminating biography based upon a collection of diaries and letters that he had inherited. Much of the information hereinafter relies upon this informative source. 3

Adolescence

Moritz Wilhelm Drobisch's father, Karl Wilhelm Drobisch, was a religious and patriotic man.4 Raised in a rather poor family, due to his diligence and persistent nature, he, nevertheless, attained the respected position of Stadtschreiber (city clerk) in the city of Leipzig. At that time, Leipzig was an important and wealthy city in the Kingdom of Saxony. He was content in his marriage, even though he and his wife suffered the loss of four of their six children. Only two of their daughters lived beyond childhood. In 1790, at the age of thirty-five, his beloved wife passed away. He recuperated from this loss, however, and was able to remarry four years later. He married Renata Dorothee Wilhelmine Klotz, the thirty-three year old daughter of a state judicial employee from Grimma, a small town located southeast of Leipzig on the Mulde River. After seven years of a childless marriage, on the sixteenth of August 1802 the couple was blessed with their first child, Moritz Wilhelm Drobisch. On Christmas Eve the following year, Karl Ludwig was bom. Karl Ludwig was later to become a well-known music teacher, conductor, and composer. At a young age, the two boys lost their twenty-eight year old stepsister in 1809. This was a tragic loss and one that their father never fully recovered from. Nevertheless, as the boys grew older, their father taught them many things. Consequently, they could already read, write, and were performing some basic mathematical calculations long before they entered primary schooling at the Nicolaischule (Nikolai School) in Leipzig. Education played an important role in the Drobisch household and, as a result, the boys came to view their father's study as if it were some kind of "holy" room. Their minds were quite engaged during these formative childhood years as they eagerly embraced many new ideas. For example, Moritz Wilhelm and his younger brother enjoyed stargazing late into the night from the roof of their home. The two boys became very interested in astronomy and began calculating the phases of the moon, the movement of the planets, and soon they had learned the names of the most famous constellations. From October the 16 th to the 19 th , 1813, the Völkerschlacht bei Leipzig (Battle of the Nations) took place on the outskirts of the city. Moritz Wilhelm was eleven years old at the time and watched the ensuing battle from the roof with his telescope. At its conclusion, he was fortunate enough, from a short distance, to witness King Friedrich I of Saxony offer the defeated Napoleon Bonaparte a glass of wine. Napoleon hastily drank the wine and, in a mood of desperation, angrily threw the empty glass to the ground. What he had just seen made a strong and lasting impression upon young Moritz Wilhelm's mind. To him it symbolized how the unrealized ambitions of Napoleon and those of his ally the Kingdom of Saxony were now laid at their feet like the glistening shards of the shattered wine glass. He decided then and there to fully develop his physical and mental skills and to become a strong contributor for his fatherland. 4

Apart from where otherwise noted, this section is based upon the biography by Neubert-Drobisch (1902: 1-20).

680 • L. von Auer

Regrettably, however, this idyllic childhood, one that played such an important role in the boys' early intellectual development, came to an abrupt end in 1815 with the passing of their father. As a result, Moritz Wilhelm's mother sent him away to the prestigious Fürstenschule St. Augustin zu Grimma (secondary school) and Karl Ludwig followed two years thereafter. The money their father had bequeathed was intended for this schooling and also for their subsequent university education. Moritz Wilhelm enjoyed his three years at the Fürstenschule and the fertile academic environment that existed there nurtured his preexisting interests in mathematics and astronomy. This was in part due to his interactions with a gifted teacher who engaged the young man in inspiring debates. To some extent, this teacher played the roll of a surrogate father for Moritz Wilhelm. In his free time, he continued to engross himself in his most important hobby, astronomy. As evening approached, he would eagerly look forward to secretly observing the stars with the equipment he purchased with the pocket money he had saved. During these adolescent years he returned to Leipzig infrequently, only to visit his mother during school vacations in the summer months. On March 28 t h , 1820, at the age of seventeen, Moritz Wilhelm began studying mathematics and physics at the Universität Leipzig. While most of his peers were indulging themselves in the newly discovered freedom called "student life", Moritz Wilhelm gave his academic responsibilities first priority. It was said that more often than not, he refrained from drinking beer and instead drank a glass of milk. As a result, after just one year he was already giving private lessons in mathematics. With the money he earned he increased his personal library and added to his accumulation of astronomical instruments. He politely declined the private invitations that were offered by his professors for he feared that such engagements would inhibit his personal and intellectual independence. During these initial years at the university, he was an extremely purposeful and resolute young man. At the age of twenty, Moritz Wilhelm lost his dear mother on March 6 t h , 1822. Consequently, he and his brother departed the family home in search of a new place to live. They found a suitable room and rented it from the widow Anna Maria Leichsenring. Two month later in May, Moritz Wilhelm, his brother, and another fellow student hiked to the scenic Sächsische Schweiz via Grimma, Meißen, and Dresden. At the conclusion of this enjoyable trip, he returned to Leipzig a bit perplexed. He had decided, in the meanwhile, to turn his focus towards more cultural and worldly things and not any longer to devote himself solely to his scholarly activities. He became increasingly captivated by the arts. His appreciation for music and literature, especially the works of William Shakespeare, grew. It was during this period of uncertainty that Moritz Wilhelm became acquainted with Emilie Charlotte Leichsenring, one of his landlady's three daughters. He had begun to write poetry and during this courtship, which ultimately lasted five years, he wooed her with many of his delightful poems. He enjoyed writing poetic prose and did so for the remainder of his life. This interlude into belletristic literature, however, was brief and lasted only a year and a half. He stated at its conclusion, "The reading of novels had corrupted me." After some soul searching, he recommitted himself to the study of mathematics and the sciences. The appreciation for music and the arts that he had gained, however, never left him.

Drobisch's Legacy to Price Statistics • 681 4

Adulthood

Moritz Wilhelm Drobisch returned to being an exceedingly focused and studious young man. 5 He completed his studies at the Universität Leipzig in the Faculty of Philosophy in four years. In 1824, at the young age of twenty-two, he received the academic degree of Dr. phil. in Mathematik (PhD in Mathematics) with a dissertation entitled: Praemissae ad theoriam organismi generalem, theoriae analyseos geometricae prolusio (An Introduction to the Theory of Analytical Geometry). During that same year he was granted his Habilitation in Philosophy (postdoctoral lecture qualification) and became a Privatdozent (qualified external lecturer). It had always been his ambition to become a teacher at a höhere Schule (secondary school) and so he felt very fortunate to be able to teach at the university level. Before long he was giving lectures in several mathematical subjects, physical geography, and popular astronomy. In 1826, at the age of twenty-four, he was appointed außerordentlicher Professor (Adjunct Professor) in the Faculty of Philosophy and by the end of that same year, following a proposal of the government and not the university, he was selected to be ordentlicher Professor (Professor) of Mathematics. Unfortunately, the appointment was not without its controversy because some of the older faculty members voiced strong opposition to the appointment. They believed that he was too young and lacked the required experience for the position. One of his supporters, however, came to his rescue by sarcastically suggesting that Dr. Drobisch's "mistakes" would surely diminish with the passage of time. The meteoric ascent of his academic career, culminating in the attainment of a professorship, offered him the prospect of entering into a prearranged marriage of convenience with one of the most influential families in Leipzig. Like his father before him, however, he resisted the temptation. He had no great craving for one of those frivolous, overly satiated rich daughters, instead he desired a woman who would be a proper mother for his yet to be born children and a skillful homemaker. In 1827, on the thirteenth day of September he married Emilie Charlotte Leichsenring. A marriage that was to last some forty-four years and one that was to be blessed with five daughters and three sons. Their initial happiness, however, was not destined last. As parents, they were forced to endure the tragic loss of five of their beloved children. Only three of their daughters survived. Consequently, as he grew older he became increasingly spiritual and devout in his beliefs. He cautioned that scientists should not be condescending towards religious faith and that they should not attempt to place themselves above it.6 He delivered his first lecture on philosophy in 1832. This event proved to be a milestone in his professional career for it marked the start of a long process where he shifted his intellectual emphasis away from mathematics towards philosophy. Ten years later in 1842, he was appointed ordentlicher Professor (Professor) of Philosophy.7 He resigned his professorship in mathematics in 1868, a position he held for forty years. From that point on he devoted his scholarly activities to the fields of philosophy and psychology.8

5 6 7 8

Apart from where Neubert-Drobisch Neubert-Drobisch Neubert-Drobisch

otherwise noted, this section is based upon Neubert-Drobisch (1902: 21—49). (1902: 64) (1902: 79) (1902: 123)

682 • L. von Auer

JL JL yV .

|

JL, yjr„ »—

— - i ,

Mjk^'ij /./Oi&ti

JM/C^fi 3-J -&

•

JLJLifj.

Figure 2 Professor M.W. Drobisch, 1841 Source: Neubert-Drobisch (1902: inner cover).

The person who had the greatest scientific influence upon him was the philosopher, psychologist, and pedagogue, Johann Friedrich Herbart (1776-1841), Professor of Philosophy at Universität Göttingen. Professor Herbart recognized his unique capacity to integrate philosophy and mathematics and appreciated it as an outstanding scientific trait. To commemorate his death in 1841, Professor Drobisch delivered a poignant eulogy in his honor. Shortly afterwards he was approached by a group of students who asked his permission to commission a portrait of his likeness. Pleased by the gesture, he agreed and prepared a comment for inscribing at the base of the lithograph (see Figure 2). "The flame of bona fide science is kindled by the spark of inspiration, it nourishes itself from the fuel of hard work and burns undimmed only in the breath of freedom." („Die Flamme der echten Wissenschaft entzündet sich an dem Funken der Begeisterung, nährt sich von dem Öl des Fleißes und brennt ungetrübt nur in der Lebensluft der Freiheit.")

Drobisch's Legacy to Price Statistics • 683 Moritz Wilhelm Drobisch seldom left his much-loved city of Leipzig. He felt completely at home there and was intimately linked to its university. Occasionally, however, he was called upon to travel. In these instances he complied dutifully and did so either in the company of his wife or alone. 9 He was a reluctant traveler, unless of course, the trip was one of his periodic "escapes" into the scenic countryside that surrounded Grimma. There he could find the solace and happiness he sought as he nostalgically immersed himself in the memories of his adolescent school days at the Fürstenschule. Afterwards, when he returned to his everyday responsibilities in Leipzig, he always felt a sense of renewal and invigoration. 10 The death of his beloved wife Emilie Charlotte in 1871 was a hard blow. 11 In difficult moments, however, he quietly spoke words of encouragement to himself and found solace in his work. Many years before, he had written the following poetic lines: 12 "Never mind the transition from day to night, Heed not the cyclic change of the seasons, Engross yourself in the depth of knowledge, Produce things of significance, worth, and veracity; Then you will be contented for the moment, And the days will tacitly run their course." („ Vergiß den Wechsel von Tag und Nacht, Vergiß die Wechsel des Jahres, Vertiefe Dich in des Wissens Schacht, Schaff Hohes, Edles und Wahres; Dann bist Du glücklich im Augenblick, Und still trägt jeder Tag sein Geschick, "j

5

Academic

Moritz Wilhelm Drobisch placed great emphasis on his teaching assignments. 13 An extremely hard working and conscientious professor, he took great pride in meticulously preparing his lectures. 14 In addition to his customary lectures in mathematics and philosophy, he also conducted lectures in such diverse fields as psychology, popular astronomy, physical geography, logic, and the philosophy of religion. His lectures were clearly presented, well organized, and scientifically precise. The solutions to the problems he posed were presented in a logical and lucid manner. His lively lecturing style was highly valued by the students at the Universität Leipzig and his personal vitality allowed him to continue teaching well into his octogenarian years. Finally, at the age of eighty-four, he requested to be released from his responsibilities as a lecturer due to an eye complaint. 15 His style of writing, as evidenced by the books and numerous scientific articles he published, was characterized by a high degree of finesse, clarity, and precision. 16 Initially, his

9 10 11 12 13 14 15 16

Neubert-Drobisch (1902: 58ff., 67ff., 109ff., 119) Neubert-Drobisch (1902: 35f., 54) Neubert-Drobisch (1902: 125) Neubert-Drobisch (1902: 49) Apart from where otherwise noted, this section is based upon Neubert-Drobisch (1902: 7 3 - 1 2 8 ) . Heinze (1896: 713ff.) Heinze (1896: 699) Heinze (1896: 699f.)

684 • L. von Auer

Figure 3 Professor M. W. Drobisch, cir. 1877

Source: Reproduced from Wiemers (2003) courtesy of Sächsische Akademie der Wissenschatten zu Leipzig.

scholarly work dealt solely with mathematically related subjects. As he aged, however, his primary interest began to shift ever more towards the philosophical and psychological bodies of thought. Subsequently, these were the areas where he received considerable acclaim. Nevertheless, during the later part of the 1860s he began to have an increased interest in statistical topics. Regrettably, however, his two articles in 1871 were his only foray into the field of price statistics. In spite of his teaching and extensive research activities, over the years he served in numerous administrative capacities for the university. He did not solicit these additional academic responsibilities because he viewed them as a hindrance to his primary duties. Nevertheless, he served as Dekan (Dean) of the Faculty of Philosophy on eight separate occasions and as a member of the faculty senate. He was the Rektor der Universität Leipzig (president) during 1841 and 1842. As Rektor he officially represented the university at numerous governmental meetings, scientific congresses, and at formal social gatherings. He did so dutifully but would have much preferred the solitude of his hiking tours in Saxony's scenic countryside. 17 He remarked later, however, that the experiences he had as Rektor helped save him from becoming a complete scientific recluse and forced him to live the life of a man of the world. During the meager amount of free time that he allowed himself, however, he especially enjoyed attending concerts at the Gewandhaus zu Leipzig (Concert Hall) where Felix Mendelssohn Bartholdy (1809-1847) was the Gewandhauskapellmeister (conductor) from 1835 to 1847. 17

Neubert-Drobisch (1902: 54)

Drobisch's Legacy to Price Statistics • 685 In addition to all this, he was a Mitglied (member since 1834) as well as later Präsident (Chairman, 1848-1863) and Ehrenmitglied (honorary member since 1877) of the Fürstlich Jablonowskische Gesellschaft der Wissenschaften zu Leipzig (Prince Jablonowski Scientific Society of Leipzig), an organization that encouraged scientific work through the awarding of prizes. Laspeyres ( 1862) won such a prize for his study of the economics' literature and the history of economic thought in the Netherlands during the period 1600 to 1785. As Chairman of the society, he advocated its modernization and wanted to transform it into a scientific society that published original research. He failed in this attempt, however, because he could not change the constitution of the organization. As a result, he founded the independent Königliche Sächsische Gesellschaft der Wissenschaften (Royal Scientific Society of Saxony) and on July 1 s t , 1846, he delivered the inaugural address on the occasion of the 200th anniversary of the birth of Gottfried Wilhelm Leibniz. He wrote the constitution himself and firmly established the independence of the new society from the Universität Leipzig. He made it clear that the main objective of the society was the publication of research and one such publication was Drobisch's (1871a) own treatise on price statistics. 18 To commemorate the fiftieth anniversary of his professorship in 1876, Moritz Wilhelm Drobisch was ceremonially proclaimed Ehrenbürger der Stadt Leipzig (honorary citizen) at the Rathaus (City Hall). He received the title of the highest official of the royal court, Königlich Sächsischer Hofrat, 1862, and Geheimer Hofrat, 1866, (Court Counselor). He was decorated with numerous medals including Ritterkreuz des Königlich Sächsischen Civil-Verdienstordens, 1844, and the Comthurkreuz des sächsischen Albrechtordens, 1877. He accepted all of these honors gratefully and with humility, but was of the opinion that he had earned none of them. The Königliche Sächsische Gesellschaft der Wissenschaften recognized Professor Drobisch for his role as their founding father with the creation of the Moritz-Wilhelm-Drobisch-Medaille in 1971. 1 9

6

Eulogy

Moritz Wilhelm Drobisch was considered to be a very honorable and trustworthy person. 20 It was said that he personified the kategorische Imperativ (Categorical Imperative) formulated by Immanuel Kant (1724-1804), the great 18 th century German philosopher. This fundamental ethical principle guided his actions as he determined what was morally right throughout the course of his life. A devout Christian who followed the Protestant faith, he lived by a set of moral guidelines that led him to be a respectful, honest, and unassuming man. Consequently, on the occasion of the fiftieth anniversary of his professorship, these virtuous character traits were revealed when he donated all of the monetary gifts he received to benefit financially disadvantaged students.21 In his dealings with strangers, he was always very considerate and extremely polite. Nevertheless, on these occasions he would often appear quite serious. This was sometimes misconstrued and it was said that he distrusted those he did not know well. This was not the case. At social functions such as birthday parties, confirmations, weddings,

18 19 20 21

Wiemers ( 2 0 0 3 : 9ff.) Wiemers ( 2 0 0 3 : 1 6 ) Apart from where otherwise noted, this section is based upon Neubert-Drobisch ( 1 9 0 2 : 1 2 7 - 1 3 1 ) . Heinze ( 1 8 9 6 : 7 1 8 )

686 • L. von Auer

or other such festive occasions, he would display a quite amiable demeanor. He would amuse others with his cordial, albeit slightly sarcastic personality. Sometimes he would even entertain guests with his own poetry, poems of both a serious and a humorous nature. 2 2 Towards the end of his long and fruitful life, Moritz Wilhelm Drobisch remained intellectually active and spry. He became, however, more and more reclusive. Having outlived his dearly beloved wife Emilie Charlotte by a quarter of a century as well as a substantial number of his closest university colleagues, while experiencing acutely the associated loneliness, he displayed his characteristic humility and devout spirituality by leaving behind this moving testimonial: 23 "I have lived long and strived relentlessly, Attempted much, alas I achieved little. More than my worth, I was venerated, Unmerited fortune was bestowed upon me. Only this attestation may I make, That I endeavored to live a dutiful life. Whenever I strayed from the path of righteousness, I humbly trust in the grace of God And into the loving hands of the Lord I entrust my final days and my demise." („Lange hab' ich gelebt und gestrebt, Viel gesponnen, doch wenig gewebt. Mehr als ich wert war, ward ich geehrt, Mehr als Verdienteren Glück mir beschert. Nur das Zeugnis darf ich mir geben, Daß ich bemüht war, pflichttreu zu leben. Wo ich gewichen vom rechten Pfade, Hoff' ich in Demut auf Gottes Gnade Und in des liebenden Vaters Hände Leg' ich den Lebensrest und mein Ende.")

Moritz Wilhelm Drobisch died in Leipzig shortly before the seventieth anniversary of his professorship on September 30 t h , 1896. He was ninety-four years old. 7

Conclusion

Moritz Wilhelm Drobisch was a multifaceted scholar who held the academic positions of Professor of Mathematics and Professor of Philosophy, both separately and concurrently, at the Universität Leipzig over a tenure lasting some sixty-eight years. His scientific contributions were insightful and varied, for he was equally at home in a wide variety of scholarly disciplines. In 1871, his scientific curiosity led him into the field of price statistics. It remains to this day an unresolved mystery, however, regarding the question: Why was his interest in price statistics so brief, lasting only one year? One can only speculate. Was it his scholarly dispute with Etienne Laspeyres? Certainly that robbed him of much of the satisfaction associated with his scientific contributions in this area. He had reached the age of sixty-

22 23

Neubert-Drobisch (1902: 49) Neubert-Drobisch (1902: 131)

Drobisch's Legacy to Price Statistics • 687

Figure 4 Moritz Wilhelm Drobisch in His Study at the Age of Ninety Source: Neubert-Drobisch (1902: 96b).

nine and the bereavement associated with the death of his beloved w i f e during that same year certainly added a great emotional burden. The professorship he held in mathematics f o r f o r t y - t w o years had previously been resigned three years earlier. As a result, his scholarly interests were shifting a w a y f r o m mathematically related subjects t o w a r d s those of a philosophical and psychological nature. H e entered the field of price statistics at the request of a faculty colleague and after formulating the unit value index he probably felt that he had found the optimal answer to the question. Perhaps he felt that his w o r k in this area w a s complete. All of these factors probably played a role. If his contributions had been received in a more positive w a y , then w o u l d he have lingered f o r a longer period in the area of price statistical research? This intriguing mystery will remain unsolved. T h e scientific recognition f o r the discovery of the t w o most widely k n o w n price indices today w a s erroneously bestowed. M o r i t z Wilhelm Drobisch's culpability in the course of events that lead to this outcome, however, must not be overlooked. A t the time, he rejected these t w o price indices as inappropriate f o r inflation measurements and instead formulated the unit value index. T h e tragedy of this episode, however, lies in his failure to recognize the inherent inconsistencies present in his f a v o r e d index due to his fixation upon a weight-related formulation. It is certain, however, that M o r i t z Wilhelm Drobisch's brief but fruitful f o r a y into the field of price statistics yielded a quantum leap in this branch of science. H e enriched the field with the introduction of three fundamental index concepts: the Laspeyres, the Paasche, and the unit value index. With these three indices he laid the theoretical cor-

688 • L. von Auer nerstone upon which price index methodology has been built. These formulas are the key for the official inflation measurements that are made around the world today. The seminal contributions he made were significant and should be appropriately acclaimed and acknowledged.

References Auer, L. von (2010), The Generalized Unit Value Index, unpublished manuscript. Balk, B . M . (2008), Price and Quantity Index Numbers. Cambridge (New York): Cambridge University Press. Boldsen C., P. Hill (2004), Calculating Consumer Price Indices in Practice. Pp. 153-177 in: ILO, IMF, OECD, UNECE, Eurostat, The World Bank (eds.), Consumer Price Index Manual: Theory and Practice, Chapter 9. Geneva: International Labour Office. Carli, G . R . (1764), Del Valore e della Proporzione de'Metalli Monetati con i generi in Italia prima delle Scoperte dell'Indie colonfronto del Valore e della Proporzione de'Tempi nostri. Opere scelte die Carli, Milan, 1: 299-366. Davies, G. R. (1924), The Problem of a Standard Index Number Formula. Journal of the American Statistical Association 19: 180-188. Diewert, W. E. (1993), The Early History of Price Index Research. Pp. 33-65 in: W. E. Diewert, A. O. Nakamura (eds.), Essays in Index Number Theory, Vol. 1, Amsterdam: North Holland. Diewert, W. E. (2004), Basic Index Number Theory. Pp. 263-288 in: ILO, IMF, OECD, UNECE, Eurostat, The World Bank (eds.), Consumer Price Index Manual: Theory and Practice, Chapter 15. Geneva: International Labour Office. Drobisch, M.W. (1871a), Ueber Mittelgrössen und die Anwendbarkeit derselben auf die Berechnung des Steigens und Sinkens des Geldwerths. Berichte der mathematisch-physischen Classe der Königlich Sächsischen Gesellschaft der Wissenschaften 1: 25-48. Drobisch, M.W. (1871b), Ueber die Berechnung der Veränderungen der Waarenpreise und des Geldwerths. Jahrbücher für Nationalökonomie und Statistik 16: 143-156. Drobisch, M.W. (1871c), Ueber einige Einwürfe gegen die in diesen Jahrbüchern veröffentlichte neue Methode, die Veränderungen der Waarenpreis und des Geldwerthes zu berechnen. Jahrbücher für Nationalökonomie und Statistik 16: 416-427. Heinze, M. (1897), Gedächtnisrede auf Moritz Wilhelm Drobisch. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematischphysische Classe, 696-719. Jevons, S. (1863), A Serious Fall in the Value of Gold Ascertained, and Its Social Effects Set Forth. 1863. Reprinted in Jevons, Investigations in Currency and Finance. London: Macmillan, 1884. Laspeyres, E. (1862), Mitteilungen aus Pieter de la Courts Schriften. Ein Beitrag zur Geschichte der niederländischen Nationalökonomie des 17. Jahrhunderts. Zeitschrift für die gesamte Staatswissenschaft 18: 330-374. Laspeyres, E. (1871), Die Berechnung einer mittleren Warenpreissteigerung. Jahrbücher für Nationalökonomie und Statistik 16: 296-314. Lehr, J. (1885), Beiträge zur Statistik der Preise insbesondere des Geldes und des Holzes. Frankfurt a . M . : F.D. Sauerländer Verlag. Lippe, P. von der (2007), Index Theory and Price Statistics. Frankfurt a . M . : Peter Lang. Lowe, J. (1822), The Present State of England in Regard to Agriculture, Trade and Finance; with a Comparison of the Prospects of England and France, 2 n d ed. 1823, London: Longman, Hurst, Rees, Orme und Brown. Neubert-Drobisch, W. (1902), Moritz Wilhelm Drobisch. Ein Gelehrtenleben, Leipzig: Dieterich'sche Verlagsbuchhandlung. Paasche, H. (1874), Ueber die Preisentwicklung der letzten Jahre nach den Hamburger Börsennotirungen. Jahrbücher für Nationalökonomie und Statistik 23: 168-179.

Drobisch's Legacy to Price Statistics • 689

Segnitz, E. (1870), Ueber die Berechnung der sogenannten Mittel, sowie deren Anwendung in der Statistik und anderen Erfahrungswissenschaften. Jahrbücher für N a t i o n a l ö k o n o m i e und Statistik 14: 183-195. Wiemers, G. (2003), Moritz Wilhelm Drobisch und die Gründung der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, 1846. Abhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-naturwissenschaftliche Klasse 60(3): 7-16. Prof. Dr. Ludwig von Auer, Universität Trier, Fachbereich IV - Finanzwissenschaft, Universitätsring 15, 54286 Trier, Germany. E-Mail: [email protected]

Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 2010) Bd. (Vol.) 230/6

Notes on Unit Value Index Bias By W. Erwin Diewert, Vancouver, and Peter von der Lippe, Duisburg-Essen* JEL C43, C81, E01, E31 Price indexes, unit value indexes, unit values, bias, Bortkiewicz, Drobisch, Paasche, Laspeyres, Fisher, Pärniczky, Balk, von der Lippe.

Summary It is often the case that the value of a number of somewhat similar units (e. g., automobiles of a certain general type) is divided by the number of units in order to form a unit value price and these unit value prices are compared over two periods in order to form a unit value price index. This unit value price index or Drobisch price index can then be compared with other standard index number formulae and the bias in the index can be determined. The present paper presents most of the known results on this bias (and derives some new ones) in a coherent framework using a simple identity from the statistics literature. A related question first considered by Parniczky (1974) is also considered: does disaggregation of a unit value into more homogeneous subgroups reduce the unit value bias? The answer seems to be: probably yes.

1

Setting the stage

It is sometimes the case that detailed price and quantity data for a group of closely related commodities (that have the same unit of measurement, such as a class of vehicles, types of grain, containers of similar products, etc.) are not available but information on the number of units is available in each period along with the value of the products in the shipment. In this case, the value of the products can be divided by the number of units and a unit value price is obtained for the period under consideration. If unit values for the product group can be calculated for two periods, then the ratio of the two unit values can be regarded as an (approximate) price index. This price index is known in the literature as a unit value price index or a Drobisch (1871) index in honour of the German measurement economist who first introduced this type of index. It is important to recognize that a Drobisch price index cannot be used over very heterogeneous items since the resulting index is not invariant to changes in the units of measurement. Thus a unit value price index can only be used over products that are measured in the same units and are „reasonably" homogeneous. The question that we are going to address in this paper is the following one: how bad is the bias in a unit value index that is constructed over „reasonably" homogeneous items that are not completely homogeneous? A major problem with the Drobisch price index is that its axiomatic properties are not entirely satisfactory. In addition to not satisfying the invariance to changes in units test if the aggregation is over heterogeneous items, this index does not satisfy the identity test, * Our thanks to Bert Balk and Jens Mehrhoff for helpful comments. The financial support of the SSHRC of Canada is gratefully acknowledged.

Notes on Unit Value Index Bias • 691

which asks the index number to equal unity if the price vectors for the two periods under consideration remain the same. 1 However, as mentioned above, the focus of the present paper is not on the axiomatic properties of the Drobisch index; our focus is on determining the bias of the Drobisch price index when aggregating over „reasonably" homogeneous products as compared to standard indexes used in index number practice such as the Laspeyres (1871), Paasche (1874) and Fisher (1922) indexes. This research was initiated by Parniczky (1974) and significant contributions to this bias literature have been made by Timmer (1996), Balk (1998, 2008: 72 ff.), Silver (2009a,b,c, 2010) and von der Lippe (2007a,b). Our goal in this paper is to present the main results in this bias literature (and some new results) in a unified framework using a simple result from the statistics literature. Some care should be used in interpreting this unit value bias literature. Economic agents often purchase and sell the same commodity at different prices over the accounting period under consideration but a bilateral index number formula requires that these multiple transactions in a single commodity be summarized in terms of a single price and quantity for the period. If the quantity is taken to be the total number of units purchased or sold during the period and it is desired to have the product of the single price and the total quantity transacted equal to the value of the transactions during the period, then the single price must be a unit value; i.e., total value transacted divided by total quantity transacted. Thus at this very first stage of aggregation, the „correct" price to insert into a bilateral index number formula is in fact the unit value for the narrowly defined commodity. 2 This unit value price should not be regarded as having a „bias". However, if there is further aggregation over „similar" commodities using unit value prices, then there can be unit value bias. The simple result from the statistics literature is the following one. Let x = [ X I , . . . , X N ] and y = [yi,...,yN] be two N dimensional vectors and let s = [ S I , . . . , S N ] be an N dimensional share vector; i. e., s has nonnegative components (s > ON) which sum up to unity (IN-S = 5Zn=iN s n = 1 where I N is a vector of ones and ON is a vector of zeros of dimension N). Define the share weighted covariance between x and y using the share vector s as follows: Cov(x,y,s) =

s n (x n - x*)(y n - y*

(1)

where x* = s x = s n x n and y* = s y = 5Z n =i N s n y n are the share weighted means of the components of the x and y vectors respectively. A straightforward computation shows that the following covariance identity holds: 3 £ n = i N s„x n y n = Cov(x,y,s) + x'y*.

(2)

We will apply the covariance identity (2) in subsequent sections of this paper. 1

2

3

Balk (1998) and von der Lippe (2007a) looked at the axiomatic properties of the Drobisch index in a systematic way. For additional material on the axiomatic approach to index number theory, see Diewert (1992, 1995), Balk (1995, 2008) and the ILO (2004). This point was made many years ago by Walsh (1901: 96, 1921: 88) and Davies (1924, 1932) and more recently by Diewert (1995). This identity was used by von Bortkiewicz (1923) in order to establish his identity relating the Paasche and Laspeyres price indexes. Silver (2009c: 8) uses a correlation coefficient version of this identity to derive his unit value bias results. We will not cover the bias results of Silver in this paper since they generally involve two covariance effects and hence are more complex than our simpler results.

692 • W.E. Diewert and P. von der Lippe

We conclude this section by formally defining the various indexes mentioned above. The basic data are two price vectors, p ® s [piV-JPn 1 ] and two quantity vectors, qf = [qiS-.^qN1], for periods t = 0,1. For now, we assume that these price and quantity vectors have positive components and hence, the period t value aggregates, V 1 = p'-q® > 0 are positive for each period t. The Laspeyres (1871) and Paasche (1874) price indexes are defined as follows: 4 P L = p ' . q V - q 0 = £ n = i N s n °(p n Vp n °) ;

(3)

PP = P ' - Q V P ^ Q 1 = [£N=L N SNVPNVP,, 0 )- 1 ]- 1

(4)

where the period t expenditure share vector is sc s [si®,...^®] for t = 0,1 and s n f = Pn'qn1/ p'q® for n = 1,...,N and t = 0,1. The Fisher (1922) ideal price index Pp is defined as the geometric mean of the Laspeyres and Paasche price indexes: P F = [P L P P ] 1/2 .

(5)

In order to define the Drobisch or unit value price index, it is necessary to restrict the N commodities under consideration to be measured in the same units. Thus it is not meaningful to add up units of crude oil with bushels of wheat in order to obtain an aggregate quantity for each period but it is meaningful to add up various grades of crude oil with differing chemical compositions or to add up bushels of wheat of varying quality. Thus in what follows, we assume that a meaningful quantity aggregate Q [ exists for each period, where Q® is just the simple sum of the components of q®: Q< = l N - q t = E n = i N q n t ; t = 0 , l .

(6)

Once the period t quantity aggregate Q® is well defined, then we can divide the period t value aggregate, V , by Q® in order to obtain the period t unit value price P®: P® = V®/Q® = p'-qVln-q 1 = p'-S® ;t = 0,1

(7)

where the period t quantity share vector S® = [SI®,...,SN®] is defined as follows: S< = q®/Q® = q®/l N q®;t = 0 , l .

(8)

Note that the period t quantity shares add up to one; i. e., we have: INS 1 = 1 ;t = 0,1.

(9)

Thus the period t unit value P® can be regarded as a physical share weighted average 5Zn=iN Sn'pn® of the individual period t prices pn® where the period t physical share weights Sn® must be distinguished from the period t expenditure shares sn® defined earlier. With the above definitions in hand, we can define the Drobisch (1871) price index PD and the corresponding Drobisch (or Dutot) 5 quantity index QD as follows: 4

5

The corresponding Laspeyres and Paasche quantity indexes are defined as QL = p° • qVp°- q° and

Qp =

PW'PW-

Balk (2008) refers to the quantity index defined by (11) as the Dutot quantity index. Dutot (1738) did not actually introduce the quantity index (11); instead he introduced the corresponding additive price index PDMOI = IN P^IN P0-

Notes on Unit Value Index Bias • 693

P D = PVP° = [ p ^ q V p V l / t l N - q V l N - q 0 ] = pl Sl/p° S0 ;

(10)

Q d = QVQ° = W / l N - q 0 .

(11)

Note that PDQD equals the value ratio, VVV°.

2

Comparisons of the Drobisch index to the Paasche index

Using (4) and (10) above, it is straightforward to compare the Drobisch index to the corresponding Paasche index: PD/PP = {[p'-qVpO-q^/FLN-QVLN-qOD/fp^qVpO-q1] = p°-S Vp 0 -S 0

(12)

where we have used definitions (8) for the quantity shares S' in order to derive the last equality in (12). Thus the bias in the Drobisch index relative to the Paasche index can be defined as follows: [PD/PP] - 1 = [p°-S1/p° S°] - 1 = p° [S1 - S°]/P° = [p° - pO'lNl tS1 - S°]/P° = N c o v i p ^ S 1 - S°,(1/N)1 N )/P°

using (12) using (7) for t = 0 since l ^ S 1 - S°] = 0 using (9) using (1) a n d (2) (13)

where p°* = [^Zn=iN Pn°]/N is the arithmetic average of the period 0 prices. Thus the Drobisch index will have an upward bias relative to the Paasche index if products n whose quantity shares are growing (so that Sn1 is greater than Sn°) are associated with period 0 prices p„° which are above the arithmetic average of the period 0 prices p°\6 As is usual in the analysis of unit value index bias, 7 there are three cases where the bias will be zero: • All prices in the base period are equal to the same price so that p° = OIN where a > 0; • The quantity shares remain constant over the two periods under consideration, which is equivalent to q 1 = Aq° where A > 0; i.e., the two quantity vectors are proportional or • The covariance (using share weights that are equal) between the base period prices p° and the difference in the quantity share vectors, S1 — S°, is zero. The above application of the covariance identity (2) leads to a fairly simple bias formula for the Drobisch index. However, formula (13) is not equivalent to the bias formulae obtained by Parniczky (1974: 233) and Balk (1998, 2008: 74). These authors obtained bias formulae using share weighted covariance matrices rather than the above equally weighted covariance matrix. In order to obtain these alternative bias decompositions, it is necessary to make some additional definitions. Thus define the vector of growth rates in quantity shares, G = [GI,...,GN], where the components GN are defined as follows: G n = [SnVSn°] - 1 ;n = 1,...,N. 6

7

(14)

Note that the first equation in (13) is particularly easy to interpret: we simply compare two weighted averages of the period 0 prices, p°, using the quantity shares of period 1, S 1 , in the numerator and the quantity shares of period 0, S°, in the denominator. See Parniczky (1974: 234) or Balk (2008: 74).

694 • W.E. Diewert and P. von der Lippe

Note that if q 1 = /lq° so that the period 1 quantity vector is proportional to the period 0 quantity vector, then G = ON where ON is a vector of zeros of dimension N . Note also that the G n satisfy the following equation: E n = i N S„°Gn = En=i N Sn°{[SnVSn0] - 1} =0

using (14) using (9).

(15)

Using our earlier bias formula for the Drobisch index relative to the Paasche index (13), we have: [PD/PP] -

1 = p°-[S 1 -

S°]/P°

= E n = l N P„°SN0(SNVSN°) - £ n = . N

= = = =

PN

V]/P°

{ £ n . l N p„°S n 0 [(S n VS n 0 ) - 1])/P° En=i N S„ 0 p n °G„/P 0 using definitions (14) £ n = i N S n °[p n ° - P°]G n /P° using (7) for t = 0 and (15) Cov(p°,G,S°)/P 0 using (1), (2) and (15).

(16)

Comparing (16) with (13), it can be seen that in (16), the covariance uses the base period share weighted average P° of the period 0 prices p° in place of the arithmetic average of the period 0 prices p° and the covariance in (16) uses the base period share vector S° as the weighting vector as opposed to the equal weights vector (1/N)1NAs usual, there are three cases where the bias in the Drobisch index relative to the Paasche index will be zero: • All prices in the base period are equal to the same price so that p° = P°1N where P° is the period 0 unit value price; • The quantity vectors are proportional so that G = ON or • The covariance (using the base period share weights S°) between the base period prices p° and the quantity share growth rate vector G is zero. The bias formula is still not quite equal to the bias formula obtained by Parniczky. In order to obtain his formula, we need to undertake a bit more algebra. First define the vector of growth rates in quantities relative to overall growth in quantities, g = [gi,...,gN], where the components g n are defined as follows: g n = [q n Vq n 0 ] - [QVQ 0 ] ;n = 1,...,N.

(17)

From (15), we have the following equation: 0= = = =

£n=iN En-1 N En=iN £n=iN

S n °([S n VS n 0 ] - 1} S n °{([q n VQ 1 ]/[q„°/Q 0 ]) - 1} using definitions (8) S^UqnVq,, 0 ]) - [Q 1 /Q°])/[Q 1 /Q°] SnVtQVQ0] using definitions (17).

(18)

Equation (18) shows that the base period quantity share weighted average of the relative quantity growth rates g is equal to zero. Put another way, the weighted mean of the quantity relatives qnVqn0 using the base period quantity shares S n ° as weights is equal to the aggregate quantity relative, QVQ 0 , which is the Dutot quantity index; i.e., we have 0

£ n = l N S n °[q n Vq n ] = QVQ

0

= QD.

(19)

Notes on Unit Value Index Bias • 695

From the bias formula (16), we have the following expression: [PD/Pp] - 1 = (En=l N Pn'VKSn'/S,, 0 ) - 1 ]}/P° = En=l N P n V W / q , , 0 ] ) - [QVQ^VIQVQ 0 ]? 0 using the algebra in (18) = En=i N Sn^n^n/IQVQ 0 ]? 0 using definitions (17) = En=l N Sn°[p„° - P ^ g n / t Q V Q O r using (18) = Cov(p 0 ,g,S 0 )/P°Q D using (1), (2) and (18).

(20)

This is the bias formula for the Drobisch index relative to the Paasche index that was derived by Parniczky (1974: 233). We have the three usual cases where the bias will be zero; i. e., all prices in the base period are equal or the quantity vectors are proportional or the covariance Cov(p°,g,S°) is zero. In the following section, we simply adapt the above analysis in order to obtain bias formulae for the Drobisch price index relative to the Laspeyres and Fisher indexes.

3

Comparisons of the Drobisch Index to the Laspeyres and Fisher Indexes

Using (3) and (10) above, it is straightforward to compare the Drobisch index to the corresponding Laspeyres index: PD/PL

= { [ p M V p V ] / [ i N q 1 / i N q 0 M p 1 q V q 0 ] = p^sVp^s0

(2i)

1

where we have used definitions (8) for the quantity shares S in order to derive the last equality in (21). If the weighted average of period 1 prices using period 1 quantity share weights, p^S 1 , is greater than the weighted average of period 1 prices using period 0 quantity share weights, p'-S 0 , then Pp will be greater than P l and vice versa. The bias in the Drobisch index relative to the Laspeyres index can be defined as follows: [Pd/Pl] - 1 = [ p ' S V p ' S 0 ] - 1 = p ^ S 1 - S^/p^S 0 = [p 1 - p ^ I n M S 1 - S ^ / p ' S 0 = N Covfp^S

1

-

using (21) since l ^ S 1 - S°] = 0 using (9)(22) 1

S°,(1/N)1n)/p -S°

using (1) and (2) 1

N

where p * = E n = i PnM/N is the arithmetic average of the period 1 prices. Thus the Drobisch index will have an upward bias relative to the Laspeyres index if products n whose quantity shares are growing (so that S,,1 is greater than Sn°) are associated with period 1 prices p n J which are above the arithmetic average of the period 1 prices P1*There are three cases where the bias will be zero: • All prices in the current period are equal to the same price so that p 1 = o I n where a > 0; • The quantity shares remain constant over the two periods under consideration, which is equivalent to q 1 = Aq° where k > 0; i. e., the two quantity vectors are proportional or • The covariance (using share weights that are equal) between the current period prices p 1 and the difference in the quantity share vectors, S 1 — S°, is zero. In order to obtain our second bias decomposition that is an analogue to formula (16), it will be necessary to develop a formula for the Laspeyres index Pl relative to the Drobisch

696 • W.E. Diewert and P. von der Lippe

index Pd rather than for Pd/Pl- 8 As usual, some additional definitions will be required. Thus define the vector of reciprocal growth rates in quantity shares, r = where the components r „ are defined as follows: r n = [ S n V S n V - 1 ;n = 1,...,N.

(23)

Note that if q 1 = Aq0 so that the period 1 quantity vector is proportional to the period 0 quantity vector, then r = On- Note also that the T n satisfy the following equation: £ n = l N S n 1 ^ = £ n = L N ^ { [ S H 0 / ^ 1 ] - 1}

=0

using (23) using (9).

(24)

Thus the period 1 share weighted average of the r n is equal to 0. Using (21), we have Pd/ P l equal to p 1 -S 1 /p 1 -S°. Taking reciprocals of this equation and subtracting unity leads to the following equation which defines the bias of the Laspeyres index relative to the Drobisch index: [Pl/Pd] - 1 = [p1-S°/p1-S1] - 1 = p 1 [S° - S^/P 1 using (7) for t = 1 = E n = l N Pn^n^S^/S,, 1 ) - £ n = 1 N Pn^n^/P 1 = {£n=l N Pn'SnHiSn 0 ^ 1 ) - 1JJ/P1 = En-1 N S n ^ n ^ n / P 1 using definitions (23) = E n = i N Sn^Pn 1 - P ^ / V P 1 using (7) for t = 1 and (24) = Cov(p 1 ,r,S 1 )/P 1 using (1), (2) and (24).

(25)

The bias formula (25) for the Laspeyres price index relative to the Drobisch price index is the symmetric counterpart to the bias formula (16), which compared the Drobisch and Paasche price indexes. The covariance in (25) will be positive if higher than average reciprocal growth rates for quantity shares are associated with higher than average period 1 prices and under these conditions, Pl will be greater than Pp. As usual, there are three cases where the bias in the Drobisch index relative to the Laspeyres index will be zero: • All prices in period 1 are equal to the same price so that p 1 = P' I n where P 1 is the period 1 unit value price; • The quantity vectors are proportional so that the vector of reciprocal quantity growth rates r = On or • The covariance (using period 1 share weights S1) between the period 1 prices p 1 and the reciprocal quantity share growth rate vector r is zero. We will now modify the above bias formula in order to obtain a Laspeyres counterpart to the Paasche bias formula (20) obtained by Parniczky. As usual, we need to undertake a bit more algebra. Define the vector of reciprocal growth rates in quantities relative to overall reciprocal growth in quantities, r = [ri,...,rN], where the components r n are defined as follows: r„ = [q^/qn 1 ] - [ Q ^ Q 1 ] ;n = 1 , - , N . (26)

8

Balk (2008: 74) developed a formula for Pd/Pl but it is more complex than the formula that we develop.

Notes on Unit Value Index Bias • 697

It can be seen that the share weighted mean of the r n is zero if we use period 1 quantity shares S n ! as weights: En=l N S A n = £ n = l N S n M l q n ' V ] ~ [Q^QM) Using (26) = £ n = l N [qaVlN-q^Iq^/qn 1 ] - £ n = l N S n ' [ Q ^ Q 1 ] Using (8) = [lN-q^lN-q 1 ] - [ Q ^ Q 1 ] using (9) = 0

(27)

using (6).

Equation (27) shows that the period 1 quantity share weighted average of the reciprocal quantity growth rates r is equal to zero. Put another way, the weighted mean of the reciprocal quantity relatives q^/qn 1 using the period 1 quantity shares Sn1 as weights is equal to the aggregate reciprocal quantity relative, Q ° / Q \ which is the reciprocal of the Dutot quantity index; i.e., we have £ n = l N S n H q n V 1 ] = Q ^ Q 1 = Qd"1-

(28)

From the bias formula (25), we have the following expression: [Pl/PD] - 1 = (£n=l N PnVUSnO/Sn 1 ) - 1J1/P1 = £ n = l N Pn1Sn1{[qn°/qn1]) ~ [Q°/Q , ])/[Q°/Q 1 ]P 1 using (6) and (8) = En=i N Sn^n^n/tQO/QMP 1 using definitions (26) = £ n = l N SnHpn1 - P M ^ W / Q M P 1 using (27) = Cov(p 1 ,r,S 1 )/P 1 Q D ~ 1 using (1), (2) and (27).

(29)

This is the bias formula for the Laspeyres price index relative to the Drobisch price index that is the counterpart to the Parniczky (1974: 233) bias formula for the Drobisch index relative to the Paasche index. We have the three usual cases where the bias will be zero; i. e., all prices in the base period are equal or the quantity vectors are proportional or the covariance Covfp^^S 1 ) is zero. We conclude this section by comparing the Drobisch price index Pd to the Fisher ideal price index Pp. The bias formulae (29) and (20) can be rewritten as follows: [Pd/Pl] = 1 + [P 1 /Cov(p 1 ,r,S 1 )Q D ] ;

(30)

P D /P p ] = 1 + [Cov(p 0 ,g,S 0 )/P°Q D ].

(31)

Using the above formula and definition (5) for the Fisher price index, it can be seen that the bias of the Drobisch index relative to the Fisher index is: 9 [PD/PF] - 1 = {1 + [P 1 /Cov(p 1 ,r,S 1 )Q D ]} ,/2 {l + [Cov(p 0 ,g,S 0 )/P°Q D ]) 1/2 - 1.

(32)

In a similar fashion, we can rewrite the bias formulae (13) and (22) as follows: [P D /P p ] = [p°-S1/p°-S°] ; 0

[PD/PL] = [ p ^ S V p ' S ] . 9

(33) (34)

This is similar to the bias formula developed by Balk (2008: 74) but our components are a bit different due to our use of reciprocal rates of growth which Balk did not use.

698 • W.E. Diewert and P. von der Lippe

Thus using definition (5), we have [PD/PF]

= [p°-S1/p°-S0]1/2[p1-S1/p1-S0]1/2 using (33) and (34) = [pO-S1 p1.S1]1/2/[p°-S0 p ^ S T 2 « [(1/2) pO-S1 + (1/2) p 1 -S 1 ]/[(l/2) p°-S° + (1/2) pi-S0] (35) where we have approximated the geometric means by arithmetic means = p*-s v - s °

where the vector of arithmetic average prices p* is defined as (l/2)p° + (l/2)p 1 . Thus the approximate bias of the Drobisch price index relative to the Fisher price index is [PD/PF] -

1

« [p'-SVp*-S°] - 1 = p'-fS 1 - S°]/p*-S° = N cov(p*,S^ - S°,(l/N)l N )/p*-S 0 using (1) and (2).

(36)

The above approximate bias formula is very similar to the earlier bias formula comparing PD to Pp and PL, formulae (13) and (22). The main difference is that in (13), the base period price vector p° appeared in the covariance and in (22) p 1 appeared in the covariance whereas in (36), the reference price vector p* is the arithmetic average of p° and p 1 . Thus the Drobisch index will (likely) have an upward bias relative to the Fisher index if products n whose quantity shares are growing (so that S n ' is greater than S„°) are associated with prices (l/2)p n ° + (l/2)p n ' which are above the arithmetic average of the period 0 and period 1 prices (l/2)p°* + (112)^*. The approximate bias on the right hand side of (36) will be zero if any one of the following three conditions is satisfied (the third condition implies the first two conditions): • All average prices (averaged over the two periods) are equal to the same price so that (l/2)p° + (l/2)p 1 = a l N where a > 0; • The quantity shares remain constant over the two periods under consideration, which is equivalent to q 1 = Aq° where A > 0; i. e., the two quantity vectors are proportional or • The covariance (using share weights that are equal) between the average prices (l/2)p° + (l/2)p J and the difference in the quantity share vectors, S1 — S°, is zero. In the next section, we derive some additional bias formulae for the Drobisch index using a technique due to von der Lippe (2007a: 415 ff., 2007b). 4

Alternative bias decompositions for the Drobisch index

The analysis in this section starts with an identity relating the Paasche and Laspeyres price indexes that was first derived by von Bortkiewicz (1923). As usual, we will require a few new definitions. Define the vector of price relatives p = [ / ? I , . . . , / > N ] where pn = p n V p n ° for n = 1,...,N and the vector of quantity relatives z = [ T I , . . . , T N ] where RN S q n Vq n ° for n = 1,...,N. Recalling definitions (3) and (4), it can be verified that a share weighted average of the price relatives is equal to the Laspeyres price index and a share weighted average of the quantity relatives is equal to the Laspeyres quantity index if we use the base period shares s n ° as weights; i.e., we have £n=I

N

s

£n=l

N

Sn°Tn = Q

n

(37)

% = PL ; L

.

(38)

Notes on Unit Value Index Bias • 699

The von Bortkiewicz (1923) identity is the following one 10 : Pp -

P L = COV(/?,T,S°)/QL

(39)

where the covariance in (39) is defined by (1). Empirically, for most value aggregates, it is found that the covariance in (39) is negative; i.e., above average growth in a price is associated with a below average growth in the corresponding quantity. Multiply both sides of (39) by Q L and note that we obtain the following identity:

PPQL

equals the value ratio, V W ° . Thus

V7V° = COV(/>,T,S°) + P l Q l .

(40)

Now divide both sides of (40) by P L Q'/Q° and noting that [VW°]/[QVQ°] equals the Drobisch price index PD, we obtain the following expression for PD relative to the Laspeyres price index PL: P D / P L = [COV(/>,T,S 0 )/P L QD] + [ Q L / Q D ]

(41)

where we have also used QD equals QVQ°. Now subtract 1 from both sides of (41) and we obtain von der Lippe's (2007b) formula for the bias of the Drobisch index relative to the Laspeyres price index: 11 [PD/PL] - 1 = [ C o v ( ^ , r , s ° ) / P L Q D ] + [ Q L / Q D ] -

1.

(42)

The first term on the right hand side of (42) can generally assumed to be negative but what can be said about the last term, [ Q L / Q D ] — 1 ? Using our definitions for the Laspeyres and Drobisch quantity indexes, we have: QL/QD = [POQV-QVUN-QVLN-Q0]

= p^SVp^S 0 = [PD/PP]

= E n = l N S n 0 p N 0 G„/P 0 ] + 1 = [Cov(p°,G,S°)/P0] + 1

using using using using

definitions (8) (16) (16) (16) again

(43)

where the G n = [SnVSn°] — 1 are the growth rates for the quantity shares. Substituting (43) into (42) leads to the following bias formula: [PD/PL] - 1 = [ C o v ( / > , r , s 0 ) / P L Q D ] + [ C o v ( p 0 , G , S ° ) / P 0 ] .

(44)

While the bias formula (44) is an interesting one, it may not be as useful as our earlier bias formulae since only one covariance is involved in our earlier formulae. 12 It is possible to develop Paasche counterparts to (41)-(44) by reversing the role of time. Define the vector of reciprocal price relatives p~l = [pi - 1 ,...,/^ - 1 ] where pn~l = p n °/p n '

10 11 12

The covariance identity (2) can be used to prove (39). See also Balk (2008: 7 3 f.). If w e substitute the middle equation in (43) into (42), w e obtain the identity [PD/PL] - [PD/ Pp] = [COV(P,T,S°)/PLQD]- Thus a negative covariance Cov(/>,R,s°) will just lead t o the conclusion that PL is greater than Pp, which is just the conclusion that w e can draw from (39).

700 • W.E. Diewert and P. von der Lippe

for n = 1,...,N and the vector of reciprocal quantity relatives z~1 = [TI~1,...,TN~1] where r n _ 1 = qn^qn 1 for n = 1,...,N. Recalling definitions (3) and (4), it can be verified that the following identities hold: £ n = l N SnVn" 1 = Pp" 1

£n=l

N

1

1

(45)

5

1

Sn ^" = Qp" .

(46)

Interchanging 0 and 1 in (40) leads to the following counterpart to (40): VO/V 1 = C O V ( / > _ 1 , T ~ \ s 1 ) + P p ^ Q p " 1 -

(47)

Again recalling that PD is equal to (V1/V°)/QD, we can use (47) in order to obtain the following formula for the reciprocal of PD relative to Pp: (PD/PP)" 1 = P P ( V ° / V 1 ) / Q D

= [ C o v t / r V ^ s ^ P p / Q c ] + [QD/QP]

using (47).

(48)

The identity (48) is the counterpart to the earlier von der Lippe identity (41). We can expect C o v i / ? - 1 ^ - 1 ^ 1 ) to be negative but again, we need an analytical formula for the second term on the right hand side of (48), the ratio of the Drobisch quantity index QD to the Paasche quantity index Qp. Using the definitions for QD and Qp, we obtain the following decomposition: QD/QP = [ L N - Q V L N - Q ^ V - Q V P V ]

= p^S'Vp'-S 1

using definitions (8)

= PL/PD

using (21)

= [£n=iN SnVTVP1] + 1 = [Covfp1,/^1)/?1] + 1

using (25) using (25) again

(49)

where the r n are defined as the reciprocal share growth rates, [SnVSn0]-1 — 1, for n = 1,...,N. Substituting (49) into (48) leads to the following bias formula for the Paasche price index relative to the Drobisch price index: [P p /P D ] - 1 = [ C o v f / r V ^ s ^ P p / Q D ] + [Cov(p 1 ,r,S 1 )/P 1 ].

(50)

Note that (48)-(50) imply the following identity: [PP/PD] -

[PL/PD] = C o v ( / r W ^ P p / Q o .

(51)

Thus if COV(/>_1,T_1,S1) is negative (the usual case), then (51) implies that Pp is less than PL, which is also implied by the usual von Bortkiewicz identity, (39). 5

D o e s the unit value bias increase with increased aggregation?

It is generally thought that constructing broader unit value prices (i. e., aggregating over more specific products to form unit value prices) will lead to a greater degree of bias in a unit value price index as compared to the underlying „true" index. Parniczky (1974) addressed this issue and showed that this is not necessarily the case. However, his analysis was somewhat brief and it will be useful to address this issue in more detail.

Notes on Unit Value Index Bias • 701

Suppose we can decompose the N products in the aggregate under consideration into M subgroups where subgroup m has N m distinct products for m = 1,...,M so that N = N i + ... + Nm- Let l m represent a vector of ones with dimension N m for m = 1 , „ . , M . Let the period t value of the products in subgroup m be V m l for t = 0,1 and m = 1,...,M. Unit value prices for subgroup m in period t, P m ', and group m total quantity in period t, Qm', can be defined as follows: Qm1 = lm-qm1 ;t = 0,1; m = 1,...,M;

(52)

Pm1

(53)

= Vtn'/Qm' = p ^ - q ^ / C V ;t = 0,1; m = 1,...,M

where the period t price and quantity vectors for subaggregate m are defined as p m l and q m ', V m ' = pm' qm1 and l m is a vector of ones of dimension N m for m = 1,...,M and t = 0,1. The overall period t unit value price for the entire aggregate, P\ is defined in the usual way as follows: F = £ m = l M p.n' q . n V E m , ^ lm-qm'

t = 0,1

= E m = l M PmMqmVlm-qmV,/ = E m = l M Pm^Sm' a m f

(54)

M -— V¿1^1111 ' rr um ' =1 P1 m

where the period t within group m quantity share vector S^ is defined as follows Smf = qm'/lm-qm' ;t = 0,1; m = 1,...,M; and the period t between subgroup C

(55)

m quantity share aml is defined as follows:

= Q m l / E i = i M Qi1 = lm q m V E , = i M l i - q i ' ;

t = 0,1; m = 1 , . . , M .

(56)

N o t e that the various quantity shares sum to unity; i.e., we have lm-Sm^l;

t = 0,1; m = 1,...,M;

(57)

Em-1 M ffm1 = 1 ;

t = 0,l.

(58)

The final equation in (54) shows that the overall period t unit value price Pc is equal to a quantity share weighted average of the period t subgroup unit value prices, E m = i M Pm' Om- N o t e also that (52), (53) and (55) imply that the period t, group m unit value price Pm' can be expressed as the inner product of the period t subgroup m price vector p m ' and the corresponding subgroup share vector Smc: Pm( = Pm' Sn,' ;

t = 0,1; m = 1,...,M.

Using our new notation, the Drobisch

(59)

price index, Pd, can be defined as follows:

P D s P'/P 0 = E m = i M P m ^ S j a m VErn=i M p m °-S m ° = £m=iM PmVm1/Em=1M P m V m °

using (54) using (59).

(60)

Thus the Drobisch price index is equal to a quantity share weighted average of the period 1 subgroup unit values Pm1 (using the period 1 between group shares a m x ) divided by a

702 • W.E. Diewert and P. von der Lippe

quantity share weighted average of the period 0 subgroup unit values P m ° (using the period 0 between group shares crm°). Again using our new notation, the Paasche price index, Pp, can be defined as follows: Pp = Em=l M P m 1 -q m 1 /Em=l M Pm^qm1 = Em=i M P m ' Q m V E m - ^ P n ^ V Q ™ 1 = Em=l M PmV m VEm=l M P m ^ V m 1

using (52), (53) and (55) Using (56).

(61)

Note that the numerators in the final equations in (60) and (61) are equal. Recall that in section 2, w e defined the bias in the Drobisch index relative to the Paasche index as [Pq/Pp] — 1. In this section, w e will find it convenient to define the bias using a reciprocal measure. Thus define the bias of the Paasche relative to the Drobisch index as B i a s ( P P / P D ) = [P P /P D ] -

1

= [Em=i M PmV m °/Em=i M P m 0 ^ 1 ^ 1 ] - 1 using (61) and (62) = [Em=l M Pm°-S m °a m 0 /Em=l M P r ^ V O - 1 using (59) = IEm=l M Pm 0 -[S m Vn 0 - S J O l / E m ^ P m ^ n ^ 1 . (62) The analysis in section 2 could be reworked at this point in order to analyze the reciprocal bias in terms of a covariance of the base period price vector p° and the vector of base period quantity shares less the vector of period 1 shares. Thus the reciprocal bias will be negative if products which have above average prices in the base period have growing quantity shares over the t w o periods under consideration. Parniczky (1974: 235) introduced the idea of aggregating over all of the N individual product classes in two stages where unit value aggregation would be used in the first stage and normal index number theory would be used in the second stage. Thus if we use the Paasche formula in the second stage of aggregation, the basic price and quantity data, P m ' and Qm', that are used in the second stage formula are the unit values and subgroup total quantities that are defined by (52) and (53). Thus the second stage Paasche index or hybrid Paasche price index can be defined as follows: PHP = Em=l M P j Q m V E m ^ P ^ Q m ' = [Em=l M P m ' Q m V E m ^ Q m ^ E m ^ Pm°QmVEm=l M Q j ] - V , M P if! W 1 M P °(T 1

(63)

Note that the numerators in the final formulae for the true Paasche index Pp defined by (61) and for the hybrid Paasche index Php defined by (63) are equal. Define the bias of the Paasche index relative to the hybrid Paasche index as follows: Bias(P P /P H p) = [Pp/Php] - 1 = [Em=l M P m ^ m ' ^ m ; ^ Pm°-Sm10'm1] — 1 using (61) and (63) - V m=l, M Fm r> °.rs d °-S 1 uam 1 — LJm0 - JSm Ma Jum V' y m=li M Fm using (59) = NCov(p°, S°* - S l *,(l/N)l N )/j:m=I M PJ-SJCTJ using (1), (2) and (57)

(64)

where p° is the N dimensional vector of base period prices and the components of the vector S°* — S r are the vectors [S m ° — S m 'jffm 1 stacked up into a single N dimensional

Notes on Unit Value Index Bias • 703

vector. We have the following three sets of conditions which will imply that the hybrid Paasche is equal to the true Paasche index: • All base period subaggregate price vectors p m ° are equal to the corresponding subaggregate unit values; i.e., p m ° = P m 0 l m for m = 1,...,M; • The subaggregate quantity shares remain constant going from period 0 to period 1; i.e., S m ° = Sm1 for m = 1,...,M or • The covariance between the base period price vector p° and the adjusted share difference vector S°* — S 1 * defined above is zero; i.e., Cov(p°, S°* — S ' ^ I / N J I n ) = 0. Of course, the first two sets of conditions are special cases of the third condition. The first set of conditions is perhaps the most important for choosing how to construct the subaggregates: in order to minimize bias (relative to the Paasche price index), use unit value aggregation over products that sell for the same price in the base period. We now address the question that was first considered by Parniczky (1974); i.e., if instead of having only one unit value over a large number of products, we disaggregate the data into M subgroups and calculate unit value prices and the corresponding quantities for these M subgroups and apply normal index number theory (in this case, we use the Paasche formula), do we reduce unit value bias? Our suspicion is that disaggregation will help reduce the bias; i.e., we expect that the Bias(Pp/Po) defined by (62) will be greater in magnitude than the magnitude of the Bias(Pp/Pnp) defined by (64). We can use the algebra developed above in order to obtain an exact relationship between these two bias measures. From (62), we have: Bias(P P /P D ) = { £ m = i M p m ° [S m 0 ff m 0 - ( V i M in °-S ° 0. N o w consider a two stage aggregation procedure where the first stage consists of unit value aggregation but the second stage uses the Laspeyres formula applied to the first

Notes on Unit Value Index Bias • 705

stage unit value prices and quantities. Thus define the hybrid Laspeyres price index, P h l , as follows: PHL = £ m = l M P m 1 Q m ° / E m = l M Pm°Q m ° = [ £ m = l M P m 1 Q m ° / E m = l M Q m ° ] / [ E m - l M Pm°Q,n 0 /£m»l M Qm°] - V . M P xa 1 M P °(j 0

(70)

Note that the denominators in the final formulae for the true Laspeyres index P l defined by (68) and for the hybrid Laspeyres index P h l defined by (70) are equal. Define the bias of the hybrid Laspeyres index relative to the Laspeyres index as follows: Bias(P H L /PL) = [PHL/PL] - 1 = [ £ m = l M P „ > m ° / £ m = l M Pn^SmVn0] " 1 using (68) and (70) - V " 1 M D J rs 1 - S °lfT ° / V 1 M n V 0 using (59) = NCovfp1^1"- S°*\(l/N)lN)/£m=1M p ^ S M using (1), (2) and (57)

(71)

where p 1 is the N dimensional vector of base period prices and the components of the vector S ° " — S 1 " are the vectors [Sn,1 - S m °]c7 m 0 stacked up into a single N dimensional vector. As usual, we have the following three sets of conditions which will imply that the hybrid Paasche is equal to the true Paasche index: • All period 1 subaggregate price vectors pn,1 are equal to the corresponding subaggregate unit values; i.e., p,,,1 = P , , , 1 ^ for m = 1,...,M; • The subaggregate quantity shares remain constant going from period 0 to period 1; i.e., S m ° = Sn,1 for m = 1,...,M or • The covariance between the current period price vector p 1 and the adjusted share difference vector S1** — S°" defined above is zero; i.e., Covfp^S 1 ** — S°**,(1/N)1N) = 0. O f course, the first two sets of conditions are special cases of the third condition. The first set of conditions important for choosing how to construct the subaggregates: in order to minimize bias (relative to the Laspeyres price index), use unit value aggregation over products that sell for the same price in the current period. Looking at the bias formulae (69) and (71), it can be seen that if Pd is greater (less) than P l so that Covfp^S 1 — S°,(1/N)1n) is positive (negative), then it is likely that P h l is also greater (less) than P l so that Covtp^S 1 **— S°**,(1/N)1n) is also positive (negative). We can use the algebra developed above in order to obtain an exact relationship between the two bias measures (69) and (71). From (69), we have: Bias(P D /P L ) = [ £ m = l M P m V J - E m = l M Pm1 ' S m V ] / E r a = l M p J ^ V . 0 = Bias(P HL /PL) - ( E m = l M Pm1 k m ' " 0 / £ m = l M P m ^ m V , , 0 } using (71) and (59) (72) = Bias(P H L/P L ) + M C o v ( P \ o l - a ° , ( V M n M ) / j : m ^ M P m l S m ° a m ° using (2) and (58) where P 1 = [ P i V . ^ P m 1 ] is the vector of period 1 unit value prices for the M subaggregates and a 1 = [a\,...,a\] is the vector of period t subaggregate quantity shares where the a m ' were defined by (56).

706 • W.E. Diewert and P. von der Lippe

Suppose that PD is greater than PL SO that the bias, BiasfPo/PiJ, is greater than 1. Then from our analysis in section 3, we know that products which have above average prices in period 1 are positively correlated with growing quantity shares. It is likely (but not certain) that this positive correlation persists to the subaggregates so that subaggregates m that have above average unit value prices Pm1 in period 1 are associated with a positive Cm1 — fm° and hence the covariance in (72) will be positive under this hypothesis. It is also likely that the hybrid Laspeyres PHL is equal to or greater than the true Laspeyres index PL under these circumstances so that the bias, Bias(PHi/PiJ, is equal to or greater than 1. Under these hypotheses, using (72), it can be seen that Bias(P D /P L ) > Bias(PHL/PL) > 1.

(73)

Hence the magnitude of the bias of the hybrid Laspeyres index is less than the bias of the Drobisch index. Thus in this case, disaggregation of the single stage unit value price index into a two stage index where the second stage uses a Laspeyres formula does lead to a price index which is closer to the true Laspeyres index. A similar analysis can be made for the case where PD is less than PL SO that the bias, Bias(P[)/PL), is less than 1. In this case, the covariance in (69) is negative and it is likely that the covariances in (71) and (72) will also be negative. Under these conditions, using (72), we can deduce that: Bias(P D /P L ) < Bias(PHL/PL) < 1.

(74)

Hence again, the magnitude of the bias of the hybrid Laspeyres index is less than the bias of the Drobisch index. The techniques used in section 3 can be adapted to the present context in order to obtain bias formulae if the target index is a Fisher index. 6

Conclusion

In this paper, we have taken a systematic look at the existing literature on unit value biases and extended it somewhat. Our results indicate that it will usually be the case that the use of finer commodity classifications to generate unit value prices and quantities that are then inserted into a bilateral index number formula will generate closer approximations to an underlying preferred index. It should be noted that some use of unit value aggregation is inevitable; i. e., it will always be necessary to aggregate household or establishment purchases or sales of individual products over time in order to obtain total purchases or sales of the unit under consideration and then these (within the period time aggregated) prices and quantities are used as inputs into a bilateral index number formula. Data limitations will generally lead to more unit value aggregation where there could be aggregation over households, establishments, geographical areas or products. 1 4 However, it is generally felt that as a theoretical target, more narrowly defined unit values will generally lead to more accurate price indexes and the present paper seems to reinforce this view. 14

The scope of unit value aggregation was discussed in Diewert (1995) without any clear resolution. The ILO (2004) and IMF (2009) Manuals recommended „narrowly defined unit values" at the first stage of aggregation but the exact meaning of this advice is ambiguous. Silver (2010: S216ff.) discusses this issue in more detail.

Notes on Unit Value Index Bias • 707

But there is a problem with the analysis in this paper that needs to be addressed. The problem is that we have assumed that all N prices and quantities for the two periods under consideration are positive. But this condition can be far from being satisfied if we make the scope of our first stage unit values narrower and narrower. As the number of separate commodities N in the aggregate grows, it will generally be the case that more and more zero prices and quantities occur. This is due to the sporadic nature of shipments and purchases, particularly if the time period is relatively short. In limiting cases with a very large N, there can be an extreme lack of matching of products, leading to nonsensical target indexes. For example, using the notation in the previous section, suppose that we have M = 2 so that there are two subaggregates. In period 0, suppose that the subaggregate 1 price and quantity vectors, pi 0 and qi° are positive so that the subaggregate 1 unit value price, Pi 0 = pi°-qi°/liqi 0 , is positive and that the subaggregate 2 price and quantity vectors, p2° and q2° are vectors of zeros. Suppose that in period 1, the subaggregate 2 price and quantity vectors, P21 and q25 are positive so that the subaggregate 2 unit value price in period 1, P 2 1 = Pi q i l ^ r ^ , is positive and that the subaggregate 1 price and quantity vectors, pi 1 and qi 1 are vectors of zeros. Under these conditions, it can be seen that the true Laspeyres and hybrid Laspeyres price indexes, PL and PHL> are both equal to 0 and the true Paasche and hybrid Paasche price indexes, Pp and PHB are both equal to +00. Note that the Drobisch price index, PD, is equal to P21/Pi° and this will be a much more reasonable measure of price change, particularly if the products in the two subaggregates are all fairly „similar". Thus some caution is required in applying the results derived in this paper when there are zeros in the data. 15 References Balk, B. M. (1995), Axiomatic Price Index Theory: A Survey. International Statistical Review 63: 69-93. Balk, B.M. (1998), On the Use of Unit Value Indices as Consumer Price Subíndices. In: W. Lane, (ed.), Proceedings of the Fourth Meeting of the International Working Group on Price Indices. Washington, DC: Bureau of Labour Statistics. Balk, B.M. (2005), Price Indexes for Elementary Aggregates: The Sampling Approach. Journal of Official Statistics 21: 675-699. Balk, B. M. (2008), Price and Quantity Index Numbers. N e w York: Cambridge University Press. Davies, G.R. (1924), The Problem of a Standard Index Number Formula. Journal of the American Statistical Association 19: 180-188. Davies, G.R. (1932), Index Numbers in Mathematical Economics. Journal of the American Statistical Association 27: 58-64. Diewert, W. E. (1992), Fisher Ideal Output, Input and Productivity Indexes Revisited. Journal of Productivity Analysis 3: 211-248. Diewert, W.E. (1995), Axiomatic and Economic Approaches to Elementary Price Indexes. Discussion Paper N o . 95-01, Department of Economics, University of British Columbia, Vancouver, Canada, http://faculty.arts.ubc.ca/ediewert/axio.pdf Drobisch, M.W. (1871), ?ber die Berechnung der Veränderungen der Waarenpreise und des Geldwerths. Jahrbücher für Nationalökonomie und Statistik 16: 143-156.

15

Silver (2010: S216) notes some other situations where it would be more appropriate to use a unit value price index rather than a superlative index. Basically, the use of a unit value index is appropriate when purchasers of a product buy essentially the same product from suppliers at different prices over the time period under consideration.

708 • W.E. Diewert and P. von der Lippe

D u t o t , C. (1738), Reflections politiques sur les finances et le commerce. Volume 1, La Haye: Les frères Vaillant et N . Prevost. Fisher, I. (1922), The M a k i n g of Index N u m b e r s . Boston: Houghton-Mifflin. ILO, IMF, O E C D , UNECE, Eurostat and World Bank (2004), Consumer Price Index M a n u a l : Theory and Practice. Ed. by P. Hill, ILO: Geneva. IMF, ILO, O E C D , U N E C E and World Bank (2009), Export and Import Price Index M a n u a l . Ed. by M . Silver, IMF: Washington, D. C. Laspeyres, E. (1871), Die Berechnung einer mittleren Waarenpreissteigerung. Jahrbücher für N a t i o n a l ö k o n o m i e und Statistik 16: 296-314. Paasche, H . (1874), Uber die Preisentwicklung der letzten Jahre nach den H a m b u r g e r Börsennotirungen. Jahrbücher für Nationalökonomie und Statistik 12: 168-178. Pärniczky, G. (1974), Some Problems of Price Measurement in External Trade Statistics. Acta Oeconomica 12: 229-240. Silver, M . (2009a), D o Unit Value Export, Import, and Terms of Trade Indices Represent or Misrepresent Price Indices? Washington D . C . , IMF Staff Papers 56: 297-322. Silver, M . (2009b), Unit Value Indices. Chapter 2 in IMF, ILO, O E C D , U N E C E and World Bank (2008), Export and Import Price Index M a n u a l . Ed. by M . Silver, IMF: Washington, D . C . Silver, M . (2009c), An Index N u m b e r Formula Problem: The Aggregation of Broadly Comparable Items. Washington D . C . , I M F Working Paper WP/09/19. Silver, M . (2010), The Wrongs and Rights of Unit Value Indices. Review of Income and Wealth Series 56, Special Issue 1: S206-S223. Timmer, M . (1996), O n the Reliability of Unit Value Ratios in International Comparisons. Research M e m o r a n d u m GD-31, University of Groningen, Dec. 1996. von Bortkiewicz, L. (1923), Zweck und Struktur einer Preisindexzahl. Nordisk Statistisk Tidsskrift 2: 369-408. von der Lippe, P.M. (2007a), Index Theory and Price Statistics. Frankfurt a . M . : Peter Lang, von der Lippe, P. M . (2007b), Price Indices and Unit Value Indices in G e r m a n Foreign Trade Statistics, unpublished Paper University of Duisburg-Essen. http://mpra.ub.uni-muenchen.de/5525/ Walsh, C . M . (1901), The Measurement of General Exchange Value. N e w York: Macmillan and Co. Walsh, C . M . (1921), The Problem of Estimation. London: P.S. King & Son. Prof. Dr. W. Erwin Diewert, Department of Economics, University of British Columbia, Vancouver, B.C., C a n a d a , V 6 T 1Z1. E-Mail: [email protected] Prof. Dr. Peter von der Lippe, Universität Duisburg-Essen Campus Duisburg, Mercator School of business, Lotharstrasse 65, LB 146, 4 7 0 5 7 Duisburg, Germany. E-Mail: [email protected]

Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 2010) Bd. (Vol.) 230/6

Aggregate Indices and Their Corresponding Elementary Indices By Jens Mehrhoff, Frankfurt a.M.*

JEL C43, D 1 1 , E31

Generalised mean, quadratic mean, log-normal distribution, partial adjustment model, price elasticity, internal consistency.

Summary "Which index formula at the elementary level, where no expenditure share weights are available, corresponds to a desired aggregate index?" To answer this question, this paper develops a statistical approach. It proposes a theoretical framework which makes it possible to achieve numerical equivalence of an elementary index with the Laspeyres, Paasche or Fisher price index. Depending on the price elasticity, different elementary indices should be applied to different groups of goods in order to approach the desired aggregate index as closely as possible. Furthermore, the new statistical approach assures internal consistency between price and volume measurement.

1

Introduction

1.1 Motivation It is customary in official statistics, although often neglected in theoretical papers, for most price indices to be calculated in two stages. At the first stage, elementary indices are calculated on the basis of prices or their relatives, without having information on quantities or expenditures. At the second stage, the aggregate index is calculated on the basis of the elementary indices from the first stage, using aggregate expenditure share weights. In general, the question of "what should be measured?" directly yields the optimal index formula at the second stage: for measuring genuine price movements, a Laspeyres price index is used; for deflation purposes, a Paasche price index is preferred; and for the "cost of living", a Fisher price index, among others, is the formula of choice. However, it is less clear which index formula should be used at the first stage, where no expenditure share weights are available. The existing approaches to index numbers including but not re* This paper represents the author's personal opinions and does not necessarily reflect the views of the Deutsche Bundesbank or its staff. Detailed results and descriptions of methodology are available on request from the author. The author would like to thank Erwin Diewert, Bert Balk, Peter von der Lippe, Hans-Albert Leifer, Robert Kirchner, Karl-Heinz Toedter, Johannes Hoffmann and Sophia Mueller-Spahn for valuable comments. All remaining errors are, of course, the author's sole responsibility.

710 • J. Mehrhoff

stricted to the axiomatic approach are of little guidance in choosing the elementary index corresponding to the characteristics of the index at the second stage. The point in question is "how can the corresponding elementary index be selected?" The answer to this question is found by the proposition of a statistical approach. A single comprehensive framework, known as "generalised means", unifies the aggregate and elementary levels. With the aid of this approach, theoretical conditions under which a particular index formula at the elementary level exactly equals the desired aggregate index are identified. This makes it possible to secure the internally consistent calculation of price indices. The remainder of the paper is organised as follows. It continues with a review of a selection of the existing literature on elementary indices. Section 2 introduces basic concepts and approaches in index theory along with a more thorough explanation of the problem at the elementary level. Both the theoretical foundations of generalised means as well as the application to the Laspeyres, Paasche and Fisher price indices and their corresponding elementary indices are presented in detail in Section 3. The final section concludes. 1.2 Literature review

After a long period of research into aggregate formulae and an almost equally long policy debate in Europe and the US on whether the Laspeyres or Fisher formula should be used for a consumer price index (cf. Boskinet. al. 1996,1998, and Schultze/Mackie 2002), the focus of attention has recently moved more to the question of which index formula should be used at the elementary level. Nowadays, the capabilities of modern computers and the increasing coverage of data, first and foremost, through the advent of scanner data, enables statistical offices to calculate more refined price indices even at the elementary level (cf. Silver 1995, Silver/Webb 2002, Feenstra/Shapiro 2003, Diewert 2004, and papers presented at the meetings of the Ottawa Group at www.ottawagroup.org). Diewert (2004), and Diewert and Silver (2004, 2010) devote whole chapters in the CPI, PPI and XMPI manuals to elementary indices. They deal with virtually all topics that arise around the calculation of price indices at the elementary level. Theoretical issues, such as the problem of aggregation, are covered as well as practical questions, such as numerical relationships between different elementary indices. They continue by outlining the classical approaches in index theory, i. e. the economic, axiomatic, stochastic and sampling approaches (cf. Koniis 1924 and Diewert 1976; Eichhorn 1978 and Diewert 1995; Selvanathan/Prasada Rao 1994; and Balk 2005, 2008, respectively), and discuss the use of scanner data. Currently, there is an active ongoing discussion at Eurostat's Working Group on Harmonisation of Consumer Price Indices - more specifically, in the Task Force on Sampling - on which index formula is to be used at the elementary level (cf. EC 2001, Section I). The Commission Regulation (EC 1996, Article 7 in conjunction with Annex II) abandons the use of the Carli index but allows the use of either the Jevons or Dutot index. More precisely, the Carli index is not prohibited de jure but de facto as it would have to be shown that the results do not differ by more than one-tenth of a percentage point from either the Jevons or Dutot index (cf. the next-but-one paragraph for empirical evidence). Balk (1994) discusses the index formula problem at the elementary level. He poses the question whether a ratio of average prices or an average of price relatives, and which type of average, i.e. arithmetic, geometric or harmonic, should be used. Turvey (1996) ad-

Aggregate Indices arid Their Corresponding Elementary Indices • 7 1 1

dresses the same problem. He also presents empirical evidence that calculations of elementary indices with different index formulae give significant changes in aggregate CPIs, annually by more than two percentage points, in Finland, Sweden, Canada and France. The use of unit values at the lowest level in a price index is analysed by Balk (1998), which is commonly taken for granted to be an appropriate method of aggregation for prices of homogeneous goods. He tries to answer the questions of the conditions under which a group of goods is sufficiently homogeneous to warrant the use of unit values, and if one needs to restrict the use of unit values to homogenous goods alone. In the context of foreign trade, Silver (2009) criticises the use of aggregate indices which are calculated from unit values at the elementary level. He advocates pure price indices and reveals substantial biases of customs-based unit values: they depend on the structure of quantities and hence, cannot be considered surrogates for survey-based prices. Szulc (1989) describes the fact that biases at the elementary level are more severe than the possible bias of the formula at the aggregate level. In particular, this might result in surprisingly low differences between different aggregate indices if the same elementary indices are used as building blocks - no matter which aggregate index should be calculated. This is because the indices at the elementary level might be ignoring the particularities of the index formula at the aggregate level. In his 1994 paper he presents numerical evidence for the Canadian CPI that the choice of the elementary index matters the most, particularly in the short term. Dalen ( 1 9 9 2 , 1 9 9 5 ) discusses the impact of the choice of an inappropriate index formula at the elementary level in the Swedish CPI. Statistics Sweden switched over to the Carli index in January 1990. As soon as April it was replaced by a variant of the geometric index due to the well-known severe upward bias of the Carli index - of more than half a percent in these three months. Using Swedish and Finnish data, he shows in his 1998 paper that the Carli index consistently gives results which are year-on-year two index points and more larger than the Dutot and Jevons indices, while the latter two indices are fairly close to each other. Fenwick (1999) presents evidence that the UK HICP, which is based on the Jevons index at the elementary level, is annually about half a percentage point lower than the national equivalent, the Retail Prices Index, which uses a combination of the Dutot and Carli indices, only because of the different formulae. His main argument for this notable difference is the relative broad item description, leading to aggregation of highly heterogeneous items. Silver and Heravi (2007) show that the difference between the Jevons and Dutot indices is due to changing variances in the observed prices at different points in time alone, i. e. these indices will differ if prices exhibit dispersion. From a hedonic regression they derive a heterogeneity-controlled Dutot index and successfully test their approach empirically with scanner data.

2

Aggregate indices

2.1 First principles At the aggregate level, the target of measurement determines the index concept to be used. This is either the cost of goods (COGI) or the cost of living (COLI). In general, the former case leads to Laspeyres (1871) and Paasche (1874) price indices, while the latter results inter alia in the Fisher (1922) price index. Other formulae include the price indices of Walsh (1901) and Tornqvist (Tornqvist/Tornqvist 1937). The Laspeyres price index is the arithmetic mean of price relatives with base period expenditure share weights. This is the only price index which ensures the principle of pure

7 1 2 • J. Mehrhoff

price comparison (cf. von der Lippe 2 0 0 7 ) . Here, p,b and q,b denote the price and quantity, respectively, of the j * good at time b 6 {0, t). pL _ y - * Pn

_ Y!¡= i P'ttm

P'OliQ

PÍO E"=1 PMtO

^

1 Pi04i0

For volume measurement, one would opt for the Laspeyres quantity index Q* - , with QL = V/PF, where V i s the ratio of expenditures at times t and 0 or the value index and Pp is the Paasche price index. The Paasche price index is the harmonic mean of price relatives with current period expenditure share weights. pP_f\^(PitY1 l ¿ f W

Pitftt YLMu)

\

_ E " = i Puq,t EUP,

01«

m { )

The Fisher price index, among others, is a superlative index. It is defined as the geometric mean of the Laspeyres and Paasche price indices. This is the most famous price index approximating the change in the minimum expenditures, which preserve utility at a constant level, owing to changes in (relative) prices (cf. Allen 1975).

Pt

=

PiOlio E n.p -q»p

^p.q q»q

L

(6)

Upon this, an explicit formula is derived by which the order of the generalised mean can be computed directly from the log-normal distribution parameters. In Subsection 3 . 1 . 3 , these distribution parameters will be linked to the price elasticity. The assumption of a quadrivariate log-normal distribution of prices and quantities seems reasonable and predecessors are found in the literature. Moulton ( 1 9 9 3 ) , and Dalen (1999) use the log-normal distribution assumption for price relatives, while Silver and Heravi (2007) use it for prices in their own right. Note that the latter assumption is a generalisation of the former one. Log-normal distribution of price relatives is a direct consequence of log-normal distribution of prices. Ehemann (2007) goes one step further and assumes log-normal distribution of both prices and quantities, which will also be assumed here. The link between the generalised mean on the one side and the log-normal distribution parameters on the other side is built in the following theorem.

716 • J. Mehrhoff Theorem 2. The generalised mean in Equation (4) corresponds to the i*h root of the rth raw moment of the (joint) marginal distribution of price relatives, which is also the lognormal distribution. It follows that E(P™(r))

= exp

-

Mpo

+


0), is an increasing function of the variance of the price relatives. Hence, a mathematical argument for the upward bias of the Carli index compared with the Jevons index is given through this: the more heterogeneous the goods become at the elementary level, the higher will be the bias (cf. Subsection 4.2 for a discussion of the Carli index' upward bias). Theorem 3 establishes the link between the Laspeyres and Paasche price indices and the log-normal distribution parameters (cf. Subsection 3.3 for the solution in the case of the Fisher price index). Moreover, Theorem 4 gives an exact expression for the generalised mean corresponding to either of the two price indices. Theorem 3. The Laspeyres price index corresponds to the ratio of the first raw product moment of the joint marginal distribution of current period prices and base period quantities, and the first raw product moment of the joint marginal distribution of base period prices and quantities. It turns out that r,VL\

)=

ex

Pl

(

—

Ap0 +

,

a

P
\

(18)

From Theorem 6, the general results for the generalised mean are as follows. A generalised mean of order r equal to minus the price elasticity (—/?*) yields approximately the same result as the Laspeyres price index. Hence, if the price elasticity is minus one, for example, r must equal one and the Carli index (cf. Table 1) at the elementary level will correspond to the Laspeyres price index as target index. This can be seen in the simplest form from the following example: from q® = qo/pio, where qo is an arbitrary constant, follows a price elasticity of minus one and PL = []T"=i pit(qo/pio)}/ c Ru=iP;o(4o/P>o)] = TH=i(pit/pio)/n = P . However, if the Paasche price index should be replicated, the order of the generalised mean must equal the price elasticity, in the above example minus one. Thus, the harmonic index gives the same result and therefore, in this case it should be used at the elementary level. Unfortunately, this simple exposition works only with unrealistically restrictive assumptions about the data generating process.

Aggregate Indices and Their Corresponding Elementary Indices • 719

pQM(q)

p j

PQM(

oo)

Q

Figure 2 Quadratic mean of price relatives 3.3 Fisher price index The Fisher price index is derived from the Laspeyres and Paasche price indices as their geometric mean. Owing to the symmetry of the generalised means which correspond to the Laspeyres and Paasche price indices, a quadratic mean corresponds to the Fisher price index. In Definition 4 the properties of quadratic means in general are presented. Definition 4. A quadratic mean of price relatives of order q is defined as follows:

(19)

The index defined by Equation (19) is symmetric, i. e. P®M(q) = P®M(—q). Furthermore, it is either increasing or decreasing in \q\, depending on the data. Both characteristics can also be seen from Figure 2. Note that a quadratic mean of order q, P®M(q), should not be mistaken for the quadratic index, PGM(T) (cf. Table 1). Dalen (1992), and Diewert (1995) show via a Taylor series expansion that all quadratic means approximate each other to the second order. However, as Hill (2006) demonTable 2 Quadratic Means and Their Formulae q

Quadratic Mean

Formula

0*

Jevons

1

BMW

pBMW

2

CSWD

P C S W D = y/EiLi (Pit/Pio)/y/E!Li (Pio/Pit)

3

cubic

P QM (3) =

v/(p jt /pi0) 3 /

4

quartic

p qm (4) =

(Pit/P») 2 /

=

E

n=i

YLI

* The Jevons index is the limit of P Q M (q) as q approaches zero.

v « S j

vAPio/Pi.) 3 (Pio/Pit)2

720 • J. Mehrhoff

strates, the limit of PQM(q) if q diverges is pQM(oo) = \/P mm P max . He concludes that quadratic means are not necessarily numerically similar. For q —> 0 the quadratic mean becomes the Jevons index. For q = 1 an index results, which was first described by Balk (2005, 2008) as the unweighted Walsh price index and independently devised by Mehrhoff (2007) as a linear approximation to the Jevons index. Hence, this index is referred to as the BMW index. Lastly, one arrives at the CSWD index (Carruthers et. al. 1980, and Dalen 1992) for q = 2, which is the geometric mean of the Carli and harmonic indices. Table 2 contrasts these indices. Applying the preceding definitions gives the final result which is stated in Theorem 7. Theorem 7. A quadratic mean of order two times the absolute sponds to the Fisher price index:

\

im) ) (¡sis i

tfVP.0

^(pu trvp.o

price elasticity

= PQM(2\p*\).

corre-

(20)

The approximate equality in Equation (20) follows from Equations (17) and (18) in conjunction with Equation (4) - with r being equal to ±/T - or (19) - with q/2 being equal to |/F|. 4

Conclusion

4.1 Summary This paper addresses the problem of index calculation at the elementary level, where no expenditure share weights are available. The question of "which index formula at the elementary level corresponds to the characteristics of the index at the aggregate level ?" is dealt with. A statistical approach is proposed which theoretically allows the achievement Laspeyres (Generalised Mean: — ft*) Paasche (Generalised Mean: 3*) ,-2.50 quadratic reciprocal quadratic >-1.50 Carli harmonic -0.50

a ' -,

Fisher (Quadratic Mean: 2 \/3* quartic

-

CSWD BMW Jevons

Jevons 0.50 harmonic Carli reciprocal quadratic quadratic

cubic

BMW CSWD

1.50

cubic quartic

2.50 Figure 3 Overview of corresponding elementary indices

Aggregate Indices and Their Corresponding Elementary Indices • 721

of numerical equivalence of an elementary index with the desired aggregate index - in this instance, the Laspeyres, Paasche or Fisher price index. Based on "generalised means" and the assumption of joint log-normal distribution of prices and quantities, it is shown that the solution depends on the price elasticity alone, which is derived from a partial adjustment model. Thus, a feasible framework is provided which aids the choice of the corresponding elementary index. The results are graphically produced in Figure 3. If, for example, the price elasticity /T is minus one, the Carli index corresponds to the Laspeyres price index, the harmonic index to the Paasche price index and the CSWD index to the Fisher price index. Two possible applications of the approach outlined in this paper arise immediately after a decision has been taken on which aggregate index is desired. Firstly, index calculation can be rendered more precise if different elementary indices are applied to each group of goods, reflecting their specific price elasticities. At least for prominent groups of goods with high expenditure shares, studies on the price elasticity should be available. This will drive down biases of official price indices. In fact, the desired aggregate index can be approximated by using appropriate elementary indices. Secondly, for different purposes - either price or volume measurement - different elementary indices should be calculated for the same data. This means that if the Carli index is applied as the single formula at the elementary level of a Laspeyres price index, implying a price elasticity of minus one, the harmonic index must be used at the elementary level of a Paasche price index. Still, this is in contrast to the current practice as regards, for example, foreign trade in Germany, where the Carli index is used at the elementary level in both price statistics and volume measurement in national accounts. The former task is achieved via the Laspeyres price index, while the latter results in an implicit deflator in the form of the Paasche price index. 4.2 Outlook In an earlier version of this paper (Mehrhoff 2009), an empirical application to German foreign trade statistics was presented. The results indicate that, depending on the price elasticity, a range of elementary indices should be applied in the calculation of price indices. In particular, the Carli index performs remarkably well at the elementary level of a Laspeyres price index, questioning the argument of the upward bias of the Carli index in comparison with the Jevons index. As regards internal consistency, notable empirical differences between different elementary indices and aggregate indices formed from them are found. After a discussion with one of the guest editors of this issue, W. Erwin Diewert, this part of the earlier paper will form the basis for a follow-up paper where the methodology outlined here is compared with a modified version of the approach of Shapiro and Wilcox. It is to be shown which method performs best given the structure of the data that is observed at statistical institutes. First, data are sampled for a few goods in each group (this is the problem of low degrees of freedom). Second, for a limited number of groups of goods continual and simultaneous information on both prices and quantities is provided (only the new statistical approach is applicable when this is not the case).

7 2 2 • J. Mehrhoff

Appendix: Proof of Theorems P r o o f o f T h e o r e m 1 . The proof is outlined in Subsection 3 . 1 . 1 . For details cf. Eichhorn and Voeller (1976), and Hardy et al. (1934). • P r o o f o f T h e o r e m 2. The k t h raw moment of a log-normally distributed random variable is given by e x p ( k f i + ¿ 2 o 2 /2). After taking natural logarithms it applies that tflnXifclnY ~ 9{(afix ± bfiy, a 2 ° x + ± l a b a x y ) - Using this and the definition of the population counterpart of the sample generalised mean, one finds the following result.

£(pGM(r) ^/£((ë)r)=exp[;

, o l . + O Î - 2a„

By reducing the terms, the proposition follows.

•

P r o o f o f T h e o r e m 3 . Using the definitions of the population counterparts of the sample Laspeyres and Paasche price indices, the expectations are as follows.

E(PL)

E(Pitqx>)

E(Pp

The proposition follows by reducing the terms.

•

P r o o f o f T h e o r e m 4 . The corresponding generalised means are found by solving the equations for r . m P r o o f o f T h e o r e m 5 . Stationarity in covariance of the processes, i.e. 0 < p < 1 and 0 < )»! < 1 , implies that the covariance between any two observations depends only on the lag between them. For the covariance of logarithmic prices, it follows that it is an exponentially decreasing function.

°p«,P, = y ] i ~ e ] a l Using the lag operator and inverting the lag polynom in the function of logarithmic quantities, it can be written as follows. oo

l n ^ = a+ / T £ > M n 5 > r=0

/ oo x

i b - r

\T=0

\

/

/

\

U\-p)6 + U

+

oo

\

T=0

/

Taking the expectation and subtracting it on both sides yields the following expression. OO

00

1=0

T=0

Inqih-nq = ff Y,PT(^P.b-r-Vp) + Y.PXEl-r

Aggregate Indices and Their Corresponding Elementary Indices • 723

Multiplying this expression with In p,c- — p p and taking the expectation results in the desired covariances. 00

= F

oo

= f t T=0

Y,pxyltib~t)l 1=0

Substituting the appropriate expressions for £ and b, either 0 or t, the proposition follows by applying the formula for the sum of a geometric series. • Proof of Theorem 6. Substituting the respective expressions into the equations directly yields the stated results. Under the stationarity in covariance assumption, the difference of (co-)variances at different points in time vanishes and approaches zero. For the generalised means corresponding to the Laspeyres and Paasche price indices, ri and rp, respectively, it is assumed that the product of the autoregressive parameters is sufficiently small to be negligible, i. e. the sluggishness of adjustment of quantities or the persistence of the process of prices is low: pyx —+ 0. The generalised mean corresponding to the Paasche price index is derived under the additional assumption of sufficiently large t in order for the serial correlations to converge to zero: p' —> 0 and y\ —> 0. • Proof of Theorem 7. The proposition follows directly by reducing the terms. •

References Allen, R. G. D. (1975), Index Numbers in Theory and Practice. London, United Kingdom: Macmillan. Balk, B.M. (1994), On the First Step in the Calculation of a Consumer Price Index. In: L.M. Ducharme (ed.), Proceedings of the First Meeting of the International Working Group on Price Indices. Ottawa, Canada: Statistics Canada. Balk, B. M. (1998), On the Use of Unit Value Indices as Consumer Price Subindices. In: W. Lane (ed.), Proceedings of the Fourth Meeting of the International Working Group on Price Indices. Washington, DC: Bureau of Labor Statistics. Balk, B.M. (2005), Price Indexes for Elementary Aggregates: The Sampling Approach. Journal of Official Statistics 21: 675-699. Balk, B.M. (2008), Price and Quantity Index Numbers. New York, NY: Cambridge University Press. Boskin, M.J., E.R. Dulberger, Z. Griliches (1996), Toward a More Accurate Measure of the Cost of Living. Darby, PA: Diane Publishing. Boskin, M.J., E.R. Dulberger, R.J., Gordon, Z. Griliches, D.W. Jorgenson (1998), Consumer Prices, the Consumer Price Index, and the Cost of Living. Journal of Economic Perspectives 12: 3-26. Carli, G.R. (1764), Del Valore e della Proporzione de' Metalli Monetati con I Generi in Italia Prima delle Scoperte dell' Indie col Confronto del Valore e della Proporzione de' Tempi Nostri. Opere Scelte di Carli 1. Carruthers, A.G., D.J. Sellwood, P.W. Ward (1980), Recent Developments in the Retail Prices Index. The Statistician 29: 1-32. Coggeshall, F. (1886), The Arithmetic, Geometric, and Harmonic Means. Quarterly Journal of Economics 1: 83-86. Dalen, J. (1992), Computing Elementary Aggregates in the Swedish Consumer Price Index. Journal of Official Statistics 8: 129-147. Dalen, J. (1995), Quantifying Errors in the Swedish Consumer Price Index. Journal of Official Statistics 11: 261-275. Dalen, J. (1998), Studies on the Comparability of Consumer Price Indices. International Statistical Review 66: 83-113.

724 • J. Mehrhoff

Dalen, J. (1999), A Note on the Variance of the Sample Geometric Mean. Department of Statistics. Stockholm University Research Report 1. Diewert, W.E. (1976), Exact and Superlative Index Numbers. Journal of Econometrics 4: 115145. Diewert, W. E. (1978), Superlative Index Numbers and Consistency in Aggregation. Econometrica 46: 883-900. Diewert, W. E. (1995), Axiomatic and Economic Approaches to Elementary Price Indexes. NBER Working Paper 5104. Diewert, W.E. (2004), Elementary Indices. Pp. 355-372 in: P. Hill (ed.), ILO, IMF, OECD, UNECE, Eurostat, and World Bank, Consumer Price Index Manual: Theory and Practice. Geneva, Switzerland: International Labour Organization. Diewert, W.E., M. Silver (2004), Elementary Indices. Pp. 508-524 in: P. Armknecht (ed.), IMF, ILO, OECD, UNECE, and World Bank, Producer Price Index Manual: Theory and Practice. Washington, DC: International Monetary Fund. Diewert, W.E., M . Silver (2010), Elementary Indices. Pp. 501-518 in: M . Silver (ed.), IMF, ILO, OECD, UNECE, and World Bank, Export and Import Price Index Manual: Theory and Practice. Washington, DC: International Monetary Fund. Eichhorn, W. (1978), Functional Equations in Economics. Reading, MA: Addison-Wesley. Eichhorn, W., J. Voeller (1976), Theory of the Price Index: Fisher's Test Approach and Generalizations. Lecture Notes in Economics and Mathematical Systems 140. European Commission (1996), Commission Regulation (EC) N o 1749/96. Official Journal of the European Communities L 229: 3-10. European Commission (2001), Compendium of HICP Reference Documents. Luxembourg: Office for Official Publications of the European Communities. Ehemann, C. (2007), Evaluating and Adjusting for Chain Drift in National Economic Accounts. Journal of Economics and Business 59: 256-273. Feenstra, R. C., M. D. Shapiro, (eds.) (2003), Scanner Data and Price Indexes. Proceedings of the Conference on Research in Income and Wealth. Arlington,VA, 15-16 September 2000, Chicago, IL: University of Chicago Press. Fenwick, D. (1999), The Impact of Choice of Base Month and Other Factors on the Relative Performance of Different Formulae Used for Aggregation of Consumer Price Index Data at an Elementary Aggregate Level. In: R. Guönason, I>. Gylfadottir (eds.), Proceedings of the Fifth Meeting of the International Working Group on Price Indices. Reykjavik, Iceland: Statistics Iceland. Fisher, I. (1922), The Making of Index Numbers. Boston, MA: Houghton Mifflin. Hardy, G . H . , J. E. Littlewood, G. Polya (1934), Inequalities. Cambridge, United Kingdom: Cambridge University Press. Hill, R.J. (2006), Superlative Index Numbers: Not All of Them Are Super. Journal of Econometrics 130: 25-43. Jevons, W. S. (1863), A Serious Fall in the Value of Gold Ascertained, and its Social Effects Set Forth. London, United Kingdom: Stanfords. Konüs, A.A. (1924), The Problem of the True Index of the Cost of Living. Economic Bulletin of the Institute of Economic Conjuncture (in Russian) 9-10: 64-71; Econometrica 7 (1939): 10-29. Laspeyres, E. (1871), Die Berechnung einer mittleren Waarenpreissteigerung. Jahrbücher für Nationalökonomie und Statistik 16: 296-314. von der Lippe, P. (2001), Chain Indices: A Study in Price Index Theory. In: Federal Statistical Office (ed.), Spectrum of Federal Statistics, Vol. 16. Stuttgart, Germany: Metzler-Poeschel. von der Lippe, P. (2007), Index Theory and Price Statistics. Frankfurt a. M., Germany: P. Lang. Lloyd, P.J. (1975), Substitution Effects and Biases in Nontrue Price Indices. American Economic Review 65: 301-313. Mehrhoff, J. (2007), A Linear Approximation to the Jevons Index. Pp. 45-46 in: P. von der Lippe, Index Theory and Price Statistics, Frankfurt a . M . , Germany: P. Lang.

Aggregate Indices and Their Corresponding Elementary Indices • 725

M e h r h o f f , J. (2009), Aggregate Indices and Their Corresponding Elementary Indices. In: C. Becker Vermeulen (ed.), Proceedings of the Eleventh Meeting of the International Working G r o u p on Price Indices. Neuchätel, Switzerland: Swiss Federal Statistical Office. M o u l t o n , B. R. (1993), Basic C o m p o n e n t s of the CPI: Estimation of Price Changes. Monthly Labor Review, December. M o u l t o n , B. R. (1996), Constant Elasticity Cost-of-Living Index in Share Relative Form, mimeo, Washington, D C : Bureau of Labor Statistics. Paasche, H . (1874), Ueber die Preisentwicklung der letzten Jahre nach den H a m b u r g e r Börsennotirungen. Jahrbücher für Nationalökonomie und Statistik 23, 168-178. Schultze, C. L., C. Mackie, (eds.) (2002), At W h a t Price? Conceptualizing and Measuring Costof-Living and Price Indexes. Washington, D C : National Academy Press. Selvanathan, E. A., D. S. Prasada Rao (2002), Index N u m b e r s : A Stochastic Approach. London, United Kingdom: Macmillan. Shapiro, M . D . , D.W. Wilcox (1997), Alternative Strategies for Aggregating Prices in the CPI. Federal Reserve Bank of St. Louis Review, May/June: 113-125. Silver, M . (1995), Elementary Aggregates, Micro-Indices, and Scanner Data: Some Issues in the Compilation of Consumer Price Indices. Review of Income and Wealth 41: 427-438. Silver, M . (2009), D o Unit Value Export, Import, and Terms-of-Trade Indices Misrepresent Price Indices? IMF Staff Papers 56: 297-322. Silver, M . , S. Heravi (2007), W h y Elementary Price Index N u m b e r Formulas Differ: Evidence on Price Dispersion. Journal of Econometrics 140: 874-883. Silver, M . , B. Webb (2002), The Measurement of Inflation: Aggregation at the Basic Level. Journal of Economic and Social Measurement 28: 21-35. Szulc, B.J. (1989), Price Indices Below the Basic Aggregation Level. Pp. 167-178 in: R. Turvey, Consumer Price Indices. Geneva, Switzerland: International Labour Office. Szulc, B.J. (1994), Choice of Price Index Formulae at the Micro-Aggregation Level: The Canadian Empirical Evidence. In: L . M . Ducharme (ed.), Proceedings of the First Meeting of the International Working G r o u p on Price Indices. O t t a w a , Canada: Statistics C a n a d a . Törnqvist, L., E. Törnqvist (1937), Vilket är Förhallandet Mellan Finska M a r k e n s och Svenska Kronans Köpkraft? Ekonomiska Samfundets Tidskrift 39: 1-39. Turvey, R. (1996), Elementary Aggregate (Micro) Indexes. In: European Commission, Improving the Quality of Price Indices: CPI and PPP. Proceedings of the Eurostat International Seminar, Florence, 18-20 December 1995, Luxembourg: Office for Official Publications of the European Communities. Walsh, C. M . (1901), The Measurement of General Exchange-Value. N e w York, NY: Macmillan. Dr. Jens M e h r h o f f , Statistics D e p a r t m e n t and Research Centre, Deutsche Bundesbank, WilhelmEpstein-Strasse 14, 6 0 4 3 1 F r a n k f u r t a . M . , Germany. Email: [email protected]; Homepage: www.bundesbank.de

Jahrbücher f. Nationalökonomie u. Statistik (Lucius & Lucius, Stuttgart 2010) Bd. (Vol.) 230/6

Lowe and Cobb-Douglas Consumer Price Indices and their Substitution Bias By Bert M. Balk, Rotterdam* JEL C43 Index number, cost-of-living index, Lowe index, Cobb-Douglas index, Geometric Young index.

Summary Catching the effect of substitution behaviour in a Consumer Price Index (CPI) as good as possible is a goal pursued by statistical agencies throughout the world. The difference between a CPI and a certain target cost-of-living index is called substitution bias. Balk and Diewert (2003) considered the substitution bias of a Lowe Consumer Price Index; see also CPI Manual (2004: Chapter 17). The present paper considers the substitution bias of a Cobb-Douglas (or Geometric Young) CPI, and compares the two price indices with respect to their substitution bias. It appears difficult to draw a clear-cut conclusion.

1

Introduction

Most if not all statistical agencies calculate their short-term Consumer Price Index (CPI) as a weighted arithmetic mean of price relatives of commodities (or, more generally, elementary aggregates). 1 Thus, formally, the aggregate price change between the most recent reference month 0 and the current month t is defined as a weighted average of price relatives «=1

where £ > „ = 1. n=1

(1)

The weights, ideally, reflect the current importance of the various commodities in the expenditure of the representative consumer. Because the processing of expenditure data is a time-consuming undertaking, the most recent set of expenditure shares usually refers to some year (or combination of years) b prior to month 0, — (n = 1, ...,N). sb„ = i>b b \ > > > Z-m=l PnXvn

(2) w

In order to come closer to the current expenditure pattern it is advised (for example, in the framework of the European Harmonized Index of Consumer Prices) that these expenditure shares be price-updated to month 0. Thus, the CPI weights are defined as * The views expressed in this paper are those of the author and do not necessarily reflect any policy of Statistics Netherlands. The first version was presented at the 11 th Meeting of the Ottawa Group, Neuchatel, 27-29 May 2009. Two referees are thanked for their valuable comments. 1 The longer-term CPI is usually defined as a chained index.

Lowe and Cobb-Douglas Consumer Price Indices and their Substitution Bias • 727

Wn

=^N

/i hi 0 / h\

~

*

N

>'

which implies that the functional form of the CPI becomes that of a so-called Lowe index: 2 E m=1

= S

1

PnXn

rLtV,

*")•

(4)

Here the prices of month t are compared with those of month 0, using the quantities of some year b prior to month 0. One of the practical virtues of this index is its transitivity; that is, PLo(p',p0-,xb)/Plo(p",p0-,xb) = PLo(p',p"-,xb). (5) Transitivity implies that the index comparing month t to month 0 can be expressed as a chained index, which is a distinct interpretative advantage. But why this price-updating? Thus, let us consider using the plain period b expenditure shares as weights. This would give a Young price index, PY(p\p°;sb)

= jrsbM/p°n). «=i

(6)

Unfortunately, this index is not transitive. Instead we consider the geometric analogue of the Young index, pcD(p',po-,sb)^f[(p>n/p°y-, n=1

(?)

which is known as a Cobb-Douglas price index. This index is transitive. Which of the two indices, the Lowe or the Cobb-Douglas, is better? To answer this question we need a yardstick. Here it is assumed that the Lowe index as well as the CobbDouglas index are used as proxies of a cost-of-living index. The difference between a certain price index and a target cost-of-living index, both covering the same time interval, is called substitution bias. The lower the substitution bias, the better the index. Balk and Diewert (2003) considered the substitution bias of a Lowe CPI; see also CPI Manual (2004: Chapter 17). In this paper I consider a Cobb-Douglas (or Geometric Young) CPI, and compare the two price indices with respect to their substitution bias. The lay-out of the paper is as follows. To provide for the necessary context, section 2 introduces the basic elements of the theory of cost-of-living indices. In section 3 we consider, without assuming optimizing behavior, the relation of the Lowe and Cobb-Douglas price indices. In section 4 an expression for the substitution bias of the Lowe price index is derived. The same is done for the Cobb-Douglas price index in section 5. Section 6 is devoted to a comparison of the relative bias of these two price indices. Finally, in section 7 some empirical evidence is presented. Section 8 concludes. 2

For the name and provenance of all the indices discussed in this paper the reader is referred to Balk (2008).

728 • B.M. Balk

2

The basic cost-of-living index theory

We consider a single consumer and assume that this consumer has a stable preference ordering 3 over a set of commodities labelled 1,... ,N. Under suitable regularity conditions such an ordering can be represented by a utility function U(x), that is a function such that U(x/) > U(x) if and only if the consumer prefers the quantity vector x' over x. Quantity vectors x are nonnegative and it is assumed that U(x) is non-decreasing in the components of x. A set {x|l/(%) = u} for u e Range U(x) is called a standard of living. Suppose that the consumer faces the positive price vector p. Then, neoclassically, the consumer's decision problem can be formulated as 4 min p • x subject to U(x) > u.

(8)

This means that, given a certain utility level u and prices p, the consumer minimizes the cost of achieving this level. The (Hicksian) quantities demanded, x(p, u), are obtained as solution of this minimization problem, and their cost is C(p,u) = p-x{p,u)

= min{p • x\U(x) > u}.

(9)

We call C(p, u) the cost (or expenditure) function. Under suitable regularity conditions this function is a dual representation of the consumer's preference ordering. 5 The cost function is nondecreasing in u, concave in p, and linearly homogeneous in p. The last property means that C{Xp, u) = XC{p, u) (k > 0).

(10) 1

We now consider the price vectors p° and p pertaining to periods or situations 0 and 1 and we let u be some reference utility level. It is convenient to think of period 0 as an earlier and period 1 as a later period. The Konus price index or cost-of-living index is defined by

'v^-esw

(11)

This is the minimum cost of achieving utility level u when the prices are p1 relative to the minimum cost of achieving this level when the prices are p°. The cost-of-living index PK(p1 ,P°;M) thus conditions on the standard of living given by u. We assume that the consumer acts cost-minimizing in periods 0 and l 6 ; that is, for the observed price and quantity vectors we assume that x° = x(p°, U(x0)) and x1 = x(p1,U(x1)), so that p0-x°

3

= C(p°,U(x"))

(12)

The case of changing preferences is considered in Balk (1989).

P*xn-

4

Notation: p • x =

5

See Diewert (1993). This is frequently called 'rational behaviour'. O n the limitations of this concept of rationality see Sen (1977).

6

Lowe and Cobb-Douglas Consumer Price Indices and their Substitution Bias • 729

and (13)

(14) and qp^u^1))

0), then C(p,u) = uC(p, 1), and PK(.) becomes independent of the reference utility level u.7 In this case PF(p\x\p°,X0)

< PK(p\p0;u)


s o that P C D (.) is a weighted geometric mean of price relatives. For b = 0, the Cobb-Douglas index reduces to the Geometric Laspeyres index. For b = 1 the Cobb-Douglas index reduces to the Geometric Paasche index. Before proceeding to the discussion of their substitution bias, that is, the relation of the Lowe and Cobb-Douglas price indices to some cost of living index PK(p1 ,p°;u), it is interesting to consider their mutual relation. Define the hybrid period 0 prices period b quantities expenditure shares by

=

(23)

The Lowe price index can then be written as a weighted arithmetic mean P L o {p\p°-,x b ) = ^ { p \ / p l ) .

(24)

n=1

Because an arithmetic mean is always greater than or equal to a geometric mean, we obtain the result that PLo(p1,p°\xb)

>

(25) n= 1

But what can be said about the relation between the index at the right-hand side of this inequality and the Cobb-Douglas index as defined by expression (22)? Consider their logarithmic difference InPCD(p\p°-,sb)

- h / n ^ / r f )

Vr=l

5

^ =

/

(Pi/Pn)-

(26)

n= 1

This expression measures the covariance of share differences between the periods b and 0 and price changes between the periods 0 and 1. When b < 0 < 1, there is no reason to expect that this covariance will be different from zero. The conclusion then is that expectedly the Lowe price index is greater than or equal to the Cobb-Douglas index. A second argument runs as follows. Using the logarithmic mean L(.) 9 , it is easily checked that the logarithm of the Lowe price index can be written as 9

The logarithmic mean of any two strictly positive real numbers a and b is defined by L(a,b) = (a - b)/ln(a/b) if a ^ b and L(a,a)=a. It has the following properties: (1) min(a, b) < L(a, b) < max(a, b); (2) L(a,b) is continuous; (3) L(Xa, kb) = XL(a, b)(X > 0); (4) L(a,b) = L(b,a); (5) ( a b ) v l < L{a,b) < (a + b)/2; (6) L(a, 1) is concave.

732 • B.M. Balk

In PLo(p\p°-,xb)

=Y

fL(pL0(PX>P°->xh)>P1JPn)

\n{pl/pOy

(27)

The logarithm of the Cobb-Douglas price index is lnPCD(p\p°-,sb)

= jrsbJn(pl/p0n). n= 1

(28)

Subtracting (28) from (27) and making use of the linear homogeneity logarithmic mean we obtain lnPLo(p1 ,p°;xb) — lnPcD(p^,p°]Sb) f

v

M

*

l W

property of the

= ri/rf

)

(29)

where = L(PLo{p1,p°; xb)pl/pb,pl/pb) (n = 1 ,...,N) are price-update factors measuring the price change from period b to period 1. The right-hand side of expression (29) has the structure of a weighted covariance, namely between relative price-update factors and relative price changes. Going from period b to period 1 the expenditure shares sb are price-updated by the factors a]^ b . The relative price-update factors are then a n° b / Yln= l snan°b^ their weighted mean being equal to 1. The covariance is between these factors, covering the time span from period b to 1, and relative price changes, covering the time span from period 0 to 1. When the time interval is short relative to [0,1] this covariance is likely positive. Without empirical material, however, not much more can be said about the sign of the covariance, and hence about the relative position of the Lowe and the Cobb-Douglas price indices. 4

Substitution bias of the Lowe price index

For defining the substitution bias of a price index we must pick the cost-of-living index that is to serve as our target. Since the Lowe price index PLo(p1 ,p°-,xh) conditions on the quantity vector xb it is natural to compare this index to the cost-of-living index PK(p1,p°; U(xb)) which conditions on the standard of living represented by x . Thus, the substitution bias of the Lowe price index is here defined as the difference PL°(p\p°-,xb)-PK(p\p°-,ub)

(30)

where ub = U(xb). Balk and Diewert (2003) employed second-order Taylor series approximations to explore the substitution bias. Instead of repeating this, I am using here their exact counterparts. 10 The first concerns the numerator of the cost-of-living index and reads

10

Following the suggestion of a referee of the earlier paper.

Lowe and Cobb-Douglas Consumer Price Indices and their Substitution Bias • 733

C(p\tf)

+(1/2) £

£

n=1 «'=1

=

^ K f r

1

M -

-

&

+ (1/2)B,

n=l

(31)

where p* € \pb ,px\ and

„=1 n>=i

°Pn°Pn>

is a temporary shorthand notation. The expression after the second equality sign was obtained by using Shephard's L e m m a 1 1 and the assumption that the consumer acts cost minimizing in period b, that is, xb = x(pb,ub) and therefore pb-xb

= C(p»,U(xb)).

(33)

Likewise, the denominator of the cost-of-living index can be expanded as

n= 1

°P"

N

= £ r f * ' + (l/2)A, n=l b „01 a n d e\pb,p°]

w h e r e p**

^

(34)

t

t

^

„=1 „") = C(P>U) and J2nP»d C(p,u)/dp„dp„i = 0 («' = 1, N ) . Since C(p,u) is concave in p, the square matrix of second-order partial derivatives d2C(p,u)/dp„dp^ is negative semidefinite.

734 • B.M. Balk

P

(P

,P

>

X

) -

C

( 3 6 )

( p ° , u » ) - ( 1 / 2 ) A >

whereas the target cost-of-living index reads

(37)

Then, by straightforward simplification we obtain for the relative substitution bias: PL°{p',

p°-xb)

-

K

P

K

(p',p°-,u

b

)_ 1 a f a

b

P (p\p°-,u )

2

-

c P )

(38)

l - l 2 C(p°,ub)

The concavity of the cost function implies that — A is non-negative, which in turn implies that 1 — j qJo ut) > 1- Thus, the relative substitution bias is positive if and only if 1S positive, or, more formally, C(p0|Mt) - q p v o PLo(p1,p°-,xb)

> PK(p\p°-,ub) 32C(pV), 1 8p„8jy 'Px

y>N V N ¿^«=1

if and only if few

1

PJIP«'

fe,

P„'j

K

,

Q

fe

The expression Oi - P*) can be interpreted as a squared distance j^etween the price" vectors p1 and pb; likewise, the expression - EiLi - P* pb (n = 1, ...,N) and that p n ~ p t

=

a

(Pn

~ Pn)

with a > 1 (« = 1,

...,N).

(40)

This implies that plJpl

=

a +

(1 -

a){p

b

J p l ) < a (» =

1 , N ) ,

(41)

which in turn implies that P

K

{ p \ p ° - , u

b

) < a .

(42)

Suppose also that d2C(p*,ub) dp„dp„> 12

81C{p**,ub dprfipn*

(«, «' = 1, ...,N).

(43)

For t w o r a n d o m c o l u m n vectors a and b f r o m the same distribution with covariance matrix S the M a h a l a n o b i s d i s t a n c e is d e f i n e d a s [(a - b)rS~'i{a

—

fc)]1'2.

Lowe and Cobb-Douglas Consumer Price Indices and their Substitution Bias • 735

Assumptions (40) and (43) immediately imply that Yln= 1 H2n'=1 0 6p„ep'" ^ (Pn ~ Pn)(Pn> ~ Pn') _

2

(44)

which means that inequality (39) is satisfied.

5

Substitution bias of the Cobb-Douglas price index

The relative substitution bias of the Cobb-Douglas price index is defined as the difference lnPCD(pl,p°;sb)-\nPK(p1,p°-,ub)

(45)

where uh = U(xb). For the logarithm of the numerator of the cost-of-living index we obtain the following second-order Taylor series expression: In

C(p\ub)

+(V2)

- I n p S ) ( b p i - Inpfr)

= [nC(pb,Hb) +

jrsbMpl/ph„)

n= 1

where p* e \pb,p1]. The expression after the second equality sign was obtained by using Shephard's Lemma and the assumption that the consumer acts cost minimizing in period b. Similarly, for the denominator we obtain In C(/>",»») = inc(p',»')

+

-

=.|nC»,«v£sJln

736 • B.M. Balk

where p** E \p b,p 0]- Subtracting (47) from (46) and applying the definitions of the Koniis and Cobb-Douglas price indices we obtain lnP

c l

V , i > V ) - In I* (p\ p V ) =

N

N

-(1/2)VV { ' ] h h

62lnC(p *,u b KV ' >\n(p l!p b ) l n ( p M ) .

(48)

l n [ P J P n

Hence,

P CD{p\p 0-,s h)

> P K{p\p°-,u b)

if and only if

Expression (49) looks starkly like (39). The t w o terms can also be interpreted as distance measures, albeit that the matrices of second-order partial derivatives are not necessarily negative semidefinite.

6

Comparison

The relative substitution bias of the Lowe price index is given by expression (38) and that of the Cobb-Douglas price index by (48). It is not possible to compare these expressions without making some assumptions. To start with, consider the second-order partial derivatives 0 2 In C(p,u)/dInp„dInp n > ( « , « ' = 1, ...,N). Straightforward computation delivers 9 2 In C(p, u) _ 6 In p„ 5 In pn